(666d) Comparison of Different Methods for Predicting Customized Drug Dosage in Superovulation Stage of In-Vitro Fertilization

Comparison
of Different Methods for Predicting Customized Drug Dosage in Superovulation
stage of In-vitro fertilization

Kirti M. Yenkie, ^a,b andUrmila
M. Diwekar ^a,b 

^aDepartment
of Bio Engineering, 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
one of the highly pursued assisted reproductive technologies worldwide. The IVF
procedure is divided into four stages: Superovulation, Egg-retrieval,
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
non-linear 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, non-linear 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.

μi=njr,trjiDrj
      
                               (1)

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) ;

G=kCfshα
                                                     (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).

μ0=constant
                                               (3)                                                                                   

dμidt=iGtμi-1t; (i=1,2,?6)    
      (4)                                                     

Here, G - follicle growth rate

           μi
 - i^th moment

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.

x=μ1μ0
                                                (5)

CV= μ2μ0μ12-1
                            (6)                       

Here, x
 - mean follicle size

CV - coefficient of variation

µ₀, µ₁andµ₂- 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
non-linear 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

                                    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.