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
Here, we derive an extended model by making the following assumptions. If x and y represent
the differences in the plasma insulin and glucose concentrations
from their
respective fasting
basal concentrations, and z represents the den- sity of the pancreatic b-cells in the proliferative
phase, then the rate of change of x satisfies the following rate equation.
dx dt
= R
1
y − R
2
x + C
1
1 where the first term describes the increase in the
insulin concentration due to the increase in the plasma glucose concentration; the second term
describes the reduction in x due to its access amount, while C
1
, is the rate of increase in x in the absence of y and x.
Since b-cells function in a negative feedback loop has been established to take a predominant
role in regulating both the basal plasma glucose and insulin concentrations Bajaj and Rao, 1987;
Lenbury et al., 1996, it is reasonable to assume that the rates R
1
, R
2
and C
1
in Eq. 1 should depend on the density of b-cells in the prolifera-
tive phase z. Eq. 1 thus becomes dx
dt =
r
1
zy − r
2
zx + c
1
z 2
where linear dependence on z is assumed while r
1
, r
2
and c
l
are positive constants. The system is thus considered to be a self-regulative one in a normal
subject, where basal plasma glucose level is pre- dominantly determined by insulin secretion from
normal b-cells. A decreased number of cells would maintain a normal basal plasma insulin concen-
tration by operating at an increased secretory rate per cell. This is reflected by the second term in Eq.
2, which becomes smaller in magnitude as z decreases so that x will be reduced at a slower rate
in the event that the number of cells becomes smaller.
As for the plasma glucose level y, if there is a decrease in insulin secretion due to a reduction to
1N of the normal number, n, of b-cells, the basal plasma glucose will increase until nearly normal
basal insulin levels are obtained Bajaj and Rao, 1987. Thus, the plasma glucose level is a function
of the b-cell capacity, Nn. Therefore, plasma glucose is assumed to satisfy the following rate
equation.
dy dt
= R
3
N z
− R
4
x + C
2
+ wt
3 where the first term accounts for the hyperbolic
increase in y due to a reduction in the number of b-cells, supported by the study of Turner et al.,
1979 mentioned previously. The second term accounts for the reduction in y due to the regula-
tive effect of plasma insulin, while C
2
accounts for the change in y in the absence of x and z. The last
term in Eq. 3 corresponds to the variation in the gastrointestinal absorption with time, accounting
for the absorption by the intestine and the subse- quent release of glucose into the bloodstream.
Finally, the density of pancreatic b-cells in the proliferative phase satisfies the following equation
used in the work of Rao et al. 1990: dz
dt =
R
5
y − yˆT − z + R
6
zT − z − R
7
z 4
where yˆ is the difference between the normal glucose level and its basal concentration, T is the
total density of dividing and non-dividing b-cells, while R
5
, R
6
, and R
7
are rate constants. The first term accounts for the increase in z due to the
interaction between the plasma glucose above the basal level and the non-dividing b-cells, while the
second term represents the increase in the dividing b-cells from the interaction between the dividing
and non-dividing cells. The last term is the rate of reduction in the cells density, proportional to its
current level with a rate constant R
7
. Thus, our reference model consists of Eqs. 2 –
4.
3. Single oral glucose administration
