BioSystems 59 2001 15 – 25
Modeling insulin kinetics: responses to a single oral glucose administration or ambulatory-fed conditions
Yongwimon Lenbury , Sitipong Ruktamatakul, Somkid Amornsamarnkul
Department of Mathematics, Faculty of Science, Mahidol Uni6ersity, Rama
6
Rd., Bangkok
10400
, Thailand Received 24 August 2000; accepted 28 August 2000
Abstract
This paper presents a nonlinear mathematical model of the glucose – insulin feedback system, which has been extended to incorporate the b-cells’ function on maintaining and regulating plasma insulin level in man. Initially, a
gastrointestinal absorption term for glucose is utilized to effect the glucose absorption by the intestine and the subsequent release of glucose into the bloodstream, taking place at a given initial rate and falling off exponentially
with time. An analysis of the model is carried out by the singular perturbation technique in order to derive boundary conditions on the system parameters which identify, in particular, the existence of limit cycles in our model system
consistent with the oscillatory patterns often observed in clinical data. We then utilize a sinusoidal term to incorporate the temporal absorption of glucose in order to study the responses in the patients under ambulatory-fed conditions.
A numerical investigation is carried out in this case to construct a bifurcation diagram to identify the ranges of parametric values for which chaotic behavior can be expected, leading to interesting biological interpretations.
© 2001 Elsevier Science Ireland Ltd. All rights reserved.
Keywords
:
Insulin kinetics; Mathematical model; Periodicity; Chaotic dynamics www.elsevier.comlocatebiosystems
1. Introduction
It has been established Bajaj and Rao, 1987 that the secretion of insulin and its biological
effectiveness in response to a glucose load is deter- mined mostly by the number and function of
b-cells in the pancreas, and the peripheral resis- tance to insulin action. Diabetes can arise from
insulin deficiency or insulin resistance and several studies have been carried out to determine their
relative contributions toward a patient’s hyper- glycemia Turner et al., 1979. A number of math-
ematical models have been proposed Ackerman et al., 1964; Geevan et al., 1990 to explain the
relationships between the concentrations of glu- cose and insulin in plasma in response to a glu-
cose load.
These models are usually too complex to be fitted to the small amount of clinical data avail-
able. Moreover, the data is generally of glucose concentrations only and has been collected over a
time scale of an experiment, which is too short for the model’s verification Bajaj and Rao, 1987.
Corresponding author. E-mail address
:
scylbmahidol.ac.th Y. Lenbury. 0303-264701 - see front matter © 2001 Elsevier Science Ireland Ltd. All rights reserved.
PII: S 0 3 0 3 - 2 6 4 7 0 0 0 0 1 3 6 - 2
More importantly, most models prove not to be consistent enough with known physiological data.
On comparing mathematical models, it is no longer sufficient to match numerical simulations
to experimental data and choose the best fitting model. We are in a better position if the model
can be shown to admit the same dynamic behav- ior exhibited by real data. Specifically, if sustained
oscillatory patterns are observed in the system under study, a model which does not admit such
a qualitative behavior must be discarded. As Ack- erman et al. 1964 have observed, body or natu-
ral rhythms range from very fast ones, such as those of the electro-encephalogram, to very slow
ones, such as seasonal variations. Some of these rhythms are induced by the external environment,
while others are characteristic of the living organ- isms. Apart from the oscillations of long periodic-
ity such as those of about 24 h, several physiological variables also show shorter rhythms,
which are often regarded as a complication that somehow must be avoided or compensated for in
the
design of
experiments in
biology and
medicine. Instead, these rhythms should be stud- ied and used as qualitative parameters of the
biological system. It was proposed by Ackerman et al. 1964 that the analysis of cyclic variations
and periodicity in a system could prove to be a useful tool to help distinguish health from disease.
A recent model for insulin kinetics, studied by Geevan et al. 1990, involves four variables, in-
corporating b-cell kinetics, a glucose – insulin feed- back system and a gastrointestinal absorption
term. Analysis of the model Bajaj and Rao, 1987 showed that only damped oscillations were admit-
ted in response to an instantaneous oral glucose administration. Several studies have, however, re-
ported persistent cyclic patterns in the plasma glucose and insulin concentrations in man and
monkeys Turner et al., 1979; Tasaka et al., 1994. In another study by Molnar et al. 1972, where
immunoreactive insulin measurements were made from ambulatory-fed subjects, plasma insulin and
glucose patterns in diabetics showed chaotic be- havior during the 48-h observation period.
We, therefore, extend the above mentioned model to take into account the role of the number
of b-cells in the regulation of plasma insulin level, following the results presented by Turner et al.
1979 in which the b-cells function in the negative feedback loop appears to have a predominant role
in regulating both the basal plasma glucose and insulin concentrations.
We show, by a singular perturbation analysis, that the model will admit sustained oscillations
for certain ranges of the system parameters. Moreover, when a term is incorporated to simu-
late the responses to an ambulatory-fed condition, we discover that chaotic behavior can be expected
in the system for certain ranges of the system parameters, yielding irregular patterns consistent
with the physiological data reported by Molnar et al. 1972.
2. Model development
