A solvable hyperbolic free boundary problem modelling tumour regrowth

Abstract

Recently, Tian and Friedman et al. developed a mathematical model on brain tumour recurrence after resection [J.P. Tian, A. Friedman, J. Wang and E.A. Chiocca, Modeling the effects of resection, radiation and chemotherapy in glioblastoma, J. Neuro-Oncol. 91(3) (2009), pp. 287–293]. The model is a free boundary problem with a hyperbolic system of nonlinear partial differential equations. In this article, we conduct a rigorous analysis on this hyperbolic system and prove the local and global existence and uniqueness of the solution. It is well known that most nonlinear free boundary problems are impossible to solve in terms of explicit analytical solutions. In contrast, the free boundary problem in this study is solvable, and the explicit solution is found using the backward characteristic curve method. This explicit solution is then validated by numerical simulation results. An interesting finding in this study is that the problem can be treated as a hyperbolic system defined on an infinite domain where the initial condition has a first-type discontinuity.

Keywords:

• free broundary problem
• tumour growth
• characteristic curve method

AMS Subject Classifications::

• 35L50
• 35L60
• 35Q92

Acknowledgements

The authors would like to thank Avner Friedman for useful discussion. D. Yang was supported by Monroe Scholarship of undergraduate summer research from the Roy R. Charles Center at the College of William and Mary.