abnormally high compared to those encountered when animals are fed by University of Iowa, Iowa City, Iowa, U.S.A.

Transcription

1 J. Phy8iol. (1965), 181, pp With 5 text-ftgure8 Printed in Great Britain THE ANALYSIS OF GLUCOSE MEASUREMENTS BY COMPUTER SIMULATION* BY R. G. JANES "D J. 0. OSBURN From the Departments of Anatomy and Chemical Engineering, University of Iowa, Iowa City, Iowa, U.S.A. (Received 2 February 1965) The homeostasis of blood glucose is known to be affected by several factors, which work simultaneously to increase or decrease blood-sugar levels. Only when organs or enzyme systems are altered, either experimentally or pathologically, is it possible to isolate with some facility one or more of the factors which control blood-sugar levels. Two recent papers have stimulated new interest in studies on the homeostasis of blood glucose. In these papers computer techniques have been used to match physiological results to assumed models so as to allow a more completely mechanistic analysis. Bolie (1960), without making any measurements of glucose, contributed data on the glucose-insulin feedback theory in which he formulated and applied a set of differential equations necessary for the quantitative description of the mechanisms involved in the homeostatic control of glyeaemia. More recently, Seed, Acton & Stunkard (1962) constructed a model for the appraisal of glucose metabolism in the human in which glucose homeostasis was divided into three parts: first, a rate of glucose dispersion into its volume of distribution; second, utilization rate; and third, production rates. Both these investigators were concerned with glucose levels obtained after intravenous injection of glucose. This resulted in blood-glucose levels abnormally high compared to those encountered when animals are fed by mouth. At these high levels, mechanisms presumably come into play which are ordinarily of negligible significance. Because of the possible physiological alterations which may occur with intravenous feeding, oral feeding was used in the present study. With the gavage method, the glucose does not enter the blood-stream immediately as it does with injection, but rather enters in proportion to the rate of absorption. For this reason, absorption studies for glucose were also made. An analog computer was used to relate the results of blood-glucose measurements with those from the intestinal absorption measurements. * This work was supported in part by grant NB 00237, U.S.P.H.S. and by the Charles F. Kettering Foundation.

2 60 R. G. JANES AND J. 0. OSBURN Not only is glucose passing from the intestine to the blood-stream at a constantly changing rate, but glucose is also being transferred to and from the liver and other organs at rates which in turn depend on the glucose level in the blood. To evaluate the transfer rates under these changing conditions, a simple model was developed and simulated on the computer. The model was adjusted until the glucose levels predicted by the computer approached the experimental data points as closely as possible. METHODS Biological Adult rats of the Long-Evans strain and New Zealand or California strain rabbits were used. The animals were fasted overnight (16-20 hr) and after blood samples had been taken for glucose determination they were fed by stomach tube with 2 g glucose/kg body weight. The rats were bled from a tail vein, the rabbits from the marginal ear vein. In addition, blood for some sugar determinations in the rabbit was obtained by heart puncture. Blood samples were taken at 0, i, i, 1, 2, and 3 hr. When duplicate samples of rat blood were taken it was found that if only three, or at the most four, samples of 0-2 ml. each were taken over the 3 hr period, the values were more accurate than if the total of six different samples were withdrawn. In certain instances urine was collected for sugar determinations but no sugar was detected. Blood-sugar determinations were carried out with Nelson's modification of the Somogyi technique. Absorption of glucose from the gastro-intestinal tract was determined by the method of Cori (1925). At appropriate times after feeding, the rats were decapitated and the rabbits were killed with a blow on the skull. The stomach and small intestine were removed quickly, washed out thoroughly with cold saline and the glucose content of the wash was determined. This method of estimating the rate of absorption of glucose from the intestine has been criticized by Crane (1960), but since it does not involve anaesthesia and surgery, as many other methods do, it probably gives a good indication of the actual amount of glucose that is absorbed. In order to test the effect of the feeding procedure on blood-sugar levels, a series of rats was fed 2 ml, and rabbits 10 ml., of water and blood samples were withdrawn at intervals up to and including 2 hr. Analog computer The computer used was the Heathkit Model no. ES-201 with a Varian recorder, and a Donner Model (3732/P) function multiplier. RESULTS In Table 1 are listed the measured values for absorption of glucose from the alimentary tract of rats and of rabbits. For both animals the disappearance is rapid, with roughly half the glucose being absorbed in the first half hour. After 2 hr most of the glucose was gone from the alimentary tract. It was found that these data can be represented satisfactorily by an exponential equation P= (1-eat), (1) where P is the fraction absorbed, a is a constant and t is time. For the

3 GLUCOSE MEASUREMENTS 61 data on rabbits, it appears that there is a delay of about 5 mi between the time of feeding and the beginning of the disappearance of glucose. Data for the rats did not show this delay. The variation of P with time is found by differentiating dp = _ aeat = _-a (1 -P). (2) dt This derivative, multiplied by m, the weight of glucose fed per kg body weight, gives the rate at which glucose leaves the alimentary tract. TABLE 1. Disappearance of glucose from alimentary tract of rats and rabbits (fasted animals fed on 2 g of glucose/kg body weight) TABLE 2. Time No. of % absorbed No. of % absorbed (hr) rats (±S.E. of mean) rabbits (+S.E. of mean) ± Blood glucose values (± s.e. of means; number of determinations in parentheses) after glucose and water-tolerance tests Time (hr) Animal Fed Rat Rat Water Glucose (43) (18) (27) 107 ±5-0 (14) 93 ± 1-4 (29) (14) Rabbit Rabbit Water Glucose (25) (20) 88±2-9 (15) (12) 94±2-1 (15) (14) Animal Fed Rat Water (31) (22) (20) Rat Glucose (16) (17) (9) Rabbit Water (22) (11) (19) Rabbit Glucose (19) (14) (11) The effect of this glucose input on the blood glucose level is shown by the data listed in Table 2. Note that even for water feeding, the glucose level rises about % fairly quickly. This is thought to be a reaction of the animals to stresses induced by forced feeding. In contrast to glucose levels encountered after intravenous feeding, which rise rapidly to rather high values, the levels observed in the feeding experiments reached a peak in about 1 hr at about twice the normal levels. In all experiments, 2000 mg of glucose was fed per kg of body weight. The amount of glucose leaving the alimentary tract in 1 hr is about 70 % of 2000, or 1400 mg. For an average value of 80 ml. of blood per kg of body weight, this amount of glucose should raise the glucose level in the blood by 140 x 100 or 1750 mg/decilitre. This would be reduced by glucose

4 62 R. G. JANES AND J. 0. OSBURN utilization and storage, but it is evident that nothing even remotely approaching this amount of glucose shows up in the blood stream. It is very efficiently removed by the liver before it reaches the point where samples are taken. The preceding rough glucose balance can be improved by using the data to infer the rates of glucose utilization. Certain well-known facts about glucose homeostasis are used to construct a model of the process which takes into account the effect of glucose level on utilization rate. The model takes into account the following observations: (1) steady glucose level in fasting animal; (2) changes in glucose level for water feeding; (3) changes in glucose level for glucose feeding. In addition, the model assumes these effects which were not observed directly: (1) a rate of production of glucose in the liver which is inversely proportional to the insulin concentration; (2) a loss of glucose from the blood at a rate which is proportional to the level in the blood; (3) a rate of production of insulin which is proportional to the blood glucose level; and (4) a loss of insulin at a rate which is proportional to its level in the blood. No provision is made for a term for spillage of glucose into the urine, because this did not occur at the low levels obtained. This is essentially a one-compartment, two-component model. Attention is centred on the vascular compartment, with input from the liver and alimentary tract, and output to various unspecified organs. Because of the disagreement in the literature of the effect of insulin on the rate of glucose utilization, this effect has not been included. Further elaboration of the model to include cellular or other compartments was not considered feasible, since no data were obtained in other parts of the animals. Computer simulation Based on the model described in the preceding section, the following equations can be written dy/dt = ki+12+ kl/x, (3) dx/dt = k3y-k4x. (4) LAst of symbols a = constant I1 = rate of glucose absorption from intestines, mg/hr. This is variable, and is the derivative of the curve of amount absorbed vs. time.

5 GLUCOSE MEASUREMENTS I2 = rate of glucose release from liver due to stress of handling, mg/hr assumed to decrease with time according to a falling exponential k relation. = fraction of glucose fed which directly affects measured blood-sugar level. k1 = rate constant for glucose input from liver, dependent on insulin concentration (mg/hr) (insulin unit). k2 = rate constant for glucose removal, hr-l. k3 = rate constant for hypoglycaemic factor on blood glucose, insulin unit/hr per mg glucose. k4 = rate constant for hypoglycaemic factor decay, hr-1. m = amount of glucose fed, mg. P = fraction of glucose fed which is absorbed at time, t. formation as dependent t = time, hours. x = insulin concentration in blood, units/decilitre. y = glucose concentration in blood, mg/decilitre. The analog computer circuit for the solution of these equations is shown in Fig. 1. The circuit is divided into four parts: glucose level in the blood, insulin level in the blood, glucose fed, and an extra input from liver caused by the stress of feeding di o _o I Stress I I I 12/00 100k2I001 xy I i F727F771IA-1 5 -Y I K½YLt~-1tK 63 I ~L 21-> 71~ I - / - l l~~~~~~~~~~~~ KJ -~~~~~~~~~~~~ I \w1 Lill" I I LTime delay Glucose feeding Fig. 1. Circuit for one-compartment, two-component model used in the analog computer. For list of symbols and equations see text.

6 64 R. G. JANES AND J. 0. OSBURN The computer circuit for the absorption of glucose, eqn. (2), is shown as a part of Fig. 1. After a time delay produced by amplifier 2 and the associated relay, a signal is connected to amplifier 3. This amplifier generates -P according to eqn. (2). Amplifier 4 gives - dp/dt, which is related to input I1 by I1 = m (dp/dt). (5) 100 >0 r- 0-0 l Hours Fig. 2. Absorption of glucose from the alimentary tract of the rabbit, after oral feeding of 2 g glucose per kg body weight. Comparison of observed data (circles) with computer correlation (full line). Vertical lines through data points show the standard error of the mean. Potentiometers S3 and S4 and the time delay were adjusted until the computer curves matched the experimental data as closely as possible, as shown in Figs. 2 and 4. To simulate the glucose release from liver due to stress, it was assumed

7 GLUCOSE MEASUREMENTS 65 that the rate of input due to this factor can be described mathematically as a falling exponential, with a high initial value at the instant of feeding. The computer was operated as described in the next section, with the coefficient potentiometer mk set at zero. This represents the situation when only water was fed. The height of the 'stress pulse' was adjusted by _ 0~~~ Hours Fig. 3. Blood glucose in the rabbit after feeding. Each point is an average of measurements on several animals. Glucose feeding (open circles) was 2 g/kg body weight. Water feeding (filled circles) was 10 ml. Vertical lines through data points show the standard error of the mean. adjusting the initial condition on amplifier 1, and its duration changed by adjusting S, until computer curves of blood glucose versus time matched the data as shown in Fig. 3 for rabbits, and Fig. 5 for rats, and compared with the experimental data as shown. The total amount of glucose released by stress was then measured by integrating the rate curve. Using the computer The time delay and glucose-feeding circuit was first adjusted to match the absorption data. Values for insulin half-life and coefficient k2 were assumed, and the corresponding potentiometers were set. With the x and y amplifiers operating, potentiometers k3 and k1 were adjusted until the steady state values of x and y were at the correct values for the fasting animal..s Physiol. 181

8 66 R. G. JANES AND J. 0. OSBURN At this point, mk was set at zero, and the stress pulse was adjusted until the water feeding data were matched. Then the mk potentiometer was adjusted until a best fit of the glucose data was obtained. Different halftimes were assumed, and repeated trials were made until the computed curve of blood glucose versus time matched the data, within 1 standard error in most cases. Results of this curve-fitting are shown in Fig. 3 for rabbits and in Fig. 5 for rats, where the computed curves are compared to the experimental data ~-z 0 ~0-40 to bo Hours Hours Fig. 4 Fig. 5 Fig. 4. Absorption of glucose from the alimentary tract of the rat, after oral feeding of 2 g glucose (circles) per kg body weight. Comparison of observed data with computer correlation (full line). Fig. 5. Blood glucose in the rat after feeding. Each point is an average of measurements on several animals. Glucose feeding (open circles) was 2 g/kg body weight. Water feeding (filled circles) was 10 ml.

9 GLUCOSE MEASUREMENTS 67 CONCLUSIONS From the potentiometer readings, values of the parameters in the model were calculated. The values obtained led to these conclusions: First, only a small amount of the glucose fed appears in the peripheral vascular system immediately. This fraction was 0 05 for the rabbit, 0-03 for the rat. The remainder is presumably stored in the liver, to be released later as needed. This assumption is only true when glucose is not spilled into the urine. None was found in the urine in the present experiment. Based on the glucose level in the fasting animal and the observed glucose-utilization rate, the rate of glucose release in the fasting animal was 57 mg/hr per 100 ml. of blood. This was the same for both animals. The half-life for glucose in the blood was 1 hr for the rabbit, 1*5 hr for the rat. The half-life of insulin in the blood was 2 hr for the rabbit, 1 hr for the rat. Since insulin was not measured, but inferred from its action, this value could not be obtained with any degree of precision. The stress of forced feeding caused a total of 14 mg of glucose per 100 ml. to be released to the blood. This is the amount normally released in about 15 min when the animal is resting and was about the same for both animals. SUMMARY 1. An analog computer has been used to simulate glucose level changes after feeding glucose to rats and rabbits. 2. Blood-glucose determinations were made in these animals. Some were sham-fed with water. 3. The absorption of glucose from the intestines was measured. 4. Besides these experimental data on blood-sugar and glucose levels, assumed values were used for glucose production in the liver and insulin concentrations in the blood. 5. The computer aided in evaluating the transfer rates of glucose with changing blood levels and in studying the homeostasis of blood sugar in these animals. REFERENCES Bous, V. W. (1960). Glucose-insulin feedback theory. Third intern. Conf. med. Electronic8, Proc CoRI, C. F. (1925). Fate of sugar in the animal body. 1. Rate of absorption of hexoses and pentoses from the intestinal tract. J. biol. Chem. 66, CRANE, R. K. (1960). Intestinal absorption of sugars. Physiol. Rev. 40, SEED, J. C., ACTON, F. S. & STUNKARD, A. J. (1962). A model for the appraisal of glucose metabolism. C(in. Pharm. Therap. 3,