Modeling the propagation of tumor fronts with shortest path and diffusion models -- implications for the definition of the clinical target volume

TL;DR

This study compares the shortest path and reaction-diffusion models for defining tumor spread beyond visible tumors, aiming to improve the objectivity and scientific basis of clinical target volume delineation in radiation therapy.

Contribution

It introduces a mathematical comparison of two tumor spread models, highlighting the reaction-diffusion model's potential as an alternative to traditional margin-based methods.

Findings

Both models produce similar tumor front shapes.

Diffusion model yields smoother tumor fronts.

Anatomical barriers affect diffusion-based fronts.

Abstract

Objective: The overarching objective is to make the definition of the clinical target volume (CTV) in radiation oncology less subjective and more scientifically based. The specific objective of this study is to investigate similarities and differences between two methods that model tumor spread beyond the visible gross tumor volume (GTV): 1. The shortest path model, which is the standard method of adding a geometric GTV-CTV margin, and 2. The reaction-diffusion model. Approach: These two models to capture the invisible tumor "fire front" are defined and compared in mathematical terms. The models are applied to example cases that represent tumor spread in non-uniform and anisotropic media with anatomical barriers. Main Results: The two seemingly disparate models bring forth traveling waves that can be associated with the front of tumor growth outward from the GTV. The shape of the fronts is similar for both models. Differences are seen in cases where the diffusive flow is reduced due to anatomical barriers, and in complex spatially non-uniform cases. The diffusion model generally leads to smoother fronts. The smoothness can be controlled with a parameter defined by the ratio of the diffusion coefficient and the proliferation rate. Significance: Defining the CTV has been described as the weakest link of the radiotherapy chain. There are many similarities in the mathematical description and the behavior of the common geometric GTV-CTV expansion method, and the definition of the CTV tumor front via the reaction-diffusion model. Its mechanistic basis and the controllable smoothness make the diffusion model an attractive alternative to the standard GTV-CTV margin model.