TEMPLATE DESIGN © A Mathematical Model of Estrogen Production and Diffusion in Ovarian Follicles Zachary Miller, Mentor: Dr. Karin Leiderman Duke University, RTG Math Bio Summer Program Abstract. Forward Euler numerical analysis was used to graphically assess the behavior of the mathematical model. Three variables are graphed at different time point up to t=1000: LH (yellow), androgen (blue), and estrogen (green). The x-axis represents the radial distance from the center of the follicle at r=0 to the outer membrane of the follicle at r=10. The y-axis corresponds to the concentration profile. Physiology 1. Follicle/Menstrual Cycle Overview The ovarian follicle, the basic unit of the ovary, is the central estrogen producing structure within the female, tasked with both oocyte development and maintenance of the menstrual cycle. It consists of several highly differentiated cell layers surrounding the oocyte that pass nutrients to the egg and produce estrogen from cholesterol. As seen in the above image, the outer cell layer is the theca, the inner cell layer is the granulosa, and within the center of the follicle is the oocyte. The follicle's ability to synthesize and release estrogen allows it to establish a two way communication via hormones with the hypothalamus-pituitary structure in the brain. In the absence of estrogen, the pituitary gland releases gonadotropins: FSH and LH that act on the follicles to up-regulate estrogen production. Estrogen feeds-back and inhibits gonadotropin secretion. This excitation/inhibition is an ideal example of a physiological control system, and ultimately accounts for the emergence of the menstrual cycle. This is better seen below: Several mathematical models exist that examine the overall menstrual cycle from a control theoretic viewpoint, but ignore the complex dynamics within the pituitary and follicle. This paper focuses on estrogen dynamics with the ovarian follicle 2. Estrogen Biosynthesis Estrogen production within the follicle, the result of gonadotropin stimulation of the theca and granulosa cell layers, is a highly compartmentalized process. The granulosa, sensitive only to FSH, is the cell layer that actually produces estrogen. However, if the theca, sensitive only to LH, is removed, estrogen production quickly drops. Why? The granulosa only contains enzymes capable of aromatizing an intermediate androgen that must be provided by the external environment. This cell type does not transcribe enzymes capable of producing estrogen from cholesterol de novo. Theca cells provide an elegant solution to this problem. These cells contain enzymes capable of converting esterified cholesterol to androgen that the granulosa cells can then use to produce estrogen. Thus theca stimulation by LH leads to production of androgen which diffuses into the granulosa layer and in the presence of FSH is aromatized to estrogen. This much simplified set of reactions is shown below. Mathematical Model 1.Introduction A mathematical model of estrogen production and diffusion was developed based on a set of reaction-diffusion PDEs describing the movement and reaction kinetics of the gonadotropins, and steroidal hormones: androgen, estrogen. The reaction-diffusion PDEs are written below: This set of PDEs work under the following assumptions: 1.Domain: The reaction-diffusion equations are defined on a horizontal section of the follicle ie a circular domain best described in axially-symmetric polar coordinates (one dimensional radial coordinates). 2.Variables: All substances that can possibly diffuse are tracked: steroidal hormones: estrogen and androgen, and gonadotropins: FSH and LH. 3.Boundary/Continuity and Initial Conditions : Material is expected to flux into or out of the follicle across the outer- boundary. Robin boundary conditions describe this motion by relating the flux to the difference in concentration inside and outside the follicle. On the other hand, no net movement of material should occur across the center of the follicle otherwise axial symmetry would be broken, hence the no-flux continuity condition. Finally, the initial condition assumes a follicle with no reacting/diffusing substance inside it, that is, a state before the gonadotropins have entered the follicle. 4.Reaction term: Below are the piecewise reaction term equations A mathematical model has been developed describing estrogen production and diffusion within ovarian follicles. This model was based on a series of time-dependent reaction-diffusion equations that tracked the movement and reaction kinetics of the pituitary gonadotropins, androgen production and diffusion within the follicle, and the ultimate result of the reactions: estrogen production and diffusion within the follicle. Initial results demonstrate pulsatile gonadotropin movement into the follicle, the reactions leading to androgen production localized to the follicular theca layer, and androgen movement into the granulosa layer leading to the production of estrogen. These preliminary results are not yet robust enough to match data from the biological literature, however it appears that the approach used in developing this model crudely fits the behavior of the hypothalamo-pituitary gonadal axis. Further model refinement and incorporation of more detailed reaction kinetics will allow not only an accurate description of normal pre-ovulatory estrogen production within ovarian follicles, but also should capture follicular behavior in pathological ammenorrheic states.. written ” Results Reactions are described following the estrogen biosynthesis model seen in image 1. LH is pulsed across the outer boundary, and reacts with the theca located in the region: 8<r<10 forming androgen. Androgen then diffuses into the granulosa layer: 6<r<8, and its reacts to form estrogen which then diffuses throughout the rest of the follicle. Early Diffusion: 0<t<50 At t=10, LH pulses (yellow) across the outer membrane, and has just begun to be consumed to form androgen in the region 8<r<10. Following from the initial condition, there is no androgen or estrogen within the follicle, these substances are produced through the reaction of LH within the theca. Later at t=50, the production of androgen (blue) can clearly be seen within the theca layer, however androgen has not diffused far enough yet to initiate production of estrogen. Late Diffusion: 100<t<1000 At t=400, it is clear that the LH pulse across the outer membrane has stopped, and remaining LH is consumed to form androgen. Both androgen and LH have diffused into the granulosa region: 6<r<8, this is seen by comparing the radial width of these substances at the different time points. This diffusion of androgen has resulted in the production of estrogen (green). At t=1000, estrogen concentration has increased significantly, Both androgen and LH have diffused significantly, and without any more LH boundary pulses, both androgen and LH eventually go to 0 while estrogen begins to diffuse out of the follicle.