Table 1.
Menstrual cycle state variables and initial conditions.
Fig 1.
The Follicular and Luteal Phases of the menstrual cycle.
The figure shows the transition of a follicle, from its growth in the follicular phase to its rupture during ovulation, as well as its transformation to a corpus luteum and degradation in the luteal phase. The blue arrow indicates the control of the hypothalamus in the secretion of pituitary hormones. The black arrows represent the influence of the pituitary system on the ovarian system through FSH and LH, and the orange arrows show the response of the ovarian system through E2, P4, and Inh.
Fig 2.
Pituitary and ovarian hormone levels in a normal menstrual cycle.
Data digitized from the study by Welt et al. [15] is interpolated by cubic splines. The hormones LH, FSH, and E2 peak in the late follicular phase while P4 and Inh reach maximum value in the luteal phase.
Fig 3.
The diagram (adapted from [13]) shows the 13 states (RPLH, RPFSH, LH, FSH, RcF, GrF, DomF, Sc1, Sc2, Lut1, Lut2, Lut3, Lut4) in the menstrual cycle model. H+ denotes stimulation by hormone H while H− denotes inhibition. The red arrow presents the release of the pituitary hormone from the reserve pool to the blood. The black arrow indicates the influence of the pituitary system on the ovarian system while the orange arrow denotes the ovary’s feedback. The green dashed arrow denotes inhibition of P4 in the RcF stage. The blue arrow describes the transition of a follicle from one ovarian stage to the next. The gold dashed arrow denotes the state contributing to the ovarian hormone production. The pink arrow represents the infusion of exogenous hormone to the ovarian system.
Fig 4.
The blue curves describe the dynamics of the pituitary and ovarian hormones predicted by the model without exogenous estrogen and progesterone. The vertical lines partition the red curves into four cycles. Each partition presents the 28-day normal cycle hormone data extracted from Welt et al. [15].
Table 2.
Sum of squared residuals between the model output and Welt data, and model output peaks.
Fig 5.
Model output with constant dose of exogenous hormone.
(black curve) or
(full green curve) is the hormone H model output in case of 20 pg/mL per day of exogenous E2 or 1.4 ng/mL per day of exogenous P4 is administered for 28 days, respectively. The stipulated red curve represents the data for the 28-day normal cycle extracted from Welt et al. [15]. The addition of exogenous E2 or P4 reduces the peak of each of the five hormones.
Fig 6.
The vertical axes in (A) and (B) present the maximum (full black curve) and minimum (full red curve) values over a 28-day cycle reached by LH and P4, respectively, when the corresponding amount of exogenous E2 is given. Panel (B) is similar to (A) but shows that increasing dosage of causes decreasing amplitude of the variation in P4 level. Anovulation is attained when
pg/mL.
Fig 7.
Shown in (A) and (B) are the maximum (full black curve) and minimum (full red curve) values attained by LH and P4 over a 28-day cycle resulting from the administration of the corresponding dosage of . Panel (B) illustrates diminishing fluctuation in P4 value. Anovulation is achieved between
ng/mL and
ng/mL.
Table 3.
Percentage decrease in model output peak with the administration of exogenous estrogen/progesterone.
Fig 8.
Constant dose combination therapy.
(A) The black curve represents the P4 model output for a combination treatment with ng/mL per day and
pg/mL per day administered for 28 days. The red circles with the interpolated curve represents the 28-day normal cycle hormone data extracted from Welt et al. [15]. The treatment suppresses the P4 concentration achieving anovulatory state. In (B) the curves composed of points (
), correspond to a combined dosage of
and
resulting in a P4 maximum value of k ng/mL. Anovulation is attained when k < 5. The yellow region (
ng/mL) corresponds to ovulation. On the lower left portion, an almost straight line with slope −0.1 separates the regions of ovulation (in gray) and anovulation (in pink).
Fig 9.
Model output with application of optimal control u1.
The full black curve is the model output when the optimal control u1 (magenta in panel (A)) is applied. (full blue curve) denotes the hormone model output without the influence of u1. This is the normal cycle solution. The Welt data for a normal cycle is presented in red circles with the interpolated curve. The maximum P4 value is 4.43 ng/mL.
Fig 10.
Follicular mass with application of optimal control u1.
The full black curve describes the follicular mass when the optimal control u1 is applied. The full blue curve shows the follicular mass without the application of u1. In (A), the steep decline in RcF is evident on the interval of FSH level drop in Fig 9(F). The inhibition of RcF subsequently contributes to the reduced development of GrF and DomF in panels (B) and (C).
Fig 11.
Model output with application of optimal control u2.
The full black curve is the model output when the optimal control u2 (cyan in panel (A)) is used. The normal cycle solution (full blue curve) is the hormone model output when u2 is not administered. The red circles with the interpolated curve denote the Welt data for a normal cycle. P4 reaches a maximum value of approximately 4.66 ng/mL.
Fig 12.
Model output with application of optimal controls u1 and u2.
The full black curve is the model output when the optimal controls u1 and u2 (magenta and cyan in panel (A)) are administered. The normal cycle solution Hwo (full blue curve) is the hormone model output when u1 and u2 are not applied. The red circles with the interpolated curve denote the Welt data for a normal cycle. The maximum P4 concentration is approximately 4.31 ng/mL.
Fig 13.
Constant dosage and nonconstant dosage comparison.
The shaded regions in Panels (A), (C), and (E) indicate the minimum total constant dosage of exogenous estrogen and/or progesterone over 28 days that lowers maximum P4 concentration to 4.99 ng/mL. The shaded region below u1 (area under the curve or AUC) in Panel (B) is the total nonconstant dosage of exogenous E2 which suppresses the P4 level to 4.43 ng/mL, a reduction by about 92% of the total dosage in (A). Panel (D) illustrates the total nonconstant dosage of exogenous P4 that reduces maximum P4 to 4.66 ng/mL, a reduction by about 43% of the total dosage in (C). Panel (F) shows the combined nonconstant doses of exogenous E2 and P4 that gives a maximum P4 level of 4.31 ng/mL.
Table 4.
Maximum progesterone level throughout a menstrual cycle caused by the indicated total dose of the optimal exogenous hormone.
Fig 14.
Multiple application of optimal control u1.
The black curve is the model output when multiple u1 (magenta curve) is applied. In panel (A), the application of u1 from day 35 to day 280 in equal intervals is unable to sustain anovulation. Panel (B) shows a scheme for administration of multiple u1 which continuously blocks ovulation.
Fig 15.
Optimal control u1 on different cycle lengths.
Panel (A) shows model output curves with different cycle lengths. Panel (B) presents the optimal control u1 obtained by applying the objective function for estrogen monotherapy. The effect of u1 administration on P4 peak is described in Panel (C).