The main goal of this talk is two-fold: (1) rigorous development of a general framework for synergy theories and (2) study of mathematical properties of one important synergy theory, proposed by David Hand in a little-known article published in 2000 and here called Hand Incremental Effect Additivity (HIEA).

Synergy theories are widely used in many biomedical sciences including pharmacology, toxicology and radiation biology. In spite of this, they are plagued by vagueness, misconceptions and controversies. This calls for development of a more rigorous conceptual and mathematical foundation of synergy theories. In general, synergy theory describes effects of joint action of multi-component agent combinations based on 1-agent dose-effect relations (DERs), known from analyzing previous 1-agent experiments.

I will discuss three specific synergy theories: Simple Effect Additivity, Loewe-Berenbaum Additivity and HIEA with an emphasis on the latter. In particular, I will show that HIEA combination DER is in general not well defined in that it can blow up" at finite total dose. I will also formulate necessary and sufficient conditions on 1-agent DERs preventing such a pathology. Using the technique of weighted harmonic means I will study the betweenness property of HIEA synergy theory and demonstrate by means of theorems and examples that Hand's combination DER has a systematic tendency for betweenness violation. On the positive side, I will introduce the concept of a strongly dominant agent and show that the presence of such an agent ensures the betweenness property.

My emphasis in this talk will be on synergy theories that can handle any number of agents. This is motivated by applications to the study of toxic effects of the galactic cosmic ray on astronauts during interplanetary travel.