Modeling insulin kinetics: responses to a single oral glucose administration or ambulatory-fed conditions
Catherine
Lloyd
Auckland Bioengineering Institute, The University of Auckland
Model Status
Please note that this particular variant of the model is an extension of the basic core model. To the three variables described in the core model: plasma insulin concentration (x), glucose concentration (y) and the density of the pancreatic beta cells (z), a fourth variable (u) is added which describes the temporal glucose absorption by the gastrointestinal tract (equations 30-34). Parameter values have been taken from the legend of figure 3. The model runs in COR and OpenCell and the units are consistent throughout, however the CellML model does not recreate the published results.
Model Structure
ABSTRACT: This paper presents a nonlinear mathematical model of the glucose-insulin feedback system, which has been extended to incorporate the beta-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.
model diagram
Schematic diagram of the pancreatic beta-cells. Glucose production is by beta-cells and uptake is by gastrointestinal cells. Beta-cell formation and loss represent the rates at which beta-cells replicate and die.
The original paper reference is cited below:
Modeling insulin kinetics: responses to a single oral glucose administration or ambulatory-fed conditions. Yongwimon Lenbury, Sitipong Ruktamatakul, and Somkid Amornsamarnkul, 2001, Mathematical Biosciences, 59, 15-25. PubMed ID: 11226623
insulin
pancreas
beta cell
endocrine
2001-00-00 00:00
Modeling insulin kinetics: responses to a single oral glucose administration or ambulatory-fed conditions (Extended Model)
The University of Auckland, Auckland Bioengineering Institute
The University of Auckland
Auckland Bioengineering Institute
Sitipong
Ruktamatakul
I've added a piecewise equation for "z" such that it can never equal zero. This now solves the NaN problem generated by having a zero as a denominator.
Catherine Lloyd
Lenbury et al.'s 2001 mathematical model of insulin kinetics.
pancreatic beta cell
The model now runs in PCEnv without producing "NaNs" but I'm still not sure it can recreate the published results.
Also, please note that this particular variant of the model is an extension of the basic core model. To the three variables described in the core model: plasma insulin concentration (x), glucose concentration (y) and the density of the pancreatic beta cells (z), we are adding a forth variable (u) which describes the temporal glucose absorption by the gastrointestinal tract.
Yongwimon
Lenbury
BioSystems
2007-09-03T13:59:03+12:00
glucose absorption
u
pancreatic beta cell density
z
Modeling insulin kinetics: responses to a single oral glucose administration or ambulatory-fed conditions
59
15
25
insulin
x
glucose
y
This is the CellML description of Lenbury et al.'s 2001 mathematical model of insulin kinetics.
11226623
Catherine
Lloyd
May
Catherine Lloyd
c.lloyd@auckland.ac.nz
Somkid
Amornsamarnkul
keyword
Catherine
Lloyd
May
2007-06-18T00:00:00+00:00
$\frac{d x}{d \mathrm{time}}=z(\mathrm{r1}y-\mathrm{r2}x+\mathrm{c1})$
$\frac{d y}{d \mathrm{time}}=\mathrm{epsilon}(\frac{\mathrm{r3}}{z}-\mathrm{r4}x+\mathrm{c2}+\mathrm{c3}u)$
$\frac{d z}{d \mathrm{time}}=\mathrm{epsilon}\mathrm{delta}(\mathrm{r5}(y-\mathrm{y\_})(\mathrm{z\_}-z)+\mathrm{r6}z(\mathrm{z\_}-z)-\mathrm{r7}z)$
$\frac{d u}{d \mathrm{time}}=-\mathrm{omega}v\frac{d v}{d \mathrm{time}}=\mathrm{omega}u$