  In this paper, we address the modeling issues of cell movement and division
with a special focus on the phenomenon of volume exclusion in a lattice-based,
exact stochastic simulation framework. We propose a new exact method, called
Reduced Rate Method -- RRM, that is substantially quicker than the previously
used exclusion method, for large number of cells. In addition, we introduce
three novel reaction types: the contact-inhibited, the contact-promoted, and
the spontaneous reactions. To the best of our knowledge, these reaction types
have not been taken into account in lattice-based stochastic simulations of
cell cultures. These new types of events may be easily applied to complicated
systems, enabling the generation of biologically feasible stochastic cell
culture simulations. Furthermore, we show that the exclusion algorithm and our
RRM algorithm are mathematically equivalent in the sense that the next reaction
to be realized and the corresponding sojourn time both belong to the same
reaction and time distributions in the two approaches -- even with the newly
introduced reaction types.
  Exact, agent-based, stochastic methods of cell culture simulations seem to be
undervalued and are mostly used as benchmarking tools to validate deterministic
approximations of the corresponding stochastic models. Our proposed methods are
exact, they are easy to implement, have a high predictive value, and can be
conveniently extended with new features. Therefore, these approaches promise a
great potential.
