A novel proposition of radiation energy conservation in radiation dose deformation using deformable image registration

<i>Objective</i>. The purpose of this study is to analytically derive and validate a novel radiation energy conservation principle for dose mapping via deformable image registration (DIR).<i>Approach</i>. A radiation energy conservation principle for the DIR-based dose-deforming process was theoretically derived with a consideration of the volumetric Jacobian and proven using synthetic examples and a patient case. Furthermore, an energy difference error was proposed that can be used to evaluate the DIR-based dose accumulation uncertainty. For the analytical validation of the proposed energy conservation principle, a synthetic isotropic deformation was considered, and artificial deformation uncertainties were introduced. For the validation with a patient case, a ground truth set of computed tomography images and the corresponding deformation was generated. Radiation energy calculation was performed using both the ground truth deformation and another deformation with uncertainty.<i>Main results</i>. The suggested energy conservation principle was preserved with uncertainty-free deformation, but not with error-containing deformations using both the synthetic examples and the patient case. For a synthetic example with a tumor volume reduction of 27.1% (10% reduction in length in all directions), the energy difference error was calculated to be -29.8% and 37.2% for an over-deforming and under-deforming DIR uncertainty of 0.3 cm. The energy difference error for the patient case (tumor volume reduction of 37.6%) was 2.9% for a displacement vector field with a registration error of 2.0 ± 3.2 mm.<i>Significance</i>. A novel energy conservation principle for DIR-based dose deformation and the corresponding energy difference error were mathematically formulated and successfully validated using simple synthetic examples and a patient example. With a consideration of the volumetric Jacobian, this investigation proposed a radiation energy conservation principle which can be met only with uncertainty-free deformations.