Regulation of Plasma Creatinine Concentration by Use of a Servo Control System

Abstract

A method for controlling the plasma creatinine concentration is described. This method uses continuous determination of the plasma creatinine concentration and a servomechanism which drives an infusion pump at a rate proportional to the difference between the actual and desired creatinine concentrations. The rate and volume of creatinine infusion necessary to maintain the desired plasma concentration provide information related to kidney function and the accumulation of creatinine in the body. Experiments using this technic and the method of data analysis are described. It has been shown that under certain conditions this technic may be used to estimate the renal clearance of creatinine and the size and nature of the space into which creatinine is distributed in the dog.

THIS PAPER DESCRIBES a servo control system for maintaining the plasma creatinine concentration at a constant (greater than normal) level. Measurement of the responses that were made by the control system to maintain the constant creatinine concentration provided information relevant to renal function and to the movement of creatinine within the body.

To achieve control, a method was devised for continuous measurement of the plasma creatinine concentration. A servomechanism was developed, which produced an error signal proportional to the difference between the desired creatinine concentration and the actual concentration. An infusion pump, driven at the speed determined by the error signal, discharged a creatinine solution directly into a vein. The rate of infusion necessary to maintain the selected creatinine concentration was used to estimate the volume of distribution and the rate of excretion of creatinine. With such a controlled system of infusion, the rate of infusion necessary to achieve an increased concentration will initially represent mainly the rate of blood dilution of the infused creatinine solution and the rate of movement of creatinine out of the vascular system into the extravascular fluids; these rates will decrease vith time as a steady state in the apparent volume of distribution of creatinine is approached. The rate of loss via renal excretion will not necessarily decrease with time at constant blood creatinine concentration. The rate of infusion of creatinine will, with time, approach the rate of excretion of creatinine. These data were compared with simultaneous determinations of creatinine and inulin urinary excretions.

Dominguez (1) and Sapirstein et al. (2) studied the disappearance of creatinine after a single intravenous injection. In such experiments, creatinine is either entering or leaving its volume of distribution. Only in the brief interval when vascular and extravascular concentrations are equal does net transfer cease. The loss from the vascular system via renal excretion can therefore he masked by the more rapid transfer into or out of body fluids.

One method of slowing the changes in blood concentration is to maintain a continuous infusion of the test substance after administration of the primary dose. Gaudino and Levitt (3) and Schwartz (4) described this method. If one is able to estimate the excretion rate and nearly balance it with constant infusion, then the blood concentration changes are slow, and the infusion rate can be used to predict the excretion rate. A bad guess ruins the determination, and any change in excretion rate would make the measurements invalid.

The regulating system described in this report partially circumvents these difficulties.

DeFares (5) recently reviewed the principles of feedback control as applied to a physiologic system. Kadish (6) described a controller in which blood glucose concentration is regulated by the infusion of glucose and insulin. The regulating system he described is of the discontinuous type—i.e., the change in manipulated variable (rate of glucose or insulin infusion) is a discontinuous function of the actuating signal.

Methods

Servo System

The control system described herein is of the continuous type. It is a nearly linear system in which the output (rate of creatinine infusion) is proportional to the actuating signal or error (desired creatinine concentration in plasma minus actual concentration). In the design of the control system an attempt was made to minimize steady-state errors and to obtain fast response. The unavoidable dead time of the system proved to be the most troublesome factor in the design. Specific steps were taken to improve the stability of the system and reduce the amplitude of the oscillations in creatininie concentration to tolerable limits.

Continuous Measurement of Creatinine

Continuous measurement of the plasma creatinine concentration was accomplished by adapting the method of Benedict and Behre (7) to the AutoAnalyzer (Fig. 1). The color reagent was prepared by adding 132 gm. of 3,5-dinitrobenzoic acid (Fisher A-114) to an aqueous solution of 54 gm. of Na₂CO₃ and 2 ml. of BRIJ-35 (Technicon Chemical Corp.). The volume was then adjusted to 4.0 L., and the pH was reduced to 7.0 ± 0.05 by adding concentrated HCl. It was necessary to warm and vigorously stir this solution to eliminate CO₂ before use. Aqueous creatinine standards in the range of 0–40 mg./100 ml. are useful in determining the relative sensitivity, base line, and drift of the AutoAnalyzer system. However, these aqueous standards do not give a valid calibration for determination of plasma creatinine concentrations in whole arterial blood. Blood is only about 85% H₂O. About one-third of the water is in the erythrocytes, and the creatinine is distributed in the water of the plasma and of the erythrocytes (8). Apparently, the permeability of the erythrocyte membrane to creatinine is so low that the creatinine does not have time to diffuse out completely during passage of the blood through the dialyzer. To obtain adequate calibrations, we drew arterial blood samples every 10 min. during the experiment. The samples were centrifuged, and plasma creatinine concentrations determined by a similar analytic procedure. A calibration curve was then constructed by plotting the measured plasma creatinine concentration against the simultaneous deflection of the recorder in the servo system sampling whole blood (Fig. 2).

Fig. 1.

Diagram of chemical method used for continuous measurement of creatinine concentration. Femoral artery cannula, 0.045 in. I.D.; femoral vein cannula, 0.10 in. I.D. These two cannulae are connected to form arteriovenous fistula. SS coil had a volume of 4 ml. Pump, dialyzer, colorimeter, and recorder are standard parts of AutoAnalyzer system.

Fig. 2.

Relationship of AutoAnalyzer recorder pen deflection to independently measured plasma creatinine concentration.

Adequate control of blood creatinine concentration required that the time delay involved in the measurement of the plasma creatinine concentration be minimized. Several features were incorporated to reduce this delay: (1) Sampling from an arteriovenous fistula eliminated delay of passage between the subject and the dialyzer. (2) Passing the blood directly through the dialyzer (bypassing the pump) eliminated the time delay in the pump. (3) Use of a high flow rate of the recipient stream reduced the time involved in moving the sample to the colorimeter. (4) Heating the color-development coil to 56° increased the rate of color development. (5) All tubing used in the transmission of the sample was cut as short as possible. As shown in Fig. 3, the AutoAnalyzer recorder pen began responding 1.1 min. after a sample was introduced at the sampling site and reached full response by 2.9 min. The shape of this curve was the same in response to step changes of different magnitude.

Fig. 3.

Response time of system to step change in creatinine concentration at input.

The AutoAnalyzer recorder used for the creatinine determination was unique in two respects. First, the pen deflection was linearly, rather than logarithmically, related to the concentration of creatinine. Second, a one-turn 5000-ohm potentiometer was mechanically linked to the recorder by connecting the potentiometer shaft to an extension of the wiper shaft of the recorder slide wire. This potentiometer provided the signal (input) to the servomechanism.

Controller

Diagrams of the servomechanism are presented in Fig. 4. Two controls are provided. First, the Span Switch 1 (Fig. 1) determines the range around the set point in which the servomechanism acts as a proportional controller. The range used was from 5 to 15% transmission on the recorder chart. If the recorder pen was above the upper limit of the range, the infusion pump was turned off; if it was below the lower limit, the pump was driven at its maximal rate. Second, the set point, Potentiometer 2 (Fig. 1), determines the approximate position on the AutoAnalyzer chart at which the plasma creatinine concentration will be maintained.

Fig. 4.

Top, block diagram of servo system. Bottom, circuit diagram of controller. Remote potentiometer was mechanically linked to recorder. #1, span switch; #2, set point potentiometer. The 1/20-hp motor drives infusion roller pump through a variable reduction gear.

Infusion System

The infusion solution was prepared by dissolving 10 or 15 gm. of creatinine and 234 gm. of mannitol in water and adjusting the volume to 2 L. The creatinine concentration of each infusion solution used was determined colorimetrically.

The volume of fluid infused was measured by a volume meter consisting of a vertical 3-ft. section of 0.6-in. (I.D.) plastic tubing with a Statham pressure transducer at the bottom. This column was refilled periodically from a stock bottle. The amplified output of the transducer was directly proportional to the height of the fluid in the column. The volume was photographically recorded on a Heiland oscillograph. The volume meter was calibrated by adding successive 20-ml. aliquots of infusion solution to the column. This volume meter was found to be stable and reproducible, and was read with an accuracy of 0.5 ml. in 150 ml. The rate of infusion was also read from the volume meter by measuring the slope of the recorded volume as a function of time.

System Stability

Several factors influenced the stability and accuracy of the system. Time lags in the AutoAnalyzer system led to overshoot and oscillation in creatinine concentration. Some degree of time lag was inescapable with this system, because the sensitivity of the colorimetric procedure depended upon adequate time for dialysis and color development. In addition, the time lag due to circulation within the dog was unavoidable.

The desired change in creatinine concentration was a step change. In order to mimic the step change as closely as possible, it was necessary to infuse creatinine at a very high rate initially. An infusion rate at least 50 times the excretion rate was desirable, but a 50-fold change in infusion rate was very difficult to achieve with good control by changing the motor armature current alone. Additional change in infusion rate was achieved by changing the gear ratio between the pump motor and the roller by a factor of eight in four steps during the experiment. Initially we used a span setting corresponding to a creatinine concentration of 5 mg./100 ml. By changing the gear ratio of the pump, it was possible to reduce the span to 1.5 mg./100 ml. without crossing the limits of the span and initiating large oscillations. This resulted in a decrease in the steady-state error of the system.

For given settings of the span and set point of the controller, gear-ratio setting of the roller pump, calibration of the AutoAnalyzer system, and concentration of the infusion solution, there was a fixed relationship between arterial creatinine concentration and creatinine infusion rate. A graph of this relationship is shown (Fig. 5) for data acquired at 20-sec. intervals during one experiment. The highest gear ratio setting was used at the beginning of the experiment when there was a high rate of diffusion of creatinine from blood into tissues. As this rate diminished, it was found convenient to reduce the gain of the servo system in order to prevent continued oscillation in the pump rate and concentration. This was achieved most simply by reducing the gear ratio of the roller pump, resulting in the three lower curves in Fig. 5. This figure may be considered to represent the transfer function of the portion of the servo system from the AutoAnalyzer recorder pen position (concentration) to the tip of the infusion catheter (infusion rate). The third dimension in the transfer function, time, may be ignored because the delay time for the pump rate to change from zero to maximum was less than 1 sec., which is small compared to the delays in the remainder of the closed-loop system. The concentration range over which this system operates under nearly proportional control is indicated by the shaded area in Fig. 5.

Fig. 5.

Steady-state relationship between creatinine concentration in plasma and infusion rate. Dog 1484.

The delays due to circulatory transport in the dog and in the Auto-Analyzer system are highly important and may lead to oscillation in the system. Reduction in the system gain (Δmg./min. ÷ Δ Cr_p, the slope of the relationship between infusion ratio and concentration) increases the system stability, but this is accomplished at the expense of an increase in steady-state error. The less the gain, the greater the effect of changes in renal excretion rate on the steady-state concentration. A practical reason for decreasing the system gain, via the gear ratio, was that the infusion rate required late in the experiment, when it represented primarily the renal excretion rate, was only 1–2% of that required initially when creatinine moved rapidly into tissues.

Indirect Control of Inulin Concentration

It is possible to use the servo system to control the blood concentration of a second substance indirectly if there is a fixed relationship between the excretion of the substance regulated by the servo system and the second substance. In four of the experiments reported, tritiated inulin (New England Nuclear Corp.) was added to the infusion solution. Because the excretion clearances of inulin and creatinine are nearly identical (9), the inulin concentration in the blood was regulated indirectly. Inulin excretion and plasma concentration were measured by liquid scintillation counting technics, so the same calculations could be made for inulin as for creatinine.

Preparation of Dogs

The servo system was used in experiments to measure the apparent volumes of distribution and renal clearances in nine mongrel dogs. The protocols for these experiments were similar. The dogs were weighed and anesthetized with 4% (w/v) pentobarbital (Nembutal). The right femoral vein was cannulated for the infusion of the creatinine solution, and the right femoral artery for monitoring the blood pressure and withdrawing arterial blood samples. The volume meter was calibrated. The dog's bladder was catheterized and emptied. The animal was heparinized, and then a left femoral artery-femoral vein fistula was connected to the dialyzer for continuous sampling. The set point and span of the servomechanism were set to the desired values and a timer was started. Starting at 5 mm., 6-ml. samples were drawn at 10-min. intervals from the right femoral artery. These were centrifuged and the plasma was used for calibrating the AutoAnalyzer. Every 10 min. the urine was removed from the bladder, and the bladder was washed with 30 ml. of 0.9% (w/v) saline. The wash solution and urine were mixed.

Technic of Data Analysis

The data were analyzed on an IBM 1620 computer. One program was written to convert the records to concentration and volume units. The chart deflection was read every 20 sec. directly from the AutoAnalyzer record. The calibration of the AutoAnalyzer was accomplished by pairing the plasma creatinine concentration measured directly in arterial samples with the AutoAnalyzer reading made 2 min. earlier (to account for the delay time of the AutoAnalyzer system and dog). The best calibration line was computed by the method of least squares. Correlation coefficients of 0.98 were routinely obtained for this calibration. From the calibration line, the creatinine concentration was computed for each 20 sec. The infusion volume record made during the experiment was also measured each 20 sec., and the best calibration line was computed by the method of least squares. The calibration graphs, as well as concentration and total volume infused as a function of time, were plotted by a CALCOMP on-line plotter.^*

A second program was written for the purpose of comparing the clearances calculated from the servomechanism infusion data with the clearances calculated in the classic method using the data from the urine collections. Clearances were calculated from the rate of infusion and arterial plasma concentration of creatinine according to the equation:

$$\text{Infusion clearance}=\frac{\text{infusate flow}(\mathrm{ml}.∕\min .)\times \text{infusate conc}.}{\text{plasma conc}.}$$

The Classic renal clearance equation is:

$$\text{Renal clearance}=\frac{\text{urine flow}(\mathrm{ml}.∕\min .)\times \text{urine conc}.}{\text{plasma conc}.}$$

The program computed the renal clearance value, using the average arterial plasma creatinine concentration obtained from the AutoAnalyzer for the time interval during which the urine was collected. The infusion clearance was computed each 20 sec., and then the values obtained within the interval of each urine collection were averaged for ease of comparison with the renal clearance data.

The total creatinine accumulation in the dog was calculated as the integral of the difference between infusion rate and urinary excretion rate (urine flow × concentration). The estimate of accumulation at the end of the experiment was checked against the difference between the total amount of creatinine infused (infusate concentration × volume removed from stock bottle) and the total excreted into the urine. The volume of distribution of the creatinine was computed by dividing the total ereatinine accumulated by the arterial plasma concentration.

Results and Discussion

When the controller is turned on, the system attempts to regulate plasma creatinine concentration at a new level, as shown in Fig. 6. After the initial overshoot, the system oscillations are rapidly damped. However, the system continued to oscillate with a period of about 5 min., which was approximately twice the dead time of the system including the dog. In spite of the continued oscillation, the mean concentration remained nearly constant.

Fig. 6.

Creatinine concentration in 2 dogs during servo control of infusion rate. Top, Dog 2374; bottom, Dog 1384.

The relationships among the plasma creatinine concentration, the total amount of creatinine infused, the total amount excreted, and the difference between amount infused and excreted (amount accumulated) are also shown in Fig. 6. The total amount of creatinine infused and the amount excreted were larger than the amount accumulated at the end of the experiment. This relationship tends to amplify any non-random errors in estimating the accumulation from the difference between infused and excreted amounts.

The ability to predict apparent glomerular filtration rate on the basis of clearance calculated from infusion rate is shown in Fig. 7. The infusion clearance of creatinine is higher than urinary creatinine clearance at the beginning of the experiment. When the body stores have become more nearly equilibrated at constant plasma creatinine concentration, the two clearances become similar.

Fig. 7.

Comparison of infusion and excretion clearances of creatinine with excretion clearance of inulin in two dogs. Top, Dog 23,74; bottom, Dog 13,84.

Information regarding the distribution of creatinine in the body can be obtained from a comparison of the apparent volume of distribution of creatinine and of inulin (Fig. 8).

Fig. 8.

Comparison of apparent volumes of distribution of creatinine (V_DCr) and inulin (V_DIn) expressed as fraction of body weight (F.B.W.) in 4 dogs. Numbers given in key are these assigned to dogs.

Comparison of creatinine and inulin data shows nearly identical clearances, but the kinetics of distribution and volume of distribution of creatinine and inulin are quite different (Fig. 7–9). It is apparent that creatinine has a much larger volume of distribution than does inulin.

Fig. 9.

Indirect servo control of inulin infusion in three dogs, showing total amounts of inulin infused, exereted, and accumulated; plasma concentration; and infusion (servo) and urinary clearances. Left, Dog 2374; center, Dog 1484; right, DOG 2084.

The last volume of distribution of creatinine determined on each of 8 dogs was less than expected (total body water), but was larger than the volume of distribution of inulin. It is possible that eventually creatinine would be distributed in a volume corresponding to total body water. Schloerb (10) concluded that creatinine administered to nephrectomized dogs is distributed in the same volume (58% of body weight) as is tritiated water, with an equilibrium time of about 4 hr. We did not continue the experiments until there was no further increase in the amount of creatinine accumulated. Another factor which would tend to make our results appear low is the interruption of the circulation to both hind limbs in these dogs. The near constancy of infusion rate and excretion rate at constant blood concentration over long periods indicates that the amount of creatinine being metabolically synthesized or destroyed is small compared to the amount of creatinine infused.

In these experiments no information was obtained regarding the identity of the apparent volumes of distribution. It is not justifiable to assume homogeneity of concentration and the occurrence of diffusion equilibria, or to characterize the anatomic representation of these volumes.

Other Applications of Feedback Control of Concentration

Our system can be described in terms of a closed loop: subject (dog) → detector (AutoAnalyzer) → controller (Thyratron circuit) → effector (infusion pump) → subject. Many variations on the theme are possible simply by changing the nature of the components.

One variation we have used is the infusion of a solution of p-aminohippurate (PAH) and the continuous estimation of its concentration in the dog's plasma by a modification (11) of the method of Bratton and Marshall in the AutoAnalyzer system. The method provides data on the tubular secretion of PAH and renal plasma flow. It requires the sequential addition of three reagents to the recipient stream from the dialyzer, with time allowed for nearly complete reaction with each. The resultant long delay in the AutoAnalyzer (more than 10 mm.) caused a delayed response in the controller and infusion rate and, therefore, resulted in instability of the whole closed-loop system and continuous oscillations in the plasma PAH level. The oscillation was less when the concentration of the infusion solution was reduced and when the dog was larger. Both of these circumstances effectively reduced the relative gain of the proportional controller.

In other similar experiments, the concentration of sodium sulfate infused into the dogs was measured by a modified turbidimetric procedure which had a time delay of about 1 min. With this short delay, the gain of the controller could be set quite high, and quick, accurate, stable control of the concentration resulted. Plasma sulfate concentrations were controlled at 10 mM, and it was possible to estimate the glomerular filtration rate and the volume of distribution of sulfate (which may be considered as extracellular fluid volume).

More recent experiments (12) have used a similar control system in the investigation of the effect of concentration of indocyanine green on its hepatic clearance in humans. The dye was infused into the venous circulation at a variable rate, and the light absorption of the blood was detected by a compensated dichromatic earpiece densitometer whose output voltage was proportional to dye concentration. Use of a combination of derivative, integral, and proportional controls (using analog computer circuits) to govern the infusion rate resulted in rapid (1½–3 min.) achievement of desired concentration values and nonoscillating levels. By using this system, the hepatic clearance of indocyanine green (which is excreted solely via the liver) could be estimated about 5 min. after the desired concentration level was attained. This time lag was required for mixing within the blood volume which, besides the liver, is the volume of distribution of this indicator. With this method, an estimate of hepatic clearance at a different concentration level could be obtained every 8–10 min. The integral control allowed precise setting of the concentration independently of the excretion rate. This is an improvement over purely proportional control under which the concentration will always decrease when clearance increases, and vice versa.