  Purpose: To present a methodology to analyze the variation of RBE with
fractionation from clinical data of tumor control probability (TCP) and to
apply it to study the response of prostate cancer to proton therapy.
  M&M: We analyzed the dependence of the RBE on the dose per fraction by using
the LQ model and the Poisson TCP formalism. Clinical TCPs for prostate cancer
treated with photon and proton therapy for conventional fractionation (2
Gy(RBE)x37 fractions), moderate hypofractionation (3 Gy(RBE)x20 fractions) and
hypofractionation (7.25 Gy(RBE)x5 fractions) were obtained from the literature
and analyzed.
  Results: The theoretical analysis showed three distinct regions with RBE
monotonically decreasing, increasing or staying constant with the dose per
fraction, depending on the change of ({\alpha}, \{beta}) values between photon
and proton irradiation (the equilibrium point being
at({\alpha}_p/\{beta}_p)=({\alpha}_X/\{beta}_X)({\alpha}_X/{\alpha}_p)). An
analysis of the clinical data showed RBE values that decline with increasing
dose per fraction: for low risk RBE=1.124, 1.119, and 1.102 for 1.82 Gy, 2.73
Gy and 6.59 Gy per fraction (physical proton doses), respectively; for
intermediate risk RBE=1.119, and 1.102 for 1.82 Gy, and 6.59 Gy per fraction
(physical proton doses), respectively. These values are nonetheless very close
to the nominal 1.1 value.
  Conclusions: We presented a methodology to analyze the RBE for different
fractionations, and we used it to study clinical data for prostate cancer. The
analysis shows a monotonically decreasing RBE with increasing dose per
fraction, which is expected from the LQ formalism and the changes in ({\alpha},
\{beta}) between photon and proton irradiation. However, the calculations in
this study have to be considered with care as they may be biased by limitations
in the modeling and/or by the clinical data set used for the analysis.
