A Mathematical Model of Estrogen Production and Diffusion in Ovarian
Follicles
written”
Zachary Miller, Mentor: Dr. Karin Leiderman
D
Duke University, RTG Math Bio Summer Program
Abstract
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.
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.
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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
Results
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.
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:
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.
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 outerboundary. 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
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.
.
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.