How might spatial nonuniformity of dose in a homogeneous biological systemaffect its total response?

The total response of a homogeneous biological system to a fixed total dose of a biological agent is modeled by dividing the system into N cubical vox els, each of which can be associated with an individual dose D-n and an ind ividual response R-n = F(D-n). Among the results shown are the following: A. (Voxel Theorem). Let the average dose D-avg be held fixed as the dose di stribution is shifted from uniform u to arbitrary a. Then, if F' greater th an or equal to 0 over [D-min, D-max] and R = Sigma (N)(n=1) R-n, a sufficie nt condition that NF(D-avg) = R(u) less than or equal to R(a) is that F be a concave-upwards function of dose; that is, F" greater than o r equal to 0 over [D-min, D-max]. B. If F' is constant over [D-min, D-max], then R(a) = R(u). That is, the total response is a function of D-avg only. The applications of these (and other) results are illustrated by examples from bioelectromagnetics. Bioelectromagnetics 22:66-70, 2001. (C) 2001 Wile y-Liss, Inc.