Evaluating the biologically effective dose (BED) concept using a dynamic tumor simulation model

Purpose: To evaluate three different formulae for calculating the biologically effective dose (BED) by use of a multipopulation reaction–diffusion simulation to determine whether these formulae produce equivalent effects for different treatment regimes. Methods: The standard BED formula, BED_s, was updated to account both for spacial nonuniformity in dose and for cellular regrowth between fractions, by creating two new formulae: BED_ϕ and BED_ϕT. These BED formulae were used to calculate dose per fraction values for two, three, and five fraction treatments and to compare the tumor volumes of those treatments to those of a single fraction. A spherical tumor model based on the reaction–diffusion equation was used to calculate the final volume of each tumor 185 days after the delivery of the first fraction. The percent difference in volume between single-fraction and multiple-fraction treatments was used as a measure to test the accuracy of each BED formula. Results: Percent differences in volume between single- and multiple-fraction treatment regimes varied up to approximately 18.5% if the dose per fraction was calculated using BED_s but the delivered dose was nonuniform. Proper application of spacial nonuniformity in dose and tumor regrowth correction factors modified the dose per fraction values by no more than 5%, but resulted in the improvement of simulated tumor volumes down to around 2% or lower difference in volume. Conclusions: Treatment regimes with the same BED value should have the same effect. However, small changes in the dose per fraction delivered in multiple-fraction treatments can have a large effect on the tumor volume of a treatment when the dose is delivered nonuniformly or when tumor regrowth between fractions is ignored. Inclusion of these correction factors is important for the underlying assumption that treatments with equal BED will have equal effects on the clinically observed tumor volume.