Response Surface Modelling of Drug Interactions
Dr. Charles Minto, MB, ChB, FFANZCA
Senior Lecturer, Department of Anaesthesia and Pain Management, University of Sydney, Australia.
Introduction
The
name response surface methodology (RSM)
has been given to the statistical methodology concerned with; (1) the design of
studies to estimate response surfaces, (2) the actual estimation of response
surfaces, and (3) the interpretation of the results.
Response surface methodology is generally employed for two principal purposes: (1) to provide a description of the response pattern in the region of the observations studied, and (2) to assist in finding the region where the optimal response occurs (i.e., where the response is at a maximum or a minimum).
In figure 1, a polynomial function has been fit to hypothetical data showing the effects of water and sunshine on plant growth. The fitted response surface showed can be used to determine the optimal amount of sun and water for plant growth. The use of polynomial response functions to approximate complex response surfaces is common in many experimental situations.
A description of methods used to estimate and evaluate a fitted response surface, and of tests used to decide whether interaction effects are required is beyond the scope of this presentation. Those seeking further information will find a brief description of RSM in Neter et al.,[1] and a full description by Box and Draper.[2]
Experimental Design.
A large variety of experimental designs have been developed for estimating response surfaces efficiently.[3] Examples for two variables are illustrated in figure 2. The “ray” design studies the response of two variables present in a number of fixed ratios. In the case of two drugs, each ratio can be considered as a single drug, which permits an analysis based on the same principles as that associated with single drug experiments. The “full factorial” design studies the response of all combinations of two variables at a number of different levels (“fractional factorial” designs are also employed). In the case of two drugs, the different levels usually represent different doses.[4],[5],[6] The “central composite design” was developed to provide enough treatment combinations to permit the estimation of the parameters in a quadratic predictive model, while using considerable fewer treatment groups.
When there are three variables, they should be; (1) studied alone, (2) studied in three pairs, and (3) studied in the triple combination.[7]
Figure 2: Graphic representation
of various experimental designs for two drugs. Ray design (left), full
factorial design (middle), and central composite design (right). Each circle
represents a dose combination that is investigated.3
Modelling Drug
Interactions
Many different approaches have been used to model drug interactions.[8] A major strength of response surface approaches is that they can help explain the similarities and differences among other approaches used to study drug interactions.[9] A response surface is a mathematical equation or the graph of that equation that relates a dependent variable (such as a drug effect) to inputs (such as two drug concentrations). A response surface illustrating a synergistic interaction for the hypnotic effect of two drugs illustrated in figure 3.
The Optimal Response
In many experimental situations, response surfaces are used to determine optimal response conditions. Although the optimal conditions are readily determined from figure 1 (the highest point), they are not so easily determined in figure 3. Either drug alone or any ratio of the two drugs is capable of achieving the maximum effect (if this is the goal). The optimum cannot be determined without considering the pharmacokinetic and side effect profiles of the individual drugs.