(393d) Stochastic Optimal Control for Prediction of Robust Drug Dosing Policies in Superovulation Stage of in-Vitro Fertilization

Stochastic optimal control for prediction of robust
drug dosing policies in superovulation stage of in-vitro fertilization

Kirti M. Yenkie ^b and Urmila Diwekar^a,b*

^aDepartment of Bioengineering, University of Illinois,
Chicago, IL

^bCenter for Uncertain Systems: Tools for Optimization
& Management (CUSTOM), Vishwamitra Research Institute, Crystal Lake, IL

*urmila@vri-custom.org

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 [1],
a deterministic model for prediction of multiple follicle growth was built by
drawing similarities from batch crystallization [2]. This model was validated
with clinical data from 50 IVF cycles. The model fitted very well and almost 60%
predictions had very low error [3]. 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 [2] by assuming them
to be spherical in shape. The eq. (1) is used for converting follicle size to
moments.

Here, µ_i - i^th moment, n_j(r,
t) - number of follicles in j^th bin with mean radius as r
at time t, r_j - mean radius of j^th
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 (ΔC_fsh) 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 [4],
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 [5]. 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, µ_i - ith moment, σ_i -
standard deviation terms for each moment equation corresponding to noise in the
clinical data, є_t - 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),

                                                                              
  (6)                   

Thus, objective of superovulation can is to minimize
the coefficient of variation at the final time (CV(t_f)),
the control variable is FSH dose with time (C_fsh (t)),
subjected to stochastic superovulation model. This is solved using stochastic
maximum principle.

  3.      Results

            Table
1. The results for Patient-A from stochastic optimal
control

  4.      Conclusions

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.

  5.     
References

[1] Yenkie, K. M., Diwekar, U.,
Bhalerao, V., 2013, "Modelling the superovulation stage in in-vitro
fertilization", IEEE Trans. Biomed. Engg., 60(11): 3003-3008.

[2] Randolph, A. D., and M.
A. Larson., 1988, Theory of Particulate Processes: Analysis  and  Techniques 
of  Continuous  Crystallization. San Diego, CA, Academic Press.

[3] 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.

[4] Baird, D. T., 1987, "A model
for follicular selection and ovulation: Lessons from Superovulation", J.
steroid Biochem., 27, 15-23.

[5] Diwekar, U. 2008. Introduction to
Applied Optimization, 2nd ed.; Springer NY.