Stochastic optimal control for prediction of robust drug dosing policies in superovulation stage of in-vitro fertilization
Kirti M. Yenkie b and Urmila Diwekara,b*
aDepartment of Bioengineering, University of Illinois, Chicago, IL
bCenter for Uncertain Systems: Tools for Optimization & Management (CUSTOM), Vishwamitra Research Institute, Crystal Lake, IL
Keywords: infertility, multiple ovulation, uncertainty, optimal control
1. Introduction and Motivation
The WHO has estimated that about 8-10% couples experience some kind of infertility problems. Medical science has come up with Assisted Reproductive Technologies (ARTs) like IVF for treatment. IVF is a four stage medical procedure and the success of superovulation, its first stage, is crucial to proceed with the next stages. Also, superovulation requires maximum medical attention, investment of time and money. The major hormone responsible for follicle growth is follicle stimulating hormone (FSH). In our previous work , a deterministic model for prediction of multiple follicle growth was built by drawing similarities from batch crystallization . This model was validated with clinical data from 50 IVF cycles. The model fitted very well and almost 60% predictions had very low error . However, some results had deviations from the expected behavior. Thus, the objective of this work is to develop a robust model which can provide a better fit for the patients with deviations in the deterministic model and then come up with a method for predicting robust hormone dosing policy.
2. Methodology 2.1. Determinstic Model
The concept of moment model in batch crystallization was used for modeling superovulation in IVF [1, 3]. The growth term in batch crystallization is temperature dependent and hence temperature is controlling variable. Similarly in IVF, follicle growth is dependent upon hormonal doses. The follicle size is converted into mathematical moments  by assuming them to be spherical in shape. The eq. (1) is used for converting follicle size to moments.
Here, µi - ith moment, nj(r, t) - number of follicles in jth bin with mean radius as r at time t, rj - mean radius of jth bin and Δr - range of radii variation in bins. Each moment corresponds to a feature of the follicles, like the zeroth moment represents the number, first the size, etc. The follicle growth term (G) is dependent on the amount of FSH injected (ΔCfsh) to the patient at time (t) and is represented in eq. (2). Here, k and α are kinetic constants.
It was suggested that number of follicles activated for growth during a superovulation cycle are always constant for a patient , hence zeroth moment is a constant. The moment equations for the follicle dynamics can be written as in eqs. (3) and (4).
2.2. The Stochastic Model
The stochastic model for superovulation is developed in terms of Ito processes. Depending upon the behavior of the system it can modeled as one of the suitable Ito processes . It was found that the uncertainties could be represented using the simple brownian motion type of Ito processes, resulting in stochastic differential equations (SDEs), i varies from 1 to 6 (eq. 5).
Here, µi - ith moment, σi - standard deviation terms for each moment equation corresponding to noise in the clinical data, єt - random numbers from unit normal distribution, dt - time interval.
2.3. Stochastic Optimal Control
Superovulation objective is to obtain eggs or follicles in high number within a specific size range (18-22mm diameter) with the aid of external hormones. Mathematically, it can be interpreted as 'minimization of the variation'. The follicle size data follows normal distribution hence, the expression for coefficient of variation (CV) is (eq. 6),
Thus, objective of superovulation can is to minimize the coefficient of variation at the final time (CV(tf)), the control variable is FSH dose with time (Cfsh (t)), subjected to stochastic superovulation model. This is solved using stochastic maximum principle.
Table 1. The results for Patient-A from stochastic optimal control
No. of Follicles
(9 ≤ Mean size ≥ 12)
% reduction in FSH
Follicle count - 18
Stochastic model for superovulation stage is used for predicting robust hormone dosing policies using stochastic control methods. Stochastic maximum principle strategy provides dosing policies which can result in better follicle size distribution on the final FSH dosing day.
 Yenkie, K. M., Diwekar, U., Bhalerao, V., 2013, "Modelling the superovulation stage in in-vitro fertilization", IEEE Trans. Biomed. Engg., 60(11): 3003-3008.
 Randolph, A. D., and M. A. Larson., 1988, Theory of Particulate Processes: Analysis and Techniques of Continuous Crystallization. San Diego, CA, Academic Press.
 Yenkie K.M., Diwekar U., Bhalerao V., 2014, "Modeling and prediction of outcome for the superovulation stage in In-Vitro Fertilization (IVF)", JFIV Reprod Med Genet 2: 122.
 Baird, D. T., 1987, "A model for follicular selection and ovulation: Lessons from Superovulation", J. steroid Biochem., 27, 15-23.
 Diwekar, U. 2008. Introduction to Applied Optimization, 2nd ed.; Springer NY.
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