The Development of a Glucose Clamp K. H. Norwich, G. Fluker, J. Anthony, I. Popescu, B. Pagurek, and G. Hetenyi, Jr. A reliable control system was developed for clamping (i.e., holding at a steady level) the concentration of blood glucose in the hyperglycemic region of a normal nonanesthetized dog. The system is based upon, and hence largely validates, a mathematical model of the canine glucoregulatory mechanism which was assembled from the results of experiments
in which radioisotopes were used. The op eration of the clamp, however, does not require the use of radioisotopes. The glucose clamp is a tool which may be applicable clinically in the measurement of glucose turnover and in the laboratory in the study of the effects of glucose turnover of variable insulin with constant glucose levels.
D
EFINITION. The term glucose clamp analogous to the term voltage clamp has been suggested’ for the system which can maintain the plasma glucose concentration of an intact animal at any specified level by means of the regulated automatic intravenous infusion of appropriate biochemical substances. For example, an investigator may wish to raise the plasma glucose level of a subject from the fasting value of 1.O mgjml to a new constant level of 1.4 mg/ml. A proficient clamping system should be capable of infusing glucose intravenously at such rates that plasma glucose concentration is raised to and held at the value 1.4 mg/ml without manual assistance. We have designated the fasting plasma glucose by C, and the desired new plasma concentration by C, (“set-point”). The completed clamping system should be capable of achieving any desired value of C,, whether in the hypo- or hyperglycemic region. The system to be described here is designed only to attain C, values in the hyperglycemic range. Purposes
The experiments were designed with three purposes in mind. The initial aim was to validate in the dog a mathematical model of glucose kinetics which employs the functional dependence of rates of production and utilization of glucose upon the plasma glucose concentration.2*3 A valid model will have predictive value, and this predictive value should be applicable in the design of regulators such as the glucose clamp. If the model is invalid, the clamping system based upon it is very likely to fail. The second purpose was to develop a tool which is applicable in the study of metabolism. The glucoregulatory system of an animal is simplified when the glucose concentration in bood is clamped, just as the neuron membrane system
From the Department of Physiology and Institute of Biomedical Engineering University of Toronto, the Department of Physiology and Computing Center, University of Ottawa, and the Department of Systems Engineering, Carleton University, Ottawa. Receivedforpublication December 9. 1974. Supported by the Medical and National Research Councils of Canada. Reprint requests should be addressed to Dr. G. Hetenyi. Jr.. Chairman, Department of Physiology. Faculty of Medicine, University of Ottawa. 275 Nicholas Street. Ottawa, Ontario, Canada KIN 6N5. 0 1975 by Grune & Stratton, Inc.
Metabolism, Vol. 24, No. 11 (November), 1975
1221
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ET AL.
is simplified when the transmembrane potential is held constant. Thus, for example, it should be possible to study in the intact animal the effect of insulin on the rate of utilization of glucose while holding the concentration of glucose in the plasma constant. Finally, since the system works (i.e., the glucose level is clamped) only if the correct rates of utilization of glucose are discovered, it may potentially be applied in the clinic to estimate the rate of utilization of glucose in patients without the use of radioisotopes. THEORY
The essence of the theory is quite simple and may be summarized in six brief paragraphs as follows. 1. The change in the size of the glucose pool of the fasted animal is the sum of the rate of endogenous glucose release, R,(C), and the rate of exogenous infusion, Rinf, minus the rate of peripheral utilization, R,,(C). Expressed mathematically, {change in the amount
of circulating
glucose] = R,(C)
+ Rinl - Rd(C) [mg.min-‘1
(1)
where the rates of release and utilization are shown as functions of the plasma glucose concentration, C. 2. The rate of change of plasma glucose concentration with time may be estimated by dividing the change in the amount of circulating glucose by the effective distribution space of glucose, V (the dilution principle), or mathematically, dC -=;[h dt
c an g e in the amount
of circulating
glucose]
[mg.ml-‘.min-‘1.
(2)
3. The exact manner in which the rates of endogenous glucose release, R,(C), and peripheral utilization, Rd(C), depend upon plasma glucose level, C, can be obtained from a previous set of experiments performed on animals of the same species using radioisotopes. These two functions are known in algebraic form for the dog, but R, depends upon two constants, (r and /3, which must be evaluated in each animal since they may vary over a range in the species2 Then combining equations (1) and (2) dC ai
= ;
[R,(C)
+ Rinf - R,,(C)1
(3)
where the value of V is approximately 20% of the total body volume, and R,,(C) may be obtained when the two constants have been evaluated. Equation (3) thus defines C as a function of time, t, and of the two parameters (Yand ,f?. 4. Suppose that an intravenous glucose infusion is given at any nominal rate, R&t) [mg.min-‘I. The plasma glucose concentration, C, will change with time. We then ask the question: What values of LYand @ wili permit equation (3) to generate the observed concentration-time curve? The computer is capable of answering this question by searching for the “best” values for cy and p. When these two parameters have been assigned numerical values, equation (3) will then define C as a function oft only. 5. Now the idea behind the (hyperglycemic) clamp is the find the “optimal”
1223
GLUCOSE CLAMP
rate of glucose infusion, Ri,r, such that the concentration, C, will attain and hold the value prescribed for the set-point, C,. Since we desire a new steady state at C = C,, we then require the condition,
Making this substitution in (3), we can then solve for the desired rate of infusion, (5) The infusion pump is then set at this value Of Ri"r. 6. Setting the infusion pump on a single occasion at the rates suggested by equation (5) is not expected to set the plasma glucose concentration into a smooth glide for C, because the values of aand /3 drift slightly with time. Therefore, the curve-fitting process-the matching of C from equation (3) to the observed concentration-time curve-must be repeated frequently to update the We chose to evaluate LYand p over a sliding values of LYand /3 and hence of Ri,r. 35-min window. That is, the first evaluation used the C-versus-f curve between the times 0 and 35 min, and so the first estimate of Rinf came for t = 35 min. The second evaluation used the C-versus-r curve between the times 7 and 42 min, and so the second estimate of Rinf came for t = 42 min, etc. In this way, the plasma glucose level was guided to C,. The mathematical development is given in more detail in the Appendix. One can now appreciate an overview of the clamping procedure. Blood glucose is measured continuously by an automatic process (Auto Analyzer). At zero time, a glucose infusion is begun at a more or less arbitrary rate. The continuous concentration-time curve is sampled every 7 min, and the value of blood glucose concentration is relayed to a remote digital computer (IBM System 360). Computation begins 35 min after the start of the glucose infusion. Following the receipt of every blood glucose value (t >_35 min), the computer fits a mathematical function very much like that of equation (3) (actually equation (A4)) to the observed C-versus-t curve over a 35-min window and estimates the best values for (Yand fi using a least squares procedure. A best value for Rinf is then calculated using equation (5). The computer’s estimate of Rinl is then relayed back to the laboratory, where the glucose infusion pump is set to this value. This process continues for the duration of the experiment. The entire process may be carried out automatically. The mathematical details are given in the Appendix. MATERIALS
AND METHODS
Eight experiments were carried out on four mongrel dogs weighing between 9.0 and 19.4 kg. The animals were trained to stand quietly in Pavlov harness and were fasted for 14 hr prior to an experiment. On the day of the experiment, polyethylene cannulae were inserted under local anesthesia into one cephalic and one saphenous vein. The cephalic cannula was connected to a syringe containing an aqueous solution of glucose which was mounted on a Sage infusion apparatus. The cannula inserted into the saphenous vein had two lumina and consisted of an inner tubing of a small diameter (PE 50 Becton-Dickinson) and an outer larger one (PE 205 Becton-Dickinson). The tip of the inner tubing lay by 0.5 cm beyond the opening of the outer (larger) one. A solution of heparin was infused through the larger tubing at a rate of 0.025
1224
NORWICH
ET AL.
ml/min, corresponding to 8.3 IJ/min during the first hour of the experiment and 2.5 U/min later. Blood was withdrawn through the inner tubing at a rate of 0.051 ml/min into the Auto Analyzer. The construction of the cannula permitted sampling of the blood at selected times, This cannula was advanced via the saphenous vein until it lay in the iliac vein. After the cannulation was completed, the dog was placed in the harness and allowed about half an hour before the start of the experiment. The Auto Analyzer was set to measure blood glucose concentration. In order to estimate plasma glucose concentration, we made the assumption that plasma
glucose
concentration
= m.(blood
glucose
concentration)
(6)
where m is a constant determined by taking blood samples at two different times when 0 < t < 25 min and plotting plasma glucose concentration (obtained by the glucose oxidase method using a Beckman Glucose Analyzer) against blood glucose concentration (obtained from the Auto Analyzer). This conversion of blood to plasma glucose was performed by the computer. The time delay from the instant a change in blood glucose was induced until the time it was recorded on the Auto Analyzer’s chart recorder was about 7 min. The blood glucose levels measured by the Auto Analyzer were transmitted acoustically to an IBM 360 located in another building. There the data were analyzed in the manner described above, and the recommended value of Rinf was transmitted back to the laboratory. An analogto-digital converter was available to set the Sage infusor at the required rate of infusion of glucose. Thus, the feedback loop was completed.4 Although the equipment available was quite capable of carrying out this whole process automatically (and was tested on several occasions), the experimental results to be reported below were obtained using a human link for two parts of the chain. Bather than relaying the Auto Analyzer’s output automatically to the computer, we relayed the output manually using a MUMAC keyboard. And rather than relying on the analog-to-digital converter to set the Sage infusor directly, the infusor was set manually. The reason for these manual interruptions was essentially conservation of time and money: it was not worthwhile losing a costly experiment to a trivial mechanical or electrical failure. We were, therefore, content to demonstrate that the clamp can work completely automatically, but we operated instead in a semiautomatic mode. While our experiments were controlled using a specially-written program for parameter estimation and the IBM system 360. they could certainly have been performed on any digital computer, given access to a library tape “hill-climbing” subprogram, of which they are many. One need only estimate the 3 parameters a, @. and D contained in Appendix equations (A2) and (A4).
RESULTS
The course of a typical experiment is shown on Fig. 1. Figure 2 shows the “running mean” of C plotted against time. All results are summarized in Table 1. As evident from the Table, in all the experiments the concentration of plasma glucose attained was within f 5.0 mg/dl of the set-point designed. In all but one experiment, the variation of C around the set-point was small, as indicated by the standard deviations of the readings of C taken. In no experiment was the regression of the C-versus-t line significantly different from zero. The time necessary to reach the set-point was about 1 hr in all experiments. Once the set-point had been reached, it was kept for another 60-100 min. During this period, the rate of glucose infusion, equal or nearly equal the overall rate of glucose utilization, varied as shown on Fig. 1. In one dog (experiment 16), after the set-point had been reached and held for 76 min, the concentration of plasma glucose inexplicably started to decrease and reached 0.97 mg/ml at t = 168 min. Continuing the experiment, however, the set-point was re-established by t = 203 min. One experiment, number 15,
GLUCOSE CLAMP
1225
ii
1.6
2 365
1.4
;
15 .
:
1.3
U-J
1.2
5
11
110
g 3
c
100
; 'i -.& ZE
90
g
70
2 d
603
80
0
7
21
35
A9
63
77
TIME
91
105
119
133
(min)
Fig. 1. The glucose clamp, experiment 11. Abscissa: time in minutes. lower panel: rate of glucose infusion (mg/min); upper panel: concentration of glucose in plasma (ma/ml). A steady glucose infusion was begun at t = 0; the computer assumed control at t = 49 min. It should be noted that the computer program is not written in such a way that the infusion rate will respond to each individual change in plasma glucose, but rather a smooth curve is fitted through the past six points. In this way, future behavior is predicted. For example, a potential overshoot of glucose (which occurred at 91 min) was foreseen by the computer at 70 min, and the infusion rate was decremented in anticipation.
SET POINT
-i cn
; 2 z
= 1.4
mg/ml
0.9 4 0
I 21
I 42
I 63 TIME
I a4 (min)
I 105
I 126
I 149
Fig. 2. The “running mean” plasma glucose concentration plotted against time for the same experiment shown in Fig. 1. Each value of the ordinate is the arithmetic mean of all preceding plasma glucose values. This reflects the ultimate goal of the exercise: to hold the plasma glucose concentration at a new mean level of 1.4 mg/ml.
NORWICH
1226
ET AL
Table 1. Summay of Eight Experiments. the Attained Concantmtion of Glucose in Plasma, 2, was Calculated a, the Mean SuccessiveReadings Made at ‘I-min lntervalr Between the Time limits in Column 2
concentration Time Limits (min)of Set-Point ExperimentSet-PointRecorded mg/dl
Attained (c) w/df
SD *
Regression Coeficient(b) (Cat) x 10-4
7
63-161
140
141
6.6
-5.0
10
63-147
140
145
8.5
-15.8
11
56133
140
142
6.6
-6.8
12
84-154
140
135
5.7
+9.5
13
70-147
120
119
6.5
-2.6
14
70-126
140
135
4.0
+ 1.9
16
70-147
140
139
18.9
-24.5
17
63-161
140
139
11.3
+4.5
Remarks “Window”
= 50 min
Anesthetized. Set-point lost after t = 126 min. Set-point lost after t = 140 min but regained by t = 210 min.
had to be abandoned because of the breakdown ment is not shown on the Table.
of the computer.
This experi-
DISCUSSION
The concept of raising and holding constant the concentration of a substance in the blood or plasma of an intact organism is not new. For example Harvey et al.5 constructed a servomechanism for producing an constant plasma creatinine level. Norwich6 discussed the applications of holding a constant level of indicator in the blood as a means of measuring pulsatile blood flow. In the area of glucose metabolism, Kadish,’ and later Kline et al.,’ demonstrated how glucose concentration could be held automatically at the basal level in the presence of greater-than-basal levels of insulin. Pagurek et a1.9 designed a system which very successfully clamped the glucose level of human subjects in the hyperglycemic range. Insel et al.’ utilized a glucose clamp in the formulation of a model of insulin-glucose interaction. All of the above papers have one element in common: they utilize a mathematical model of the physiologic system, and they are designed to achieve control within a period of minutes. The introduction of a simple control algorithm which gives the optimal glucose infusion rate as a function of the displacement, integral, and derivative of the blood glucose concentration (P.1.D algorithm) is very appealing. We have not been completely successful with the use of such an algorithm, but if the delay in measurement of blood glucose can be reduced, this method may well become the method of choice. The challenge in designing a glucose clamp is largely one of avoiding uncontrolled oscillations in the concentration of blood glucose. The larger the delay between sampling blood and recording the concentration, the greater the probability of oscillation and loss of control. As faster methods of measuring blood glucose become available,‘“~” this problem is expected to lessen. Even if measurement of blood glucose level could be made nearly instantaneously, the problem of delay would not be totally overcome. There is a physiologic delay during which newly infused glucose disperses throughout its ultimate volume of distribution. This physiologic delay has been approximated by the term “D” in Appendix equation (A4). Thus, some mod-
GLUCOSE
1227
CLAMP
elling will probably be necessary for control of glucose levels even with the ultimate glucose monitor. The mathematical model used in our control system is not completely valid during the entire duration of the experiment. During the first 30-40 min of a glucose infusion, the presence of an additional transient element is clearly indicated.2*‘2 We have deliberately ignored this transient in the computer program used in the experiments reported here. Thus, equation (5) cannot really be expected to give an exact estimate of the desired infusion rate, Rinf, until the lower end of a window passes t + 35 min-i.e., until about 35 min after the beginning of a glucose infusion. It should be observed that the glucose infusion rate which we sought was not the rate which would be expected to achieve and hold the set-point in minimum time, but rather the best constant infusion rate. For a truly optimal infusion rate, we should have to consider, for example, the optimization principle of Pontriagin. l3 The optimal infusion would probably involve two components: first an infusion rate greater than the final maintenance rate, and then the maintenance rate. The potential applications of the glucose clamp are intriguing. In the clinic it might be used as a means of estimating the steady state turnover rate of glucose without the use of radioisotopes-i.e., estimating the rate of disappearance, R,, of glucose at steady state. Acting upon the assumption that the human glucoregulator does not differ in principle from the canine, the procedure might be implemented in the following ways. Cannulae can be inserted into two veins of a fasted patient in much the same manner as done in the dog, and a glucose clamp applied at some arbitrary hyperglycemic level below the renal threshold, e.g., at 1.5 mg/ml. In the hyperglycemic range, the endogenous glucose output is largely suppressed, so that in Appendix equation (A4) the value of R, is substantially smaller than the value of R, or Rinc and therefore does not influence dC
the rate of change of C to any degree. Hence, the derivative, dt,
is determined
largely by a balancing of the input rate Rinl with the output rate RI(C,a,/3). Achieving the glucose clamp at the desired level must then almost certainly imply correct evaluation of the constants LYand fi in the only unknown function, RJC,a,P). The rate of disappearance at the fasting level, C,, which is equal to the basal or fasting rate of appearance, is then given by R,(C,,a$). This function is obtained explicitly from Appendix equation (A2): R,(Cor~,P)
= a + K,
(7)
A critical matter in assessing the confidence to be placed in the calculation of of hepatic glucose output when blood sugar is elevated. If hepatic output were not suppressed, then one might suspect that the glucose clamp had been attained by virtue of compensatory errors: e.g., overestimating both R,(C) and RI(C) in such a way that their difference gave a correct value. In this case, basal R, as given by equation (7) would also be in error. But if endogenous hepatic output is largely suppressed in hyperglycemic in the human being, *4~‘s then we may place a good deal of confidence in the uniqueness and correctness of the result given by equation (7). R, by this method is the extent of suppression
NORWICH
1228
ET AL.
APPENDIX MATHEMATICAL
DEVELOPMENT
Let R, be the rate of endogenous appearance or entry of glucose and Rd the rate of disappearance or utilization. Let C be the plasma glucose concentration. It was shown by analysis of a number of experiments in dogs’ that at steady state, R, = R~(0)e-MC-Co) and Rd = a + PC
(AZ)
where C is the steady state plasma glucose concentration, Co is the fasting plasma glucose concentration and (Y, ,8, and X are constants. During the early minutes following the termination of a fast by a steady intravenous glucose infusion, the rate of change of R, with time is a function of an additional parameter /1.3 But after about 40-min infusion, the value of R, at any time is given very nearly by its steady state value from (A2). R, at any time may be estimated from (A 1) with X = 3.3 ml/mg without introducing serious error, primarily because the ratio of R, to R, is small in hyperglycemia. If the rate of intravenous infusion is given by Rinf, then, to a first approximation, dC dt=
’ [Ra + Rinf I/
RdI
(A3) (text Eq (3))
where V is the total distribution volume of glucose. However, newly infused or released glucose does not disperse instantly throughout all of V, and so the finite dispersion time is allowed for by modifying (A3), dC dt=
’ v
(R, + Rinl)
where D > 0 is a dispersion
l
1 + D ~
- Rd
(A4)
I coefficient
in units
of [time].
d(lnC) As -d7
-
0, Eq
(A3) comes to approximate Eq (A4) quite closely. Equations (A 1) and (A2) can now be introduced into (A4) to give, finally, a nonlinear differential equation in C and t. The clamping experiment is always begun by giving a glucose infusion of arbitrary magnitude* in order to displace the glucose system from steady state. Equation (A4) was then solved by finite element methods (Euler’s method), *Actually using statistics compiled from a number of earlier experiments, we were able to make an educated guess at the magnitude of the optimal infusion rate, and we often adopted this rate for the initial infusion. The value of @ (ml/min) is of the order of magnitude of 10 times total body weight (kg); Rd(CO) + 2.8 times total body weight (kg). Substituting the above estimates for fi and R&CO) into Eq (A2) gives an initial estimate for a. Finally, applying equations (5). (Al), and (A2) with the estimates for a and @ and C equal to C, gives the initial estimate for Ri,f, the rate of infusion of glucose.
GLUCOSE CLAMP
1 + D ‘9
AC(t) = $
)
1
- Rd At.
C(0) or in general, C(t,,,), was always taken as equal to the experimental at that time. (Y,j3, and D were selected so that,
value
window
was minimized. The optimization of the three parameters was carried out on an IBM S 360 using a searching or “hill-climbing” procedure similar to that suggested by Hazelrig et al. I6 The data points were weighted by their ordinal number: the second point in the window was given weight 2, the third weight 3 I--., and the nth, weight n. The more recent the data point, the greater its weight. Once (Yand /3 had been determined, Rinf (the constant rate of infusion necessary to achieve and hold the set-point, C,) was calculated from text Eq (5), Ri”r = Rd(Cs) - R,(Cs) with R, and Rd evaluated from (A 1) to (A2). The fitted value of D was always close to 10 min, and the values of (Yand B were nearly always in the range found in the radioisotope studies.* ACKNOWLEDGMENT The impetus for our development of this system followed several discussions with Dr. P. A. Inset and Dr. K. J. Kramer who informed us of the successful results of Drs. J. D. Tobin and R. Andres in experiments of this type. We are most grateful to these investigators and acknowledge their fundamental studies. We are also indebted to Dr. J. B. R. McKendry for helpful discussions and for the loan of the Auto Analyzer. REFERENCES 1. Insel PA, Kramer KJ, Sherwin RS, Liljenquist JE, Tobin JD, Andres R, Berman M: Modeling the insulin-glucose system in man. Fed Proc 33:1865-1868.1974 2. Hetenyi G Jr, Norwich KH, Zelin S: Analysis of the glucoregulatory system in dogs. Am J Physiol224635642, 1973 3. Zehn S, Norwich KH, Hetenyi G Jr: The glucose control mechanism viewed as a regulator. Med Biol Eng (in press) 4. Norwich KH, Hetenyi G Jr, Fluker G, Pagurek B: Development of the glucose clamp. Digest, Fifth Canadian Medical and Biological Engineering Conference, Montreal, 1.3a, 1974 5. Harvey RB, Bassingthwaighte JB, Heppner RL: Regulation of plasma creatinine concentration by use of a servo control system. Clin Chem 14944-959, 1968 6. Norwich KH: Determination of pulsatile blood flow by indicator-dilution methods. J Theor Biol50:353-361, 1975
7. Kadish AH: Automation control of blood sugar. Am J Med Elec 3:82-86, 1964 8. Kline NS, Shimano E, Stearns H, McWilliams C, Kohn M, Blair JH: Technique for automatic in vivo regulation of blood sugar. Med Res Eng 14-19, 1968 9. Pagurek B, Riordon JS, Mahmoud S: Adaptive control of the human glucose regulatory system. Med Biol Eng 10:752-761, 1972 10. Chang KW, Aisenberg S, Soeldner JS, Hiebert JM: Validation and bioengineering of an implantabie glucose sensor. Tram Am Sot Int Organs 19:352-360, 1973 11. Bessman .SP, Schultz RD: Progress toward a glucose sensor for the artificial pancreas. Digest of the Tenth International Conference on Medical and Biological Engineering 34-IO,1973 12. Moorhouse JA, Smithen CS, Houston ES: A study in glucose transport kinetics in man
1230 by means of continuous glucose infusion. J Clin Endocrinol Meta 27:256-264, 1967 13. Pontriagin LS, Boltyanskii VG, Gamkrelidze RV, Mishchenko EF: The Mathematical Theory of Optimal Processes. New York, Interscience, 1962 14. Franckson JRM, Ooms HA, Bellens R, Conard V, Bastenie PA: Physiologic signihcance of the intravenous glucose tolerance test. Metabolism 1I :482-500, I %2
NORWICH ET AL. 15. Forbath NF, Hetenyi G Jr: Glucose dynamics in normal subjects and diabetic patients before and after a glucose load. Diabetes 15:778-789,1966 16. Hazelrig JB. Ackerman E, Rosevear JW: An iterative technique for conforming mathematical models to biomedical data. Proceedings of the Sixteenth Annual Conference on Medical and Biological Engineering, Baltimore, 5:8-9, 1963
in which radioisotopes were used. The op eration of the clamp, however, does not require the use of radioisotopes. The glucose clamp is a tool which may be applicable clinically in the measurement of glucose turnover and in the laboratory in the study of the effects of glucose turnover of variable insulin with constant glucose levels.
D
EFINITION. The term glucose clamp analogous to the term voltage clamp has been suggested’ for the system which can maintain the plasma glucose concentration of an intact animal at any specified level by means of the regulated automatic intravenous infusion of appropriate biochemical substances. For example, an investigator may wish to raise the plasma glucose level of a subject from the fasting value of 1.O mgjml to a new constant level of 1.4 mg/ml. A proficient clamping system should be capable of infusing glucose intravenously at such rates that plasma glucose concentration is raised to and held at the value 1.4 mg/ml without manual assistance. We have designated the fasting plasma glucose by C, and the desired new plasma concentration by C, (“set-point”). The completed clamping system should be capable of achieving any desired value of C,, whether in the hypo- or hyperglycemic region. The system to be described here is designed only to attain C, values in the hyperglycemic range. Purposes
The experiments were designed with three purposes in mind. The initial aim was to validate in the dog a mathematical model of glucose kinetics which employs the functional dependence of rates of production and utilization of glucose upon the plasma glucose concentration.2*3 A valid model will have predictive value, and this predictive value should be applicable in the design of regulators such as the glucose clamp. If the model is invalid, the clamping system based upon it is very likely to fail. The second purpose was to develop a tool which is applicable in the study of metabolism. The glucoregulatory system of an animal is simplified when the glucose concentration in bood is clamped, just as the neuron membrane system
From the Department of Physiology and Institute of Biomedical Engineering University of Toronto, the Department of Physiology and Computing Center, University of Ottawa, and the Department of Systems Engineering, Carleton University, Ottawa. Receivedforpublication December 9. 1974. Supported by the Medical and National Research Councils of Canada. Reprint requests should be addressed to Dr. G. Hetenyi. Jr.. Chairman, Department of Physiology. Faculty of Medicine, University of Ottawa. 275 Nicholas Street. Ottawa, Ontario, Canada KIN 6N5. 0 1975 by Grune & Stratton, Inc.
Metabolism, Vol. 24, No. 11 (November), 1975
1221
1222
NORWICH
ET AL.
is simplified when the transmembrane potential is held constant. Thus, for example, it should be possible to study in the intact animal the effect of insulin on the rate of utilization of glucose while holding the concentration of glucose in the plasma constant. Finally, since the system works (i.e., the glucose level is clamped) only if the correct rates of utilization of glucose are discovered, it may potentially be applied in the clinic to estimate the rate of utilization of glucose in patients without the use of radioisotopes. THEORY
The essence of the theory is quite simple and may be summarized in six brief paragraphs as follows. 1. The change in the size of the glucose pool of the fasted animal is the sum of the rate of endogenous glucose release, R,(C), and the rate of exogenous infusion, Rinf, minus the rate of peripheral utilization, R,,(C). Expressed mathematically, {change in the amount
of circulating
glucose] = R,(C)
+ Rinl - Rd(C) [mg.min-‘1
(1)
where the rates of release and utilization are shown as functions of the plasma glucose concentration, C. 2. The rate of change of plasma glucose concentration with time may be estimated by dividing the change in the amount of circulating glucose by the effective distribution space of glucose, V (the dilution principle), or mathematically, dC -=;[h dt
c an g e in the amount
of circulating
glucose]
[mg.ml-‘.min-‘1.
(2)
3. The exact manner in which the rates of endogenous glucose release, R,(C), and peripheral utilization, Rd(C), depend upon plasma glucose level, C, can be obtained from a previous set of experiments performed on animals of the same species using radioisotopes. These two functions are known in algebraic form for the dog, but R, depends upon two constants, (r and /3, which must be evaluated in each animal since they may vary over a range in the species2 Then combining equations (1) and (2) dC ai
= ;
[R,(C)
+ Rinf - R,,(C)1
(3)
where the value of V is approximately 20% of the total body volume, and R,,(C) may be obtained when the two constants have been evaluated. Equation (3) thus defines C as a function of time, t, and of the two parameters (Yand ,f?. 4. Suppose that an intravenous glucose infusion is given at any nominal rate, R&t) [mg.min-‘I. The plasma glucose concentration, C, will change with time. We then ask the question: What values of LYand @ wili permit equation (3) to generate the observed concentration-time curve? The computer is capable of answering this question by searching for the “best” values for cy and p. When these two parameters have been assigned numerical values, equation (3) will then define C as a function oft only. 5. Now the idea behind the (hyperglycemic) clamp is the find the “optimal”
1223
GLUCOSE CLAMP
rate of glucose infusion, Ri,r, such that the concentration, C, will attain and hold the value prescribed for the set-point, C,. Since we desire a new steady state at C = C,, we then require the condition,
Making this substitution in (3), we can then solve for the desired rate of infusion, (5) The infusion pump is then set at this value Of Ri"r. 6. Setting the infusion pump on a single occasion at the rates suggested by equation (5) is not expected to set the plasma glucose concentration into a smooth glide for C, because the values of aand /3 drift slightly with time. Therefore, the curve-fitting process-the matching of C from equation (3) to the observed concentration-time curve-must be repeated frequently to update the We chose to evaluate LYand p over a sliding values of LYand /3 and hence of Ri,r. 35-min window. That is, the first evaluation used the C-versus-f curve between the times 0 and 35 min, and so the first estimate of Rinf came for t = 35 min. The second evaluation used the C-versus-r curve between the times 7 and 42 min, and so the second estimate of Rinf came for t = 42 min, etc. In this way, the plasma glucose level was guided to C,. The mathematical development is given in more detail in the Appendix. One can now appreciate an overview of the clamping procedure. Blood glucose is measured continuously by an automatic process (Auto Analyzer). At zero time, a glucose infusion is begun at a more or less arbitrary rate. The continuous concentration-time curve is sampled every 7 min, and the value of blood glucose concentration is relayed to a remote digital computer (IBM System 360). Computation begins 35 min after the start of the glucose infusion. Following the receipt of every blood glucose value (t >_35 min), the computer fits a mathematical function very much like that of equation (3) (actually equation (A4)) to the observed C-versus-t curve over a 35-min window and estimates the best values for (Yand fi using a least squares procedure. A best value for Rinf is then calculated using equation (5). The computer’s estimate of Rinl is then relayed back to the laboratory, where the glucose infusion pump is set to this value. This process continues for the duration of the experiment. The entire process may be carried out automatically. The mathematical details are given in the Appendix. MATERIALS
AND METHODS
Eight experiments were carried out on four mongrel dogs weighing between 9.0 and 19.4 kg. The animals were trained to stand quietly in Pavlov harness and were fasted for 14 hr prior to an experiment. On the day of the experiment, polyethylene cannulae were inserted under local anesthesia into one cephalic and one saphenous vein. The cephalic cannula was connected to a syringe containing an aqueous solution of glucose which was mounted on a Sage infusion apparatus. The cannula inserted into the saphenous vein had two lumina and consisted of an inner tubing of a small diameter (PE 50 Becton-Dickinson) and an outer larger one (PE 205 Becton-Dickinson). The tip of the inner tubing lay by 0.5 cm beyond the opening of the outer (larger) one. A solution of heparin was infused through the larger tubing at a rate of 0.025
1224
NORWICH
ET AL.
ml/min, corresponding to 8.3 IJ/min during the first hour of the experiment and 2.5 U/min later. Blood was withdrawn through the inner tubing at a rate of 0.051 ml/min into the Auto Analyzer. The construction of the cannula permitted sampling of the blood at selected times, This cannula was advanced via the saphenous vein until it lay in the iliac vein. After the cannulation was completed, the dog was placed in the harness and allowed about half an hour before the start of the experiment. The Auto Analyzer was set to measure blood glucose concentration. In order to estimate plasma glucose concentration, we made the assumption that plasma
glucose
concentration
= m.(blood
glucose
concentration)
(6)
where m is a constant determined by taking blood samples at two different times when 0 < t < 25 min and plotting plasma glucose concentration (obtained by the glucose oxidase method using a Beckman Glucose Analyzer) against blood glucose concentration (obtained from the Auto Analyzer). This conversion of blood to plasma glucose was performed by the computer. The time delay from the instant a change in blood glucose was induced until the time it was recorded on the Auto Analyzer’s chart recorder was about 7 min. The blood glucose levels measured by the Auto Analyzer were transmitted acoustically to an IBM 360 located in another building. There the data were analyzed in the manner described above, and the recommended value of Rinf was transmitted back to the laboratory. An analogto-digital converter was available to set the Sage infusor at the required rate of infusion of glucose. Thus, the feedback loop was completed.4 Although the equipment available was quite capable of carrying out this whole process automatically (and was tested on several occasions), the experimental results to be reported below were obtained using a human link for two parts of the chain. Bather than relaying the Auto Analyzer’s output automatically to the computer, we relayed the output manually using a MUMAC keyboard. And rather than relying on the analog-to-digital converter to set the Sage infusor directly, the infusor was set manually. The reason for these manual interruptions was essentially conservation of time and money: it was not worthwhile losing a costly experiment to a trivial mechanical or electrical failure. We were, therefore, content to demonstrate that the clamp can work completely automatically, but we operated instead in a semiautomatic mode. While our experiments were controlled using a specially-written program for parameter estimation and the IBM system 360. they could certainly have been performed on any digital computer, given access to a library tape “hill-climbing” subprogram, of which they are many. One need only estimate the 3 parameters a, @. and D contained in Appendix equations (A2) and (A4).
RESULTS
The course of a typical experiment is shown on Fig. 1. Figure 2 shows the “running mean” of C plotted against time. All results are summarized in Table 1. As evident from the Table, in all the experiments the concentration of plasma glucose attained was within f 5.0 mg/dl of the set-point designed. In all but one experiment, the variation of C around the set-point was small, as indicated by the standard deviations of the readings of C taken. In no experiment was the regression of the C-versus-t line significantly different from zero. The time necessary to reach the set-point was about 1 hr in all experiments. Once the set-point had been reached, it was kept for another 60-100 min. During this period, the rate of glucose infusion, equal or nearly equal the overall rate of glucose utilization, varied as shown on Fig. 1. In one dog (experiment 16), after the set-point had been reached and held for 76 min, the concentration of plasma glucose inexplicably started to decrease and reached 0.97 mg/ml at t = 168 min. Continuing the experiment, however, the set-point was re-established by t = 203 min. One experiment, number 15,
GLUCOSE CLAMP
1225
ii
1.6
2 365
1.4
;
15 .
:
1.3
U-J
1.2
5
11
110
g 3
c
100
; 'i -.& ZE
90
g
70
2 d
603
80
0
7
21
35
A9
63
77
TIME
91
105
119
133
(min)
Fig. 1. The glucose clamp, experiment 11. Abscissa: time in minutes. lower panel: rate of glucose infusion (mg/min); upper panel: concentration of glucose in plasma (ma/ml). A steady glucose infusion was begun at t = 0; the computer assumed control at t = 49 min. It should be noted that the computer program is not written in such a way that the infusion rate will respond to each individual change in plasma glucose, but rather a smooth curve is fitted through the past six points. In this way, future behavior is predicted. For example, a potential overshoot of glucose (which occurred at 91 min) was foreseen by the computer at 70 min, and the infusion rate was decremented in anticipation.
SET POINT
-i cn
; 2 z
= 1.4
mg/ml
0.9 4 0
I 21
I 42
I 63 TIME
I a4 (min)
I 105
I 126
I 149
Fig. 2. The “running mean” plasma glucose concentration plotted against time for the same experiment shown in Fig. 1. Each value of the ordinate is the arithmetic mean of all preceding plasma glucose values. This reflects the ultimate goal of the exercise: to hold the plasma glucose concentration at a new mean level of 1.4 mg/ml.
NORWICH
1226
ET AL
Table 1. Summay of Eight Experiments. the Attained Concantmtion of Glucose in Plasma, 2, was Calculated a, the Mean SuccessiveReadings Made at ‘I-min lntervalr Between the Time limits in Column 2
concentration Time Limits (min)of Set-Point ExperimentSet-PointRecorded mg/dl
Attained (c) w/df
SD *
Regression Coeficient(b) (Cat) x 10-4
7
63-161
140
141
6.6
-5.0
10
63-147
140
145
8.5
-15.8
11
56133
140
142
6.6
-6.8
12
84-154
140
135
5.7
+9.5
13
70-147
120
119
6.5
-2.6
14
70-126
140
135
4.0
+ 1.9
16
70-147
140
139
18.9
-24.5
17
63-161
140
139
11.3
+4.5
Remarks “Window”
= 50 min
Anesthetized. Set-point lost after t = 126 min. Set-point lost after t = 140 min but regained by t = 210 min.
had to be abandoned because of the breakdown ment is not shown on the Table.
of the computer.
This experi-
DISCUSSION
The concept of raising and holding constant the concentration of a substance in the blood or plasma of an intact organism is not new. For example Harvey et al.5 constructed a servomechanism for producing an constant plasma creatinine level. Norwich6 discussed the applications of holding a constant level of indicator in the blood as a means of measuring pulsatile blood flow. In the area of glucose metabolism, Kadish,’ and later Kline et al.,’ demonstrated how glucose concentration could be held automatically at the basal level in the presence of greater-than-basal levels of insulin. Pagurek et a1.9 designed a system which very successfully clamped the glucose level of human subjects in the hyperglycemic range. Insel et al.’ utilized a glucose clamp in the formulation of a model of insulin-glucose interaction. All of the above papers have one element in common: they utilize a mathematical model of the physiologic system, and they are designed to achieve control within a period of minutes. The introduction of a simple control algorithm which gives the optimal glucose infusion rate as a function of the displacement, integral, and derivative of the blood glucose concentration (P.1.D algorithm) is very appealing. We have not been completely successful with the use of such an algorithm, but if the delay in measurement of blood glucose can be reduced, this method may well become the method of choice. The challenge in designing a glucose clamp is largely one of avoiding uncontrolled oscillations in the concentration of blood glucose. The larger the delay between sampling blood and recording the concentration, the greater the probability of oscillation and loss of control. As faster methods of measuring blood glucose become available,‘“~” this problem is expected to lessen. Even if measurement of blood glucose level could be made nearly instantaneously, the problem of delay would not be totally overcome. There is a physiologic delay during which newly infused glucose disperses throughout its ultimate volume of distribution. This physiologic delay has been approximated by the term “D” in Appendix equation (A4). Thus, some mod-
GLUCOSE
1227
CLAMP
elling will probably be necessary for control of glucose levels even with the ultimate glucose monitor. The mathematical model used in our control system is not completely valid during the entire duration of the experiment. During the first 30-40 min of a glucose infusion, the presence of an additional transient element is clearly indicated.2*‘2 We have deliberately ignored this transient in the computer program used in the experiments reported here. Thus, equation (5) cannot really be expected to give an exact estimate of the desired infusion rate, Rinf, until the lower end of a window passes t + 35 min-i.e., until about 35 min after the beginning of a glucose infusion. It should be observed that the glucose infusion rate which we sought was not the rate which would be expected to achieve and hold the set-point in minimum time, but rather the best constant infusion rate. For a truly optimal infusion rate, we should have to consider, for example, the optimization principle of Pontriagin. l3 The optimal infusion would probably involve two components: first an infusion rate greater than the final maintenance rate, and then the maintenance rate. The potential applications of the glucose clamp are intriguing. In the clinic it might be used as a means of estimating the steady state turnover rate of glucose without the use of radioisotopes-i.e., estimating the rate of disappearance, R,, of glucose at steady state. Acting upon the assumption that the human glucoregulator does not differ in principle from the canine, the procedure might be implemented in the following ways. Cannulae can be inserted into two veins of a fasted patient in much the same manner as done in the dog, and a glucose clamp applied at some arbitrary hyperglycemic level below the renal threshold, e.g., at 1.5 mg/ml. In the hyperglycemic range, the endogenous glucose output is largely suppressed, so that in Appendix equation (A4) the value of R, is substantially smaller than the value of R, or Rinc and therefore does not influence dC
the rate of change of C to any degree. Hence, the derivative, dt,
is determined
largely by a balancing of the input rate Rinl with the output rate RI(C,a,/3). Achieving the glucose clamp at the desired level must then almost certainly imply correct evaluation of the constants LYand fi in the only unknown function, RJC,a,P). The rate of disappearance at the fasting level, C,, which is equal to the basal or fasting rate of appearance, is then given by R,(C,,a$). This function is obtained explicitly from Appendix equation (A2): R,(Cor~,P)
= a + K,
(7)
A critical matter in assessing the confidence to be placed in the calculation of of hepatic glucose output when blood sugar is elevated. If hepatic output were not suppressed, then one might suspect that the glucose clamp had been attained by virtue of compensatory errors: e.g., overestimating both R,(C) and RI(C) in such a way that their difference gave a correct value. In this case, basal R, as given by equation (7) would also be in error. But if endogenous hepatic output is largely suppressed in hyperglycemic in the human being, *4~‘s then we may place a good deal of confidence in the uniqueness and correctness of the result given by equation (7). R, by this method is the extent of suppression
NORWICH
1228
ET AL.
APPENDIX MATHEMATICAL
DEVELOPMENT
Let R, be the rate of endogenous appearance or entry of glucose and Rd the rate of disappearance or utilization. Let C be the plasma glucose concentration. It was shown by analysis of a number of experiments in dogs’ that at steady state, R, = R~(0)e-MC-Co) and Rd = a + PC
(AZ)
where C is the steady state plasma glucose concentration, Co is the fasting plasma glucose concentration and (Y, ,8, and X are constants. During the early minutes following the termination of a fast by a steady intravenous glucose infusion, the rate of change of R, with time is a function of an additional parameter /1.3 But after about 40-min infusion, the value of R, at any time is given very nearly by its steady state value from (A2). R, at any time may be estimated from (A 1) with X = 3.3 ml/mg without introducing serious error, primarily because the ratio of R, to R, is small in hyperglycemia. If the rate of intravenous infusion is given by Rinf, then, to a first approximation, dC dt=
’ [Ra + Rinf I/
RdI
(A3) (text Eq (3))
where V is the total distribution volume of glucose. However, newly infused or released glucose does not disperse instantly throughout all of V, and so the finite dispersion time is allowed for by modifying (A3), dC dt=
’ v
(R, + Rinl)
where D > 0 is a dispersion
l
1 + D ~
- Rd
(A4)
I coefficient
in units
of [time].
d(lnC) As -d7
-
0, Eq
(A3) comes to approximate Eq (A4) quite closely. Equations (A 1) and (A2) can now be introduced into (A4) to give, finally, a nonlinear differential equation in C and t. The clamping experiment is always begun by giving a glucose infusion of arbitrary magnitude* in order to displace the glucose system from steady state. Equation (A4) was then solved by finite element methods (Euler’s method), *Actually using statistics compiled from a number of earlier experiments, we were able to make an educated guess at the magnitude of the optimal infusion rate, and we often adopted this rate for the initial infusion. The value of @ (ml/min) is of the order of magnitude of 10 times total body weight (kg); Rd(CO) + 2.8 times total body weight (kg). Substituting the above estimates for fi and R&CO) into Eq (A2) gives an initial estimate for a. Finally, applying equations (5). (Al), and (A2) with the estimates for a and @ and C equal to C, gives the initial estimate for Ri,f, the rate of infusion of glucose.
GLUCOSE CLAMP
1 + D ‘9
AC(t) = $
)
1
- Rd At.
C(0) or in general, C(t,,,), was always taken as equal to the experimental at that time. (Y,j3, and D were selected so that,
value
window
was minimized. The optimization of the three parameters was carried out on an IBM S 360 using a searching or “hill-climbing” procedure similar to that suggested by Hazelrig et al. I6 The data points were weighted by their ordinal number: the second point in the window was given weight 2, the third weight 3 I--., and the nth, weight n. The more recent the data point, the greater its weight. Once (Yand /3 had been determined, Rinf (the constant rate of infusion necessary to achieve and hold the set-point, C,) was calculated from text Eq (5), Ri”r = Rd(Cs) - R,(Cs) with R, and Rd evaluated from (A 1) to (A2). The fitted value of D was always close to 10 min, and the values of (Yand B were nearly always in the range found in the radioisotope studies.* ACKNOWLEDGMENT The impetus for our development of this system followed several discussions with Dr. P. A. Inset and Dr. K. J. Kramer who informed us of the successful results of Drs. J. D. Tobin and R. Andres in experiments of this type. We are most grateful to these investigators and acknowledge their fundamental studies. We are also indebted to Dr. J. B. R. McKendry for helpful discussions and for the loan of the Auto Analyzer. REFERENCES 1. Insel PA, Kramer KJ, Sherwin RS, Liljenquist JE, Tobin JD, Andres R, Berman M: Modeling the insulin-glucose system in man. Fed Proc 33:1865-1868.1974 2. Hetenyi G Jr, Norwich KH, Zelin S: Analysis of the glucoregulatory system in dogs. Am J Physiol224635642, 1973 3. Zehn S, Norwich KH, Hetenyi G Jr: The glucose control mechanism viewed as a regulator. Med Biol Eng (in press) 4. Norwich KH, Hetenyi G Jr, Fluker G, Pagurek B: Development of the glucose clamp. Digest, Fifth Canadian Medical and Biological Engineering Conference, Montreal, 1.3a, 1974 5. Harvey RB, Bassingthwaighte JB, Heppner RL: Regulation of plasma creatinine concentration by use of a servo control system. Clin Chem 14944-959, 1968 6. Norwich KH: Determination of pulsatile blood flow by indicator-dilution methods. J Theor Biol50:353-361, 1975
7. Kadish AH: Automation control of blood sugar. Am J Med Elec 3:82-86, 1964 8. Kline NS, Shimano E, Stearns H, McWilliams C, Kohn M, Blair JH: Technique for automatic in vivo regulation of blood sugar. Med Res Eng 14-19, 1968 9. Pagurek B, Riordon JS, Mahmoud S: Adaptive control of the human glucose regulatory system. Med Biol Eng 10:752-761, 1972 10. Chang KW, Aisenberg S, Soeldner JS, Hiebert JM: Validation and bioengineering of an implantabie glucose sensor. Tram Am Sot Int Organs 19:352-360, 1973 11. Bessman .SP, Schultz RD: Progress toward a glucose sensor for the artificial pancreas. Digest of the Tenth International Conference on Medical and Biological Engineering 34-IO,1973 12. Moorhouse JA, Smithen CS, Houston ES: A study in glucose transport kinetics in man
1230 by means of continuous glucose infusion. J Clin Endocrinol Meta 27:256-264, 1967 13. Pontriagin LS, Boltyanskii VG, Gamkrelidze RV, Mishchenko EF: The Mathematical Theory of Optimal Processes. New York, Interscience, 1962 14. Franckson JRM, Ooms HA, Bellens R, Conard V, Bastenie PA: Physiologic signihcance of the intravenous glucose tolerance test. Metabolism 1I :482-500, I %2
NORWICH ET AL. 15. Forbath NF, Hetenyi G Jr: Glucose dynamics in normal subjects and diabetic patients before and after a glucose load. Diabetes 15:778-789,1966 16. Hazelrig JB. Ackerman E, Rosevear JW: An iterative technique for conforming mathematical models to biomedical data. Proceedings of the Sixteenth Annual Conference on Medical and Biological Engineering, Baltimore, 5:8-9, 1963
