# Sequential Search of an Optimal Dosage

• Shelemyahu Zacks
Chapter
Part of the Statistics for Industry, Technology, and Engineering book series (SITE)

## Abstract

New drugs are subjected to clinical trials before being approved for usage. These clinical trials consist of several phases. In phase I the objective is to establish the maximal tolerated dose (MTD). The assumption is that the effectiveness of a drug is a monotone increasing function of the dosage applied. On the other hand, there are always side effects due to the toxicity of the drugs. The level of toxicity tolerated by different subjects varies at different dosages. There is a tolerance distribution in a population of subjects (patients), which is a function of the dosage. There are cases in which the level of toxicity can be measured on a linear scale and is represented by a continuous random variable Y (x), where x is the dosage. The first part of the present chapter deals with such a model, following the paper of Eichhorn and Zacks (J Am Stat Assoc, 68:594–598, 1973). The second part of the chapter deals with cases in which the level of toxicity is represented by a binary variable J(x), where J(x) = 1, if the level of toxicity is dangerous to the life of the patient, and J(x) = 0, otherwise. A dangerous dose is called “lethal doese, LTD.” An MTD is a maximal dosage such that the probability of an lethal doese, LTD is smaller than a prescribed threshold γ (in many cases γ = 0.3). A sequential search for an MTD applies the drug to individuals, one by one, starting from a “safe” dosage, and increasing or decreasing the dosage according to the observed toxicity levels.

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