MATHEMATICAL FORMULATION
The key concept underlying EWOC is that one can select dose levels for use in a phase I trial so that the predicted proportion of patients who receive an overdose is equal to a specified value 
, called the feasibility bound. This is accomplished by computing, at the time of each dose assignment, the posterior cumulative distribution function (cdf) of the MTD. For the k-th dose assignment the posterior cdf of the MTD is the function 
given by
,where the data at the time of treatment for the k-th patient would include, for each previously treated patient, the dose administered, the highest level of toxicity observed and any relevant covariate measurements. 
is the conditional probability that 
is an overdose given the data currently available. Based on this, EWOC selects for the k-th patient the dose level 
such that 
. That is, the dose for each patient is selected so that the predicted probability it exceeds the MTD is equal to 
.
Let 
denote the minimum and maximum dose levels available for use in the trial. The dose to be given the first patient is taken to be 
and only dose levels between 
will be selected for use in the trial. Thus, if n is the total number of patients to be accrued to the trial and xi denotes the dose level selected for the i-th patient, 
, then 
and 
The dose-toxicity relationship is modeled as
- (1)
where F is a specified distribution function, called a tolerance distribution, and 
are unknown. It is assumed that 
so that the probability of a DLT is a monotonic increasing function of dose. The MTD is the dose level, denoted 
, such that the probability of a DLT is 
. It follows that
or, equivalently,
,where 
denotes the probability of a DLT at the starting dose 
. Figure 1 illustrates a typical dose-toxicity model.
The binomial response of the i-th patient is denoted 
and assumes the value 
if a DLT is manifest and the value 
, otherwise. The data after k patients have been observed is 
and the likelihood function of 
given 
is
Prior information about 
and 
is incorporated through a prior probability density function 
defined on
- (2)
.After an application of Bayes theorem, the joint posterior distribution of 
given the data 
is found to be
- (3)
,where
and 
denotes the indicator function for the set 
. The posterior cumulative distribution function of the MTD given 
can be derived from (3) through the transformation 
. Denoting the image of 
under the transformation T by 
, it follows from (2) that
.The inverse transformation is given by
,where the functions f1
and f2 are defined on 
by
and
.The joint posterior probability density function (pdf) of 
given 
can now be written as
,where
.
Note that g is the prior pdf induced for 
by the choice of h as the prior pdf of 
. Elicitation of prior information can be through specification of the pdf g directly, rather than through the choice of h. This might be advantageous since 
is the parameter of interest and preliminary studies are often conducted at or near the starting dose so that a meaningful informative prior can be selected for 
. Letting
,the marginal posterior pdf of the MTD given 
can be written as
.The marginal posterior cdf of the MTD given 
is then given by
.EWOC can now be described as follows. The first patient, or cohort of patients, receives the dose 
. The dose for each subsequent patient is selected so that on the basis of all the available data the posterior probability it exceeds the MTD is equal to the feasibility bound 
. Hence, the k-th patient receives the dose
where m(k) denotes the number of observations available at the time the k-th patient is to be treated.
Upon completion of the trial the MTD can be estimated by minimizing the posterior expected loss with respect to some choice of loss function l. Thus, the dose recommended for use in a subsequent phase II trial would be the estimate 
given by
Note that the dose 
selected by EWOC for the k-th patient corresponds to the estimate of the MTD having minimal risk with respect to the asymmetric loss function
- (4)
The loss function 
implies that for any 
, the loss incurred by treating a patient at 
units above the MTD is 
times greater than the loss associated with treating the patient at 
units below the MTD.
This interpretation might provide a meaningful basis for the selection of the feasibility bound.
REFERENCES
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- Sinha R, Kaufman JL, Khoury HJ, King N, Shenoy PJ, Lewis C, Bumpers K, Hutchison-Rzepka A, Tighiouart M, Lonial S, Lechowicz MJ, Heffner LT, and Flowers CR. 2012. A Phase 1 Dose Escalation of Bortezomib Combined with Rituximab, Cyclophosphamide, Doxorubicin, Modified Vincristine, and Prednisone for Untreated Follicular Lymphoma and other Low Grade B-cell Lymphomas. Cancer, in press.
Samuel Oschin Comprehensive Cancer Institute
Cedars-Sinai Medical Center
116 N. Robertson Blvd Suite 500
Los Angeles, CA 90048, USA
