Biomedical Engineering Reference
In-Depth Information
Load
Stimulus (
y ≡
J
2
)
Granulation tissue
volume (V
s
)
Stem cells (c
s
)
Proliferation
c
s
= c
smax
c
s
< c
smax
Differentiation
Death
Migration
Callus size
c
s
↓
c
s
↑
c
s
↑
Fibrous tissue volume (V
f
)
Fibroblast (c
f
)
Matrix cartilage volume (V
c
)
Chodrocytes (c
c
)
Callus stiffness
Endochondral
ossification
Matrix bone volume (V
b
)
Osteoblasts (c
b
)
Internal bone remodeling
Figure 4.6
Scheme of the mechanobiological model of
fracture healing proposed by Gomez-Benito
et al
. [109-111].
They used finite differences to study the healing process on a rat tibia [106];
later, they incorporated the influences of the mechanical factors to the previous
growth factor-based model [107]. Recently, Geris
et al
. [108] adopted the model of
Bail on-Plaza and van der Meulen incorporating angiogenesis to simulate normal
and pathological bone healing with the finite volume method.
Gomez-Benito
et al
. [109-111] developed a continuum mathematical model of
the bone healing process (Figure 4.6). It was successfully applied to simulate
the influence of gap size [109], amount of interfragmentary movement [111], and
external fixator stiffness [110] on the fracture healing process, and distraction osteo-
genesis [112]. The model simulates tissue regulation and callus growth, taking into
account different cellular events (i.e., mesenchymal cell migration and proliferation;
mesenchymal cells' differentiation into chondrocytes, fibroblasts, and osteoblasts,
death of mesenchymal stem cells, and also the endochondral ossification process),
and matrix synthesis, degradation, damage, calcification and remodeling over time.
The evolution of the main components of the extracellular matrix of the different
tissues (i.e., different collagen types, proteoglycans, minerals, and water) was also
analyzed to determine the mechanical properties and permeability according to the
particular composition.
In order to define all these processes, the fundamental variables were the
concentration (
c
i
) of each cell type, with subscripts s, b, f, and c indicating stem
cells, osteoblasts, fibroblasts, and chondrocytes, respectively, and the mechanical
stimulus. Cell concentration can change because of proliferation (
pr
i
β
·
c
i
), migration
2
c
i
), and differentiation (
f
differentiation
(
(
D
·∇
ψ
,
t
)) for each specialized cell type
