Purpose: There is growing interest about biological effective dose (BED) and its application in treatment plan evaluation due to its stronger correlation with treatment outcome. An approximate biological effective dose (BED_A) equation was introduced in order to simplify BED calculations by treatment planning systems in multi-phase treatments. The purpose of this work is to reveal its mathematical properties relative to the true, multi-phase BED (BED_T) equation.

Methods: The BED_T equation was derived and used to reveal the mathematical properties of BED_A. MATLAB (MathWorks, Natick, MA) was used to simulate and analyze common and extreme clinical multi-phase cases. In those cases, percent error and Bland-Altman analysis were used to study the significance of the inaccuracies of BED_A for different combinations of total doses, numbers of fractions, doses per fractions and α/β values. All the calculations were performed on a voxel-basis in order to study how dose distributions would affect the accuracy of BED_A.

Results: When the voxel dose-per-fractions (DPF) delivered by both phases are equal, BED_A and BED_T are equal (0% error). In heterogeneous dose distributions, which significantly vary between the phases, there are fewer occurrences of equal DPFs and hence the imprecision of BED_A is greater. It was shown that as the α/β ratio increased the accuracy of BED_A would improve. Examining twenty-four cases, it was shown that the range of DPF ratios for 3% P_error varied from 0.32 to 7.50Gy, whereas for P_error of 1% the range varied from 0.50 to 2.96Gy.

Conclusion: The DPF between the different phases should be equal in order to render BED_A accurate. OARs typically receive heterogeneous dose distributions hence the probability of equal DPFs is low. Consequently, the BED_A equation should only be used for targets or OARs that receive uniform or very similar dose distributions by the different treatment phases.

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Cite this article as: Kauweloa KI, Gutierrez AN, Bergamo A, Stathakis S, Papaniko-laou N, Mavroidis P. Mathematical analysis of approximate biological effective dose (BED) calculation for multi-phase radiotherapy treatment plans. Int J Cancer Ther Oncol 2014; 2(2):020226. DOI: 10.14319/ijcto.0202.26