The migration and proliferation of populations of cells underpin a variety of physiological and pathological processes, for example embryogenesis, wound healing and cancer progression. Mathematical models have offered insight into how individual-level cell behaviour scales up to population-level outcomes, such as invasion speed or population growth rate. A key aspect of a successful model is to accurately capture the effect of local interactions between cells, for example in their competition for space or other resources.

The majority of mathematical models use a pre-defined lattice on which the cells can move. An alternative is to use a lattice-free model, which allows the cell to move through continuous space. I will describe some examples of lattice-free modelling and show that the use of a lattice can affect the outcomes of cell-cell interactions. I will also show how the lattice-free model can be extended, via the use of spatial moment dynamics, to include cell crowding effects and directional bias.