Efficiency improvement in proton dose calculations with an equivalent restricted stopping power formalism
Bibliographic record
Abstract
Abstract The equivalent restricted stopping power formalism is introduced for proton mean energy loss calculations under the continuous slowing down approximation. The objective is the acceleration of Monte Carlo dose calculations by allowing larger steps while preserving accuracy. The fractional energy loss per step length ϵ was obtained with a secant method and a Gauss–Kronrod quadrature estimation of the integral equation relating the mean energy loss to the step length. The midpoint rule of the Newton–Cotes formulae was then used to solve this equation, allowing the creation of a lookup table linking ϵ to the equivalent restricted stopping power L eq , used here as a key physical quantity. The mean energy loss for any step length was simply defined as the product of the step length with L eq . Proton inelastic collisions with electrons were added to GPUMCD , a GPU-based Monte Carlo dose calculation code. The proton continuous slowing-down was modelled with the L eq formalism. GPUMCD was compared to Geant4 in a validation study where ionization processes alone were activated and a voxelized geometry was used. The energy straggling was first switched off to validate the L eq formalism alone. Dose differences between Geant4 and GPUMCD were smaller than 0.31% for the L eq formalism. The mean error and the standard deviation were below 0.035% and 0.038% respectively. 99.4 to 100% of GPUMCD dose points were consistent with a 0.3% dose tolerance. GPUMCD 80% falloff positions ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:msub> <mml:mi>R</mml:mi> <mml:mrow> <mml:mn>80</mml:mn> </mml:mrow> </mml:msub> </mml:mstyle> </mml:math> ) matched Geant’s <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:msub> <mml:mi>R</mml:mi> <mml:mrow> <mml:mn>80</mml:mn> </mml:mrow> </mml:msub> </mml:mstyle> </mml:math> within 1 μ m. With the energy straggling, dose differences were below 2.7% in the Bragg peak falloff and smaller than 0.83% elsewhere. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:msub> <mml:mi>R</mml:mi> <mml:mrow> <mml:mn>80</mml:mn> </mml:mrow> </mml:msub> </mml:mstyle> </mml:math> positions matched within 100 μ m. The overall computation times to transport one million protons with GPUMCD were 31–173 ms. Under similar conditions, Geant4 computation times were 1.4–20 h. The L eq formalism led to an intrinsic efficiency gain factor ranging between 30–630, increasing with the prescribed accuracy of simulations. The L eq formalism allows larger steps leading to a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mrow> <mml:mi mathvariant="script">O</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>c</mml:mi> <mml:mi>o</mml:mi> <mml:mi>n</mml:mi> <mml:mi>s</mml:mi> <mml:mi>t</mml:mi> <mml:mi>a</mml:mi> <mml:mi>n</mml:mi> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mstyle> </mml:math> algorithmic time complexity. It significantly accelerates Monte Carlo proton transport while preserving accuracy. It therefore constitutes a promising variance reduction technique for computing proton dose distributions in a clinical context.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".