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Uncertainty in clinical data and stochastic model for in-vitro fertilization.

Kirti M. Yenkiea,b and Urmila Diwekara,b

aDepartment of Bioengineering, University of Illinois, Chicago, IL 60607 - USA

bCenter for Uncertain Systems: Tools for Optimization & Management (CUSTOM), Vishwamitra

Research Institute, Clarendon Hills, IL 60514 – USA

Abstract

In-vitro Fertilization (IVF) is the most common technique in Assisted Reproductive Technology (ART). It has been divided into four stages; (i) superovulation, (ii) egg retrieval, (iii) insemination/fertilization and (iv) embryo transfer. The first stage of superovulation is a drug induced method to enable multiple ovulation, i.e., multiple follicle growth to oocytes or matured follicles in a single menstrual cycle. IVF being a medical procedure that aims at manipulating the biological functions in the human body is subjected to inherent sources of uncertainty and variability. Also, the interplay of the hormones with the natural functioning of the ovaries to stimulate multiple ovulation as against a single ovulation in a normal menstrual cycle makes the procedure dependent on several factors like the patient's condition in terms of cause of infertility, actual ovarian function, responsiveness to the medication. The treatment requires continuous monitoring and testing and this can give rise to errors in observations and reports. These uncertainties can be observed in the form of measurement noise in the available data. Thus, it becomes essential to look at the process noise and think of a way to account for it and build better representative models for follicle growth. The purpose of this work is to come up with a robust model which can project the superovulation cycle outcome based on the hormonal doses and patient response and hence provide a treatment guideline to enhance the success rate of the procedure.

Keywords-infertility, assisted reproduction, multiple ovulation, uncertainty, stochastic differential equations

1. Introduction

Infertility is the inability of a couple to achieve conception or to bring a pregnancy to term after a year or more of regular, unprotected intercourse. The World Health Organization has estimated that about 8-10% couples experience some kind of infertility problems. Medical science has come up with ARTs like IVF to treat these problems. IVF involves fertilization of oocytes by a sperm in an IVF laboratory simulating similar conditions as in the human body. It is
a four stage medical procedure and the success of superovulation, its first stage, is critical to proceed with the next stages in the cycle. Also, superovulation requires maximum medical attention, investment of time and money as compared to the other stages.
The major hormonal drug involved in superovulation is the follicle stimulating hormone (FSH), which is responsible for follicle growth and development. In our previous work (Yenkie et. al. (2013)), a deterministic model for prediction of multiple follicle growth, dependent on the amount of FSH dosage was built by drawing similarities with the particulate process of batch crystallization. This model was validated with clinical data from 50 IVF cycles, available from our collaborating hospital in India. The model fitted very well to most of the data and almost
85% predictions were within 0 to 30% range of the modeling error. However, some of the projected results had deviations from the expected behavior and hence could be used to model the uncertainties in the process.

2. The Determinstic Model

Thus, the concept of the moment model in batch crystallization was used for modeling superovulation in IVF (Yenkie et. al. (2013)). The growth term in batch crystallization is temperature dependent and hence it becomes the decision variable for controlling the process. On similar lines in IVF the follicle growth is dependent upon the doses of hormones injected to the patient.
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.

(1)

Here, µ i - 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 ?r - range of radii variation in bins. Each moment corresponds to a feature of the follicles, like the zeroth moment represents the number, first moment represents
the size, etc.
The follicle growth term (G) is dependent on the amount of follicle stimulating hormone (FSH) injected (Cfsh) to the patient at the particular time (t) in the cycle and is represented in eq. (2). Here, k and a are kinetic constants.

(2)

In the literature by Baird (1997) it was suggested that the number of follicles activated for growth during that particular superovulation cycle are always constant for a patient, hence the
zeroth moment has a constant value. The moment equations for the follicle dynamics can be written as in eqs. (3) and (4).

(3)

(4)
The data obtained from our collaborating hospital is converted to moments as shown earlier in eq. (1) and used for model fitting and validation. The validation method requires the conversion of predicted moments back to the follicle size and number and this was done by adapting the methodology proposed by Flood (2002).

3. The Stochastic Model

The aim is to develop a stochastic model for the superovulation stage in terms of Ito processes. Initially, these Ito processes were mostly applied in financial field and stock price modeling to model uncertainties. Since they can characterize time-dependent uncertainties and can be integrated and differentiated using the rules of Ito’s stochastic calculus, they can prove beneficial in modeling biological processes. The simplest example of an Ito process is the Brownian motion or Wiener process. For any stochastic process to be characterized as the Wiener process it must follow three important properties (Diwekar (2008)).
1. It should follow the Markov property: probability distribution for all future values of the process depend only on its current value.
2. It should have independent increments in time. probability distribution for changes in the process over any time interval is independent of any other time interval.
3. Changes in the process over a finite time interval should be normally distributed, with variance linearly dependent on the length of time interval (i.e., N (0, vdt) for all t > 0).
We check whether superovulation classifies as a Wiener process. As per the current protocols, hormonal dosage is decided based on daily monitoring of patient’s response and the follicle size observed. Thus, the next follicle size distribution (FSD) after medication is dependent on FSD at that time, hence it is reasonable to categorize it as a Markov process.
The follicle growth term developed in the determinstic model and which satisfies the data is dependent on the amount of FSH injected (Cfsh) to the patient at the particular day (t) in the cycle (eq. (2)). Thus it satisfies the second property, where FSH dosage have independent increments.
The follicle growth and number is represented as the FSD, where the follicle size is divided into equispaced bins and number of follicles are represented as the discrete markers in each of the size bins as shown in Fig. 1. When these markers are connected with a smooth curve,
the data tends to follow a normal distribution.

7

Day 5 Day 7

6

5

4

3

2

1

0

0 3 6 9 12

Mean Follicle size (mm)

Figure 1 Follicle size showing normal distribution on Day 5 and Day 7.

The Ito processes are of various types whose basic building block is a Wiener process. Depending upon the behavior of the system under random influences 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 a (eq. (7) to eq. (12)) set of stochastic
differential equations (SDEs).

(7) (8) (9) (10)

(11)

(12)

Here, µ i - ith moment, si - standard deviation terms for each moment equation corresponding to noise in the clinical data, ?t - set of random numbers derived from unit normal distribution and dt - time interval.

4. Results and Discussions

The results for the FSDs are shown in Fig 2A and 2B for day 5 and day 9 respectively. The plots show observed clinical data as discrete data points (O), deterministic model (D) projections as dotted profiles and stochastic model (S) projections as continuous profiles. The results projected by the stochastic model match the data better when compared to deterministic
results.

18 A

16

14

12

10

8

6

4

2

0

Day5(O) Day5(D) Day5(S)

18 B

16

14

12

10

8

6

4

2

0

Day9(O) Day9(D) Day9(S)

0 3 6 9 12

Mean size (mm)

0 3 6 9 12

Mean size (mm)

Figure 2. Comparison of FSDs from observed clinical data (O), deterministc (D) and stochastic
(S) model predictions for Patient-I.

5. Conclusions

The approach to develop a stochastic model for the superovulation stage in IVF in terms of Ito form of stochastic differential equations proves promising. The approach used for stochastic model development involving a novel parameter estimation procedure is an added contribution in this work. The model predictions obtained from the stochastic model are inclusive of all possible noise in the data and hence match better as compared to deterministc results. The robust predictions from the SDE model can be used for predicting robust drug dosing policies using stochastic control methods for achieving the desired outcome in superovulation for enhancing the success rate of the overall IVF cycle.