Mathematical Formulation of DMH-Based Inverse Optimization

Purpose: To introduce the concept of dose–mass-based inverse optimization for radiotherapy applications. Materials and Methods: Mathematical derivation of the dose–mass-based formalism is presented. This mathematical representation is compared to the most commonly used dose–volume-based formulation used in inverse optimization. A simple example on digitally created phantom is presented. The phantom consists of three regions: a target surrounded by high- and low-density regions. The target is irradiated with two beams through those regions and inverse optimization with dose–volume and dose–mass-based objective functions is performed. The basic properties of the two optimization types are demonstrated on the phantom. Results: It is demonstrated that dose–volume optimization is a special case of dose–mass optimization. In a homogenous media, dose–mass optimization turns into dose–volume optimization. The dose calculations performed on the digital phantom show that in this very simple case dose–mass optimization tends to penalize more the dose delivery through the high-density region and therefore it results in delivering more dose through the low-density region. Conclusion: It was demonstrated that dose–mass-based optimization is mathematically more general than dose–volume-based optimization. In the case of constant density media, dose–mass optimization transforms into dose–volume optimization.