Dragalin V.
Adaptive Designs
ADAPTIVE DESIGNS FOR DOSE-FINDING BASED ON EFFICACY-TOXICITY RESPONSE Dragalin V.* and Fedorov V. Research Statistics Unit, SDS-BDS, GlaxoSmithKline 1250 S. Collegeville Rd, Collegeville, PA 19426-0989, U.S.A. Email:
[email protected] [email protected] Phone: 1-610-917 6242 ; Fax: 1-610-917 7494 Corresponding author: Dragalin V. Keywords: Adaptive design; Dose-…nding; Gumbel model; Phase I/II clinical trials Topic area of the submission: Design of dose responses studies for both safety and e¢ cacy Introduction. Most designs for dose-…nding in Phase I clinical trials determine a maximum tolerable dose (MTD) based on toxicity alone, while ignoring e¢ cacy response. Most Phase II designs assume that a toxicity acceptable dose range has been determined and aim to establish treatment e¢ cacy at some dose in this range, with early stopping if response rate is too low. However, under a variety of circumstances, it is useful to address safety and e¢ cacy simultaneously. Here is just a sample of references with such examples: graft-versus-host disease vs rejection in transplant complications [1]; biologic agent IL-12 in malignant melanoma [2]; antiretroviral treatment for children with HIV [3]; allogeneic stem cell transplantation for leukemia [4]; and many on-going trials of cytostatic and biologic agents. We present a class of models that can be used in early phase clinical trials in which patient response is characterized by two dependent binary outcomes, one (Y ) for e¢ cacy and one (Z) for toxicity. We are interested in response-adaptive designs with both dose allocation rules and early stopping rules in terms of response and toxicity, thus combining elements of more typical Phase I and Phase II trials. They are based on model-oriented design of experiment with constraints, used to penalize sampling at doses that are toxic or non-e¢ cacious. The designs are adaptive in the sense that decisions are made sequentially throughout the trial, with each decision based on the data accumulated thus far. Such an approach is e¢ cient from the perspective of both time and patient resources. Dose-Response Model. Let Y be the binary 0=1 indicator of e¢ cacy response, and let Z be the binary 0=1 indicator of toxicity response. Patients have a staggered entry in the trial and are allocated to one of the available doses from some …xed set of doses D = fd1 ; d2 ; : : : ; dk g. The doses are ordered in terms of their toxic potential. The response (Y; Z) can be modeled as a bivariate quantal response using Gumbel model for bivariate logistic regression. The standard Gumbel distribution function is given by G(y; z) = F (y)F (z)f1 + [1
F (y)][1
where F (y) =
F (z)]g; 1 1+e
y
j j < 1; 1 < y; z < +1;
(1)
Dragalin V.
Adaptive Designs
is the standard logistic distribution function. A straightforward generalization is to replace F ( ) in (1) with any distribution function. A further generalization might be to replace the survival function [1 F ( )] in (1) with an arbitrary positive non-increasing function g( ): This is so-called Farlie-Gumbel-Morgenstern class of bivariate distributions, see [5], Sect. 44.13 for details. In the univariate case, the probability of a response given a dose is expressed as the logistic cumulative distribution function of dose. A natural extension in the bivariate case is to express each of the four cell probabilities as an integral, over the corresponding region of the plane, of a bivariate logistic density function. If the dose d is transformed by the location parameter ( E = ED50 ; T = T D50 ) and scale parameter to the ”standard doses” dE =
d
E
for e¢ cacy Y ,
and
dT =
d
E
T
for safety Z,
T
then individual cell probabilities pyz (d) = pyz (d; ) = Pr(Y = y; Z = z j D = d) can be derived: p11 (d) p10 (d) p01 (d) p00 (d)
= = = =
G(dE ; dT ); G(dE ; 1) G(dE ; dT ) = F (dE ) G(dE ; dT ); G(1; dT ) G(dE ; dT ) = F (dT ) G(dE ; dT ); 1 F (dE ) F (dT ) + G(dE ; dT );
where = ( E ; E ; T ; T ; ): Likelihood Function and Information Matrix. The log-likelihood function of a single observation (Y; Z) at dose d is l(y; z; d; ) = yz log p11 + y(1
z) log p10 + (1
y)z log p01 + (1
y)(1
z) log p00 :
The Fisher information matrix of a single observation can be written as I( ; d) =
@p @p I(p; ) | ; @ @
where p = (p11 ; p10 ; p01 )| and 0 1 1 p11 0 0 I(p; ) = @ 0 p101 0 A + 1 0 0 p011
p11
1 p10
p01
``| ;
`| = (1; 1; 1):
Criteria. The criterion is based on …xed trial-speci…c standards for the minimum acceptable response rate qE and the maximum acceptable toxicity rate qT . De…ne dE as the minimum dose d for which Pr(Y = 1 j D = d)
qE
(2)
qT:
(3)
and dT as the maximum dose d for which Pr(Z = 1 j D = d)
Dragalin V.
Adaptive Designs
1.0
P(Y=1) P(Z=1) P(Y=1,Z=0)
0.8 0.6 0.4
qT qE
0.2 0.0
-3 dE* 54th Session 2003 Berlin
-2
-1d* dT*
0
1
2
3
x
Figure 1: Dose-response relationship and the best acceptable dose
See Fig 1. The best acceptable dose is the dose d in interval [dE ; dT ] maximizing the probability of success Pr(Y = 1; Z = 0 j D = d): (4) The goals are to select the one best acceptable dose for Phase III clinical trial, to estimate e¢ cacy and toxicity probabilities at that dose, and to stop the trial early if it becomes unlikely that any dose is safe and e¤ective, i.e. [dE ; dT ] = ?: Adaptive Design. Our adaptive procedure is based on optimal design methods for convex objective functions and cost constraints proposed in [6], [7]. However, while these authors considered the static locally optimal designs, i.e. designs derived for the whole experiment, we apply similar ideas in adaptive setting. Each new cohort of patients is allocated to the doses that maximize the expected increment of information (in terms of selected criterion), given the current interim data Ft . The maximization is carried over the whole range of possible doses D with additional constraints that involve the probability of toxicity at those doses, accommodating the MTD mentality: dose escalate cautiously starting from the lowest dose. Since the model is nonlinear, the information matrix depends on the values of unknown parameters and the design has to be performed in stages. Initial design is chosen and preliminary parameter estimates are obtained. Then, the next dose is selected from the available range of doses that satisfy the e¢ cacy and toxicity constraints (2-3) and provide the maximal improvement of the design with respect to the selected criterion of optimality and current parameter estimates. The next available cohort of patients is allocated to this dose. The estimates of unknown parameters
Dragalin V.
Adaptive Designs
are re…ned given these additional observations. This reiteration is repeated until either the available resources are exhausted or the set of acceptable doses is empty. For example, for the D-criterion, the dose dt+1 maximizing (d; bt ) = trfI(bt ; d)I
1
(bt ;
t
j Ft )g;
subject to given constraints on e¢ cacy toxicity (2-3), will be selected at stage t: Closely related, but computationally simpler, approaches are to select as the dose for the next cohort the one that maximizes b =
1 (d; t )
(d; bt )=C(d; bt j Ft )g; or
b =
2 (d; t )
(d; bt )
C(d; bt j Ft )g;
where the function C(d; bt j Ft ) is a current estimate of cost function, which may depend on unknown parameters, and re‡ects ethical and economical burden associated with that dose. It can be shown that the proposed adaptive procedure converge to the corresponding locally optimal design. The approach can also be implemented in Bayesian framework. In this case, the prior on p at two-three di¤erent doses will be solicited, similarly as in [2]. The …rst procedure then will select the next dose as the one that maximizes the …rst-order approximation of the Shannon information increment. Adaptive dosing is more e¤ective than is the standard design at identifying the optimal dose. Accounting for toxicity and response addresses the ethical concern that, as much as possible, subjects be allocated at or near doses that are both safe and e¢ cacious.
References [1] Gooley, T., Martin, P., Fisher, L. and Pettinger, M. (1994). Simulation as a design tool for phase I/II clinical trials: An example from bone marrow transplantation. Controlled Clinical Trials 15: 450–462.
[2] Thall, P. and Russell, K. (1998). A strategy for dose-…nding and safety monitoring based on e¢ cacy and adverse outcomes in phase I/II clinical trials. Biometrics 54: 251–264. .
[3] O’Quigley, J., Hughes, M., and Fenton, T. (2001) Dose …nding designs for HIV studies. Biometrics 57: 1018–1029.
[4] Braun, T. (2002). The bivariate continual reassessment method: extending the CRM to phase I trials of two competing outcomes. Controlled Clinical Trials 23: 240–256.
[5] Kotz, S., Balakrishnan, N. and Johnson, N.L. (2000). Continuous Multivariate Distributions, Vol. 1, Wiley.
[6] Fedorov, V.V. and Hackl, P. (1997). Model-Oriented Design of Experiments. Lecture Notes in Statistics, 125. Springer.
[7] Cook, D. and Fedorov, V. (1995). Constrained optimization of experimental design. Statistics, 26: 129–178.
