Abstract ID: 104 Efficiency improvement in proton dose calculations with an equivalent restricted stopping power formalism

To maintain the dose accuracy of Monte Carlo simulations, the mean energy loss calculation usually requires a step restriction (d max ). It leads to a O(n) algorithmic time complexity, where n is the subdivision number imposed by d max . A new formalism is proposed to accelerate Monte Carlo dose calculations, allowing the removal of d max in the step selection leading to a O(1) algorithmic time complexity. In this formalism, the midpoint rule of the Newton–Cotes formulae was used to solve the integral equation relating the mean energy loss to the step. The fractional energy loss was obtained with a secant method and a Gauss–Kronrod quadrature, revealing within the midpoint rule the equivalent restricted stopping power (L eq ), used here as a key physical quantity. For any step, the mean energy loss was simply defined as the product of the step 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. First, the dose and time impacts of d max were studied within Geant4. Second, in voxelized geometries, GPUMCD was compared to Geant4 using a high accuracy simulation setup (d max  = 10 μm). The ionization processes alone were activated and the energy straggling was first switched off to validate alone the L eq formalism. The default settings (d max  = 1 mm) in Geant4 led to an error of up to 16.5% in the falloff region, up to 4.8% elsewhere and the computation times were inversely proportional to the maximal step length allowed. Dose differences between Geant4 and GPUMCD were smaller than 0.31% in the Bragg peak for the L eq formalism. GPUMCD 80% falloff positions (R 80 ) matched Geant R80 within 1 μm. With the energy straggling, dose agreements were within 2.7% in the falloff, below 0.83% elsewhere and R 80 positions matched within 100 μm. The overall computation times per million transported protons with GPUMCD were 31–173 ms. Under similar conditions, Geant4 computation times were 1.4–20 h. The L eq formalism allows larger steps while preserving the accuracy. It significantly accelerates Monte Carlo proton transport. The L eq formalism constitutes a promising variance reduction technique for computing proton dose distributions in a clinical context.