Toward an optimal contraception dosing strategy

Abstract

Anovulation refers to a menstrual cycle characterized by the absence of ovulation. Exogenous hormones such as synthetic progesterone and estrogen have been used to attain this state to achieve contraception. However, large doses are associated with adverse effects such as increased risk for thrombosis and myocardial infarction. This study utilizes optimal control theory on a modified menstrual cycle model to determine the minimum total exogenous estrogen/progesterone dose, and timing of administration to induce anovulation. The mathematical model correctly predicts the mean daily levels of pituitary hormones LH and FSH, and ovarian hormones E2, P4, and Inh throughout a normal menstrual cycle and reflects the reduction in these hormone levels caused by exogenous estrogen and/or progesterone. Results show that it is possible to reduce the total dose by 92% in estrogen monotherapy, 43% in progesterone monotherapy, and that it is most effective to deliver the estrogen contraceptive in the mid follicular phase. Finally, we show that by combining estrogen and progesterone the dose can be lowered even more. These results may give clinicians insights into optimal formulations and schedule of therapy that can suppress ovulation.

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. Model diagram.

The diagram (adapted from [13]) shows the 13 states (RP_LH, RP_FSH, 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. Normal cycle solution.

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].

Fig 5. Model output with constant dose of exogenous hormone.

$H_{P_4^{\text{exo}}}$ (black curve) or $H_{E_2^{\text{exo}}}$ (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. Varying E2exo 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 $E_2^{\text{exo}}$ causes decreasing amplitude of the variation in P₄ level. Anovulation is attained when $E_2^{\text{exo}}>34.7$ pg/mL.

Fig 7. Varying P4exo dose.

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 $P_4^{\text{exo}}$. Panel (B) illustrates diminishing fluctuation in P₄ value. Anovulation is achieved between $P_4^{\text{exo}}=3.1$ ng/mL and $P_4^{\text{exo}}=3.7$ ng/mL.

Fig 8. Constant dose combination therapy.

(A) The black curve represents the P₄ model output for a combination treatment with $P_4^{\text{exo}}=1.4$ ng/mL per day and $E_2^{\text{exo}}=20$ 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 ($E_2^{\text{exo}},P_4^{\text{exo}}$), correspond to a combined dosage of $E_2^{\text{exo}}$ and $P_4^{\text{exo}}$ resulting in a P₄ maximum value of k ng/mL. Anovulation is attained when k < 5. The yellow region ($P_4^{\text{exo}}>3.7$ 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. $H^{\text{wo}}$ (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 $H^{\text{wo}}$ (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 H^wo (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.

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).