Pith. sign in

REVIEW 3 major objections 4 minor 2 cited by

Efficient Proton Transport Modelling for Proton Beam Therapy and Biological Quantification

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper claims that a one-dimensional continuous-slowing-down transport model with a closed-form fluence reproduces Monte Carlo depth-dose curves for proton beams and makes LET-weighted biological-dose optimization practical.

desk verdict A clean, reproducible 1D dose/LET framework with a fixable but central sign error in the transport equation; worth refereeing after correction. read the letter →

arxiv 2411.16735 v1 pith:B6BO7H7B submitted 2024-11-23 physics.med-ph

Ben S. Ashby , Veronika Chronholm , Daniel K. Hajnal , Alex Lukyanov , Katherine MacKenzie , Aaron Pim , Tristan Pryer This is my paper · ORCID
classification physics.med-ph
keywords protonbeamtherapytransportBragg-Kleemanruleclosed-formfluencelinearenergytransferrelativebiologicaleffectivenessdoseoptimizationMonteCarlovalidation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proposes that proton transport relevant to beam therapy can be described by a one-dimensional deterministic balance equation for particle fluence, and that under the Bragg-Kleeman stopping-power law this equation has an explicit closed-form solution. From that solution the authors obtain depth-dose, track- and dose-averaged LET, cell survival fraction, spatially variable RBE, and a linear biological-dose metric, and they validate the depth-dose curve against MCsquare and TOPAS for a 62 MeV beam with reported high fidelity. Treatment planning is then posed as a weighted least-squares problem over Gaussian basis beams, with examples covering uniform target dose, organ-at-risk sparing, and LET-weighted biological dose. If the model holds, it would give near-instant evaluations of physical and biological quantities that currently require Monte Carlo simulation.

What carries the argument

The load-bearing object is the one-dimensional transport equation $\partial u/\partial z + \partial(S u)/\partial E = 0$ on a track-length coordinate $z$, with inflow spectrum $g(E)$ and the Bragg-Kleeman stopping power $S(E) = (\alpha p)^{-1} E^{1-p}$. The equation is hyperbolic, and along characteristic curves $E^p = E_0^p - z/\alpha$ the fluence is transported with an explicit density factor, yielding Equation (15). This closed-form fluence is the engine of the paper: Equation (16) integrates it against stopping power for dose, Equations (18)-(19) take weighted moments for track- and dose-averaged LET, Equation (25) evaluates cell survival, and Equation (29) defines the linear biological dose that makes LET-aware optimization a least-squares problem.

What would settle it

Compare the analytical dose from Equation (16) with a TOPAS or MCsquare simulation of the same 62 MeV beam with nuclear interactions turned on, measuring the 80%-to-20% distal falloff distance and the dose tail beyond the Bragg peak: if the analytical curve is visibly narrower or ends more sharply than the Monte Carlo curve beyond statistical uncertainty, the no-straggling assumption is falsified. A second decisive check is the dose-averaged LET peak, which weights the square of stopping power and therefore magnifies any difference in the low-energy tail.

Watch

Extended reading notes

Core claim

The central claim is that the continuous-slowing-down approximation plus the Bragg-Kleeman rule $S(E) = (\alpha p)^{-1} E^{1-p}$ turns proton transport into a hyperbolic PDE whose method-of-characteristics solution is the closed-form fluence $u(z,E) = (E^p + z/\alpha)^{(1-p)/p}\, g\big((E^p + z/\alpha)^{1/p}\big)\, E^{p-1}$. Dose is the stopping-power-weighted integral of this fluence (Equation 16), and LET averages are its first and second moments in stopping power (Equations 18 and 19). The authors compare the resulting depth-dose curve to MCsquare and TOPAS for a 62 MeV beam, with and without nuclear interactions, and state that the analytical model captures the depth-dose behaviour with high fidelity. The same fluence feeds a LET-dependent linear-quadratic survival model, a spatially variable RBE defined by matching the equivalent photon dose, and a linear biological dose $BD(z) = D(z)(1 + (\lambda/c_{\rm X-ray}) L_D(z))$ used for optimization. On this basis the paper presents the model as a complement to Monte Carlo for rapid, biologically informed treatment planning.

Load-bearing premise

The model assumes protons slow down continuously and deterministically, with no random energy jitter (straggling), almost no sideways scattering, and very few nuclear collisions; if energy straggling meaningfully broadens the distal dose falloff at clinical energies, the closed-form match to Monte Carlo will not hold beyond the single 62 MeV water-slab benchmark.

Editorial extensions

If this is right

  • Depth-dose and LET profiles for any energy spectrum can be evaluated by direct quadrature of closed forms, removing Monte Carlo cost for one-dimensional water-like geometries.
  • Because dose is linear in beam weights, LET-weighted biological dose optimization becomes a weighted least-squares problem that can be solved rapidly, allowing fast exploration of organ-at-risk trade-offs.
  • The framework yields spatially resolved RBE and cell survival predictions showing RBE near 1.1 before the Bragg peak and larger values in the distal falloff, matching the clinical picture.
  • Uncertainty in the Bragg-Kleeman parameters $\alpha$ and $p$ produces roughly $\pm 1$ cm 95% confidence intervals for the Bragg peak depth at the studied energy, a level the paper flags as clinically relevant for plan robustness.
  • The biological-dose optimization example produces a tapered distal dose with reduced healthy-tissue dose while keeping the biological effect uniform in the target, illustrating a concrete planning consequence.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural next step beyond the paper is to concatenate one-dimensional slabs with different material parameters, which would extend the closed-form fluence to layered heterogeneous geometries while keeping the speed advantage.
  • Because the model omits energy-loss straggling, it likely overpredicts the sharpness of the distal falloff; convolving Equation (16) with a Gaussian straggling kernel is a testable extension that would quantify the bias in the no-straggling assumption.
  • The same characteristic solution generalizes to heavier ions by changing the exponent $p$, which could connect this framework to carbon-ion LET-weighted planning, though the linear-quadratic survival model would need the heavy-ion corrections the paper explicitly sets aside.
  • The convex linearity of the biological dose in beam weights suggests the optimization could be embedded directly in existing inverse-planning solvers, a clinical integration the paper demonstrates in one dimension but does not develop for three-dimensional pencil-beam scanning.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper develops a one-dimensional analytical model for proton transport in water-like geometries under continuous-slowing-down assumptions, yielding closed-form expressions for fluence, absorbed dose, track- and dose-averaged LET, and LET-dependent biological quantities such as survival fraction, RBE, and biological dose. The model is validated against MCsquare and TOPAS for a 62 MeV pristine Bragg peak, an uncertainty analysis of the Bragg-Kleeman parameters is carried out, and the framework is applied to treatment-planning optimisation with absorbed-dose and biological-dose objectives. The code used to generate the figures is made available via Zenodo.

Significance. If the internal sign inconsistency is corrected and the Monte Carlo comparison is made quantitative, the closed-form formulas would provide a genuinely fast and interpretable tool for exploring LET-weighted optimisation and variable RBE in one-dimensional geometries. The availability of code, the clear treatment of parameter uncertainty, and the explicit falsifiable predictions for dose and LET are strengths. However, as printed the central transport equation is inconsistent with the claimed solution, and the validation is single-case and qualitative, so the significance cannot be fully assessed without revision.

major comments (3)
  1. [§2.2, §2.5, Eqs (7), (14)–(15)] The sign in Eq (7) is inconsistent with the characteristic solution (14) and the closed-form fluence (15). Since Eq (13) defines S(E) > 0, physical energy loss requires dE/dz = -S(E), and the phase-space continuity equation is ∂_z u - ∂_E(S u) = 0. Equation (14), E^p = E_max^p - z/α, is exactly the characteristic of the minus-sign equation, and direct substitution of Eq (15) into Eq (7) leaves a nonzero residual; the proposed solution solves the minus-sign equation instead. Equation (1) carries the same plus-sign error. If the implementation behind Fig 6 uses the minus-sign equation, Eqs (1) and (7) should be corrected and the derivation revisited; if the implementation follows the printed plus-sign equation, the agreement in Fig 6 cannot be reproduced. This must be resolved before the transport model can be accepted.
  2. [§2.9, Fig 6] The Monte Carlo validation is a single-energy, qualitative comparison. Only the 62 MeV pristine Bragg peak is shown, no quantitative agreement metric (gamma index, range difference, dose difference) or Monte Carlo statistical uncertainty is reported, and LET—which the biological sections depend on—is not validated at all. The analytical model uses α = 2.2 × 10^-3 and p = 1.77 from [19], while Table 1 lists α = 0.00246 ± 0.00025 and p = 1.75 ± 0.02 for water; this discrepancy is not explained. The claim of high fidelity needs multi-energy comparisons with explicitly stated parameter choices and a quantitative agreement metric.
  3. [§2.5–2.6, Eq (16)] The energy interval I = [E_min, E_max] used in Eq (16) and in Figs 5–6 is never specified. At depth z > 0, characteristics with initial energies near E_min produce protons with energies below E_min, so the dose integral is truncated at E_min; depending on the chosen E_min this can remove a non-negligible low-energy contribution near the Bragg peak. The authors should report E_min and E_max and demonstrate that the dose and LET results are insensitive to E_min, for example by a convergence study in E_min.
minor comments (4)
  1. [§4.1, Fig 11 caption] The caption states σα = 0.01µα = 0.0175 and σp = 0.01µp = 0.0000246, which swaps the two parameters; the correct values should be σα = 0.0000246 and σp = 0.0175.
  2. [§5.1, Fig 17] The text for Examples 1 and 2 sets beam ranges equally spaced in [z_prox, z_dist + 1/4], while the Fig 17 caption states that ranges extend to z_dist + 0.15; the two choices should be reconciled.
  3. [§2.3, Eq (21)] Equation (21) is valid only for z ≤ α E_0^p; for larger depths the argument (E_0^p - z/α)^{1/p} is not real. The domain of validity should be stated.
  4. [§2.3, Eq (17)] Equation (12) defines S(z) := -dE/dx as a positive quantity, while Eq (17) writes L_Δ = dE_Δ/dz without a sign; since LET is a positive energy-deposition quantity, the sign convention should be clarified.

Circularity Check

0 steps flagged · score 0.0 of 10

No material circularity: the model uses externally calibrated empirical parameters and is validated against independent Monte Carlo codes; no prediction reduces to a fitted value or to the authors' own prior work.

full rationale

The paper's central claims are a closed-form fluence and dose from a continuous-slowing-down transport model, and biological metrics built on literature LQ parameters. The transport model takes the Bragg-Kleeman range-energy power law with alpha and p taken from published sources (Table 1 and [19]) as inputs; the closed-form solution (15) is obtained by the method of characteristics, and the dose (16) is the standard stopping-power-weighted fluence integral. Validation in Section 2.9 is against independent MCsquare and TOPAS simulations, not against the same data used to fit the model. The biological quantities use LQ coefficients, lambda, c_X-ray, and beta from Chaudhary et al. [8], which are external experimental fits; equations (24)-(27) and (29) are explicit definitions and standard modelling choices, not derivations whose conclusions are hidden in an assumption. No self-citation is load-bearing: the authors do not cite their own prior work as the basis for any premise, and no uniqueness theorem is invoked. The apparent sign inconsistency between Eq. (7) and the characteristic Eq. (14) is a mathematical-correctness concern, not a circular reduction, so it does not affect the circularity score.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The model's output is dominated by empirical inputs (α, p from range-energy fits; LQ parameters from cell survival fits) and by four physical simplifications: no straggling, no scatter, rare nuclear collisions, and power-law stopping. The paper is transparent about most of these, but the straggling omission is never stated or justified.

free parameters (4)
  • Bragg-Kleeman coefficient α for water = 0.00246 cm MeV^-p (Table 1); 0.0022 used in Section 2.9
    Empirical range-energy factor fitted to range data in references [17-19]; enters stopping power Eq 13 and dose Eq 16.
  • Bragg-Kleeman exponent p for water = 1.75 (Table 1); 1.77 in Section 2.9
    Empirical exponent in the range-energy rule Eq 9; controls the depth of the Bragg peak.
  • LQ parameters c_X-ray, β, λ for AG01522 and U87 = AG01522: 0.54, 0.051, 0.0451; U87: 0.11, 0.059, 0.0127
    Fitted to clonogenic survival data in reference [8]; used in survival fraction, RBE, and biological dose calculations in Section 3.
  • Basis beam widths σ_i^2 = 1 for all i in Examples 1-3
    Chosen by hand in Section 5; the optimization results depend on this choice, and the paper does not study its sensitivity.
assumptions (4)
  • domain assumption Continuous energy loss without straggling
    Implicit in Eq 7; no stochastic term appears, and the paper does not address energy-loss straggling.
  • domain assumption Angular scattering is minimal and nonelastic collisions are rare
    Stated in Section 2.1 just before Eq 1; this reduces full proton transport to a one-dimensional energy-continuity equation.
  • domain assumption Stopping power follows the Bragg-Kleeman power law over the whole energy interval
    Eq 13; Remark 2.4 acknowledges large deviations from Bethe-Bloch at low energies, exactly where the Bragg peak is formed.
  • domain assumption The linear-quadratic survival model holds, with α linear in LET and β constant
    Eqs 24-25 and Table 2; the paper notes some studies report LET-dependent β, so this is a modeling choice.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Efficient Proton Transport Modelling for Proton Beam Therapy and Biological Quantification." pith.science (2026). https://pith.science/paper/B6BO7H7B

@misc{pith2026241116735,
  author       = {Pith},
  title        = {Pith review of: Efficient Proton Transport Modelling for Proton Beam Therapy and Biological Quantification},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B6BO7H7B}},
  note         = {Machine review of arXiv:2411.16735}
}
read the original abstract

In this work, we present a fundamental mathematical model for proton transport, tailored to capture the key physical processes underpinning Proton Beam Therapy (PBT). The model provides a robust and computationally efficient framework for exploring various aspects of PBT, including dose delivery, linear energy transfer, treatment planning and the evaluation of relative biological effectiveness. Our findings highlight the potential of this model as a complementary tool to more complex and computationally intensive simulation techniques currently used in clinical practice.

Figures

Figures reproduced from arXiv: 2411.16735 by the authors.

Figure 1
Figure 1. Simulated dose profile of a proton beam illustrating the Bragg peak. The simulation was performed using MCsquare with 1.21 · 107 particles, a beam width of 2 mm, without nuclear interactions. The initial proton energy is 150 MeV with a 1% energy spread, and the dose is integrated along the plane orthogonal to the beam axis. a fundamental mathematical perspective to address some of these challenges in PBT treatment p… view at source ↗
Figure 2
Figure 2. The three main interactions of a proton with matter. A nonelastic proton-nucleus collision, an inelastic Coulomb interaction with atomic electrons and elastic Coulomb scattering with the nucleus. 2.1. A model for proton transport. In this section, we introduce a simplified model for proton transport that builds upon the fundamental principles shown in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. gives a visualisation of the domain and inflow boundaries. 0 zmax Emin Emax ∂X− ∂X− [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (19 more)
Figure 4
Figure 4. Figure 4: Stopping power as a function of energy for the Bragg-Kleeman and Bethe-Bloch models. The two models show good agreement in the intermediate energy range relevant to proton therapy however differ dramatically in the low energy range. u(z, E) is u(z, E) = E p + z α  1−…
Figure 5
Figure 5. Figure 5: A visualisation of dose calculation. Left: the initial Gaussian energy profile of a 62 MeV proton beam with a 1% energy spread. Middle: the fluence in depth-energy space, illustrating how the beam evolves as it travels through the medium. Bottom: the resultant dose as …
Figure 6
Figure 6. Figure 6: Comparison of depth-dose curves for a 62 MeV mono-energetic proton beam in water obtained from the one-dimensional analytical model (black), MC￾square (green) and TOPAS (pink). Left: nuclear interactions are excluded in the Monte Carlo simulations. Right: nuclear inter…
Figure 7
Figure 7. Figure 7: Left: depth-dose profiles for 62 MeV proton beams of two different intensities, Right: corresponding survival fractions of cells against depth assuming a homogeneous medium of cells, computed using the model (25). 3.3. Relative Biological Effectiveness. Relative Biolog…
Figure 8
Figure 8. Figure 8: Left: RBE-weighted dose curves for a 62 MeV mono-energetic proton beam in water phantom, calculated using the TDRA model for cell survival and parameters from [8] for AG01522 and U87 cell lines. Right: Corresponding RBE. Dose curve illustrates RBE behaviour along the B…
Figure 9
Figure 9. Figure 9: Positions for the active subspace analysis. Point A is halfway between the start and the Bragg peak, point B is at the peak, and point C is at the position with the steepest gradient. The initial beam has an energy of 62MeV, with a spread of 5%. 0.00230 0.00235 0.00240…
Figure 10
Figure 10. Figure 10: Contour plot of dose at points A, B, and C when α and p are assumed to follow normal distributions with absolute standard deviations σα = σp = 0.0001. in p has a greater impact on D(x; α, p) than a 1% change in α. Finally, [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]
Figure 11
Figure 11. Figure 11: Contour plot of dose at points A, B, and C when α and p are assumed to follow normal distributions with relative standard deviations σα = 0.01µα = 0.0175 and σp = 0.01µp = 0.0000246. 0.0023 0.0024 0.0025 0.0026 0.0027 1.730 1.735 1.740 1.745 1.750 1.755 1.760 1.765 1.…
Figure 12
Figure 12. Figure 12: Contour plot of dose at points A, B, and C when α and p are assumed to follow normal distributions with empirical standard deviations σα = 0.000128 and σp = 0.0102. distributed random variables with means (µα, µp) and standard deviations (σα, σp). Specifically, we tak…
Figure 13
Figure 13. Figure 13: Sensitivity of D(z; α, p) to under- and overestimation of α and p by one or two standard deviations. The nominal dose curve D∗ (z; µα, µp) is shown for reference. On the left, only α has been under- or overestimated; in the centre only p; and on the right both α and p…
Figure 14
Figure 14. Figure 14: Confidence intervals for the variation in dose D(z; α, p) when α and p are normally distributed with means µα = 0.00246, µp = 1.75 and standard deviations σα = 0.000128, σp = 0.0102. The nominal dose curve D(z; µα, µp), resulting from the assumed parameter values µα a…
Figure 15
Figure 15. Figure 15: Confidence intervals for the variation in peak depth zpeak when α and p are normally distributed with means µα = 0.00246, µp = 1.75 and standard deviations σα = 0.000128, σp = 0.0102. The nominal dose curve D(z; µα, µp), resulting from the assumed parameter values µα …
Figure 16
Figure 16. Figure 16: Visualisation of the input beam, fluence, and dose profile for Example 1. Left: the optimised input beam intensities across different energies. Middle: the fluence in depth-energy space, showing how the superposition of beams evolves through the medium. Right: the res…
Figure 17
Figure 17. Figure 17: Spread-out Bragg peaks (left) and dose-averaged LET curves (right) resulting from the optimisation problem 5.2. Dashed lines: beam ranges are equally spaced between zprox and zdist. Solid lines: beam ranges are equally spaced be￾tween zprox and zdist + 0.15. Allowing …
Figure 18
Figure 18. Figure 18: Left: dose profile resulting from the optimisation in Example 1. Right: corresponding dose-averaged LET. A total of 30 energy levels are used, with ener￾gies chosen such that their ranges are equally spaced and cover the target region. Uniform dose delivery to the tum…
Figure 19
Figure 19. Figure 19: Left: dose profile resulting from the optimisation in Example 2. Right: corresponding dose-averaged LET. Penalising dose in the OAR (shaded green) reduces dose penetration into healthy tissue but introduces slight under-dosing and oscillation in the tumour’s distal re…
Figure 20
Figure 20. Figure 20: Confidence intervals for a spread-out Bragg peak and the correspond￾ing dose-averaged LET. The confidence intervals are obtained as in section 4, by introducing uncertainty in the stopping power parameters α and p, and estimating the empirical confidence intervals fro…
Figure 21
Figure 21. Figure 21: Left: dose profile resulting from the optimisation in Example 3. Right: corresponding dose-averaged LET. A total of 30 energy levels are used, with ener￾gies chosen such that their ranges are equally spaced and cover the target region. 0 1 2 3 4 5 6 7 8 depth (cm) 0.0…
Figure 22
Figure 22. Figure 22: Comparison of relative dose (left) and survival fraction profiles (right) resulting from optimisation for absorbed dose (dashed line) and biological dose (solid line). The survival fraction was computed using the linear-quadratic model, with the α parameter accounting…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A nodally bound-preserving finite element method for hyperbolic convection-reaction problems

    math.NA 2025-01 conditional novelty 6.0 of 10

    A SUPG finite element method with nodal bound constraints, solved as a variational inequality, is analyzed for hyperbolic convection-reaction problems and is shown to converge at the expected rates in the linear case.

  2. Optimal control of a kinetic equation

    math.NA 2024-12 conditional novelty 6.0 of 10

    Introduces a hypocoercive finite element framework for optimal control of the Kolmogorov equation, proving temporal stability and error bounds for the regularized problem.

Reference graph

Works this paper leans on

47 extracted references · 47 canonical work pages · cited by 2 Pith papers

  1. [19]

    An analytical approximation of the Bragg curve for therapeutic proton beams

    T. Bortfeld. “An analytical approximation of the Bragg curve for therapeutic proton beams”. In: Medical physics 24.12 (1997), pp. 2024–2033

    work page 1997

  2. [1]

    Treatment planning for proton therapy: what is needed in the next 10 years?

    H. Nystrom et al. “Treatment planning for proton therapy: what is needed in the next 10 years?” In: The British Journal of Radiology 93.1107 (2020), p. 20190304

    work page 2020

  3. [2]

    A generic algorithm for Monte Carlo simulation of proton transport

    F. Salvat. “A generic algorithm for Monte Carlo simulation of proton transport”. In: Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms 316 (2013), pp. 144–159

    work page 2013

  4. [3]

    A fast Monte Carlo code for proton transport in radiation therapy based on MCNPX

    K. Jabbari et al. “A fast Monte Carlo code for proton transport in radiation therapy based on MCNPX”. In: Journal of Medical Physics 39.3 (2014), pp. 156–163

    work page 2014

  5. [4]

    Reoptimization of intensity modulated proton therapy plans based on linear energy transfer

    J. Unkelbach et al. “Reoptimization of intensity modulated proton therapy plans based on linear energy transfer”. In: International Journal of Radiation Oncology* Biology* Physics 96.5 (2016), pp. 1097–1106

    work page 2016

  6. [5]

    Using multi-centre data to train and validate a knowledge-based model for planning radiotherapy of the head and neck

    M. Frizzelle et al. “Using multi-centre data to train and validate a knowledge-based model for planning radiotherapy of the head and neck”. In: Physics and Imaging in Radiation Oncology 21 (2022), pp. 18–23

    work page 2022

  7. [6]

    Implementation, dosimetric assessment, and treatment validation of knowledge-based planning (KBP) models in VMAT head and neck radiation oncology

    A.-M. Fanou et al. “Implementation, dosimetric assessment, and treatment validation of knowledge-based planning (KBP) models in VMAT head and neck radiation oncology”. In: Biomedicines 11.3 (2023), p. 762

    work page 2023

  8. [7]

    An empirical method for deriving RBE values associated with electrons, photons and radionuclides

    M. Bellamy et al. “An empirical method for deriving RBE values associated with electrons, photons and radionuclides”. In: Radiation protection dosimetry 167.4 (2015), pp. 664–670

    work page 2015

Show all 47 references
  1. [8]

    Relative biological effectiveness variation along monoenergetic and mod- ulated Bragg peaks of a 62-MeV therapeutic proton beam: a preclinical assessment

    P. Chaudhary et al. “Relative biological effectiveness variation along monoenergetic and mod- ulated Bragg peaks of a 62-MeV therapeutic proton beam: a preclinical assessment”. In: International Journal of Radiation Oncology* Biology* Physics 90.1 (2014), pp. 27–35. 27

    2014

  2. [9]

    Relative biological effectiveness of proton beams in clinical therapy

    L. E. Gerweck et al. “Relative biological effectiveness of proton beams in clinical therapy”. In: Radiotherapy and oncology 50.2 (1999), pp. 135–142

    1999

  3. [10]

    Variable proton relative biological effectiveness: how do we move for- ward?

    T. Underwood et al. “Variable proton relative biological effectiveness: how do we move for- ward?” In: International Journal of Radiation Oncology, Biology, Physics 95.1 (2016), pp. 56– 58

    2016

  4. [11]

    Linear energy transfer-guided optimization in intensity modulated pro- ton therapy: feasibility study and clinical potential

    D. Giantsoudi et al. “Linear energy transfer-guided optimization in intensity modulated pro- ton therapy: feasibility study and clinical potential”. In: International Journal of Radiation Oncology* Biology* Physics 87.1 (2013), pp. 216–222

    2013

  5. [12]

    Report of the AAPM TG-256 on the relative biological effectiveness of proton beams in radiation therapy

    H. Paganetti et al. “Report of the AAPM TG-256 on the relative biological effectiveness of proton beams in radiation therapy”. In: Medical physics 46.3 (2019), e53–e78

    2019

  6. [13]

    Difference in the relative biological effectiveness and DNA damage repair processes in response to proton beam therapy according to the positions of the spread out Bragg peak

    H. Hojo et al. “Difference in the relative biological effectiveness and DNA damage repair processes in response to proton beam therapy according to the positions of the spread out Bragg peak”. In: Radiation oncology 12 (2017), pp. 1–9

    2017

  7. [14]

    Relative biological effectiveness (RBE) values for proton beam therapy. Vari- ations as a function of biological endpoint, dose, and linear energy transfer

    H. Paganetti. “Relative biological effectiveness (RBE) values for proton beam therapy. Vari- ations as a function of biological endpoint, dose, and linear energy transfer”. In: Physics in Medicine & Biology 59.22 (2014), R419

    2014

  8. [15]

    Optimal control for mathematical models of cancer therapies

    H. Sch¨ attler et al. “Optimal control for mathematical models of cancer therapies”. In: An application of geometric methods (2015)

    2015

  9. [16]

    Adaptive radiotherapy for anatomical changes

    J.-J. Sonke et al. “Adaptive radiotherapy for anatomical changes”. In: Seminars in radiation oncology. Vol. 29. 3. Elsevier. 2019, pp. 245–257

    2019

  10. [17]

    Accuracy of parameterized proton range models; a comparison

    H. E. S. Pettersen et al. “Accuracy of parameterized proton range models; a comparison”. In: Radiation Physics and Chemistry 144 (2018), pp. 295–297

    2018

  11. [18]

    Dosimetry and quality control of scanning proton beams

    S. N. Boon. “Dosimetry and quality control of scanning proton beams”. In: (1998)

    1998

  12. [20]

    Theoretical aspects of energy–range relations, stopping power and energy strag- gling of protons

    W. Ulmer. “Theoretical aspects of energy–range relations, stopping power and energy strag- gling of protons”. In: Radiation physics and chemistry 76.7 (2007), pp. 1089–1107

    2007

  13. [21]

    Review of particle physics

    S. Navas et al. “Review of particle physics”. In: Physical Review D 110.3 (2024), p. 030001

    2024

  14. [22]

    D. J. Thomas. ICRU report 85: fundamental quantities and units for ionizing radiation . 2012

    2012

  15. [23]

    A systematic review on the usage of averaged LET in radiation biology for particle therapy

    F. Kalholm et al. “A systematic review on the usage of averaged LET in radiation biology for particle therapy”. In: Radiotherapy and Oncology 161 (2021), pp. 211–221

    2021

  16. [24]

    Fast multipurpose Monte Carlo simulation for proton therapy using multi-and many-core CPU architectures

    K. Souris et al. “Fast multipurpose Monte Carlo simulation for proton therapy using multi-and many-core CPU architectures”. In: Medical physics 43.4 (2016), pp. 1700–1712

    2016

  17. [25]

    The TOPAS tool for particle simulation, a Monte Carlo simulation tool for physics, biology and clinical research

    B. Faddegon et al. “The TOPAS tool for particle simulation, a Monte Carlo simulation tool for physics, biology and clinical research”. In: Physica Medica 72 (2020), pp. 114–121

    2020

  18. [26]

    Intensity modulated proton therapy

    H. Kooy et al. “Intensity modulated proton therapy”. In: The British journal of radiology 88.1051 (2015), p. 20150195

    2015

  19. [27]

    Biophysical characteristics of HIMAC clinical irradiation system for heavy- ion radiation therapy

    T. Kanai et al. “Biophysical characteristics of HIMAC clinical irradiation system for heavy- ion radiation therapy”. In: International Journal of Radiation Oncology* Biology* Physics 44.1 (1999), pp. 201–210

    1999

  20. [28]

    Reformulation of a clinical-dose system for carbon-ion radiotherapy treat- ment planning at the National Institute of Radiological Sciences, Japan

    T. Inaniwa et al. “Reformulation of a clinical-dose system for carbon-ion radiotherapy treat- ment planning at the National Institute of Radiological Sciences, Japan”. In: Physics in Medicine & Biology 60.8 (2015), p. 3271

    2015

  21. [29]

    RBE and related modeling in carbon-ion therapy

    C. P. Karger et al. “RBE and related modeling in carbon-ion therapy”. In: Physics in Medicine & Biology 63.1 (2017), 01TR02. 28

    2017

  22. [30]

    A simple method to assess clonogenic survival of irradiated cancer cells

    I. Serrano-Mendioroz et al. “A simple method to assess clonogenic survival of irradiated cancer cells”. In: Methods in Cell Biology . Vol. 174. Elsevier, 2023, pp. 127–136

    2023

  23. [31]

    A generalized formulation of dual radiation action

    A. M. Kellerer et al. “A generalized formulation of dual radiation action”. In: Radiation research 75.3 (1978), pp. 471–488

    1978

  24. [32]

    Cell-survival probability at large doses: an alternative to the linear-quadratic model

    L. Hanin et al. “Cell-survival probability at large doses: an alternative to the linear-quadratic model”. In: Physics in Medicine & Biology 55.16 (2010), p. 4687

    2010

  25. [33]

    Chadwick et al

    K. Chadwick et al. Molecular theory of cell survival. Tech. rep. Instituut voor Toepassing van Atoomenergie in de Landbouw, Wageningen . . ., 1973

    1973

  26. [34]

    A microdosimetric-kinetic theory of the dependence of the RBE for cell death on LET

    R. B. Hawkins. “A microdosimetric-kinetic theory of the dependence of the RBE for cell death on LET”. In: Medical physics 25.7 (1998), pp. 1157–1170

    1998

  27. [35]

    Direct comparison between protons and alpha-particles of the same LET: I. Irradiation methods and inactivation of asynchronous V79, HeLa and C3H 10T1 /2 cells

    D. Goodhead et al. “Direct comparison between protons and alpha-particles of the same LET: I. Irradiation methods and inactivation of asynchronous V79, HeLa and C3H 10T1 /2 cells”. In: International journal of radiation biology 61.5 (1992), pp. 611–624

    1992

  28. [36]

    Inactivation and mutation induction in V79 cells by low energy protons: re- evaluation of the results at the LNL facility

    M. Belli et al. “Inactivation and mutation induction in V79 cells by low energy protons: re- evaluation of the results at the LNL facility”. In: International journal of radiation biology 63.3 (1993), pp. 331–337

    1993

  29. [37]

    A phenomenological model for the relative biological effectiveness in ther- apeutic proton beams

    J. Wilkens et al. “A phenomenological model for the relative biological effectiveness in ther- apeutic proton beams”. In: Physics in Medicine & Biology 49.13 (2004), p. 2811

    2004

  30. [38]

    Radiobiological investigations of proton and light ion therapy

    N. Tilly. “Radiobiological investigations of proton and light ion therapy”. PhD thesis. Stock- holm University, 2002

    2002

  31. [39]

    Range uncertainty in proton therapy due to variable biological effectiveness

    A. Carabe et al. “Range uncertainty in proton therapy due to variable biological effectiveness”. In: Physics in Medicine & Biology 57.5 (2012), p. 1159

    2012

  32. [40]

    The incorporation of the concept of minimum RBE (RBE min) into the linear-quadratic model and the potential for improved radiobiological analysis of high-LET treatments

    A. Carabe-Fernandez et al. “The incorporation of the concept of minimum RBE (RBE min) into the linear-quadratic model and the potential for improved radiobiological analysis of high-LET treatments”. In: International journal of radiation biology 83.1 (2007), pp. 27–39

    2007

  33. [41]

    Variations in linear energy transfer within clinical proton therapy fields and the potential for biological treatment planning

    C. Grassberger et al. “Variations in linear energy transfer within clinical proton therapy fields and the potential for biological treatment planning”. In: International Journal of Radiation Oncology* Biology* Physics 80.5 (2011), pp. 1559–1566

    2011

  34. [42]

    A systematic review of let-guided treatment plan optimisation in proton therapy: identifying the current state and future needs

    M. McIntyre et al. “A systematic review of let-guided treatment plan optimisation in proton therapy: identifying the current state and future needs”. In: Cancers 15.17 (2023), p. 4268

    2023

  35. [43]

    T. J. Sullivan. Introduction to uncertainty quantification / [internet resource] . eng. 1st ed. 2015.. Texts in applied mathematics ; v. 63. Cham: Springer, 2015. isbn: 9783319233956

    2015

  36. [44]

    A fast optimization algorithm for multicriteria intensity modulated proton therapy planning

    W. Chen et al. “A fast optimization algorithm for multicriteria intensity modulated proton therapy planning”. In: Medical physics 37.9 (2010), pp. 4938–4945

    2010

  37. [45]

    Beyond bixels: Generalizing the optimization parameters for intensity modulated radiation therapy

    J. Markman et al. “Beyond bixels: Generalizing the optimization parameters for intensity modulated radiation therapy”. In: Medical Physics 29.10 (2002), pp. 2298–2304

    2002

  38. [46]

    Variable RBE in proton therapy: comparison of different model predic- tions and their influence on clinical-like scenarios

    G. Giovannini et al. “Variable RBE in proton therapy: comparison of different model predic- tions and their influence on clinical-like scenarios”. In: Radiation Oncology 11 (2016), pp. 1– 16

    2016

  39. [47]

    Clinical commissioning of a pencil beam scanning treatment planning system for proton therapy

    J. Saini et al. “Clinical commissioning of a pencil beam scanning treatment planning system for proton therapy”. In: International journal of particle therapy 3.1 (2016), pp. 51–60. 1Institute of Mathematical Innovation, University of Bath 29 2Department of Mathematical Scienc...

    2016

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.