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Controlled Optimal Designs for Dose Response Studies

---- and a Website for Optimal Designs

Xiangfeng Wu1, Wei Zhu1, Holger Dette2, Weng Kee Wong3

1Department of Applied Mathematics and Statistics, State University of New York, Stony Brook, NY 11794-3600, USA; 2Ruhr-Universität Bochum, Fakultät für Mathematik, 44780 Bochum, Germany;3Department of Biostatistics, University of California, Los Angeles, CA, 90095-1772, USA

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Introduction

We present the numerical algorithms for finding Bayesian multiple-objective optimal designs for dose-response studies subject to certain constraints. Dose-response studies are routinely conducted in drug design process. Due to safety, efficacy, and experimental design considerations, practical constraints are often imposed on dose range, dose levels, dose numbers, dose proportions and potential missing trials. The resulting controlled optimal designs satisfying these constraints can be readily adopted for optimal estimation of the parameters of interest such as the median effective dose or the threshold dose. In addition, we describe the methodologies and the implementation of a web based interactive optimal design platform for the practitioners. We demonstrate our results and methodology through the widely used logit dose response model.

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Optimal Experimental Design

The fundamental idea of Optimal Experimental Design is to find the best allocation rule such that the underlying model or other parameters of interest can be estimated or predicted with the best precision for a given sample size. The variances of parameter estimates and predictions depend upon the experimental design and should be made as small as possible. Unnecessarily large variances and imprecise predictions resulting from a poorly designed experiment could render an experiment inconclusive.

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Quantal Dose Response Model

The study of drug toxicity and efficacy is a crucial part of the drug development process. Quantal dose-response experiment is routinely conducted to study the relationship between the dose level of a drug and the probability of a response (1 if the subject is responsive at the given dose and 0 otherwise). A commonly used model is the simple logit model:

)()(1

)(log

xx

x

Here is the probability of a response at dosage , where is the dose interval. The parameter is the slope in the logit scale and is the dose level at which the response probability is 0.5 and often referred to as the “median effective dose”, and denoted by . More generally, we let denote the dose level x at which the probability of observing a response is . This means that the percentile or is equal to , where .

))(exp(1/1)( xx x

50ED 100ED

100ED

/ 1/log

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Optimal Design Criteria –(1)

2

( , ) log | ,ij

i j

M y x dx

Fisher’s Information Matrix. Let y be the indicator of a response at dose x and let θ=(α,β). Since the probability of observing a response at dose level x is π(y=1|x,θ)=π(x), it follows that the (i,j)th element of the observed Fisher information matrix for design ξ is

where design is a probability measure with weights on design support point for

iw

ix

m

i ii wwmi1

1,0,,...,1

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Optimal Design Criteria – (2)

),( M

1log ,M

cMc T ,1

An optimal design is one that minimizes some properly chosen convex function of the information matrix . In particular, we introduce the following optimality criteria.1. D-optimality. The D-optimal design would

minimize the generalized variance of the parameter estimates:

2. C-optimality. Here our interest lies in estimating the parameters

c(θ) with minimum variance. Thus the c-optimal design would

minimize the function:

where is the gradient of c(θ). c

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Bayesian Approach – (1)

An immediate issue relating to the construction of optimal designs for the dose-response model is that the information matrix M or the design criteria function depends on the unknown parameters of that model.

A common approach to the problem is to plug in the best guess of the parameter values, and thus the name ‘locally optimal design’.

Alternatively, one can impose distribution assumptions on the unknown parameters from previous studies or by elicitation, and use a Bayesian approach.

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Where denotes the prior probability density, can be constructed .

Bayesian Approach – (2)

1 1log , log ,D E M M g d

dgcMccMcE TT

c ,, 11

The Bayesian Approach is optimizing the average of a function of the information matrix over a prior distribution placed on the unknown parameters. Specifically, the Bayesian D-optimal design minimizes:

And the Bayesian c-optimal design minimizes:

g

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Suppose there are m objectives in the study and each of them is represented by a convex criterion

Two equivalent approaches have been suggested for constructing a multiple-objective optimal design. One way is to find a compound optimal design and the other is to find a constrained optimal design.

We consider compound optimal design. The compound optimal design is the design which minimizes the convex combination

where each is user-selected and

For each convex combination, the compound optimal design is found using the same numerical algorithm for generating a single-objective optimal design.

Multiple-Objective Optimal Design

mii ,...,1,

i

m

i i 1

1,0i 11

m

i i

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The design efficiency of an arbitrary design relative to the optimal design is defined as

For the c-optimality criterion, this ratio compares the expected variances for the estimated given by the two designs. Designs with high efficiencies are desirable because they provide more accurate estimates for a given sample size.

Design Efficiency

*

*( ) ( ) / ( )E

)(c

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Restricted Design Space

In the framework of does-response studies accomplished on human subjects such as clinical trials prior to the launch of new drugs, we often encounter the problem that the support points of an optimal design lie outside a reasonable dosage range, i.e. dose levels are either below zero or exceed safety levels such as the maximum tolerated dose of the drug.

Practitioners are therefore in need of efficient designs that take the restricted design spaces into account in response to the drug toxicity and/or efficacy considerations.

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More Design Constraints Constraints on the number of dose levels (e.g. because of the usual modest sample

size, one can not accommodate too many dose levels). Potential missing trials (e.g. potential missing trials due to toxicity/side effect or

lack of efficacy). Constraints on dose levels (e.g. certain dose such as the placebo must be

included). Constraints on dose proportions (e.g. each dose level must have at least 10% of

the sample). We refer to optimal designs satisfying one ore more types of these practical

constrains as the controlled optimal designs.

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Numerical Approach

Using advanced math theory, one can sometimes, derive the optimal designs analytically. With the development of advanced computing software and hardware, however, we can always derive the optimal designs numerically.

Computationally, the optimization of the design criterion function is a nonlinear programming problem(NLP). In Practice, the controlled optimal designs based on a fixed number of design points can be found directly using an appropriate constrained nonlinear optimization algorithm.

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The Kuhn-Tucker (KT) equation. The KT equations are necessary conditions for optimality for a constrained optimization problem. If the problem is a so-called convex programming problem, that is, and are convex functions, then the KT equations are both necessary and sufficient for a global solution point.

Constrained Nonlinear Optimization

)(xf mixGi ,2,1),(

0)( *

1

**

xGxf i

m

ii

0** xGii mi ,...,1

The solution of the KT equations forms the basis to many nonlinear programming algorithms. These algorithms attempt to compute the Lagrange multipliers directly.

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Sequential Quadratic Programming (SQP)-(1)

The principal idea is the formulation of a QP sub problem on a quadratic approximation of the Lagrangian function.

m

iii xgxfxL

1

,

The Quadratic Programming (QP) Sub Problem

dxfdHdd

imize Tkk

Tn

2

1min

0 kiT

ki xgdxg

0 kiT

ki xgdxg

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Sequential Quadratic Programming (SQP)-(2)

Updating the Hessian Matrix. At each major iteration a positive definite quasi-Newton approximation of the Hessian of the Lagrangian function, H, is calculated using the BFGS method.

kkTk

kTk

kTk

Tkk

kk sHs

HH

sq

qqHH 1

kkk xxs 1

n

ikiik

n

ikiikk xgxfxgxfq

1111 )(

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Sequential Quadratic Programming (SQP)-(3)

At each major iteration of the SQP method, a QP problem of the following form is solved, where Ai refers to the ith row of the m-by-n matrix A.

dcHdddqd

imize TTn

2

1min

ii bdA emi ,...,1

ii bdA mmi e ,...,1

The method used to solve the problem is an active set strategy (also known as a projection method).

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The step length parameter is determined in order to produce a sufficient decrease in a merit function.

Line Search and Merit Function. The solution to the QP sub problem produces a vector , which is used to form a new iterate.

Sequential Quadratic Programming (SQP)-(4)

kd

kkk adxx 1

ka

m

miii

m

iii

e

e

xgrxgrxfx11

,0max

iiki

iiki rrr 2

1,max1

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Web Based Design Software

Why Web Based? For one thing, we would like to enable the users to perform statistical design and analysis from any location where an internet connection is available, without any complicated software installation and setup procedures.

Web Server

Design Software

Internet

Internet

Clie

nt

com

pu

ters

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System Architecture

Clie

nt w

eb

Bro

wse

r (IE

, etc

.)

IIS W

eb

Serve

r

Inte

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eb

A

pp

licatio

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Op

timal D

esig

n

En

gin

e

Matla

b S

ystem

HTML

HTML

ACTIVE SERVER PAGE

COMMON GATEWAY

M-FILES

BINARYSTREAM

BINARYSTREAM

CLIENT SIDE SEVER SIDE

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Implementation Technologies

Matlab Programming. MATLAB® is a high-performance language for technical computing. It integrates computation, visualization, and programming in an easy-to-use environment where problems and solutions are expressed in familiar mathematical notation.

ASP. Net C Sharp Programming. ASP.NET is a technology for building powerful, dynamic Web applications. ASP.NET makes building real world Web applications dramatically easier.

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The Optimal Experimental Design Website

www.optimal-design.org is the interface to the interactive optimal experimental design software. We also provide information of optimal design theories and useful resources on this website.

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Example: Consider the dose-response model on restricted dose space [0, 3.5] with at least 10% of the subjects allocated to each dose level. Suppose that independent uniform priors for alpha on the interval [1.5, 2.5] and beta on [0.9, 1.1] are selected. Furthermore, the experimenter wish to find a 2-point D-optimal design.

D-optimal design for alpha ~ U[1.5, 2.5] beta ~U[0.9, 1.1]

CONSTRAINTS

Restricted dose

Range: [0, 3.5]

Dose level: ≥ 0.1 2.233

An Example of Controlled Optimal Design – (1)

*D *DD

2/12/1

217.3784.0

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An Example of Controlled Optimal Design – (2)

Inp

uttin

g D

esig

n

Para

mete

rs

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An Example of Controlled Optimal Design – (3)

Th

e O

utp

ut

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Future Work

Improving Optimization Algorithms Incorporating Different types of Prior distributions Optimal Designs for Other Dose Response Models Incorporating Analysis of Experiments Exploration of theoretical properties of controlled optimal designs such

as the derivation of the general equivalence theorem.

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Reference

A.C. Atkinson and A.N. Donev (1992). Optimum Experimental Designs. Oxford Statistical Science.V.V. Fedorov (1972). Theory of Optimal Experiments. Academic Press New York and London.W. Zhu and W.K. Wong. Bayesian optimal designs for estimating a set of symmetrical quantiles. Statistics in Medicine. 2001;20:123-137M.C. Biggs. "Constrained Minimization Using Recursive Quadratic Programming," Towards Global Optimization (L.C.W. Dixon and G.P. Szergo, eds.), North-Holland, pp 341-349, 1975.M.J.D. Powell. "A Fast Algorithm for Nonlinearly Constrained Optimization Calculations," Numerical Analysis, G.A.Watson ed., Lecture Notes in Mathematics, Springer-Verlag, Vol. 630, 1978. S. Biedermann, H. Dette and W. Zhu. Optimal Designs for Dose-Response Models with Restricted Design Spaces. Journal of the American Statistical Association. 101(474), 747 -759.

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Acknowledgement

This work is supported by the US National Institute of Health grant no: 1R01 GM072876-01A1 “Cost Effective Designs for Practitioners ”.