This document discusses a mathematical model of the insulin-glucose feedback system. The model aims to describe the dynamic relationships between blood glucose concentration, insulin, and other hormones under various metabolic conditions. It represents insulin secretion from pancreatic beta cells as bursting oscillations driven by bistability in the cells. The model predicts ultradian oscillations in glucose and insulin levels consistent with real data. Analysis of the model's dynamics reveals two Hopf bifurcations that could explain physiological oscillations. Future work will refine the model and estimate parameter values to better predict glucose variability.

1. Modelling the Variability in the Insuline-
Glucose Feedback System
Vicent Ribas

2. Introduction
• Diabetes Mellitus (DM) can be considered as the 21st century pandemic and it
therefore represents an alarming threat to public health with rising trends and
severity worldwide.
• DM comprises a set of metabolic disorders, which share the phenotype of
hyperglycaemia (increment in blood glucose concentration).
• There are different types of DM, which are the result of the complex interaction
between genetic factors with other environmental and way of life factors
(sedentarism, diet, and so on).
• Depending on the etiology of DM, the factors that contribute to hyperglycaemia
may include:
• the reduction of the secretion of insulin,
• insufficiency in the expense of glucose at a metabolic level
• an incremented glucose production by the organism.

3. Introduction
• The disorders associated with DM suppose a serious compromise in the
organism whilst imposing a great burden on the National Health System. In
developed countries DM is the primary cause for renal failure, amputations
not associated to trauma of inferior limbs and blindness in adults. It has
been documented that 1.7% of the population worldwide suffers DM and this
incidence is suspected to grow in the mid and long term. DM is an important
cause for morbidity and mortality.
• There is an urgent need for improved diagnostic methods that provide more
precise clinical assessments and sensitive detection of symptoms
associated to DM (such as glucose variability). This critical task may be
enabled by the utilization of mathematical models that reliably describe the
dynamic interrelationships among key physiological variables in the
underlying physiology (i.e. blood glucose concentration and various hormones
such as insulin, glucagon, epinephrine, norepinephrine, cortisol and so on) under a
variety of metabolic and behavioral conditions (e.g. pre/postpandrial, exercise/rest,
stress/relaxation, treatment/no-treatment).

4. Physiology
• The Pancreas is composed by two different kinds of tissue:
1. The acini, which secrete gastric juices into the duodenum.
2. The Langerhans Islets, which secrete insuline, glucagon and
somatostatine.
•The human pancreas consists of about 1-2 million Langerhans Islets (0.3 mm
diameter). These islets are organized around small capillaries into which they
convey their hormones (i.e. the Islets have no connection whatsoever with the
gastric system).
• The islets contain three different kinds of cells: alpha, beta and delta, which
differentiate by their morphology and tinction properties. Here we are only
interested in the pancreatic beta-cells, which account for about 60% of the cell
population in the Langerhans Islets, and are in charge of the secretion of
insuline and amiline. The function of the amiline hormone is not yet fully
understood.

5. Physiology

6. Physiology
• Insuline mediates the uptake and metabolism of glucose.
• During most of the day, the energy available for the muscle tissue comes
from the fatty acids and not glucose. This is due to the fact that muscular
membranes are not very permeable to glucose unless they are stimulated by
insuline. However, there are two situations where muscle consumes glucose:
moderate/intense exercise and during digestion.
• In response to glucose, beta-cells of the Langerhans islet secrete insuline,
which causes the increased use or uptake of glucose in target tissues (like muscle,
liver and adipose tissue as it has been stated above).
• When blood levels of glucose decline, insuline secretion stops, and the tissues
begin to use their energy stores instead (fatty acids).
• Of course, interruption of this control system results in Diabetes Mellitus (DM),
which may result in heart disease, kidney failure, amputations and death if left
untreated.

7. Physiology
• A sudden increase in plasma glucose results in the immediate opening of the
insuline reservoirs in the pancreatic islets (beta-cells) during the first 5-10 minutes.
In a second phase, a part from this release the beta-cells start an enzymatic
reaction to produce and release more insuline into the bloodstream (note that the
rate now is much higher than in the first phase).
• This behavior is due to bursting in the pancreatic beta-cells (Islets) as it will be
discussed below (i.e. opening of Ca2+ and K+ channels).

8. Bursting in Pancreatic Beta-Cells
• Bursting in Beta-Cells begin at a saddle-node bifurcation and ends at a
homoclinic bifurcation. The fast subsystem is bistable, and requires only
one slow variable. The spike period tends to increase monotonically through
the active phase.
• In other words, Bursting arises from hysteresis and bistability in the beta-cell
model.

9. Main Hypothesis
•The frequency analysis of the ultradian glucose recording of 24 presented in
figure shows that there are oscillations at 64.12 min, 32 min and 21.4 min.

10. Frequency Analysis
• The frequency analysis of the ultradian glucose recording of 24 presented here
shows that there are oscillations at 64.12 min, 32 min and 21.4 min.

11. Frequency Analysis
• Oscillations occur during constant intravenous glucose infusion and are not
dependent on periodic nutrient absorption from the gut.
• Damped oscillations occur after a meal. Second, glucose and insuline
concentrations are highly correlated, with the glucose peak occurring at about 10-
20 minutes earlier than that of insuline.
• The amplitude of the oscillations is an increasing function of glucose
concentration, while the frequency is not.
• The oscillations do not appear to depend on glucagon.

12. Pulsatile Insuline Secretion Model

13. Pulsatile Insuline Secretion Model (Dynamics)

14. Pulsatile Insuline Secretion Model (Dynamics)
• Plasma insuline is produced at a rate f1(G) that is dependent on plasma
glucose.
• Insuline exchange with the remote pool Ii is a linear function of the concentration
difference between the pools (Ip/Vp-Ii-Vi) with constant rate E and Vp is the
distribution volume for insuline in plasma. Vi is the effective volume of the
intercellular space.

15. Pulsatile Insuline Secretion Model (Dynamics)
• Pancreatic insuline production is controlled by:
• Insuline dependent glucose utilization (uptake by the brain and nerve cells) is
controlled by:
• Glucose utilization by the muscle and fat cells is modelled by:
• And the insuline dependent term is given by:

16. Numerical Study (Phase-Space and Nullclines)

17. Numerical Study (Frequency and Bifurcation
Analysis)
• HB1: Ig=99.94 mg/dl.
• HB2: Ig=253.9 mg/dl.
• Oscillation Period: [60,120] min.

18. Numerical Study (Bi-parametric Analysis)

19. Conclusions
• It is quite likely that bursting oscillations are modulated by ultradian Glucose
intake.
• Insuline production is oscillatory by nature (both from the cell standpoint and
ultradian).
• Intermediate oscillations in Insuline-Glucose are yet not fully understood
although they become apparent in IVGTT and Induced Hypoglycaemic tests.
• The Insuline-Glucose Feedback system results in oscillations with physiologic
relevance (i.e. almost equivalent to those observed in real data).
• The presence of two hopf-bifurcations for most of the system settings has been
shown by means of a bi-parametric plot. However, it may be interesting to update
the model in order to account for intermediate oscillations (for example, by
changing the glucose input Ig and changing parameters).
• As future work, it may interesting to study the parameter tying and estimation in
order to draw more physiologically relevant conclusions for the model and, also, be
able to make predictions in glucose variability.