On the existence conditions of interaction indices in response surface models.

Response-surface models (RSMs) are widely used to characterize the combined effects of two agents and to classify their interaction as additivity, synergy, or antagonism. In this context, an interaction index is frequently used as a quantitative measure. However, estimation of the interaction index outside feasible ranges may result in complex-valued or biologically implausible predictions in certain regions of the input domain, thereby limiting the validity of the model. This paper introduces a feasibility analysis framework that distinguishes between isobole-level feasibility (i.e. the existence of well-defined isoboles at a given effect level) and global feasibility (i.e. well-posedness across the entire domain). The analysis explicitly characterizes the singular sets that arise when feasibility conditions are violated, thereby explaining when and how models fail. The approach is demonstrated on three canonical models, i.e. Greco, Minto, and Finney, and supported by numerical illustrations, offering practical guidance for systematic and robust parameter selection in drug combination studies, toxicology, and process engineering.