305638 Comparison of Different Methods for Predicting Customized Drug Dosage in Superovulation Stage of InVitro Fertilization
Comparison of Different Methods for Predicting Customized Drug Dosage in Superovulation stage of Invitro fertilization
Kirti M. Yenkie, ^{a,b }and^{ }Urmila M. Diwekar ^{a,b }
^{a}Department of Bio Engineering, University of Illinois, Chicago, IL 60607  USA
^{b}Center for uncertain Systems: Tools for Optimization & Management (CUSTOM),
Vishwamitra Research Institute, Clarendon Hills, IL 60514 – USA
Abstract
In vitro fertilization (IVF) is one of the highly pursued assisted reproductive technologies worldwide. The IVF procedure is divided into four stages: Superovulation, Eggretrieval, Insemination/Fertilization and Embryo transfer. Superovulation is the most crucial stage in IVF, since it involves external injection of hormones to stimulate development and maturation of multiple oocytes. The maximum amount of effort and money for IVF procedure goes into superovulation. Although numerous advancements have been made in IVF procedures, little attention has been given to modifying the standard protocols based on a predictive model. Currently, the same protocol is followed for every patient undergoing the IVF superovulation procedure. In reality every patient reponds differently and hence the proposition to modify the amounts of drug administered based on the patient's initial treatment response is a reasonable approach. The modification of drug dose if based on a well developed mathematical model which takes into account the variability in the follicle growth dynamics as well as the desired outcome thus increasing the predictive value of the method.
A model for the follicle growth dynamics and number as a function of the injected hormones and patient characteristics has been developed and validated. The modeling basics have been adapted from batch crystallization moment model, since moments are representatives of specific properties like number, shape and size of the particles under consideration. Based on this model, the dosage of the hormones to stimulate multiple ovulation or follicle growth is predicted by using the theory of optimal control. The objective of successful superovulation is to obtain maximum number of mature oocytes/follicles within a particular size range. Using the mathematical model involving follicle growth dynamics and the optimal control theory, optimal dose and frequency of medication customized for each patient is predicted for obtaining the desired result.
The optimal control problem is solved by different methods like the maximum principle and dicretized nonlinear programming. The problem is solved with and without constraints to check the variation in the dosage amounts and size of follicles at the retrieval time. The results from different approaches are compared. Thus, a systematic comparison of the different methods will help in deciding the best solution strategy for customized drug dosage. It will also provide information about the sensitivity of the model parameters and hence the source of uncertainty in the system.
Keywords: multiple ovulation, follicle growth, maximum principle, nonlinear programming
1. Introduction:
Fig. 1 shows a schematic diagram of the overall IVF procedure. Superovulation is a method to retrieve multiple eggs using drug induced stimulation of the ovaries. 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.
Figure 1. Schematic diagram of the IVF procedure
The success of superovulation is critical in proceeding with the further stages of the IVF. The fertility drugs used for inducing multiple ovulations are external hormones which are very expensive. The superovulation protocols follow a standard for hormonal dosage and after the initial dose of follicle stimulating hormone the patient requires daily testing and monitoring to keep a check on the patient's response and thus modify the dose. Improper dose may result into life threatening complications like ovarian hyperstimulation syndrome (OHSS) or at times may not cause any response in the patient. Thus, existing protocols lack planned treatment initiation and are highly dependent upon the equipments for monitoring.
The current work aims at developing a model predictive method for the drug dosage in superovulation. The drug dosage prediction method will take into account the treatment which is highly cost intensive and lacks individualized treatment variations will get a strong base. The comparisons between different method will help in deciding the best predictive method. The information regarding the single or multiple solutions possible by the different methods used will also provide insight into the system uncertainties.
2. Model Equations
Superovulation model is developed on the lines of batch crystallization modeling using the method of moments. The follicle size is converted into mathematical moments by assuming the follicles to be spherical in shape. The eq. 1 is used for converting follicle size to moments. Each moment has its own significance and corresponds to a feature of the follicles. The zeroth moment is a representative of the number, first moment represents the size, second the area, etc.
Here, µ_{i} = i^{th} moment
n_{j}(r,t) = number of follicles in bin 'j' of mean radius 'r' at time 't'.
r_{j} = mean radius of j^{th} bin
Δr = range of radii variation in each bin
It is assumed that follicle growth is dependent on follicle stimulating hormone (FSH) administered. The growth term is written as shown in equation (2) ;
Here, G  follicle growth rate
k  rate constant
C_{fsh}  amount of FSH injected
α  rate exponent
From the literature by Baird, 1997 it can be assumed that the number of follicles activated for growth are always constant (for a particular protocol initiation) for a particular patient, hence the zeroth moment has a constant value. We use the 0^{th} to 6^{th} order moments since they help in better prediction of moment values as well as help in efficient recovery of the size distributions as against the lower order moments. The moment equations for the follicle dynamics can be written as in equations (3) and (4).
dμidt=iGtμi1t; (i=1,2,…6) (4)
Here, G  follicle growth rate
3. Optimal Control Problem
The moment model for follicle size distribution prediction and the method for deriving normal distribution parameters have been used as the basis for deriving expressions for the mean (eq. 5) and coefficient of variation (eq. 6) for the follicle size distribution. The paper by John et. al. 2007 highlights the techniques for reconstructing distributions from moments and they obtain good approximation of realistic distributions using some finite order of moments. Since, the data clearly reflects a normal distribution it is quite reasonable to assume it as an apriori distribution for follicles.
CV  coefficient of variation
µ_{0}, µ_{1}_{ }and_{ }µ_{2}  zeroth, first and second order moment respectively.
Thus, the objective of superovulation can be stated as; to minimize the coefficient of variation at the final time (CV(t_{f}))and the control variable for this shall be the dosage of FSH with time (C_{fsh }(t)).
4. Solution Methods
The optimal control problem is solved by unconstrained and constrained maximum principle and dicretized nonlinear programming. Initial guess for the optimization variables are varied to check for multiple solutions. The results from all the methods are compared for better understanding of the parametric uncertainty as well as the best solution strategy. The fig. 2 shows the flowchart for the maximum priciple method.
Figure 2. Flowchart for optimal FSH dosage evaluation using maximum principle
5. Some Results
The optimal control results show the comparison of optimized prediction with the actual observations on the final day of the follicle size and number measured. The figures for two patients B and C are shown and observations for 4 patients are summarized in the Table 1.
Table 1. Summary of Results for Patient A, B, C & D




 






A (8)
 1650
 1388
 4
 6
 18.88

B (23)
 525
 463.5
 12
 15
 13.02

C (18)
 750
 617
 10
 13
 17.73

D (5)
 2250
 1855
 0
 1
 17.56

Figure 3a. Optimal FSH dose against the administered dose
3b. Final day follicle number prediction (Patient B)
Figure 4a. Optimal FSH dose against the administered dose
4b. Final day follicle number prediction (Patient C)
6. Conclusions
The moment model developed for IVF superovulation predicts the follicle size distribution which is in good agreement with the actual size distribution. The optimal control theory application to superovulation stage provides a new approach for model predictive drug delivery in IVF. The mathematical formulation of the objective function in terms of the coefficient of variation by utilizing the concepts of normal distribution provides a reasonably good measure of the final outcome. The predetermined dosage saves the cost of excess medicines and also the requirements of daily monitoring and testing. It can be said that the applications of control principles to a medical treatment procedure like IVF which was previously based on trial and error gets a good basis for planned treatment initiation. Also the current work has been done in collaboration with clinicians and real patient data has been used; which makes the study more emphatic when compared to theoretical work.
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