Biomedical Engineering Reference
In-Depth Information
about six to eight weeks. Six to eight months of healing is called delayed union and
non-union if healing is delayed even further.
Mathematical models of the fracture healing process have been developed in the
recent decades. The mechanical excitation and stimulation of the callus region has
been the main focus of the majority of these models [
1
-
5
]. But also the biochemical
influences have been highlighted in some of the simulations, like the re-establishing
of blood vessels and nutrient supply [
6
]. In [
7
] a sophisticated model of biochemi-
cal fracture healing was presented, which was further refined in [
8
]. A mechanical
stimulation was added to both models in [
9
,
10
], respectively.
Here, a finite element approach for the model presented in [
8
] is proposed and
coupled with a mechanical stimulus based on the strains in the callus area. This
approach is investigated on an osteotomy model of sheep.
2 Mathematical Fracture Healing Model
The investigated model of biochemical fracture healing was developed by Bailón-
Plaza and van der Meulen [
7
] and complemented in [
8
] by Geris et al., where ad-
ditional equations for fibroblasts, connective tissue and angiogenesis were added.
It will be reviewed here in abbreviated form, the interested reader is referred to the
articles cited above and the authors own work [
11
] for a more detailed representation.
The biochemical model describes the fracture healing process by incorporating five
cell concentrations. The local change of concentration with time is given, in general,
by an advection-diffusion-reaction equation
∂
c
i
∂
=∇
(
D
i
(
m
)
∇
c
i
)
−∇
(
C
i
(
g
k
,
m
j
)
c
i
)
+
R
i
(
m
j
,
g
k
),
(1)
t
where
D
i
is a diffusion parameter dependent on the total extracellular matrix den-
sity (ECM)
m
j
)
,
C
i
are advection coefficients representing chemo- and
haptotactical movement controlled by growth factor concentrations and ECM den-
sities. The index
i
(
m
=
indicates the different cell types, stem cells,
fibroblasts, chondrocytes, osteoblasts and endothelial cells, respectively. Further-
more, proliferation, differentiation and apoptosis are realized by reactive terms
R
i
,
which are regulated by the biochemical milieu, i.e. matrix densities and growth factor
concentrations.
The tissue development inside the callus is given by four extracellular matrix
densities
m
j
, where
j
=
m
,
f
,
c
,
b
,v
denotes fibrous, cartilage, immature bone and
vascular tissue, respectively. The production and, in case of connective tissue and
cartilage the resorption, is represented by reaction equations
=
f
,
c
,
b
,v
∂
m
j
∂
=
R
pr od
,
j
(
−
)
−
R
deg
,
j
(
m
j
)
c
i
.
1
m
c
j
(2)
t
