Mathematical analysis of approximate biological effective dose (BED) calculation for multi-phase radiotherapy treatment plans

Kevin I Kauweloa, Alonso N Gutierrez, Angelo Bergamo, Sotirios Stathakis, Niko Papanikolaou, Panayiotis Mavroidis


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 (BEDA) 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 (BEDT) equation.

Methods: The BEDT equation was derived and used to reveal the mathematical properties of BEDA. 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 BEDA 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 BEDA.

Results: When the voxel dose-per-fractions (DPF) delivered by both phases are equal, BEDA and BEDT 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 BEDA is greater. It was shown that as the α/β ratio increased the accuracy of BEDA would improve. Examining twenty-four cases, it was shown that the range of DPF ratios for 3% Perror varied from 0.32 to 7.50Gy, whereas for Perror 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 BEDA accurate. OARs typically receive heterogeneous dose distributions hence the probability of equal DPFs is low. Consequently, the BEDA equation should only be used for targets or OARs that receive uniform or very similar dose distributions by the different treatment phases.


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

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International Journal of Cancer Therapy and Oncology (ISSN 2330-4049)

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