**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 *x _{i}* 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 *f*_{1}

and *f*_{2} 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.

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