  Continuum models for the spatial dynamics of growing cell populations have
been widely used to investigate the mechanisms underpinning tissue development
and tumour invasion. These models consist of nonlinear partial differential
equations that describe the evolution of cellular densities in response to
pressure gradients generated by population growth. Little prior work has
explored the relation between such continuum models and related
single-cell-based models. We present here a simple stochastic individual-based
model for the spatial dynamics of multicellular systems whereby cells undergo
pressure-driven movement and pressure-dependent proliferation.We show that
nonlinear partial differential equations commonly used to model the spatial
dynamics of growing cell populations can be formally derived from the branching
random walk that underlies our discrete model. Moreover, we carry out a
systematic comparison between the individual-based model and its continuum
counterparts, both in the case of one single cell population and in the case of
multiple cell populations with different biophysical properties. The outcomes
of our comparative study demonstrate that the results of computational
simulations of the individual-based model faithfully mirror the qualitative and
quantitative properties of the solutions to the corresponding nonlinear partial
differential equations. Ultimately, these results illustrate how the simple
rules governing the dynamics of single cells in our individual-based model can
lead to the emergence of complex spatial patterns of population growth observed
in continuum models.
