The qualitative physiological model
The first task when designing a mathematical model is to take the abstraction step from the transition of a physiological model to a compartment model. As the figures below show, our model for the female menstrual cycle includes the compartments hypothalamus, pituitary gland and ovaries, connected by the bloodstream. The model delivers a qualitative description of the following regulatory circuit:
In the hypothalamus, the hormone GnRH (gonadotropin-releasing hormone) is formed, which reaches the pituitary gland through a portal system in the form of impulses and stimulates the release of the gonadotropins LH (luteinizing hormone) and FSH (follicle-stimulating hormone) into the bloodstream. The gonadotropins regulate the processes in the ovaries, i.e. the multi-stage maturation process of the follicles, ovulation and the development of the corpus luteum, which control the synthesis of the steroids progesterone and estradiol and of the hormone inhibin. Through the blood, these hormones then reach the hypothalamus and pituitary gland, where they again influence the formation of GnRH, LH, and FSH.
The quantitative physiological model
From the mathematical point of view, a medical phenomenon (in healthy or ill organisms) is not sufficiently understood until a model has been obtained that permits medically sound quantitative predictions. To arrive at statements of this kind, the time sequences of the physiological processes are modeled with the aid of differential equations that describe the concentrations of the substances involved over time. Time delays are a typical phenomenon in physiological processes. They are either caused by transport through the blood (as a time difference between the site of their formation and the site of effect), by the delayed start of an effect due to biochemical processes, or by the recovery time of receptors.
To formulate the differential equations of the mathematical model, the physiological and biological processes that occur must be known very accurately. However, the exact chemical reaction mechanisms are often not understood in sufficient detail; often one only knows whether certain hormones have a stimulating or inhibiting effect on other hormones. In semi-quantitative modeling of such switch behavior, Hill functions are used.
If all processes are included, that generally results in a "large" system. We are not frightened of large systems since this is exactly our field of expertise, which is now indispensable in the new scientific area of systems biology. Examples are the program LIMEX for large scale differential-algebraic or descriptor systems and LARKIN for the simulation of LARge chemical KINetics. Both programs are freely available at ZIB (under the web address www.zib.de/nowak/codes.html).
From a mathematical perspective, the difficulty today is no longer to simulate such systems, but to identify the arising model parameters. Only few of them can be measured, and while the approximate range of value is known for some parameters, not even the orders of magnitude are known for others. The aim is to identify interpretable parameter values, so that not only the modeled concentration curves match the given measured data well, but also predictions can be made about the parameter range not covered by measurements (see ethical motivation above). Subtle mathematical techniques are needed to measure the quality of the applied approximations, for example, via affine covariant Gauss- Newton methods [P. Deuflhard: Newton Methods for Nonlinear Problems, 2004] as implemented in the program NLSCON for moderately large systems (accessible via the web address www.zib.de/Numerik/numsoft/CodeLib/nonlin.en.html). Our objective for the future is to not just model the idealized cycle of an "idealized woman", but to describe individual "virtual" patients by reliable models.