Mathematical Formulation
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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

  • Babb J, Rogatko A, Zacks S. 1998. Cancer phase I clinical trials: efficient dose escalation with overdose control. Stat. Med. 17:1103-1120.
  • Zacks S, Rogatko A, Babb J. 1998. Optimal Bayesian-feasible dose escalation for cancer phase I trials. Stat. and Prob. Ltrs. 38:215-220.
  • Babb JS, Rogatko A. 2001. Patient specific dosing in a cancer phase I clinical trial. Stat. Med. 20:2079-2090.
  • Cheng J, Babb JS, Langer C, Aamdal S, Robert F, Engelhardt LR, Fernberg O, Schiller J, Forsberg G, Alpaugh RK, Weiner LM, Rogatko A. 2004. Individualized patient dosing in phase I clinical trials: the role of EWOC in PNU-214936. J. Clin. Oncol. 22(4):602-9.
  • Babb JS, Rogatko A. 2004. Bayesian Methods for Cancer Phase I Clinical Trials. In: Advances in Clinical Trial Biostatistics, edited by Nancy Geller, New York, Marcel Dekker, pp. 1-40.
  • Tighiouart M, Rogatko A, Babb JS. 2005. Flexible Bayesian methods for cancer phase I clinical trials. Dose escalation with overdose control. Statistics in Medicine, 24, 2183-2196.
  • Tighiouart M, Rogatko A. 2006. Dose Finding in Oncology - Parametric Methods. In: Dose Finding in Drug Development, ed. N. Ting. Springer, New York, pp 59-72.
  • Tighiouart M, Rogatko A. 2006. Dose-escalation with overdose control. In: Statistical Methods for Dose-Finding Experiments, ed. S. Chevret. John Wiley and Sons, pp 173-188.
  • Rogatko A, Tighiouart M. 2007. Novel and Efficient Translational Clinical Trial Designs in Advanced Prostate Cancer, ed. L. Chung, W. Isaacs, and Simons, J. Humana Press, New Jersey.
  • Xu Z, Tighiouart M, Rogatko A. 2007. EWOC 2.0: Interactive Software for Dose Escalation in Cancer Phase I Clinical Trials. Drug Information Journal, 41(2):221-228.
  • Tighiouart M, Rogatko A. 2010. Dose Finding with Escalation with Overdose Control (EWOC) in Cancer Clinical Trials. Statistical Science 25(2)217-226.
  • Lonial S, Kaufman J, Tighiouart M, Nooka A, Langston AA, Heffner LT, Torre C, McMillan S, Renfroe H, Lechowicz MJ, Khoury HJ, Flowers CR, Waller EK. A Randomized Phase I trial Combining High Dose Melphalan and Autologous PBSC Transplant with Escalating Doses of Bortezomib for Multiple Myeloma: A Dose and Schedule Finding Study. 2010. Clinical Cancer Research, 16 (20), 5079-5086.
  • 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

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