Mathematical Modeling and the Design of Studies in Drug Research

Abstract

In the most general sense, a drug can be thought of as having two dose response curves:

P₁(d) = Proportion of patients who “respond” at dose ≤ d.

P₂(d) = Proportion of patients who suffer adverse reactions at dose ≤ d.

The ideal drug is one with the two curves widely separated, where the first curve is steep (so a small range of doses are adequate to most patients) and the second curve is shallow. Both pre-clinical animal studies and human clinical studies are used to estimate the form and placement of those curves. However, the curves can never be fully characterized, and there are important definitional problems that have to be resolved, such as

What is a “response”?

Who are the patients?

What adverse reactions?

Time dependent patterns of response?

Etc.

As a drug is developed, segments of this overall mathematical model are formulated and studies are used to estimate parameters or test different submodels. This paper will try to present a coherent picture of how the designs of traditional studies fit this process and explore alternative designs that may be more useful.