  We formulate haptotaxis models of cancer invasion wherein the infiltrating
cancer cells can occupy a spectrum of states in phenotype space, ranging from
`fully mesenchymal' to `fully epithelial'. The more mesenchymal cells are those
that display stronger haptotaxis responses and have greater capacity to modify
the extracellular matrix (ECM) through enhanced secretion of matrix-degrading
enzymes (MDEs). However, as a trade-off, they have lower proliferative capacity
than the more epithelial cells. The framework is multiscale in that we start
with an individual-based model that tracks the dynamics of single cells, which
is based on a branching random walk over a lattice representing both physical
and phenotype space. We formally derive the corresponding continuum model,
which takes the form of a coupled system comprising a partial
integro-differential equation for the local cell population density function, a
partial differential equation for the MDE concentration and an
infinite-dimensional ordinary differential equation for the ECM density.
Despite the intricacy of the model, we show, through formal asymptotic
techniques, that for certain parameter regimes it is possible to carry out a
detailed travelling wave analysis and obtain invading fronts with spatial
structuring of phenotypes. Precisely, the most mesenchymal cells dominate the
leading edge of the invasion wave and the most epithelial (and most
proliferative) dominate the rear, representing a bulk tumour population. As
such, the model recapitulates similar observations into a front to back
structuring of invasion waves into leader-type and follower-type cells,
witnessed in an increasing number of experimental studies over recent years.

Analysis of PDEs