|Speciale, modelbygger-variant, 3. modul, 2009, id:389|
|Findes på RUb:||ja|
In this thesis two mathematical models of the hypothalamic-pituitary-adrenal-axis (HPA-axis) are build using well known physiological mechanisms. The HPA-axis controls the secretion of the hormones CRH, ACTH and cortisol. The regulation of these hormones are important to human health. These hormones are the variables in two systems of coupled non-linear differential equations that constitute the models. The models includes a negative feedback of cortisol on ACTH. The first model has a negative feedback from cortisol on CRH corresponding to the "standard biology textsbook" description of the HPA-axis. The second model allows a feedback from cortisol on CRH to be positive og negative depending on the cortisol concentration by including mechanisms from hippocampus. For parameter values in a physiologically relevant range it is investigated if the models are capable of guaranteeing solutions with reasonable levels in hormone concentration. It is investigated if the models are capable of producing the ultradian oscillations that are observed in data of hormone concentrations. It is investigated if an external imposed function on the differential equation governing the CRH concentration can cause the circadian rhythm that is seen in the concentrations of ACTH and cortisol. Previous papers of the HPA-axis  and  claim to make models showing ultradian oscillations. We analyze the two models and find significant drawbacks that must be elaborated for a successful model taking care of the physiological mechanisms of the HPA-axis. Results of analytical investigation of our models For both models the results of the investigation is that all solutions end in a trapping region in the positive octant of (R3), thus guaranteeing reasonable levels in hormone concentration. Within this trapping region there exists at least on fixed point. The first model has a unique fixed point. The unique fixed point is locally stable for all physiological choices of parameters. Therefore no Hopf bifurcation is possible as an explanation for the ultradian oscillations in data. For the second model more than one fixed point is possible. The stability of a fixed point is categorized depending on the sign of the feedback on CRH at the fixed point. A sufficient, easily applicable criteria for a unique, globally stable fixed point is formulated for a more general model. This can be applied on the two specific models. Results of numerical investigation of our models In the case of a unique fixed point this is asymptotically stable for all reasonable parameter values and initial conditions. Perturbating the parameters in the second model makes the system undergo a bifurcation where two new fixed points emerge. In the case of three fixed points there is one unstable fixed point and two asymptotically stable fixed points. For all reasonable values of parameters and initial conditions the solutions converge towards one of the two stable fixed points. Thus for reasonable parameter values neither of the models are capable of producing the ultradian oscillations. The analytical criteria for a globally stable fixed point is fulfilled for some set of parameters within physiologically relevant ranges for both models. An external input in the differential equation governing CRH is capable of showing circadian oscillations in ACTH and cortisol concentration.