Model output with application of optimal control <i>u</i>₂.

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.</p

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.</p

Supplementary information file.

This file presents the data used in our study, model parameters, standard deviation of model parameters, result of the sensitivity analysis performed on the model, effect of various weights on the optimal control results, optimal control results from different forms of the objective function, and computation time and optimal cost for the optimal control simulations. (PDF)</p

Maximum progesterone level throughout a menstrual cycle caused by the indicated total dose of the optimal exogenous hormone.

Maximum progesterone level throughout a menstrual cycle caused by the indicated total dose of the optimal exogenous hormone.</p

Model diagram.

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.</p

Varying dose.

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.</p

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).</p

Model output with application of optimal controls <i>u</i>₁ and <i>u</i>₂.

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.</p

Optimal control <i>u</i>₁ 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).</p

Sum of squared residuals between the model output and Welt data, and model output peaks.

Sum of squared residuals between the model output and Welt data, and model output peaks.</p