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

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 geometric 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.

The influence of anisotropy on the clinical target volume of brain tumor patients

“…Let S(r) measure the shortest distance of any point r = (x, y, z) from the GTV surface. Following Bortfeld and Buti (2022), S(r) is the solution to the equation:…”

“…r ( ) contains 6 independent tensor elements which, in our framework, describe relative 'resistance coefficients' to tumor spread in the brain in xx, yy, zz, xy = yx, xz = zx, and yz = zy directions. The metric tensor is proportional to the inverse diffusion tensor used in the FKPP formalism (Bortfeld and Buti 2022). The default value for the resistance coefficients is r…”

“…for more resistant tissues) (Shusharina et al 2020). Equation (1) can be transformed into a form similar to an anisotropic Eikonal equation enabling fast numerical computation using the Fast Marching Method which is a numerical scheme dedicated to solving a Riemannian Eikonal equation (Bortfeld and Buti 2022). It takes as input (a) the GTV mask surface defining the origin (r = 0), (b) the brain mask defining the domain interior Ω, (c) barrier structure masks (including the skull) to define the Neumann boundary conditions, and finally (d) the metric tensor r ( ) defined for all discretization points r ä Ω.…”

“…The warped metric tensor image served as an input to the Hamiltonian Fast Marching Python library (Mirebeau and Portegies 2019) to solve the Eikonal equation (1). The output is the shortest path in each voxel from the GTV surface considering the metric in warped space and pre-defined barrier structures (Bortfeld and Buti 2022). Figure 1 shows an example of the segmentation and resulting shortest path distance calculation.…”

“…Appendix. Obtaining the metric tensor from a diffusion tensor atlas Bortfeld and Buti (2022) shows the equivalence between the diffusion tensor mm s…”