Abstract:
A method and system are disclosed for more accurate administration of a composition being infused into a body fluid having formed elements. The improved accuracy is due to a correction factor that corrects for the relative impermeability of the formed elements to the infused composition. In one embodiment, the infusion system is a cardioplegia system in which a controller receives information pertaining to the hematocrit and initial blood potassium concentration and adjusts blood flow rate and crystalloid (KCl) solution flow rate so as to obtain the desired resulting potassium concentration in the cardioplegia. The invention further includes an infusion system that comprises a controller which uses an equation that relates the correction factor to an index of formed elements in the body fluid, such as the hematocrit. The equation is determined empirically from a series of experiments in which the resulting concentration of the substance infused into the body fluid is correlated to a formed element index of the body fluid, such as hematocrit.

Description:
FIELD OF THE INVENTION 
     The subject invention pertains to the controlled infusion of a composition into a body fluid containing formed elements. More particularly, it pertains to the controlled infusion of cardioplegia solution into blood. 
     BACKGROUND OF THE INVENTION 
     During open heart surgery, a system for the extracorporeal circulation of fluids through a number of fluid circuits is required. This system is referred to as a perfusion control system or heart-lung machine. The fluid circuits of the system typically include a cardiopulmonary circuit, a cardioplegia circuit, a cardiotomy circuit and a ventricular vent circuit. 
     The cardiopulmonary circuit, which is designed to functionally replace or supplement the heart and lungs during heart surgery, comprises tubing, one or more pumps for blood circulation and an oxygenation device. Blood is received from a major vessel entering the heart (e.g., the vena cava) by a venous line. The venous line transports blood optionally to a reservoir, and then to an oxygenator. Oxygenated blood is transported back to the patient via the arterial line and enters the patient in a major vessel leaving the heart (e.g., aorta). 
     The cardioplegia circuit delivers cardioplegia to the heart. Cardioplegia discontinues the beating of the heart in a manner that will minimize damage to the myocardium and provides a motionless heart on which the surgeon can operate. Cardioplegia can also supply other ingredients to provide for myocardial protection. Cardioplegia may be the crystalloid solution alone or may also include oxygenated blood diverted from the arterial line. The crystalloid solution typically contains potassium chloride (KCl), sugars and magnesium. The potassium (K+) concentration in the cardioplegia is initially elevated (e.g., 25 mmol/l) during induction of arrest and reduced (8.5-9 mmol/l). during maintenance. Other cations such as magnesium (Mg++) can be used as heart arresting agents. It is preferred to include oxygenated blood in the cardioplegia so that the cardioplegia is buffered and can oxygenate the myocardium. Where cardioplegia includes arterial blood, the cardioplegia circuit comprises the oxygenated blood line containing blood diverted from the arterial blood line, the crystalloid solution bag and line, the cardioplegia delivery line (containing the mixture of blood and crystalloid solution) and one or more pumps. The crystalloid solution line and the oxygenated blood line may both be threaded through the same pump or through different pumps. The pumps can be peristaltic or other pumps known in the art. There is typically a device for controlling and recording the total volume of crystalloid solution and oxygenated blood that are combined to form the cardioplegia. The cardioplegia is delivered to the coronary arterial network or coronary sinus for distribution throughout the myocardium. The cardioplegia is then distributed through the circulatory system, or may occasionally be drawn out the chest cavity and discarded or directed via the cardiotomy line to the cardiopulmonary circuit, as discussed immediately below. 
     The cardiotomy circuit is used to withdraw or suction blood or blood mixed with other fluids from the opened heart or the chest cavity and deliver it to the cardiopulmonary circuit at a point upstream of the oxygenator. 
     The ventricular vent circuit functions to drain the left ventricle of blood that returns via the bronchial artery and pulmonary veins. The vent line collects blood from the left ventricle and delivers it to the cardiopulmonary circuit at a point upstream of the oxygenator. 
     Existing systems for extracorporeal circulation commonly have pumps, reservoir(s), an oxygenator and monitoring devices mounted on a console. The system can also include a controller that regulates pump speeds and receives information from patient monitoring devices. The controller may also cause the collected information to be displayed on a monitor. A description of perfusion control systems can be found in U.S. Ser. No. 08/304,725, filed Sep. 12, 1994, pending, which is incorporated herein in its entirety by reference. 
     The infusion of KCl or other arresting agents into the patient&#39;s blood is critical to induce and maintain arrest during surgery. It is generally considered desirable to avoid excessive dilution of the blood with crystalloid solution so as to maintain adequate oxygenation of tissues and minimize the need for heat exchange (Houerou, D. et al. (1992) Ann. Thorac. Surg. 54:809-16). Thus, the trend is to use crystalloid solutions of high concentration. It is also considered desirable to avoid an excessive dose of KCl as it can result in increased systemic K+ concentration and delayed resumption of heart function at the conclusion of surgery. 
     To obtain a desired K+ concentration in the cardioplegia, cardiologists select the crystalloid solution concentration and the flow rates of oxygenated blood and crystalloid solution. See, e.g., Fried, D. W. &amp; Mohamed, H. (1993) Perfusion 8:401-407. U.S. Pat. No. 5,385,540 (1995) describes a cardioplegia pump system for controlling the ratio of blood and crystalloid solution that are mixed. 
     A prior art equation that has been used to predict the K+ concentration in the cardioplegia is: ##EQU1## where C o  is the desired K+ concentration in the cardioplegia, Q b  is the flow rate of the oxygenated blood, C s  is the initial K+ concentration in the oxygenated serum, Q k  is the flow rate of crystalloid solution, and C k  is the crystalloid K+ concentration. Equation (1) suffers from at least one major setback: it does not correct for the effect of the formed elements that are not substantially permeable to the infused cations, such as K+. Therefore, equation (1) consistently under-predicts the therapeutic K+ concentration. 
     Thus, in the prior art, practitioners believed the unfractionated cardioplegia K+ concentration to be the therapeutically effective dose. The subject invention recognizes that the actual therapeutic concentration is the cardioplegia plasma K+ concentration, which invariably is greater than the unfractionated cardioplegia K+ concentration. The subject invention remedies the prior art erroneous prediction method by recognizing the reduced volume in cardioplegia that is actually available for solubilizing K+. Further, the subject invention provides accurate methods for determining the actual volume in cardioplegia available for solubilizing K+ based on an empirical relationship between plasma volume and hematocrit. 
     SUMMARY OF THE INVENTION 
     The subject invention comprises a method for infusion of a composition into a body fluid having formed elements or cells. Because the method corrects for the presence of formed elements in the body fluid, it more accurately produces the desired composition concentration in the body fluid. The correction is necessary because the formed elements are not substantially permeable to the infused composition and the formed elements represent a significant fraction of the body fluid volume. 
     In one embodiment, the body fluid is whole blood, the formed elements are red blood cells, and the composition is K+ ion. The infusion system is a perfusion control system in which the K+ ion is infused into oxygenated blood in an extracorporeal circuit that returns the blood and cation mixture (cardioplegia) to the myocardium, where it arrests the beating of the heart. 
     In other embodiments, the body fluid can be white blood cell or platelet concentrates, and the substance can be any material that does not substantially permeate the formed elements. 
     As mentioned in the Background, the cation concentration in cardioplegia was incorrectly predicted in the prior art according to equation (1). This formula consistently calculates resultant cation concentrations in cardioplegia that are lower than measured cation concentrations because it fails to correct Q b  for the formed element volume that is unavailable to the added K+. If Q s  or the serum volume is used instead of Q b , then a more accurate resultant cation concentration can be calculated. 
     As a practical matter, Q s  cannot be easily measured in an infusion system because complete separation of serum from formed elements on-line is not feasible. Moreover, Q s  cannot be accurately calculated solely from the hematocrit. Hematocrit, or the volume percent of red blood cells, is determined by sedimenting the red blood cells in whole blood by centrifugation. The conditions of centrifugation, such as radius length, rpm, g force and duration, determine the degree of separation of plasma from cells. Because the sedimented cells retain a residual layer of plasma around each cell and between cells, the hematocrit is always larger than the actual red cell volume. The red cell volume fraction is actually between 50% and 70% of the measured hematocrit, depending on the method of measuring hematocrit. Further, even for a given hematocrit measuring method, the actual red cell volume can vary as a function of the hematocrit value and the relationship between actual red cell volume and hematocrit value must be determined empirically. That is, the relationship between the red cell volume and the hematocrit value is not necessarily a constant proportion. 
     In accordance with the subject invention, a corrected formula for calculating the resulting concentration of cation in the cardioplegia is: ##EQU2## where Z i  is the red cell volume correction factor at a given hematocrit, i, that converts Q b  to Q s . Z i  =1/(1-RCVF) where RCVF is the fraction of the blood volume excluded by the red cell membranes. Z i  can vary as a function of hematocrit; e.g., the higher the hematocrit, the greater the correction factor. 
     &#34;Z&#34; refers to the correction factor variable without regard to a specific hematocrit value. To obtain an equation relating Z to a range of hematocrit values, the following protocol can be used. A blood sample of known hematocrit and volume, V b , is measured for the cation concentration, D s  (e.g., using a Radiometer™ sodium/potassium analyzer having an ion selective electrode); then an aliquot (V k ) of cation solution of known concentration (D k ) is added to the blood sample, and the resulting concentration, D r , is measured. This process is repeated several times with blood samples having the same hematocrit to yield several Z i  values at the first hematocrit value. Z i  is calculated from measurable concentration and volume data: ##EQU3## 
     Blood having a second hematocrit value is then tested in the same manner to obtain a second series of Z i  values. The process can be repeated for multiple blood samples each having different known hematocrits. The collected Z i  values are plotted against their respective hematocrit values and an equation relating Z to hematocrit is obtained from the plot. 
     Alternatively, an equation relating Z to hematocrit can be obtained by using an analogous protocol with blood and crystalloid flow rates rather than volumes. Z i  is calculated from the measured concentration and flow rate data according to equation (3a): ##EQU4## where Q b  is the blood flow rate, Q k  is the crystalloid flow rate, C r  is the resulting cardioplegia concentration, C k  is the crystalloid solution concentration, and C s  is the initial serum concentration of cation. 
     As is described hereinbelow in Example 1, the aforementioned protocol relating to equation (3) was used with bovine blood to empirically determine an equation relating Z to hematocrit. The following equation was derived: ##EQU5## 
     The invention further comprises a method for correcting C r  for the effect of differences in osmolarity between the crystalloid solution and the intracellular plasma. In whole blood, the osmolarity of the blood plasma is usually maintained at a slightly lower value than that of the intracellular plasma. When crystalloid solution is added to the blood, water flows out of the red blood cells. The blood plasma ion concentration is reduced by the additional water. 
     The foregoing equation (4) which was obtained by a least squares exponential fit to data obtained in Example 1 below, takes into account the differential osmolarity effect for a crystalloid solution of 400 mEq/l and an initial blood K+ concentration of approximately 5 mEq/l. However, when crystalloid solution concentration is higher than 400 mEq/l, it may be desirable to adjust formula (2) to further accommodate the greater differential osmolarity effect. The differential osmolarity effect at crystalloid concentrations over 400 mEq/l may result in a curve that deviates significantly from the exponential or linear curve observed for lower crystalloid concentrations. While it is possible to derive an equation that fits the curve over a range of crystalloid concentration from 0 to values in excess of 400 mEq/l, clinical application of the subject method does not require such a comprehensive equation. It could be sufficient at higher crystalloid concentrations to use both the Z correction factor derived for lower crystalloid concentrations and a volume correction factor that reflects the volume of water contributed by the red blood cells when using crystalloid solution having a concentration in excess of 400 mEq/l. The adjusted formula that includes the water contributed by red blood cells is: ##EQU6## where ΔQ rc  is the change in the red cell volume. ##EQU7## where Q rci  is the initial red cell intracellular plasma volume, which is about 0.667 of the total red cell volume, RCVF. As mentioned above, RCVF can be determined from the Z i  for a particular hematocrit from Z i  =1/(1-RCVF). O s  is the initial serum osmolarity, and O r  is the resultant osmolarity of the blood after addition of the crystalloid solution. 
     Assuming that the osmolarity of the blood and intracellular serum are equal, the O r  can be obtained by the following equation: ##EQU8## where Q rci  is the initial red blood cell plasma volume (about 0.667 of the RCVF), Q b  is the initial blood volume, Z i  is the correction factor that converts Q b  to Q s  or initial serum volume, O s  is the initial serum osmolarity (measured by a sodium/potassium analyzer), Q k  is the volume of crystalloid added to the blood, and O k  is the osmolarity of the crystalloid solution. 
     As the trend is to use crystalloid solutions of greater concentration so as to minimize blood dilution and attendant disadvantages, practitioners using such solutions can correct C r  for Q rc  per equation (5), rather than redetermining the relationship between Z and hematocrit for crystalloid solution concentrations in excess of 400 mEq/l. 
     Thus, the subject invention comprises an infusion method for obtaining a desired concentration of a composition in a body fluid having formed elements that are not substantially permeable by using an empirically determined equation relating a correction factor, Z, to the hematocrit or other measurable index of the formed elements. It further comprises an infusion system comprising a controller that receives information pertaining to the body fluid&#39;s initial composition concentration and hematocrit or formed element index, and controls one or more of the composition solution flow rate and concentration, and the body fluid flow rate to obtain a desired resulting composition concentration in the mixture of body fluid and composition. In particular, the infusion system can be a concentration, C r . cardioplegia infusion system in which a controller receives C s  and hematocrit information, and controls one or more of C k , Q k  and Q b  to produce cardioplegia having a desired cation concentration, C r . 
     More generally, the invention comprises a method for producing a mixture comprising a first fluid having a dissolved substance in a first concentration and blood having a volume of formed elements, where the formed elements are not substantially permeable to the dissolved substance, and the mixture has a predetermined concentration of the dissolved substance. The method comprises the steps of: establishing a volume of the blood; correcting the volume of the blood for the effects of the volume and permeability to the dissolved substance of the formed elements; establishing a volume of the first fluid to achieve the predetermined concentration of the dissolved substance in the mixture; and combining the volume of the first fluid with the volume of the blood to form the mixture. The correcting step involves determining the portion of the blood volume that comprises the formed elements. Because the formed elements respond to the presence of the dissolved substance in the mixture by transferring fluid from within the formed elements to a portion of the mixture outside the formed elements, the volume of the blood can further be adjusted for the effect of the transferred fluid. 
     The invention also generally comprises an apparatus for producing a mixture comprising a first fluid having a dissolved substance in a first concentration and blood having a volume of formed elements, with the formed elements being not substantially permeable to the dissolved substance, and the mixture having a predetermined concentration of the dissolved substance. The apparatus comprises: means for establishing a blood volume; means for correcting the blood volume for the effects of the volume and permeability to the dissolved substance of the formed elements; means for establishing a volume of the first fluid to achieve the predetermined concentration of the dissolved substance in the mixture; and means for combining the volume of the first fluid with the volume of the blood. The means for correcting the blood volume for the effects of volume and permeability of the formed elements can be a controller. The means for establishing the volume of the first fluid can be a first pump controlled by the controller. The means for establishing the blood volume can be a second pump controlled by the controller. The means for correcting the blood volume for the effects of volume and permeability of the formed elements can include a means for determining the ratio of the volume of formed elements to the total blood volume. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a pictorial representation of a perfusion control system. 
     FIG. 2 is a schematic representation of the fluid circuits for the apparatus of FIG. 1. 
     FIG. 3 is a schematic representation of the fluid circuits used in a preferred embodiment of the subject invention. 
     FIG. 4 is a schematic representation of a control system for the subject invention. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     A more complete understanding of the invention can be obtained from a description of the drawings which illustrate a heart-lung machine, the COBE Perfusion Control System™, sold by the assignee of the subject invention. As of the filing date of the subject application, the COBE Perfusion Control System has been sold without a control function and/or operator instructions that would effect the subject invention. 
     FIG. 1 illustrates a horizontal row of six modules: five are pump assemblies 12 comprising peristaltic pump heads 14 and an instrument control panel 16; the sixth is a central control module 18. An assembly 20 of the oxygenator 44, heat exchanger and reservoir 38 (&#34;oxygenator assembly&#34;) is mounted on a mast 22 with a swing arm 24. At the top of each of two masts 22, 26 are hooks 28 for hanging of crystalloid solution or other fluids. A display monitor 30 is mounted on mast 26, and is electronically connected to the central control unit 18. Pump assemblies 12 can be under the local control of their instrument panels 16 or may be under the control of the central controller 18. The central controller 18 can receive information from an arterial bubble sensor 56, a blood level sensor (not shown) in the reservoir 38, temperature sensors disposed in arterial or cardioplegia lines or in the patient (not shown), and pressure sensors disposed in the arterial, cardioplegia or left ventricle lines (not shown). The controller 18 processes such information and can communicate it to the monitor 30. The controller can respond to information by controlling the cardiopulmonary, cardioplegia and other pump heads 14. The central controller 18 and monitor 30 have keypads for perfusionist control of such functions or monitoring systems. The COBE Computerized Perfusion Controller Operator&#39;s Manual (May, 1990), incorporated herein in its entirety by reference, provides a complete description of the functions of the prior art central controller 18. 
     FIG. 2 is a pictorial schematic illustrating only the prior art tubing and oxygenator assembly of FIG. 1. A venous line 60 which is connected to a major vessel entering the heart (not shown) transports blood from the patient to the venous port 36 of the reservoir 38. The cardiopulmonary pump loop 64 comprises a line connected to a second reservoir port 40 which is threaded through a peristaltic pump head 14 of a pump assembly 12 (FIG. 1), and which connects with a first oxygenator port 42 to transport blood from the reservoir 38 to the oxygenator 44. The arterial line 62, connected to a &#34;Y&#34; adaptor off a second oxygenator port 46, transports the oxygenated blood to a major vessel exiting the heart (not shown). The crystalloid solution supply line 66, connected to a crystalloid solution supply bag 48, together with the cardioplegia blood line 68, connected to the other stem of the &#34;Y&#34; adaptor off oxygenation port 46, are threaded through another peristaltic pump head 14 of a pump assembly 12 (FIG. 1), and then combined into one line to produce a cardioplegia delivery line 70, which transports the cardioplegia to the heart (not shown). Alternatively, the crystalloid solution supply line 66 and the cardioplegia blood line 68 can be threaded through different pump heads 14 of different pump assemblies 12. Additionally, a single pump assembly can contain a double pump head (not shown), such that lines 66 and 68 are threaded through separate pump heads located on the same pump assembly. Cardiotomy line 72, which is threaded through another peristaltic pump head 14 (FIG. 1), suctions blood and other fluids from the chest cavity (not shown) and transports it to a third reservoir port 52, whereby the fluid joins the cardiopulmonary circuit. Finally, a left ventricular vent line 74, which drains the left ventricle (not shown), is threaded through a fourth pump head 14 (FIG. 1), and transports blood to a fourth reservoir port 54, thereby relieving pressure in the ventricle. 
     In the subject invention, the flow through lines 66 and 68 is regulated by selecting tubing diameters and/or selecting pump speeds. In a preferred embodiment, the lines are threaded through different pump heads (either on different pump assemblies or on a single pump assembly having a double pump head), and flow rate through each is independently regulated by independently controlled pump heads. Alternatively, the lines are threaded through the same pump head, but have diameters selected to produce a predetermined flow rate for a preselected pump speed. Thus, in either embodiment, the flow rates in the crystalloid supply line and the oxygenated blood line are independently controlled such that a desired C r  is obtained. FIG. 3 illustrates the preferred embodiment in which the crystalloid supply line 66 and the cardioplegia blood line 68 are threaded through independent pump heads 14 (FIG. 1). 
     FIG. 4 illustrates a control system for the subject invention. The controller 18 receives information from an operator 82 and may also receive information from a hematocrit monitor 80. The hematocrit monitor 80 measures the patient&#39;s hematocrit in the cardioplegia blood line 68 or at a point upstream (e.g., in the reservoir 38 (FIG. 1)) via a fiber optic cable 84. In one embodiment, the monitor 80 is the COBE Saturation Hematocrit Monitor sold by the assignee of the subject application and described in U.S. Pat. No. 5,356,593, issued Oct. 18, 1994, incorporated herein in its entirety by reference. Other hematocrit monitors known in the art can be used. The hematocrit monitor 80 can supply hematocrit values to the controller 18 once or on a periodic or continuous basis. Alternatively, patient hematocrit can be determined by a hematocrit monitor 80 or other means and supplied to the controller 18 via the operator 82. 
     The operator 82 can also supply other information such as the initial patient serum K+ concentration (C s ) or blood flow rate (Q b ), the desired resulting cardioplegia concentration (C r ) or flow rate (Q r ), the crystalloid solution concentration (C k ) or flow rate (Q k ), to the controller 18. The controller calculates Z i  from the hematocrit and the empirically derived equation relating Z to hematocrit. Using Z i  and the information supplied by the operator, the controller can calculate the Q b  :Q k  ratio, or Q b , Q k  or C k  values that will produce the desired C r . These variables are calculated using equations (2) and (3a). Thus, depending on the information supplied to the controller 18, it can calculate other variables of equations (2) and (3a). For example, the controller can calculate the Q b  :Q k  ratio from C s , C r  and C k , and then regulate pump heads 14 or specify diameters for the cardioplegia blood line 68 and the crystalloid solution line 66 to produce the desired Q b  :Q k  ratio and C r  in cardioplegia line 70. Additionally, the controller 18 could calculate the C k  if it is supplied with C s , C r , Q b  and Q k . Also, the controller 18 can calculate Q k  if it is supplied with C s , C r , C k  and Q b . Further, the controller 18 can calculate Q b  if it is supplied with C s , C r , C k  and Q k . These and other calculations that can be performed by the controller 18 using equations (2) and (3a) will be readily apparent to those of skill in the art. 
     For example, once the Q b  :Q k  ratio is calculated according to a rearranged equation (2), i.e., ##EQU9## the crystalloid solution flow rate, Q k , can be determined by selecting a desired infusion mixture flow rate, Q r , and calculating the Q k  according to the equation: ##EQU10## 
     Additionally, once the Q b  :Q k  ratio is calculated according to equation (2a), the body fluid flow rate, Q b , can be determined by selecting a desired infusion mixture flow rate, Q r , and calculating the Q b  according to the equation: ##EQU11## 
     Further, once the Q b  :Q k  ratio is calculated according to equation (2a), the crystalloid solution concentration, C k , can be determined according to the equation: ##EQU12## 
     The subject invention is further detailed by reference to the following Examples. These Examples are provided for the purpose of illustrating the invention and are not intended to be limiting thereof. 
     EXAMPLE 1 
     Determination of Z, the Hematocrit Correction Factor 
     Bovine blood of a known hematocrit of 42 was treated with beef lung heparin. The blood was diluted with sterile normal saline solution (0.9% NaCl) to produce samples having hematocrit values of 40, 30 and 20. Aliquots of 200 mls of the blood samples were dispensed in beakers, such that there were three beakers of 40 hematocrit, two beakers of 30 hematocrit, and two beakers of 20 hematocrit blood. In Table 1, beakers 1-3 contain blood having hematocrit equal to 40; beakers 4-5 have hematocrits equal to 30 and beakers 6-7 have hematocrits equal to 20. Two hundred mls of saline was dispensed in beaker 8 as a control. K+ concentration was measured for each sample using a sodium/potassium analyzer. Tare and loaded weight were measured for each beaker. 
     About 1-3 mls of stock KCl solution (about 400 mEq/l) were added to each beaker, except beaker no. 1, which also served as a control. After KCl addition, the blood mixture was stirred and the K+ concentrations and weights were measured. 
     A second dose of KCl was added to each beaker (except beaker no. 1), and K+ concentrations and weights were measured. A total of 5 KCl aliquots were added to beakers 2-8, and weight and ion measurements were taken each time. 
     From the tare, initial and subsequent beaker weights, the blood mixture (cardioplegia) weights were calculated. Using density values (p, rho), the total blood or cardioplegia volumes were calculated. The density of the cardioplegia was determined by adding the densities of the component parts, i.e., of the red blood cells (1.098 g/ml), plasma (1.024 g/ml), and saline diluent (1.0046 g/ml): 
     (Hct=40)=1.098*0.4+1.024*0.6=1.0536 
     (Hct=30)=1.0098*0.3+1.024*0.45+1.0046*0.25=1.041 
     (Hct=20)=1.0098*0.2+1.024*0.3+1.0046*0.5=1.029 
     Table 1 presents the densities of blood and blood mixtures, the blood or blood mixture volume (V b ), the crystalloid solution volume (V k ), the initial concentration of potassium in the blood or mixture (D s  or D r ), the resultant concentration of potassium after the aliquot of KCl is added (D r ), and the crystalloid concentration (D k ). The correction factor, Z i , was calculated as according to equation (3). The Z i  values were plotted against the hematocrit values and a least square exponential fit of the data yielded: 
     
         Z=0.906427e.sup.0.0099497*Hct                              (4) 
    
     
                                           TABLE 1__________________________________________________________________________Beaker 1 Beaker 2          Beaker 3                Beaker 4                      Beaker 5                            Beaker 6                                  Beaker 7                                        Beaker 8__________________________________________________________________________InitialTare    199.8    214.9 216.3 222.7 215.2 216   217.5 187.7Hct 40   40    40    30    30    20    20    0Density    1.05 1.05  1.05  1.04  1.04  1.03  1.03  1.0046V.sub.b    190.2857    204.6667          206   214.1346                      206.9231                            209.7087                                  211.165                                        186.8405D.sub.s    5.3  5.35  5.4   3.6   3.6   2.2   2.2   0After 0.94 ml (V.sub.k) of 397.34 mEq/l KCl (D.sub.k)Hct 40   40    40    30    30    19.5  19.5  0D.sub.r    5.4  7.85  7.85  5.7   5.8   4.2   4.15  2V.sub.b    190.2857    205.6067          206.94                215.0746                      207.8631                            210.6487                                  212.105                                        187.7805Z.sub.i  1.397536          1.378508                1.221494                      1.23688                            1.134936                                  1.114106                                        1.005547After 1.88 ml (V.sub.k) of 397.34 mEq/l KCl (D.sub.k)Hct 40   39.5  39    29    29    19.5  19.5  0D.sub.r    5.4  12.75 12.65 9.8   10.1  8.05  8     5.7V.sub.b    190.2857    207.4867          208.82                216.9546                      209.7431                            212.5287                                  213.985                                        189.6605Z.sub.i  1.375875          1.356537                1.197645                      1.21426                            1.097273                                  1.104718                                        0.934811After 1.88 ml (V.sub.k) of 397.34 mEq/l KCl (D.sub.k)Hct 40   39    39    28.5  29    19    18.5  0D.sub.r    5.5  17.55 17.3  13.95 14.35 11.8  11.8  9.35V.sub.b    190.2857    209.3667          210.7 218.8346                      211.6231                            214.4087                                  215.865                                        191.5405Z.sub.i  1.377449          1.34263                1.235784                      1.224443                            1.088975                                  1.110913                                        0.94021After 2.82 ml (V.sub.k) of 397.34 mEq/l KCl (D.sub.k)Hct 40   38    38    28    28    19    19    0D.sub.r    5.55 24.1  24    19.9  20.45 17.4  17.35 14.8V.sub.b    190.2857    212.1867          213.52                221.6546                      214.4431                            217.2287                                  218.685                                        194.3605Z.sub.i  1.280431          1.317228                1.204323                      1.195243                            1.104363                                  1.101937                                        0.954086After 2.82 ml (V.sub.k) of 397.34 mEq/l KCl (D.sub.k)Hct 40   37    37    27    27    18    18.5  0D.sub.r    5.6  30.45 30.4  25.6  26.25 22.7  22.65 20.1V.sub.b    190.2857    215.0067          216.34                224.4746                      217.2631                            220.0487                                  221.505                                        197.1805Z.sub.i  1.280131          1.297971                1.187011                      1.170244                            1.074556                                  1.081617                                        0.9549__________________________________________________________________________ 
    
     EXAMPLE 2 
     Illustration of the Effect of the Correction Factor on Resulting Concentration 
     Determinations of corrected C r  and uncorrected C r  (C 0 ) were performed for a hypothetical cardioplegia system having a blood supply line and a crystalloid solution supply line combining their respective volumes in a ratio of Q b  :Q k  =4:1. The C s  was assumed to be 5 mEq/l and the hematocrit of the blood samples was assumed to range from 15 to 35%. The C k  was set at 30 mEq/l for maintenance of the arrested state and 100 mEq/l for induction of the arrested state. 
     For blood samples having hematocrit values of 15, 16, 17 35, C r  was calculated using equation (2): ##EQU13## where Z i  =0.906427*e 0 .00995*Hct. 
     C 0 , the resultant concentration that would have been obtained without a hematocrit correction factor, was also calculated by using the prior art equation (1): ##EQU14## The E i  or error, C r  -C 0 , is set forth below. 
     
         ______________________________________  Maintenance C.sub.k                    Induction C.sub.kHct      C.sub.r E.sub.i     C.sub.r                              E.sub.i______________________________________15       10.21   0.21        24.799                              0.79916       10.252  0.252       24.956                              0.95617       10.293  0.293       25.114                              1.11418       10.335  0.335       25.273                              1.27319       10.377  0.377       25.433                              1.43320       10.419  0.419       25.594                              1.59421       10.462  0.462       25.756                              1.75622       10.505  0.505       25.918                              1.91823       10.548  0.548       26.082                              2.08224       10.591  0.591       26.246                              2.24625       10.635  0.635       26.412                              2.41226       10.678  0.678       26.578                              2.57827       10.722  0.722       26.745                              2.74528       10.767  0.767       26.914                              2.91429       10.811  0.811       27.083                              3.08330       10.856  0.856       27.253                              3.25331       10.901  0.901       27.423                              3.42332       10.946  0.946       27.595                              3.59533       10.992  0.992       27.768                              3.76834       11.037  1.037       27.941                              3.94135       11.083  1.083       28.116                              4.116______________________________________ 
    
     These data indicate that the error increases with increasing hematocrit value and with increased crystalloid concentrations. 
     EXAMPLE 3 
     Illustration of the Effect of the Correction Factor on Resulting Concentration 
     Determinations of corrected C r  and uncorrected C r  (C 0 ) were performed for a hypothetical cardioplegia system having a higher crystalloid concentration of 400 mEq/l. Higher concentrations are generally preferred to reduce blood dilution. The resulting cardioplegia flow, Q r , was assumed to be 100 ml/min and C s  was assumed to be 5 mEq/l. Q b  and Q k  values that would accommodate the high C r  were calculated. However, the Q b  and Q k  values were not corrected for hematocrit so that the error that is generated by prior art methods could be assessed. The Q b  and Q k  values were calculated using the following equations, assuming that the C r  =10 mEq/l during maintenance and C r  =25 mEq/l during induction, and Hct=0: ##EQU15## 
     Equation (11) was derived as follows. Given that Q k  +Q b  =Q r , Q b  can be expressed in terms of the Q b  :Q k  ratio as: ##EQU16## Equation (3a) can be rearranged to: ##EQU17## Since no correction is made for hematocrit, Hct=0 and Z i  =1, resulting in the Z i  term being dropped out of the rearranged (3a) equation. If the Q b  /Q b  equation (3a) is then substituted into equation (13), equation (11) is obtained. 
     Equation (12) is a rearrangement of Q k  +Q b  =Q r . The calculated Q b  and Q k  values at maintenance and induction crystalloid and blood flow rates were: 
     
         ______________________________________      Maintenance              Induction______________________________________Q.sub.b      98.734    95.19Q.sub.k      1.266     4.81______________________________________ 
    
     Using these uncorrected Q b  and Q k  values, the actual C r  values corrected for a range of hematocrit values (15 to 35), were calculated according to equation (2): ##EQU18## where Z i  =0.906427*e 0 .00995*Hct. 
     Additionally, corresponding C 0  values which did not reflect corrections for hematocrit, were calculated using equation (1): ##EQU19## The E i  or error, C r  -C 0 , is set forth below. 
     
         ______________________________________  Maintenance C.sub.k                    Induction C.sub.kHct      C.sub.r E.sub.i     C.sub.r                              E.sub.i______________________________________15       10.262  0.262       24.958                              0.95816       10.314  0.314       25.149                              1.14917       10.367  0.367       25.341                              1.34118       10.42   0.42        25.534                              1.53419       10.474  0.474       25.73 1.7320       10.528  0.528       25.927                              1.92721       10.583  0.583       26.126                              2.12622       10.638  0.638       26.327                              2.32723       10.694  0.694       26.53 2.5324       10.75   0.75        26.734                              2.73425       10.807  0.807       26.941                              2.94126       10.865  0.865       27.149                              3.14927       10.923  0.923       27.359                              3.35928       10.982  0.982       27.571                              3.57129       11.041  1.041       27.784                              3.78430       11.1    1.1         28.   4.31       11.161  1.161       28.218                              4.21832       11.222  1.222       28.437                              4.43733       11.283  1.283       28.659                              4.65934       11.345  1.345       28.882                              4.88235       11.408  1.408       29.107                              5.107______________________________________ 
    
     As with Example 2, these data indicate that the error increases with increasing hematocrit value and with increasing crystalloid concentration. 
     EXAMPLE 4 
     Clinical Study 
     A prototype cardioplegia administration system designed for variable potassium concentration control and more accurate C r  calculation was employed in a clinical study conducted at St. Paul&#39;s Hospital in Vancouver, Canada. There were 30 patients in the study. 
     Thirteen were treated according to St. Paul&#39;s standard procedure (a prior art procedure without hematocrit correction) which employed a 4:1 Q b  :Q k  ratio, and which used an induction C k  of 100 mEq/l, a maintenance C k  of 30 mEq/l and a desired C r  of 24 mEq/l at induction or 10 mEq/l at maintenance (&#34;Standard Group&#34;). 
     Seventeen were treated using a variable potassium dosage protocol which employed a single C k  of 400 mEq/l (&#34;Variable Group&#34;). The variable potassium dosage prototype apparatus consists of two peristaltic pumps with a control/display panel; one pump meters the crystalloid solution and the second pump meters the oxygenated blood into the cardioplegia line. The prototype contains software that calculates the Q b  :Q k  ratio necessary to obtain the desired C r , and regulates the crystalloid solution pump and the oxygenated blood pump accordingly. The patient C s  and hematocrit were measured after the institution of bypass, but before the cardioplegic arrest, and the C s  and C k  values were input into the prototype software. The software, using equation (2) and an assumed patient hematocrit i=25, calculates the Q b  :Q k  ratio necessary to obtain the desired C r . After the desired Q r  is input into the software, the Q b  and Q k  can be calculated from the Q b  :Q k  ratio. In the variable potassium protocol, the C r  was initially set at 25 mEq/l with Q r  =300 ml/min, and C r  was decreased in increments of 3 mEq/l to 8 or 10 mEq/l. If cardiac activity resumed during the stepwise decrease in C r , the crystalloid concentration was increased to 15 mEq/l until activity stopped, and then incrementally decreased to 8-13 mEq/l. Once the lowest, steady state, heart-inactivating C r  was achieved, cardioplegia hematocrit and cardioplegia potassium concentration were measured at intervals of about 3 minutes (T 1 ), 33 minutes (T 2 ) and 63 minutes (T 3 ). 
     Table 2 presents general information about the patients in the Standard and Variable Groups. Table 3 presents C r  measured at established times T 1 , T 2  and T 3  after initiation of crystalloid infusion minus the predicted cardioplegia K+ concentration. 
     
                       TABLE 2______________________________________Parameter      Standard Gp Avr                      Variable Gp Avr______________________________________Age (yrs)      58.1        63.8Sex            2 f, 11 m   7 f, 10 mWeight (kg)    79.7        79.7Pre-Op systemic K+          3.97        3.92Operative initial          4.5         4.2systemic K+ (mEq/1)Operative systemic Hct (%)          23.4        25.1Total K+ dose (mEq)          73          61Operative Avr Systemic K+          6.1         5.3(mEq/1)Cardioplegia Hct (%)          19.5        24.5Crystalloid Vol (ml)          1700        152Total Cardioplegia          8500        6510Volume (ml).sup.1Cardioplegia O.sub.2 cap/min          6.4         9.6(cc O.sub.2 /min).sup.1______________________________________ .sup.1 Assumes 100% cardioplegia blood oxygen saturation. 
    
     
                       TABLE 3______________________________________Measured minus target C.sub.r                Average    AveragePat Id T.sub.1 T.sub.2   T.sub.3                          T.sub.2 &amp; T.sub.3.sup.2                                   T.sub.1, T.sub.2,______________________________________                                   T.sub.3STANDARD GROUP 3     7.8     4.3       NA    4.3      6.1 6     -3.6    1         1.1   1.1      -0.5 9     -2.5    4.2       1.8   3.0      1.212     9.8     4         NA    4.0      6.914     9.8     2         NA    2.0      5.915     13.1    2.7       NA    2.7      7.917     -1.7    3.2       0.3   1.8      0.618     7.6     2.3       1.6   2.0      3.819     7.7     9.9       -0.1  4.9      5.821     3.1     0.4       NA    0.4      1.823     -1      0.3       0.6   0.5      0.025     2.4     -2.8      NA    -2.8     -0.229     4.1     2.6       NA    2.6      3.4VARIABLE GROUP 1     1       2         0.5   1.3      1.2 2     0.6     NA        NA    NA       0.6 4     1.8     0         1     0.5      0.9 5     1.9     2.5       NA    2.5      2.2 7     1.6     -0.1      NA    -0.1     0.8 8     3.2     0.4       -0.4  0.0      1.110     0.9     2.1       1.7   1.9      1.611     2.5     1         1.8   1.4      1.813     2.3     1.5       NA    1.5      1.916     -2.1    -1.8      NA    -1.8     -2.020     0.6     1.5       0     0.8      0.722     3.9     1.1       NA    1.1      2.524     5.4     -0.1      0.8   0.4      2.026     -0.5    1.5       0.7   1.1      0.627     4.1     4         0.4   2.2      2.828     2.5     1.6       NA    1.6      2.130     .sup. NA.sup.1          NA        NA    NA       NA______________________________________ TOTAL AVERAGE K+ 2.18 STANDARD GROUP K+ AVR 3.27 VARIABLE GROUP K+ AVR 1.29 TTEST (not = var) 0.02 TTEST (= var) -0.04 FTEST 0.00 .sup.1 Data not available. .sup.2 Due to some variation in the actual time at which T.sub.1 values were taken, the T.sub.2 &amp; T.sub.3 average may be considered more reliable than the T.sub.1, T.sub.2 &amp; T.sub.3 average. 
    
     Table 3 indicates that the difference between the measured C r  and the expected cardioplegia K+ concentration was reduced in the Variable Group. Table 2 indicates that higher cardioplegia hematocrit, hemoglobin and oxygen content were also demonstrated in the Variable Group. The Variable Group&#39;s improved oxygen content, and higher hematocrit and hemoglobin reflect the reduced dilution achieved with the subject method. 
     From the foregoing, it will be appreciated that numerous variations and modifications may be effected without departing from the spirit and scope of the invention. The appended claims are intended to cover all such modifications and variations.