Analytical convolution/superposition modelling of one-dimensional dose profiles with asymmetric energy deposition kernels in homogeneous media.

Abstract

ObjectiveAccurate modelling of dose deposition remains a central challenge in external beam radiotherapy, particularly for modern treatment systems involving complex beam configurations and magnetic field effects. While convolution/superposition (CS) algorithms are widely used in clinical treatment planning systems, they typically rely on numerical implementations and symmetric energy deposition kernels (EDKs), limiting both analytical tractability and the representation of asymmetric dose distributions. The objective of this work is to develop a fully analytical framework for the calculation of one-dimensional (1D) dose profiles based on the CS formalism.

Approach. The total energy released per unit mass (TERMA) was modelled as a polynomial function of lateral position, while the EDK was described using an asymmetric generalized Lorentzian distribution. This parameterization yields a closed-form expression of the convolution integral in terms of Gaussian hypergeometric functions, providing an analytical representation of dose profiles. The model was evaluated on multiple clinical linear accelerators, including Varian TrueBeam and Halcyon, Elekta Versa HD and Elekta Unity, over a range of field sizes and beam energies. Model parameters were fitted to reference dose profiles computed with advanced treatment planning system algorithms and validated against independent measurements. Additional validation was performed at multiple depths for the Unity system.

Main results. Agreement was assessed using γ-index analysis, yielding passing rates above 95% for all configurations under 2%/1 mm criteria against dose calculation algorithms and 2%/2 mm against measurements. The proposed model accurately reproduced symmetric and asymmetric dose profiles, including those observed in MR-guided radiotherapy, while maintaining execution times below 1.5 s.

Significance.
The proposed analytical formulation enables rapid dose evaluation and provides a physically interpretable separation between primary fluence and lateral energy transport. This approach offers a computationally efficient complement to conventional numerical CS implementations, with potential applications in quality assurance, rapid dose estimation, inter-machine comparison and the study of magnetic field effects on dose deposition.