Cellular automata (CA) may be viewed as simple models of self-organizing complex systems. Here, we focus on an important class of CA, the socalled lattice-gas cellular automata (LGCA), which have been proposed as models of spatio-temporal pattern formation in biology. As an example, we introduce a LGCA model for a simple biological growth process based on randomly moving and proliferating agents. We demonstrate how a mean-field approximation can yield insight into the formation of spatial patterns and calculate important macroscopic observables for the biological growth process. In particular, we address the role of the diffusion strength in the approximation by distinguishing well-stirred and spatially distributed cases. Finally, we discuss the potential and limitations of the mean-field description in analyzing biological pattern formation.