Applying E_max model and bivariate thin plate splines to assess drug interactions

We review the semiparametric approach previously proposed by Kong and Lee and extend it to a case in which the dose-effect curves follow the Emax model instead of the median effect equation. When the maximum effects for the investigated drugs are different, we provide a procedure to obtain the additive effect based on the Loewe additivity model. Then, we apply a bivariate thin plate spline approach to estimate the effect beyond additivity along with its 95% point-wise confidence interval as well as its 95% simultaneous confidence interval for any combination dose. Thus, synergy, additivity, and antagonism can be identified. The advantages of the method are that it provides an overall assessment of the combination effect on the entire two-dimensional dose space spanned by the experimental doses, and it enables us to identify complex patterns of drug interaction in combination studies. In addition, this approach is robust to outliers. To illustrate this procedure, we analyzed data from two case studies.