Patent Publication Number: US-2022218886-A1

Title: Therapy prediction and optimization for renal failure blood therapy

Description:
PRIORITY CLAIM 
     This application claims priority to and the benefit as a continuation application of U.S. patent application Ser. No. 16/525,058, filed Jul. 29, 2019, entitled “Therapy Prediction and Optimization for Renal Failure Blood Therapy”, which claims priority to and the benefit as a continuation application of U.S. patent application Ser. No. 15/454,387, filed Mar. 9, 2017, entitled “Therapy Prediction and Optimization for Renal Failure Blood Therapy, Especially Home Hemodialysis”, now U.S. Pat. No. 10,363,351, issued Jul. 30, 2019, which claims priority to and the benefit as a continuation application of U.S. patent application Ser. No. 14/472,757, filed Aug. 29, 2014, entitled “Therapy Prediction and Optimization for Renal Failure Blood Therapy, Especially Home Hemodialysis”, now U.S. Pat. No. 9,629,949, issued Apr. 25, 2017, which claims priority to and the benefit as a continuation application of U.S. patent application Ser. No. 13/088,064, filed Apr. 15, 2011, entitled “Therapy Prediction and Optimization for Renal Failure Blood Therapy, Especially Home Hemodialysis”, now U.S. Pat. No. 8,845,570, issued Sep. 30, 2014, which claims priority to and the benefit of U.S. Provisional Patent Application Ser. No. 61/438,002 filed on Jan. 31, 2011, U.S. Provisional Patent Application Ser. No. 61/388,315 filed on Sep. 30, 2010 and U.S. Provisional Patent Application Ser. No. 61/325,113 filed on Apr. 16, 2010, the entire disclosures of each of which are hereby incorporated by reference and relied upon. 
    
    
     BACKGROUND 
     The present disclosure generally relates to dialysis systems. More specifically, the present disclosure relates to therapy prediction and optimization systems and methods for hemodialysis, especially home hemodialysis (“HHD”). 
     Hemodialysis (“HD”) in general uses diffusion to remove waste products from a patient&#39;s blood. A diffusive gradient that occurs across the semi-permeable dialysis membrane between the blood and an electrolyte solution called dialysate causes diffusion. Hemofiltration (“HF”) is an alternative renal replacement therapy that relies on a convective transport of toxins from the patient&#39;s blood. This therapy is accomplished by adding substitution or replacement fluid to the extracorporeal circuit during treatment (typically ten to ninety liters of such fluid). That substitution fluid and the fluid accumulated by the patient in between treatments is ultrafiltered over the course of the HF treatment, providing a convective transport mechanism that is particularly beneficial in removing middle and large molecules (in hemodialysis there is a small amount of waste removed along with the fluid gained between dialysis sessions, however, the solute drag from the removal of that ultrafiltrate is not enough to provide convective clearance). 
     Hemodiafiltration (“HDF”) is a treatment modality that combines convective and diffusive clearances. HDF uses dialysate to flow through a dialyzer, similar to standard hemodialysis, providing diffusive clearance. In addition, substitution solution is provided directly to the extracorporeal circuit, providing convective clearance. 
     Home hemodialysis (“HHD”) has to date had limited acceptance even though the clinical outcomes of this modality are more attractive than conventional hemodialysis. There are benefits to daily hemodialysis treatments versus bi- or tri-weekly visits to a treatment center. In certain instances, a patient receiving more frequent treatments removes more toxins and waste products than a patient receiving less frequent but perhaps longer treatments. 
     In any of the blood therapy treatments listed above, there is an art that goes along with the science. Different patients will respond differently to the same therapy. In centers where most dialysis takes place, a patient&#39;s therapy is honed over time with the aid of staff clinicians or nurses. With home therapy, the patient will visit a clinician&#39;s or doctor&#39;s office on a regular, e.g., monthly basis, but will not typically have a nurse or clinician at home to help optimize the therapy. For this reason, a mechanism to aid in optimizing a hemodialysis or other blood therapy treatment early on after beginning dialysis is desirable. 
     Home therapy or HHD also provides the patient with therapy options that the in-center patient does not have. For example, HHD can perform therapy at night if desired, using a single or double needle modality. Any therapy, including a nighttime therapy can be performed over an amount of time that the patient can elect. Because the patient does not have to commute to a center, the patient can perform therapy on days that are convenient for the patient, e.g., weekend days. Similarly, the patient can choose a therapy frequency, or number of therapies per week, that is most convenient and/or most effective. With the added flexibility comes questions, for example, the patient may wonder whether it is better to run six therapies a week at 2.5 hours per therapy or five therapies a week at three hours per therapy. For this additional reason, not only is a way to help optimize a hemodialysis or other blood therapy treatment upfront desirable, it is also desirable to know what will happen when therapy parameters for the optimized therapy are changed. 
     Optimizing hemodialysis therapies for a hemodialysis patient can also be done by knowing the serum phosphorus levels of the hemodialysis patient during and outside of a hemodialysis treatment session. However, serum phosphorus levels will vary depending on the type of hemodialysis patient and the characteristics of the hemodialysis treatment sessions. 
     Plasma or serum (the two terms will be used interchangeably) phosphorus kinetics during HD treatments cannot be explained by conventional one or two compartment models. Previous approaches have been limited by assuming that the distribution of phosphorus is confined to classical intracellular and extracellular fluid compartments. More accurate kinetic models able to describe phosphorus kinetics during HD treatments and the post-dialytic rebound period during short HD (“SHD”) and conventional HD (“CHD”) treatment sessions can be used to predict steady state, pre-dialysis serum phosphorus levels in patients treated with HD therapies. This information can be useful in determining optimal treatment regimens for hemodialysis patients. 
     SUMMARY 
     The present disclosure sets forth two primary embodiments, a first primary embodiment, under Roman Numeral I below, the present disclosure sets forth systems and methods for a renal failure blood therapy, such as, hemodialysis, hemofiltration, hemodiafiltration, and in particular for home hemodialysis (“HHD”). The systems and methods of the present disclosure have in one embodiment three primary components, which can be stored on one or more computer, such as the computer for a doctor, clinician or nurse (referred to herein collectively as “doctor” unless otherwise specified). The doctor&#39;s computer, can be in data-networked communication with the renal failure therapy machine, e.g., via an internet, a local or a wide area network. Or, the output of the doctor&#39;s computer can be stored on a portable memory device such as a universal serial bus (“USB”) drive that is taken and inserted into the renal failure therapy machine. The first component is an estimation component. The second component is a prediction component. The third component is an optimization component. The output of the estimation component is used as an input to both the prediction and optimization components. The prediction and optimization components can both be used to determine therapy prescriptions that will yield suitable solute clearances, e.g., for urea, beta 2-microglobulin (“β2-M”) and phosphorus or phosphate (the two terms will be used interchangeably). The doctor then consults with the patient to pick one or more chosen prescription that the patient believes best fits the patient&#39;s lifestyle. 
     As shown below, the inputs to the optimization component are the therapy outcomes desired by the doctors. The output of the optimization component is one or more suitable therapy prescription for the patient to be run on a renal blood treatment machine, such as a hemodialysis, hemofiltration, or hemodiafiltration machine. The therapy prescription can set therapy parameters, such as (i) therapy frequency, e.g., number of therapies per week, (ii) therapy duration, e.g., one hour to eight hours, (iii) therapy type, e.g., single needle versus dual needle, (iv) dialysate flowrate and/or overall volume of fresh dialysate used during therapy, (v) blood flowrate, (vi) ultrafiltration rate and/or overall volume of ultrafiltrate removed during therapy, (vii) dialysate composition, e.g., conductivity, and (viii) dialyzer or hemofilter parameters, such as dialyzer clearance capability or flux level. 
     The initial or estimation component includes a test that is run on the patient while the patient is undergoing the therapy of choice, e.g., hemodialysis, hemofiltration, or hemodiafiltration. The test uses a set of therapy prescription parameters, such as, treatment time, blood flowrate and dialysate flowrate. While the patient is undergoing treatment, blood samples are taken at various times over the treatment, e.g., at half-hour, forty-five minute or hour intervals. The samples are analyzed to determine the level of certain therapy markers, such as urea concentration (small molecule), beta 2-microglobulin (“β2-M”) (middle molecule), and phosphate (certain dialysis therapies can remove too much phosphate, so it is desirable to know if the patient may be predisposed to this phenomenon). In general, the concentration of each of the markers will lower over time as the urea, β2-M and phosphate are cleared from the patient&#39;s blood. 
     The concentration or clearance results are then fed into a series of models or algorithms for the estimation component to determine a set of estimated patient parameters for (i) the particular patient, (ii) the particular molecule and (iii) its corresponding algorithm. For example, one of the parameters is G, which is the generation rate for the particular solute or molecule produced as a result of dietary intake, e.g., food and drink. K D , another estimated patient parameter, is dialyzer clearance for the molecule. K IC , a further estimated parameter, is the patient&#39;s inter-compartmental mass transfer-area coefficient for the molecule or solute. K M , another estimated parameter, is the mobilization clearance of phosphorus determining the rate at which phosphorus is released into the extracellular space. V, yet another parameter that may be estimated, is the distribution volume of phosphorus. Other estimated parameters are discussed below. 
     The prediction component uses the estimated patient parameters fed into the modules or algorithms from the estimation component or module to calculate clearance results for one or more solute, e.g., urea, β2-M and/or phosphate over a varied set of therapy prescription parameters. The prediction component also uses assumed patient parameters. For example, K NR , is the patient&#39;s residual kidney coefficient for the molecule or solute (non-renal clearance of the solute is also often included into this term), which can be considered a constant, such that it does not have to be estimated on an individual basis. Shown in detail below are graphs illustrating the output of the prediction component, in which a combination of therapy duration and therapy frequency is graphed along the x-axis and a solute concentration, e.g., urea, β2-M or phosphate is graphed along the y-axis. The graphs provide (i) a visual cue to the average concentration level for the solute and (ii) estimate the maximum concentration level that the solute will reach. The graphs can be used to determine one or more clinically acceptable parameters, such as standard Kt/V of urea and mean pre-treatment plasma concentration (“MPC”) of β2-M and pre-dialysis plasma concentration of phosphorus, all of which in turn help identify the appropriate and customized therapy prescription. The doctor can communicate the acceptable prescriptions to the patient, who then picks one or more chosen prescription for downloading to the HHD machine. 
     The optimization component operates the reverse of the predication component and instead inputs desired solute concentration levels into the models of the estimation component, using the estimated parameters of the estimation component to determine an optimized set of therapy prescription parameters for the patient. The optimization takes into account the doctor&#39;s desired solute concentration clearance for the patient for one or more solutes and the patient&#39;s preference, e.g., as to therapy frequency and duration. In one example, the optimization component inputs a doctor&#39;s requirement for urea, β2-M and phosphate clearances, which yields a plurality of therapy prescriptions that meet the clearance requirements. The doctor and patient can then view the acceptable therapy prescriptions and select one or more chosen prescription for loading into the renal failure blood therapy, e.g., HHD, machine. 
     Thus, the prediction and optimization components can both lead to chosen therapy prescriptions that are downloaded to the HHD machine. The optimization component may be easier to use to choose a suitable therapy prescription than the prediction components because it is less iterative (for the doctor) than the prediction component. However, the prediction component can provide more detailed information for the doctor and for a particular therapy prescription. Thus, in one particularly useful implementation of the present disclosure, the system estimates, optimizes and allows for choice and then predicts detailed results for the optimized and chosen therapy prescription. 
     It is contemplated that the patient&#39;s estimated patient parameters, e.g., G, V D , K IC  and K M  be updated periodically to adjust for a changing condition of the patient and to adjust for actual data obtained from past therapies. For example, the patient can have blood work done periodically, such that the downloaded prescription can be changed if the results of the blood work warrant such a change. Thus, the three components can be cycled or updated periodically, e.g., once or twice a year or as often as the doctor finds necessary. 
     In the second primary embodiment under Roman Numeral II below, the present disclosure sets forth methods of predicting serum phosphorus concentrations in a patient during hemodialysis. In one embodiment, the method includes measuring serum phosphorus concentrations (“C”) of the patient over a hemodialysis treatment session time and an ultrafiltration rate (“Q UF ”) calculated by a difference between pre- and post-dialytic body weight of the patient during an initial hemodialysis treatment session divided by a total treatment time of the treatment session, estimating a phosphorous mobilization clearance (“K M ”) and a pre-dialysis distribution volume of phosphorus (“V PRE ”) for the patient, and predicting C of the patient at any time during any hemodialysis treatment session with the estimated K M  and V PRE  of the patient. C of the patient can be measured every 15 or 30 minutes during the hemodialysis treatment session. 
     In another embodiment, a computing device for predicting serum phosphorus concentrations in a patient during hemodialysis includes a display device, an input device, a processor, and a memory device that stores a plurality of instructions, which when executed by the processor, cause the processor to operate with the display device and the input device to: (a) receive data relating to C of a hemodialysis patient over a hemodialysis treatment session time and an Q UF  calculated by a difference between pre- and post-dialytic body weight of the hemodialysis patient during a hemodialysis treatment session divided by a total treatment time of the treatment session, (b) estimate K M  and V PRE  for the patient, and (c) predict C of the patient at any time during hemodialysis. The processor can operate with the display device and the input device to receive data relating to at least one of K R , K D  or a sampling time for collecting the serum phosphorus concentration. 
     In yet another embodiment, a method of determining steady state, pre-dialysis serum phosphorus levels in a hemodialysis patient includes obtaining a net generation of phosphorus (“G”) of the hemodialysis patient, determining steady state, pre-dialysis serum phosphorus levels (“C SS-PRE ”) of the hemodialysis patient, and simulating the effect of at least one of a patient parameter or a treatment parameter on C SS-PRE  of the hemodialysis patient. 
     In an embodiment, a computing device for determining steady state, pre-dialysis serum phosphorus levels in a hemodialysis patient includes a display device, an input device, a processor, and a memory device that stores a plurality of instructions, which when executed by the processor, cause the processor to operate with the display device and the input device to: (a) receive data relating to G from at least a dietary phosphorus intake of a hemodialysis patient or a urea kinetic modeling of the hemodialysis patient, (b) determine C SS-PRE  of the patient, and (c) simulate the effect of at least one of a patient parameter or a treatment parameter on the C SS-PRE  of the hemodialysis patient. The processor can operate with the display device and the input device to receive data relating to at least one of K R , K D , K M , V PRE , t tx , F, C PRE  about a month before a hemodialysis treatment session or a sampling time for collecting the serum phosphorus concentration. The computing device can display a treatment regimen of the hemodialysis patient so that C SS-PRE  is within a desired range. 
     An advantage of the present disclosure is accordingly to provide improved renal failure blood therapy systems and methods. 
     Another advantage of the present disclosure is to provide renal failure blood therapy systems and methods having a therapy prediction tool with which clinicians may adjust renal failure blood therapies, e.g., HHD therapies, for specific patients with respect to the key solute clearance measures. 
     Yet another advantage of the present disclosure is to provide renal failure blood therapy systems and methods that offer clinicians multiple choices to achieve desired target goals. 
     Still another advantage of the present disclosure is to provide renal failure blood therapy systems and methods that employ a clinically viable and practical test procedure to help characterize the patient&#39;s response to a particular renal blood therapy. 
     A further advantage of the present disclosure is to provide renal failure blood therapy systems and methods that help to reduce the amount of trial and error in optimizing a blood therapy for the patient. 
     Still a further advantage of the present disclosure is to provide renal failure blood therapy systems and methods that aid the patient in attempting to optimize lifestyle choices for therapy. 
     Another advantage of the present disclosure is to accurately predict plasma phosphorus levels in a patient during short, conventional daily and nocturnal hemodialysis. 
     Yet another advantage of the present disclosure is to accurately predict steady state, pre-dialysis plasma phosphorus serum levels in a patient who is maintained by a hemodialysis therapy for a specified time. 
     Still another advantage of the present disclosure is to develop or modify hemodialysis treatment regimens so that the plasma phosphorus serum levels of a hemodialysis patient are maintained within a desired range. 
     A further advantage of the present disclosure is provide systems that develop or modify hemodialysis treatment regimens so that the plasma phosphorus serum levels of a hemodialysis patient are maintained within a desired range. 
     Yet a further advantage of the present disclosure is to conserve dialysate and other dialysis supplies, streamlining the therapy and lowering therapy cost. 
     Additional features and advantages are described herein, and will be apparent from, the following Detailed Description and the figures. 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
         FIG. 1  is a schematic overview of one embodiment of a renal failure blood therapy system and method of the present disclosure. 
         FIG. 2  is an example of a system component selection screen of one embodiment of a renal failure blood therapy system and method of the present disclosure. 
         FIG. 3  is an example of a patient parameter estimation input screen of one embodiment of a renal failure blood therapy system and method of the present disclosure. 
         FIG. 4  is an example of a patient parameter estimation output screen of one embodiment of a renal failure blood therapy system and method of the present disclosure. 
         FIG. 5  is an example of a therapy prediction input screen of one embodiment of a renal failure blood therapy system and method of the present disclosure. 
         FIG. 6  is an example of a therapy prediction output screen of one embodiment of a renal failure blood therapy system and method of the present disclosure. 
         FIG. 7  is an example of a therapy optimization input screen of one embodiment of a renal failure blood therapy system and method of the present disclosure. 
         FIG. 8A  is an example of a therapy optimization routine screen of one embodiment of a renal failure blood therapy system and method of the present disclosure. 
         FIG. 8B  is another example of a therapy optimization routine screen of one embodiment of a renal failure blood therapy system and method of the present disclosure. 
         FIGS. 9, 9A and 9B  are a schematic flow diagram summarizing some of the features for the renal failure blood therapy system and method of the present disclosure. 
         FIG. 10  illustrates an embodiment of a computing device of the present disclosure. 
         FIG. 11  is a conceptual description of the pseudo one-compartment model. 
         FIG. 12  shows the modeled and measured plasma phosphorus concentrations for patient  1  during short HD and conventional HD treatments. 
         FIG. 13  shows the modeled and measured plasma phosphorus concentrations for patient  2  during short HD and conventional HD treatments. 
         FIG. 14  shows the modeled and measured plasma phosphorus concentrations for patient  3  during short HD and conventional HD treatments. 
         FIG. 15  shows the modeled and measured plasma phosphorus concentrations for patient  4  during short HD and conventional HD treatments. 
         FIG. 16  shows the modeled and measured plasma phosphorus concentrations for patient  5  during short HD and conventional HD treatments. 
         FIG. 17  is a conceptual model used to describe steady state phosphorus mass balance over a time-averaged period. 
         FIG. 18  illustrates the effect of treatment frequency per se on pre-dialysis serum phosphorus concentration as a function of dialyzer phosphate clearance. 
         FIG. 19  shows the effects of an increase in treatment time and frequency with reference to nocturnal forms of hemodialysis. 
         FIG. 20  shows the effect of increasing treatment frequency and treatment time relevant to short daily hemodialysis on pre-dialysis serum phosphorus concentration—K M =50 ml/min. 
         FIG. 21  shows the effect of increasing treatment frequency and treatment time relevant to short daily hemodialysis on pre-dialysis serum phosphorus concentration—K M =50 ml/min. 
         FIG. 22  shows the effect of increasing treatment frequency and treatment time relevant to short daily hemodialysis on pre-dialysis serum phosphorus concentration—K M =150 ml/min. 
         FIG. 23  shows the effect of increasing treatment frequency and treatment time relevant to short daily hemodialysis on pre-dialysis serum phosphorus concentration—K M =150 ml/min. 
         FIG. 24  shows the relationships between urea stdKt/v, β2-M MPC and therapy duration. 
     
    
    
     DETAILED DESCRIPTION 
     I. Therapy Estimation, Prediction and Optimization 
     Referring now to the drawings and in particular to  FIG. 1 , system  10  illustrates one optimization system and method of the present disclosure for implementing a therapy prescription into a renal failure blood therapy machine  100 , such as a hemodialysis machine, and in particular home hemodialysis (“HHD”) machine. One particularly well suited HHD machine to operate the systems and methods of the present disclosure is set forth in the following United States patent applications: (i) U.S. Pub. No. 2008/0202591, (ii) U.S. Pub. No. 2008/0208103, (iii) U.S. Pub. No. 2008/0216898, (iv) U.S. Pub. No. 2009/0004033, (v) U.S. Pub. No. 2009/0101549, (vi) U.S. Pub. No. 2009/0105629, U.S. Pub. No. 2009/0107335, and (vii) U.S. Pub. No. 2009/0114582, the contents of each of which are incorporated herein expressly by reference and relied upon. The HHD machine includes at least one processor and at least one memory that are specifically modified or programmed to accept a therapy prescription to run, which is prescribed by a doctor, clinician or nurse (for convenience referred to collectively hereafter as a doctor  14  unless otherwise specified). The therapy is downloaded to the HHD machine (for convenience “HHD machine” refers collectively hereafter to a home hemodialysis, home hemofiltration, home hemodiafiltration or a continuous renal replacement therapy (“CRRT”) machine unless otherwise specified), e.g., via a memory storage device, such as a flash drive or universal serial bus (“USB”) drive, or via an internet or other local or wide area data network. 
     A patient  12  may be provided with multiple suitable therapies and be allowed to choose between the therapies, e.g., based upon the patient&#39;s schedule that day or week. United States patent applications that disclose the provision of and selection from multiple peritoneal dialysis therapies include: (i) U.S. Pub. No. 2010/0010424, (ii) U.S. Pub. No. 2010/0010423, (iii) U.S. Pub. No. 2010/0010426, (iv) U.S. Pub. No. 20100010427, and (v) U.S. Pub. No. 20100010428, each assigned to the assignee of the present disclosure, and the entire contents of each of which are incorporated herein by reference and relied upon. 
     System  10  includes three components, namely, an estimation component  20 , a prediction component  40  and an optimization component  60 . 
     Estimation Component 
     Estimation component  20  includes a patient test  22 , which involves actually performing the therapy of choice, e.g., a hemodialysis, hemofiltration, hemodiafiltration or CRRT therapy on patient  12  and taking blood samples at different times, e.g., at five different times, during the actual therapy. The test therapy can for example be a four-hour therapy, in which the sample times are, for example, at beginning of the test, ½ hour, one hour and two hours; or the test therapy can for example be an eight-hour therapy, in which the samples are taken at the beginning of therapy, and at one hour, two hours, four hours, five hours, six hours and eight hours. The test therapy durations and number of samples can be varied as desired. 
     The blood samples are each analyzed to determine the level of concentration of certain marker solutes, such as, urea  24  (small molecule), beta 2-microglobulin (“β2-M”)  26  (middle molecule), and phosphate  28 . It is known that for certain patients running certain dialysis therapies, e.g., longer therapies, that too much phosphate can be removed from the patient, to the point that in some instances phosphate has to be returned to the patient. System  10  contemplates the determination of whether the patient may be predisposed to experiencing low phosphate levels. 
     The concentration levels determined at known times during therapy are fed into a series of models or algorithms  30 , one model for each solute of concern, e.g., a first model  30   a  for urea  24 , a second model  30   b  for β2-M  26  and a third model  30   c  for phosphate  28 . Models  30  (referring collectively to models  30   a ,  30   b ,  30   c , . . .  30   n ) can each include one or more algorithm. Suitable models  30  are discussed herein. 
     The output of estimation component  20  includes a multitude of estimated patient parameters  32  for each solute  24 ,  26 , and  28  of concern, which are based on the patient&#39;s blood results using models  30  and are therefore specific to the patient. While it is important to the present disclosure that the estimated patient parameters  32  are tailored to the patient&#39;s physiologic make-up, the doctor may feel that a blood test is too strenuous or invasive. Thus, system  10  also contemplates the estimating parameters  32  instead using empirical data, e.g., typical parameter values for the patient based on for example age, weight, sex, blood type, typical blood pressure, height, therapy duration, nutritional status, and disease information related to the kinetics of the solute(s) of interest. It is believed that this data can be developed over time using system  10 . Important parameters for models  30  of system  10 , include estimated patient parameters  32  and known, assumed or calculated (outside of the model) values  42 , such as: 
     K IC , which is the patient&#39;s inter-compartmental diffusion coefficient for the molecule or solute and is an estimated parameter  32 ; 
     K D , which is a known dialyzer clearance for the particular molecule or solute and can be a parameter  42  calculated outside the model; 
     K M , which is the patient&#39;s phosphorus mobilization clearance and is an estimated parameter  32 ; 
     K NR , which is the patient&#39;s residual kidney coefficient for the particular molecule or solute and may be a parameter  42  assumed to be a constant; 
     V is the distribution volume of phosphorus and is an estimated parameter  32 ; 
     G, which is the generation rate for the particular solute or molecule produced by patient&#39;s intake and is an estimated parameter  32  or may be an assumed parameter  42 ; 
     V P , which is the perfused or extracellular volume, is an estimated parameter  32 ; 
     V NP , which is the non-perfused or intracellular volume, is an estimated parameter  32 ; 
     V D , which is the solute distribution volume in the body, equal to V P +V NP  for urea and beta2-microglobulin, is an estimated parameter  32 ; 
     C P , which is the extracellular concentration of the solute, is a parameter  42  determined from test  22  and is therefore a known in the models  30  of estimation component  20  (it should be noted that not only can C P  be a measured estimation component  20 , C P  can also be predicted along with Cnp by a prediction module, after which one can compare measured and predicted versions of C P , etc., to gauge the performance of system  10 ); 
     C NP , which is the intracellular concentration of the solute and cannot be measured, is not a result of test  22  and is not an input to the models, it is instead a predicted output from the prediction and optimization components; 
     Φ NP , which is a ratio of intracellular compartment volume to the total distribution volume and Φ P , which is the ratio of extracellular compartment volume to the total distribution volume, both of which are known parameters  42  from literature; and 
     α, which is an interdialytic fluid intake, i.e., water intake, is a parameter  42  calculated outside of models  30  based on average fluid intake, or weight gain. 
     As shown above, there are at least seven estimated parameters  32 , namely, K IC , K M , V, G, V P , V NP , and V D , where V NP  and V D  are related to V D  through Φ P  and Φ NP . For convenience, only three of them are illustrated in  FIG. 1 , namely, K IC , V P  and V NP . It should be appreciated that any one, some or all of the seven parameters can be estimated via estimation component  20  and models  30 . It is contemplated to allow desired parameters for estimation to be chosen by the doctor, e.g., via selection boxes like those shown below in  FIG. 5  for the solutes under the prediction component  40 . 
     There are also operational parameters  44  discussed herein, such as, blood flowrate, dialysate flowrate, dialysate total volume, ultrafiltration flowrate and ultrafiltration volume, and also operational parameters  44  affecting the patient&#39;s life style, such as: 
     T, which is in one instance the time duration at which each sample is taken in estimation component  20 , and is therefore known parameter  42  for the test  22  of estimation component  20 , and is in another instance the duration of dialysis in the prediction  40  and optimization  60  components; and 
     F, which is the frequency of therapy, is taken to be one for the single therapy of test  22 , but is varied in the prediction  40  and optimization  60  components. 
     Prediction Component 
     The estimated patient parameters  32  are then fed back into the models in prediction component  40 , particularly into the personalized solute flux and volume flux routine  50 . Personalized solute flux and volume flux routine  50  uses in essence the same models or algorithms  30  of component  20 , but here using the estimated patient parameters  32  of estimation component  30 , as inputs, making the parameters  32  knowns instead of variables. 
     As shown in  FIG. 1 ., the patient prediction component  40  accepts other known, assumed or calculated (outside of the model) values  42  for the particular solute, such as dialyzer clearance K D , into personalized solute flux and volume flux routine  50 . What is left as unknown with routine  50  are (i) variable prescription operational parameters  44 , such as dialysis duration (“T”) and dialysis frequency (“F”) and (iii) solute concentrations  52  (C for both intracellular and extracellular) for solutes  24 ,  26  and  28 . Other machine operating parameters  44  that may be inputted into prediction component  40  and varied include blood flowrate, dialysate flowrate, dialysate total volume, ultrafiltration flowrate and ultrafiltration volume. The solute distribution volume and total body water volume are not constant throughout therapy. System  10  accordingly uses a variable-volume model allowing system  10  to change during the simulated therapy duration. Prediction component  40  then computes C P , C NP , V P , and V NP  based on the given input variables such F, T, K D , etc. Patient prediction component  40  plugs in different realistic values for operational parameters  44 , e.g., as x-axis values of a graph, and outputs solute concentrations  52 , e.g., as y-axis values of the graph. 
     The graphs allow the doctor to view how the concentration  52  of a certain solute varies over the course of, e.g., a week, for a particular set of prescription operational parameters  44 . The doctor can see an average value or other accepted measure of quantification for the concentration  52 , e.g., a standardized Kt/v for the clearance of urea. The doctor can also see the peak value for the concentration  52  over the course of a therapy cycle. 
     The doctor can vary therapy duration and frequency input parameters to develop multiple sets of graphs of the outputted solute concentrations, for example, set  1  of graphs for urea, β2-M and phosphate for therapy duration 1 and therapy frequency  1 , and set  2  of graphs for urea, β2-M and phosphate for therapy duration 2 and therapy frequency  2 . If desired, each set of graphs can be merged onto a single graph, e.g., urea, β2-M and phosphate concentration on one single graph for therapy duration 1 and therapy frequency  1 . The therapy duration(s) and frequency(ies) that yield suitable solute concentrations can then be communicated to the patient, who in turn applies life style preferences  54  to yield one or more chosen therapy prescription  56  for downloading to an HHD machine  100 . The patient or doctor then selects one of the prescriptions, e.g., on a weekly basis to run for treatment. In other examples, blood and dialysate flowrates may also be adjusted to reach certain clearance goals or to suit the patient&#39;s needs. 
     It is also expressly contemplated to optimize the visual outlay and functionality for the doctor, that is, to optimize the look and operation of the graphs and tables, for example, to only allow values for desired adequacy parameters to be manipulated. System  10  can manipulate these values, so as to be customized to each doctor&#39;s needs and preferences. The screens shown herein are accordingly intended to be examples. The examples are not intended to limit the invention. 
     Optimization Component 
     Optimization component  60  inputs a plurality of therapy targets  62 , such as target removal of urea  24 , target removal of β2-M, target removal of phosphate  28 , and target removal of ultrafiltration (“UF”) or excess water that has built inside the patient between treatments. Therapy targets  62  are entered into an optimization routine  70 . In one embodiment, optimization routine  70  uses in essence the kinetic models or equations  30  discussed above for estimation component  20 , which like with routine  50 , have entered the estimated patient parameters  32  obtained from estimation component  20 . Then, calculations for each solute are made in the reverse of prediction component  40 . That is, instead of entering prescription operational parameters  44  and calculating solute concentration  52 , a desired solute concentration  52  is entered and operational parameters  44 , which will satisfy the desired or optimized solute concentration  52  are calculated. Here, the results  72  of optimization component  60  are independent of, or more precisely the reverse of, the results of prediction component  40 . Optimization routine  70  identifies one or more therapy prescription  72  that meets the desired or optimized solute concentration  52  for each designated solute. 
     In particular, the computational techniques using optimization routine  70  of optimization component  60  have been found to be robust and stable procedures that identify the therapy conditions that achieve the aimed input target values (e.g. β2-M pre-dialysis serum concentration (“MPC”), urea standard Kt/v (std Kt/v), and phosphate steady state pre-dialysis serum concentration) by the clinician. The computational techniques identify multiple optimized therapy prescriptions and attempt to do so by performing the least number of iterative simulations. Outputted therapy parameters from optimization component  60  can include therapy duration (“T”), therapy frequency (“F”), blood and dialysate flow rates (“Q B ” and “Q D ”, respectively). 
     In one example, the target (i.e., input) urea stdKt/v and β2-M MPC may be set to be 2.0 and 27.5 mg/L respectively.  FIG. 24  shows the relationships (e.g., curves) between urea stdKt/v, β2-M MPC and therapy duration. In the example, the input target values are indicated by dotted lines. Optimization routine  70 , following a relatively easy and iterative procedure, varies therapy duration T (for a given set of F, Q B , and Q D ) until both urea stdKt/v and β2-M MPC target values are satisfied at the minimum necessary T, which is presumed to be the optimal T for both the patient and the hemodialysis machine because the time the patient is connected to the machine and the time the machine needs to run and consume dialysate components is minimized. 
     In a first sample iteration, the optimization routine  70  performs a simulation at T 1 =600 minutes, a therapy duration generally long enough to produce adequacy parameters much better than desired. In a second sample iteration, optimization routine  70  performs a simulation at T 2 =600/2=300 minutes, producing satisfactory results. In the third step, optimization routine  70  performs a simulation at T 3 =300/2=150 minutes, this time producing unsatisfactory results for both stdKt/v and β2-M MPC. In a fourth iteration, optimization routine  70  performs a simulation at an increased time T 4 =(150+300)/2=225 minutes, producing satisfactory result for stdKt/v only. In a fifth iteration, optimization routine  70  performs a simulation at a further increased time T 5 =(225+300)/2=263 minutes, producing satisfactory results for both stdKt/v and β2-M MPC. 
     At the end of each step, if both target parameters are achieved, the optimization routine  70  in one embodiment calculates the difference between the target and achieved values. If the difference for at least one of the target parameters is greater than a threshold value, then optimization routine  70  performs yet another iteration to achieve results closer to the target values, further lessening and optimizing duration T. Using the above procedure, optimization routine  70  performs a final simulation at T=(263+225)/2=244 minutes (bold vertical line), where both stdKt/v and β2-M MPC targets are satisfied and the differences between achieved stdKt/v and β2-M MPC and target stdKt/v and β2-M MPC are small. 
     As illustrated, the optimal therapy duration T of 244 minutes is found in only six iterations, again for a given set of F, Q B  and Q D . Multiple optimized therapy prescriptions can then be identified, e.g., varying therapy duration, frequency, blood and/or dialysate flowrates, to allow the patient a choice based on lifestyle as discussed below. 
     Patient  12  and doctor  14  review the therapy prescriptions that meet the desired or optimized solute concentration  52  and factor in the patient&#39;s life style preferences  74 . Perhaps patient  12  prefers a short daily therapy during the day when the patient&#39;s spouse is awake for assistance. Or perhaps the patient works out on Monday, Wednesday and Friday and has less UF due to sweat on those days, preferring then to run treatments on other days. 
     Applying lifestyle preferences  74  to the therapy prescriptions  72  that meet the desired or optimized solute concentration  52  yields a chosen one or more therapy prescription  76 . Chosen therapy prescription  56  and  76  can be downloaded to machine  100 , e.g., via manual entry into machine  100 , a download from a memory storage device, e.g., flash drive or universal serial bus (“USB”) drive, or a download from a data network such as an internet. 
     It is contemplated to modify a chosen one or more therapy prescription  56  or  76  from time to time, e.g., due to regular and periodic blood testing  78 , which the patient has performed from time to time. The patient may lose residual renal function over time causing the chosen one or more therapy prescription  56  or  76  to need to be modified. The blood work may in any case indicate that the chosen one or more therapy prescription  56  or  76  is not removing one or more solute effectively enough, prompting a change. The patient may lose weight or undergo a lifestyle change, which allows for a less rigorous one or more therapy prescription  56  or  76  to be used instead. In any case, it is contemplated that lifestyle preferences  74  will continue to play a role in potentially modifying the selected one or more therapy prescription  76 . 
     Sample Screen Shots 
       FIGS. 2 to 8B  are sample screenshots further illustrating system  10  described in connection with  FIG. 1 . The screen shots of  FIGS. 2 to 8B  can be custom generated per the request of the doctor and can be implemented on the processing and memory of one or more computer used by the doctor, clinician or nurse, which can be in data networked communication with the HHD machine  100 , e.g., via an internet, local or wide area network. It is also contemplated, especially for in-center machines, to implement system  10  and the screen shots of  FIGS. 2 to 8B  at the one or more processing and memory of machine  100 . 
       FIG. 2  illustrates a sample startup screen that allows the doctor to choose whether to enter or work with the patient parameter estimation component  20 , the therapy prediction component  40  or the therapy optimization component  60 . System  10  was described above from start to finish. It may be however that patient parameter estimation component  20  has already been performed, such that the doctor can jump directly to either the prediction component  40  or the therapy optimization component  60 . As discussed above, therapy optimization component  60  can operate independent of prediction component  40 . Thus, it may be that the doctor only uses one of prediction component  40  and optimization component  60  at a particular time or application of system  10 . 
     Prediction component  40  and therapy optimization component  60  rely on information from patient parameter estimation component  20 , however, it should be noted that if the patient  12  does not wish to undergo test  22 , or the doctor  14  does not want the patient  12  to undergo test  22 , it may be possible, although not preferred, to use standardized values based on the patient&#39;s information, such as age, sex, blood type, residual renal function if known, etc. It is also expressly contemplated to maintain a database of estimated patient parameters  32  developed over time using system  10 , which may provide viable standardized estimated patient parameters  32  based on patient category. 
       FIG. 3  illustrates a sample data entry and test  22  result screen for parameter estimation component  20 . The screen at the left accepts patient information, such as name, age, gender and weight. It is contemplated for system  10  to be able to search for a file under any of this inputted data. The doctor enters data for test  22  into the middle of the screen of  FIG. 3 , such as total treatment time, blood flowrate, dialysate flowrate (or total volume) and UF rate or volume (not illustrated). The patient then undergoes a test therapy that is run according to this inputted data. The screen of  FIG. 3  at the right then accepts the results of the blood testing done for urea  24 , β2-M  26  and phosphate  28  at various times over the course of the treatment, forming a time-based profile for each of the analyzed solutes. The sample times show in  FIG. 3  include a starting time, and one hour, two hours, four hours, five hours, six hours and eight hours from the starting time. Other intervals including more or less time entries can be used alternatively. 
       FIG. 4  illustrates a sample estimated patient parameters  32  display screen. Estimated patient parameters  32  can include, for example, generation rate G, intracellular clearance K IC , phosphorus mobilization clearance K M , and distribution volume V D , (V D =V P +V NP , where V P  is the perfused or extracellular volume, and V NP  is the non-perfused or intracellular volume). The values for G, K IC , and V D , are the outputs of models  30  of estimation components  20 , and are then used as inputted data in components  40  and  60  as estimated patient parameters  32 . 
       FIG. 5  illustrates a sample input screen for prediction component  40 . In the illustrated example, the doctor at the left of the screen chooses to run the prediction routine for urea  24  and β2-M (boxes checked), but not for phosphate  28  (boxes not checked). The doctor also enters operational inputs  44 , namely, the doctor wishes to model a therapy that is run five days a week (i.e., F) for three hours (i.e., T) per session. As discussed above, other machine operating parameters that may be entered (and varied) alternatively or additionally to F and T include blood flowrate, dialysate flowrate, dialysate total volume, ultrafiltration flowrate and ultrafiltration total volume. “Back” and “Run” buttons allow the doctor to navigate through each of the components  20 ,  40  or  60  when the particular component are selected. 
       FIG. 6  illustrates a sample output screen for prediction component  40 , which shows the results after running the personalized solute flux and volume equations  50 . If desired, the concentration results can be merged onto a single graph, for example, with the urea concentration scale along the left and β2-M scale along the right. Solute concentrations  52  could alternatively be displayed in spreadsheet form but are shown in  FIG. 6  in graphical form, with days from a start of using the particular therapy prescription along the x-axis. This way, the doctor can readily see the predicted solute profile for a given frequency and duration, and for the patient&#39;s personalized estimated parameters. Solute concentrations  52  are also shown in an average or standardized form, e.g., as a standard Kt/v, which is understood by those of skill in the art. Knowing peak concentration and the average or standardized concentration, the doctor can quickly determine if the proposed frequency and duration are adequate for the selected solutes, here, urea  24  and β2-M  26 . As delineated in  FIG. 6 , P is for perfused or extracellular, NP is for non-perfused or intracellular. If the solute, for example urea, is in the extracellular or blood volume then the dialyzer can readily clear the solute. If the solute is in the intracellular volume, then the solute has to first pass in to the extracellular volume overcoming the resistance defined by K IC . 
       FIG. 6  illustrates concentration values for a particular therapy duration T and frequency F. It is contemplated for the doctor to re-run prediction component  40  to vary T and F. The doctor can then choose one or more sets of graphs, e.g., from (i) T 1 , F 1 ; (ii) T 2 , F 2 ; (iii) T 3 , F 3 ; etc., that are clinically acceptable. The acceptable graphs or their corresponding therapy prescriptions can then be reviewed with the patient, who selects one or more graph or prescription that best meets the patient&#39;s lifestyle needs and/or requirements. 
       FIG. 7  illustrates a sample input screen for optimization component  60 . Running opposite to prediction component  40 , the doctor in optimization component  60  enters desired values for therapy results, e.g., a desired value for urea, e.g., via standard Kt/v, a desired value for phosphorus in, e.g., pre-dialysis phosphorus plasma concentration in milligrams per liter, a desired value for β2-M in, e.g., milligrams per liter, and a desired ultrafiltrate (“UF”) removal value, e.g., in liters. UF is generally a machine-controlled function but can affect solute removal, so the input of UF is desirable for optimization. 
       FIG. 8A  shows an example of an optimization routine  70 , which for inputted urea and β2-M (and phosphate if desired) requirements, shows a spreadsheet of frequency in Days Per Week (along the side) and Therapy Duration in hours (along the top), and places an “X” in the cell corresponding to a treatment that will meet the requirement for that solute. In the twelve possible combinations shown in  FIG. 8 , two (four days of three hour treatments and vice versa) meet the requirements for urea and β2-M requirements. The patient can then decide which option fits his/her lifestyle better. Or, both prescriptions can be machine entered or chosen prescriptions  72 . The patient then decides, for example on a weekly basis, which of the two approved and chosen prescriptions is a better fit for that week. 
       FIG. 8A  also shows example inputted parameters  72  for optimization component  60 , here, resulting blood flowrate, total solution volume, dialysate flowrate (or total volume) and UF rate (or volume), which are used in the equations for all of the frequency combinations in the urea and β2-M optimization routines  70 . The doctor, independent of system  10 , may calculate blood flowrate and dialysate flowrate, etc., to achieve desired K D  values. However, those calculations are independent of the calculations taking place as part of the prediction  40  and/or optimization  60  components. For system  10 , flowrates are inputs to determine K D , which in turn is an input to the prediction and/or optimization components  40  and  60 . 
       FIG. 8B  shows another example of an optimization routine  70 , which for inputted urea and β2-M (and phosphate if desired) requirements, shows a spreadsheet of Therapy Frequency (i.e., F) in days of the week (along side) and Therapy Duration (i.e., T) in hours (along top). Here, the actual days of the week are shown. Optimization component  60  can discern between different combinations of the same number of days, e.g., three days Monday/Wednesday/Friday versus three days Monday/Wednesday/Saturday. In one embodiment, system  10  assumes certain preset days when therapy frequency values are entered. For instance, for an F of three days per week, system  10  would assume, e.g., Monday/Wednesday/Friday. System  10 , however, allows doctor  14  to enter specific days (as opposed to entering F). System  10  makes the calculations according to the days entered. The ability of simulating custom therapy days can be important because system  10  can then more accurately track the accumulation of solutes within the body. 
     In  FIG. 8B , each cell is then color coded or otherwise designated into one of three categories (for example) for the clearance of not just a particular solute, but for the analyzed solutes as a group. The desired standardized Kt/v values for urea located at the upper right of the chart of routine  70  show the group of solute cleaners. Each cell of solute clearances is labeled in the example as inadequate, borderline or adequate. For example, adequate can mean meets all requirements, borderline can mean meets some requirements or almost meets all requirements, while inadequate means misses most or all requirements. More or less than three classifications having different meanings can be used alternatively. The patient can then choose from one of the adequate therapy prescription cells, for example, choose the least rigorous therapy prescription. 
     Referring now to  FIG. 9 , method  110  illustrates the relationship between components  20 ,  40  and  60  of system  10  discussed herein. Method  110  is meant to help understand the interrelationship between components  20 ,  40  and  60  and is in no way meant to describe all the alternatives for the components, which have been described in detail above. 
     At oval  112 , method  110  begins. At block  114 , the test therapy is performed on the patient to determine concentration levels for various solutes, such as urea, β2-M and phosphate. The solutes for system  10  and method  110  are not limited to those three solutes and could include others, such as calcium, parathyroid hormone (“PTH”), and certain protein-bound solutes such as p-cresol sulfate. It is expressly contemplated to include these and other solutes in system  10  and method  110 . The additional solutes can at least be tracked via the testing in estimation component  20  for example as they relate to general adequacy and/or to correlate with phosphate clearance/mobilization. The additional solutes can eventually be predicted via prediction component  40  and optimized via component  60  when models are devised in the future for the additional solutes, as has been done below for phosphate. 
     At blocks  116   a ,  116   b , and  116   c , the concentration levels for urea, β2-M and phosphate are fed into the corresponding kinetic model to determine at least one patient-specific parameter. All of the test concentrations of the solute, less than all of the test concentrations of the solute, or some averaged test concentration for the solute may be entered into the corresponding kinetic model. 
     At blocks  118   a ,  118   b , and  188   c , the at least one patient-specific estimated parameter is fed along with at least one machine operational parameter, such as therapy duration (i.e., T) and therapy frequency (i.e., F), into the corresponding kinetic model for urea, β2-M and phosphate to determine clearance volumes or solution levels for the solute. 
     At blocks  120   a ,  120   b , and  120   c , the solute clearance values for urea, β2-M and phosphate are graphed (could be single combined graph) or tabulated for the doctor&#39;s evaluation. At diamonds  122   a ,  122   b  and  122   c , it is determined whether blocks  118   a  to  118   c  and  120   a  to  120   c  are repeated for another set of inputted operational therapy parameters. If not, at block  124  the doctor determines which graph(s), tabulation(s), prescription(s) are clinically acceptable. 
     At blocks  126   a  to  126   c , the patient-specific estimated parameters for urea, β2-M and phosphate are fed into the corresponding kinetic model along with a desired solute removal level or level for the solute to determine one or more machine operational parameter that will satisfy the equation and achieve the desired level for the one or more solute. 
     At block  128 , the machine operating parameters that achieve the desired solute level (or some of the desired solute levels) are tabulated for the doctor. The doctor can then hone in on the best clinically acceptable therapies for the patient. 
     At block  130 , which is fed from both the predicting block  124  and the optimization block  128 , the doctor consults with the patient to determine which one or more chosen therapy prescription best suits the patient&#39;s personal needs. As discussed above, machine operational parameters include T, F and others, such as fluid flowrates and/or volumes. These other parameters are likely to be mandated by the doctor and not be as negotiable with the patient. To a certain degree, T and F will drive the other parameters. For example, a shorter therapy will likely require higher flowrates. 
     At block  132 , the chosen one or more therapy prescription is downloaded to the patient&#39;s (or clinic&#39;s) renal failure therapy machine (e.g., HHD machine  100 ). If multiple chosen prescriptions are downloaded, the patient may be empowered to choose, e.g., weekly, which prescription to run. Alternatively, the doctor may dictate, at least initially, which prescription to run. The prescription download may be via a data network or data storage device such as USB or flash drive. 
     It should be appreciated that system  10  in alternative embodiments can incorporate any of the methods, models and equations discussed below and in detail in section II. 
     Kinetic Modeling 
     (i) Urea and β2-M Modeling 
     Suitable kinetic models  30  for urea and β2-M for system  10  are shown below and described in detail in by: Ward (Ward et al.,  Kidney International,  69: 1431 to 1437 (2006)), the entire contents of which are incorporated expressly herein by reference and relied upon, for a two compartment model, 
     
       
         
           
             
               
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     and Clark (Clark et al.,  J. AM Soc. Nephrol.,  10: 601-609 (1999)), the entire contents of which are incorporated expressly herein by reference and relied upon, 
     
       
         
           
             
               
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     The above equations are applicable to both urea and β2-M. The same model is used for both solutes, with the parameter values being different, such as, generation rate, non-renal clearance, distribution volume, etc. 
     (ii) Mass Balance Modeling 
     One suitable model for electrolyte balance for system  10 , e.g., for sodium, potassium, etc., is a three compartment model, and is described in detail by Ursino et al., (Ursino M. et al., Prediction of Solute Kinetics, Acid-Base Status, and Blood Volume Changes During Profiled Hemodialysis,  Annals of Biomedical Engineering , Vol. 28, pp. 204-216 (2000)), the entire contents of which are incorporated expressly herein by reference and relied upon. 
     (iii) Modeling Modifications for Replacement Fluid 
     As described herein, system  10  is not limited to dialysis and can be applied to other renal failure blood treatments such as hemofiltration and hemodiafiltration. Hemofiltration and hemodiafiltration both involve the use of a replacement fluid, which is pumped directly into the blood lines for convective clearance used in place of (hemofiltration) or in addition to (hemodiafiltration) the osmotic clearance of dialysis. 
     The equations below show modifications to the kinetic models that Applicants have found can be made to all for use of replacement fluid. The first equation below shows the effect on mass balance. In particular, the factor Q R *C s,R  is added for replacement fluid. The second equation below shows the effect of convective clearance (JO) on dialyzer clearance. 
     
       
         
           
             
               
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     II. Phosphate Modeling 
     Phosphate Prediction Methods in Hemodialysis Patients and Applications Thereof 
     In light of the systems discussed herein, it is contemplated to provide methods of predicting serum or plasma phosphorus concentrations or levels in a hemodialysis patient before, during and after hemodialysis therapies. Being able to predict serum phosphorus levels can be useful in determining optimal treatment regimens for hemodialysis patients. These methods can be incorporated into any of the systems and computing devices described herein to optimize hemodialysis therapies for the patient. 
     Elevated levels of serum phosphorus in end stage renal disease patients have been associated with greater risk of mortality, primarily due to cardiac-related causes. Such associations have been demonstrated among various countries throughout the world and over time. Although the physiological mechanisms involved remain incompletely understood, inadequate control of serum phosphorus levels and the use of calcium-based phosphate binders have been linked to rapid progression of coronary calcification, increased stiffness of the arterial wall and high blood pressure. 
     Control of serum phosphorus concentrations in most hemodialysis (“HD”) patients requires both the daily use of oral phosphate binders to inhibit intestinal absorption of phosphate and the removal of phosphate by HD treatments. Despite this dual approach, hyperphosphatemia often occurs because typical western diets contain high phosphate content. Calcium-based oral phosphate binders are still used extensively because of their low cost, and more effective binders are under active development. Other efforts have attempted to increase dialytic removal of phosphorus during thrice-weekly therapy by various methods, often without substantial improvements. The only HD prescription parameter that has been shown to consistently reduce serum phosphorus concentrations is the use of longer treatments, both during thrice-weekly HD and during HD treatments applied more frequently. 
     Methods of predicting or determining serum phosphorus levels of a patient undergoing hemodialysis using a robust and practical phosphorus kinetic model allow for effectively modifying new HD treatment modalities on an individual patient basis. In an embodiment, a method of predicting serum phosphorus concentration in a patient during hemodialysis is provided. The method includes measuring serum phosphorus concentrations (“C”) of the patient over a hemodialysis treatment session time (using any suitable methods of measuring such as, e.g., fluorometric and colorimetric assays) and an ultrafiltration or fluid removal rate (“Q UF ”) calculated by a difference between pre- and post-dialytic body weight of the patient during an initial hemodialysis treatment session divided by the total treatment time of the treatment session and estimating K M  and V PRE  for the patient using a non-linear least squares fitting to the governing transport equations having analytical solutions of the following form: 
     
       
         
           
             
               
                 
                   
                     
                       C 
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         C 
                         
                           P 
                           ⁢ 
                           R 
                           ⁢ 
                           E 
                         
                       
                       [ 
                       
                         
                           
                             K 
                             M 
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   K 
                                   D 
                                 
                                 + 
                                 
                                   K 
                                   R 
                                 
                                 - 
                                 
                                   Q 
                                   
                                     U 
                                     ⁢ 
                                     F 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   
                                     V 
                                     ⁡ 
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                   
                                     V 
                                     
                                       P 
                                       ⁢ 
                                       R 
                                       ⁢ 
                                       E 
                                     
                                   
                                 
                                 ) 
                               
                               
                                 
                                   
                                     K 
                                     M 
                                   
                                   + 
                                   
                                     K 
                                     D 
                                   
                                   + 
                                   
                                     K 
                                     R 
                                   
                                   - 
                                   
                                     Q 
                                     UF 
                                   
                                 
                                 
                                   Q 
                                   UF 
                                 
                               
                             
                           
                         
                         
                           
                             K 
                             M 
                           
                           + 
                           
                             K 
                             D 
                           
                           + 
                           
                             K 
                             R 
                           
                           - 
                           
                             Q 
                             
                               U 
                               ⁢ 
                               F 
                             
                           
                         
                       
                       ] 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   and 
                 
               
               
                 
                   ( 
                   A 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       C 
                       ⁡ 
                       
                         ( 
                         T 
                         ) 
                       
                     
                     = 
                     
                       
                         C 
                         
                           P 
                           ⁢ 
                           R 
                           ⁢ 
                           E 
                         
                       
                       - 
                       
                         
                           ( 
                           
                             
                               C 
                               
                                 P 
                                 ⁢ 
                                 R 
                                 ⁢ 
                                 E 
                               
                             
                             - 
                             
                               C 
                               
                                 P 
                                 ⁢ 
                                 O 
                                 ⁢ 
                                 S 
                                 ⁢ 
                                 T 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           e 
                           
                             ( 
                             
                               - 
                               
                                 
                                   
                                     K 
                                     M 
                                   
                                   ⁢ 
                                   T 
                                 
                                 
                                   
                                     V 
                                     PRE 
                                   
                                   - 
                                   
                                     
                                       Q 
                                       UF 
                                     
                                     ⁢ 
                                     
                                       t 
                                       tx 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   B 
                   ) 
                 
               
             
           
         
       
     
     wherein t is a time during the hemodialysis treatment session, T is a time after an end of the hemodialysis treatment session, t tx  is a total duration of the hemodialysis treatment session, C PRE  is a pre-dialysis plasma phosphorus concentration, C POST  is a post-dialytic plasma phosphorus concentration, K M  is a phosphorous mobilization clearance of the patient, K R  is a residual renal clearance of phosphate, K D  is a dialyzer phosphate clearance, V PRE  is a pre-dialysis distribution volume of phosphorus of the patient, and 
         V ( t )= V   PRE   −Q   UF   ×t   (C).
 
     C (i.e., serum phosphorus concentrations) of the patient can then be predicted at any time during any hemodialysis treatment session by using the equations A and B, for the previously estimated set of K M  and V PRE  of the patient. Alternative to the non-linear least squares fitting, V PRE  can also be estimated as a certain percentage of body weight or body water volume of the patient. 
     In another embodiment, a method of predicting serum phosphorus concentration in a patient during hemodialysis is provided when the ultrafiltration rate is assumed to be negligible (i.e., Q UF =0). The method includes measuring C of the patient during an initial hemodialysis treatment session (using any suitable methods of measuring such as, e.g., fluorometric and colorimetric assays) and estimating K M  and V PRE  for the patient using a non-linear least squares fitting to the governing transport equations having analytical solutions of the following form: 
     
       
         
           
             
               
                 
                   
                     
                       C 
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         C 
                         PRE 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               K 
                               M 
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     K 
                                     D 
                                   
                                   + 
                                   
                                     K 
                                     R 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   
                                     - 
                                     t 
                                   
                                   ⁢ 
                                   
                                     
                                       ( 
                                       
                                         
                                           K 
                                           M 
                                         
                                         + 
                                         
                                           K 
                                           D 
                                         
                                         + 
                                         
                                           K 
                                           R 
                                         
                                       
                                       ) 
                                     
                                     
                                       V 
                                       
                                         P 
                                         ⁢ 
                                         R 
                                         ⁢ 
                                         E 
                                       
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               K 
                               M 
                             
                             + 
                             
                               K 
                               D 
                             
                             + 
                             
                               K 
                               R 
                             
                           
                         
                         ] 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   and 
                 
               
               
                 
                   ( 
                   D 
                   ) 
                 
               
             
             
               
                 
                   
                     C 
                     ⁡ 
                     
                       ( 
                       T 
                       ) 
                     
                   
                   = 
                   
                     
                       C 
                       
                         P 
                         ⁢ 
                         R 
                         ⁢ 
                         E 
                       
                     
                     - 
                     
                       
                         ( 
                         
                           
                             C 
                             
                               P 
                               ⁢ 
                               R 
                               ⁢ 
                               E 
                             
                           
                           - 
                           
                             C 
                             
                               P 
                               ⁢ 
                               O 
                               ⁢ 
                               S 
                               ⁢ 
                               T 
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         e 
                         
                           ( 
                           
                             - 
                             
                               
                                 
                                   K 
                                   M 
                                 
                                 ⁢ 
                                 T 
                               
                               
                                 V 
                                 
                                   P 
                                   ⁢ 
                                   R 
                                   ⁢ 
                                   E 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   E 
                   ) 
                 
               
             
           
         
       
     
     C of the patient can be predicted at any time during any hemodialysis treatment session by using the equations D and E for a given set of previously estimated parameters, K M  and V PRE , of the patient. Alternatively, V PRE  can be further estimated as a certain percentage of body weight or body water volume of the patient. In an embodiment, K M  may be estimated using data from a case where Q UF ≠0 and used in equation D where Q UF =0. 
     In any of the methods of predicting serum phosphorus concentrations in a patient during hemodialysis described herein, K D  can be determined using the equation: 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         K 
                         D 
                       
                       = 
                       
                         
                           Q 
                           B 
                         
                         ⁢ 
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     0 
                                     . 
                                     9 
                                   
                                   ⁢ 
                                   4 
                                 
                                 - 
                                 
                                   Hct 
                                   × 
                                   1 
                                   ⁢ 
                                   0 
                                   ⁢ 
                                   0 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   e 
                                   z 
                                 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                           
                             ( 
                             
                               
                                 e 
                                 z 
                               
                               - 
                               
                                 
                                   
                                     ( 
                                     
                                       0.94 
                                       - 
                                       
                                         H 
                                         ⁢ 
                                         c 
                                         ⁢ 
                                         t 
                                         × 
                                         1 
                                         ⁢ 
                                         0 
                                         ⁢ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     Q 
                                     B 
                                   
                                 
                                 
                                   Q 
                                   D 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     wherein 
                   
                 
               
               
                 
                   ( 
                   F 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       Z 
                       = 
                       
                         
                           K 
                           O 
                         
                         ⁢ 
                         A 
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 Q 
                                 D 
                               
                               - 
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         0 
                                         . 
                                         9 
                                       
                                       ⁢ 
                                       4 
                                     
                                     - 
                                     
                                       H 
                                       ⁢ 
                                       c 
                                       ⁢ 
                                       t 
                                       × 
                                       1 
                                       ⁢ 
                                       0 
                                       ⁢ 
                                       0 
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   Q 
                                   B 
                                 
                               
                             
                             ) 
                           
                           
                             ( 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       0 
                                       . 
                                       9 
                                     
                                     ⁢ 
                                     4 
                                   
                                   - 
                                   
                                     H 
                                     ⁢ 
                                     c 
                                     ⁢ 
                                     t 
                                     × 
                                     1 
                                     ⁢ 
                                     0 
                                     ⁢ 
                                     0 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 Q 
                                 B 
                               
                               × 
                               
                                 Q 
                                 D 
                               
                             
                             ) 
                           
                         
                       
                     
                     , 
                   
                 
               
               
                 
                   ( 
                   G 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         K 
                         O 
                       
                       ⁢ 
                       A 
                     
                     = 
                     
                       
                         
                           
                             ( 
                             
                               
                                 
                                   0 
                                   . 
                                   9 
                                 
                                 ⁢ 
                                 4 
                               
                               - 
                               
                                 H 
                                 ⁢ 
                                 c 
                                 ⁢ 
                                 t 
                                 × 
                                 1 
                                 ⁢ 
                                 0 
                                 ⁢ 
                                 0 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             Q 
                             
                               B 
                               , 
                               M 
                             
                           
                           × 
                           
                             Q 
                             
                               D 
                               , 
                               M 
                             
                           
                         
                         
                           
                             Q 
                             
                               D 
                               , 
                               M 
                             
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   
                                     0 
                                     . 
                                     9 
                                   
                                   ⁢ 
                                   4 
                                 
                                 - 
                                 
                                   H 
                                   ⁢ 
                                   c 
                                   ⁢ 
                                   t 
                                   × 
                                   1 
                                   ⁢ 
                                   0 
                                   ⁢ 
                                   0 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               Q 
                               
                                 B 
                                 , 
                                 M 
                               
                             
                           
                         
                       
                       × 
                       
                         ln 
                         ⁡ 
                         
                           ( 
                           
                             
                               1 
                               - 
                               
                                 
                                   K 
                                   
                                     D 
                                     , 
                                     M 
                                   
                                 
                                 / 
                                 
                                   Q 
                                   
                                     D 
                                     , 
                                     M 
                                   
                                 
                               
                             
                             
                               1 
                               - 
                               
                                 
                                   K 
                                   
                                     D 
                                     , 
                                     M 
                                   
                                 
                                 / 
                                 
                                   [ 
                                   
                                     
                                       ( 
                                       
                                         
                                           0. 
                                           ⁢ 
                                           9 
                                           ⁢ 
                                           4 
                                         
                                         - 
                                         
                                           H 
                                           ⁢ 
                                           c 
                                           ⁢ 
                                           t 
                                           × 
                                           1 
                                           ⁢ 
                                           0 
                                           ⁢ 
                                           0 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       Q 
                                       
                                         B 
                                         , 
                                         M 
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   H 
                   ) 
                 
               
             
           
         
       
     
     Q B  and Q D  are the blood and dialysate flow rates at which the desired dialyzer clearance K D  is calculated using equations F and G. K O A is a dialyzer mass transfer area coefficient for phosphate obtained as a result of a previous measurement where the set of blood and dialysate flow rates Q B,M  and Q D,M  resulted in dialyzer clearance K D,M , and Hct is hematocrit count measured from patient&#39;s blood sample. Alternatively, K D  can be determined at any time t using the equation: 
     
       
         
           
             
               
                 
                   
                     K 
                     D 
                   
                   = 
                   
                     
                       
                         
                           C 
                           D 
                         
                         ⁡ 
                         
                           ( 
                           
                             t 
                             s 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           Q 
                           D 
                         
                         ⁡ 
                         
                           ( 
                           
                             t 
                             s 
                           
                           ) 
                         
                       
                     
                     
                       C 
                       ⁡ 
                       
                         ( 
                         
                           t 
                           s 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   I 
                   ) 
                 
               
             
           
         
       
     
     wherein t s  is a sampling time and C D (t s ) is a concentration of phosphorus in a dialysate outflow at time t s , Q D (t s ) is a dialysate flowrate at time t s , and C(t s ) is a serum phosphorus concentration at time t s . 
     Alternative to non-linear least squares fitting, K M  can be determined using the following algebraic equation: 
     
       
         
           
             
               
                 
                   
                     
                       K 
                       M 
                     
                     = 
                     
                       
                         C 
                         
                           P 
                           ⁢ 
                           O 
                           ⁢ 
                           S 
                           ⁢ 
                           T 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               K 
                               D 
                             
                             - 
                             
                               Q 
                               
                                 U 
                                 ⁢ 
                                 F 
                               
                             
                           
                           
                             
                               C 
                               
                                 P 
                                 ⁢ 
                                 R 
                                 ⁢ 
                                 E 
                               
                             
                             - 
                             
                               C 
                               
                                 P 
                                 ⁢ 
                                 O 
                                 ⁢ 
                                 S 
                                 ⁢ 
                                 T 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                   . 
                 
               
               
                 
                   ( 
                   J 
                   ) 
                 
               
             
           
         
       
     
     C of the patient can be measured at any suitable time during the hemodialysis treatment session, for example, such as every 15 or 30 minutes. t tx  can be any suitable amount of time such as, for example, 2, 4 or 8 hours. T can be any suitable time, for example, such as 30 minutes or 1 hour. 
     V POST  is a measure of the distribution volume of phosphorus at the end of the hemodialysis treatment when the patient is considered to be normohydrated. This parameter approximates the volume of extracellular fluids. Thus, V POST  is a clinically relevant patient parameter that can be used to evaluate the patient&#39;s hydration status. In an application from knowing the previously determined V PRE , V POST  can be determined using the equation: 
         V   POST   =V   PRE   −Q   UF   ×t   tx   (K)
 
     and a suitable therapy can be provided to the patient based on the value of V POST . As seen from equation K, if Q UF =0, then V POST =V PRE . 
     Specific steps of the methods of predicting phosphorus mobilization in a patient during hemodialysis can be performed using a computing device. Such a computing device can include a display device, an input device, a processor, and a memory device that stores a plurality of instructions, which when executed by the processor, cause the processor to operate with the display device and the input device to (a) receive data relating to C of a hemodialysis patient over a hemodialysis treatment session time and a Q UF  calculated based on a difference between pre- and post-dialytic body weight of the hemodialysis patient during a hemodialysis treatment session divided by the total treatment time of the treatment session; (b) estimating K M  and V PRE  for the patient using a non-linear least squares fitting to the governing transport equations having analytical solutions of the following form: 
     
       
         
           
             
               
                 
                   
                     C 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         C 
                         
                           P 
                           ⁢ 
                           R 
                           ⁢ 
                           E 
                         
                       
                       [ 
                       
                         
                           
                             K 
                             M 
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   K 
                                   D 
                                 
                                 + 
                                 
                                   K 
                                   R 
                                 
                                 - 
                                 
                                   Q 
                                   
                                     U 
                                     ⁢ 
                                     F 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   
                                     V 
                                     ⁡ 
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                   
                                     V 
                                     
                                       P 
                                       ⁢ 
                                       R 
                                       ⁢ 
                                       E 
                                     
                                   
                                 
                                 ) 
                               
                               
                                 
                                   
                                     K 
                                     M 
                                   
                                   + 
                                   
                                     K 
                                     D 
                                   
                                   + 
                                   
                                     K 
                                     R 
                                   
                                   - 
                                   
                                     Q 
                                     UF 
                                   
                                 
                                 
                                   Q 
                                   UF 
                                 
                               
                             
                           
                         
                         
                           
                             K 
                             M 
                           
                           + 
                           
                             K 
                             D 
                           
                           + 
                           
                             K 
                             R 
                           
                           - 
                           
                             Q 
                             
                               U 
                               ⁢ 
                               F 
                             
                           
                         
                       
                       ] 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     and 
                   
                 
               
               
                 
                   ( 
                   L 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         C 
                         ⁡ 
                         
                           ( 
                           T 
                           ) 
                         
                       
                       = 
                       
                         
                           C 
                           
                             P 
                             ⁢ 
                             R 
                             ⁢ 
                             E 
                           
                         
                         - 
                         
                           
                             ( 
                             
                               
                                 C 
                                 
                                   P 
                                   ⁢ 
                                   R 
                                   ⁢ 
                                   E 
                                 
                               
                               - 
                               
                                 C 
                                 
                                   P 
                                   ⁢ 
                                   O 
                                   ⁢ 
                                   S 
                                   ⁢ 
                                   T 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             e 
                             
                               ( 
                               
                                 - 
                                 
                                   
                                     
                                       K 
                                       M 
                                     
                                     ⁢ 
                                     T 
                                   
                                   
                                     
                                       V 
                                       PRE 
                                     
                                     - 
                                     
                                       
                                         Q 
                                         UF 
                                       
                                       ⁢ 
                                       
                                         t 
                                         tx 
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     ; 
                   
                 
               
               
                 
                   ( 
                   M 
                   ) 
                 
               
             
           
         
       
     
     and (c) predict C of the patient at any time during hemodialysis by using the equations L and M for a given set of estimated parameters, K M  and V PRE , of the patient. It should be appreciated that the variables for equations L and M can be determined using any of the equations set forth herein. The information/data obtained for the hemodialysis patient can be displayed/printed out and used by the healthcare provider to provide improved treatment and nutritional regimens for the hemodialysis patient. Any of the unknown factors can be determined using the appropriate equations or measurements discussed previously for the methods of determining phosphorus mobilization in a patient during hemodialysis. 
     
       
         
           
             
               
                 
                   
                     
                       
                         If 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           Q 
                           UF 
                         
                       
                       = 
                       0 
                     
                     , 
                     then 
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       C 
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           C 
                           
                             P 
                             ⁢ 
                             R 
                             ⁢ 
                             E 
                           
                         
                         [ 
                         
                           
                             
                               K 
                               M 
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     K 
                                     D 
                                   
                                   + 
                                   
                                     K 
                                     R 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   
                                     - 
                                     t 
                                   
                                   ⁢ 
                                   
                                     
                                       ( 
                                       
                                         
                                           K 
                                           M 
                                         
                                         + 
                                         
                                           K 
                                           D 
                                         
                                         + 
                                         
                                           K 
                                           R 
                                         
                                       
                                       ) 
                                     
                                     
                                       V 
                                       
                                         P 
                                         ⁢ 
                                         R 
                                         ⁢ 
                                         E 
                                       
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               K 
                               M 
                             
                             + 
                             
                               K 
                               D 
                             
                             + 
                             
                               K 
                               R 
                             
                           
                         
                         ] 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   N 
                   ) 
                 
               
             
           
         
       
     
     The computing device can also be preprogrammed or run according to software that causes the processor to operate with the display device and the input device to receive data relating to at least one of K R , K D  or a sampling time for collecting the serum phosphorus concentration. In an embodiment, the computing device can be system  10  described in section I. 
     Along with the previously described methods of determining phosphorus mobilization in a patient during hemodialysis, a mass balance model to predict steady state, pre-dialysis serum phosphorus levels (“C SS-PRE ”) in patients treated with HD therapies has also been developed. A mass balance model was used in combination with a pseudo one-compartment model for intradialytic and rebound periods to determine steady state pre-dialysis serum phosphorus levels in individual patients. Using this model, the effect of specific therapy parameters (e.g., dialyzer phosphate clearances, weekly therapy frequency, therapy duration, etc.) on individual hemodialysis patients&#39; serum phosphorus levels can be evaluated. 
     The disclosed steady state, mass balance model combines the intradialytic phosphorus kinetics with dietary intake, use of phosphate binders, and residual renal clearance to predict steady state, pre-dialysis serum phosphorus levels. Unlike those with previous models, the predictions with this model involve simplified calculations; hence, this model can easily be integrated in daily clinical practice. Furthermore, the model involves patient-specific parameters enabling individualized predictions. This model can eventually be used to optimize therapies with a HHD device to remove adequate amounts of phosphorus using minimum necessary volumes of dialysate (i.e., minimized water consumption). Alternatively, the model can be used to determine the amount of required phosphate salt supplements in dialysate. 
     In an application of the kinetic model, a method of determining the C SS-PRE  in a hemodialysis patient is provided. The method includes obtaining a net generation of phosphorus (“G”) from at least a dietary phosphorus intake of the patient or a urea kinetic modeling of the patient and determining C SS-PRE  of the hemodialysis patient using the equation: 
     
       
         
           
             
               
                 
                   
                     C 
                     
                       
                         S 
                         ⁢ 
                         S 
                       
                       - 
                       
                         P 
                         ⁢ 
                         R 
                         ⁢ 
                         E 
                       
                     
                   
                   = 
                   
                     
                       
                         ( 
                         G 
                         ) 
                       
                       ⁢ 
                       
                         ( 
                         
                           1 
                           ⁢ 
                           0 
                           ⁢ 
                           080 
                           ⁢ 
                           
                             / 
                           
                           ⁢ 
                           F 
                         
                         ) 
                       
                     
                     
                       
                         
                           ( 
                           
                             
                               K 
                               D 
                             
                             + 
                             
                               K 
                               R 
                             
                           
                           ) 
                         
                         ⁢ 
                         n 
                         ⁢ 
                         
                           
                             C 
                             _ 
                           
                           
                             t 
                             ⁢ 
                             x 
                           
                         
                         ⁢ 
                         
                           t 
                           
                             t 
                             ⁢ 
                             x 
                           
                         
                       
                       + 
                       
                         
                           K 
                           R 
                         
                         ⁢ 
                         n 
                         ⁢ 
                         
                           
                             
                               C 
                               _ 
                             
                             i 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 10080 
                                 ⁢ 
                                 
                                   / 
                                 
                                 ⁢ 
                                 F 
                               
                               - 
                               
                                 t 
                                 
                                   t 
                                   ⁢ 
                                   x 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   O 
                   ) 
                 
               
             
           
         
       
     
     wherein F is a frequency of treatments per week, t tx  is a treatment time for one hemodialysis treatment session (e.g., in units of minutes per treatment session), K D  is a dialyzer phosphate clearance, K R  is a residual renal clearance of phosphate, n C   tx  is the normalized time averaged plasma phosphorus concentration during a dialysis treatment, and n C   i  is the normalized time averaged plasma phosphorus concentration for an interdialytic interval. The effect of at least one of a patient parameter or a treatment parameter on C SS-PRE  of the patient can be simulated so as to obtain an optimal range of C SS-PRE  for the patient. For example, the patient parameter can be G, K M  or V PRE , and the treatment parameter can be tix, K D  (e.g., Q B , Q D ) or F. 
     In an alternative embodiment, a method of predicting the C SS-PRE  in a hemodialysis patient is provided. The method includes determining a net generation of phosphorus (“G”) using the equation: 
     
       
         
           
             
               
                 
                   G 
                   = 
                   
                     
                       C 
                       
                         
                           S 
                           ⁢ 
                           S 
                         
                         - 
                         
                           P 
                           ⁢ 
                           R 
                           ⁢ 
                           E 
                         
                         - 
                         
                           I 
                           ⁢ 
                           N 
                         
                       
                     
                     × 
                     
                       [ 
                       
                         
                           
                             
                               ( 
                               
                                 
                                   K 
                                   D 
                                 
                                 + 
                                 
                                   K 
                                   R 
                                 
                               
                               ) 
                             
                             ⁢ 
                             n 
                             ⁢ 
                             
                               
                                 C 
                                 _ 
                               
                               tx 
                             
                             ⁢ 
                             
                               t 
                               tx 
                             
                           
                           + 
                           
                             
                               K 
                               R 
                             
                             ⁢ 
                             n 
                             ⁢ 
                             
                               
                                 
                                   C 
                                   _ 
                                 
                                 i 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     10080 
                                     ⁢ 
                                     
                                       / 
                                     
                                     ⁢ 
                                     F 
                                   
                                   - 
                                   
                                     t 
                                     tx 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         
                           1 
                           ⁢ 
                           0 
                           ⁢ 
                           080 
                           ⁢ 
                           
                             / 
                           
                           ⁢ 
                           F 
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   P 
                   ) 
                 
               
             
           
         
       
     
     wherein C SS-PRE-IN  is an initial, measured, steady state, pre-dialysis serum phosphorus level of the hemodialysis patient who is maintained by a hemodialysis therapy (e.g., identified by K D , F and t tx ) for a specified time prior to the calculation of G using equation P. The specified time can be, for example, at least one week, two weeks, three weeks, 1 month, 2 months, 3 months, 4 months or more prior to the time when G is calculated. F is a frequency of treatments per week, t tx  is a treatment time for one hemodialysis treatment session (e.g., in units of minutes per treatment session), K D  is a dialyzer phosphate clearance, K R  is a residual renal clearance of phosphate, n C   tx  is the normalized time averaged plasma phosphorus concentration during a dialysis treatment, and n C   i  is the normalized time averaged plasma phosphorus concentration for an interdialytic interval. 
     Once G has been calculated using equation P or estimated by other methods, it can be used in predicting the effect of changes in hemodialysis treatment parameters on the steady state serum phosphorus concentration. For example, once G of the hemodialysis patient is known, C SS-PRE  of the patient under different hemodialysis treatment conditions can be predicted by rearranging equation P to form equation 0 and utilizing the known G to solve for C SS-PRE  of the hemodialysis patient. The effect of at least one of a patient parameter or a treatment parameter on C SS-PRE  of the patient can be simulated, and a treatment regimen of the hemodialysis patient can then be modified so that C SS-PRE  is within a desired range. 
     In general, there is an optimal range of steady state, pre-dialysis serum phosphorus levels in patients with end stage renal disease. Optimal prescription/regimen/nutritional therapies resulting in steady state, pre-dialysis phosphorus levels within the desired optimal ranges can be determined using equations 0 and P, for example, in the Optimization Component of the HHD system previously described herein. Because changes in the hemodialysis prescription or in patient behavior (e.g., changes in diet) could lead to changes in G, optimization of home hemodialysis therapies based on equations O and P to maintain C SS-PRE  within a desired range is advantageous. 
     In any of the methods of determining the G or the C SS-PRE  in a hemodialysis patient, n C   tx  and n C   i  can be determined using the equations: 
     
       
         
           
             
               
                 
                   
                     n 
                     ⁢ 
                     
                       
                         C 
                         _ 
                       
                       tx 
                     
                   
                   = 
                   
                     
                       1 
                       
                         t 
                         tx 
                       
                     
                     ⁢ 
                     
                       { 
                       
                         
                           
                             [ 
                             
                               
                                 
                                   K 
                                   M 
                                 
                                 ⁢ 
                                 
                                   t 
                                   
                                     t 
                                     ⁢ 
                                     x 
                                   
                                 
                               
                               
                                 
                                   K 
                                   M 
                                 
                                 + 
                                 
                                   K 
                                   R 
                                 
                                 + 
                                 
                                   K 
                                   D 
                                 
                                 - 
                                 
                                   Q 
                                   
                                     U 
                                     ⁢ 
                                     F 
                                   
                                 
                               
                             
                             ] 
                           
                           + 
                           
                             
                               [ 
                               
                                 1 
                                 - 
                                 
                                   
                                     K 
                                     M 
                                   
                                   
                                     
                                       K 
                                       M 
                                     
                                     + 
                                     
                                       K 
                                       R 
                                     
                                     + 
                                     
                                       K 
                                       D 
                                     
                                     - 
                                     
                                       Q 
                                       
                                         U 
                                         ⁢ 
                                         F 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                             ⁢ 
                             
                                 
                               
                                 
                                   [ 
                                   
                                     
                                       V 
                                       
                                         P 
                                         ⁢ 
                                         R 
                                         ⁢ 
                                         E 
                                       
                                     
                                     
                                       
                                         K 
                                         M 
                                       
                                       + 
                                       
                                         K 
                                         R 
                                       
                                       + 
                                       
                                         K 
                                         D 
                                       
                                     
                                   
                                   ] 
                                 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     1 
                                     - 
                                     
                                       
                                         ( 
                                         
                                           
                                             V 
                                             
                                               P 
                                               ⁢ 
                                               O 
                                               ⁢ 
                                               S 
                                               ⁢ 
                                               T 
                                             
                                           
                                           
                                             V 
                                             
                                               P 
                                               ⁢ 
                                               R 
                                               ⁢ 
                                               E 
                                             
                                           
                                         
                                         ) 
                                       
                                       
                                         
                                           ( 
                                           
                                             
                                               K 
                                               M 
                                             
                                             + 
                                             
                                               K 
                                               R 
                                             
                                             + 
                                             
                                               K 
                                               D 
                                             
                                           
                                           ) 
                                         
                                         ⁢ 
                                         
                                           / 
                                         
                                         ⁢ 
                                         
                                           Q 
                                           
                                             U 
                                             ⁢ 
                                             F 
                                           
                                         
                                       
                                     
                                   
                                   ] 
                                 
                               
                               } 
                             
                           
                         
                         , 
                         
                           
 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         and 
                       
                     
                   
                 
               
               
                 
                   ( 
                   Q 
                   ) 
                 
               
             
             
               
                 
                   
                     n 
                     ⁢ 
                     
                       
                         C 
                         _ 
                       
                       i 
                     
                   
                   = 
                   
                     
                       1 
                       
                         
                           1 
                           ⁢ 
                           0 
                           ⁢ 
                           080 
                           ⁢ 
                           
                             / 
                           
                           ⁢ 
                           F 
                         
                         - 
                         
                           t 
                           
                             t 
                             ⁢ 
                             x 
                           
                         
                       
                     
                     ⁢ 
                     
                       { 
                       
                         
                           
                             [ 
                             
                               
                                 
                                   K 
                                   M 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       1 
                                       ⁢ 
                                       0 
                                       ⁢ 
                                       080 
                                       ⁢ 
                                       
                                         / 
                                       
                                       ⁢ 
                                       F 
                                     
                                     - 
                                     
                                       t 
                                       tx 
                                     
                                   
                                   ) 
                                 
                               
                               
                                 
                                   K 
                                   M 
                                 
                                 + 
                                 
                                   K 
                                   R 
                                 
                                 + 
                                 
                                   Q 
                                   WG 
                                 
                               
                             
                             ] 
                           
                           + 
                           
                             [ 
                             
                               
                                 
                                   K 
                                   M 
                                 
                                 
                                   
                                     K 
                                     M 
                                   
                                   + 
                                   
                                     K 
                                     R 
                                   
                                   + 
                                   
                                     K 
                                     D 
                                   
                                   - 
                                   
                                     Q 
                                     UF 
                                   
                                 
                               
                               + 
                               
                                   
                                 
                                   
                                     
                                       
                                         [ 
                                         
                                           1 
                                           - 
                                           
                                             
                                               K 
                                               M 
                                             
                                             
                                               
                                                 K 
                                                 M 
                                               
                                               + 
                                               
                                                 K 
                                                 R 
                                               
                                               + 
                                               
                                                 K 
                                                 D 
                                               
                                               + 
                                               
                                                 Q 
                                                 UF 
                                               
                                             
                                           
                                         
                                         ] 
                                       
                                       ⁡ 
                                       
                                         [ 
                                         
                                           
                                             V 
                                             POST 
                                           
                                           
                                             V 
                                             PRE 
                                           
                                         
                                         ] 
                                       
                                     
                                     
                                       
                                         ( 
                                         
                                           
                                             K 
                                             M 
                                           
                                           + 
                                           
                                             K 
                                             D 
                                           
                                           + 
                                           
                                             K 
                                             R 
                                           
                                           - 
                                           
                                             Q 
                                             UF 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         / 
                                       
                                       ⁢ 
                                       
                                         Q 
                                         UF 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       
                                         K 
                                         M 
                                       
                                       
                                         
                                           K 
                                           M 
                                         
                                         + 
                                         
                                           K 
                                           R 
                                         
                                         + 
                                         
                                           Q 
                                           WG 
                                         
                                       
                                     
                                     × 
                                     
                                       [ 
                                       
                                         
                                           V 
                                           POST 
                                         
                                         
                                           
                                             K 
                                             M 
                                           
                                           + 
                                           
                                             K 
                                             R 
                                           
                                         
                                       
                                       ] 
                                     
                                     ⁢ 
                                     1 
                                   
                                   - 
                                   
                                     
                                       ( 
                                       
                                         
                                           V 
                                           POST 
                                         
                                         
                                           V 
                                           PRE 
                                         
                                       
                                       ) 
                                     
                                     
                                       
                                         ( 
                                         
                                           
                                             K 
                                             M 
                                           
                                           + 
                                           
                                             K 
                                             R 
                                           
                                         
                                         ) 
                                       
                                       / 
                                       
                                         Q 
                                         WG 
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                             } 
                           
                         
                         , 
                       
                     
                   
                 
               
               
                 
                   ( 
                   R 
                   ) 
                 
               
             
           
         
       
     
     wherein K M  is a phosphorus mobilization clearance of the patient, Q WG  is a constant rate of fluid gain by the patient during the interdialytic time interval (calculated by Q WG =(t tx Q UF )/(10080/F)), Q UF  is a constant rate of fluid removed from the patient, V PRE  is a pre-dialysis distribution volume of phosphorus of the patient prior to a hemodialysis treatment session, and V POST  is a post-dialysis distribution volume of phosphorus of the patient at the end of a hemodialysis treatment session. 
     In any of the methods of determining G or C SS-PRE  in a hemodialysis patient when there is negligible net ultrafiltration or fluid removal from the patient during hemodialysis therapies and no weight gain between hemodialysis therapies, n C   tx  and n C   i  can be determined using the equations: 
     
       
         
           
             
               
                 
                   
                     n 
                     ⁢ 
                     
                       
                         C 
                         _ 
                       
                       
                         t 
                         ⁢ 
                         x 
                       
                     
                   
                   = 
                   
                     
                       1 
                       
                         t 
                         tx 
                       
                     
                     ⁢ 
                     
                       { 
                       
                         
                           
                             [ 
                             
                               
                                 
                                   K 
                                   M 
                                 
                                 ⁢ 
                                 
                                   t 
                                   tx 
                                 
                               
                               
                                 
                                   K 
                                   M 
                                 
                                 + 
                                 
                                   K 
                                   R 
                                 
                                 + 
                                 
                                   K 
                                   D 
                                 
                               
                             
                             ] 
                           
                           + 
                           
                             
                               
                                 [ 
                                 
                                   1 
                                   - 
                                   
                                     
                                       K 
                                       M 
                                     
                                     
                                       
                                         K 
                                         M 
                                       
                                       + 
                                       
                                         K 
                                         R 
                                       
                                       + 
                                       
                                         K 
                                         D 
                                       
                                     
                                   
                                 
                                 ] 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     V 
                                     PRE 
                                   
                                   
                                     
                                       K 
                                       M 
                                     
                                     + 
                                     
                                       K 
                                       R 
                                     
                                     + 
                                     
                                       K 
                                       D 
                                     
                                   
                                 
                                 ] 
                               
                             
                             ⁢ 
                             
                                 
                               
                                 [ 
                                 
                                   1 
                                   - 
                                   
                                     exp 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           - 
                                           
                                             ( 
                                             
                                               
                                                 K 
                                                 D 
                                               
                                               + 
                                               
                                                 K 
                                                 R 
                                               
                                               + 
                                               
                                                 K 
                                                 M 
                                               
                                             
                                             ) 
                                           
                                         
                                         ⁢ 
                                         
                                           t 
                                           tx 
                                         
                                         ⁢ 
                                         
                                           / 
                                         
                                         ⁢ 
                                         
                                           V 
                                           PRE 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ] 
                               
                               } 
                             
                           
                         
                         , 
                         
                           
 
                         
                         ⁢ 
                         and 
                       
                     
                   
                 
               
               
                 
                   ( 
                   S 
                   ) 
                 
               
             
             
               
                 
                   
                     n 
                     ⁢ 
                     
                       
                         C 
                         _ 
                       
                       i 
                     
                   
                   = 
                   
                     
                       1 
                       
                         
                           10080 
                           ⁢ 
                           
                             / 
                           
                           ⁢ 
                           F 
                         
                         - 
                         
                           t 
                           tx 
                         
                       
                     
                     ⁢ 
                     
                       { 
                       
                         
                           [ 
                           
                             
                               
                                 K 
                                 M 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     10080 
                                     ⁢ 
                                     
                                       / 
                                     
                                     ⁢ 
                                     F 
                                   
                                   - 
                                   
                                     t 
                                     tx 
                                   
                                 
                                 ) 
                               
                             
                             
                               
                                 K 
                                 M 
                               
                               + 
                               
                                 K 
                                 R 
                               
                             
                           
                           ] 
                         
                         + 
                         
                             
                           
                             [ 
                             
                               
                                 
                                   
                                     K 
                                     M 
                                   
                                   
                                     
                                       K 
                                       M 
                                     
                                     + 
                                     
                                       K 
                                       R 
                                     
                                     + 
                                     
                                       K 
                                       D 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                       
                                     
                                       
                                         
                                           [ 
                                           
                                             1 
                                             - 
                                             
                                               
                                                 K 
                                                 M 
                                               
                                               
                                                 
                                                   K 
                                                   M 
                                                 
                                                 + 
                                                 
                                                   K 
                                                   R 
                                                 
                                                 + 
                                                 
                                                   K 
                                                   D 
                                                 
                                               
                                             
                                           
                                           ] 
                                         
                                         × 
                                         
                                           exp 
                                           ⁡ 
                                           
                                             [ 
                                             
                                               
                                                 - 
                                                 
                                                   ( 
                                                   
                                                     
                                                       K 
                                                       D 
                                                     
                                                     + 
                                                     
                                                       K 
                                                       R 
                                                     
                                                     + 
                                                     
                                                       K 
                                                       M 
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                               ⁢ 
                                               
                                                 t 
                                                 tx 
                                               
                                               ⁢ 
                                               
                                                 / 
                                               
                                               ⁢ 
                                               
                                                 V 
                                                 PRE 
                                               
                                             
                                             ] 
                                           
                                         
                                       
                                       - 
                                       
                                         
                                           K 
                                           M 
                                         
                                         
                                           
                                             K 
                                             M 
                                           
                                           + 
                                           
                                             K 
                                             R 
                                           
                                         
                                       
                                     
                                     ] 
                                   
                                   × 
                                   
                                     [ 
                                     
                                       
                                         V 
                                         POST 
                                       
                                       
                                         
                                           K 
                                           M 
                                         
                                         + 
                                         
                                           K 
                                           R 
                                         
                                       
                                     
                                     ] 
                                   
                                   ⁢ 
                                   
                                       
                                     
                                       [ 
                                       
                                         1 
                                         - 
                                         
                                           exp 
                                           ⁡ 
                                           
                                             ( 
                                             
                                               
                                                 - 
                                                 
                                                   ( 
                                                   
                                                     
                                                       K 
                                                       R 
                                                     
                                                     + 
                                                     
                                                       K 
                                                       M 
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                               ⁢ 
                                               
                                                 ( 
                                                 
                                                   
                                                     10080 
                                                     ⁢ 
                                                     
                                                       / 
                                                     
                                                     ⁢ 
                                                     F 
                                                   
                                                   - 
                                                   
                                                     t 
                                                     tx 
                                                   
                                                 
                                                 ) 
                                               
                                               ⁢ 
                                               
                                                 / 
                                               
                                               ⁢ 
                                               
                                                 V 
                                                 POST 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       ] 
                                     
                                     } 
                                   
                                 
                               
                               , 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   T 
                   ) 
                 
               
             
           
         
       
     
     It should be appreciated that the variables for equations S and T can be determined using any of the equations set forth herein. 
     In any of the methods of determining the G or the C SS-PRE  in a hemodialysis patient, K D  can be determined using the equation: 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         K 
                         D 
                       
                       = 
                       
                         
                           Q 
                           B 
                         
                         ⁢ 
                         
                           
                             
                               ( 
                               
                                 0.94 
                                 - 
                                 
                                   Hct 
                                   × 
                                   100 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   e 
                                   z 
                                 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                           
                             ( 
                             
                               
                                 e 
                                 z 
                               
                               - 
                               
                                 
                                   
                                     ( 
                                     
                                       0.94 
                                       - 
                                       
                                         Hct 
                                         × 
                                         100 
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     Q 
                                     B 
                                   
                                 
                                 
                                   Q 
                                   D 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     wherein 
                   
                 
               
               
                 
                   ( 
                   U 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       Z 
                       = 
                       
                         
                           K 
                           O 
                         
                         ⁢ 
                         A 
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 Q 
                                 D 
                               
                               - 
                               
                                 
                                   ( 
                                   
                                     0.94 
                                     - 
                                     
                                       Hct 
                                       × 
                                       100 
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   Q 
                                   B 
                                 
                               
                             
                             ) 
                           
                           
                             ( 
                             
                               
                                 ( 
                                 
                                   0.94 
                                   - 
                                   
                                     Hct 
                                     × 
                                     100 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 Q 
                                 B 
                               
                               × 
                               
                                 Q 
                                 D 
                               
                             
                             ) 
                           
                         
                       
                     
                     , 
                   
                 
               
               
                 
                   ( 
                   V 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         K 
                         O 
                       
                       ⁢ 
                       A 
                     
                     = 
                     
                       
                         
                           
                             ( 
                             
                               0.94 
                               - 
                               
                                 Hct 
                                 × 
                                 100 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             Q 
                             
                               B 
                               , 
                               M 
                             
                           
                           × 
                           
                             Q 
                             
                               D 
                               , 
                               M 
                             
                           
                         
                         
                           
                             Q 
                             
                               D 
                               , 
                               M 
                             
                           
                           - 
                           
                             
                               ( 
                               
                                 0.94 
                                 - 
                                 
                                   Hct 
                                   × 
                                   100 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               Q 
                               
                                 B 
                                 , 
                                 M 
                               
                             
                           
                         
                       
                       × 
                       
                         ln 
                         ( 
                         
                           
                             1 
                             - 
                             
                               
                                 K 
                                 
                                   D 
                                   , 
                                   M 
                                 
                               
                               ⁢ 
                               
                                 / 
                               
                               ⁢ 
                               
                                 Q 
                                 
                                   D 
                                   , 
                                   M 
                                 
                               
                             
                           
                           
                             
                               1 
                               - 
                               
                                 
                                   K 
                                   
                                     D 
                                     , 
                                     M 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     / 
                                   
                                   [ 
                                   
                                     0.94 
                                     - 
                                     
                                       Hct 
                                       × 
                                       100 
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   Q 
                                   
                                     B 
                                     , 
                                     M 
                                   
                                 
                               
                             
                             ] 
                           
                         
                         ) 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   W 
                   ) 
                 
               
             
           
         
       
     
     Q B  and Q D  are the blood and dialysate flow rates at which the desired dialyzer clearance K D  is calculated using equations U and V. K O A is a dialyzer mass transfer area coefficient for phosphate obtained as a result of a previous measurement where the set of blood and dialysate flow rates Q B,M  and Q D,M  resulted in dialyzer clearance K D,M , and Hct is hematocrit count measured from patient&#39;s blood sample. Alternatively, K D  can be determined at any time t using the equation: 
     
       
         
           
             
               
                 
                   
                     K 
                     D 
                   
                   = 
                   
                     
                       
                         
                           C 
                           D 
                         
                         ⁡ 
                         
                           ( 
                           
                             t 
                             s 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           Q 
                           D 
                         
                         ⁡ 
                         
                           ( 
                           
                             t 
                             s 
                           
                           ) 
                         
                       
                     
                     
                       C 
                       ⁡ 
                       
                         ( 
                         
                           t 
                           s 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   X 
                   ) 
                 
               
             
           
         
       
     
     wherein t s  is a sampling time and C D (t s ) is a concentration of phosphorus in a dialysate outflow at time t s  and Q D (t s ) is a dialysate flowrate at time t s  and C(t s ) is a serum phosphorus concentration at time t s . 
     Alternative to non-linear least squares fitting, K M  can be determined using the following algebraic equation: 
     
       
         
           
             
               
                 
                   
                     K 
                     M 
                   
                   = 
                   
                     
                       C 
                       
                         P 
                         ⁢ 
                         O 
                         ⁢ 
                         S 
                         ⁢ 
                         T 
                       
                     
                     ( 
                     
                       
                         
                           K 
                           D 
                         
                         - 
                         
                           Q 
                           
                             U 
                             ⁢ 
                             F 
                           
                         
                       
                       
                         
                           C 
                           
                             P 
                             ⁢ 
                             R 
                             ⁢ 
                             E 
                           
                         
                         - 
                         
                           C 
                           
                             P 
                             ⁢ 
                             O 
                             ⁢ 
                             S 
                             ⁢ 
                             T 
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   Y 
                   ) 
                 
               
             
           
         
       
     
     wherein C POST  is a post-dialytic plasma phosphorus concentration, and C PRE  is a pre-dialysis plasma phosphorus concentration. G can be determined using the equation: 
     
       
         
           
             
               
                 
                   G 
                   = 
                   
                     
                       
                         
                           I 
                           P 
                         
                         ⁢ 
                         
                           A 
                           P 
                         
                       
                       - 
                       
                         
                           I 
                           B 
                         
                         ⁢ 
                         
                           P 
                           B 
                         
                       
                     
                     
                       1 
                       ⁢ 
                       0 
                       ⁢ 
                       0 
                       ⁢ 
                       8 
                       ⁢ 
                       0 
                     
                   
                 
               
               
                 
                   ( 
                   Z 
                   ) 
                 
               
             
           
         
       
     
     wherein I P  is a weekly dietary intake of phosphorus of the hemodialysis patient, A P  is a percent phosphorus absorption of the hemodialysis patient, IB is a weekly binder intake of the hemodialysis patient, and PB is a binding power of the binder. 
     In an embodiment, K M  and V PRE  can be determined using the methods of predicting phosphorus mobilization in a patient during hemodialysis as previously discussed. In this case, K M  and V PRE  are determined by measuring C of the hemodialysis patient over a hemodialysis treatment session time and Q UF  calculated by a difference between pre- and post-dialytic body weight of the patient during an initial hemodialysis treatment session divided by the total treatment time of the treatment session, and estimating K M  and V PRE  for the hemodialysis patient using a non-linear least squares fitting to the governing transport equations having analytical solutions as follows: 
     
       
         
           
             
               
                 
                   
                     C 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         C 
                         
                           P 
                           ⁢ 
                           R 
                           ⁢ 
                           E 
                         
                       
                       [ 
                       
                         
                           
                             K 
                             M 
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   K 
                                   D 
                                 
                                 + 
                                 
                                   K 
                                   R 
                                 
                                 - 
                                 
                                   Q 
                                   
                                     U 
                                     ⁢ 
                                     F 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   
                                     V 
                                     ⁡ 
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                   
                                     V 
                                     
                                       P 
                                       ⁢ 
                                       R 
                                       ⁢ 
                                       E 
                                     
                                   
                                 
                                 ) 
                               
                               
                                 
                                   
                                     K 
                                     M 
                                   
                                   + 
                                   
                                     K 
                                     D 
                                   
                                   + 
                                   
                                     K 
                                     R 
                                   
                                   - 
                                   
                                     Q 
                                     UF 
                                   
                                 
                                 
                                   Q 
                                   UF 
                                 
                               
                             
                           
                         
                         
                           
                             K 
                             M 
                           
                           + 
                           
                             K 
                             D 
                           
                           + 
                           
                             K 
                             R 
                           
                           - 
                           
                             Q 
                             
                               U 
                               ⁢ 
                               F 
                             
                           
                         
                       
                       ] 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     and 
                   
                 
               
               
                 
                   ( 
                   AA 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       C 
                       ⁡ 
                       
                         ( 
                         T 
                         ) 
                       
                     
                     = 
                     
                       
                         C 
                         
                           P 
                           ⁢ 
                           R 
                           ⁢ 
                           E 
                         
                       
                       - 
                       
                         
                           ( 
                           
                             
                               C 
                               
                                 P 
                                 ⁢ 
                                 R 
                                 ⁢ 
                                 E 
                               
                             
                             - 
                             
                               C 
                               
                                 P 
                                 ⁢ 
                                 O 
                                 ⁢ 
                                 S 
                                 ⁢ 
                                 T 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           e 
                           
                             ( 
                             
                               - 
                               
                                 
                                   
                                     K 
                                     M 
                                   
                                   ⁢ 
                                   T 
                                 
                                 
                                   
                                     V 
                                     PRE 
                                   
                                   - 
                                   
                                     
                                       Q 
                                       UF 
                                     
                                     ⁢ 
                                     
                                       t 
                                       tx 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   BB 
                   ) 
                 
               
             
           
         
       
     
     wherein t is a time during the hemodialysis treatment session, T is a time after an end of the hemodialysis treatment session, t tx  is a total duration of the hemodialysis treatment session, C PRE  is a pre-dialysis plasma phosphorus concentration, C POST  is a post-dialytic plasma phosphorus concentration, K M  is a phosphorus mobilization clearance of the patient, K R  is a residual renal clearance of phosphate, K D  is a dialyzer phosphate clearance, V PRE  is a pre-dialysis distribution volume of phosphorus of the patient, and V(t)=V PRE −Q UF ×t (CC). 
     The methods of determining G or C SS-PRE  in a hemodialysis patient can also be used to determine or modify the appropriate treatments/dietary changes to meet a desired phosphorus serum level in the hemodialysis patient over a period of time. For example, the methods can be used to determine or modify a level of phosphorus intake so that C SS-PRE  of the hemodialysis patient ranges between about 3.6 mg/dl and 5.0 mg/dl. The methods can be used to determine or modify a phosphorus binder administered to the patient so that C SS-PRE  of the hemodialysis patient ranges between about 3.6 mg/dl and 5.0 mg/dl. The methods can further be used to determine or modify an amount of phosphorus salt supplements added to the dialysate so that C SS-PRE  of the hemodialysis patient ranges between about 3.6 mg/dl and 5.0 mg/dl. 
     The methods can be used to determine or modify the total hemodialysis treatment session time so that C SS-PRE  of the hemodialysis patient ranges between about 3.6 mg/dl and 5.0 mg/dl. The methods can be used to determine or modify the frequency F so that C SS-PRE  of the hemodialysis patient ranges between about 3.6 mg/dl and 5.0 mg/dl. The methods can be used to determine or modify a required blood flowrate and/or a dialysate flowrate so that C SS-PRE  of the hemodialysis patient ranges between about 3.6 mg/dl and 5.0 mg/dl. It should be appreciated that the preferred range of C SS-PRE  can be patient specific. 
     Specific steps of determining C SS-PRE  of a hemodialysis patient can be performed using a computing device. Such a computing device can include a display device, an input device, a processor, and a memory device that stores a plurality of instructions, which when executed by the processor, cause the processor to operate with the display device and the input device to: (a) receive data relating to G from at least a dietary phosphorus intake of a hemodialysis patient or a urea kinetic modeling of the hemodialysis patient; (b) determine the C SS-PRE  of the patient using the equation: 
     
       
         
           
             
               
                 
                   
                     C 
                     
                       
                         S 
                         ⁢ 
                         S 
                       
                       - 
                       
                         P 
                         ⁢ 
                         R 
                         ⁢ 
                         E 
                       
                     
                   
                   = 
                   
                     
                       
                         ( 
                         G 
                         ) 
                       
                       ⁢ 
                       
                         ( 
                         
                           1 
                           ⁢ 
                           0 
                           ⁢ 
                           080 
                           ⁢ 
                           
                             / 
                           
                           ⁢ 
                           F 
                         
                         ) 
                       
                     
                     
                       
                         
                           ( 
                           
                             
                               K 
                               D 
                             
                             + 
                             
                               K 
                               R 
                             
                           
                           ) 
                         
                         ⁢ 
                         n 
                         ⁢ 
                         
                           
                             C 
                             _ 
                           
                           tx 
                         
                         ⁢ 
                         
                           t 
                           tx 
                         
                       
                       + 
                       
                         
                           K 
                           R 
                         
                         ⁢ 
                         n 
                         ⁢ 
                         
                           
                             
                               C 
                               _ 
                             
                             i 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 10080 
                                 ⁢ 
                                 
                                   / 
                                 
                                 ⁢ 
                                 F 
                               
                               - 
                               
                                 t 
                                 tx 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   DD 
                   ) 
                 
               
             
           
         
       
     
     wherein F is a frequency of treatments per week, t tx  is a treatment time for one hemodialysis treatment session (e.g., in units of minutes per treatment session), K D  is a dialyzer phosphate clearance, K R  is a residual renal clearance of phosphate, n C   tx  is the normalized time averaged plasma phosphorus concentration during a dialysis treatment, and n C   i  is the normalized time averaged plasma phosphorus concentration for an interdialytic interval; and (c) simulate the effect of at least one of a patient parameter or a treatment parameter on C SS-PRE  of the hemodialysis patient. It should be appreciated that the variables for equation DD can be determined using any of the appropriate equations set forth herein. 
     Another such computing device can include a display device, an input device, a processor, and a memory device that stores a plurality of instructions, which when executed by the processor, cause the processor to operate with the display device and the input device to: (a) determine a net generation of phosphorus (“G”) using the equation: 
     
       
         
           
             
               
                 
                   G 
                   = 
                   
                     
                       C 
                       
                         
                           S 
                           ⁢ 
                           S 
                         
                         - 
                         
                           P 
                           ⁢ 
                           R 
                           ⁢ 
                           E 
                         
                         - 
                         
                           I 
                           ⁢ 
                           N 
                         
                       
                     
                     × 
                     
                       [ 
                       
                         
                           
                             
                               ( 
                               
                                 
                                   K 
                                   D 
                                 
                                 + 
                                 
                                   K 
                                   R 
                                 
                               
                               ) 
                             
                             ⁢ 
                             n 
                             ⁢ 
                             
                               
                                 C 
                                 _ 
                               
                               tx 
                             
                             ⁢ 
                             
                               t 
                               tx 
                             
                           
                           + 
                           
                             
                               K 
                               R 
                             
                             ⁢ 
                             n 
                             ⁢ 
                             
                               
                                 
                                   C 
                                   _ 
                                 
                                 i 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     10080 
                                     ⁢ 
                                     
                                       / 
                                     
                                     ⁢ 
                                     F 
                                   
                                   - 
                                   
                                     t 
                                     tx 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         
                           1 
                           ⁢ 
                           0 
                           ⁢ 
                           080 
                           ⁢ 
                           
                             / 
                           
                           ⁢ 
                           F 
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   EE 
                   ) 
                 
               
             
           
         
       
     
     wherein C SS-PRE-IN  is an initial, measured, steady state, pre-dialysis serum phosphorus level of the hemodialysis patient who is maintained by a hemodialysis therapy (e.g., identified by K D , F, and t tx ) for a specified time prior to the calculation of G using equation EE. The specified time can be, for example, at least one week, two weeks, three weeks, 1 month, 2 months, 3 months, 4 months or more prior to time when G is calculated. F is a frequency of treatments per week, t tx  is a treatment time for one hemodialysis treatment session, K D  is a dialyzer phosphate clearance, K R  is a residual renal clearance of phosphate, n C   tx  is the normalized time averaged plasma phosphorus concentration during a dialysis treatment, and n C   i  is the normalized time averaged plasma phosphorus concentration for an interdialytic interval; (b) predict steady state, pre-dialysis serum phosphorus levels (“C SS-PRE ”) of the hemodialysis patient using the equation: 
     
       
         
           
             
               
                 
                   
                     
                       C 
                       
                         
                           S 
                           ⁢ 
                           S 
                         
                         - 
                         
                           P 
                           ⁢ 
                           R 
                           ⁢ 
                           E 
                         
                       
                     
                     = 
                     
                       
                         
                           ( 
                           G 
                           ) 
                         
                         ⁢ 
                         
                           ( 
                           
                             1 
                             ⁢ 
                             0 
                             ⁢ 
                             080 
                             ⁢ 
                             
                               / 
                             
                             ⁢ 
                             F 
                           
                           ) 
                         
                       
                       
                         
                           
                             ( 
                             
                               
                                 K 
                                 D 
                               
                               + 
                               
                                 K 
                                 R 
                               
                             
                             ) 
                           
                           ⁢ 
                           n 
                           ⁢ 
                           
                             
                               C 
                               _ 
                             
                             
                               t 
                               ⁢ 
                               x 
                             
                           
                           ⁢ 
                           
                             t 
                             
                               t 
                               ⁢ 
                               x 
                             
                           
                         
                         + 
                         
                           
                             K 
                             R 
                           
                           ⁢ 
                           n 
                           ⁢ 
                           
                             
                               
                                 C 
                                 _ 
                               
                               i 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   10080 
                                   ⁢ 
                                   
                                     / 
                                   
                                   ⁢ 
                                   F 
                                 
                                 - 
                                 
                                   t 
                                   tx 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   FF 
                   ) 
                 
               
             
           
         
       
     
     and 
     (c) simulate the effect of at least one of a patient parameter or a treatment parameter on C SS-PRE  of the hemodialysis patient. It should be appreciated that the variables for equations EE and FF can be determined using any of the appropriate equations or methods set forth herein. 
     In any of the computing devices described herein, the information/data obtained for the hemodialysis patient can be displayed/printed out and used by the healthcare provider to provide improved treatment and nutritional regimens for the hemodialysis patient. Any of the unknown factors can be determined using any of the appropriate equations or measurements discussed herein for the methods of determining the steady state, pre-dialysis serum phosphorus levels of a hemodialysis patient. 
     The computing devices can also be preprogrammed or run according to software that causes the processor to operate with the display device and the input device to receive data relating to at least one of K R , K D , K M , V PRE , t tx , F, C PRE  about a month before a hemodialysis treatment session or a sampling time for collecting the serum phosphorus concentration. The computer device utilizes this information to simulate the effect of one or more of these patient parameters or treatment parameters on C SS-PRE  of the hemodialysis patient, for example, using equations DD or FF (e.g., seeing how a change in one or more of the patient parameters or treatment parameters impacts C SS-PRE ). The computing device can be preprogrammed to display a treatment regimen of the hemodialysis patient so that C SS-PRE  is within a desired range using any of the methods disclosed herein. In an embodiment, the computing device can be system  10  described in section I. 
     Any of the computer devices described herein (including any portions of system  10  described in section I) can be a device having a processor capable of receiving data and performing calculations based on that data. Such computing device can be, for example, a handheld client device, personal computer client device, database server, etc.). A more detailed block diagram of the electrical systems of the computing devices described herein is illustrated in  FIG. 10 . Although the electrical systems of these computing devices may be similar, the structural differences between these devices are well known. For example, a typical handheld client device is small and lightweight compared to a typical database server. 
     In  FIG. 10 , an example computing device  202  preferably includes one or more processors  204  electrically coupled by an address/data bus  206  to one or more memory devices  208 , other computer circuitry  210 , and one or more interface circuits  212 . Processor  204  may be any suitable processor, such as a microprocessor from the INTEL PENTIUM® family of microprocessors. Memory  208  preferably includes volatile memory and non-volatile memory. Preferably, memory  208  stores a software program (e.g., Matlab, C++, Fortran, etc.) that can perform the calculations necessary according to embodiments described herein and/or that interacts with the other devices in a hemodialysis system. This program may be executed by processor  204  in any suitable manner. Memory  208  may also store digital data indicative of documents, files, programs, web pages, etc. retrieved from another computing device and/or loaded via an input device  214 . 
     Interface circuit  212  may be implemented using any suitable interface standard, such as an Ethernet interface and/or a Universal Serial Bus (“USB”) interface. One or more input devices  214  may be connected to the interface circuit  212  for entering data and commands into computing device  202 . For example, input device  214  may be a keyboard, mouse, touch screen, track pad, track ball, isopoint, and/or a voice recognition system. 
     One or more displays, printers, speakers, and/or other output devices  216  may also be connected to computing device  202  via interface circuit  212 . Display  216  may be a cathode ray tube (“CRT”), a liquid crystal display (“LCD”), or any other type of display. Display  216  generates visual displays of data generated during operation of computing device  202 . The visual displays may include prompts for human input, run time statistics, measured values, calculated values, data, etc. 
     One or more storage devices  218  may also be connected to computing device  202  via interface circuit  212 . For example, a hard drive, CD drive, DVD drive, and/or other storage devices may be connected to computing device  202 . Storage devices  218  may store any type of suitable data. 
     Computing device  202  may also exchange data with other network devices  220  via a connection to a network  230 . The network connection may be any type of network connection, such as an Ethernet connection, digital subscriber line (“DSL”), telephone line, coaxial cable, etc. This allows computing device  202  to communicate with a suitable dialysis machine, a patient database and/or a hospital network depending on the desired applications. 
     EXAMPLES 
     By way of example and not limitation, the following examples are illustrative of various embodiments of the present disclosure and further illustrate experimental testing conducted with the systems and methods in accordance with embodiments of the present disclosure. 
     Example 1 
     Objective 
     The objective of this analysis was to demonstrate the non-linear least squares fitting procedure for estimating patient specific parameters (e.g., K M  and V PRE ) from a pseudo one-compartment model using clinical data, and to evaluate the validity of parameter estimates over different HD treatment modalities. 
     Pseudo One-Compartment Model 
     A conceptual description of the pseudo one-compartment model is shown in  FIG. 11 . In this model, phosphorus is removed by the dialyzer from a compartment of volume V, also called the distribution volume, and phosphorus concentration C. The distribution volume is assumed to be in equilibrium with plasma. Phosphorus mobilization into this compartment occurs from phosphorus pools in the body that are inaccessible to the dialyzer. These pools are represented as a large compartment with a constant phosphorus concentration equal to the pre-dialytic plasma phosphorus concentration (“C PRE ”). The rate of phosphorus mobilization into the distribution volume is described as the difference between pre-dialytic and instantaneous plasma phosphorus levels multiplied by the phosphorus mobilization clearance (“K M ”). K M  is analogous to an intercompartmental mass transfer coefficient, and is assumed to be constant during the treatment and the post-dialytic rebound periods. Residual renal clearance of phosphate is neglected in this example. 
     Changes in the volume and phosphorus concentration of phosphorus distribution volume during and shortly after an HD treatment session are represented by equations E-A1 and E-A2, 
     
       
         
           
             
               
                 
                   
                     
                       d 
                       ⁡ 
                       
                         ( 
                         VC 
                         ) 
                       
                     
                     
                       d 
                       ⁢ 
                       t 
                     
                   
                   = 
                   
                     
                       
                         K 
                         M 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             C 
                             
                               P 
                               ⁢ 
                               R 
                               ⁢ 
                               E 
                             
                           
                           - 
                           C 
                         
                         ) 
                       
                     
                     - 
                     
                       Θ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         K 
                         D 
                       
                       ⁢ 
                       C 
                     
                   
                 
               
               
                 
                   E 
                   - 
                   A1 
                 
               
             
             
               
                 
                   
                     
                       d 
                       ⁡ 
                       
                         ( 
                         V 
                         ) 
                       
                     
                     
                       d 
                       ⁢ 
                       t 
                     
                   
                   = 
                   
                     
                       - 
                       Θ 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       Q 
                       UF 
                     
                   
                 
               
               
                 
                   E 
                   - 
                   A2 
                 
               
             
           
         
       
     
     where, Θ is a variable that indicates whether dialysis treatment is taking place (Θ=1) or not (Θ=0), K D  is the dialyzer phosphate clearance, and Q UF  is the ultrafiltration (“UF”) rate. The kinetic model described above also assumes that all fluid removed during the treatment is from the distribution volume of phosphorus. 
     Closed form analytical solutions to the time-dependent plasma phosphorus concentration can be obtained by integrating equations 1 and 2. For the intradialytic (Θ=1) and rebound (Θ=0) periods, the time dependence of phosphorus concentration can be expressed as shown by equations E-A3 and E-A4 respectively: 
     
       
         
           
             
               
                 
                   
                     C 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       C 
                       PRE 
                     
                     [ 
                     
                       
                         
                           K 
                           M 
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 K 
                                 D 
                               
                               - 
                               
                                 Q 
                                 UF 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   
                                     V 
                                     PRE 
                                   
                                   - 
                                   
                                     
                                       Q 
                                       UF 
                                     
                                     × 
                                     t 
                                   
                                 
                                 
                                   V 
                                   PRE 
                                 
                               
                               ) 
                             
                             
                               
                                 
                                   K 
                                   M 
                                 
                                 + 
                                 
                                   K 
                                   D 
                                 
                                 - 
                                 
                                   Q 
                                   UF 
                                 
                               
                               
                                 Q 
                                 UF 
                               
                             
                           
                         
                       
                       
                         
                           K 
                           M 
                         
                         + 
                         
                           K 
                           D 
                         
                         - 
                         
                           Q 
                           UF 
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   E 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   A3 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       C 
                       ⁡ 
                       
                         ( 
                         T 
                         ) 
                       
                     
                     = 
                     
                       
                         C 
                         PRE 
                       
                       - 
                       
                         
                           ( 
                           
                             
                               C 
                               PRE 
                             
                             - 
                             
                               C 
                               POST 
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           e 
                           
                             ( 
                             
                               - 
                               
                                 
                                   
                                     K 
                                     M 
                                   
                                   ⁢ 
                                   T 
                                 
                                 
                                   
                                     V 
                                     PRE 
                                   
                                   - 
                                   
                                     
                                       Q 
                                       UE 
                                     
                                     ⁢ 
                                     
                                       t 
                                       tx 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   E 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   A4 
                 
               
             
           
         
       
     
     wherein V PRE  is the pre-dialytic distribution volume of phosphorus, t time during the treatment, T is time after the end of the treatment, and t tx  is total duration of treatment prior to the rebound period. It is also assumed that distribution volume of phosphorus remains constant during the post-dialytic rebound period. 
     Methods 
     Clinical data was obtained from 5 chronic hemodialysis patients who participated in a crossover trial. Patients underwent a short HD (“SHD”) treatment session and a conventional HD (“CHD”) treatment session one week apart. Blood samples were collected at t=0, 60, 90 min during SHD and, t=0, 30, 60, 120, 180 min during CHD treatments. Dialysate samples were collected 60 min after the start of treatments to determine dialyzer phosphate clearance. Additional blood samples were collected at t=10 seconds, 2, 10, 30, 60 min after the end of treatments. Plasma and dialysate samples were assayed for phosphorus. 
     Patient specific parameters (K M  and V PRE ) were estimated by non-linear least squares fitting to clinical data using equations 3 and 4. Least squares fitting was performed using a scientific computational software (MATLAB v2008a, Mathworks, Natick, Mass., USA). The model was fit to SHD and CHD data separately, resulting in two sets of K M  and V PRE  estimates for each patient. Q UF  was calculated by the difference between pre-dialytic and post-dialytic body weight of the patient divided by total treatment time. Dialyzer phosphate clearance was calculated according to equation E-A5, where C D  is the concentration of phosphorus in the dialysate outflow and Q D  is the dialysate flow rate. 
     
       
         
           
             
               
                 
                   
                     K 
                     D 
                   
                   = 
                   
                     
                       
                         
                           C 
                           D 
                         
                         ⁡ 
                         
                           ( 
                           
                             t 
                             = 
                             
                               60 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               min 
                             
                           
                           ) 
                         
                       
                       × 
                       
                         Q 
                         D 
                       
                     
                     
                       C 
                       ⁡ 
                       
                         ( 
                         
                           t 
                           = 
                           
                             60 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             min 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   E 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   A5 
                 
               
             
           
         
       
     
     Non-linear regression fits to clinical data from each patient are presented in  FIGS. 12-16 .  FIG. 12  shows modeled and measured plasma phosphorus concentrations for patient  1  during SHD and CHD treatment sessions.  FIG. 13  shows modeled and measured plasma phosphorus concentrations for patient  2  during SHD and CHD treatment sessions.  FIG. 14  shows modeled and measured plasma phosphorus concentrations for patient  3  during SHD and CHD treatment sessions.  FIG. 15  shows modeled and measured plasma phosphorus concentrations for patient  4  during SHD and CHD treatment sessions.  FIG. 16  shows modeled and measured plasma phosphorus concentrations for patient  5  during SHD and CHD treatment sessions. There was good agreement between modeled and measured plasma phosphorus concentration during SHD and CHD treatment sessions. 
     A detailed summary of parameter estimates are presented in Table 11.1. Parameter estimates varied considerably among patients, but for each patient, estimates obtained from SHD and CHD treatment sessions were similar. Low values for standard errors (“SE”) indicate high precision in parameter estimates. 
     These results suggest that K M  and V PRE  are patient specific parameters independent of HD treatment time. Therefore extending kinetic model predictions from thrice-weekly conventional (3-4 hours) to short-daily (2-3 hours) and nocturnal HD (6-10 hours) therapies may be feasible using K M  and V PRE  values estimated from conventional HD treatments. 
     
       
         
           
               
             
               
                 TABLE II.1 
               
             
            
               
                   
               
               
                 Estimated values of patient parameters 
               
               
                 from SHD and CHD treatment sessions. 
               
            
           
           
               
               
               
               
               
            
               
                   
                 K M  (SHD) 
                 K M  (CHD) 
                 V PRE  (SHD) 
                 V PRE  (CHD) 
               
               
                 Patient ID 
                 (ml/min) 
                 (ml/min) 
                 (L) 
                 (L) 
               
               
                   
               
               
                 1 
                  66 ± 10 
                 55 ± 5 
                 11.2 ± 1.4  
                 11.8 ± 1.0 
               
               
                 2 
                 78 ± 5 
                 84 ± 6 
                 8.1 ± 0.5 
                  9.6 ± 0.7 
               
               
                 3 
                 67 ± 8 
                  96 ± 12 
                 14.6 ± 1.3  
                 14.0 ± 2.2 
               
               
                 4 
                 104 ± 18 
                 101 ± 11 
                 7.3 ± 1.1 
                  8.9 ± 0.9 
               
               
                 5 
                 58 ± 5 
                 50 ± 5 
                 9.0 ± 0.7 
                 10.7 ± 1.1 
               
               
                   
               
               
                 Parameter estimates are expressed as estimated value ± standard error 
               
            
           
         
       
     
     Example 2 
     Objective 
     The objective of this study was to demonstrate the application of a simple method for estimating the patient parameter K M  from a pseudo one-compartment model using data from conventional 4-hour hemodialysis treatments, and to evaluate the accuracy of estimated K M  values via comparison to results obtained using non-linear least squares fitting. 
     Methods 
     Clinical data was obtained from 5 chronic hemodialysis patients who underwent CHD treatments. Blood samples were collected at t=0, 30, 60, 120, 180 min during the treatments and 10 seconds, 2, 10, 30, 60 min after the end of treatments. Dialysate samples were collected 60 min from the start of treatments to determine dialyzer phosphate clearance. Plasma and dialysate samples were assayed for phosphorus. 
     K M  was computed for each patient using equation E-B1, where C POST  is the post-dialytic plasma phosphorus concentration, K D  is the dialyzer phosphate clearance, Q UF  is the ultrafiltration rate or net fluid removal rate, and C PRE  is the pre-dialytic plasma phosphorus concentration. 
     
       
         
           
             
               
                 
                   
                     K 
                     M 
                   
                   = 
                   
                     
                       C 
                       POST 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             K 
                             D 
                           
                           - 
                           
                             Q 
                             UF 
                           
                         
                         
                           
                             C 
                             PRE 
                           
                           - 
                           
                             C 
                             POST 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   E 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   B1 
                 
               
             
           
         
       
     
     Q UF  was calculated by the difference between pre-dialytic and post-dialytic body weight of the patient divided by total treatment time. Dialyzer phosphate clearance was calculated according to equation E-B2, where C D  is concentration of phosphorus in the dialysate outflow and Q D  is the dialysate flow rate. 
     
       
         
           
             
               
                 
                   
                     K 
                     D 
                   
                   = 
                   
                     
                       
                         
                           C 
                           D 
                         
                         ⁡ 
                         
                           ( 
                           
                             t 
                             = 
                             
                               60 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               min 
                             
                           
                           ) 
                         
                       
                       × 
                       
                         Q 
                         D 
                       
                     
                     
                       C 
                       ⁡ 
                       
                         ( 
                         
                           t 
                           = 
                           
                             60 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             min 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   E 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   B2 
                 
               
             
           
         
       
     
     To evaluate the accuracy of equation E-B1, computed K M  values were compared with estimates obtained using non-linear least squares fitting to 10 measured intradialytic and post-dialytic rebound concentrations of plasma phosphorus, as described in Example 1. 
     Results 
     K M  values for individual patients, computed using equation E-B1 and estimated from non-linear least squares fitting to frequent measurements are presented in Table 11.2 along with the pre-dialytic and post-dialytic plasma phosphorus concentrations, ultrafiltration rate, and the dialyzer phosphate clearance. There was good agreement between K M  values obtained using equation E-B1, and non-linear least squares fitting. 
     These results suggest that equation E-B1 can be used as an alternative to performing non-linear least squares fitting to frequent measurements of plasma phosphorus concentrations for the estimation of patient specific K M . Its simple algebraic form and utilization of only pre-dialysis and post-dialysis blood samples makes it a practical method to study kinetics of phosphorus mobilization during HD treatments on an individual patient basis. 
     
       
         
           
               
             
               
                 TABLE II.2 
               
             
            
               
                   
               
               
                 K M  values for individual patients computed from equation E-B1, and 
               
               
                 estimated using non-linear least squares fitting (“NLSQ”) 
               
            
           
           
               
               
               
               
               
               
               
            
               
                   
                 C PRE   
                 C POST   
                 Q UF   
                 K D   
                 K M  (E-B1) 
                 K M  (NLSQ) 
               
               
                 Patient ID 
                 (mg/dl) 
                 (mg/dl) 
                 (ml/min) 
                 (ml/min) 
                 (ml/min) 
                 (ml/min) 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 1 
                 8.4 
                 2.3 
                 8 
                 154 
                 55 
                 56 
               
               
                 2 
                 4.4 
                 1.8 
                 7 
                 131 
                 85 
                 84 
               
               
                 3 
                 6.7 
                 3.2 
                 8 
                 129 
                 110 
                 96 
               
               
                 4 
                 7.3 
                 3.2 
                 12 
                 135 
                 96 
                 102 
               
               
                 5 
                 4.2 
                 1.5 
                 12 
                 117 
                 58 
                 51 
               
               
                   
               
               
                 Abbreviations: SD, standard deviation 
               
            
           
         
       
     
     Example 3 
     Steady State Phosphorus Mass Balance Model 
     Objectives 
     As previously discussed, the inventors have proposed a kinetic model for describing changes in serum or plasma phosphorus concentrations during hemodialysis (more generally extracorporeal treatments) and the post-dialytic rebound period. That kinetic model allows prediction of intradialytic phosphorus concentrations as a function of time and total phosphate removal from the knowledge of: 1) the pre-dialysis concentration of phosphorus in plasma or serum, 2) the dialyzer clearance of phosphate, 3) the volume of distribution of phosphorus, 4) the amount of fluid removed during the treatment, and 5) a patient-specific phosphorus mobilization clearance. The steady state phosphorus mass balance model described below will be used in combination with the previous kinetic model to allow determination of pre-dialysis serum phosphorus concentration for individual patients under any hemodialysis treatment prescription when the above parameters 2-5 are established, the frequency of hemodialysis treatments per week and the hemodialysis treatment duration are prescribed, and the net generation of phosphorus (defined below), and residual kidney or renal phosphorus clearance are all known. Alternatively, the steady state phosphorus mass balance model in combination with the previous kinetic model can be used to determine the net generation of phosphorus for a given patient when the above parameters 1-5 are established and the frequency of hemodialysis treatments per week, the hemodialysis treatment duration and the residual kidney phosphorus clearance are known. As in other mass balance models, the patient is assumed to be in steady state. 
     Steady State Mass Balance Model 
     The model used to describe steady state phosphorus mass balance over a time averaged period, i.e., a week, for a patient treated by hemodialysis is shown schematically in  FIG. 17 . This model is a generalized version of the kinetic model described previously characterizing phosphorus kinetics during treatments and the post-dialytic rebound period. The model assumes that phosphorus is distributed in a single, well-mixed compartment. 
     There are several pathways that result in changes in the concentration of phosphorus (“C”) within its distribution volume (“V”). Dietary intake of phosphorus is derived mostly from dietary protein; however, food additives can also contain significant amounts of phosphate. The amount of dietary phosphorus intake often exceeds the amount of phosphate that can be removed by conventional thrice weekly hemodialysis; therefore, dialysis patients are frequently prescribed oral phosphate binders to control serum phosphorus concentrations. Dietary intake of phosphorus minus the amount of phosphate that is bound and not absorbed intestinally can be combined and is defined as the net generation of phosphorus (“G”); this parameter will be assumed to be a constant in this model. Phosphorus can be removed directly from its distribution volume by dialyzer clearance (“K D ”) or residual renal or kidney clearance (“K R ”). As described previously in the kinetic model, phosphorus can also be mobilized from other compartments at a rate proportional to the difference between the instantaneous and the pre-dialysis (“C PRE ”) phosphorus concentrations. The proportionality parameter has been termed the phosphorus mobilization clearance (“K M ”). Phosphorus can also be deposited into tissues depending on a critical tissue concentration (“C t ”); this process has been modeled as a tissue deposition clearance (“Kt”). 
     The model shown in  FIG. 17  was devised to identify all major pathways for phosphorus distribution in hemodialysis patients; however, it is likely too complex to be useful clinically and requires simplification. Specifically, tissue deposition of phosphorus is likely to be small compared with the other pathways and can be neglected as a first approximation in this mass balance model. 
     A mass balance differential equation for the model shown in  FIG. 17  (neglecting tissue deposition of phosphorus) is the following: 
     
       
         
           
             
               
                 
                   
                     
                       d 
                       ⁡ 
                       
                         ( 
                         CV 
                         ) 
                       
                     
                     
                       d 
                       ⁢ 
                       t 
                     
                   
                   = 
                   
                     G 
                     - 
                     
                       
                         K 
                         D 
                       
                       ⁢ 
                       C 
                     
                     - 
                     
                       
                         K 
                         R 
                       
                       ⁢ 
                       C 
                     
                     + 
                     
                       
                         K 
                         M 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             C 
                             PRE 
                           
                           - 
                           C 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   E 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   C 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   1 
                 
               
             
           
         
       
     
     Assuming that hemodialysis treatment sessions are symmetrically placed throughout the week, integrating equation E-C1 over a week is equivalent to integrating it over one complete cycle of a treatment (with a treatment time of t tx ) and an interdialytic interval between treatments (with a time of T i ). Note that lower case t indicates time during the treatment and it varies from 0 to t tx , whereas T indicates the time during the interdialytic interval and it varies from 0 to T i . The values of t tx  and T i  are related and dependent on the number of treatments per week (the above mathematical analysis is general and applies to an arbitrary number of hemodialysis treatment sessions per week; here, F denotes the number of treatments per week). If t and T are reported in units of hours, then T i =168/F−t tx . If t and T are reported in units of minutes, then T i =10080/F−t tx . This integration results in equation E-C2 after some rearrangement. 
     
       
         
           
             
               
                 
                   
                     
                       Δ 
                       ⁡ 
                       
                         ( 
                         CV 
                         ) 
                       
                     
                     - 
                     
                       
                         K 
                         M 
                       
                       ⁢ 
                       
                         
                           ∫ 
                           0 
                           
                             
                               t 
                               tx 
                             
                             + 
                             
                               T 
                               i 
                             
                           
                         
                         ⁢ 
                         
                           
                             [ 
                             
                               
                                 C 
                                 PRE 
                               
                               - 
                               
                                 C 
                                 ⁡ 
                                 
                                   ( 
                                   τ 
                                   ) 
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           d 
                           ⁢ 
                           τ 
                         
                       
                     
                   
                   = 
                   
                     
                       G 
                       ⁡ 
                       
                         ( 
                         
                           
                             t 
                             tx 
                           
                           + 
                           
                             T 
                             i 
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         ( 
                         
                           
                             K 
                             D 
                           
                           + 
                           
                             K 
                             R 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         
                           ∫ 
                           
                             t 
                             = 
                             0 
                           
                           
                             t 
                             tx 
                           
                         
                         ⁢ 
                         
                           
                             C 
                             ⁡ 
                             
                               ( 
                               τ 
                               ) 
                             
                           
                           ⁢ 
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           τ 
                         
                       
                     
                     - 
                     
                       
                         K 
                         R 
                       
                       ⁢ 
                       
                         
                           ∫ 
                           
                             T 
                             = 
                             0 
                           
                           
                             T 
                             i 
                           
                         
                         ⁢ 
                         
                           
                             C 
                             ⁡ 
                             
                               ( 
                               τ 
                               ) 
                             
                           
                           ⁢ 
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           τ 
                         
                       
                     
                   
                 
               
               
                 
                   E 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   C 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   2 
                 
               
             
           
         
       
     
     where Δ(CV) indicates the change in phosphorus mass within its volume of distribution. Note that the second term on the left hand side of this equation indicates (minus) the mass of phosphorus transported into V via the mobilization pathway. 
     Assuming further that the patient is in steady state and total phosphorus mass in the body (i.e., left hand side of equation E-C2) does not change during one complete cycle of a treatment and an interdialytic interval, the left hand side of equation E-C2 must be zero; thus, equation E-C2 is reduced in steady state to the following: 
     
       
         
           
             
               
                 
                   0 
                   = 
                   
                     
                       G 
                       ⁡ 
                       
                         ( 
                         
                           
                             t 
                             tx 
                           
                           + 
                           
                             T 
                             i 
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         ( 
                         
                           
                             K 
                             D 
                           
                           + 
                           
                             K 
                             R 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         
                           ∫ 
                           
                             t 
                             = 
                             0 
                           
                           
                             t 
                             tx 
                           
                         
                         ⁢ 
                         
                           
                             C 
                             ⁡ 
                             
                               ( 
                               τ 
                               ) 
                             
                           
                           ⁢ 
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           τ 
                         
                       
                     
                     - 
                     
                       
                         K 
                         R 
                       
                       ⁢ 
                       
                         
                           ∫ 
                           
                             T 
                             = 
                             0 
                           
                           
                             T 
                             i 
                           
                         
                         ⁢ 
                         
                           
                             C 
                             ⁡ 
                             
                               ( 
                               τ 
                               ) 
                             
                           
                           ⁢ 
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           τ 
                         
                       
                     
                   
                 
               
               
                 
                   E 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   C 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   3 
                 
               
             
           
         
       
     
     To use this integrated mass balance equation, it is necessary to calculate both the integrals in equation E-C3. Note that the first integral is over the time period during the treatment and the second is over the time period during the interdialytic interval. 
     To evaluate the integrals in equation E-C3, the inventors make two additional assumptions. First, the inventors assume that changes in phosphorus concentration during a treatment and the post-dialytic rebound period can be described by the previously proposed kinetic model where net generation of phosphorus can be neglected. Second, the inventors assume that this same kinetic model describes changes in phosphorus concentration during the entire interdialytic interval, not just the post-dialysis rebound period. The equation describing the kinetic model is the following 
     
       
         
           
             
               
                 
                   
                     
                       d 
                       ⁡ 
                       
                         ( 
                         
                           C 
                           ⁢ 
                           V 
                         
                         ) 
                       
                     
                     
                       d 
                       ⁢ 
                       t 
                     
                   
                   = 
                   
                     
                       
                         - 
                         
                           K 
                           D 
                         
                       
                       ⁢ 
                       C 
                     
                     - 
                     
                       
                         K 
                         R 
                       
                       ⁢ 
                       C 
                     
                     + 
                     
                       
                         K 
                         M 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             C 
                             PRE 
                           
                           - 
                           C 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   E 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   C4 
                 
               
             
           
         
       
     
     Equation E-C4 can be solved analytically; the time dependence of the serum phosphorus concentration during the treatment can be described by 
     
       
         
           
             
               
                 
                   
                     
                       C 
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     
                       C 
                       PRE 
                     
                   
                   = 
                   
                     
                       
                         K 
                         M 
                       
                       
                         
                           K 
                           M 
                         
                         + 
                         
                           K 
                           R 
                         
                         + 
                         
                           K 
                           D 
                         
                         - 
                         
                           Q 
                           
                             U 
                             ⁢ 
                             F 
                           
                         
                       
                     
                     + 
                     
                       
                         
                           [ 
                           
                             1 
                             - 
                             
                               
                                 K 
                                 M 
                               
                               
                                 
                                   K 
                                   M 
                                 
                                 + 
                                 
                                   K 
                                   R 
                                 
                                 + 
                                 
                                   K 
                                   D 
                                 
                                 - 
                                 
                                   Q 
                                   UF 
                                 
                               
                             
                           
                           ] 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               V 
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             
                               V 
                               PRE 
                             
                           
                           ] 
                         
                       
                       
                         
                           ( 
                           
                             
                               K 
                               M 
                             
                             + 
                             
                               K 
                               D 
                             
                             + 
                             
                               K 
                               R 
                             
                             - 
                             
                               Q 
                               UF 
                             
                           
                           ) 
                         
                         / 
                         
                           Q 
                           UF 
                         
                       
                     
                   
                 
               
               
                 
                   E 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   C5 
                 
               
             
           
         
       
     
     where it has been assumed that fluid is removed from the patient at a constant rate (“Q UF ”), such that the distribution volume decreases linearly from its initial, pre-dialysis value (“V PRE ”) throughout the treatment. Stated in mathematical terms, 
         V ( t )= V   PRE   −Q   UF   ×t   E-C6
 
     Thus, it is assumed that all fluid removed during the treatment is removed from the distribution volume of phosphorus. 
     During the rebound period (and the entire interdialytic interval), equation E-C4 remains valid except that K D  is zero. Assuming that the patient gains fluid at a constant rate (“Q WG ”) during the interdialytic interval, such that the distribution volume increases linearly from its initial post-dialysis value (“V POST ”), the analytical solution describing the time dependence of the serum phosphorus concentration during the interdialytic interval is 
     
       
         
           
             
               
                 
                   
                     
                       C 
                       ⁡ 
                       
                         ( 
                         T 
                         ) 
                       
                     
                     
                       C 
                       PRE 
                     
                   
                   = 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 C 
                                 POST 
                               
                               
                                 C 
                                 PRE 
                               
                             
                             - 
                             
                               
                                 K 
                                 M 
                               
                               
                                 
                                   K 
                                   M 
                                 
                                 + 
                                 
                                   K 
                                   R 
                                 
                                 + 
                                 
                                   Q 
                                   WG 
                                 
                               
                             
                           
                           ] 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               V 
                               ⁡ 
                               
                                 ( 
                                 T 
                                 ) 
                               
                             
                             
                               V 
                               POST 
                             
                           
                           ] 
                         
                       
                       
                         
                           - 
                           
                             ( 
                             
                               
                                 K 
                                 M 
                               
                               + 
                               
                                 K 
                                 R 
                               
                               + 
                               
                                 Q 
                                 
                                   W 
                                   ⁢ 
                                   G 
                                 
                               
                             
                             ) 
                           
                         
                         / 
                         
                           Q 
                           
                             W 
                             ⁢ 
                             G 
                           
                         
                       
                     
                     + 
                     
                       
                         K 
                         M 
                       
                       
                         
                           K 
                           M 
                         
                         + 
                         
                           K 
                           R 
                         
                         + 
                         
                           Q 
                           WG 
                         
                       
                     
                   
                 
               
               
                 
                   E 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   C7 
                 
               
             
           
         
       
     
     where the time dependence of the distribution volume during the interdialytic interval is described by 
         V ( T )= V ( T= 0)+ Q   WG   ×T=V   POST   ±Q   WG   ×T=V   PRE   −Q   UF   ×t   tx   +Q   WG   ×T   E-C8
 
     Note that it is assumed that all fluid gained during the interdialytic interval is confined to the distribution volume of phosphorus. The two integrals in equation E-C3 can now be obtained by integrating equations E-C5 and E-C7. The normalized time averaged plasma phosphorus concentration during dialysis treatments (n C   tx ) is obtained by integrating equation E-C5: 
     
       
         
           
             
               
                 
                   
                     n 
                     ⁢ 
                     
                       
                         C 
                         _ 
                       
                       tx 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           1 
                           
                             
                               t 
                               tx 
                             
                             ⁢ 
                             
                               C 
                               PRE 
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         
                           ∫ 
                           0 
                           
                             t 
                             tx 
                           
                         
                         ⁢ 
                         
                           
                             C 
                             ⁡ 
                             
                               ( 
                               τ 
                               ) 
                             
                           
                           ⁢ 
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           τ 
                         
                       
                     
                     = 
                     
                       
                         1 
                         
                           t 
                           tx 
                         
                       
                       ⁢ 
                       
                         { 
                         
                           
                             [ 
                             
                               
                                 
                                   K 
                                   M 
                                 
                                 ⁢ 
                                 
                                   t 
                                   tx 
                                 
                               
                               
                                 
                                   K 
                                   M 
                                 
                                 + 
                                 
                                   K 
                                   R 
                                 
                                 + 
                                 
                                   K 
                                   D 
                                 
                                 - 
                                 
                                   Q 
                                   UF 
                                 
                               
                             
                             ] 
                           
                           + 
                           
                             
                               [ 
                               
                                 1 
                                 - 
                                 
                                   
                                     K 
                                     M 
                                   
                                   
                                     
                                       K 
                                       M 
                                     
                                     + 
                                     
                                       K 
                                       R 
                                     
                                     + 
                                     
                                       K 
                                       D 
                                     
                                     - 
                                     
                                       Q 
                                       UF 
                                     
                                   
                                 
                               
                               ] 
                             
                             ⁢ 
                             
                                 
                               
                                 
                                   [ 
                                   
                                     
                                       V 
                                       PRE 
                                     
                                     
                                       
                                         K 
                                         M 
                                       
                                       + 
                                       
                                         K 
                                         R 
                                       
                                       + 
                                       
                                         K 
                                         D 
                                       
                                     
                                   
                                   ] 
                                 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     1 
                                     - 
                                     
                                       
                                         ( 
                                         
                                           
                                             V 
                                             POST 
                                           
                                           
                                             V 
                                             PRE 
                                           
                                         
                                         ) 
                                       
                                       
                                         
                                           ( 
                                           
                                             
                                               K 
                                               M 
                                             
                                             + 
                                             
                                               K 
                                               R 
                                             
                                             + 
                                             
                                               K 
                                               D 
                                             
                                           
                                           ) 
                                         
                                         ⁢ 
                                         l 
                                         ⁢ 
                                         
                                           Q 
                                           UF 
                                         
                                       
                                     
                                   
                                   ] 
                                 
                               
                               } 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   E 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   C9 
                 
               
             
           
         
       
     
     Integration of equation E-C7 and calculation of C POST /C PRE  from equation E-C5 when t=t tx  gives the normalized time averaged plasma phosphorus concentration for the interdialytic interval (n C   i ): 
     
       
         
           
             
               
                 
                   
                     n 
                     ⁢ 
                     
                       
                         C 
                         _ 
                       
                       i 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           1 
                           
                             
                               T 
                               i 
                             
                             ⁢ 
                             
                               C 
                               PRE 
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         
                           ∫ 
                           0 
                           
                             T 
                             i 
                           
                         
                         ⁢ 
                         
                           
                             C 
                             ⁡ 
                             
                               ( 
                               τ 
                               ) 
                             
                           
                           ⁢ 
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           τ 
                         
                       
                     
                     = 
                     
                       
                         1 
                         
                           
                             1 
                             ⁢ 
                             0 
                             ⁢ 
                             0 
                             ⁢ 
                             8 
                             ⁢ 
                             
                               0 
                               / 
                               F 
                             
                           
                           - 
                           
                             t 
                             tx 
                           
                         
                       
                       ⁢ 
                       
                         { 
                         
                           
                             [ 
                             
                               
                                 
                                   K 
                                   M 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       1 
                                       ⁢ 
                                       0 
                                       ⁢ 
                                       0 
                                       ⁢ 
                                       8 
                                       ⁢ 
                                       
                                         0 
                                         / 
                                         F 
                                       
                                     
                                     - 
                                     
                                       t 
                                       tx 
                                     
                                   
                                   ) 
                                 
                               
                               
                                 
                                   K 
                                   M 
                                 
                                 + 
                                 
                                   K 
                                   R 
                                 
                                 + 
                                 
                                   Q 
                                   WG 
                                 
                               
                             
                             ] 
                           
                           + 
                           
                             
                               [ 
                               
                                 
                                   
                                     K 
                                     M 
                                   
                                   
                                     
                                       K 
                                       M 
                                     
                                     + 
                                     
                                       K 
                                       R 
                                     
                                     + 
                                     
                                       K 
                                       D 
                                     
                                     - 
                                     
                                       Q 
                                       UF 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     
                                       [ 
                                       
                                         1 
                                         - 
                                         
                                           
                                             K 
                                             M 
                                           
                                           
                                             
                                               K 
                                               M 
                                             
                                             + 
                                             
                                               K 
                                               R 
                                             
                                             + 
                                             
                                               K 
                                               D 
                                             
                                             - 
                                             
                                               Q 
                                               UF 
                                             
                                           
                                         
                                       
                                       ] 
                                     
                                     ⁡ 
                                     
                                       [ 
                                       
                                         
                                           V 
                                           POST 
                                         
                                         
                                           V 
                                           PRE 
                                         
                                       
                                       ] 
                                     
                                   
                                   
                                     
                                       ( 
                                       
                                         
                                           K 
                                           M 
                                         
                                         + 
                                         
                                           K 
                                           D 
                                         
                                         + 
                                         
                                           K 
                                           R 
                                         
                                         - 
                                         
                                           Q 
                                           UF 
                                         
                                       
                                       ) 
                                     
                                     / 
                                     
                                       Q 
                                       UF 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     K 
                                     M 
                                   
                                   
                                     
                                       K 
                                       M 
                                     
                                     + 
                                     
                                       K 
                                       R 
                                     
                                     + 
                                     
                                       Q 
                                       
                                         W 
                                         ⁢ 
                                         G 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                             × 
                             
                                 
                               
                                 
                                   [ 
                                   
                                     
                                       V 
                                       POST 
                                     
                                     
                                       
                                         K 
                                         M 
                                       
                                       + 
                                       
                                         K 
                                         R 
                                       
                                     
                                   
                                   ] 
                                 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     1 
                                     - 
                                     
                                       
                                         ( 
                                         
                                           
                                             V 
                                             POST 
                                           
                                           
                                             V 
                                             PRE 
                                           
                                         
                                         ) 
                                       
                                       
                                         
                                           ( 
                                           
                                             
                                               K 
                                               M 
                                             
                                             + 
                                             
                                               K 
                                               R 
                                             
                                           
                                           ) 
                                         
                                         / 
                                         
                                           Q 
                                           
                                             W 
                                             ⁢ 
                                             G 
                                           
                                         
                                       
                                     
                                   
                                   ] 
                                 
                               
                               } 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   E 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   C10 
                 
               
             
           
         
       
     
     When equations E-C3, E-C9 and E-C10 are combined, the resulting equation governing phosphorus mass balance at steady state can be expressed as: 
     
       
         
           
             
               
                 
                   G 
                   = 
                   
                     
                       C 
                       
                         PRE 
                         - 
                         SS 
                         - 
                         IN 
                       
                     
                     × 
                     
                       
                         
                           
                             ( 
                             
                               
                                 K 
                                 D 
                               
                               + 
                               
                                 K 
                                 R 
                               
                             
                             ) 
                           
                           ⁢ 
                           n 
                           ⁢ 
                           
                             
                               C 
                               _ 
                             
                             
                               t 
                               ⁢ 
                               x 
                             
                           
                           ⁢ 
                           
                             t 
                             
                               t 
                               ⁢ 
                               x 
                             
                           
                         
                         + 
                         
                           
                             K 
                             R 
                           
                           ⁢ 
                           n 
                           ⁢ 
                           
                             
                               
                                 C 
                                 _ 
                               
                               i 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   100 
                                   ⁢ 
                                   8 
                                   ⁢ 
                                   
                                     0 
                                     / 
                                     F 
                                   
                                 
                                 - 
                                 
                                   t 
                                   
                                     t 
                                     ⁢ 
                                     x 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         1 
                         ⁢ 
                         0 
                         ⁢ 
                         0 
                         ⁢ 
                         8 
                         ⁢ 
                         
                           0 
                           / 
                           F 
                         
                       
                     
                   
                 
               
               
                 
                   E 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   C11 
                 
               
             
           
         
       
     
     C SS-PRE-IN  is an initial, measured, steady state, pre-dialysis serum phosphorus level of the hemodialysis patient who is maintained by a hemodialysis therapy (e.g., identified by K D , F and t tx ) for a specified time prior to the calculation of G using equation E-C11. Equation E-C11 can be used to predict G if the pre-dialysis serum concentration is measured in a patient with knowledge of various treatment and patient parameters. 
     Once G has been calculated using equation E-C11 or estimated by other methods, it can be used in predicting the effect of changes in hemodialysis treatment parameters on the steady state serum phosphorus concentration by rearranging equation E-C11 to the following: 
     
       
         
           
             
               
                 
                   
                     C 
                     
                       PRE 
                       - 
                       SS 
                     
                   
                   = 
                   
                     G 
                     × 
                     
                       
                         1 
                         ⁢ 
                         0 
                         ⁢ 
                         0 
                         ⁢ 
                         8 
                         ⁢ 
                         
                           0 
                           / 
                           F 
                         
                       
                       
                         
                           
                             ( 
                             
                               
                                 K 
                                 D 
                               
                               + 
                               
                                 K 
                                 R 
                               
                             
                             ) 
                           
                           ⁢ 
                           n 
                           ⁢ 
                           
                             
                               C 
                               _ 
                             
                             
                               t 
                               ⁢ 
                               x 
                             
                           
                           ⁢ 
                           
                             t 
                             
                               t 
                               ⁢ 
                               x 
                             
                           
                         
                         + 
                         
                           
                             K 
                             R 
                           
                           ⁢ 
                           n 
                           ⁢ 
                           
                             
                               
                                 C 
                                 _ 
                               
                               i 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   100 
                                   ⁢ 
                                   8 
                                   ⁢ 
                                   
                                     0 
                                     / 
                                     F 
                                   
                                 
                                 - 
                                 
                                   t 
                                   
                                     t 
                                     ⁢ 
                                     x 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   E 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   C12 
                 
               
             
           
         
       
     
     In general, there is an optimal range of pre-dialysis serum phosphorus levels in patients with end stage renal disease; thus, equation E-C12 can be used to optimize the prescription to obtain a desired pre-dialysis serum phosphorus concentration. It should be mentioned that changes in the hemodialysis prescription or in patient behavior (e.g., dietary intake) could lead to changes in G; thus, the iterative use of equations E-C11 and E-C12 to optimize C SS-PRE  may be necessary. 
     Equation E-C12 can be used to predict the pre-dialysis serum phosphorus concentration with knowledge of various treatment and patient parameters. Thus, equations E-C9 to E-C12 can be viewed as defining a steady state phosphorus mass balance of the hemodialysis patient. 
     Equations E-C5 to E-C10 do not apply when there is negligible net ultrafiltration or fluid removal from the patient during the treatment and no weight gain between treatments. When there is negligible ultrafiltration during the treatment and no weight gain between treatments, equations E-C5 to E-C10 become: 
     
       
         
           
             
               
                 
                   
                     
                       C 
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     
                       C 
                       PRE 
                     
                   
                   = 
                   
                     
                       
                         K 
                         M 
                       
                       
                         
                           K 
                           M 
                         
                         + 
                         
                           K 
                           R 
                         
                         + 
                         
                           K 
                           D 
                         
                       
                     
                     + 
                     
                         
                       
                         
                           [ 
                           
                             1 
                             - 
                             
                               
                                 K 
                                 M 
                               
                               
                                 
                                   K 
                                   M 
                                 
                                 + 
                                 
                                   K 
                                   R 
                                 
                                 + 
                                 
                                   K 
                                   D 
                                 
                               
                             
                           
                           ] 
                         
                         × 
                         
                           exp 
                           ⁡ 
                           
                             [ 
                             
                               
                                 - 
                                 
                                   ( 
                                   
                                     
                                       K 
                                       D 
                                     
                                     + 
                                     
                                       K 
                                       R 
                                     
                                     + 
                                     
                                       K 
                                       M 
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 t 
                                 / 
                                 
                                   V 
                                   PRE 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   E 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   C5A 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       V 
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         V 
                         ⁡ 
                         
                           ( 
                           
                             t 
                             = 
                             0 
                           
                           ) 
                         
                       
                       = 
                       
                         V 
                         
                           P 
                           ⁢ 
                           R 
                           ⁢ 
                           E 
                         
                       
                     
                   
                 
               
               
                 
                   E 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   C6A 
                 
               
             
             
               
                 
                   
                     
                       C 
                       ⁡ 
                       
                         ( 
                         T 
                         ) 
                       
                     
                     
                       C 
                       PRE 
                     
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               C 
                               POST 
                             
                             
                               C 
                               PRE 
                             
                           
                           - 
                           
                             
                               K 
                               M 
                             
                             
                               
                                 K 
                                 M 
                               
                               + 
                               
                                 K 
                                 R 
                               
                             
                           
                         
                         ] 
                       
                       × 
                       
                         exp 
                         ⁡ 
                         
                           [ 
                           
                             
                               - 
                               
                                 ( 
                                 
                                   
                                     K 
                                     R 
                                   
                                   + 
                                   
                                     K 
                                     M 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               T 
                               / 
                               
                                 V 
                                 POST 
                               
                             
                           
                           ] 
                         
                       
                     
                     + 
                     
                       
                         K 
                         M 
                       
                       
                         
                           K 
                           M 
                         
                         + 
                         
                           K 
                           R 
                         
                       
                     
                   
                 
               
               
                 
                   E 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   C7A 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       V 
                       ⁡ 
                       
                         ( 
                         T 
                         ) 
                       
                     
                     = 
                     
                       
                         V 
                         ⁡ 
                         
                           ( 
                           
                             T 
                             = 
                             0 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           V 
                           POST 
                         
                         = 
                         
                           V 
                           PRE 
                         
                       
                     
                   
                 
               
               
                 
                   E 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   C8A 
                 
               
             
             
               
                 
                   
                     n 
                     ⁢ 
                     
                       
                         C 
                         _ 
                       
                       
                         t 
                         ⁢ 
                         x 
                       
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           1 
                           
                             
                               t 
                               
                                 t 
                                 ⁢ 
                                 x 
                               
                             
                             ⁢ 
                             
                               C 
                               PRE 
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         
                           ∫ 
                           0 
                           
                             t 
                             tx 
                           
                         
                         ⁢ 
                         
                           
                             C 
                             ⁡ 
                             
                               ( 
                               τ 
                               ) 
                             
                           
                           ⁢ 
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           τ 
                         
                       
                     
                     = 
                     
                       
                         1 
                         
                           t 
                           tx 
                         
                       
                       ⁢ 
                       
                         { 
                         
                           
                             [ 
                             
                               
                                 
                                   K 
                                   M 
                                 
                                 ⁢ 
                                 
                                   t 
                                   
                                     t 
                                     ⁢ 
                                     x 
                                   
                                 
                               
                               
                                 
                                   K 
                                   M 
                                 
                                 + 
                                 
                                   K 
                                   R 
                                 
                                 + 
                                 
                                   K 
                                   D 
                                 
                               
                             
                             ] 
                           
                           + 
                           
                             
                               [ 
                               
                                 1 
                                 - 
                                 
                                   
                                     K 
                                     M 
                                   
                                   
                                     
                                       K 
                                       M 
                                     
                                     + 
                                     
                                       K 
                                       R 
                                     
                                     + 
                                     
                                       K 
                                       D 
                                     
                                   
                                 
                               
                               ] 
                             
                             ⁢ 
                             
                                 
                               
                                 
                                   [ 
                                   
                                     
                                       V 
                                       PRE 
                                     
                                     
                                       
                                         K 
                                         M 
                                       
                                       + 
                                       
                                         K 
                                         R 
                                       
                                       + 
                                       
                                         K 
                                         D 
                                       
                                     
                                   
                                   ] 
                                 
                                 ⁢ 
                                 
                                     
                                   
                                     [ 
                                     
                                       1 
                                       - 
                                       
                                         exp 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               - 
                                               
                                                 ( 
                                                 
                                                   
                                                     K 
                                                     D 
                                                   
                                                   + 
                                                   
                                                     K 
                                                     R 
                                                   
                                                   + 
                                                   
                                                     K 
                                                     M 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                             ⁢ 
                                             
                                               t 
                                               
                                                 t 
                                                 ⁢ 
                                                 x 
                                               
                                             
                                             ⁢ 
                                             l 
                                             ⁢ 
                                             
                                               V 
                                               PRE 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     ] 
                                   
                                   } 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   E 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   C9A 
                 
               
             
             
               
                 
                   
                     n 
                     ⁢ 
                     
                       
                         C 
                         _ 
                       
                       i 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           1 
                           
                             
                               T 
                               i 
                             
                             ⁢ 
                             
                               C 
                               PRE 
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         
                           ∫ 
                           0 
                           
                             T 
                             i 
                           
                         
                         ⁢ 
                         
                           
                             C 
                             ⁡ 
                             
                               ( 
                               τ 
                               ) 
                             
                           
                           ⁢ 
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           τ 
                         
                       
                     
                     = 
                     
                       
                         1 
                         
                           
                             1 
                             ⁢ 
                             0 
                             ⁢ 
                             0 
                             ⁢ 
                             8 
                             ⁢ 
                             
                               0 
                               / 
                               F 
                             
                           
                           - 
                           
                             t 
                             
                               t 
                               ⁢ 
                               x 
                             
                           
                         
                       
                       ⁢ 
                       
                         { 
                         
                           
                             [ 
                             
                               
                                 
                                   K 
                                   M 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       1 
                                       ⁢ 
                                       0 
                                       ⁢ 
                                       0 
                                       ⁢ 
                                       8 
                                       ⁢ 
                                       
                                         0 
                                         / 
                                         F 
                                       
                                     
                                     - 
                                     
                                       t 
                                       
                                         t 
                                         ⁢ 
                                         x 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               
                                 
                                   K 
                                   M 
                                 
                                 + 
                                 
                                   K 
                                   R 
                                 
                               
                             
                             ] 
                           
                           + 
                           
                             [ 
                             
                               
                                 
                                   K 
                                   M 
                                 
                                 
                                   
                                     K 
                                     M 
                                   
                                   + 
                                   
                                     K 
                                     R 
                                   
                                   + 
                                   
                                     K 
                                     D 
                                   
                                 
                               
                               + 
                               
                                 
                                     
                                   
                                     
                                       
                                         [ 
                                         
                                           1 
                                           - 
                                           
                                             
                                               K 
                                               M 
                                             
                                             
                                               
                                                 K 
                                                 M 
                                               
                                               + 
                                               
                                                 K 
                                                 R 
                                               
                                               + 
                                               
                                                 K 
                                                 D 
                                               
                                             
                                           
                                         
                                         ] 
                                       
                                       × 
                                       
                                         exp 
                                         ⁡ 
                                         
                                           [ 
                                           
                                             
                                               - 
                                               
                                                 ( 
                                                 
                                                   
                                                     K 
                                                     D 
                                                   
                                                   + 
                                                   
                                                     K 
                                                     R 
                                                   
                                                   + 
                                                   
                                                     K 
                                                     M 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                             ⁢ 
                                             
                                               
                                                 t 
                                                 tx 
                                               
                                               / 
                                               
                                                 V 
                                                 PRE 
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                     
                                     - 
                                     
                                       
                                         K 
                                         M 
                                       
                                       
                                         
                                           K 
                                           M 
                                         
                                         + 
                                         
                                           K 
                                           R 
                                         
                                       
                                     
                                   
                                   ] 
                                 
                                 × 
                                 
                                   [ 
                                   
                                     
                                       V 
                                       POST 
                                     
                                     
                                       
                                         K 
                                         M 
                                       
                                       + 
                                       
                                         K 
                                         R 
                                       
                                     
                                   
                                   ] 
                                 
                                 ⁢ 
                                 
                                     
                                   
                                     [ 
                                     
                                       1 
                                       - 
                                       
                                         exp 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               - 
                                               
                                                 ( 
                                                 
                                                   
                                                     K 
                                                     R 
                                                   
                                                   + 
                                                   
                                                     K 
                                                     M 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                             ⁢ 
                                             
                                               
                                                 ( 
                                                 
                                                   
                                                     10080 
                                                     / 
                                                     F 
                                                   
                                                   - 
                                                   
                                                     t 
                                                     tx 
                                                   
                                                 
                                                 ) 
                                               
                                               / 
                                               
                                                 V 
                                                 POST 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     ] 
                                   
                                   } 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   E 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   C10A 
                 
               
             
           
         
       
     
     Under these conditions, equations E-C11 and E-C12 can be used with these modified equations. 
     Applications 
     Equations E-C9 to E-C11 can be used calculate patient specific values of G from the data analyzed in Examples 1 and 2. The measured pre-dialysis concentrations of phosphorus (“C PRE ”) and the calculated values of G from that data during conventional 4-hour treatments are summarized in the Table 11.3 below. 
     
       
         
           
               
               
               
               
             
               
                   
                 TABLE II.3 
               
               
                   
                   
               
               
                   
                 Patient Label 
                 C PRE  (mg/dl) 
                 G (g/week) 
               
               
                   
                   
               
             
            
               
                   
                 Patient 1 
                 8.4 
                 4.13 
               
               
                   
                 Patient 2 
                 4.4 
                 1.71 
               
               
                   
                 Patient 3 
                 6.7 
                 3.64 
               
               
                   
                 Patient 4 
                 7.3 
                 3.83 
               
               
                   
                 Patient 5 
                 4.2 
                 1.68 
               
               
                   
                   
               
            
           
         
       
     
     The calculated G values are consistent with expected phosphorus net generation rates in chronic hemodialysis patients. 
     Equations E-C9 to E-C12 can also be used to simulate the effect of patient parameters (G, K M  and K R ) and treatment parameters (t tx , K D  and F) on pre-dialysis serum phosphorus concentrations. Several different simulations will be illustrated; K R  will be assumed to be zero in these simulations. These simulation examples show that the steady state mass balance model predicts results that are similar to those expected from the medical literature. 
     The importance of treatment time under conditions relevant to thrice weekly hemodialysis is of high clinical interest; therefore, the inventors examined the effect of treatment time on the pre-dialysis serum phosphorus concentration at the same dialysis dose or urea Kt/V. The inventors used the above described model to perform computer simulations of steady state serum phosphorus concentrations during thrice weekly hemodialysis. Simulations were performed for a fixed net phosphorus intake or generation rate (dietary intake minus absorption by oral binders), a urea Kt/V of 1.4 and a constant relationship between dialyzer clearances of phosphate and urea (i.e., the inventors assumed that dialyzer clearance of phosphate was one-half of that for urea and that the phosphorus distribution volume was one-third of that for urea). 
     Simulated pre-dialysis serum phosphorus concentrations (in mg/dl) are tabulated in Table II.4 below for hypothetical patients with different K M , a post-dialysis phosphorus distribution volume of 12 L, and net fluid removal per treatment of 2 L. 
     
       
         
           
               
               
               
             
               
                   
                 TABLE II.4 
               
             
            
               
                   
                   
               
               
                   
                 Treatment Time 
                 K M  (ml/min) 
               
            
           
           
               
               
               
               
               
            
               
                   
                 (min) 
                 50 
                 100 
                 200 
               
               
                   
                   
               
               
                   
                 180 
                 7.65 
                 6.91 
                 6.07 
               
               
                   
                 240 
                 7.36 
                 6.56 
                 5.75 
               
               
                   
                 300 
                 7.07 
                 6.24 
                 5.47 
               
               
                   
                   
               
            
           
         
       
     
     For a given patient, increasing treatment time at a given urea Kt/V resulted in modest decreases in pre-dialysis serum phosphorus concentration. Similar findings were obtained for other values of urea Kt/V between 1.0 and 2.0 (results not shown). These predictions show that use of urea Kt/V as the only measure of dialysis dose or dialysis adequacy does not account for differences in phosphate removal. 
       FIG. 18  illustrates the effect of treatment frequency per se on pre-dialysis serum phosphorus concentration as a function of dialyzer phosphate clearance where it was assumed that K M  was equal to 100 ml/min, V was assumed equal to 10 L with no fluid removal during the treatment, the treatment time was 630 minutes/week and the net generation of phosphorus was kept constant at 3 g/week. There is a relatively uniform decrease in pre-dialysis serum phosphorus concentration upon increasing treatment frequency from 3-times per week to 6-times per week, independent of dialyzer phosphate clearance. The uniformity of the decrease was surprising, it varied between 0.98 and 1.00 when K M =100 ml/min as shown in this figure (for dialyzer phosphate clearance between 100 and 200 ml/min). Relatively uniform decreases in concentration were also evident for K M  of 50, 150 and 200 ml/min (data not shown). The respective decreases in pre-dialysis serum phosphorus concentration were 1.67-1.84 (K M =50 ml/min), 0.60-0.63 (K M =150 ml/min) and 0.40-0.43 (K M =200 ml/min). 
       FIG. 19  illustrates the effects of an increase in treatment time and frequency with reference to nocturnal forms of hemodialysis. Doubling treatment time during three-times per week therapy produces substantial reductions in pre-dialysis serum phosphorus concentration. When these reductions are compared with those for doubling treatment frequency at the same weekly treatment time as in  FIG. 18 , it can be concluded that doubling treatment time (at the same treatment frequency) has a more substantial effect on pre-dialysis serum phosphorus concentrations than does doubling frequency (at the same weekly treatment time). Doubling both treatment time and treatment frequency reduces pre-dialysis serum phosphorus concentration even further. 
     Previous clinical studies have shown that patients treated by short daily hemodialysis often have a higher rate of protein catabolism (or protein nitrogen appearance) and a higher dietary intake of both protein and phosphorus. The increase in the protein nitrogen appearance rate has been reported to be approximately 20%. Therefore, the inventors evaluated the effect of increasing treatment frequency and treatment time relevant to short daily hemodialysis on pre-dialysis serum phosphorus concentration when net generation of phosphorus was increased up to 30% more than during conventional 3-times per week hemodialysis therapy. These results assuming post-dialytic phosphorus distribution volume of 10 L with 6 L of fluid removal per week are shown in  FIGS. 20-21  for K M =50 ml/min and  FIGS. 22-23  for K M =150 ml/min. As expected, pre-dialysis serum phosphorus concentrations were higher for lower K M  values. The interactions between K M , dialyzer phosphate clearance (“K D ”) and treatment frequency and treatment time are complex when net generation of phosphorus is increased. Some specific values of pre-dialysis serum phosphorus concentration in mg/dl during conventional 3-times per week hemodialysis (“CHD”) and during short daily hemodialysis (“SDHD”) when net generation of phosphorus during the latter therapy was increased by 20% are tabulated in Table 11.5. 
     
       
         
           
               
               
               
             
               
                   
                 TABLE II.5 
               
             
            
               
                   
                   
               
               
                   
                 K M  = 50 ml/min 
                 K M  = 150 ml/min 
               
            
           
           
               
               
               
               
               
               
               
            
               
                   
                 CHD 
                 SDHD 
                 SDHD 
                 CHD 
                 SDHD 
                 SDHD 
               
            
           
           
               
               
            
               
                   
                 Treatment Time (min) 
               
            
           
           
               
               
               
               
               
               
               
            
               
                   
                 240 
                 120 
                 180 
                 240 
                 120 
                 180 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 K D  (ml/min) 
                 80 
                 8.34 
                 8.53 
                 6.38 
                 6.98 
                 7.85 
                 5.52 
               
               
                   
                 110 
                 7.11 
                 6.97 
                 5.36 
                 5.62 
                 6.22 
                 4.42 
               
               
                   
                 140 
                 6.43 
                 6.10 
                 4.80 
                 4.86 
                 5.29 
                 3.80 
               
               
                   
               
            
           
         
       
     
     Increasing hemodialysis treatment session frequency without an increase in weekly treatment time (CHD to SDHD at a treatment time of 120 min) may result in either an increase or a decrease in pre-dialysis serum phosphorus concentration, depending on both dialyzer phosphate clearance and the patient-specific K M . Further, short daily hemodialysis with reduced dialyzer phosphate clearance does not result in reductions in pre-dialysis serum phosphorus concentration unless treatment time is increased substantially. The inventors conclude that increasing both dialyzer phosphate clearance and treatment time during short daily hemodialysis can result in clinically significant reductions in pre-dialysis serum phosphorus concentration. 
     An additional example of the use of this steady state mass balance model is its application to determining optimal hemodialysis prescriptions during frequent nocturnal hemodialysis (e.g. 6-times per week, 8 hours per treatment). During this therapy, K DS  are often empirically lowered by adding phosphate salt supplements to the dialysis solution in order to maintain pre-dialysis serum phosphorus concentrations within an optimal range; however, there are no quantitative guides to determine an optimal K D . The inventors used the above model to determine K D  that maintains pre-dialysis serum phosphorus concentration within the range recommended by the Dialysis Outcomes and Practice Patters Study (“DOPPS”) of 3.6-5.0 mg/dl. Computer simulations were performed for a given dietary phosphorus intake (e.g., assuming no use of oral binders), post-dialytic phosphorus distribution volume of 12 L, and net fluid removal per treatment of 1 L. 
     Calculated ranges for K D  (ml/min) to maintain pre-dialysis serum phosphorus concentration between 3.6 and 5.0 mg/dl for hypothetical patients with different K M  at steady state are tabulated in Table 11.6. 
     
       
         
           
               
               
             
               
                   
                 TABLE II.6 
               
             
            
               
                   
                   
               
               
                   
                 K M  (ml/min) 
               
            
           
           
               
               
               
               
               
            
               
                   
                 Dietary P Intake 
                 50 
                 100 
                 150 
               
               
                   
                   
               
               
                   
                 4 g/week 
                 40-70  
                 35-53 
                 32-48 
               
               
                   
                 5 g/week 
                 57-114 
                 46-75 
                 42-65 
               
               
                   
                 6 g/week 
                 81-200 
                  60-102 
                 54-84 
               
               
                   
                   
               
            
           
         
       
     
     These simulations demonstrate that individualization of K D , depending on both dietary intake of phosphorus and the patient-specific K M , is required during frequent nocturnal hemodialysis. 
     It should be understood that various changes and modifications to the presently preferred embodiments described herein will be apparent to those skilled in the art. Such changes and modifications can be made without departing from the spirit and scope of the present subject matter and without diminishing its intended advantages. It is therefore intended that such changes and modifications be covered by the appended claims.