Traveling wave solutions of a mathematical model for tumor encapsulation

The formation of a capsule of dense, fibrous extracellular matrix around a solid tumor is a key prognostic indicator in a wide range of cancers. However, the cellular mechanisms underlying capsule formation remain unclear. One hypothesized mechanism is the 'expansive growth hypothesis,' which suggests that a capsule may form by the rearrangement of existing extracellular matrix, without new matrix production. A mathematical model was recently proposed to study the implications of this hypothesis [Perumpanani, Sherratt, and Norbury, Nonlinearity, 10 (1997), pp. 1599-1614]. The model consists of conservation equations for tumor cells and extracellular matrix and exhibits traveling wave solutions in which a pulse of extracellular matrix, corresponding to a capsule, moves in parallel with the advancing front of the tumor. In this paper the author presents a detailed study of traveling wave behavior in the model, deriving conditions for the existence of traveling waves and their key properties. Numerical methods for solving the model equations are presented, and numerical simulations suggest that the traveling waves are stable and are the biologically relevant solution form for the model. The analytical results are extended to an improved model, which includes a saturation in the extent of matrix rearrangement per cell. Finally, the author discusses the biological implications of the model results.