Table 1.
Parameter values.
Fig 1.
The outer ring represents ovarian development starting with menstruation (black oval, top) and proceeding in a clockwise direction through the follicular phase (blue ovals, right), ovulation (black oval, bottom), and the luteal phase (red ovals, left). The ovary produces estrodial (E2), progesterone (P4), and inhibin A (InhA), while the brain produces FSH and LH. Directed solid arrows indicate the action of hormones at various stages of the cycle.
Fig 2.
The model diagram shows all states broken into two sub-models. The hypothalamus/pituitary model has four states (RPFSH,FSH,RPLH, LH) and the ovarian model has nine states (RcF, GrF, DomF, Sc1, Sc2, Lut1-4). In the hypothalamus/pituitary model, the black horizontal arrows represent hormone (E2, Papp) movement and in the ovarian model they represent movement of cells or tissue (mediated by LH) between stages. The red horizontal arrows represent output from a sub-model, and the green horizontal arrows represent input into a sub-model. A hormone H written as H+ or H− has a stimulating or inhibiting effect respectively on movement between chambers or effectiveness of a hormone within the chamber. The blue dashed-dotted lines within the ovarian model show stages contributing to ovarian hormone production in the auxiliary equations. The gray dashed lines in the ovarian model represent autocrine influence of ovarian hormones within the model. Full expression for the hypothalamus/pituitary and ovarian models can be found in Eqs (1)–(4) and Eqs (5)–(13), respectively, and parameter values and dimensions are listed in Table 1.
Fig 3.
Results for a normal cycle with no exogenous estrogen or progestin. The model output for 84 days (3 cycles) is denoted by the solid blue line and the connected points are data with error bars digitized from Figure 1 in Welt et al. [25].
Fig 4.
Model results with a low dose (pdose = 0.6 ng/mL) are plotted with a solid blue line, while the solid red horizontal red line denotes the mean maximum hormonal values resulting from the 21 day progestin treatment reported in Figure 1 and Table 3 in Obruca et al. [16] with the standard deviation represented by the horizontal dashed lines. The mid-cycle LH surge has been eliminated. With this dose we have reached biological contraception by preventing the LH surge, but we have not reached total contraception. For comparison, the normal cycling data are presented by a dashed-dotted magenta line.
Fig 5.
Model results with high a dose *pdose = 1.3 ng/mL are plotted with a solid blue line, while the solid red horizontal line represents the mean maximum hormonal value resulting from the 21 day progestin treatment reported in Figure 1 and Table 3 in Obruca et al. [16] with the standard deviation represented by the horizontal dashed lines. For P4 we note a significant difference between model predictions and the data. This likely stems from the fact that in the model P4 includes both endogenous and exogenous progestin, while the data only measure the endogenous levels. We have reached a steady state here and thus total contraception. For comparison, the normal cycling data are presented by a dashed-dotted magenta line.
Fig 6.
Model results (solid blue line) with edose = 40 pg/mL. In this simulation, LH has a small mid-cycle rise, but the large LH surge is significantly suppressed and ovulation does not occur indicating a contraceptive state, yet the hormone levels still vary during the cycle. For comparison, the normal cycling data are presented by a dashed-dotted magenta line.
Fig 7.
Model results (solid blue line) with edose = 92 pg/mL. For this high dose total contraception has been achieved. For comparison, the normal cycling data are presented by a dashed-dotted magenta line.
Fig 8.
Model results (solid blue line) with pdose = 0.6ng/mL and edose = 40 pg/mL. The dotted red line is the median maximum hormonal value during days 8-14 of combined hormonal treatment reported in Figure 1 and Table 3 in Mulders and Dieben [18]. These are the two low doses that did not reach total contraception when used individually. The application of both low doses though has achieved total contraception. For comparison, the normal cycling data are presented by a dashed-dotted magenta line.
Fig 9.
Bifurcation diagram representing location of Hopf bifurcations in the (edose, pdose) space. Solutions below the curve of Hopf bifurcations are periodic and solutions above the curve are steady state. Our total contraception as we have defined it then occurs along this curve of Hopf bifurcations. Any doses falling above the curve are totally contraceptive and any below are not. The low dose combination that we tested (used in Fig 8) is shown with a red star and falls just into the steady state region. The progestin (used in Fig 5) and estrogen (used in Fig 7) only doses can be seen approximately where the Hopf curve intersects the axes.
Fig 10.
Simulation (solid blue line) of a temporary treatment of a low dose combined hormonal contraceptive. Dosing begin at day 84 (green dashed line) and ends at day 168 (red dashed line) at which point the dose decreases exponentially due to the half-life of the drug. A nearly instant contraceptive effect after dosing is observed and, once the drug is removed, return to ovulation occurs within 1-2 cycles.