/*
 * Modeling insulin kinetics: responses to a single oral glucose
 * administration or ambulatory-fed conditions
 * 
 * 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
 */

import nsrunit;
unit conversion on;
unit rate=1 second^(-1);

math main {
	realDomain time second;
	time.min=0;
	extern time.max;
	extern time.delta;
	real x(time) dimensionless;
	when(time=time.min) x=5;
	real r1 rate;
	r1=0.15;
	real r2 rate;
	r2=0.12;
	real c1 rate;
	c1=0.1;
	real z(time) dimensionless;
	when(time=time.min) z=1;
	real y(time) dimensionless;
	when(time=time.min) y=0;
	real r3 rate;
	r3=0.05;
	real r4 rate;
	r4=0.03;
	real c2 rate;
	c2=0.1;
	real c3 rate;
	c3=0.005;
	real epsilon dimensionless;
	epsilon=0.1;
	real u(time) dimensionless;
	when(time=time.min) u=1;
	real r5 rate;
	r5=0.09;
	real r6 rate;
	r6=0.1;
	real r7 rate;
	r7=0.05;
	real z_ dimensionless;
	z_=1.01;
	real y_ dimensionless;
	y_=1.08;
	real delta dimensionless;
	delta=0.01;
	real omega rate;
	omega=0.05;
	real v(time) dimensionless;
	when(time=time.min) v=0;

	// <component name="environment">

	// <component name="x">
	x:time=(z*(r1*y+(-1)*r2*x+c1));

	// <component name="y">
	y:time=(epsilon*(r3/z+(-1)*r4*x+c2+c3*u));

	// <component name="z">
	z:time=(epsilon*delta*(r5*(y-y_)*(z_-z)+r6*z*(z_-z)-r7*z));

	// <component name="u">
	u:time=((-1)*omega*v);
	v:time=(omega*u);

	// <component name="model_constants">
}