## Ultradian Oscillations

Ultradian insulin oscillations have a number of observable features. First, oscillations occur during constant intravenous glucose infusion and are not dependent on periodic nutrient absorption from the gut. However, damped oscillations occur after a single stimulus such as a meal. Second, glucose and insulin concentrations are highly correlated, with the glucose peak occurring about 10-20 minutes earlier than that of insulin. Third, the amplitude of the oscillations is an increasing function of glucose concentra-

Figure 19.10 Oscillations of insulin and glucose. A: During the ingestion of 3 meals. B: During oral glucose. C: During continuous nutrition. D: During constant glucose infusion. Oscillations with a period of around 120 minutes occur even during constant stimulation (i.e., constant glucose infusion), and occur in a damped manner after a single stimulus such as ingestion of a meal. (Sturis et al., 1991, Fig. 1.)

Figure 19.10 Oscillations of insulin and glucose. A: During the ingestion of 3 meals. B: During oral glucose. C: During continuous nutrition. D: During constant glucose infusion. Oscillations with a period of around 120 minutes occur even during constant stimulation (i.e., constant glucose infusion), and occur in a damped manner after a single stimulus such as ingestion of a meal. (Sturis et al., 1991, Fig. 1.)

tion, while the frequency is not; and fourth, the oscillations do not appear to depend on glucagon.

Although there are many possible mechanisms that are consistent with the above observations, they can all be explained by a relatively simple model (Sturis et al., 1991) in which the oscillations are produced by interactions between glucose and insulin.

A schematic diagram of the model is shown in Fig. 19.13. There are three pools in the model, representing remote insulin storage in the interstitial fluid, insulin in the blood, and blood glucose. As we will see, two insulin pools are necessary, which is, by itself, an interesting model prediction. There are two delays, one explicit and the other implicit. Although plasma insulin regulates glucose production, it does so only after a delay of about 36 minutes. This delay is incorporated explicitly as a three-stage linear filter. An additional implicit delay arises because glucose utilization is regulated by the o

40 160

1 200

Glucose

40 160

1 200

Figure 19.11 Intermediate frequency oscillations of glucose, insulin, and glucagon in monkeys. (Goodner et al., 1977, Fig. 1A.)

remote (interstitial) insulin, and not by the plasma insulin, while glucose has a direct effect (through insulin secretion from the pancreas) on plasma insulin levels.

We let x,y,in units of mU, denote the amounts of plasma insulin and remote insulin, respectively, and we let z, in units of mg, denote the total amount of glucose. Then the model equations follow from the following assumptions:

1. Plasma insulin is produced at a rate f1(z) that is dependent on plasma glucose. The insulin exchange with the remote pool is a linear function of the concentration difference between the pools x/V1 — y/V2 with rate constant E, where V1 is the plasma volume and V2 is the interstitial volume. In addition, there is linear removal of insulin from the plasma by the kidneys and the liver, with rate constant 1/t 1. Thus, dx . . . ( x y \ x

Note that this equation and the two that follow are written in terms of total amounts of insulin and glucose, rather than concentrations. Formulations using concentrations or total quantities are equivalent, provided that the blood and interstitial volumes remain constant, which we assume.

2. Remote insulin accumulates via exchange with the plasma pool and is degraded in muscle and adipose tissue at rate 1/t2:

dt V1 V2 t2

3. Plasma glucose is produced at a rate f5 that is dependent on plasma insulin, but only indirectly, as f5 is a function of h3, the output of a three-stage linear filter. The ra

Time (min)

Time (min)

Frequency (min-i)

Figure 19.12 A: Oscillations of insulin release in perifused islets. The data indicate a slowtime scale decreasing trend (the smooth line) upon which are superimposed faster time scale oscillations. B: When the slow decrease is removed from the data, the residuals exhibit oscillations around 0. C: Spectral analysis of the residuals shows a frequency peak at about 0.07 min-1, corresponding to oscillations with a period of 14.5 minutes. The dashed and continuous lines correspond to two different filters used in the spectral analysis. (Bergstrom et al., 1989. Figs. 1A, C, and 3.)

input to the filter is x, so glucose production is regulated by plasma insulin but delayed by the filter. There is input I from the addition of glucose from outside the system, by eating a meal, say. Finally, glucose is removed from the plasma by two processes. Thus, ddZ = f5(h3) + I - f2(z) - f3(z)f4(y). (19.31)

Glucose utilization is described by two terms: f2(z) describes utilization of glucose that is independent of insulin, as occurs, for instance, in the brain, and is an increasing function that saturates quickly. The second removal term, f3(z)f4(y), describes insulin-dependent utilization of glucose. Both f3 and f4 are increasing functions, with f3 linear and f4 sigmoidal.

4. The three-stage linear filter sastifies the system of differential equations

The specific functional forms used for f1, . ..,f5 are

1 + exp(-1.772log y\y2 + M\ + m s 180 , f5(h3) = 1 + exp(0.29)/z3/V1 - 7.5), (19.39)

and these are graphed in Fig. 19.14. The remaining model parameters are given in the caption to Fig. 19.15.

Numerical solution of the model equations shows that a constant infusion of glucose causes oscillations in insulin and glucose. As I increases, the oscillation pe-

0 10 20 40x103

glucose, z(mg)

0 1000 2000 insulin in interstitial fluid, y (mU)

Figure 19.14 Graphs of f1r...,f5 in the model for ultradian insulin oscillations. The exact forms of these functions are not physiologically significant, but are chosen to give the correct qualitative behavior.

Figure 19.15 Ultradian insulin oscillations in the model. The glucose infusion rates are A: I = 108 mg/min, and B: I = 216 mg/min. Note that insulin and glucose are expressed in units of concentration. An amount is easily converted to a concentration by dividing by the volume of the appropriate space. Parameter values are V = 3 liters, t1 = 6 min, V2 = 11 liters, t2 = 100 min, V3 = 10 liters, t3 = 12 min, E = 0.2 liter/min. (Sturis et al., 1991, Fig. 5.)

Figure 19.15 Ultradian insulin oscillations in the model. The glucose infusion rates are A: I = 108 mg/min, and B: I = 216 mg/min. Note that insulin and glucose are expressed in units of concentration. An amount is easily converted to a concentration by dividing by the volume of the appropriate space. Parameter values are V = 3 liters, t1 = 6 min, V2 = 11 liters, t2 = 100 min, V3 = 10 liters, t3 = 12 min, E = 0.2 liter/min. (Sturis et al., 1991, Fig. 5.)

riod remains practically unchanged, but the amplitude increases (Fig. 19.15), in good qualitative agreement with experimental data. However, it is interesting that these oscillations disappear if the compartment of remote insulin is removed from the model. This indicates that the division of insulin into two functionally separate stores could play an important role in the dynamic control of insulin levels. Another prediction of the model is that the oscillations are dependent on the delay in the regulation of glucose

Perifusate beads with cells f

Efflux

Figure 19.16 Schematic diagram of a typical perifusion system.

f production. If the delay caused by the three-stage filter is either too large or too small, the oscillations disappear.

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