  We introduce a model for describing the dynamics of large numbers of
interacting cells. The fundamental dynamical variables in the model are
sub-cellular elements, which interact with each other through phenomenological
intra- and inter-cellular potentials. Advantages of the model include i)
adaptive cell-shape dynamics, ii) flexible accommodation of additional
intra-cellular biology, and iii) the absence of an underlying grid. We present
here a detailed description of the model, and use successive mean-field
approximations to connect it to more coarse-grained approaches, such as
discrete cell-based algorithms and coupled partial differential equations. We
also discuss efficient algorithms for encoding the model, and give an example
of a simulation of an epithelial sheet. Given the biological flexibility of the
model, we propose that it can be used effectively for modeling a range of
multi-cellular processes, such as tumor dynamics and embryogenesis.
