Efficiency improvement in proton dose calculations with an equivalent restricted stopping power formalism

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 ([Formula: see text]) matched Geant's [Formula: see text] within 1 μm. With the energy straggling, dose differences were below 2.7% in the Bragg peak falloff and smaller than 0.83% elsewhere. The [Formula: see text] positions matched within 100 μm. 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 [Formula: see text] 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.