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arxiv: 2303.09485 · v3 · submitted 2023-03-16 · ⚛️ physics.med-ph

Variation of the relative biological effectiveness with fractionation in proton therapy: analysis of prostate cancer response

Juan Pardo-Montero , Miguel Pombar , Antonio G\'omez-Caama\~no , Simona Giordanengo , Isabel Gonz\'alez-Crespo This is my paper

Pith reviewed 2026-05-24 09:24 UTC · model grok-4.3

classification ⚛️ physics.med-ph
keywords relative biological effectivenessproton therapyprostate cancerfractionationtumor control probabilitylinear-quadratic modelhypofractionation
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The pith

Clinical TCP data show proton RBE for prostate cancer declines with rising dose per fraction

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a method that combines the linear-quadratic model with Poisson tumor-control statistics to back-calculate RBE from published clinical outcomes. It applies the approach to prostate-cancer TCP values reported for photon and proton regimens at three different fraction sizes. The resulting RBE estimates decrease modestly as the dose per fraction increases, yet remain near the usual fixed value of 1.1. A reader would care because any real dependence of RBE on fractionation would alter how proton doses are prescribed and compared with photon therapy.

Core claim

Using literature TCP values, the derived RBE for low-risk prostate cancer is 1.124, 1.119 and 1.102 at physical proton doses per fraction of 1.82 Gy, 2.73 Gy and 6.59 Gy respectively; a parallel decline appears for intermediate-risk disease. This monotonic decrease is the expected outcome of the LQ formalism when the ratio α/β differs between photon and proton irradiation.

What carries the argument

The LQ-Poisson TCP formalism applied to paired photon and proton clinical TCPs, with the equilibrium condition (α_p/β_p)=(α_X/β_X)(α_X/α_p) that fixes whether RBE rises, falls or stays constant with dose per fraction.

If this is right

  • RBE is expected to be higher at smaller doses per fraction under the observed change in α/β.
  • The calculated RBE values remain close to the conventional 1.1 for all three regimens examined.
  • The same extraction method can be applied to clinical data sets for other tumor sites.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the trend is confirmed, treatment-planning systems could incorporate a modest fractionation-dependent RBE correction for prostate protons.
  • Matched-cohort trials would be the cleanest way to tighten the uncertainty that currently arises from combining heterogeneous literature reports.

Load-bearing premise

Published tumor-control probabilities from separate photon and proton studies can be inserted directly into the model without bias from differences in patient selection, follow-up length or endpoint definition.

What would settle it

A single prospective randomized trial that measures TCP in the same patient population for both photons and protons at multiple fractionation schedules would confirm or refute the reported decline in RBE.

Figures

Figures reproduced from arXiv: 2303.09485 by Antonio G\'omez-Caama\~no, Isabel Gonz\'alez-Crespo, Juan Pardo-Montero, Miguel Pombar, Simona Giordanengo.

Figure 1
Figure 1. Figure 1: Tumor control probability (TCP) versus Biologically Effective Dose (BED) for photon and proton therapy of low and intermediate risk tumors. Solid squares and circles represent the clinical data, with error bars showing the 95% confidence intervals calculated by using the binomial distribution. The curves show the modeled dose-response. For photon treatments, the curves were obtained by fitting the clinical… view at source ↗
Figure 2
Figure 2. Figure 2: Calculated RBE versus the proton dose per fraction for low and intermediate risk prostate cancer. Error bars represent 95% confidence intervals. The solid lines represent the best fits to the RBE dependence on the fractionation expected from the LQ model (equation 7). 10 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Distribution of differences between RBE at 2 Gy(RBE)/fraction and 7.25 Gy(RBE)/fraction and between (αX/βX) and (αp/βp) obtained from the Monte Carlo evaluation of uncertainties for low risk and intermediate risk prostate cancer. The dashed lines represent the 95% confidence intervals. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Theoretical calculation of the RBE versus proton dose per fraction based on the LQ formalism. Three regimes are observed: i) (αp/βp) > (αX/βX)(αX/αp) monotonically decreasing RBE; ii) (αp/βp)=(αX/βX)(αX/αp) constant RBE; iii) (αp/βp) < (αX/βX)(αX/αp) monotonically increasing RBE. The α and β values were selected to illustrate the three regimes of a variable RBE with fractionation while having RBE =1.1 at 2… view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a methodology based on the linear-quadratic model and Poisson TCP formalism to derive three analytic regimes for how RBE varies with dose per fraction, depending on the relative change in α/β between photons and protons. It then extracts numerical RBE values by inserting literature TCP estimates for prostate cancer (low- and intermediate-risk) under three fractionation schedules (2 Gy(RBE)×37, 3 Gy(RBE)×20, 7.25 Gy(RBE)×5) into the model, reporting a modest monotonic decline (low-risk: 1.124 → 1.119 → 1.102) that remains close to the nominal value of 1.1.

Significance. If the extracted trend is robust, the work supplies a transparent algebraic framework for fractionation-dependent RBE and supplies concrete numerical estimates that can be tested against future clinical data. The derivation of the three regimes (equilibrium at (α_p/β_p)=(α_X/β_X)(α_X/α_p)) is internally consistent and parameter-free once the α/β ratios are specified. The clinical extraction step, however, inherits all uncertainties of the heterogeneous TCP literature.

major comments (2)
  1. [Methods and Results] Methods (M&M) and Results: The central extraction equates photon and proton TCP values taken directly from disparate literature studies and solves the Poisson TCP equation for RBE at each fractionation schedule. No quantitative correction or sensitivity analysis is provided for differences in patient risk stratification, PSA endpoint timing, follow-up duration, or competing-risk handling between the photon and proton source cohorts. Because the reported RBE differences are small (ΔRBE ≈ 0.02), even modest systematic offsets in the input TCPs can erase or reverse the claimed monotonic decline.
  2. [Results] Results: The RBE values are obtained by requiring the LQ-Poisson model to reproduce exactly the same clinical TCP numbers that were used as input; the reported figures are therefore fitted quantities conditioned on the chosen TCP data rather than independent predictions. While the Conclusions note possible bias, the strength of the claim that “RBE values … decline with increasing dose per fraction” rests on the untested assumption that the literature TCPs are unbiased point estimates on a common scale.
minor comments (2)
  1. [Abstract] Abstract and throughout: inconsistent LaTeX rendering of Greek symbols (e.g., (α, β) appears as ( {α}, {beta} )).
  2. The paper does not tabulate the exact literature TCP values and their uncertainties that were inserted into the equations; providing these numbers would improve reproducibility.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive comments, which highlight important limitations in the clinical data extraction step. The analytic framework for the three RBE regimes is the primary contribution and remains unaffected; we will revise the manuscript to better qualify the numerical results and strengthen the caveats already present in the Conclusions.

read point-by-point responses

  1. Referee: [Methods and Results] Methods (M&M) and Results: The central extraction equates photon and proton TCP values taken directly from disparate literature studies and solves the Poisson TCP equation for RBE at each fractionation schedule. No quantitative correction or sensitivity analysis is provided for differences in patient risk stratification, PSA endpoint timing, follow-up duration, or competing-risk handling between the photon and proton source cohorts. Because the reported RBE differences are small (ΔRBE ≈ 0.02), even modest systematic offsets in the input TCPs can erase or reverse the claimed monotonic decline.

    Authors: We agree that the TCP values originate from heterogeneous literature sources and that no formal sensitivity analysis was performed. Systematic differences in cohorts, endpoints, or follow-up could plausibly alter or eliminate the small observed ΔRBE. In revision we will add a new subsection in the Discussion that (i) tabulates the key study characteristics (risk groups, PSA definitions, median follow-up) for the photon and proton cohorts and (ii) performs a simple one-at-a-time sensitivity test by shifting each input TCP by ±3–5 % (typical of reported uncertainties) and recomputing the RBE trend. This will quantify how robust the monotonic decline remains under plausible offsets. revision: yes

  2. Referee: [Results] Results: The RBE values are obtained by requiring the LQ-Poisson model to reproduce exactly the same clinical TCP numbers that were used as input; the reported figures are therefore fitted quantities conditioned on the chosen TCP data rather than independent predictions. While the Conclusions note possible bias, the strength of the claim that “RBE values … decline with increasing dose per fraction” rests on the untested assumption that the literature TCPs are unbiased point estimates on a common scale.

    Authors: The referee is correct: the reported RBE numbers are obtained by solving the model for the RBE that exactly reproduces the input TCPs; they are therefore conditioned estimates rather than a priori predictions. The manuscript already states in Conclusions that the calculations “have to be considered with care as they may be biased by … the clinical data set.” We will revise the Results section to explicitly label the values as “model-derived RBE estimates conditioned on the selected literature TCPs” and will rephrase the claim to “within the chosen data set the derived RBE declines monotonically with dose per fraction, consistent with the theoretical regimes.” This removes any implication of independent validation while preserving the illustrative application of the framework. revision: yes

standing simulated objections not resolved
  • A full quantitative correction for all listed cohort differences (risk stratification, PSA timing, competing risks) would require re-analysis of individual-patient data from the source trials, which is outside the scope of the present methodological study and not feasible with the published aggregate TCP values alone.

Circularity Check

0 steps flagged

No significant circularity; RBE extraction is explicit parameter estimation from external TCP data

full rationale

The paper obtains clinical TCP values from independent literature sources for photon and proton arms across fractionation schedules, then inverts the LQ-Poisson formalism to solve for the RBE values that reproduce those TCPs. This is a standard data-driven estimation step, not a first-principles derivation or prediction that reduces to its own inputs by construction. The theoretical section on RBE dependence regions follows directly from the LQ model equations without reference to the clinical data. No self-citations, uniqueness theorems, or ansatzes are invoked as load-bearing elements. The reported RBE decline is the direct numerical output of the inversion applied to external inputs; the paper explicitly frames the work as analysis rather than prediction and notes potential biases from data and modeling. This matches the default case of a self-contained analysis against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the applicability of the linear-quadratic model to both modalities, the Poisson TCP approximation, and the assumption that literature TCP values can be treated as comparable point estimates.

free parameters (2)
  • alpha/beta ratio for photons and protons
    Used to locate the equilibrium point and to back-calculate RBE; values are taken from prior literature or implicitly adjusted to match the observed TCPs.
  • RBE values solved from TCP data
    The reported RBE figures (1.124, 1.119, 1.102, etc.) are obtained by fitting the model equations to the chosen clinical TCP points.
axioms (2)
  • domain assumption Linear-quadratic survival model remains valid across the dose-per-fraction range examined for both photons and protons
    Invoked when the TCP formalism is applied to the three fractionation schemes.
  • domain assumption Poisson statistics adequately describe tumor control probability from the LQ cell-kill term
    Stated in the M&M section as the basis for the TCP calculation.

pith-pipeline@v0.9.0 · 5961 in / 1624 out tokens · 37111 ms · 2026-05-24T09:24:43.259674+00:00 · methodology

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Reference graph

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