SU-F-T-109: A Shortcoming of the Fisher-Kolmogorov Reaction-Diffusion Equation for Modeling Tumor Growth

Authors


Abstract

Purpose:

To demonstrate the shortcoming of the Fisher-Kolmogorov reaction-diffusion (FK) equation for modeling the exponential tumor growth through a semi-analytical method and full numerical solutions.

Methods:

The FK equation is increasingly often chosen as the mathematical equation, by which one can model the time-dependent variation of the tumor volume in multi-dimensions after a therapeutic intervention of cancer therapy. The hall-mark of the cancer growth is its exponential growth, in particular, at its early stage of the development. In this study, we solved the FK equation in a slab geometry analytically, which lead to a simple formula of the solution with one integral term. Also, a full numerical solution of the FK equation was accomplished in the spherical coordinate system. The former gave us an insight on the characteristics of the solution. The latter method provided the data for the tumor growth as the function of time with biologically sound model parameters.

Results:

The semi-analytical solution showed that the tumor volume only grows slower than the exponential function of the time. The numerical solutions clearly demonstrated that the tumor volume growth can be very well approximated by a second-order polynomial function instead of an exponential function.

Conclusion:

The FK equation is not adequate for modeling the tumor volume variation, in particular, in the early stage of the development, since the solution only can provide the growth in the second order polynomial function of the time.

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