VHEE beam parameter extraction from depth-dose and lateral profile measurements: a Monte Carlo parametric study

Very High Energy Electrons (VHEE, 50-300 MeV) are a promising modality for deep-seated tumour irradiation. Accurate Monte Carlo (MC) dose calculation requires precise determination of five primary beam parameters: Energy (E), Source size (σ₀), angular divergence (θ), energy spread (ΔE), and source-to-surface distance (SSD). This study quantified the independent influence of each parameter and derived inverse formulae enabling their reconstruction from measured percentage depth-dose (PDD) curves and lateral dose profiles. GATE v9.3 simulations were performed for 50-300 MeV electron beams in a homogeneous water phantom (30 × 30 × 30 cm³, 1 mm³ voxels). A one-factor-at-a-time approach varied E, σ₀ (0-10 mm), θ (0-80 mrad), ΔE (0-50 MeV), and SSD (0-100 cm). For each configuration, 2 × 10⁹ primary electrons were simulated, yielding uncertainties < 1% at Dmax and < 3% distally. Longitudinal observables (R100, R90, R80, R50, R20, surface dose, distal width W=R20 - R80) and lateral observables (FWHM, penumbra) were extracted. Inverse relationships - relating beam energy to R50, source size to FWHM, divergence to R90, and energy spread to the distal width W - were derived by weighted least-squares regression - expressing beam energy as a function of R50, source size as a function of the lateral field size (FWHM), divergence as a function of R90, and energy spread as a function of the distal fall-off width (W = R20 - R80) - and validated using five independent datasets. Longitudinal observables followed a quadratic model in σ₀ (adjusted R²>0.98, RMSE < 2 mm). For the reference configuration, energy was related to R50 by E=(R50 - 38.97)/0.5699. Source size was obtained from σ₀=FWHM/2.355, divergence from R90, and energy spread from W. Validation showed energy agreement within ± 4% for five datasets. The proposed inverse formulae enable rapid and accurate extraction of VHEE beam parameters from routine dosimetric measurements, supporting beam commissioning, quality assurance, and TPS validation for VHEE and FLASH-VHEE systems.