A possible experimental design for combination experiments is to compa
re the dose-response curve of a single agent with the corresponding cu
rve of the same agent using either a fixed amount of a second one or a
fixed dose ratio. No interaction is then often defined by a parallel
shift of these curves. We have performed a systematic study for variou
s types of dose-response relations both for the dose-additivity (Loewe
additivity) and for the independence (Bliss independence) criteria fo
r defining zero interaction. Parallelism between dose-response curves
of a single agent and those of the same agent in the presence of a fix
ed amount of another one is found for the Loewe-additivity criterion f
or linear dose-response relations. For nonlinear relations, one has to
differentiate between effect parallelism (parallel shift on the effec
t scale) and dose parallelism (parallel shift on the dose scale). In t
he case of Loewe additivity, zero-interaction dose parallelism is foun
d for power, Weibull, median-effect and logistic dose-response relatio
ns, given that special parameter relationships are fulfilled. The mech
anistic model of competitive interaction exhibits dose parallelism but
not effect parallelism for Loewe additivity. Bliss independence and L
oewe additivity lead to identical results for exponential dose-respons
e curves. This is the only case for which dose parallelism was found f
or Bliss independence. Parallelism between single-agent dose-response
relations and Loewe additivity mixture relations is found for examples
with a fixed dose-ratio design. However, this is again not a general
property of the design adopted but holds only if special conditions ar
e fulfilled. The comparison of combination dose-response curves with s
ingle-agent relations has to be performed taking into account both pot
ency and shape parameters. The results of this analysis lead to the co
nclusion that parallelism between zero interaction combination and sin
gle-agent dose-response relations is found only for special cases and
cannot be used as a general criterion for defining zero-interaction in
combined-action assessment even if the correct potency shift is taken
into account. (C) 1998 Society for Mathematical Biology.