E41 Reaction-diffusion models

Topic

Tumor invasion is a spatially continuous process: tumor cells migrate into the surrounding tissue while simultaneously continuing to proliferate.

This dual process—simultaneous migration and proliferation—is mathematically described by a reaction-diffusion model, in which diffusion represents tumor cell migration (whether random or directed toward a specific signal) and reaction represents the proliferation of those same tumor cells.

When tumor cell migration and proliferation combine in this way, the result is an invasion front advancing into the surrounding tissue at a constant speed, provided the tissue conditions are homogeneous; this type of constant-speed advancing front is a well-known phenomenon in physics and mathematics known as the Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) wave.

This model is particularly relevant to glioblastoma, a brain tumor with an invasion front that spreads diffusely along the brain's white matter tracts. This diffuse mode of invasion means that surgical resection—even if it removes all visible tissue—cannot completely eliminate tumor cells that have already migrated beyond the limits of what is visible or can be safely resected.

In glioblastoma, the speed of this invasion front ranges from one to ten millimeters per week—a rate that can be measured directly through serial MRI scans and predicted by this same reaction-diffusion model based on the migration and proliferation of the constituent tumor cells.

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