Biomedical Engineering Reference
In-Depth Information
at specific times of the healing process. On the other hand, there are evolutive
models that predict the evolution of tissues in the fracture site through time. In
this section, we focus on the second type of models.
One of the first analytical models of fracture healing was developed by Davy and
Connolly [101] on the basis of their experimental works of healing canine ribs and
radii. In that model they simulated the callus in simple traversal fractures, using
concepts of continuum mechanics to determine the stiffness and strength of the
fracture site through the healing process. They assumed the complex bone-callus
as a variable section beam composed by different materials with elasto-plastic
behavior and performed a parametric study.
Logvenkov [102] proposed a mathematical model of the fracture callus. The main
elements of that model were osteoprogenitor and cartilage cells, osteoblasts, extra-
cellular matrix, and blood vessels. The derived equations took into account matrix
production, dependent on the stress level in the tissues, and cell differentiation
dependent on the oxygen concentration. The callus material was considered as a
growing elastic body with growth coefficients dependent on a structural tensor and
calcium concentration. The structural tensor represented the anisotropy of fibrils'
orientation that took place during deformation. The concentrations of oxygen and
calcium were connected with the density of blood vessels whose propagation was
related to the diffusion of a hypothetical substance. The formulated boundary
problem allowed defining changes in the growing zone of the callus and to define
its size. Nevertheless, this model did not account for the load history in the healing
callus.
Ament and Hofer [103, 104] proposed a theoretical model of tissue healing and
differentiation in the fracture callus. They considered both the osteogenic factors
determined by the mechanical stimulus represented by the strain energy function.
This function was obtained by means of finite element simulation and the effect
of vascularity represented by a vascularization factor determined by means of
fuzzy logic, from rules defined by medical experts. Similarly, Simon
et al
. [90, 91]
proposed a new model based on eight rules of fuzzy logic. They also incorporated
the influences of blood supply and mechanical stimulus. The model was able to
predict the different tissue evolution in the fracture callus under compression and
shear loads.
Lacroix and Prendergast [95] formulated a finite element model considering
poroelastic material properties, which included mesenchymal stem cell origin and
external bone remodeling based on the differentiation theory proposed by Prender-
gast
et al
. [93]. The same differentiation theory was used by Isaksson
et al
. [105] to
develop a model of cell and tissue differentiation, using a mechanistic approach.
The model directly couples cellular mechanisms to mechanical stimulation during
bone healing including cell-phenotype-specific activities when modeling tissue dif-
ferentiation. It was applied to simulate fracture healing under normal and excessive
mechanical stimulation and the effect of periosteal stripping.
Bail on Plaza and van der Meulen [106] developed a mathematical framework
in which they considered for the first time a differentiation theory based on
growth factors. They incorporated chondrogenic and osteogenic growth factors.
