/*
 * On the stability of a model of testosterone dynamics
 * 
 * Model Status
 * 
 * This model is known to run in both OpenCell and COR, although
 * there are no figures in the paper with which to compare the
 * CellML model simulation output. The units have been checked
 * and they are consistent.
 * 
 * Model Structure
 * 
 * Blood testosterone levels in human males are known to oscillate
 * periodically with a period of a few hours. Testosterone secretion
 * from the testes is stimulated by luteinizing hormone (LH) and
 * the production of LH is, in turn, stimulated by the hypothalamic
 * hormone luteinizing hormone releasing hormone (LHRH) (as summarised
 * in the figure below). Interestingly, the observed fluctuations
 * in the blood levels of testosterone can be traced back to the
 * pulsatile release of LHRH. Several mathematical models have
 * been proposed to describe the pulsatile release of these three
 * hormones. Some of these models depend on an introduced delay
 * in one of the variables in order to create oscillations in an
 * otherwise stable model. Examples of such delayed mathematical
 * models include those published by Cartwright and Husain (1980)
 * which describes the control of testosterone secretion, and also
 * the model published by Lenbury and Pornsawad (2005) to characterise
 * feedback control in the hypothalamus-pituitary-adrenal axis
 * in humans. Both of these models have been translated into CellML
 * and can be found in the model repository. However, while delayed
 * models can be biologically realistic (for example there is often
 * a delay between the secretion of a hormone and the binding of
 * that hormone to its receptor and the subsequent response), introducing
 * a delay to a mathematical model often comes at the expense of
 * increased model complexity. (In particular, it should be noted
 * that in its current form, CellML is unable to express time delays).
 * 
 * In the paper described here, Enciso and Sontag critique published
 * mathematical models which describe testosterone dynamics. In
 * particular they highlight how models which use time delays to
 * create hormone oscillations can exhibit stability.
 * 
 * model diagram
 * 
 * [[Image file: enciso_2004.png]]
 * 
 * Schematic diagram of the mathematical model representing the
 * luteinizing hormone releasing hormone (LHRH), luteinizing hormone
 * (LH), and testosterone system. The model has three main components:
 * the hypothalamus which secretes LHRH, the pituitary which responds
 * to LHRH by secreting LH, and the testes which, in responds to
 * LH, secrete testosterone. Finally, LH and testosterone feedback
 * on the hypothalamus to regulate LHRH secretion.
 * 
 * The original paper reference is cited below:
 * 
 * On the stability of a model of testosterone dynamics, German
 * Enciso and Eduardo D. Sontag, 2004, Journal of Mathematical
 * Biology, 49, 627-634. PubMed ID: 15480670
 */

import nsrunit;
unit conversion on;
unit hour=3600 second^1;
unit ng_ml=1E-6 kilogram^1*meter^(-3);
unit ng_ml_hr=2.7777778E-10 kilogram^1*meter^(-3)*second^(-1);
unit first_order_rate_constant=2.7777778E-4 second^(-1);

math main {
	realDomain time hour;
	time.min=0;
	extern time.max;
	extern time.delta;
	real R(time) ng_ml;
	when(time=time.min) R=0.1;
	real b1 first_order_rate_constant;
	b1=1.0;
	real R.b2 dimensionless;
	R.b2=1.0;
	real R.b3 dimensionless;
	R.b3=1.0;
	real R.g1 dimensionless;
	R.g1=10.0;
	real R.g2 dimensionless;
	R.g2=10.0;
	real f_T(time) ng_ml_hr;
	real T(time) ng_ml;
	when(time=time.min) T=0.0;
	real L(time) ng_ml;
	when(time=time.min) L=0.0;
	real L.b2 first_order_rate_constant;
	L.b2=1.0;
	real L.g1 first_order_rate_constant;
	L.g1=10.0;
	real T.b3 first_order_rate_constant;
	T.b3=1.0;
	real T.g2 first_order_rate_constant;
	T.g2=10.0;

	// <component name="environment">

	// <component name="R">
	R:time=(f_T-b1*R);
	f_T=(b1*R.b2*R.b3*T/(R.g1*R.g2));

	// <component name="L">
	L:time=(L.g1*R-L.b2*L);

	// <component name="T">
	T:time=(T.g2*L-T.b3*T);
}