Patent Publication Number: US-10328192-B2

Title: Relative pump calibration for ultrafiltration control in a dialysis apparatus

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     The present application is a National Phase of International Application No. PCT/EP2014/064744, filed on Jul. 9, 2014, which claims priority to Sweden Patent Application No. 1350878-3, filed Jul. 15, 2013, the entire contents of each of which are incorporated herein by reference and relied upon. 
     TECHNICAL FIELD 
     The present invention relates to a technique for controlling ultrafiltration in connection with dialysis, and in particular to a technique for calibrating one or more volumetric pumps for pumping a dialysis fluid through a dialyzer in a dialysis apparatus. 
     BACKGROUND ART 
     In treating chronic renal failure, various methods of purification and treatment of blood with machinery are used to replace the function of a healthy kidney. Such methods aim at withdrawing fluid and removing substances from the blood, and they may also involve adding fluid and substances to the blood. Such purification and treatment may be performed by pumping a dialysis fluid through a blood filtration unit, commonly denoted a dialyzer, in which fluid and substances are transported over a semi-permeable membrane. Diffusive mass transport through the membrane is predominant in hemodialysis (HD), whereas hemofiltration (HF) uses mainly convective mass transport through the membrane. Hemodiafiltration (HDF) is a combination of the two methods. 
     The withdrawal of fluid in the dialyzer, also known as ultrafiltration, is given by the difference between the spent dialysis fluid pumped out of the dialyzer and the fresh dialysis fluid pumped into the dialyzer. Because of the large volume of dialysis fluid that is exposed to the membrane in the dialyzer during a dialysis treatment, there is a need for accurate control of the ultrafiltration. Taking hemodialysis as an example, typically about 200 liters of dialysis fluid are passed through the dialyzer during a treatment session. The target amount of ultrafiltrate during a treatment session is typically about 2 to 3 liters and may need to be controlled with a maximum deviation of the order of only 0.1 to 0.2 liter. Accordingly, in this example, ultrafiltration may need to be controlled with a maximum error of approximately 1:1000 in relation to the total flow of dialysis fluid. 
     There are different prior art techniques for achieving accurate control of ultrafiltration in a dialysis apparatus. 
     U.S. Pat. No. 4,267,040 discloses a dialysis apparatus having a passive balancing device. The balancing device consists of two chambers, each being subdivided by a displaceable element and having an inlet line for fresh dialysis fluid and an outlet line connected to a drain for spent dialysis fluid. Cutoff valves driven and switched by a control unit are arranged in the inlet and outlet lines. A pump is provided between the dialyzer and the balancing device to convey the dialysis fluid. The balancing device is operated in such a way that fresh dialysis fluid is supplied from a dialysis fluid source to the two balancing chambers in alternation through appropriate switching of the cutoff valves in the inlet lines. At the same time, fresh dialysis fluid is supplied from an already filled space of the other balance chamber to the dialyzer. The spent dialysis fluid from the dialyzer is pumped into the second space of the same balance chamber, from which the spent dialysis fluid then goes into an outlet. The part of the liquid circuit enclosed between the balancing device and the dialyzer behaves like a closed, constant-volume system. The ultrafiltration in the dialyzer, i.e. the amount of fluid that passes from the blood side to the dialysis fluid side of the dialyzer membrane, is controlled by a dedicated filtration pump which is connected for controlled removal of fluid from the system. 
     A dialysis apparatus equipped with this type of balancing device has a number of disadvantages. For one, the switching of the balancing chambers may produce audible noise to the discomfort of the patient undergoing dialysis and also the caretakers. Audible noise may be highly undesirable in a clinic setting as well as in a home setting. Further, the cutoff valves that control the switching will be subjected to significant mechanical load over time and may start to leak dialysis fluid as result of wear and fatigue. Any such leaks will produce errors in the resulting ultrafiltration. Still further, errors in the dedicated filtration pump will have a significant impact on the accuracy of the ultrafiltration, and it may be necessary to take measures to carefully control and supervise the operation of the filtration pump. 
     In this context, DE69007342 discloses a technique for calibrating a filtration pump, which is a volumetric pump that has either a rotor or a diaphragm for displacing the dialysis fluid. The pump is equipped with a pulse generator which emits pulses that represent a certain angle of rotation of the rotor or a certain displacement of the diaphragm and correspond to a certain quantity of dialysis fluid. During regular operation, the flow rate of the filtration pump is determined by dead reckoning, i.e. by counting the number of pulses emitted and applying a known correspondence between the number of pulses and the quantity of liquid pumped. This correspondence is determined in a calibration procedure, in which the pumped liquid is switched into a reservoir of exactly known capacity. By counting the number of pulses emitted in order to fill the reservoir, the exact correspondence between the number of impulses emitted and the quantity of liquid pumped can be determined. 
     The prior art also comprises U.S. Pat. No. 4,747,950 which is designed to balance the flow of dialysis fluid into the dialyzer, generated by an upstream pump, with the flow of dialysis fluid out of the dialyzer, generated by a downstream pump, and to achieve ultrafiltration by withdrawing a controlled amount of dialysis fluid from a location either upstream or downstream of the dialyzer. The disclosed dialysis fluid supply system contains a large number of pumps, in addition to the upstream and downstream pumps, and involves a complex distribution of dialysis fluid between and within different receptacles in the system. To ensure a balanced flow, a calibration procedure is implemented, in which the dialyzer is temporarily by-passed such that the dialysis fluid is pumped from the upstream pump through a calibration chamber to the downstream pump. The calibration chamber has a sight tube allowing the level of dialysis fluid to be visually inspected. The flow rate of the upstream pump is controlled in relation to the flow rate of the downstream pump until the level of dialysis fluid remains at a constant level in the sight tube. 
     Other techniques for ultrafiltration control by balancing the flow rates of fresh and spent dialysis fluid are e.g. known from US2012/0193290, US2011/0132838, US2008/0105600, US2002/0088752, US2012/0279910 and US2010/0016777. 
     Another approach to control the ultrafiltration during dialysis is to install one or more pumps upstream of the dialyzer and one or more pumps downstream the dialyzer in the dialysis fluid supply system and to relatively control the flow rates of the upstream and downstream pumps to achieve a desired ultrafiltration rate. This type of control requires accurate and continuous measurements of the flow rates of dialysis fluid into and out of the dialyzer, and one or more advanced flow meters are installed in the dialysis fluid supply system to provide this information. An example of a flow meter capable of continuously measuring the difference between the dialysis fluid flows into and out of the dialyzer is known from U.S. Pat. No. 4,585,552. A dialysis apparatus based on this approach may produce significantly less audible noise than a dialysis device with a balancing device. Further, since the ultrafiltration is controlled based on the readings of one or more flow meters, ultrafiltration may be largely unaffected by leaks in the dialysis fluid supply system, as long as the flow meter(s) operate correctly. However, flow meters with the required accuracy and precision may be quite costly and complex. It is also vital that the flow meters operate correctly over a long time of use. Techniques for calibrating the flow meters in this context are e.g. known from U.S. Pat. No. 6,331,252 and US2012/0145615. 
     The prior art also comprises US2007/0243990 which is unrelated to ultrafiltration control and discloses an apheresis system in which blood is removed from a donor, directed to a blood component separation device, such as a centrifuge, which collects one or more blood components (e.g. red blood cells, white blood cells, platelets or plasma), whereupon the remainder is returned to the donor. US2007/0243990 proposes a calibration procedure for relatively calibrating pairs of pumps in the apheresis system. During the calibration procedure, the pumps are alternately activated so that one pump is operated to increase the fluid level in a reservoir from an initial level, and then the other pump is operated to decrease the fluid level back to the initial level in the reservoir. A pump volume ratio between the pumps is given by the number of strokes performed by the respective pump while filling and emptying the reservoir. The pump volume ratio is stated to be useful for increasing the effectiveness and efficiency of the donation process. Another technique for relative pump calibration in the context of apheresis for blood plasma filtering and/or platelet separation is known from U.S. Pat. No. 4,769,001. 
     SUMMARY 
     It is an objective of the invention to at least partly overcome one or more limitations of the prior art. 
     One objective is to provide an alternative technique of controlling ultrafiltration in a dialysis apparatus. 
     Another objective is to provide a simple, accurate and robust technique of controlling ultrafiltration in a dialysis apparatus. 
     A still further objective is to enable a silent dialysis apparatus. 
     One or more of these objectives, as well as further objectives that may appear from the description below, are at least partly achieved by a dialysis apparatus, a method of calibrating a dialysis fluid distribution system, a computer-readable medium, and control units according to the independent claims, embodiments thereof being defined by the dependent claims. 
     A first aspect of the invention is a dialysis apparatus, comprising: a dialyzer; a dialysis fluid distribution system connected for fluid communication with the dialyzer and comprising a first pump and a second pump, the dialysis fluid distribution system being operable to selectively establish a main flow path that extends between the first and second pumps via the dialyzer, and a bypass flow path that bypasses the dialyzer and extends between the first and second pumps; and a control unit electrically connected to the dialysis fluid distribution system and being operable to control a respective frequency of the first and second pumps, wherein the first and second pumps are configured to generate a respective flow rate by repeatedly discharging, at the respective frequency, a respective stroke volume of dialysis fluid. The control unit is configured to, in a calibration mode, operate the dialysis fluid distribution system to establish the bypass flow path, control the first pump to operate at a first calibration frequency and second pump to operate at a second calibration frequency so as to balance the flow rates generated by the first and second pumps, and determine relative calibration data that represents a relation between the stroke volumes of the first and second pumps based on the first and second calibration frequencies. The control unit is further configured to, in a treatment mode, operate the dialysis fluid distribution system to establish the main flow path, and control the first and second pumps, based on said relative calibration data, to operate at a respective treatment frequency so as to generate a selected ultrafiltration rate in the dialyzer. 
     The first aspect is based on the insight that improved control of the ultrafiltration rate in the dialyzer is enabled if the relation between the stroke volumes of the pumps is known. Thus, the first aspect involves a technique for relatively calibrating the pumps in the dialysis fluid distribution system, in the sense that the relation between the actual stroke volumes of the pumps is determined or estimated. The relative calibration is simple to implement, since it only involves establishing the bypass flow path, and then controlling the first and second pumps to balance the flow rates of dialysis fluid generated by the pumps. The pumps are volumetric pumps, which are configured to generate a respective flow rate by repeatedly discharging, at a respective operating frequency, a respective stroke volume of dialysis fluid. With this type of pump, the relation between the stroke volumes of the pumps may be determined based on the frequency of the respective pump when their flow rates are balanced. 
     The balancing of the flow rates during the calibration mode may be achieved by the control unit based on the output signal of a relatively simple, robust and inexpensive flow difference meter installed in the dialysis fluid distribution system. The flow difference meter may, e.g., be embodied as a pressure sensor that measures the fluid pressure in the bypass flow path, by a level sensor that measures a level of dialysis fluid in a chamber which is connected in fluid communication with the bypass flow path, or by a flow meter that measures the flow of dialysis fluid to and from such a chamber. It is realized that the exact volume of the chamber is not of importance for the accuracy of the relative calibration of the pumps. 
     The first aspect also involves a technique of generating a selected ultrafiltration rate in the dialyzer by establishing the main flow path and setting the frequencies of the pumps, based on the relation between the stroke volumes as determined in the relative calibration. In contrast to prior art techniques for controlling ultrafiltration, the first aspect makes it possible to directly and accurately control the difference in flow rates between the pumps during the treatment mode, by setting the frequency of the respective pump. This means that it is possible to dispense with highly advanced, complex and costly flow meters as well as noisy, complex and costly balancing devices. Thereby, the first aspect enables a simplification in the design of the dialysis apparatus. In the absence of balancing chambers and advanced flow meters, the robustness of the dialysis apparatus may also be increased. Further, the dialysis apparatus of the first aspect may be switched to the calibration mode at any time to re-calibrate the relation between the stroke volumes of the pumps. Thus, by the first aspect, the dialysis apparatus may compensate for changes or drifts in the stroke volumes of the pumps. In the absence of a balancing device, the dialysis apparatus may be designed to produce significantly less audible noise. 
     It should be emphasized, though, that the present invention may be combined with a conventional advanced flow meter for measuring the flow rate of dialysis fluid or a conventional balancing device for balancing the flows of dialysis fluid to and from the dialyzer, if desired. 
     It should also be noted that the first pump may be located either upstream or downstream of the dialyzer, with the second pump consequently being located either downstream or upstream of the dialyzer. As used herein, “upstream” and “downstream” refers to a direction in relation to the flow of dialysis fluid in the main flow path. 
     The dialysis apparatus of the first aspect may be operated to control the ultrafiltration in the dialyzer by directly setting the treatment frequencies of the pumps, which are located upstream and downstream of the dialyzer in the main flow path, such that the difference between the flow rates generated by the pumps equals the selected ultrafiltration rate. Alternatively, the dialysis apparatus of the first aspect may be operated to control the ultrafiltration in the dialyzer by setting the treatment frequencies of the pumps such that their flow rates are balanced, and by controlling a dedicated filtration pump, which is connected to the main flow path downstream of the dialyzer, to generate a flow rate equal to the selected ultrafiltration rate. 
     In one embodiment, the control unit is further configured to, in the calibration mode, store the relative calibration data in an electronic memory for subsequent retrieval by the control unit in the treatment mode. 
     In one embodiment, the control unit is further configured, in the treatment mode, to assign a predefined nominal stroke volume to the first pump and set the treatment frequency of the first pump so as to generate, based on the predefined nominal stroke volume, a first flow rate of dialysis fluid, and to set the treatment frequency of the second pump, given that the stroke volume of the second pump fulfils said relation with respect to the predefined nominal stroke volume, so as to generate a second flow rate of dialysis fluid which differs from the first flow rate by the selected ultrafiltration rate. The control unit may be configured to set the treatment frequencies of the first and second pumps such that one of the first or second flow rates is equal to a selected flow rate of dialysis fluid out of or into the dialyzer, and the control unit may be further configured to receive a first set value that represents the selected flow rate of dialysis fluid out of or into the dialyzer, and a second set value that represents the selected ultrafiltration rate. 
     In one embodiment, the control unit, in the treatment mode, is configured to set the treatment frequencies of the first and second pumps (P 1 , P 2 ) according to: 
     
       
         
           
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     wherein n 1  is the treatment frequency of the first pump, Q 1  is the first flow rate of dialysis fluid generated by the first pump, S 1   n  is the nominal stroke volume assigned to the first pump, n 2  is the treatment frequency of the second pump, Q 2  is the second flow rate of dialysis fluid generated by the second pump, n 1   c  is the first calibration frequency, and n 2   c  is the second calibration frequency, and wherein the absolute difference between Q 1  and Q 2  is equal to the selected ultrafiltration rate. 
     In one embodiment, said relation between the stroke volumes of the first and second pumps is equal to the inverse of the relation between the first and second calibration frequencies. 
     In one embodiment, the dialysis fluid distribution system further comprises a flow difference meter configured to generate an output signal representative of a difference between the flow rates generated by the first and second pumps, wherein the control unit is operable, in the calibration mode, to balance the flow rates generated by the first and second pumps based on the output signal of the flow difference meter. 
     In one embodiment, the flow difference meter comprises a pressure sensor arranged to sense a pressure of the dialysis fluid in the dialysis fluid distribution system, and wherein the output signal represents said pressure. 
     In one embodiment, the bypass flow path is connected in fluid communication with a chamber in the dialysis fluid distribution system, wherein the flow difference meter comprises a level detector, which is arranged to indicate at least one level of dialysis fluid in the chamber, and preferably a range of levels of dialysis fluid in the chamber, and wherein the control unit is operable, in the calibration mode, to balance the flow rates between the first and second pumps by generating a stabilized level of dialysis fluid in the chamber as indicated by the output signal. 
     In one embodiment, the level detector comprises a pressure sensor which is arranged to sense a pressure in the dialysis fluid distribution system such that an invariant pressure indicates that the level of dialysis fluid is stabilized in the chamber. 
     In one installation, a top portion of the chamber contains a gas and is sealed during the calibration mode, wherein the pressure sensor is configured to generate the output signal to represent the pressure of the gas in the chamber. The pressure sensor may be installed in contact with the gas to directly measure the gas pressure, or in contact with the dialysis fluid to indirectly measure the gas pressure. In another installation, the pressure sensor is connected to the chamber so as to sense a hydrostatic pressure of the dialysis fluid in the chamber. In the latter installation, a top portion of the chamber may be vented to ambient atmosphere during the calibration mode. 
     In one implementation, the pressure sensor is arranged to be included in both the bypass flow path and the main flow path. Thereby, the output signal of the pressure sensor may be used by the control unit not only during the calibration mode but also during the treatment mode, e.g. to determine current relative calibration data using a calibration function (below). 
     In one embodiment, the control unit is configured to, in the calibration mode, operate the first and second pumps to balance the flow rates of dialysis fluid at different working points of the dialysis fluid distribution system; determine the relative calibration data for each of the different working points; and generate a calibration function that relates the relative calibration data to the different working points; and wherein the control unit is configured to, in the treatment mode at a current working point of the dialysis fluid distribution system, obtain current relative calibration data by use of the calibration function and the current working point; and control the first and second pumps, based on the current relative calibration data, to operate at a respective treatment frequency so as to generate a selected ultrafiltration rate in the dialyzer. In one embodiment, the control unit is configured to identify the different working points and the current working point based on an output signal from a pressure sensor arranged to sense a pressure of the dialysis fluid in the dialysis fluid distribution system. 
     In one embodiment, the control unit is operable to validate the calibration function. In the validation, the control unit is configured to operate the dialysis fluid distribution system to establish the bypass flow path, control the first and second pumps to balance their flow rates at a validation working point of the dialysis fluid distribution system, determine the relative calibration data for the validation working point, and validate the calibration function by comparing the relative calibration data for the validation working point with an output value of the calibration function at the validation working point. The control unit may be configured to take dedicated action based on a difference between the relative calibration data for the validation working point and the output value, said dedicated action comprising at least one of: initiating an alarm, and adjusting the calibration function. The control unit may be configured to adjust the calibration function by adding the difference to the calibration function. 
     In one embodiment, the control unit is configured to obtain the current relative calibration data by retrieving a current output value of the calibration function at the current working point and compensate the current output value for one or more estimated or measured pressure differences within the dialysis fluid distribution system between the treatment mode and the calibration mode. In one embodiment, the one or more pressure differences relate to at least one of: an inlet of the first pump, an outlet of the first pump, an inlet of the second pump, and an outlet of the second pump. In one general implementation of the dialysis fluid distribution system, the first pump is located upstream of the second pump in the main flow path and the bypass flow path, the treatment frequency of the first pump is invariant during the treatment and calibration modes, and the selected ultrafiltration rate is generated by controlling the treatment frequency of the second pump. In this implementation, the pressure difference between the treatment mode and the calibration mode at the outlet of the second pump may be estimated by the selected ultrafiltration rate, and/or the pressure difference between the treatment mode and the calibration mode at the inlet of the first pump may be estimated to be zero, and/or the pressure difference between the treatment mode and the calibration mode at the inlet of the second pump may be estimated to be zero. The current output value may be further compensated as a function of the flow rate generated by the first pump. 
     In one embodiment, the control unit is configured to, during the treatment mode, operate a source to generate dialysis fluid by dosing a concentrate into water, said concentrate comprising sodium bicarbonate, and the control unit is configured to, during the calibration mode, disable the dosing of said concentrate into the water. 
     In one embodiment, the control unit, in the calibration mode, is configured to assign a respective predefined nominal stroke volume to the first and second pumps, operate the first and second pumps at a first and a second start frequency, respectively, such that the first start frequency multiplied by the predefined nominal stroke volume of the first pump is essentially equal to the second start frequency multiplied by the predefined nominal stroke volume of the second pump, and modify at least one of the first and second start frequencies until the flow rates of dialysis fluid generated by the first and second pumps are balanced. 
     In one embodiment, each of the first and second pumps comprises a respective pulse generator arranged to generate one or more pulses for each stroke volume displaced by the first and second pump, respectively, and wherein the control unit, in the calibration mode, is configured to represent the first and second calibration frequencies by a first number of pulses and a second number of pulses, respectively, generated by the respective pulse generator during a test period while the flow rates of dialysis fluid generated by the first and second pumps are balanced. In an alternative embodiment, the control unit is configured to represent the first and second calibration frequencies by a respective set value for the first and second pumps while the flow rates of dialysis fluid generated by the first and second pumps are balanced. 
     In one embodiment, the first pump is located upstream of the dialyzer in the main flow path and the second pump is located downstream of the dialyzer in the main flow path. 
     In one embodiment, each of the first and second pumps is one of a diaphragm pump and a piston pump. 
     A second aspect of the invention is a method of operating a dialysis fluid distribution system that comprises first and second pumps which are connected for fluid communication on a main flow path that extends between first and second pumps via a dialyzer, wherein the first and second pumps are configured to generate a respective flow rate by repeatedly discharging, at a respective frequency, a respective stroke volume of the dialysis fluid. The method comprises a calibration procedure and a treatment procedure. The calibration procedure comprises: operating the dialysis fluid distribution system to establish a bypass flow path, which bypasses the dialyzer and extends between the first and second pumps, controlling the first pump to operate at a first calibration frequency and the second pump to operate at a second calibration frequency so as to balance the flow rates generated by the first and second pumps, and determining, based on the first and second calibration frequencies, relative calibration data that represents a relation between the stroke volumes of the first and second pumps. The treatment procedure comprises: operating the dialysis fluid distribution system to establish the main flow path, and controlling the first and second pumps, based on said relative calibration data, to operate at a respective treatment frequency so as to generate a selected ultrafiltration rate in the dialyzer. 
     A third aspect of the invention is a computer-readable medium comprising program instructions which, when executed by a processing unit, is adapted to carry out the method of the second aspect. 
     Fourth and fifth aspects of the invention concern a control unit for a dialysis fluid distribution system arranged in a dialysis apparatus to pump a dialysis fluid through a dialyzer. The dialysis fluid distribution system comprises first and second pumps which are connected for fluid communication on a main flow path that extends between the first and second pumps via the dialyzer, wherein the first and second pumps are configured to generate a respective flow rate by repeatedly discharging, at a respective frequency, a respective stroke volume of the dialysis fluid. 
     The control unit of the fourth aspect comprises: a signal communication interface for connection to the dialysis fluid distribution system; and a signal processor operable in a calibration mode and a treatment mode, wherein the signal processor, in the calibration mode and via the signal communication interface, is configured to: operate the dialysis fluid distribution system to establish a bypass flow path, which bypasses the dialyzer and extends between the first and second pumps; control the first pump to operate at a first calibration frequency and the second pump to operate at a second calibration frequency so as to balance the flow rates generated by the first and second pumps; and determine, based on the first and second calibration frequencies, relative calibration data that represents a relation between the stroke volumes of the first and second pumps; and wherein the signal processor, in the treatment mode and via the signal communication interface, is configured to: operate the dialysis fluid distribution system to establish the main flow path; and control the first and second pumps, based on said relative calibration data, to operate at a respective treatment frequency so as to generate a selected ultrafiltration rate in the dialyzer. 
     The control unit of the fifth aspect comprises: means for operating the dialysis fluid distribution system to establish a bypass flow path, which bypasses the dialyzer and extends between the first and second pumps; means for controlling the first pump to operate at a first calibration frequency and the second pump to operate at a second calibration frequency so as to balance the flow rates generated by the first and second pumps; means for determining, based on the first and second calibration frequencies, relative calibration data that represents a relation between the stroke volumes of the first and second pumps; means for operating the dialysis fluid distribution system to establish the main flow path; and means for controlling the first and second pumps, based on said relative calibration data, to operate at a respective treatment frequency so as to generate a selected ultrafiltration rate in the dialyzer. 
     Any one of the above-identified embodiments of the first aspect may be adapted and implemented as an embodiment of the second to fifth aspects. 
     Still other objectives, features, aspects and advantages of the present invention will appear from the following detailed description, from the attached claims as well as from the drawings. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
       Embodiments of the invention will now be described in more detail with reference to the accompanying schematic drawings. Throughout the description, the same reference numerals are used to identify corresponding elements. 
         FIG. 1  is a block diagram of a system for ultrafiltration according to an embodiment. 
         FIG. 2  is a flow chart of a calibration mode executed by a control unit in the system of  FIG. 1 . 
         FIG. 3  illustrate the operation of the system in  FIG. 1  during the calibration mode in  FIG. 2 . 
         FIG. 4  is a flow chart of a treatment mode executed by the control unit in the system of  FIG. 1 . 
         FIG. 5  illustrate the operation of the system in  FIG. 1  during the treatment mode in  FIG. 4 . 
         FIG. 6  is a block diagram of a system for ultrafiltration according to another embodiment. 
         FIG. 7  is an elevated side view of a calibration chamber associated with a pressure-based level detector. 
         FIGS. 8-9  are block diagrams of systems for ultrafiltration according to further embodiments. 
         FIG. 10  is a plot of a calibration function relating stroke volume ratio to dialysis fluid pressure measured by the pressure sensor in the system of  FIG. 9 . 
         FIG. 11  illustrates adjustment of the calibration function as result of a treatment calibration. 
         FIG. 12  is a flow chart of a process for achieving the adjustment in  FIG. 11 . 
         FIG. 13  is a flow chart of a process for compensating a stroke volume ratio for pressure variations between calibration and treatment modes. 
         FIG. 14  is a block diagram of a source of fresh dialysis fluid for use in embodiments of the invention. 
     
    
    
     DETAILED DESCRIPTION OF EXAMPLE EMBODIMENTS 
       FIG. 1  illustrates an embodiment of a system  10  for ultrafiltration. The system  10  may be included in a dialysis apparatus. It is understood that only components relevant to the following description are represented in  FIG. 1 , with other components being well within the purview of one skilled in the art to ascertain. The system  10  comprises a blood circuit  11 , a dialysis fluid circuit  12  and a dialyzer  13 . The dialyzer  13  is a blood filtration unit that generally has a blood side  14  and a dialysis fluid side  15  separated by a semipermeable membrane  16 . The blood circuit  11  is connected to an inlet and an outlet of the blood side  14 . One or more pumps (not shown) are arranged in the blood circuit  11  to pump blood from a source (not shown) through the dialyzer  13  to a receptacle (not shown). The blood may e.g. be pumped from the cardiovascular system of a subject and back to the subject, as is well-known in the art. The dialyzer  13  may be any of the well-known dialyzers useful for hemodialysis, such as a coil dialyzer, a parallel plate dialyzer, a hollow fiber dialyzer, etc. 
     The dialysis fluid circuit  12  extends from a source  17  of fresh dialysis fluid to a receptacle or drain  18  for spent dialysis fluid. The dialysis fluid circuit  12  includes an arrangement of fluid lines that define an inlet conduit  19  connected to an inlet of the dialysis fluid side  15  of the dialyzer  13 , an outlet conduit  20  connected to an outlet of the dialysis fluid side  15  of the dialyzer  13 , and a bypass conduit  21  that extends between the inlet and outlet conduits  19 ,  20 . A first pump P 1  (“upstream pump”) is arranged in the inlet conduit  19  to pump dialysis fluid from the source  17 , and a second pump P 2  (“downstream pump”) is arranged in the outlet conduit  20  to pump dialysis fluid towards the receptacle  18 . A first and a second three-way valve VV 1 , VV 2  are arranged in the inlet conduit  19  and the outlet conduit  20 , respectively, and are connected to the bypass conduit  21 . The first valve VV 1  is operable to establish fluid communication either between the first pump P 1  and the dialyzer  13 , or between the first pump P 1  and the bypass conduit  21 . The second valve VV 2  is operable to establish fluid communication either between the dialyzer and the second pump P 2 , or between the bypass conduit  21  and the second pump P 2 . A chamber  26 , denoted calibration chamber in the following, is arranged in fluid communication with the bypass conduit  21  via a connecting conduit  27 . A level detector  28  is installed to generate a sensor signal C L  which is indicative of at least one fluid level L in the calibration chamber  26 . The level detector  28  may be of any conventional type that allows single-point, multi-point or continuous level detection, such as an ultrasonic detector, an optical detector, a capacitive detector, a microwave sensor, a hydrostatic sensor, a magnetic float detector, etc. In a specific embodiment, described below with reference to  FIG. 7 , continuous level detection is achieved by pressure detection in the calibration chamber  26 . 
     The pumps P 1 , P 2  are “volumetric pumps”, which means that each pump is configured to generate a given flow rate by repeatedly discharging a respective stroke volume of dialysis fluid. Volumetric pumps are well-known per se and are also referred to as positive displacement pumps. A volumetric pump is configured to drive a fluid by trapping a fixed and well-defined volume of the fluid in a pump chamber and by forcing (displacing) that trapped volume through a pump outlet by a movable pumping element, e.g. a piston or a diaphragm. Volumetric pumps that may be used in the embodiment of  FIG. 1  include reciprocating pumps such as piston pumps and diaphragm pumps. One characteristic of volumetric pumps is that they have a well-defined stroke volume, and that the flow rate of a volumetric pump is controlled via the frequency of the pump, i.e. the frequency of the stroke volumes that are discharged by the pump. In the following, the “speed” of a pump is synonymous with the frequency of stroke volumes. 
     Each pump P 1 , P 2  is equipped or associated with a respective pulse generator SC 1 , SC 2  that generates one or more pulses for each stroke volume that is displaced by the pump P 1 , P 2 . Such pulse generators are well-known in the art and may e.g. be implemented as a pulse encoder or a tachometer. Each pulse may correspond to a predefined displacement of above-mentioned pumping element. As indicated in  FIG. 1 , the pulse generators SC 1 , SC 2  are configured to generate a respective pulse signal C SC1 , C SC2  that is indicative of the pulses that are generated during operation of the pumps P 1 , P 2 . The pulse signals C SC1 , C SC2  may be used by the control unit  30  (below), or a dedicated pump controller (not shown), for feedback control of the speed of the respective pump P 1 , P 2  in a manner well-known in the art. In certain embodiments, as will be further described below, the pulse signals C SC1 , C SC2  are also used by the control unit  30  for relatively calibrating the stroke volumes of the first and second pumps P 1 , P 2 . 
     As indicated, the system  10  further includes an electronic control unit  30 . The control unit  30  has a signal communication interface  31  for electrical connection to electrically responsive components in the dialysis fluid circuit  12 . Specifically, the control unit  30  is operable to receive the sensor signal C L  from the level detector  28  and the pulse signals C SC1 , C SC2  from the pulse generators SC 1 , SC 2 . The control unit  30  is further operable to generate and output control signals C P1 , C P2  for controlling the operation of the pumps P 1 , P 2  and control signals C VV1 , C VV2  for controlling the switching of the valves VV 1 , VV 2 . The control unit  30  may implement further functions of the dialysis apparatus and may be included as a part of a control system for the dialysis apparatus. The control unit  30  may be a computer, or a similar data processing device, with adequate hardware for acquiring, processing and generating signals in accordance with different embodiments of the invention. Embodiments of the invention may be implemented by software instructions that are supplied on a computer-readable medium for execution by a processor  32  in conjunction with an electronic memory  33  in the device  30 , as indicated in  FIG. 1 . The computer-readable medium may be a tangible product (e.g. magnetic medium, optical disk, read-only memory, flash memory, etc) or a propagating signal. 
     The system  10  is operable in a treatment mode or procedure, in which the control unit  30  controls the dialysis fluid circuit  12  to generate a flow of dialysis fluid through the dialysis fluid side  15  of the dialyzer  13 . During the treatment mode, the pumps P 1 , P 2  are set to generate different flow rates, such that the flow rate of the downstream pump P 2  exceeds the flow rate of the upstream pump P 1  by the desired ultrafiltration rate in the dialyzer  13 . As explained in the Background section, the ultrafiltration rate (or UF rate) is the rate of fluid removal from the blood that takes place in the dialyzer  13 . In blood treatment, ultrafiltration generally denotes a process of removing water from blood plasma. The blood is passed on the blood side  14  of the dialyzer  13 , and a gradient of pressure is created through the semipermeable membrane  16  by the difference in flow rates between the pumps P 1 , P 2 . The pressure gradient forces fluid through the pores of the membrane  16 . The pores filter electrolytes and small and middle sized molecules (up to 20,000 to 30,000 daltons) from the blood plasma. 
     The volumetric pumps P 1 , P 2  are manufactured with a nominal stroke volume. However, the actual stroke volume of an individual pump may differ from the nominal stroke volume due to tolerances in manufacture and assembly of the pump. Further, the stroke volume may change during operation of the pump, e.g. due to build-up of deposits or contaminations in the pump chamber or due to wear of the pumping element or the pump chamber. Depending on the type of pump, the stroke volume may also vary depending on the fluid pressure at the inlet and/or outlet of the pump. 
     Even small differences between the actual and nominal stroke volumes of the pumps P 1 , P 2  may lead to a large deviation between the desired UF rate and the actual UF rate that is generated by the flow rate difference between the pumps P 1 , P 2 . This sensitivity will be illustrated by reference to a numeric example. Assume that the pumps P 1 , P 2  are identical to the degree that their nominal stroke volume is 3.8 ml but that their actual stroke volumes deviate by −1% and +1%, respectively, from the nominal stroke volume. Also assume that the set value for the inlet flow of dialysis fluid to the dialyzer is 500 ml/min and that the desired UF rate is 500 ml/h. Using the nominal stroke volumes to achieve this UF rate, the speed of pump P 1  is set to 500/3.8=131.58 strokes/min and the speed of pump P 2  is set to (500+500/60)/3.8=133.77 strokes/min. However, based on the actual stroke volumes, the pump P 1  will generate a flow rate of 131.58*3.8*1.01=505.00 ml/min and pump P 2  will generate a flow rate of 133.77*3.8*0.99=503.25 ml/min. Thus, the actual UF rate is (503.25−505.00)*60=−105 ml/h. This means that instead of removing 500 ml fluid per hour from the blood, the system will pump 105 ml fluid per hour into the blood. 
     It is realized that proper control of the UF rate may require the pumps P 1 , P 2  to be manufactured with very tight tolerances so that the actual stroke volumes closely correspond to the nominal stroke volumes. To overcome this drawback, embodiments of the invention take the approach of relatively calibrating the pumps P 1 , P 2  before or even during ongoing blood processing. To this end, the system  10  is configured to be operable in a calibration mode or procedure, in which a relation between the actual stroke volumes of the pumps P 1 , P 2  is determined by clever control of the pumps P 1 , P 2  and use of the calibration chamber  26 . The calibration mode is implemented by the control unit  30 , which generates dedicated control signals C P1 , C P2 , C VV1 , C VV2  based on the sensor signal C L , and processes the pulse signals C SC1 , C SC2  for determination of the relation between the actual stroke volumes. In the following, the actual stroke volumes of the pumps P 1 , P 2  are represented as S 1 , S 2 , respectively, and the nominal stroke volumes of the pumps P 1 , P 2  are represented as S 1   n , S 2   n , respectively. 
     An embodiment of the calibration mode  100  is represented in the flow chart of  FIG. 2  and will be discussed below with further reference to  FIG. 3 . In a first step  101 , the valves VV 1 , VV 2  are controlled to establish a direct flow path between the pumps P 1 , P 2 , i.e. a flow path that bypasses the dialyzer  12 . This bypass flow path is also in fluid communication with the calibration chamber  26 . In the example of  FIG. 3 , the valves VV 1 , VV 2  are controlled to establish fluid communication between the inlet and outlet conduits  19 ,  20  via the bypass conduit  21 , while closing off the fluid communication to and from the dialyzer  13 . Thereby, all of the fresh dialysis fluid will be pumped from the source  17  through the bypass conduit  21  to the receptacle  18 . In  FIG. 3 , a closed valve passage is represented by a filled triangle and an open valve passage is represented by an open triangle. In step  102 , the pumps P 1 , P 2  are controlled to generate a stable level of dialysis fluid in the calibration chamber  26 , as indicated by the sensor signal C L . This means that the flow rates of the pumps P 1 , P 2  are balanced, i.e. the flow rate generated by pump P 1  is essentially identical to the flow rate generated by pump P 2 . Even if it is possible to balance the flow rates using a level detector  28  indicating a single predefined level L in the chamber  26 , the balancing is generally facilitated if the level detector  28  is arranged for multi-point or continuous level measurements. In one non-limiting example, the pumps P 1 , P 2  are first set to operate at start frequencies that are known to nominally generate the same reference flow rate, given the nominal stroke volumes S 1   n , S 2   n  of the pumps P 1 , P 2 . The reference flow rate or the start frequencies may be predefined and retrieved from memory  33  in step  102 . The frequency of one or both of the pumps P 1 , P 2  is then adjusted until the fluid level in the chamber  26  is stabilized. It should be noted that the absolute location of the fluid level in the chamber  26  is not important. The skilled person realizes that there are many alternative ways of controlling the pumps P 1 , P 2  based on the level signal C L  to stabilize the fluid level in the chamber  26 . 
     In step  103 , while the pumps are operated to generate a stabilized fluid level in the chamber  26 , the current speed of the respective pump P 1 , P 2  is determined. In the following, the current speeds are also denoted “calibration frequencies” and are represented as n 1   c , n 2   c . In one embodiment, the control unit  30  determines the current speeds n 1   c , n 2   c  based on the current set values for the pumps P 1 , P 2 . In an alternative embodiment, to increase accuracy, the control unit  30  counts the number of pulses in the pulse signals C SC1 , C SC2  during a predefined time period, which may but need not be the same for both pumps P 1 , P 2 . The current speeds n 1   c , n 2   c  may be represented by the respective count of pulses or by a frequency value which is generated based on the count of pulses (e.g. by dividing the count of pulses by the respective time period). 
     In step  104 , the pumps P 1 , P 2  are relatively calibrated, by calculating a ratio R between their actual stroke volumes S 1 , S 2 . The rationale behind this operation is that that balanced flows for volumetric pumps are given by:
 
 n 1 c   ·S 1 =n 2 c   ·S 2  (1)
 
     and thus the stroke volume ratio R is given by: 
     
       
         
           
             
               
                 
                   R 
                   = 
                   
                     
                       
                         S 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                       
                         S 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                       
                     
                     = 
                     
                       
                         n 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           2 
                           c 
                         
                       
                       
                         n 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           1 
                           c 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     In step  105 , the stroke volume ratio R is stored in the electronic memory ( 33  in  FIG. 1 ) for subsequent retrieval when the system  10  enters a treatment mode (see below). 
     As noted, the system  10  in  FIG. 1  is also operable in a treatment mode, in which a desired UF rate is achieved in the dialyzer  13 . An embodiment of the treatment mode  200  is represented in the flow chart of  FIG. 4  and will be discussed below with further reference to  FIG. 5 . In a first step  201 , the valves VV 1 , VV 2  are controlled to establish a flow path (“main flow path”) from the pump P 1  via the dialysis fluid side  15  of the dialyzer  13  to the pump P 2 , such that all of the fresh dialysis fluid from source  17  is pumped through the dialyzer  13  and that all of the spent dialysis fluid is pumped from the dialyzer  13  to the receptacle  18 . In the example of  FIG. 5 , the valves VV 1 , VV 2  are controlled to establish fluid communication between the inlet and outlet conduits  19 ,  20  and the dialyzer  13 , while closing off the fluid communication between the inlet and outlet conduits  19 ,  20  and the bypass conduit  21 . In step  202 , set values are obtained for the flow rate of dialysis fluid into the dialyzer  13 , denoted main flow rate and designated by Q MAIN , and the ultrafiltration rate, designated by Q UF . The set values Q MAIN , Q UF  may be pre-stored in and retrieved from the memory  33  or they may be acquired in real time via a user interface (not shown) associated with the control unit  30  or the dialysis apparatus. In step  203 , the stroke volume ratio R that was determined by the control unit  30  in the calibration mode ( FIG. 2 ) is retrieved, e.g. from the memory  33  or via the above-mentioned user interface. In step  204 , the nominal stroke volume S 1   n  of the upstream pump P 1  is retrieved from memory  33 . In step  205 , the speed n 1  of the upstream pump P 1  is calculated and set to generate a flow rate Q 1  which is equal to the main dialysis flow Q MAIN , given the nominal stroke volume S 1   n , i.e. 
     
       
         
           
             
               
                 
                   
                     n 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                   = 
                   
                     
                       
                         Q 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                       
                         S 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           1 
                           n 
                         
                       
                     
                     = 
                     
                       
                         Q 
                         MAIN 
                       
                       
                         S 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           1 
                           n 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     Thus, step  205  assumes that the actual stroke volume Si is equal to the nominal stroke volume S 1   n . As will be shown below, even if this assumption is generally incorrect and introduces an error in the flow of dialysis fluid generated by pump P 1 , the impact of this error on the UF rate is limited by proper use of the stroke volume ratio R when setting the speed of the downstream pump P 2 . In step  206 , the speed n 2  of the downstream pump P 2  is calculated and set to achieve a difference of flow rates between the pumps P 1 , P 2  equal to the ultrafiltration rate Q UF , given by the relation between the actual stroke volumes S 1 , S 2  obtained in the calibration mode  100  and represented by the stroke volume ratio R. Specifically, the downstream pump P 2  should be set to generate a flow rate Q 2  which is equal to Q MAIN +Q UF . Given the assumption that the actual stroke volume of the upstream pump P 1  is S 1   n , the actual stroke volume S 2  of the downstream pump P 2  is estimated as: 
     
       
         
           
             
               
                 
                   
                     S 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       2 
                       est 
                     
                   
                   = 
                   
                     
                       S 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         1 
                         n 
                       
                     
                     R 
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     Based on the estimated stroke volume S 2   est , the speed n 2  of the downstream pump P 2  is set to: 
     
       
         
           
             
               
                 
                   
                     n 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                   
                   = 
                   
                     
                       
                         Q 
                         2 
                       
                       
                         S 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           2 
                           est 
                         
                       
                     
                     = 
                     
                       
                         
                           
                             Q 
                             MAIN 
                           
                           + 
                           
                             Q 
                             UF 
                           
                         
                         
                           S 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             2 
                             est 
                           
                         
                       
                       = 
                       
                         
                           
                             
                               Q 
                               MAIN 
                             
                             + 
                             
                               Q 
                               UF 
                             
                           
                           
                             S 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               1 
                               n 
                             
                           
                         
                         · 
                         R 
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     Following step  206  and as indicated by thick arrows in  FIG. 5 , dialysis fluid will now be pumped through the dialysis fluid side  15  of the dialyzer  13 , while blood is being pumped through the blood side  14  of the dialyzer  13 . The difference in flow rates between the pumps P 1 , P 2  draws ultrafiltrate from the blood into the dialysis fluid, as indicated by the horizontal arrow in the dialyzer  13  in  FIG. 5 . 
     It is to be understood that the set value Q UF  may vary during a treatment, and the control unit  30  will control the speed n 2  of the downstream pump P 2  accordingly. The set value Q MAIN  is typically held constant during a treatment, although it can certainly be varied if desired. 
     The accuracy of the resulting UF rate is dependent on the accuracy of the stroke volume ratio R, which in turn is dependent on the accuracy of the calibration frequencies n 1   c , n 2   c  that are determined in step  103  in the calibration mode  100 . It should be noted that the relative calibration of the pumps is independent of the volume of the calibration chamber  26 . As noted above, a deliberate error is introduced in step  205  when the actual stroke volume of one of the pumps P 1 , P 2  is set equal to the nominal stroke volume. However, this error has limited impact on the resulting UF rate as will be shown in the following. The actual stroke volumes are represented as S 1 =S 1   n *e 1  and S 2 =S 2   n *e 2 , where e 1 , e 2  are relative errors caused by the difference between the actual and nominal stroke volumes. The actual UF rate Q UF   _   real  is given by, using Eq. (5) and Eq. (3): 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           Q 
                           
                             U 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             F 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             real 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             S 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               2 
                               · 
                               n 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                           - 
                           
                             S 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               1 
                               · 
                               n 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             S 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 2 
                                 n 
                               
                               · 
                               e 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               2 
                               · 
                               
                                 
                                   
                                     Q 
                                     MAIN 
                                   
                                   + 
                                   
                                     Q 
                                     UF 
                                   
                                 
                                 
                                   S 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     1 
                                     n 
                                   
                                 
                               
                               · 
                               R 
                             
                           
                           - 
                           
                             
                               
                                 Q 
                                 MAIN 
                               
                               · 
                               e 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     Further, since 
     
       
         
           
             
               
                 
                   R 
                   = 
                   
                     
                       
                         S 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                       
                         S 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                       
                     
                     = 
                     
                       
                         S 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             1 
                             n 
                           
                           · 
                           e 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                       
                         S 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             2 
                             n 
                           
                           · 
                           e 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     it is possible to re-write Eq. (6) as:
 
 Q   UF   _   real   =e 1·( Q   MAIN   +Q   UF )− Q   MAIN   ·e 1 =Q   UF   ·e 1  (8)
 
     As seen, the resulting error in the actual UF rate is proportional to the relative difference or error between the actual and nominal stroke volumes for the upstream pump P 1 , which is assumed to have an actual stroke volume equal to the nominal stroke volume. Reverting to the numeric example given above, where the actual stroke volumes of the pumps P 1 , P 2  in  FIG. 1  deviate by −1% and +1%, respectively, from the nominal stroke volume and the desired UF rate is 500 ml/h, the actual UF rate is 500*0.99=495 ml/h, which is significantly better that the actual UF rate of −105 ml/h that is attained without the inventive calibration. It should be noted that the stroke volume error of 1% is merely given as an example and that significantly larger stroke volume errors may be accepted. 
     It should be understood that  FIG. 4  is an example embodiment and that many alternatives and variants are conceivable. For example, the method may be modified to assume that the actual stroke volume of the downstream pump P 2  (instead of the upstream pump P 1 ) is equal to the nominal stroke volume. In such an alternative, step  204  may be modified to retrieve the nominal stroke volume S 2   n  of the downstream pump P 2 , and steps  205 - 206  may be modified to calculate the speeds according to: n 1 =Q MAIN /(R*S 2   n ) and n 2 =(Q MAIN +Q UF )/S 2   n . 
     In a further variant, the set value Q MAIN  instead represents the flow of dialysis fluid out of the dialyzer  13 . The skilled person can readily adapt steps  205 - 206  to this situation. In many dialysis systems, it is desirable to maintain a constant flow of dialysis fluid from the source  17 , e.g. when the source  17  is configured to prepare the dialysis fluid by mixing of one or more concentrates with water. The complexity of such a source  17  may increase if the source  17  need to handle momentary changes in the amount of dialysis fluid to be output. In such dialysis systems, the embodiment in  FIG. 5  may be preferable, where the UF rate is controlled by adjusting the speed of the downstream pump P 2 . 
     Reverting to the calibration mode  100  in  FIG. 2 , it should be understood that the stroke volume ratio R=n 2   c /n 1   c  is only one of many possible representations of the relation between the actual stroke volumes of the pumps P 1 , P 2 . Generally, step  104  may be seen to generate “relative calibration data”, which represents the relation between the actual stroke volumes. The relative calibration data may alternatively be given by the ratio n 1   c /n 2   c  or by the speed values n 1   c  and n 2   c  that are determined in step  103 . 
     It is also conceivable that the control unit  30 , when in calibration mode  100 , executes steps  102 - 103  to determine relative calibration data, e.g. stroke volume ratios R, for a number of different working points of the dialysis fluid circuit  12 , and that the control unit  30  in step  104  stores a calibration function in the electronic memory  33 , where the calibration function may be implemented as individual combinations of relative calibration data and the different working points, or as a mathematical function that represents the functional relationship between the relative calibration data and the different working points. Such a variant may be desirable if the actual stroke volumes are known to vary during operation of the dialysis fluid circuit  12 . For example, the actual stroke volumes may vary if there is a leakage flow of dialysis fluid past the pumping element in the respective pump P 1 , P 2  (e.g. a backflow from the outlet to the inlet of the respective pump, or a reverse backflow from the inlet to the outlet) and if this leakage flow changes with operating condition. The leakage flow may e.g. vary with the fluid pressure at the inlet and/or outlet of the respective pump. To counteract such variations in the actual stroke volumes, the control unit  30  may in step  203  retrieve from the memory  33  the relative calibration data (e.g. stroke volume ratio R) for the working point that comes closest to the current working point of the dialysis fluid circuit  12 , or by inputting the current working point to the above-mentioned mathematical function. The working point may be represented by any combination of set values and/or measurement data, such as the set value Q MAIN  or one or more measured pressure values that represent the fluid pressure at the inlet/outlet of one or both pumps P 1 , P 2 . In one example, a pressure differential between the inlet and the outlet of the respective pump P 1 , P 2  is measured and used to represent the working point of the dialysis fluid circuit  12 . In another example, which is further described below with reference to  FIG. 10 , the working point is represented by the measured pressure of the dialysis fluid in the dialysis fluid circuit  12 . 
     It is to be understood that the control unit  30  may be operated in the calibration mode at any time, e.g. at manufacture of a dialysis machine that includes the dialysis fluid circuit, at scheduled maintenance, before start of a dialysis treatment, intermittently during a dialysis treatment, or after a completed dialysis treatment. Each calibration mode results in relative calibration data that quantifies the relation between the actual stroke volumes S 1 , S 2  of the pumps P 1 , P 2 . 
       FIG. 6  illustrates another embodiment of a system  10  for ultrafiltration. The system  10  in  FIG. 6  differs from the system in  FIG. 1  by the placement of the calibration chamber  26 . The chamber  26  is arranged for fluid communication with the inlet conduit  19  between the upstream pump P 1  and the first valve VV 1  via a connecting conduit  27 . An on/off valve VV 3  is arranged in the connecting circuit  27  to selectively establish fluid communication between the chamber  26  and the inlet conduit  19 . The control unit  30  is configured to generate a control signal C VV3  for controlling the opening and closing of the valve VV 3 . The foregoing embodiments, variants and examples are equally applicable to the embodiment in  FIG. 6 , with the difference that the control unit  30  operates the valve VV 3  to open the connecting conduit  27  when the bypass flow path is established in step  101  during the calibration mode  100 , and operates the valve VV 3  to close off the connecting conduit  27  when the main flow path is established in step  201  during the treatment mode  200 . In variant (not shown), the chamber  26  is instead connected for fluid communication with the outlet conduit  20  between the second valve VV 2  and the pump P 2 . 
       FIG. 7  shows an embodiment of a level detector  28  that operates by sensing the pressure in the calibration chamber  26 . Such a level detector  28  is capable of generating a sensor signal C L  that represents a continuous range of fluid levels in the chamber  26 . The level detector  28  includes a pressure sensor  28 A which is installed to sense the gas pressure above the dialysis fluid in the chamber  26 . In the illustrated example, the top of the chamber  26  is vented via an outlet conduit  29 , and an on/off valve VV 4  is arranged in the outlet conduit  29 . The control unit  30  is arranged to generate a control signal C VV4  to control the opening and closing of the valve VV 4 . During level sensing, e.g. in the calibration mode  100 , the control unit  30  closes valve VV 4  to confine a gas (air) in the space above the dialysis fluid. Thereby, the gas pressure will vary with the volume of the space above the dialysis fluid according to Boyle&#39;s law. Thus, changes in measured gas pressure, as represented in the sensor signal C L , corresponds to changes in fluid level. In a variant, the outlet conduit  29  is omitted and the chamber  26  is permanently sealed. 
     In an alternative embodiment (not shown), the pressure sensor  28 A is instead installed at the bottom of the calibration chamber  26  to sense the hydrostatic pressure of the dialysis fluid. As is well-known, the hydrostatic pressure varies in proportion to the fluid level because of the increasing weight of dialysis fluid exerting downward force from above. To reduce the impact of changes in gas pressure in the space above the dialysis fluid, the valve VV 4  may be controlled to open during the level sensing. In a variant, the valve VV 4  is omitted and the chamber  26  is open to ambient atmosphere. 
       FIG. 8  illustrates yet another embodiment of a system  10  for ultrafiltration. The system  10  in  FIG. 8  is identical to the system in  FIG. 1  except for the equipment used for balancing the flow rates of the pumps P 1 , P 2  during the calibration mode. In  FIG. 8 , the chamber  26  and the level detector  28  are replaced by a pressure sensor  28 A, which is installed to generate a sensor signal C L  that represents the pressure of the dialysis fluid in the bypass conduit  21 . It is realized that the pressure sensor  28 A generates a time-invariant pressure value when the flow rates of the pumps P 1 , P 2  are balanced. Thus, the system in  FIG. 8  may be operated according to the flow chart in  FIG. 2 , with the difference that step  102  involves controlling the pumps P 1 , P 2  to stabilize the pressure as measured by the pressure sensor  28 A. In a variant (not shown), the pressure sensor  28 A is instead installed in the inlet conduit  19  between the pump P 1  and the first valve VV 1 , or in the outlet conduit  20  between the second valve VV 2  and the pump P 2 . 
     Since the dialysis fluid is an incompressible fluid, comparatively small differences between the flow rates of the pumps P 1 , P 2  during the calibration mode may give rise to large pressures in the dialysis fluid. Excessive pressures in the dialysis fluid may cause leaks or damage the pumps P 1 , P 2 , the valves VV 1 , VV 2 , connectors, tubings, etc in the dialysis fluid circuit  12 . Thus, it may be desirable to add “compliance” to the dialysis fluid circuit  12 , i.e. an ability of the dialysis fluid circuit  12  to expand and contract passively with changes in pressure in the dialysis fluid. Such compliance may be provided by the tubings that define the conduits  19 ,  20 ,  21 . However, a more controlled compliance may be achieved by installing a compliance chamber in fluid communication with the bypass flow path in the dialysis fluid circuit  12 . By ensuring that the compliance chamber is partially filled with dialysis fluid, differences between the flow rates of the pumps P 1 , P 2  will cause dialysis fluid to enter or leave the compliance chamber and thereby reduce the risk for excessive pressures in the dialysis fluid circuit  12 . The skilled person realizes that the chambers  26  shown in  FIGS. 1, 6 and 7  will provide such compliance in the dialysis fluid circuit  12 . 
       FIG. 9  illustrates another embodiment of a system  10  for ultrafiltration. The system in  FIG. 9  differs from the system in  FIG. 8  by the provision of a chamber  26 , which forms a compliance chamber, and by the location of the pressure sensor  28 A in the dialysis fluid circuit  12 . The chamber  26  is installed in the inlet conduit  19  between the valve VV 2  and the pump P 2 , and the pressure sensor  28 A is arranged to sense the pressure of the dialysis fluid in the inlet conduit  19  between the valve VV 2  and the chamber  26 . The chamber  26  is sealed to confine a gas (e.g. air) in the space above the dialysis fluid in the chamber  26 . Thereby, the pressure measured by the pressure sensor  28 A will approximately correspond to the gas pressure in the space above the dialysis fluid. This means that the pressure sensor  28 A effectively forms a level sensor  28  and that the sensor signal C L  reflects the level of dialysis fluid in the chamber  26 . It is thus realized that the flow rates of the pumps P 1 , P 2  may be balanced, according to step  102  in  FIG. 2 , based on the sensor signal C L . The skilled person realizes that the pressure sensor  28 A may be located in contact with the dialysis fluid anywhere along the bypass flow path, e.g. in the chamber  26 , in the outlet conduit  20  between the chamber  26  and the pump P 2 , in the bypass conduit  21  or in the inlet conduit  19  between the pump P 1  and the valve VV 1 . Alternatively, the pressure sensor  28 A may be arranged in direct contact with the gas in the chamber  26 , in analogy with  FIG. 7 . Likewise, the chamber  26  may alternatively be installed in the bypass conduit  21  or in the inlet conduit  19  between the pump P 1  and the valve VV 1 . 
     In all embodiments disclosed herein, the chamber  26  may be an existing chamber that serves another purpose in a dialysis apparatus. Such an existing chamber may be intermittently switched into fluid communication with the bypass flow path between the pumps P 1 , P 2  during the calibration mode. Alternatively, the chamber  26  may be permanently connected in fluid communication with both the main flow path and bypass flow path. In one example, the chamber  26  is part of a device for removing gas (e.g. carbon dioxide or air) from the dialysis fluid. Taking the embodiment in  FIG. 9  as an example, the chamber  26  may thus be part of a conventional gas removal device which is installed in the outlet conduit  20  to remove gases that may be introduced into the dialysis fluid in the dialyzer  13 . As included in such a gas removal device, the chamber  26  includes a vent in its top portion which is selectively opened by a dedicated on/off valve (cf. valve VV 4  in  FIG. 7 ) for removal of gases that accumulate above the dialysis fluid in the chamber  26 . Any such valve is to be closed when the flow rates are balanced during the calibration mode, such that the pressure measured by the pressure sensor  28 A represents the gas pressure in the chamber  26  and thus the dialysis fluid level in the chamber  26 . 
     As noted above, the control unit  30  may be configured to determine the relative calibration data for a number of working points of the dialysis fluid circuit  12 . These different working points may be given by different pressures measured by the pressure sensor  28 A in any of the embodiments shown in  FIGS. 7-9 , or different levels measured by the level sensor  28  in any of the embodiments shown in  FIGS. 1 and 6  provided that the top portion of the chamber  26  is sealed to confine a gas in the space above the dialysis fluid.  FIG. 10  is a plot of stroke volume ratios R, acquired according to steps  101 - 104  in  FIG. 2 , at three different pressures (working points) measured by the pressure sensor in the embodiment of  FIG. 9 . Each pair of a working point and a value of the stroke volume ratio forms a data point. The different pressures are established by changing the balanced flow rates generated by the pumps P 1 , P 2  through the bypass flow path. In the example of  FIG. 10 , a curve  50  has been fitted to the data points, as indicated by a full line. In this particular example, the stroke volume ratio R decreases approximately linearly with increasing pressure. The control unit  30  may be configured to store a calibration function that represents the data in  FIG. 10  in the memory  33 . The calibration function may be implemented as a data structure, e.g. a table, that associates stroke volume ratios R (or any other type of relative calibration data) with pressure values. Alternatively, the calibration function may be implemented as a mathematical function, linear or non-linear, that represents a factual or estimated relation between stroke volume ratio R (or any other type of relative calibration data) and measured pressure. This relation may correspond to the curve  50  in  FIG. 10 . In the example of  FIG. 9 , the control unit  30  may be operable to retrieve the sensor signal C L  also during the treatment mode and thereby measure the current pressure of the dialysis fluid pumped through the dialysis fluid circuit  12 . Specifically, in step  203  in  FIG. 4 , the control unit  30  may access the calibration function  50  based on the current pressure to acquire a corresponding value of the stroke volume ratio R. The control unit  30  then uses this value of the stroke volume ratio R to set the speeds of the pumps P 1 , P 2 , e.g. according to steps  204 - 206  in  FIG. 4 . This adaptation of the stroke volume ratio R may be done at any time interval during the treatment mode. The adaptation presumes that the pressure is measurable during both the calibration mode and the treatment mode. Thus, the pressure sensor  28 A is suitably located to be included in both the bypass flow path and the main flow path, e.g. as shown in  FIG. 9 . 
     The control unit  30  may be operable to generate and store the calibration function during manufacture of the system  10 , so that the calibration function is available for retrieval when the system  10  is subsequently operated in the treatment mode  200 . Suitably, the calibration function is generated based on stroke volume ratios R for at least three different pressures, which are well-distributed within the normal range of operating pressures for the treatment mode  200 . The control unit  30  may be further operable to execute the calibration mode  100  for re-generation of the calibration function whenever there is a change to the configuration of the system  10 , e.g. during service and maintenance. The re-generated calibration function is stored in memory  33 . In the following, the generation of the calibration function is referred to as “service calibration”. It is conceivable to execute the service calibration more frequently, e.g. at start-up of the system  10  in preparation of each treatment. The service calibration results in a new calibration function being generated and stored in the system  10 , e.g. in memory  33 . 
     In one embodiment, the control unit  30  is configured to execute the calibration mode  100  for validation of the system  10  in preparation of each treatment, and possibly at one or more time points during the treatment. In the following, this validation process is denoted “treatment calibration”. The treatment calibration is executed for the purpose of verifying that the stored calibration function adequately represents the system  10  and, if necessary, updating the calibration function, which suitably is implemented as a mathematical function and stored in the memory  33 . Both the service calibration and the treatment calibration involves establishing the bypass flow path, controlling the pumps P 1 , P 2  to balance their flow rates, and computing the relative calibration data based on the pump speeds. The treatment calibration differs from the service calibration in that it does not determine a sufficient number of data points to completely re-generate the calibration function, but rather determines one data point, or at least no more than a few different data points, to be used for validation based on an existing calibration function. Thereby, the treatment calibration is more time and processing efficient than the service calibration. 
     The rationale for performing the treatment calibration is that the relation between the stroke volumes of the pumps P 1 , P 2  may change during use of the system  10 , e.g. as a result of wear and deterioration of the pumps P 1 , P 2 , changes in the pressure of the fresh dialysis fluid supplied by the source  17 , changes to the main flow rate Q MAIN , changes in the flow resistance of the receptacle  18 , etc. Generally, any change of an operating parameter that affects the pressures at the inlet and outlet of the respective pump P 1 , P 2  may cause a change in the stroke volume ratio. 
     It may be advantageous to perform the treatment calibration at start-up of the system  10  in preparation of each treatment and/or at one or more time points during each treatment, to ensure that the stored calibration function is representative for the system  10 . This will increase the accuracy and robustness of the UF control. The treatment calibration may be governed by pre-scheduling or requested by control logics, or a combination thereof. Such control logics may initiate a treatment calibration whenever one or more specific operating parameters of the system  10  are changed, e.g. the main flow rate Q MAIN , the temperature of the fresh dialysis fluid, the UF rate Q UF , etc. The pre-scheduling may include an early treatment calibration (e.g. 5-15 minutes into treatment) followed by fixed or increasing intervals between subsequent treatment calibrations. Increasing the number of treatment calibrations during the treatment may be an option to achieve a higher UF accuracy, e.g. in the case of pediatric or acute treatments. 
     During the treatment calibration, the control unit  30  calculates a stroke volume ratio R (relative calibration data) for a validation working point, which is defined by a measured pressure or a measured level in the chamber  26  (if present). The control unit  30  then retrieves the calibration function from memory  33 , and adjusts parameters of the calibration function such that it matches the acquired data point, i.e. the calculated stroke volume ratio R and the measured pressure/level. For example, as shown in  FIG. 11 , the stored calibration function, represented by the curve  50 , may be translated in the vertical direction such that the acquired data point  55  falls on the curve. Such an adjusted calibration function  50 ′ may be generated by adding an offset value to the calibration function  50 , the offset value being computed as the difference between the calculated ratio R at the validation working point (i.e. the ratio R obtained by the treatment calibration) and the output value of the calibration function  50  at the validation working point. The control unit  30  then uses the adjusted calibration function  50 ′ for UF control, at least in the subsequent treatment mode. 
     In one implementation, the control unit  30  stores the adjusted calibration function  50 ′ in memory  33  to replace the calibration function  50 . Thus, in this implementation the adjusted calibration function  50 ′ is accessed based on the current pressure to acquire a corresponding value of the stroke volume ratio for UF control (cf. step  203  in  FIG. 2 ) and is assumed to correctly represent the system  10  until another service calibration is performed or until another adjusted calibration function  50 ′ is obtained by a forthcoming treatment calibration. In this implementation, the treatment calibration will validate the system  10  against the latest calibration function  50 ,  50 ′, be it obtained by a service calibration or a previous treatment calibration. 
     In another implementation, the control unit  30  stores, in memory  33 , both the calibration function  50  obtained by the latest service calibration and the adjusted calibration function  50 ′ obtained by the latest treatment calibration. In this implementation, the adjusted calibration function  50 ′ is used temporarily for UF control in the subsequent treatment mode (cf. step  203  in  FIG. 2 ), until the control unit  30  initiates another treatment calibration or terminates the current treatment session. The calibration function  50  obtained by the latest service calibration is used as a master or reference, and each treatment calibration is implemented to validate the system  10  against this calibration function  50 . Similarly, at start-up of the system  10 , the control unit  30  will retrieve the calibration function  50  and use it for UF control. 
     The service and treatment calibrations are further exemplified in  FIG. 12 , in which all steps are performed by the control unit  30 . In step  300 , the system  10  is set to operate in the calibration mode. In step  301 , a service calibration is executed, e.g. according to steps  101 - 104  in  FIG. 2  at a number of different pressures, and a calibration function is generated. In step  302 , the calibration function is stored, e.g. in memory  33 . Some time later (indicated by dashed arrow in  FIG. 12 ), the treatment calibration is performed. In the illustrated example, the system  10  is operated in the treatment mode (step  303 ) and switched to the calibration mode (step  304 ) for execution of a treatment calibration, e.g. according to steps  101 - 103  in  FIG. 2 , at a current pressure measured by the pressure sensor ( 28 A in  FIG. 9 ). The treatment calibration (step  305 ) results in a calculated stroke volume ratio R for the current pressure, which defines a validation working point. In principle, the treatment calibration may be executed at any current pressure, as measured by the sensor  28 A. However, it may be advantageous for the validation working point to be similar or identical to an actual working point for the system  10  during the treatment mode(s), such that the pumps P 1 , P 2  operate at reasonably similar conditions during the treatment calibration and the treatment mode. Thus, the control unit  30  may actively control one or more of the pumps P 1 , P 2  to achieve a pressure (at sensor  28 A) which is essentially the same as the pressure (at sensor  28 A) during the preceding treatment mode (cf. step  303 ). In one practical embodiment, the pumps P 1 , P 2  are furthermore controlled to maintain the main flow rate Q MAIN  used in the preceding treatment mode (cf. step  303 ), recalling that it may be desirable to avoid momentary changes to Q MAIN  at least for certain designs of the source  17 . 
     Returning to the flow chart in  FIG. 3 , the calibration function is retrieved from memory  33  in step  306  and the system  10  is then validated in step  307  using the calculated stroke volume ratio R from step  305 . The validation may involve calculating the above-mentioned offset value and comparing it to one or more thresholds. In step  308 , the control unit  30  takes action depending on the outcome of the validation in step  307 . For example, if the offset value exceeds a first threshold, the control unit  30  may issue an alarm and cause the system  10  to enter a fail safe state. If the offset value is below a second threshold, which is lower than the first threshold, the control unit  30  may refrain from adjusting the stored calibration function. If the offset value lies between the first and second thresholds, the control unit  30  may adjust the calibration function and store the adjusted calibration function in memory, e.g. as described in relation to  FIG. 11 . In a variant, only the first threshold is used in the validation, which thus aims at identifying an alarm condition. In a further variant, only the second threshold is used in the validation, which thus aims at identifying a need for adjustment of the calibration function. After step  308 , unless an alarm condition has been identified, the control unit  30  will return to the treatment mode. As noted above, steps  304 - 308  may be executed more than once during a treatment. 
     It is to be understood that shifting the calibration function by the offset value, as described above and illustrated in  FIG. 11 , generally improves the accuracy of the UF control. In practice, the true calibration function may deviate from the shifted calibration function, especially at pressures far away from the current pressure used in the treatment calibration (cf. data point  55  in  FIG. 11 ). Furthermore, the data point  55  is correct for the pressures that are present at the inlets and outlets of the pumps P 1 , P 2  during the treatment calibration. If one or more of these pressures differ significantly when the system  10  is switched between the main flow path and the bypass flow path, the data point will not be entirely correct during the treatment, and the shifted curve will be less accurate. 
     In one embodiment, errors caused by varying pressure conditions around the pumps between the calibration and treatment modes are reduced by computing a compensation value as a function of a set of operating parameters for the system  10 . The set of operating parameters estimates the changes in the pressures at the inlets and outlets of the pumps P 1 , P 2  when switching between the calibration and treatment modes. At lest part of these operating parameters may be measured in the system  10 , while others may be estimated from set values for the system  10 . In the following discussion, the pressures at the inlet and outlet of the upstream pump P 1  are denoted p 1  and p 2 , respectively, and the pressures at the inlet and outlet of the downstream pump P 2  are denoted p 3  and p 4 , respectively. 
     The following linear compensation function may be used:
 
 R′=R +( c   Δp1   ×Δp 1)+( c   Δp2   ×Δp 2)+( c   Δp3   ×Δp 3)+( c   Δp4   ×Δp 4)  (9)
 
     In the compensation function, R′ is the compensated ratio, R is the ratio that is obtained for a current pressure during a treatment mode, by use of data obtained in a calibration mode, the differential parameters Δp 1 , Δp 2 , Δp 3  and Δp 4  represent differences in pressure p 1 , p 2 , p 3  and p 4 , respectively, between the calibration mode and the treatment mode, and c Δp1 , c Δp2 , c Δp3  and c Δp4  are compensation factors. The compensation factors may be determined empirically or theoretically. The ratio R may be obtained in any of the ways described hereinabove. Thus, the ratio R may be given as an output value of a calibration function  50 ,  50 ′ which has been obtained during the calibration mode. Alternatively, if the system  10  is controlled such that the working point (pressure) is the same during treatment calibration and during the treatment mode, the ratio R may be obtained by performing a treatment calibration at the current pressure. 
     In one embodiment, the system  10  is provided with pressure sensors (not shown) that measure all of the pressures p 1 , p 2 , p 3  and p 4 . Based on this data, Eq. (9) yields a compensated ratio R′ for a given ratio R. 
     In another embodiment, assumptions are made to reduce the cost and complexity of evaluating the compensation function. 
     One assumption may be that the pressure p 1  is invariant between the calibration and treatment modes, i.e. Δp 1 =0. This assumption is valid at least if the main flow rate Q MAIN  is the same or similar in these modes. 
     Another assumption is that the pressure p 3  is invariant between the calibration and treatment modes. This assumption presumes that the control unit  30  maintains (substantially) the same working point between the treatment mode and the calibration mode (treatment calibration), as described above. The pressure p 3  in the treatment mode is governed by the pressures on the blood side (inlet and outlet of the dialyzer  13 ) and the flow conditions through the dialyzer  13  on the dialysis fluid side (through flow rates Q 1  and Q 2 ). As long as the main flow rate, Q MAIN , and the UF rate, Q UF , are constant during the treatment mode, the pressure p 3  will only change if the ultrafiltration coefficient changes (for example as a consequence of a clotted dialyzer). It is also assumed that the variations in pressure p 3  during the treatment mode will be compensated for by the calibration function. Under these assumptions, the compensation function need not compensate for the pressure difference Δp 3 . Otherwise, compensation for Δp 3  is straight-forward since the pressure p 3  may be given by, or at least estimated from, the pressure readings of the sensor  28 A. 
     Another assumption is based on the fact that the pressure p 4  will decrease when switching from the treatment mode to the calibration mode since the fluid flow rate generated by the pump P 2  will decrease with the size of the UF rate Q UF , thus lowering the flow resistance in the drain path downstream of the pump P 2 . Tests show that the pressure difference Δp 4  correlates strongly with ultrafiltration. Hence, it is assumed that Q UF  may be used instead of Δp 4  in the compensation function. 
     In total, these assumptions yield the following simplified compensation function:
 
 R′=R +( c   Δp2   ×Δp 2)+( c   Δp4   ×Q   UF )  (10)
 
     It should be noted that the UF rate Q UF  may be a small quantity in many dialysis settings. Hence, the pressure difference Δp 4  may have a minor impact on the accuracy of the ratio R. Under such circumstances, the compensation function need not compensate for the pressure difference Δp 4 . 
     When applying the compensation function, the present applicant has surprisingly found that a small error may persist between the set value Q UF  and the actual UF rate Q UF   _   real . This UF error has been found to correlate with the main flow rate Q MAIN . A corresponding compensation term may therefore be added to the compensation function according to Eq. (10):
 
 R′=R +( c   Δp2   ×Δp 2)+( c   Δp4   ×Q   UF )+( c   MAIN   ×Q   main )  (11)
 
where c MAIN  is an empirically determined compensation factor. A corresponding compensation term may be added to the compensation function according to Eq. (9). It is realized that the dependence on Q MAIN  may alternatively be overcome by calculating the ratio R in a service or treatment calibration, in which Q MAIN  is the same as in the treatment mode.
 
     Depending on the configuration and operation of the system  10 , the compensation function may differ from examples described above. For example, the pressure differentials may be included in any combination in the compensation function, or only one of the pressure differentials may be included. Furthermore, other functional relations, linear or non-linear, between the ratio R (or any other type of relative calibration data) and the pressure differential(s) are conceivable. 
     The use of a compensation function will be further described with reference to the flow chart in  FIG. 13 . in which all steps are performed by the control unit  30 . The system  10  is operated in the treatment mode (step  400 ), and the pressure p 2  at the outlet of the upstream pump P 1  is measured or estimated based on the output of a pressure sensor (step  401 ). This pressure sensor is not shown in the drawings, but may be arranged to measure the pressure in the inlet conduit  19  between the pump P 1  and the first valve VV 1 . In one implementation (not shown), the pressure sensor  28 A is connected for fluid communication with both the compliance chamber  26  and inlet conduit  19 , via a respective control valve. Thereby, the pressure sensor  28 A is operable to measure either of the pressures p 2 , p 3 , by the control unit  30  selectively opening one of these control valves. This implementation saves the cost of one pressure sensor. The system  10  is then set to operate in the calibration mode (step  402 ), the pressure p 2  at the outlet of the upstream pump P 1  is again measured or estimated, and the pressure difference Δp 2  is computed. The pressure p 3  at the inlet of the pump P 2  is also measured (step  405 ). The calibration function is retrieved from memory (step  406 ) and using the calibration function, the ratio R is obtained for the measured pressure p 3  (step  407 ). The set values for the main flow rate, Q MAIN , and the UF rate, Q UF , are acquired, if not already available (step  408 ), and the relevant compensation factors are retrieved from memory (step  409 ). Finally, in step  410 , the compensated stroke volume ratio R′ is computed using Eq. (11). 
       FIG. 14  illustrates a source  17  which is configured to prepare the fresh dialysis fluid by mixing concentrates with water. Block  40  is a water inlet that supplies water to a conditioning block  42 , which may be configured to heat the incoming water to a predefined temperature and remove gaseous substances, e.g. air, from the incoming water. The conditioned water is supplied to a mixing block  44 , which is configured to mix the water with at least two different concentrates, supplied from containers  46 ,  48 , to produce the fresh dialysate fluid which is pumped into the inlet conduit  19  by the pump P 1  (see e.g.  FIG. 1 ). For hemodialysis, one of the concentrates may contain sodium bicarbonate NaHCO 3 . Tests have revealed that the ratios R measured during a service calibration in a system  10  having this type of source  17  are slightly incorrect. After significant experimentation, the present Applicant found the cause of the error to be a carbon dioxide (CO 2 ) fluid/gas equilibrium process. Before the NaHCO 3  dosing begins, virtually no CO 2  is present in the flow path between the pumps. As the dosing begins, the following (simplified) reactions occur:
 
NaHCO 3 +CH 3 COOH&lt;--&gt;CH 3 COONa+H 2 CO 3 ( l )+CO 2 ( l ) H 2 CO 3 ( l )&lt;--&gt;CO 2 ( l )&lt;--&gt;CO 2 ( g )
 
     This means that carbonic acid and CO 2  in the fresh dialysis fluid will have a corresponding equilibrium partial pressure in the gas phase. Whether this equilibrium partial pressure is achieved or not depends on the contact time, the contact surface and flow conditions. When the fresh dialysis fluid containing CO 2  enters the compliance chamber  26 , CO 2  will be released from the dialysis fluid so that it moves towards the equilibrium partial pressure of CO 2 . Thereby, the compliance gas mass (and hence the total pressure between the pumps P 1 , P 2 ) will increase until equilibrium is reached. If pressure control is active during the calibration mode, the pump P 2  will increase its rotational speed in order to keep the pressure/level constant in the compliance chamber  26 . If the speed n 2   c  of the pump P 2  is determined before equilibrium is reached, the calculated stroke volume ratio R will be incorrect. One solution to this problem is to use a long stabilization time at each pressure set point during the service calibration. To reduce the total time for the service calibration, the control unit  30  may instead be configured to disable the dosing of sodium bicarbonate into the mixing block  44  during the service calibration. 
     While the invention has been described in connection with what is presently considered to be the most practical and preferred embodiments, it is to be understood that the invention is not to be limited to the disclosed embodiments, but on the contrary, is intended to cover various modifications and equivalent arrangements included within the spirit and the scope of the appended claims. 
     For example, the system for ultrafiltration may include more than one upstream pump and/or more than one downstream pump. The relative calibration data may be determined for these pumps as described herein, by establishing the bypass flow path between pairs of upstream and downstream pumps and performing steps  102 - 105  for each pair of upstream and downstream pumps. 
     The upstream and downstream pumps are volumetric pumps and maybe implemented as either piston pumps or diaphragm (membrane) pumps. It is to be understood that the upstream and downstream pumps may have different nominal stroke volumes. It is also possible that the upstream and downstream pumps are of different designs or types. 
     The skilled person also realizes that there are alternatives to using a pressure sensor or a level sensor for balancing the flow rates of the pumps during the calibration mode. For example, the level sensor  28  in  FIG. 1  may be replaced by simple flow sensor that measures the flow of dialysis fluid in the connecting conduit  27  when the dialysis fluid circuit  12  is set to establish the bypass flow path. The control unit  30  may adjust the frequency of one or both pumps P 1 , P 2  until the output signal of the flow meter indicates that the flow rate in the connecting conduit  27  is zero. 
     Even if it is possible to use the pressure sensor  28 A to both identify an invariant level in the compliance chamber  26  and to determine a pressure representative of a working point of the system  10 , it is quite possible to implement the system  10  with separate devices for determining level and pressure. As noted above, the pressure of the working points need not be determined at the location of the compliance chamber  26 , but may in principle be determined anywhere within the dialysis fluid circuit  12 . If the pressure is measured in the inlet conduit  19  between the pump P 1  and the valve VV 1 , Equations (9)-(11) are modified by replacing p 2  with p 3  and Δp 2  with Δp 3 . It is also to be noted that Equations (9)-(11) are also applicable if the set value Q MAIN  represents the flow of dialysis fluid out of the dialyzer  13  and the UF rate is controlled by adjusting the speed of the upstream pump P 1 . Furthermore, is may be noted that the compliance chamber  26  may instead be located in fluid communication with the inlet conduit  19  upstream of the first valve VV 1 . 
     In the foregoing description, the control unit  30  controls the UF rate exclusively by setting the speeds of the upstream and downstream pumps, using the relative calibration data obtained in the calibration mode, to achieve a desired difference in the flow rates between the upstream and downstream pumps. In an alternative embodiment, the control unit  30  sets the flow rates of the upstream and downstream pumps equal, using the relative calibration data obtained in the calibration mode, and controls the UF rate by operating a dedicated UF pump (“filtration pump”), which is connected to the outlet conduit  20  between the downstream pump P 2  and the dialyzer  13 , as is well-known in the art and described in the Background section. It is also conceivable to configure the control unit  30  to set the UF rate by a combination of controlling a separate UF pump and controlling the difference in flow rates between the upstream and downstream pumps.