An Androgen-Dependent Mathematical Model of the Menstrual Cycle

The condition PCOS (Polycystic Ovarian Syndrome) is an affliction that affects approximately $10\%$ of those who menstruate. The defining characteristics of PCOS are hyperandrogenism, irregular menstruation, and polycystic morphology in the ovaries caused by hormonal imbalances during menstruation. Since there is a focus on hyperandrogenism as a symptom, the research modeled the hormonal imbalances with a focus upon testosterone, the prevalent androgen during menstruation, and the introduction of anti-androgens. The modeling of PCOS hormones in an attempt to find an effective treatment is a relatively new endeavor, thus modeling these hormonal imbalances to find ways to lessen symptoms is imperative. Previously, research towards PCOS treatment has mainly focused upon insulin treatments, which have been shown to improve symptoms, and attempting to model the effects of an anti-androgen treatment has been less studied. In order to model these hormonal levels, we modified the differential equations, that Erica Graham put forth in her paper, to focus on pre-menstrual testosterone dependent follicle growth, in an attempt to better model testosterone levels during the menstrual cycle. A bifurcation analysis was unable to be performed on our model due to system and time limitations, which resulted in the model being unstable. Once a proper bifurcation analysis has been done, and if it produces a stable result, one can model the hormonal affects of hyperinsulinemia or of PCOS.