Modeling the Superovulation Stage in in-Vitro Fertilization (IVF)

Abstract

               In-vitro fertilization (IVF) is the most common technique in assisted reproductive technology and in most cases the last resort for infertility treatment. It has four basic stages: superovulation, egg retrieval, insemination/fertilization and embryo transfer. Superovulation is a drug induced method to enable multiple ovulation per menstrual cycle. The success of IVF majorly depends upon successful superovulation, defined by the number and similar quality of eggs retrieved in a cycle.

               The model is adapted from the theory of batch crystallization. The aim of crystallization is to get maximum crystals of similar size and purity, while superovulation aims at eggs of similar quality and size. The rate of crystallization and superovulation are both dependent on the process conditions and varies with time. Thus, model formulation for multiple ovulation is on parallel lines to crystal formation in a batch process and is modeled as such. The kinetics of follicle growth is modeled as a function of injected hormones and the follicle properties are represented in terms of the moments. Hence, modeling of this stage in terms of distribution of eggs (oocytes) obtained per cycle involving the chemical interactions of drugs used and the conditions imposed on the patient during the process provides a basis for predicting the possible outcome.  The results from the model prediction were verified with the known data from Jijamata Hospital Nanded, India. The predictions were found to be in well matched with the actual observations.

               Thus, a phenomenon currently based on trial and error will get a strong base in terms of a predictive model. It will help the patient to decide whether to undergo superovulation or start the IVF from donor eggs, which in turn would save the patient from financial loss as well as emotional distress.

1. Introduction

               Around 80 million people in the world are suffering from infertility issues. The rate of fertility is constantly declining in the developed nations due to late marriages, postponed childbearing and primary infertility. On the contrary, in the developing world the reasons for infertility involve prevalence of sexually transmitted diseases, infections which increase the rate of secondary infertility. Childlessness is often stigmatized and leads to profound social suffering for women in the developing nations (de Melo Martin, 1998).

1.1. In-vitro fertilization:

               It is a process by which oocytes or egg cells are fertilized by a sperm outside the body in a laboratory simulating similar conditions in the body and then the fertilized eggs are implanted back in the uterus for full term completion of pregnancy.

Four stages in IVF:

i. Superovulation: It is method to retrieve multiple eggs using drug induced stimulation. In normal female body only one egg is ovulated per menstrual cycle, but with the use of fertility drugs and hormones, more number of eggs can be ovulated per cycle.

ii. Egg collection (retrieval): On the maturation of the multiple eggs produced in the previous stage, the eggs are retrieved through special techniques like ultrasonically guided transvaginal oocyte retrieval.

iii. In vitro fertilization (inseminaton/fertilization): Fertilization is done in the incubator using the retrieved oocytes and sperms. The conditions are maintained so as to mimic the invivo environment.

iv. Embryo transfer: The fertilized embryos are implanted into the uterus via a non-surgical technique using ultrasound guidance.

               The major cost of IVF is associated with the superovulation stage where expensive drugs are used and almost daily monitoring is required. Success of this stage in terms of number and quality of eggs affects the outcome of IVF. In this work, we concentrated on modeling this stage and the approach is presented in the next section.

1.2. Superovulation:

In this work, we follow the analogy between batch crystallization and superovulation to model the process.

Analogy between Superovulation and Batch crystallization:

               The moment model for follicle number and size is adapted from the concept of batch crystallization (Q. Hu. et. al., 2005) based on the analogy between batch crystallization and superovulation presented in Table 1.

Table 1: Analogy between batch crystallization and IVF superovulation stage

Batch Crystallization	Superovulation (IVF stage I)
Production of multiple crystals	Production of multiple oocytes or eggs
Crystal quality is determined in terms of size distribution and purity	Oocyte quality is determined in terms of no abnormalities, similar size.
The rate of crystallization or crystal growth varies with time and process conditions	The rate of ovulation or oocyte growth varies with time and drug interactions
Process is affected by external variables like agitation, and process operating variables like temperature, pressure, etc.	Process is affected by externally administered drugs and body conditions of the patient undergoing the process

               The superovulation follicle growth model in general resembles greatly to the growth of seeded batch crystals. Thus, the moment model for both the processes remain the same the growth term which is a function of process variables like temperature and supersaturation in batch seeded crystallization will become a function of medicinal dosage in case of superovulation process.

2. Model details

               Due to ovarian stimulation using externally injected hormones the number of follicles activated to enter into the ovulation stage are more in number as compared to a single follicle in a normal menstrual cycle. From the current data on successful superovulation for patient 1, organized in Table 2; it can be observed that during FSH dosage regime, as the time progresses the size of the eggs increase.

Table 2: Variation of Follicle size (diameter) with time and FSH dose

Size range (mm) dia Sr. No	Days	Day 1	Day 2	Day 5	Day 7	Day 9
1.	0-4	15	8	6	0	0
2.	4-8	3	6	4	2	0
3.	8-12	0	4	10	14	4
4.	12-16	0	0	0	2	11
5.	16-20	0	0	0	0	3
6.	20-24	0	0	0	0	0
	FSH dose	0	300	300	300	225

2.1. Model Assumptions:

               The rate expression for follicle growth is dependent on FSH administered. Thus, we can write the growth term as;

G=kCfshα                                                      (1)

               Assuming moment model, we consider only the first seven moments; zeroth moment corresponding to follicle number, the first moment corresponding to follicle size and the other 5 moments. 1^st to 6^th moments are being used since they help in recovering the size distributions more precisely as against lower number of moments.

2.2. Model equations (Q. Hu. et. al., 2005):

μ0=constant                                                (2)

dμ1dt=Gtμ0t                                            (3)

dμ2dt=2Gtμ1t                                          (4)

dμ3dt=3Gtμ2t                                          (5)

dμ4dt=4Gtμ3t                                          (6)

dμ5dt=5Gtμ4t                                          (7)

dμ6dt=6Gtμ5t                                          (8)

               Conversion of the data available on follicle number and size to moment using the expression given in literature by Flood, 2002:

μi=nir,trinDri                                (9)

   Here, µ_i = i^th moment

               n_i(r,t) = number of follicles in bin of mean radius 'r' at time 't'.

               r_i = mean radius of i^th bin

               Δr = range of radii variation in each bin

3. Results:

               We integrate the equations 2-8 for predicting the kinetic constants in the follicle growth expression. Later we use non-linear optimization algorithm to predict the values of these kinetic constants along with the integration constants obtained after integrating the set of moment equations. In real practice, the model will be calibrated with the first two days of data and then used for prediction of the complete cycle.

3.1. Model Validation:

               The current moment model predicts the moment values, however our final output desired is the follicle size distribution, thus in model validation the approach to obtain follicle size distribution from moment values are shown. The method is adapted from the literature by Flood 2002; where he shows the method to recover particle size distribution from moments in batch crystallization. Using the model predicted moment values we evaluate n(r,t) and compare with the actual data to check the model accuracy. We plot the follicle size distribution for four patients for various days. The experimental size distribution is shown by symbols while the continuous curve shows the model predicted values after using the inversion method. 

4. Conclusion:

The moment model developed for IVF superovulation predicts the follicle size distribution which is in well agreement with the actual size distribution seen in the IVF cycle data.  The model can be used to predict the outcome. This will reduce the almost daily requirement of testing. The model can also provide a basis for predicting the optimum dosage for the desired outcome from the superovulation stage. The model used here is a very basic model and the complexities present in the patient are not considered. Later, we aim to include these complexities and model the system uncertainties, using more data for analysis, modeling and validation.

Fig 1. Follicle size distribution  for four patients

References

de Melo-Martin, I., 1998. "Ethics and Uncertainty: In Vitro Fertilization and Risks to Women's Health" Risk: Health, Safety & Environment 201.

Flood, A. E., 2002. "Thoughts on recovering particle size distributions from the moment form of the population balance." Dev. Chem. Eng. Mineral Process, Vol. 10, No. 5/6, pp. 501-519.

Q. Hu, S. Rohani, A. Jutan, 2005. “Modelling and optimization of seeded batch crystallizers” Computers and Chemical Engg. Vol. 29, pp. 911-918.