Mathematical properties of Hand Incremental Effect Additivity and other synergy theories

Synergy theories for multi-component agent combinations use 1-agent dose-effect relations (DERs), known from analyzing previous 1-agent experiments, to calculate neither-synergy-nor-antagonism combination DERs. Synergy theories are widely used in pharmacology, toxicology, and radiation biology. This article analyzes mathematical properties of one important synergy theory, proposed by John Hand in a little-known article published in 2000 and here called Hand Incremental Effect Additivity (HIEA). In 2018, Hand's approach was reinvented in a radiobiology study that inadvertently overlooked his paper. We carefully state the assumptions required for mathematically rigorous development of HIEA and other synergy theories. Under these assumptions, studying synergy/antagonism between any number of agents can be done on a single 2d plot. We show that HIEA combination DER is in general not well-defined in that it can "blow up" at finite total dose. We also formulate necessary and sufficient conditions on 1-agent DERs preventing such pathology. Using weighted harmonic means, we study the betweenness property of HIEA synergy theory and demonstrate that Hand's combination DER has a systematic tendency for betweenness violation. On the positive side, we introduce the concept of a strongly dominant agent and show that the presence of such an agent ensures the betweenness property. We show also that the betweenness property holds for another major synergy theory, Loewe-Berenbaum Additivity. Our emphasis in this article on synergy theories that can handle any number of agents was motivated by applications to the study of toxic effects of galactic cosmic rays on astronauts during interplanetary travel.