Probabilistic Proton Treatment Planning: a novel approach for optimizing underdosage and overdosage probabilities of target and organ structures

Danny Lathouwers; Jelte R. de Jong; Mischa S. Hoogeman; Sebastiaan Breedveld; Steven J. M. Habraken; Zolt\'an Perk\'o

REVIEW 3 major objections 3 minor 3 references

Probabilistic Proton Treatment Planning: a novel approach for optimizing underdosage and overdosage probabilities of target and organ structures

T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper claims that proton treatment plans can be optimized to meet explicit per-voxel underdosage and overdosage probability targets by iterating a simple $E[d] \pm \delta \cdot SD[d]$ percentile surrogate, and that on phantom…

desk verdict Solid proof-of-principle for a probabilistic proton planning method that adapts per-voxel δ-factors to non-Gaussian dose distributions; the outer-loop convergence is unproven, but the phantom evidence and PCE checks are strong enough to justify a serious referee. read the letter →

arxiv 2507.01763 v2 pith:DXSHR4D2 submitted 2025-07-02 physics.med-ph

Jelte R. de Jong , Sebastiaan Breedveld , Steven J. M. Habraken , Mischa S. Hoogeman , Danny Lathouwers , Zoltán Perkó This is my paper

classification physics.med-ph

keywords protontherapyprobabilistictreatmentplanningrobustoptimizationsetupuncertaintyrangePolynomialChaosExpansiondosepercentilesorgan-at-risksparing

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

Proton therapy is more sensitive to setup and range uncertainties than photon therapy, and the standard remedies—CTV-to-PTV margins or worst-case robust optimization—are respectively ill-suited or dependent on the chosen scenario set. This paper tries to establish a third path: optimize the plan directly against voxel-wise underdosage and overdosage probabilities chosen by the clinician. The method approximates each voxel's dose percentile as $E[d] \pm \delta \cdot SD[d]$, optimizes beam weights for that fixed $\delta$, then recomputes $\delta$ from a Polynomial Chaos Expansion sample of the true percentile, repeating until convergence. On spherical and spinal phantom geometries, the resulting plans reach the requested probability levels and, compared with composite-wise mini-max robust plans, either spare the OAR more at equal target coverage or improve target coverage at equal OAR dose. If it holds up in clinical geometries, the payoff is that uncertainty handling becomes an explicit, patient-specific probability budget instead of a margin or a worst-case scenario list.

What carries the argument

The $\delta$-factor is the signed number of standard deviations from the expected voxel dose to a target percentile, defined per voxel as $\delta_i^{\alpha} = (E[d_i] - d_i^{\alpha})/SD[d_i]$. It converts a probabilistic objective into the deterministic dose level $E[d_i] \pm \delta_i \cdot SD[d_i]$ used in quadratic penalties with analytical gradient and Hessian. The outer loop updates $\delta_i$ after each inner optimization using percentiles sampled from the PCE meta-model, with damping $\kappa = 0.2$ and a moving-average convergence check, so that the surrogate tracks the true percentile across iterations.

What would settle it

Take a geometry with a strongly non-Gaussian voxel dose distribution, such as a CTV voxel at the field edge under range uncertainty alone, run the outer loop to convergence, then evaluate the final beam weights by direct Monte Carlo sampling of $10^4$ error scenarios and count the fraction of scenarios with $d_i \le \gamma_i$. If that fraction exceeds the requested $\alpha$ for a non-negligible set of voxels, or if the $\delta$-sequence cycles instead of settling, the probability-control claim fails.

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Extended reading notes

Core claim

The central claim is that voxel-wise dose percentiles can be controlled during treatment-plan optimization by writing them as $E[d_i] \pm \delta_i \cdot SD[d_i]$ and updating the per-voxel factors $\delta_i$ in an outer loop, using Polynomial Chaos Expansion to sample the uncertainty distribution cheaply. With this machinery, the inner optimization replaces a probabilistic goal such as $P(d_i \le \gamma_i) \le \alpha$ with a deterministic dose level that can enter a quadratic objective with analytical gradient and Hessian, and the outer loop re-estimates each $\delta_i$ from the sampled percentile so the surrogate tracks the true distribution. In homogenous phantom geometries, the paper demonstrates that the resulting probabilistic plans meet their stated probability targets and outperform composite-wise mini-max robust plans on the chosen trade-off: for matched CTV coverage, $P(D_{2\%} > 30\,\mathrm{Gy})$ dropped by 10\textendash15\% in the spherical cases and spinal overdosage probability dropped by 24\textendash28\%; for matched OAR dose, $P(D_{98\%} > 57\,\mathrm{Gy})$ increased by 67.5\textendash71\% in the spherical cases and by 10\textendash15\% in the spinal plans.

Load-bearing premise

The load-bearing premise is that the outer-loop update $\delta_i = (E[d_i] - d_{\alpha,i})/SD[d_i]$, damped with $\kappa = 0.2$ and stopped by a moving-average rule, converges for every voxel to the true percentile; the paper shows this empirically for one spherical plan but gives no convergence proof.

Editorial extensions

If this is right

Treatment planners could prescribe explicit probability budgets, such as at most 10\% of uncertainty scenarios underdosing the CTV, instead of selecting a margin or a robust scenario set, and the optimizer would work toward that budget voxel by voxel.
Because probabilistic objectives weight scenarios by their probability, plans become more conformal than composite-wise mini-max plans: the margin shrinks in directions where large shifts are unlikely, sparing OARs without losing CTV coverage.
On the phantom cases, the method produces a plan at least as good as a tuned robust plan on one side of the trade-off and better on the other: matched CTV coverage with lower OAR overdosage probability, or matched OAR dose with higher target coverage probability.
The same $E \pm \delta \cdot SD$ machinery can be pointed at other statistics, including CVaR, dose-coverage objectives, and biological metrics, because $\delta$ is just a mapping from a sampled statistic to a deterministic dose level.
Plan evaluation can use the same probabilistic language as the optimization, with dose population histograms and scenario fractions replacing worst-case DVH bands.

Reading between the lines

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

If the $\delta$-loop converges reliably on clinical anatomies, a natural extension is per-patient, per-fraction re-estimation of the uncertainty distribution, turning the optimizer into a component of adaptive proton therapy rather than a one-time planning step.
The memory bottleneck from the $E[D_{ij}D_{ij'}]$ terms could be reduced by restricting probabilistic objectives to CTV edge voxels and OAR near-edge voxels; the paper mentions voxel sampling strategies, and the $\delta$-formulation would survive such a restriction.
Replacing the percentile-based $\delta$-factor with a CVaR-based one would make the inner objective convex and might remove the need for damping and moving-average convergence checks; the paper explicitly identifies CVaR as future work.
The probability-control claim is modality-independent, so the same framework is testable in photon VMAT, where uncertainty is smaller but inter-patient variation in PTV coverage is documented.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The genuinely new piece is the outer loop. Instead of fixing δ as a Gaussian-derived constant, the authors re-estimate per-voxel δ from PCE-sampled percentiles during optimization, so the E±δ·SD surrogate tracks non-Gaussian voxel dose PDFs. That is a real step beyond Chu et al. and Fabiano et al., and the paper shows it changes plans in ways that make physical sense: the margin conforms to the CTV when scenario probabilities are accounted for, and on the spinal case the method hits specified overdosage probabilities (≈5% for ν=95%, ≈2% for ν=98%).

The care taken with PCE accuracy deserves credit. They run Γ-analyses against the dose engine, report tail errors, and evaluate final plans with an independently constructed PCE, so the evaluation is not tied to the optimization surrogate. The comparison against composite-wise mini-max robust planning is honestly matched on CTV coverage or OAR dose, and the DPHs show consistent improvements.

The main soft spot is exactly what the stress-test note flags: the δ outer loop has no convergence guarantee. The update is a damped fixed-point iteration, and the stopping rule in Eq. 27 uses a moving average, with tolerances as loose as 10% for the spinal OAR. Fig. 19 shows convergence for one spherical plan only. This does not sink the empirical claims for these phantoms, but it means the probability-control guarantee is conditional on a fixed point that may not always be reached. The paper would be stronger with sensitivity analyses to the tolerance values, or a proof of convergence for a simplified setting.

Other gaps: evidence is limited to homogeneous water phantoms with systematic setup and range errors only; no random errors, no patient anatomy, and no comparison against the closest probabilistic baselines (e.g., fixed-δ optimization from Chu or Fabiano). The manual tuning of robust weights to match the probabilistic plans is a fair comparison strategy, but the tuning process is not quantified. Data and code are not released, though the method is described in enough detail to reimplement.

Overall, this is a serious proof-of-principle from an experienced group. The math setup is coherent, the PCE accuracy work is credible, and the phantom results support feasibility. The convergence gap is significant and should be addressed, but it is not fatal at this stage.

It deserves peer review. A good referee should push for more convergence evidence, a clinical geometry, and at least one direct comparison against a fixed-δ probabilistic baseline.

Referee Report

3 major / 3 minor

Summary. The paper proposes a probabilistic proton treatment planning framework in which voxel-wise dose percentiles are approximated as E[d] ± δ·SD[d], with a per-voxel and per-probability-level δ that is iteratively re-estimated in an outer loop from PCE-sampled percentiles. The inner loop minimizes a weighted sum of such E ± δ·SD objectives together with low-weight expectation-based terms, using fmincon with analytical gradient and Hessian. Uncertainties are modeled as Gaussian setup (3 mm) and range (3%) errors. The method is validated on a spherical CTV with and without an OAR and on a horseshoe-shaped CTV surrounding a cylindrical spine, and compared with composite-wise mini-max robust plans. The authors report improved OAR sparing at matched CTV coverage and improved CTV coverage at matched OAR dose, and a consistency check with the Van Herk margin recipe for systematic setup errors.

Significance. If the outer-loop convergence gap is closed, this is a useful proof-of-principle contribution: it replaces the scenario-set selection of robust optimization with voxel-specific probability control and provides a tractable PCE-based inner/outer formulation. Strengths include the independent final-evaluation PCE, the Γ-analysis benchmark against the dose engine in Appendix A, explicit reporting of convergence behavior and computation times, and use of the open-source OpenGPC toolbox. The main limitation is that the central probability-control claim rests on an unverified fixed-point assumption for the δ iteration, and the reported probability metrics are not always the same quantities that are optimized voxel-wise.

major comments (3)

[Section 2.2.2, Eqs. (25)–(27), Table 8] The central claim that the final plan realizes Eq. (4) requires the outer-loop δ-factors to match the percentiles of the delivered beam weights, but no convergence theorem or contraction argument is given for the δ update Eq. (12), and the only convergence display is for one spherical plan (Fig. 19). The stopping rule Eq. (27) compares moving averages of percentiles computed at the damped iterates x^k, not the current inner solution x^k_*, and the tolerances in Table 8 are loose (τ_OAR,ν = 0.1 for ν = 90% and 95%, and 0.05 for ν = 98%). A stale δ can therefore halt the outer loop while the inner objective Eq. (20) is still optimizing a surrogate that does not correspond to the target percentile. The paper should either provide a convergence argument or, at minimum, report for every geometry and probability level a final-plan check of Eq. (4) at x* for all voxels, evaluated with an independent PCE or dose-engine sampling, rather than only representative voxels and moving averages.
[Appendix A.2 and Section 2.5.1] The final voxel-dose PCE used for evaluation has ΔD2% up to about 2 Gy over the test scenarios and mean dose differences up to about 0.9 Gy for 95% of the voxels. Given that the probabilistic thresholds are 57 Gy, 64.2 Gy, and 30 or 54 Gy and the target probabilities are 2–10%, a 1–2 Gy tail error can shift a reported probability by several percentage points. No analysis is given of how PCE approximation error propagates into the reported probabilities, and the PCE accuracy benchmark is only demonstrated for the spherical CTV-only setupXYrange plan, not for the spinal or OAR cases. Please add a sensitivity analysis, for instance by recomputing the final-plan probabilities with dose-engine sampling on a subset of scenarios, and report PCE error bounds for each treatment site.
[Section 3.2, Tables 3 and Figures 9/10] For the spherical CTV+OAR cases, the reported OAR improvement is the DVH-metric probability P(D_2% > 30 Gy), whereas the optimized objective Eq. (14) controls the voxel-wise probability P(d_i > 30 Gy) ≤ 10% for each OAR voxel. These two quantities are not equivalent, and no voxel-wise probability map is given for the spherical OAR, in contrast to the spinal case in Fig. 13. The same remark applies to the CTV metric P(D_98% > 57 Gy) versus the optimized voxel-wise underdosage probability. To support the claim that the method controls under- and overdosage probability per voxel, please provide voxel-wise acceptance maps or summary statistics for the spherical plans as well.

minor comments (3)

[Figure 3] In the submitted version, the flowchart contains placeholder text ('Lorem ipsum') and unreadable path tokens such as '/gid00035/...'; this figure must be replaced with a legible version.
[Section 3.1.2] The 'verification against the Van Herk margin recipe' is a consistency check for one plan under ideal spherical and static-dose-cloud assumptions; the text should say 'consistent with' rather than 'probabilistically equivalent to,' since the derivation relies on a 2D Gaussian population formula.
[Discussion] The statement that 'in some probabilistic optimizations, even one or two outer loop iterations were sufficient' is not quantified; please specify which cases and how this was determined, since it bears on the convergence and stopping-rule discussion.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: delta-factors are re-estimated from PCE percentiles and the final evaluation uses an independent PCE, so the probability-control claim does not reduce to its fitted inputs; minor self-citations are not load-bearing.

full rationale

The central derivation is not circular. The inner objective (Eq. 10) uses d_alpha = E - delta*SD (Eq. 11), with delta defined as (E - d_alpha)/SD from PCE-sampled percentiles (Eq. 12). At each outer iteration this substitution is a definitional identity, not a fitted prediction: delta is the output of the percentile estimate, and the final probability claim is verified by scenario counting on an independently constructed voxel-dose PCE (Section 2.5.1), not by the E +/- delta*SD surrogate. The PCE itself is validated inside the paper against the dose engine via Gamma-analysis (Appendix A), so the cited OpenGPC toolbox and prior PCE work by the same group (Perko et al. 2014, 2016; Rojo-Santiago et al.) are supporting but not load-bearing. The robust-plan comparisons tune robust weights to match one metric (D10th_98% or D90th_2%) and then report the other, so the reported CTV/OAR probability differences are not forced by the matching procedure. The genuine weakness, acknowledged by the empirical rather than proven convergence of the outer loop (Eqs. 25-27; Fig. 19; Discussion notes that one or two iterations sometimes sufficed), is an unproven fixed-point assumption and hence a correctness/convergence risk, not a circularity.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The method introduces no new physical entities. It relies on Gaussian uncertainty modeling, PCE surrogate accuracy, hand-set objective-weight balances, and an unproven convergence of the δ outer loop. The robust comparison also depends on manually tuned robust weights that match one metric, which can favor the probabilistic plan on the other metric.

free parameters (4)

Per-voxel δ-factor (δα, δβ, δν) = Updated each outer iteration via Eq. 12; converged values not tabulated
Controls how many standard deviations below or above the mean the percentile surrogate sits. It is recalculated from PCE percentiles, but during inner optimization it is frozen, so convergence and final probabilities depend on this schedule.
Probabilistic objective weights (πα_CTV, πβ_CTV, πν_OAR, πlow_CTV, πlow_OAR, πtissue) = Table 8: e.g., CTV-only 15,15,1,1; spinal OAR 750,15
Hand-chosen priorities that determine which trade-off point on the CTV/OAR Pareto surface is reached.
Outer-loop hyperparameters (κ, ΔW, Δk, τ) = κ=0.2; ΔW=15-20; Δk=5-10; τ=5e-4 to 0.1
Damping, moving-average window, and convergence tolerances are chosen by hand; no proof shows the δ update reaches a fixed point for arbitrary cases.
Robust comparison weights (ωCTV, ωOAR, ωOARmax, ωnomCTV, ωtissue) = Table 9: e.g., {120,1,1,160} and {100,10,10,100} for XZ cases
Manually tuned to match either CTV coverage or OAR dose; this can bias the reported advantage on the unmatched metric.

assumptions (6)

domain assumption Gaussian independent setup (σ=3 mm in x,y) and range (σ=3%) uncertainties, truncated at the 99% confidence ellipsoid.
Section 2.1: all probability claims are conditional on this distribution; real clinical error distributions may differ.
domain assumption PCE of the dose-influence matrix (GO7E8PO8) and of voxel dose (GO6E7PO7) accurately represents the dose engine, including the lower and upper tails used for 2-10% percentiles.
Appendix A provides Γ-evaluation on two test beams and one plan, but no formal error bound over all geometries.
ad hoc to paper The outer-loop δ iteration converges to the true percentile for every voxel.
Section 2.2.2 and Eq. 27: convergence is checked empirically with moving-average tolerances; no theorem or example-independent guarantee is given.
ad hoc to paper During an inner optimization with fixed δ, improving E[d] ± δ·SD[d] moves the true target percentile in the intended direction.
Section 2.2.1: the surrogate and the true percentile can in principle diverge between outer iterations; monotonicity is not established.
standard math Dose is a linear function of pencil-beam weights with a precomputed dose-influence matrix.
Section 2.2, Eq. 24: standard IMPT linearity assumption.
domain assumption The static dose cloud approximation holds for the spherical margin comparison.
Section 3.1.2: used only for the Van Herk margin verification, not for the probabilistic optimization itself.

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Cite this review

Pith. "Pith review of Probabilistic Proton Treatment Planning: a novel approach for optimizing underdosage and overdosage probabilities of target and organ structures." pith.science (2026). https://pith.science/paper/DXSHR4D2

@misc{pith2026250701763,
  author       = {Pith},
  title        = {Pith review of: Probabilistic Proton Treatment Planning: a novel approach for optimizing underdosage and overdosage probabilities of target and organ structures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DXSHR4D2}},
  note         = {Machine review of arXiv:2507.01763}
}

abstract

Treatment planning uncertainties are typically managed using margin-based or robust optimization. Margin-based methods expand the clinical target volume (CTV) to a planning target volume, generally unsuited for proton therapy. Robust optimization considers worst-case scenarios, but its quality depends on the uncertainty scenario set: excluding extremes reduces robustness, while too many make plans overly conservative. Probabilistic optimization overcomes these limits by modeling a continuous scenario distribution. We propose a novel probabilistic optimization approach that steers plans toward individualized probability levels to control CTV and organs-at-risk (OARs) under- and overdosage. Voxel-wise dose percentiles ($d$) are estimated by expected value ($E$) and standard deviation (SD) as $E[d] \pm \delta \cdot SD[d]$, where $\delta$ is iteratively tuned to match the target percentile given Gaussian-distributed setup (3 mm) and range (3%) uncertainties. The method involves an inner optimization of $E[d] \pm \delta \cdot SD[d]$ for fixed $\delta$, and an outer loop updating $\delta$. Polynomial Chaos Expansion (PCE) provides accurate and efficient dose estimates during optimization. We validated the method on a spherical CTV abutted by an OAR in different directions and a horseshoe-shaped CTV surrounding a cylindrical spine. For spherical cases with similar CTV coverage, $P(D_{2\%} > 30 Gy)$ dropped by 10-15%; for matched OAR dose, $P(D_{98\%} > 57 Gy)$ increased by 67.5-71%. In spinal plans, $P(D_{98\%} > 57 Gy)$ increased by 10-15% while $P(D_{2\%} > 30 Gy)$ dropped 24-28%. Probabilistic and robust optimization times were comparable for spherical (hours) but longer for spinal cases (7.5 - 11.5 h vs. 9 - 20 min). Compared to discrete scenario-based optimization, the probabilistic method offered better OAR sparing or target coverage depending on the set priorities.

Figures

Figures reproduced from arXiv: 2507.01763 by the authors.

**Figure 2.** Probability density functions f of the voxel dose di(x (k) , ξ) for two different pencil-beam weights (e.g., at iterations k) during the optimization. Target underdosage probability is optimized for P(di(x, ξ) ≤ γi) ≤ α, or equivalently d α% i (x) ≥ γi . Target and OAR overdosage can be optimized by P(di(x, ξ) ≤ ϵi) ≥ β, i.e., d β% i (x) ≤ ϵi . The PDF shape has changed for iteration 2, resulting in sufficient targe… view at source ↗

**Figure 3.** The proposed probabilistic optimization approach has a nested structure: the [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (25 more)

**Figure 4.** Figure 4: The robust error scenario set used during optimization. For every setup error [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: Illustration of the three-dimensional homogeneous (water) phantom geometries [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 6.** Figure 6: Illustration of dose population histograms for the CTV metric [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

**Figure 7.** Figure 7: Nominal dose distributions in the XY-plane through the CTV center for the [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]

**Figure 8.** Figure 8: Comparison of nominal XZ-displaced dose distributions for the (left) proba [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]

**Figure 9.** Figure 9: Dose population histograms of various DVH metrics ( [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]

**Figure 10.** Figure 10: Dose volume histogram distributions for the CTV (top) and OAR (bottom), [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]

**Figure 11.** Figure 11: Comparison of nominal dose distributions for the (left) probabilistic and [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]

**Figure 12.** Figure 12: Cross sections through the spine center along the [PITH_FULL_IMAGE:figures/full_fig_p022_12.png]

**Figure 13.** Figure 13: The probability of CTV underdosage (top), CTV overdosage (middle) and [PITH_FULL_IMAGE:figures/full_fig_p023_13.png]

**Figure 14.** Figure 14: Dose population histograms of various DVH metrics ( [PITH_FULL_IMAGE:figures/full_fig_p023_14.png]

**Figure 15.** Figure 15: Dose volume histogram distributions for the CTV (left) and spine (right) for [PITH_FULL_IMAGE:figures/full_fig_p024_15.png]

**Figure 16.** Figure 16: b. We analyse two different pencil-beams that have pencil-beam spots located at (x, y, z) = (4.5 mm, 4.5 mm, 89.5 mm) and (x, y, z) = (40.5 mm, 40.5 mm, 125.5 mm) (in the spherical geometry). We refer to these pencil-beam spots as Test pencil-beam 1 and Test pencil-be…

**Figure 17.** Figure 17: The PCE accuracy analysis for the (top) Dij matrix and (bottom) voxel dose distributions, showing (left) the voxel fraction for which the mean dose difference over all scenarios (∆D) exceeds the dose value, and (right) the scenario fraction for which the minimum dose …

**Figure 18.** Figure 18: An accuracy analysis of E[Dij ]GO, comparing different grid orders (GO3, GO4, GO5) with GO8 (considered true). The fraction of pencil-beam spots that have a) a ∆D2% and b) mean dose difference (over all voxels) exceeding a dose value is shown. A.3 Accuracy of the expe…

**Figure 19.** Figure 19: The convergence behavior of (10th and 90th) CTV and (90th) OAR voxel dose percentiles (dashed) for the XZ-displaced setupXYrange probabilistic optimization, with their moving average (MA15, solid). Only representative outer voxels of both structures are shown, along t…

**Figure 20.** Figure 20: Inner optimization times of the probabilistic spherical and spinal plans. [PITH_FULL_IMAGE:figures/full_fig_p039_20.png]

**Figure 21.** Figure 21: Comparison of nominal X-displaced dose distributions for the (left) probabilis [PITH_FULL_IMAGE:figures/full_fig_p041_21.png]

**Figure 22.** Figure 22: Dose population histograms of various DVH metrics ( [PITH_FULL_IMAGE:figures/full_fig_p042_22.png]

**Figure 23.** Figure 23: Dose volume histogram distributions for the CTV (top) and OAR (bottom), [PITH_FULL_IMAGE:figures/full_fig_p043_23.png]

**Figure 24.** Figure 24: Comparison of the nominal dose distributions corresponding to the probabilis [PITH_FULL_IMAGE:figures/full_fig_p043_24.png]

**Figure 25.** Figure 25: Dose population histograms of various DVH metrics ( [PITH_FULL_IMAGE:figures/full_fig_p044_25.png]

**Figure 26.** Figure 26: The probability of CTV underdosage (top), CTV overdosage (middle) and [PITH_FULL_IMAGE:figures/full_fig_p045_26.png]

**Figure 27.** Figure 27: Nominal dose distribution for probabilistically optimizing the horseshoe [PITH_FULL_IMAGE:figures/full_fig_p045_27.png]

**Figure 28.** Figure 28: Probability density functions of some representative CTV and spinal voxels [PITH_FULL_IMAGE:figures/full_fig_p046_28.png]

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Works this paper leans on

3 extracted references · 2 canonical work pages

[1]

E., Bues, M

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

E., Bues, M

An, Y., Liang, J., Schild, S. E., Bues, M. and Liu, W. (2017). Robust treatment planning with conditional value at risk chance constraints in intensity-modulated proton therapy, Medical Physics44(1): 28–36. Biggs, S., Jennings, M., Swerdloff, S., Chlap, P., Lane, D., Rembish, J., McAloney, J., King, P., Ayala, R., Guan, F., Lambri, N., Crewson, C. and Sob...

work page 2017

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and Mohan, R

Wu, Q. and Mohan, R. (2000). Algorithms and functionality of an intensity modulated radiotherapy optimization system,Medical Physics27(4): 701–711. Xiu, D. and Karniadakis, G. E. (2002). The wiener–askey polynomial chaos for stochastic differential equations,SIAM Journal on Scientific Computing24(2): 619–644. 33 A Cost-accuracy analyses of the dose approx...

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Bortfeld, T. (1997). An analytical approximation of the bragg curve for therapeutic proton beams,Medical Physics24(12): 2024–2033. Chan, T. C., Mahmoudzadeh, H. and Purdie, T. G. (2014). A robust-CVaR optimization approach with application to breast cancer therapy,European Journal of Operational Research238(3): 876–885. Chen, W., Unkelbach, J., Trofimov, ...

1997