3.1 The Lacker/Peskin Model of Mammalian Ovulation

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1 Mammalian Ovulation 3.1 The Lacker/Peskin Model of Mammalian Ovulation Reference: Regulation of Ovulation Number in Mammals A large reserve pool of follicles is formed before birth in many female mammals. Each follicle in the reserve pool consists of an oocyte (egg) and a few surrounding supporting cells. Follicles continually initiate growth from this reserve pool. This growth is driven by follicle-stimulating hormone (FSH) secreted from the pituitary gland of the hypothalamus. Although hundreds of follicles develop each menstrual cycle, only a few follicles, typically one in humans, fully mature. The developing follicles release estradiol, which stimulates the pituitary gland to release more FSH, which in-turn causes the follicles to grow and release more estradiol. This positive feedback loop gives rise to dramatic growth in the follicles. A small number of follicles end up being dominant as relatively small follicles have their growth inhibited. In the dominant follicles, the release of the oocyte is triggered by a surge in another pituitary hormone, the luteinizing hormone (LH). This egg release is known as ovulation. The ruptured follicle becomes a corpus lutem that is responsible for the release of the hormone progesterone, which among other functions, serves to stop the release of LH and FSH. We will refer to the number of eggs released during ovulation as the ovulation number. See Fig for a diagram of show the blood serum concentrations of the four hormones discussed vary over the menstrual cycle. Figure 3.13 shows how the follicles communicate with each other and the pituitary gland via the blood stream. We will formulate a model of the follicular phase of the menstrual cycle. Consider a system of N interacting follicles. Each follicle is characterized by it estradiol secretion rate s i (t), i = 1,..., N. Assume that estradiol is distributed in the plasma volume V at concentration ξ(t) and that it is removed at a rate proportional to its concentration, γξ. 28

2 Figure 3.12: A diagram of the essential hormones of the menstrual cycle. From Martin How Female Hormones Affect Your Diet. Therefore, V dξ N dt = s i (t) γξ. (3.43) Supposing that s i (t) slowly varies on the time scale given by V/γ, ξ(t) = 1 γ i=1 N s i (t) = i=1 N ξ i, (3.44) so that ξ i (t) is the contribution that each follicle makes to ξ(t). Follicle estradiol secretion rate is a marker of follicle maturity and so ξ i (t) should be interpreted as both the maturity of the i th follicle and its estradiol secretion rate. We further assume that the blood serum concentrations of estradiol controls the release of FSH and LH and that the instantaneous serum concentration of FSH and LH, in turn, regulate the rate of follicle maturation. Thus, at each instant, the response of each follicle depends on the follicle s maturity as, i dξ i dt = f(ξ i, ξ), (3.45) for some function f(ξ i, ξ) that accounts for the i th follicle s interaction with all other follicles. 29

3 Figure 3.13: A diagram of the interaction between the follicles and pituitary gland via the blood stream. From Lacker Regulation of ovulation number in mammals. A follicle interaction law that controls maturation. It will be easier to understand f(ξ i, ξ) as an exponential growth rate and so we write the model as dξ i dt = ξ iφ(ξ i, ξ), i = 1,..., N, (3.46) N ξ = ξ i, (3.47) i=1 φ(ξ i, ξ) = 1 (ξ M 1 ξ i )(ξ M 2 ξ i ), (3.48) ξ i (0) U(0, ɛ). (3.49) The choice of the form of φ(ξ i, ξ) has been made to reproduce (no pun intended) what is known about this biological system. Most importantly, as the follicle matures it becomes more sensitive to FSH so that both development of less mature follicles is inhibited during later stages of growth and FSH can initiate early growth for immature follicles. This choice of φ promotes the growth of follicles whose maturities lie in a certain range. Initially the ξ i values are small and the φ function appears flat over that small range. Follicles with sizes 30

4 in the preferred range only have a slight increase in growth rate. However, this advantage is enough to lead to the dominance of only a few follicles. As ξ becomes larger, the preferred sizes are given more of a growth advantage. See Fig The follicles compete with each other for growth-inducing FSH leading to a run-away process culminating in ovulation φ ξ i Figure 3.14: φ plotted against ξ i for at a sequence of times. Follicles with preferred sizes are given a positive growth rate. This advantage is made more significant at later times when ξ is large. The follicle sizes and the φ values are plotted with circles. Six follicles eventually dominant, with nearly the same follicle sizes, while the rest of the follicles are driven to zero size. The parameters are N = 1000, ɛ = 10 4, M 1 = 5.5, and M 2 = Each follicle is treated the same in its interaction with rest of the population of follicles. If we wanted the interaction to be different for each follicle then we would have made φ vary with the follicle index i. This symmetry will turn out to be useful. Without loss of generality, we will assume M 1 M 2. These two parameters will need to be selected to tailor the model to the particular mammalian species of interest. We will begin with values of M 1 = 5.5 and M 2 = Dormant follicles are continually beginning to develop at random times throughout the cycle. As a model, we suppose that there is a specific number of follicles N of various sizes that begin developing at the start of each cycle. Therefore, the initial follicle sizes have been selected to be independent and identically distributed uniformly on the interval [0, ɛ]. 31

5 The additional parameter ɛ is small and whose specific value that is not very consequential for the model outcomes. We use the condition ɛ 1/N so that E[ξ(0)] 1. The value for ɛ does affect the time until ovulation occurs. Since the initial condition is random, the solution is random. In particular, the ovulation number is a random variable. Note that since ξ i (0) are independent and identically distributed, ξ(0) is very nearly normally distributed according to the central limit theorem. A typical solution to the model is represented in Fig Figure 3.15: A simulation resulting in an ovulation number of 6 for N = 1000, M 1 = 5.5, and M 2 = For the dominant follicles, ξ i(t) is plotted in blue. The first-to-third quartiles of the distribution of follicle sizes is the shaded region with the black curve showing the median. The dashed curve is a plot of ξ(t). A few comments will help you appreciate the behavior of this model. The order of the follicle sizes is preserved in time. If ξ 1 (0) ξ 1 (0)... ξ N (0) then ξ 1 (t) ξ 1 (t)... ξ N (t) for t 0. This follows from a simple contradiction argument that if they are to change order then at some time two follicles will have to have the size and the previously smaller follicle with have to be increasing in size faster than the other one. However, two follicles of the same size must have the same growth rate from the differential equation. This makes it impossible for the order of follicle sizes to change. The model typically exhibits finite time blow-up. It is possible, and common, for many follicles to have their sizes approach infinity. When this occurs for multiple follicles, their sizes all asymptote to infinity at the same time, while the remaining follicles have their sizes driven to zero. This blow-up gives a clear indication that ovulation has occurred. The number of eggs released is assumed to be equal to the number of follicles that have 32

6 sizes that approach infinity even though in reality not all large follicles release an egg. At this finite time blow-up, the amount of LH being released by the pituitary gland is also becoming infinitely large. This very large release of LH can readily be detected in the blood serum. This LH surge is what home ovulation predictor kits detect. Since the model variable ξ i can also be interpreted as follicle maturity, a blow-up in ξ i corresponds nicely to the remarkable change in size undergone by follicles, growing to the impressive size of around 20mm. Observe that φ = 0 and ξ i = 0 implies that ξ = 1. Initially, when ξ < 1, all but exceptionally large follicles will have a positive growth rate. Once the follicles have grown enough so that ξ > 1, the smallest follicles will have a negative growth rate and will be driven to zero. With continued growth, more follicles will fall below the threshold, the smaller root of φ = 0, ξ SS below, which increases with ξ. This process is how the follicles compete for dominance within the model. For simplicity, we will consider a special initial condition for which M follicles have identical sizes and the remaining N M follicles remain dormant. The differential equation then becomes dξ ( 1 dt = ξ 1φ(ξ 1, Mξ 1 ) = ξ 1 1 ξ1 2 ˆM ), ˆM = (M M1 )(M M 2 ), (3.50) ξ i (t) = ξ 1 (t), i = 2,..., M, (3.51) ξ i (t) = 0, i = M + 1,..., N. (3.52) When ˆM > 0, the differential equation has three real equilibria, ξ 1 = 1/ ˆM, 0, 1/ ˆM, which are stable, unstable, and stable respectively. When ˆM < 0, ξ 1 = 0 is the only real equilibrium and it is unstable. Therefore, the solution goes to infinity when M 1 < M < M 2 for all non-zero initial conditions and asymptotes to 1/ ˆM otherwise. Additionally, the solution is explicitly ξ 1 (0) ξ 1 (t) = e 2t + (1 e 2t ) ˆMξ (3.53) 1 (0) 2 and we see that the blow-up time is t = 1 2 log (1 1 ˆMξ 1 (0) 2 ). We have learned from the symmetrical initial condition that the ovulation numbers are restricted to be between M 1 and M 2. Consider the equilibria of the differential equation. In equilibrium, the follicles sizes are either zero ξ0 SS = 0 or one of ξ SS ± = M 1 + M 2 2M 1 M 2 ξ ± (M2 M 1 2M 1 M 2 ) 2 ξ M 1 M 2. (3.54) 33

7 Note that ξ SS < 0 for ξ < 1. To determine the steady-states of the system, we are seeking two non-negative integer numbers M SS 0 and M+ SS 0 so that M SS + ξ SS + + M SS ξ SS = ξ, (3.55) M+ SS + M SS N, (3.56) ξ < 1 = M SS = 0. (3.57) Since φ is a downward opening parabola for any value of ξ, when M SS 0 the equilibrium solution will be unstable. Seeking stable equilibria, we take M SS = 0, in which case ξ = M+ SS (M 1 M+ SS)(M 2 M+ SS so that M SS + = 0,..., min(n, ), M 1 M 2 M 1 +M 2 ) give valid solutions. We have only considered stability with regard to perturbations in each ξ i holding ξ fixed. However, we jump to the conclusion that there will be some stable equilibria with some non-zero follicle sizes. Thus, the long-run behavior of the system will be either an approach to a stable equilibrium or a blow-up to infinity for some number (the range determined by M 1 and M 2 ) of dominant follicles. The former is known as an anovulatory state and the later is ovulation. Having introduced and studied a model of mammalian ovulation, it is natural to attempt to control the process as we will do in the computer lab. 3.2 Computer Lab: Optimal Control of Mammalian Ovulation In this lab you will use Matlab to solve the ordinary differential equation associated with the Lacker/Peskin model of ovulation. You will modify the model to account for one form of birth control and solve an optimal control problem to prevent ovulation. 1. Download simulate LackerPeskin.m and ovulation.m. Run the latter. The ordinary differential equation solver ode23s is used to integrate Eqs. (3.46) (3.49). An event is used to detect ovulation and stop the simulation when a follicle achieves a size above xiimax. In the plot, the first-to-third quartiles of the values of ξ i (t) is shaded and the median is plotted with a black line. The sizes of the dominant follicles are plotted in blue. Note the logarithmic scale. Repeated runs of the code should give different results since the initial condition is random. Change N = 200 as that is a more reasonable for the number of follicles that develop each cycle in young and healthy human females. 34

8 2. Write a function, called ovulation number distribution.m, that takes as input the parameter structure, uses a for-loop to solve the differential equation for many initial conditions, and returns as output estimates of 1) the distribution of ovulation numbers; 2) the average ovulation time (ignoring those for which ovulation did not occur); and 3) the probability of ovulation. Use histc to compute the probability mass function for the ovulation number from the samples. This function will be useful for the next three steps. 3. In the model, a variable number of eggs are released depending on the parameters M 1 and M 2. For a grid of these parameters (each on [0,25]), produce contour plots for the probability of ovulation, the probability of an ovulation number of one, and the probability of an ovulation number of two. Use contourf. A template is provided in M1M2contours.m. For each value of M 1 and M 2, this will require many hundreds of simulations to get an accurate estimate of the probability. This might take a long time to run. Try using parfor to compute your independent for-loop iterates in parallel. 4. Make a choice of parameters M 1 and M 2 so as to be consistent with the estimated human probabilities of 93% and 7% for ovulation numbers of one and two. The release of two eggs accounts for the approximately 3% occurrence rate of fraternal twins. 5. In the model development, we did not specify the units of time. Since the differential equation is autonomous, we are free to scale time, t λt, without altering the solution behavior. Scale time so that, on average, ovulation occurs at T = 14 days. In particular, multiply the right-hand side of each differential equation by λ so that on average ovulation occurs at T = 14. Note that this is mostly a cosmetic change to the model. Implement the scaling in the indicated location in simulate LackerPeskin.m. 6. We need to modify the model so as to account for birth control, which acts primarily by changing the progesterone level in the blood stream. We first need to describe the natural release of progesterone in the body. During a natural cycle, progesterone can be found in high levels during the lateral phase, after ovulation. Leading up to and mostly after ovulation, the ovaries release progesterone to shutoff the positive feedback loop that drove the development of the follicles. We introduce an additional variable for the blood progesterone concentration, P (t). Since progesterone in the blood serum is quickly degraded, we will assume that P (t) is proportional to the total instantaneous rate of progesterone increase. These sources will be both natural N(t) and artificial D(t). We model the natural release of progesterone by mature follicles as N(t) ξ. We model the affect of progesterone on the positive feedback cycle driving the growth 35

9 of the follicles as a shift in the φ function. Thus, we modify the φ function as φ = φ(ξ i, ξ) αξ βd(t). The function D(t) is left to our choosing. Modify the right-hand side of the differential equation in simulate LackerPeskin.m. Remember to pass a function handle for D(t) to the rhs function. Start with the parameters α = 0.5 and β = 2, but you may need to adjust them so that the behavior is reasonable. 7. To specify an optimization problem, we must pick an objective and some constraints. We will consider only time from 0 to T = 14 days. The drug release rate must be non-negative, D(t) 0. So that ovulation does not occur, we will impose that ξ(t ) = 0. We could impose that each follicle size at time T is zero, ξ i (T ) = 0, but that would be N conditions instead of one and since ξ i 0, these conditions are equivalent. Remember that since the model has a random initial condition, we are really requiring Pr{ξ i (T ) = 0} = 1. There are several possible objectives. Here are some possibilities: (a) min max t [0,T ] D(t) so that maximum amount of drug used is as small as possible (b) min T 0 D(t)dt so that the total amount of drug used is as small as possible (c) min t [0,T ] D (t) so that changes in the drug are kept to a minimum Choose the one objective that you think is most reasonable. 8. We will take a direct approach to numerical optimal control. Choose one of the following families of functions. They each represent a subset of admissible functions for D(t) and we will select the best within that subset by optimizing for its parameters. Note that some families are incompatible with some of the above objectives. (a) a polynomial of degree K, D(t) = K k=0 a kt k (b) a Fourier series with K frequencies, D(t) = a 0 + K k=1 [ ak sin ( 2πk t ) T + bk cos ( 2πk t )] T (c) a bang-bang control with K switch times, D(t) = D max K k=1 ( 1)k+1 H(t t k ) 9. Solve the constrained optimization problem using fminsearch. Impose a penalty for violating the constraint ξ(t ) = 0 by adding κξ(t ) to your objective for some large value of κ. You will need to run multiple simulations (for each evaluation of the objective function) to account for the random initial condition. What parameters did the optimization procedure return? What is the value of the objective function? 36