- Email: [email protected]
Models for the insulin response to intravenous glucose
Models for the Insulin Response to Intravenous Glucose PER HAGANDER, Department
of Surgety, Lund Universiv,
KARL-G&4N Department
LunCr: Sweden
TRANBERG,
of Nuclear Medicine, General Hospital, Malmij, Sweden
JAN THORELL, Biocybernetics
Laboratory,
Uniwrsi@ of California, Los Angeles, USA
AND
JOSEPH DISTEFANO III Department
Received
of Automatic
I4 November
Control, Lund Institute of Technology, LUG
Sweden
1977; revised 2 May 1978
ABSTRACT The Grodsky packet storage model describes many features of insulin release, but at present more or less arbitrary simplifications are necessary. The consequences of various simplifications are discussed, especially with regard to identification of parameters thought to be of importance for glucose tolerance. In particular, the insulin release dynamics of the ordinary intravenous glucose tolerance test is examined. The proposed model contains the following features: It considers arterial rather than venous blood glucose concentration as the stimulus, it takes the glucose injection time into account, and it contains a positive derivative term during the rise of the glucose concentration. When the insulin elimination-rate time constant is fixed to an a priori value, model fitting gives a clear quantification of the sensitivity of early and late insulin release to glucose.
INTRODUCTION The primary defect in maturity-onset diabetes mellitus-impaired insulin secretion or impaired tissue sensitivity to insulin-remains unknown. There is a growing tendency to favor a subnormal insulin response as the basic defect in the diabetic syndrome, but this conception has also been seriously challenged [ 1,2]. The difficulty in detecting the initial lesion rests mainly on the following. Among subjects with normal glucose tolerance there are wide variations in insulin release and insulin sensitivity, and the correlation MATHEMATICAL
BIOSCIENCES
0Elsevier North-Holland,
Inc., 1978
42, 15-29 (1978)
15
0025-5564/78/090015 + 15+ 15t02.25
PER HAGANDER
16
ET AL.
between these two factors is very weak. Blood glucose and plasma insulin levels are continuously modifying each other, making it difficult to establish the dynamic relationship between glucose and insulin in vivo. Most earlier modeling studies have been based on time-invariant linear differential equations [3-51, whereas it has been shown that the insulin secretion rate is both nonlinear and time-dependent [6,7]. Many dynamic characteristics of insulin release can be described by the Grodsky packet storage model [S], which has a solid in vitro experimental background. Recently, LiEko and Silvers [9] presented a modified Grodsky model, suitable for application to the intravenous glucose tolerance test (IVGTT) in man. We have investigated the insulin secretion of the IVGTT by means of the Grodsky model and modifications thereof. This has led to a model with a complexity adjusted to the limited amount of information and focused on those aspects of insulin release that are most important for glucose homeostasis. THE GRODSKY
PACKET
HYPOTHESIS
In the packet storage model [8] the insulin release system contains two pools: the labile pool, which releases insulin to the circulation, and the storage pool, which replenishes the labile pool (see Fig. 1). The secretion rate of insulin from the labile pool depends on the glucose concentration. Insulin in the labile pool is stored in a large number of “packets”, releasing
FIG. 1. Block diagram of the Grodsky packet storage model for insulin secretion. The glucose concentration G controls the release of insulin from the labile pool. The provisionary factor P controls the replenishment of insulin from the storage pool to the labile pool.
MODELS FOR INSULIN RESPONSE TO I.V. GLUCOSE
17
insulin at different preset glucose levels called thresholds. by the function &0, t): 5(&t)=
This is described
Amount of insulin in the labile pool at time t, having 1. ( the glucose threshold 4
All the insulin with a threshold lower than the glucose level G is assumed to be secreted at a rate proportional to 5 with proportionality constant m:
(1) Also, replenishment of the labile pool from the storage pool is assumed to be controlled by the glucose concentration, but indirectly via the production and utilization of a so-called provisionary factor P. The insulin replenishment rate to threshold f7 is assumed to be proportional to the provisionary factor P(t), with the proportionality coefficient Redistribution insulin different is in model. packets threshold insulin assumed be to threshold o a proportion [(e,t), proportionalcoefficient being function the level From assumptions mass equation the pool described [8]:
$p,t)=-sR(o)+
mqt)-
loXv’(w)do.Ke,r)+v’(e )~mt(~~+h (4
The production rate of the provisionary factor function h of the glucose concentration, so that ;P(t)=-a[P(r)-h(G(r))]. Grodsky
was assumed
to be a
(3)
assumed that
(4) where f andf’ are two constants. For his perfusion experiments Grodsky [8] suggested explicit expressions for the functions [(e,O) and h(G):
PER HAGANDER
18 where x,,,,
c, and k are constants,
f.X,,X. h(G)=
ET AL.
and OSG”
8.875x1~‘+2.25x10’5G3+3.5x106G7+G’o’
(6)
The Grodsky model for insulin release can thus be summarized by the equations (lH6). The dynamics is described by one ordinary differential equation for p(t), Eq. (3) while Eq. (2) can be considered as an infinite number of differential equations for E(r9,t), 0<8< 00, weakly coupled together via the integral J F&o, t)do. The model was originally designed to describe results from perfused rat pancreas, and computer simulations in [8] showed good agreement with measured data. For special glucose perfusion experiments such as step or staircase functions, the model can actually be reduced to a system of ordinary differential equations. The model then depends on the actual levels used, and it cannot be applied to other input levels. These are the cases simulated by Grodsky [8]. Another situation in which the Grodsky model may be simplified is for short observation periods like the 18-min ramp stimulation in 181. At time t redistribution and replenishment of insulin with a threshold higher than the glucose level G(t) may then be disregarded, and &0,0) may be used instead of 5(0,1). The model described by Eqs. (lH6) requires an enormous amount of experimentation for its validation, and even for simulation purposes it is too complicated. But conceptually the model is appealing, and it should be useful as a platform for further simplifications in different directions. APPLICATION OF THE (LICK0 AND SILVERS)
PACKET
HYPOTHESIS
TO THE
IVGTT
In the IVGTT a brief infusion of glucose is given into a peripheral vein, and blood samples are taken during the following hour. Figure 2 shows the results of a typical IVGTT in a normal man. LiEko and Silvers modeled the kinetics of distribution and elimination of plasma insulin as a well-mixed plasma compartment with volume V and elimination rate constant n: dI -=-nz+E dt The glucose kinetics was
was
as simple
V’
(7)
The glucose injection go, elimination was with rate constant k, from the
MODELS FOR INSULIN
RESPONSE
19
TO I.V. GLUCOSE
300
300 Patient + 20
_
200 2
200
z 5
g
L3
F " 100 % 2 3 01
$ 100 2 .1E cl-
B 0
0 0 Time
FIG. 2.
LO
20 Cminl
60
A typical normal IVGTT’. Venous glucose and plasma insulin
initial concentration
2
concentrations.
G(0) = G, + go: t
t>O.
(8)
G, represents the basal level. LiEko and Silvers thus assumed that there is no detectable feedback influence of insulin on the glucose elimination during the one hour period of the test. Although there is evidence against this hypothesis [IO], it is not necessary to include such an effect in a model of the insulin response. LiEko and Silvers’s model for insulin secretion is based on Grodsky’s. Certain simplifications were made, thus eliminating the complexity of Eq. (2). The time constant for insulin release, l/m, was assumed to be zero. Thus, when the glucose concentration instantaneously jumps from G, to G(O), insulin is released from the labile pool as an impulse, and the insulin concentration is instantaneously raised from I = I, to I = I, + i,,. Thus Z(0) = I,
+ i(0) = I,
+ io= I,
+ +
JG(o)&f3,0)d9 G,
describes the early phase of insulin secretion. LiEko and Silvers also suggested the function &Q,O) defined by Eq. (5). Furthermore, no redistribution is considered, i.e., f’= 0 in Eq. (4).
PER HAGANDER
20
ET AL.
The replenishment from the storage pool makes further secretion possible. With the assumptions l/m = 0 and f’ = 0, Eqs. (1) and (2) simply give
SR(@,t) = Thus, the so-called
0”’ )P(% 1 9
“late” insulin secretion
@;z;i;?)Y response would be
SR(r)=P(QjG(f)y(+#=P(QI(G(t)),
(10)
0
with the notation
introduced in [8] and [9]. However, LiEko and Silvers instead assume that all the insulin available for replenishment is also available for immediate release, i.e., SR(r)=P(t)I(co)=SR,+y(Q
t>O,
where SR, is the basal secretion rate. They also assume structure of the function h in Eq. (3): P(G-G,)+SR,/I’(oo),
G-G,>O, otherwise,
(12) a simple linear
(13)
so that Eq. (3) results in
!!L -Q-/&)9 dt y(0)
=o.
g>o,
(‘4)
The LiEko-Silvers model [Eqs. (7) (8) (9) (12) (14)] has in fact an alternative physical interpretation, shown in Fig. 3. In this interpretation the late response y(t) is elicited directly from the storage pool; no replenishment or redistribution in the labile pool is included in the model. Packet release from the labile pool is controlled instantaneously by the glucose concentration, while release from the storage pool is controlled by “old” glucose concentrations. By assuming the form (12) instead of the form (lo), the nonlinear function I(B) or &S,O) only enters into the “early” insulin response, i,, and the number of parameters necessary to quantify the IVGTT is drastically reduced. This is a fundamental departure from the original Grodsky model,
21
MODELS FOR INSULIN RESPONSE TO I.V. GLUCOSE
FIG. 3. Alternative block diagram of the LiEko-Silvers model for insulin secretion. The glucose concentration G controls the release of insulin from the labile pool, and the provisionary factor P controls the release of insulin from the storage pool.
and it may result in qualitatively different behavior of the two models. Specifically, there is no instant effect of glucose at time t >0 on insulin secretion at time I > 0 in the LiEko-Silvers model, not even if g(t) = 0. In the Grodsky model g(t) = 0 would give almost no secretion rate. An experiment that would distinguish between the two models, Eq. (10) and Eq. (12), is shown in Fig. 4. Both models are simulated during a
z
(-J
I
0 Time
20
I
I
LO
I
60
I
I
80
I
10
Cminl
FIG. 4. Two-pulse glucose perfusion experiment. Simulation showing the difference between the Grodsky model (a) and the LiEko-Silvers model (b).
PER HAGANDER
22
ET AL.
constant glucose perfusion interrupted by a period during which glucose is maintained at its basal level. During a low-glucose interval Eq. (12) gives almost no change in insulin secretion, while Eq. (10) gives a secretion rate slightly above the basal rate. The corresponding insulin is stored in the labile pool and later released as an early response to the second pulse. For Eq. (12) there is no such response. It should be noted that the LiEko-Silvers model was not intended to be valid for a glucose perfusion. The literature contains some results from similar experiments. Grodsky [8, Fig. 31 describes a perfusion experiment with the low glucose level being zero. In the pilot gland experiments by Bergman and Urquhart [7] a basal glucose concentration of 150 mg/lOO ml was used. In both experiments there is a pronounced early response to the second pulse and a substantial decrease of the secretion during the low glucose interval, favoring Grodsky’s original assumption represented by Eq. (10). Some in uiuo studies (Cerasi et al. [ 111, Koncz et al. [ 121) show similar results, although less transparent. A delayed early insulin release after the second pulse was noted in [12], and a similar phenomenon can in fact be seen in [8]. As will be demonstrated below, it is not possible to distinguish between (10) and (12) during an IVGTT. ALTERNATIVE
MODEL
The early insulin release during the first few minutes of the IVGTT should be depicted as accurately as possible [l, lo]. The large amount of glucose, usually 25 g, infused in an IVGTT requires that the injection time be taken into consideration. A minimum of two minutes is required to ensure that the test is not unduly uncomfortable to the patient. The equations taking the injection time into account are given in Table 1. The blood supporting the pancreas is arterial. Between arterial and venous blood concentrations there is a substantial “damping”, especially of the initial peak, during an IVGTT. In order to include this distribution effect it is necessary to use a two-compartment model. The elimination would then occur from both compartments, which is impossible to quantify from measurements in only one of them. A simple “black box” (input-output) approach is therefore taken: Eq. (6) where the variables x1 and x1 have no physiological meaning. Actually, any linear two-compartment system with a single infusion site and a single measurement site can be described by such a model. The parameter Vg is the initial distribution volume, while the parameters b, k,, and k, have no immediate physiological meaning. The parameter k, corresponds to what is clinically called k,. G, is the basal glucose concentration, T is the injection time and D is the total glucose dose given.
MODELS
FOR INSULIN
RESPONSE
TO I.V. GLUCOSE
TABLE 1 Equations of the Alternative
23
Model
64
$x,(r)=-k,x,+b+
8
&X*(1)=4*x*+(1-b)+ g
(b)
(4 Early insulin release:
v,(t)=
E(g(W)&W. O
(4
t>T
0, Late insulin release: d ;iiYz’
Insuliu distribution
-ay,+aSg(0,
o
-
OthfXWiSe
aY2,
(4
and elimination: Y=YI+Yz
(0
d. --I=-ni+$ dt
(g)
Z(t)-Z,+i(t)
(h)
Plasma insulin concentration:
The influence of insulin on glucose elimination is disregarded. The glucose model only represents the glucose response in a certain experiment. It is not required that differently large glucose loads give the same decay rate constants. The glucose dynamics is known to be complicated. There are nonlinearities both in the liver storage of glucose [13] and via the insulin feedback on the glucose elimination, so there is probably no hope for an explanatory model of the glucose dynamics baaed on the data from the
PER HAGANDER
24
ET AL.
IVGTT alone. The basic assumption is instead that all glucose effects on insulin are mediated by the measured blood glucose concentration alone. For insulin distribution and elimination the model due to LiEko and Silvers in Table l(g) is used. The choice of one well-mixed plasma compartment has experimental support [14], but there is strong evidence for a nonlinear and glucose-dependent insulin elimination by the liver [ 141. These effects are not sufficiently visible for quantification from the simple IVGTT with only posthepatic measurements. In order to describe the release also during the glucose infusion time, reconsider the Grodsky model, i.e., Eqs. (1) and (2) for the labile packet storage pool and Eq. (3) for the provisionary factor. Neglect the redistribution (f’ = 0), and assume that the release is immediate, (1 /m = 0). The Grodsky model is designed for glucose perfusion in vitro with the basal level reaching zero. As in [9], the basal in vivo glucose concentration G, is subtracted to give a “stimulating” glucose level g(t). The basal secretion rate SR, is assumed to give the basal plasma insulin level I,. The replenishment is called y,. Because of the monotonic increase and decrease of the glucose concentration, it is not necessary to keep track of what happens with the insulin storage at the different threshold levels, so the difficult equation (2) can be simplified. When the glucose concentration rises from g(t) to g(t) + dg, the quantity & g(t), t) dg of insulin stored between the thresholds 0 = g(t) and 0 =g(t) + dg will be secreted. The “density” .$(g(t), t) is composed of the initial storage plus the part of the replenishment that was accumulated:
Part of the replenishment, y2(t). r( g(t))/r(oo), is of course directly released, but the rest is accumulated in the labile pool. When the glucose concentration is decreasing, no insulin stored in the labile pool will be released; only the direct release of replenished insulin remains. The secretion rate y during an IVGTI can thus be described by:
y(t)=y,(t)+
~Y?(Ih
rdt)=
OT
where an initial zero steady state was assumed.
(16)
MODELS FOR INSULIN RESPONSE TO I.V. GLUCOSE
The function
25
h in Eq. (3) is linearized:
(17) which gives the equations (e) of Table 1 for the replenishment y2. In Table 1 the simpler equations (d) and (f) are given instead of Eq. (16). Equations (d) and (f) follow from the LiEko and Silvers’s assumption that all the replenished insulin is released. Both possibilities will be investigated in the next section. The kinetics of the insulin release and distribution was assumed to be fast. When venous insulin should be described, it is reasonable to include a release-distribution time constant:
!!L dt
The model described following respects:
in Table
-
m(_!J
-Y1
-Yz).
(18)
1 differs from the LiEko-Silvers model in the
(1) the glucose infusion time is included, (2) a two-compartment model is used to describe arterial glucose, (3) a “positive derivative term” is included during the rise of the glucose concentration, while the original Grodsky release mechanism (16) would add a fourth difference: (4) there is an “immediate” glucose control of the release, and the insulin replenishment may accumulate in the labile pool. SIMULATION
STUDIES
In modeling the insulin release during an IVGTT it is important to let the arterial glucose concentration control the release. Although arterial sampling is not feasible for a clinical test, it is of greatest importance for research purposes. Tranberg et al. [14] report the results from more than fifty such tests. The arterial curves of a typical test are shown in Fig. 5 together with a manual fit to the new model in Table 1. The time constants l/k, and l/ k2 in the glucose model [Table l(b)] were chosen to be 60 min and 2 min respectively. The initial distribution volume V, was chosen as 5 liters, in good agreement with the blood volume, while the relative contribution of the dose to the slow and fast pools was a and : respectively. The corresponding venous glucose curve, Fig. 6, could be described by the same model structure with l/k, = 60 min l/k2 = 3 min, V8= 14 liters (approximately the extracellular volume), and the relative dose contribution to the slow pool was f.
26
PER HAGANDER
ET AL.
300
300
Time
Cminl
FIG. 5. Arterial IVGTT data and model fittings for two different sets of parameter values for Table l(d) and one set for Eq. (16). Table l(d): (a) a=0.5 min-‘, /?=0.065 mU/mgmin, x,,=240 mu/ml, n=0.33 mir-l; (b) a=0.04 mir-‘, /3 = 0.0022 mU/mgmin, xIIIm = 170 mu/ml, n = 0.14 min-*. Equation (16): (c) a=O.l min-‘, /3=0.03 mU/mgmin, x,,- -240 mu/ml, n=0.33 min-‘, I,=8 mu/ml, G, =300 mg/lOO ml.
Of the two alternatives for early insulin release, (16) is in better agreement with a two-pulse experiment, while Table l(d) is simpler. Both of the models contain the function 5(0,0), which could not be determined from a single IVGTT. Even the parametrization (5) is too complicated to estimate. A simple linear approximation is proposed:
which would give an almost identical fit to the data. Different glucose doses may require different values of the parameter x,,,, but for a single IVGTT Eq. (19) could be used instead of (5).
MODELS FOR INSULIN RESPONSE TO I.V. GLUCOSE
Patient
Time FIG. 6.
l/m.
asO.2
27
# 20
Cminl
Venous IVG’IT data and a fit of the model including a release time constant min-I, /3=0.004 mU/mgmin, x,,=600 mu/ml, n=0.33 min-‘, m=0.25
mill-‘.
The two release alternatives Table l(d) and (16) also give identical fits. (See curve c in Fig. 5.) The lower release of replenished insulin in (16) for the lower glucose values could be compensated for by a slower replenishment time constant l/a and a higher “replenishment gain” p. The simplest version of the new model, given by Table 1 and (19), thus contains five parameters for glucose: G,, k,, k,, b, Vg; and five parameters for insulin: Zm,n,~,/3,xmax. The insulin distribution volume V, is impossible to obtain from this kind of experiment, so the parameters x,, and p are normalized with respect to Vi. The time constant for the late release, l/o, is difficult to estimate. It is to some extent hidden in the effects of the early release, and the relative accuracy of the insulin measurements at the level of about 25 pU/ml has to be high for a good quantification of (Y. As can be seen from Fig. 5, it is mainly the interplay with the insulin elimination time constant l/n that causes trouble. Two very different values of cxand n give about the same fit. It is reasonable to assume an u priori value of the parameter n: 1/n = 3 min, and estimate the other parameters, giving (r =OS mm-‘, p =0.0065 mU/mgmin, x,, = 240 Z~U/rnl. For the venous data Fig. 6 shows that the new model with a release-distribution time constant l/m =4 min would give a good fit. Clearly the
28
PER HAGANDER
ET AL.
interpretation of the parameters p and x,_ is difficult when venous glucose concentration is assumed to control the release. As seen from Fig. 6, a much larger x,,, is required. CONCLUDING
REMARKS
Different models have been advocated for the insulin release of the IVGTT, e.g. [7-9, 111. The mechanisms are complicated, and the suggested models are verified only partly by experiments. The information content of glucose infusion tests is limited, and it is therefore necessary to reduce the modeling. In the Grodsky model this can be done by neglecting redistribution, which simplifies the equations considerably. However, as pointed out by LiEko and Silvers [9], further simplifications are necessary for the IVGTT. The simplifications chosen may be critical and should be adjusted to the purpose of the study. Insulin distribution and elimination was described by the simplest linear model. Such a model is probably not valid for pancreatic insulin release, whereas it should be more appropriate for posthepatic insulin molecules. It also proved meaningless to model potentiation and/or inhibition, and likewise to model insulin effect on blood glucose. The main advantage of the model is that it can quantify the influence of glucose on the early and late insulin responses. Such quantification and separation is obligatory for proper estimation of the effect of an abnormal early or late insulin release in response to glucose. Other inputs might give more information about the secretory dynamics, but would require more complex models. It is possible that such in uiuo experiments will prove equally insufficient with respect to parameter identification. For long experiments new aspects must be considered, e.g. the synthesis of insulin [15, 161. This work was supported by grants from the Swedish Diabetic Association, The Royal Physiographic Society Lund, and the Swedish Medical Research Council. REFERENCES
1 K. Johansen, 2 3 4 5
J. S. Soeldner, and R. E. Gleason, Insulin, growth hormone, and glucagon in prediabetes mellitus-a review, Mefabolism 23: 1185-l 199 (1974). G. M. Reaven, R. Bernstein, B. Davis, and J. M. Olefsky, Nonketotic diabetes mellitus: insulin deficiency or insulin resistance? JAMA 60: 8k88 (1976). v. W. Bolie, Coefficients of normal blood glucose regulation, J. Appl. Physiol. 16: 783-788 (1961). E. Ackerman, J. W. Rosevear, and W. F. McGuckin, A mathematical model of the glucose-tolerance test, Phys. Med. Biol. 9: 203-213 (1964). G. Segre, G. L. Turco, and G. Vercellone, Modeling blood glucose and insulin kinetics in normal, diabetic and obese subjects, Diabetes 22: 94103 (1973).
MODELS FOR INSULIN RESPONSE TO I.V. GLUCOSE
29
6 G. M. Grodsky, D. Curry, H. Landahl, and L. Bennett, Further studies of the dynamic aspects of insulin release in who, with evidence for the two-compartmental storage system, Actu Diabet. Lotim. 6 (Suppl. 1): 554-579 (1969). 7 R. N. Bergman and J. Urquhart, The pilot gland approach to the study of insulin secretory dynamics, Recent Progr. Hormone Res. 27: 583605 (1971). 8 G. M. Grodsky, A threshold distribution hypothesis for packet storage of insulin and its mathematical modeling, J. CZin Znoest. 51: 2047-2059 (1972). 9 V. Lick0 and A. Silvers, Open-loop glucose-insulin control with threshold secretory mechanism: analysis of intravenous glucose tolerance tests in man, Math. Biosci. 27: 319-332 (1975). 10 J. I. Thorell, Effect of transient elevation of plasma insulin within physiologic levels
(simulated early insulin response) on blood glucose, J. CZin Endocrin Metab. 37: 423430 (1973). 11 E. Cerasi, G. Fick, and M. Rudemo, A mathematical model for the glucose induced insulin release in man, Europ. J. Clin. Znoest. 4: 267-278 (1974). 12 L. Koncz, R. E. Gleason, and H. Otto, Insulin secretory dynamics after two consecutive intravenous stimulations, Diabetes 25: 370 (1976). 13 M. El-Refai and R. N. Bergman, Simulation study of control of hepatic glycogen synthesis by glucose and insulin, Am J. Physiology 231: 1608-1619 (1976). 14 K.-G. Tranberg and H. Dencker, Plasma disappearance of unlabeled insulin following short, intraportal and peripheral infusions in man. II. Physiologic aspects, submitted for publication. 15 H. S. Sando and G. M. Grodsky, Dynamic synthesis and release of insulin and proinsulin from perfused islets, Diabetes 22: 354360 (1973). 16 D. Porte, Jr., and A. A. Pupo, Insulin responses to glucose: evidence for a two pool system in man, J. Ch. Znwst. 48: 2309-2319 (1969).
of Surgety, Lund Universiv,
KARL-G&4N Department
LunCr: Sweden
TRANBERG,
of Nuclear Medicine, General Hospital, Malmij, Sweden
JAN THORELL, Biocybernetics
Laboratory,
Uniwrsi@ of California, Los Angeles, USA
AND
JOSEPH DISTEFANO III Department
Received
of Automatic
I4 November
Control, Lund Institute of Technology, LUG
Sweden
1977; revised 2 May 1978
ABSTRACT The Grodsky packet storage model describes many features of insulin release, but at present more or less arbitrary simplifications are necessary. The consequences of various simplifications are discussed, especially with regard to identification of parameters thought to be of importance for glucose tolerance. In particular, the insulin release dynamics of the ordinary intravenous glucose tolerance test is examined. The proposed model contains the following features: It considers arterial rather than venous blood glucose concentration as the stimulus, it takes the glucose injection time into account, and it contains a positive derivative term during the rise of the glucose concentration. When the insulin elimination-rate time constant is fixed to an a priori value, model fitting gives a clear quantification of the sensitivity of early and late insulin release to glucose.
INTRODUCTION The primary defect in maturity-onset diabetes mellitus-impaired insulin secretion or impaired tissue sensitivity to insulin-remains unknown. There is a growing tendency to favor a subnormal insulin response as the basic defect in the diabetic syndrome, but this conception has also been seriously challenged [ 1,2]. The difficulty in detecting the initial lesion rests mainly on the following. Among subjects with normal glucose tolerance there are wide variations in insulin release and insulin sensitivity, and the correlation MATHEMATICAL
BIOSCIENCES
0Elsevier North-Holland,
Inc., 1978
42, 15-29 (1978)
15
0025-5564/78/090015 + 15+ 15t02.25
PER HAGANDER
16
ET AL.
between these two factors is very weak. Blood glucose and plasma insulin levels are continuously modifying each other, making it difficult to establish the dynamic relationship between glucose and insulin in vivo. Most earlier modeling studies have been based on time-invariant linear differential equations [3-51, whereas it has been shown that the insulin secretion rate is both nonlinear and time-dependent [6,7]. Many dynamic characteristics of insulin release can be described by the Grodsky packet storage model [S], which has a solid in vitro experimental background. Recently, LiEko and Silvers [9] presented a modified Grodsky model, suitable for application to the intravenous glucose tolerance test (IVGTT) in man. We have investigated the insulin secretion of the IVGTT by means of the Grodsky model and modifications thereof. This has led to a model with a complexity adjusted to the limited amount of information and focused on those aspects of insulin release that are most important for glucose homeostasis. THE GRODSKY
PACKET
HYPOTHESIS
In the packet storage model [8] the insulin release system contains two pools: the labile pool, which releases insulin to the circulation, and the storage pool, which replenishes the labile pool (see Fig. 1). The secretion rate of insulin from the labile pool depends on the glucose concentration. Insulin in the labile pool is stored in a large number of “packets”, releasing
FIG. 1. Block diagram of the Grodsky packet storage model for insulin secretion. The glucose concentration G controls the release of insulin from the labile pool. The provisionary factor P controls the replenishment of insulin from the storage pool to the labile pool.
MODELS FOR INSULIN RESPONSE TO I.V. GLUCOSE
17
insulin at different preset glucose levels called thresholds. by the function &0, t): 5(&t)=
This is described
Amount of insulin in the labile pool at time t, having 1. ( the glucose threshold 4
All the insulin with a threshold lower than the glucose level G is assumed to be secreted at a rate proportional to 5 with proportionality constant m:
(1) Also, replenishment of the labile pool from the storage pool is assumed to be controlled by the glucose concentration, but indirectly via the production and utilization of a so-called provisionary factor P. The insulin replenishment rate to threshold f7 is assumed to be proportional to the provisionary factor P(t), with the proportionality coefficient Redistribution insulin different is in model. packets threshold insulin assumed be to threshold o a proportion [(e,t), proportionalcoefficient being function the level From assumptions mass equation the pool described [8]:
$p,t)=-sR(o)+
mqt)-
loXv’(w)do.Ke,r)+v’(e )~mt(~~+h (4
The production rate of the provisionary factor function h of the glucose concentration, so that ;P(t)=-a[P(r)-h(G(r))]. Grodsky
was assumed
to be a
(3)
assumed that
(4) where f andf’ are two constants. For his perfusion experiments Grodsky [8] suggested explicit expressions for the functions [(e,O) and h(G):
PER HAGANDER
18 where x,,,,
c, and k are constants,
f.X,,X. h(G)=
ET AL.
and OSG”
8.875x1~‘+2.25x10’5G3+3.5x106G7+G’o’
(6)
The Grodsky model for insulin release can thus be summarized by the equations (lH6). The dynamics is described by one ordinary differential equation for p(t), Eq. (3) while Eq. (2) can be considered as an infinite number of differential equations for E(r9,t), 0<8< 00, weakly coupled together via the integral J F&o, t)do. The model was originally designed to describe results from perfused rat pancreas, and computer simulations in [8] showed good agreement with measured data. For special glucose perfusion experiments such as step or staircase functions, the model can actually be reduced to a system of ordinary differential equations. The model then depends on the actual levels used, and it cannot be applied to other input levels. These are the cases simulated by Grodsky [8]. Another situation in which the Grodsky model may be simplified is for short observation periods like the 18-min ramp stimulation in 181. At time t redistribution and replenishment of insulin with a threshold higher than the glucose level G(t) may then be disregarded, and &0,0) may be used instead of 5(0,1). The model described by Eqs. (lH6) requires an enormous amount of experimentation for its validation, and even for simulation purposes it is too complicated. But conceptually the model is appealing, and it should be useful as a platform for further simplifications in different directions. APPLICATION OF THE (LICK0 AND SILVERS)
PACKET
HYPOTHESIS
TO THE
IVGTT
In the IVGTT a brief infusion of glucose is given into a peripheral vein, and blood samples are taken during the following hour. Figure 2 shows the results of a typical IVGTT in a normal man. LiEko and Silvers modeled the kinetics of distribution and elimination of plasma insulin as a well-mixed plasma compartment with volume V and elimination rate constant n: dI -=-nz+E dt The glucose kinetics was
was
as simple
V’
(7)
The glucose injection go, elimination was with rate constant k, from the
MODELS FOR INSULIN
RESPONSE
19
TO I.V. GLUCOSE
300
300 Patient + 20
_
200 2
200
z 5
g
L3
F " 100 % 2 3 01
$ 100 2 .1E cl-
B 0
0 0 Time
FIG. 2.
LO
20 Cminl
60
A typical normal IVGTT’. Venous glucose and plasma insulin
initial concentration
2
concentrations.
G(0) = G, + go: t
t>O.
(8)
G, represents the basal level. LiEko and Silvers thus assumed that there is no detectable feedback influence of insulin on the glucose elimination during the one hour period of the test. Although there is evidence against this hypothesis [IO], it is not necessary to include such an effect in a model of the insulin response. LiEko and Silvers’s model for insulin secretion is based on Grodsky’s. Certain simplifications were made, thus eliminating the complexity of Eq. (2). The time constant for insulin release, l/m, was assumed to be zero. Thus, when the glucose concentration instantaneously jumps from G, to G(O), insulin is released from the labile pool as an impulse, and the insulin concentration is instantaneously raised from I = I, to I = I, + i,,. Thus Z(0) = I,
+ i(0) = I,
+ io= I,
+ +
JG(o)&f3,0)d9 G,
describes the early phase of insulin secretion. LiEko and Silvers also suggested the function &Q,O) defined by Eq. (5). Furthermore, no redistribution is considered, i.e., f’= 0 in Eq. (4).
PER HAGANDER
20
ET AL.
The replenishment from the storage pool makes further secretion possible. With the assumptions l/m = 0 and f’ = 0, Eqs. (1) and (2) simply give
SR(@,t) = Thus, the so-called
0”’ )P(% 1 9
“late” insulin secretion
@;z;i;?)Y response would be
SR(r)=P(QjG(f)y(+#=P(QI(G(t)),
(10)
0
with the notation
introduced in [8] and [9]. However, LiEko and Silvers instead assume that all the insulin available for replenishment is also available for immediate release, i.e., SR(r)=P(t)I(co)=SR,+y(Q
t>O,
where SR, is the basal secretion rate. They also assume structure of the function h in Eq. (3): P(G-G,)+SR,/I’(oo),
G-G,>O, otherwise,
(12) a simple linear
(13)
so that Eq. (3) results in
!!L -Q-/&)9 dt y(0)
=o.
g>o,
(‘4)
The LiEko-Silvers model [Eqs. (7) (8) (9) (12) (14)] has in fact an alternative physical interpretation, shown in Fig. 3. In this interpretation the late response y(t) is elicited directly from the storage pool; no replenishment or redistribution in the labile pool is included in the model. Packet release from the labile pool is controlled instantaneously by the glucose concentration, while release from the storage pool is controlled by “old” glucose concentrations. By assuming the form (12) instead of the form (lo), the nonlinear function I(B) or &S,O) only enters into the “early” insulin response, i,, and the number of parameters necessary to quantify the IVGTT is drastically reduced. This is a fundamental departure from the original Grodsky model,
21
MODELS FOR INSULIN RESPONSE TO I.V. GLUCOSE
FIG. 3. Alternative block diagram of the LiEko-Silvers model for insulin secretion. The glucose concentration G controls the release of insulin from the labile pool, and the provisionary factor P controls the release of insulin from the storage pool.
and it may result in qualitatively different behavior of the two models. Specifically, there is no instant effect of glucose at time t >0 on insulin secretion at time I > 0 in the LiEko-Silvers model, not even if g(t) = 0. In the Grodsky model g(t) = 0 would give almost no secretion rate. An experiment that would distinguish between the two models, Eq. (10) and Eq. (12), is shown in Fig. 4. Both models are simulated during a
z
(-J
I
0 Time
20
I
I
LO
I
60
I
I
80
I
10
Cminl
FIG. 4. Two-pulse glucose perfusion experiment. Simulation showing the difference between the Grodsky model (a) and the LiEko-Silvers model (b).
PER HAGANDER
22
ET AL.
constant glucose perfusion interrupted by a period during which glucose is maintained at its basal level. During a low-glucose interval Eq. (12) gives almost no change in insulin secretion, while Eq. (10) gives a secretion rate slightly above the basal rate. The corresponding insulin is stored in the labile pool and later released as an early response to the second pulse. For Eq. (12) there is no such response. It should be noted that the LiEko-Silvers model was not intended to be valid for a glucose perfusion. The literature contains some results from similar experiments. Grodsky [8, Fig. 31 describes a perfusion experiment with the low glucose level being zero. In the pilot gland experiments by Bergman and Urquhart [7] a basal glucose concentration of 150 mg/lOO ml was used. In both experiments there is a pronounced early response to the second pulse and a substantial decrease of the secretion during the low glucose interval, favoring Grodsky’s original assumption represented by Eq. (10). Some in uiuo studies (Cerasi et al. [ 111, Koncz et al. [ 121) show similar results, although less transparent. A delayed early insulin release after the second pulse was noted in [12], and a similar phenomenon can in fact be seen in [8]. As will be demonstrated below, it is not possible to distinguish between (10) and (12) during an IVGTT. ALTERNATIVE
MODEL
The early insulin release during the first few minutes of the IVGTT should be depicted as accurately as possible [l, lo]. The large amount of glucose, usually 25 g, infused in an IVGTT requires that the injection time be taken into consideration. A minimum of two minutes is required to ensure that the test is not unduly uncomfortable to the patient. The equations taking the injection time into account are given in Table 1. The blood supporting the pancreas is arterial. Between arterial and venous blood concentrations there is a substantial “damping”, especially of the initial peak, during an IVGTT. In order to include this distribution effect it is necessary to use a two-compartment model. The elimination would then occur from both compartments, which is impossible to quantify from measurements in only one of them. A simple “black box” (input-output) approach is therefore taken: Eq. (6) where the variables x1 and x1 have no physiological meaning. Actually, any linear two-compartment system with a single infusion site and a single measurement site can be described by such a model. The parameter Vg is the initial distribution volume, while the parameters b, k,, and k, have no immediate physiological meaning. The parameter k, corresponds to what is clinically called k,. G, is the basal glucose concentration, T is the injection time and D is the total glucose dose given.
MODELS
FOR INSULIN
RESPONSE
TO I.V. GLUCOSE
TABLE 1 Equations of the Alternative
23
Model
64
$x,(r)=-k,x,+b+
8
&X*(1)=4*x*+(1-b)+ g
(b)
(4 Early insulin release:
v,(t)=
E(g(W)&W. O
(4
t>T
0, Late insulin release: d ;iiYz’
Insuliu distribution
-ay,+aSg(0,
o
-
OthfXWiSe
aY2,
(4
and elimination: Y=YI+Yz
(0
d. --I=-ni+$ dt
(g)
Z(t)-Z,+i(t)
(h)
Plasma insulin concentration:
The influence of insulin on glucose elimination is disregarded. The glucose model only represents the glucose response in a certain experiment. It is not required that differently large glucose loads give the same decay rate constants. The glucose dynamics is known to be complicated. There are nonlinearities both in the liver storage of glucose [13] and via the insulin feedback on the glucose elimination, so there is probably no hope for an explanatory model of the glucose dynamics baaed on the data from the
PER HAGANDER
24
ET AL.
IVGTT alone. The basic assumption is instead that all glucose effects on insulin are mediated by the measured blood glucose concentration alone. For insulin distribution and elimination the model due to LiEko and Silvers in Table l(g) is used. The choice of one well-mixed plasma compartment has experimental support [14], but there is strong evidence for a nonlinear and glucose-dependent insulin elimination by the liver [ 141. These effects are not sufficiently visible for quantification from the simple IVGTT with only posthepatic measurements. In order to describe the release also during the glucose infusion time, reconsider the Grodsky model, i.e., Eqs. (1) and (2) for the labile packet storage pool and Eq. (3) for the provisionary factor. Neglect the redistribution (f’ = 0), and assume that the release is immediate, (1 /m = 0). The Grodsky model is designed for glucose perfusion in vitro with the basal level reaching zero. As in [9], the basal in vivo glucose concentration G, is subtracted to give a “stimulating” glucose level g(t). The basal secretion rate SR, is assumed to give the basal plasma insulin level I,. The replenishment is called y,. Because of the monotonic increase and decrease of the glucose concentration, it is not necessary to keep track of what happens with the insulin storage at the different threshold levels, so the difficult equation (2) can be simplified. When the glucose concentration rises from g(t) to g(t) + dg, the quantity & g(t), t) dg of insulin stored between the thresholds 0 = g(t) and 0 =g(t) + dg will be secreted. The “density” .$(g(t), t) is composed of the initial storage plus the part of the replenishment that was accumulated:
Part of the replenishment, y2(t). r( g(t))/r(oo), is of course directly released, but the rest is accumulated in the labile pool. When the glucose concentration is decreasing, no insulin stored in the labile pool will be released; only the direct release of replenished insulin remains. The secretion rate y during an IVGTI can thus be described by:
y(t)=y,(t)+
~Y?(Ih
rdt)=
O
where an initial zero steady state was assumed.
(16)
MODELS FOR INSULIN RESPONSE TO I.V. GLUCOSE
The function
25
h in Eq. (3) is linearized:
(17) which gives the equations (e) of Table 1 for the replenishment y2. In Table 1 the simpler equations (d) and (f) are given instead of Eq. (16). Equations (d) and (f) follow from the LiEko and Silvers’s assumption that all the replenished insulin is released. Both possibilities will be investigated in the next section. The kinetics of the insulin release and distribution was assumed to be fast. When venous insulin should be described, it is reasonable to include a release-distribution time constant:
!!L dt
The model described following respects:
in Table
-
m(_!J
-Y1
-Yz).
(18)
1 differs from the LiEko-Silvers model in the
(1) the glucose infusion time is included, (2) a two-compartment model is used to describe arterial glucose, (3) a “positive derivative term” is included during the rise of the glucose concentration, while the original Grodsky release mechanism (16) would add a fourth difference: (4) there is an “immediate” glucose control of the release, and the insulin replenishment may accumulate in the labile pool. SIMULATION
STUDIES
In modeling the insulin release during an IVGTT it is important to let the arterial glucose concentration control the release. Although arterial sampling is not feasible for a clinical test, it is of greatest importance for research purposes. Tranberg et al. [14] report the results from more than fifty such tests. The arterial curves of a typical test are shown in Fig. 5 together with a manual fit to the new model in Table 1. The time constants l/k, and l/ k2 in the glucose model [Table l(b)] were chosen to be 60 min and 2 min respectively. The initial distribution volume V, was chosen as 5 liters, in good agreement with the blood volume, while the relative contribution of the dose to the slow and fast pools was a and : respectively. The corresponding venous glucose curve, Fig. 6, could be described by the same model structure with l/k, = 60 min l/k2 = 3 min, V8= 14 liters (approximately the extracellular volume), and the relative dose contribution to the slow pool was f.
26
PER HAGANDER
ET AL.
300
300
Time
Cminl
FIG. 5. Arterial IVGTT data and model fittings for two different sets of parameter values for Table l(d) and one set for Eq. (16). Table l(d): (a) a=0.5 min-‘, /?=0.065 mU/mgmin, x,,=240 mu/ml, n=0.33 mir-l; (b) a=0.04 mir-‘, /3 = 0.0022 mU/mgmin, xIIIm = 170 mu/ml, n = 0.14 min-*. Equation (16): (c) a=O.l min-‘, /3=0.03 mU/mgmin, x,,- -240 mu/ml, n=0.33 min-‘, I,=8 mu/ml, G, =300 mg/lOO ml.
Of the two alternatives for early insulin release, (16) is in better agreement with a two-pulse experiment, while Table l(d) is simpler. Both of the models contain the function 5(0,0), which could not be determined from a single IVGTT. Even the parametrization (5) is too complicated to estimate. A simple linear approximation is proposed:
which would give an almost identical fit to the data. Different glucose doses may require different values of the parameter x,,,, but for a single IVGTT Eq. (19) could be used instead of (5).
MODELS FOR INSULIN RESPONSE TO I.V. GLUCOSE
Patient
Time FIG. 6.
l/m.
asO.2
27
# 20
Cminl
Venous IVG’IT data and a fit of the model including a release time constant min-I, /3=0.004 mU/mgmin, x,,=600 mu/ml, n=0.33 min-‘, m=0.25
mill-‘.
The two release alternatives Table l(d) and (16) also give identical fits. (See curve c in Fig. 5.) The lower release of replenished insulin in (16) for the lower glucose values could be compensated for by a slower replenishment time constant l/a and a higher “replenishment gain” p. The simplest version of the new model, given by Table 1 and (19), thus contains five parameters for glucose: G,, k,, k,, b, Vg; and five parameters for insulin: Zm,n,~,/3,xmax. The insulin distribution volume V, is impossible to obtain from this kind of experiment, so the parameters x,, and p are normalized with respect to Vi. The time constant for the late release, l/o, is difficult to estimate. It is to some extent hidden in the effects of the early release, and the relative accuracy of the insulin measurements at the level of about 25 pU/ml has to be high for a good quantification of (Y. As can be seen from Fig. 5, it is mainly the interplay with the insulin elimination time constant l/n that causes trouble. Two very different values of cxand n give about the same fit. It is reasonable to assume an u priori value of the parameter n: 1/n = 3 min, and estimate the other parameters, giving (r =OS mm-‘, p =0.0065 mU/mgmin, x,, = 240 Z~U/rnl. For the venous data Fig. 6 shows that the new model with a release-distribution time constant l/m =4 min would give a good fit. Clearly the
28
PER HAGANDER
ET AL.
interpretation of the parameters p and x,_ is difficult when venous glucose concentration is assumed to control the release. As seen from Fig. 6, a much larger x,,, is required. CONCLUDING
REMARKS
Different models have been advocated for the insulin release of the IVGTT, e.g. [7-9, 111. The mechanisms are complicated, and the suggested models are verified only partly by experiments. The information content of glucose infusion tests is limited, and it is therefore necessary to reduce the modeling. In the Grodsky model this can be done by neglecting redistribution, which simplifies the equations considerably. However, as pointed out by LiEko and Silvers [9], further simplifications are necessary for the IVGTT. The simplifications chosen may be critical and should be adjusted to the purpose of the study. Insulin distribution and elimination was described by the simplest linear model. Such a model is probably not valid for pancreatic insulin release, whereas it should be more appropriate for posthepatic insulin molecules. It also proved meaningless to model potentiation and/or inhibition, and likewise to model insulin effect on blood glucose. The main advantage of the model is that it can quantify the influence of glucose on the early and late insulin responses. Such quantification and separation is obligatory for proper estimation of the effect of an abnormal early or late insulin release in response to glucose. Other inputs might give more information about the secretory dynamics, but would require more complex models. It is possible that such in uiuo experiments will prove equally insufficient with respect to parameter identification. For long experiments new aspects must be considered, e.g. the synthesis of insulin [15, 161. This work was supported by grants from the Swedish Diabetic Association, The Royal Physiographic Society Lund, and the Swedish Medical Research Council. REFERENCES
1 K. Johansen, 2 3 4 5
J. S. Soeldner, and R. E. Gleason, Insulin, growth hormone, and glucagon in prediabetes mellitus-a review, Mefabolism 23: 1185-l 199 (1974). G. M. Reaven, R. Bernstein, B. Davis, and J. M. Olefsky, Nonketotic diabetes mellitus: insulin deficiency or insulin resistance? JAMA 60: 8k88 (1976). v. W. Bolie, Coefficients of normal blood glucose regulation, J. Appl. Physiol. 16: 783-788 (1961). E. Ackerman, J. W. Rosevear, and W. F. McGuckin, A mathematical model of the glucose-tolerance test, Phys. Med. Biol. 9: 203-213 (1964). G. Segre, G. L. Turco, and G. Vercellone, Modeling blood glucose and insulin kinetics in normal, diabetic and obese subjects, Diabetes 22: 94103 (1973).
MODELS FOR INSULIN RESPONSE TO I.V. GLUCOSE
29
6 G. M. Grodsky, D. Curry, H. Landahl, and L. Bennett, Further studies of the dynamic aspects of insulin release in who, with evidence for the two-compartmental storage system, Actu Diabet. Lotim. 6 (Suppl. 1): 554-579 (1969). 7 R. N. Bergman and J. Urquhart, The pilot gland approach to the study of insulin secretory dynamics, Recent Progr. Hormone Res. 27: 583605 (1971). 8 G. M. Grodsky, A threshold distribution hypothesis for packet storage of insulin and its mathematical modeling, J. CZin Znoest. 51: 2047-2059 (1972). 9 V. Lick0 and A. Silvers, Open-loop glucose-insulin control with threshold secretory mechanism: analysis of intravenous glucose tolerance tests in man, Math. Biosci. 27: 319-332 (1975). 10 J. I. Thorell, Effect of transient elevation of plasma insulin within physiologic levels
(simulated early insulin response) on blood glucose, J. CZin Endocrin Metab. 37: 423430 (1973). 11 E. Cerasi, G. Fick, and M. Rudemo, A mathematical model for the glucose induced insulin release in man, Europ. J. Clin. Znoest. 4: 267-278 (1974). 12 L. Koncz, R. E. Gleason, and H. Otto, Insulin secretory dynamics after two consecutive intravenous stimulations, Diabetes 25: 370 (1976). 13 M. El-Refai and R. N. Bergman, Simulation study of control of hepatic glycogen synthesis by glucose and insulin, Am J. Physiology 231: 1608-1619 (1976). 14 K.-G. Tranberg and H. Dencker, Plasma disappearance of unlabeled insulin following short, intraportal and peripheral infusions in man. II. Physiologic aspects, submitted for publication. 15 H. S. Sando and G. M. Grodsky, Dynamic synthesis and release of insulin and proinsulin from perfused islets, Diabetes 22: 354360 (1973). 16 D. Porte, Jr., and A. A. Pupo, Insulin responses to glucose: evidence for a two pool system in man, J. Ch. Znwst. 48: 2309-2319 (1969).