Mechanobiological Modelling of Angiogenesis: Impact on Tissue Engineering and Bone Regeneration - Computational Modeling in Tissue Engineering - page 375

densities of the most important cell types, generic growth factor families and
tissues. The process of angiogenesis is modelled through the spatiotemporal
evolution of an endothelial cell concentration and a vascular density. Shefelbine
et al. [ 112 ] proposed a fuzzy logic model to simulate fracture healing based on the
mechanoregulation theory proposed by Claes and Heigele [ 25 ] and the vascularity.
They developed fuzzy rules and considered a single variable representing the
vascularity. Using a similar model, Simon et al. [ 113 ] included blood perfusion as
a spatio-temporal state variable to simulate the revascularisation process. Chen
et al. [ 22 ] modified the set of fuzzy logic rules and simulated nutrition supply
instead of the vascularity.

3.2 Discrete Models

Discrete models aim to determine the system behavior at the cellular level, instead
of averages responses of continuum variables. This has the advantage of enabling
microscopic properties of the capillary network, such as vessel branching and
looping, to be captured. They are based on phenomenological descriptions of
single cells either by solving systems of differential equations at discrete locations
(i.e. points on a lattice) or by ''if-then'' computational algorithms [ 94 ]. There are
several different types of discrete models: cellular automata, agent-based, cellular
potts and cell force-based models.

3.2.1 Cellular Automata Models

Cellular automata (CA) models capture complex biological phenomena by
defining a series of simple rules that are easy to quickly compute and that can even
be parallelized [ 4 ]. Most cellular automata models reduce the simulation process
into a set of discrete states, discrete time cycles and space coordinates [ 4 ]. CA are
usually divided in three main categories: deterministic automata, lattice-gas

models and solidification models [ 31 ].
In the deterministic automata, the space is divided in a fixed lattice and each
lattice point has a state associated with it. The state at the next time step is only
function of the earlier states of the cell and its neighbors. Lattice gas models
consist of a discrete spatial grid in which particles, in contrast to the deterministic
automata, are driven by random events. Therefore they contain a stochastic ele-
ment in the sense that they can allow the computation of different outcomes across
different simulations. Solidification models are very similar to lattice ones except
that moving particles may be irreversibly bound at grid points, or cells may
undergo irreversible state transitions.
Within the field of tumour induced angiogenesis, Anderson and Chaplain [ 2 ]
derived a discrete model using a cellular automaton approach. In particular they
proposed a lattice based model in which the endothelial cell migration depends on
chemotactic and haptotactic probabilistic rules. An extension to three dimensions
is presented in Chaplain [ 19 ].