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Purpose: For high linear energy transfer (LET) radiations, the relative biologic effect (RBE) changes with dose per fraction. Methods for calculating the optimum dose per fraction for high LET radiations should therefore include an allowance for RBE.Methods and Materials: The linear-quadratic (LQ) model, and the associated biologic effective dose (BED) concept, has previously been extended to incorporate the RBE effect. Differential calculus is now used to calculate the optimum dose per fraction (z), when high-LET radiation is used, which is given by the solution for z of (g-(LATE)(α/β)(L)(TUM)(α/β)(L)) ·RBE(M)z2-2·f·g·K·z-(LATE)(α/β)(L)·f·K·RBE(M)=0 where g is the normal tissue sparing factor, RBE(M) is the maximum RBE value, f the mean interfraction interval, K the daily low-LET BED equivalent dose for clonogen repopulation and (LATE)(α/β)(L) and (TUM)(α/β)(L) are the respective late reacting normal tissue and tumor fractionation sensitivities for low-LET radiation.Results: The optimum dose per fraction for proton therapy is generally lower than that calculated for photons but there is not a simple relationship between the magnitude of the reduction and the assumed value of RBE(M.) Thus, generic values of RBE(M) cannot always be used in such calculations. In some cases, where tumor α/β ratios are low (around 5-6 Gy) and where there is good normal tissue sparing, the optimum dose per fraction is relatively large, typically 4-8 Gy.Conclusion: BED equations that include the RBE parameter, together with low-LET α/β ratios and repopulation dose equivalents, constitute a rational model of high-LET radiotherapy. In the case of proton beam therapy, a wide range of optimum dose per fraction is predicted. Copyright (C) 2000 Elsevier Science Inc.

Original publication




Journal article


International Journal of Radiation Oncology Biology Physics

Publication Date





1549 - 1557