Biomedical Engineering Reference
In-Depth Information
i
(osteoblast, chondrocyte, fibroblast, and mesenchymal stem cell) following a
similar equation:
Dc
i
(
x
,
t
)
pr
i
2
=
β
·
c
i
−
D
·∇
c
i
−
f
differentiation
(
ψ
,
t
)
(4.1)
Dt
As a first approach, migration and proliferation of chondrocytes, osteoblasts, and
fibroblasts are assumed to be negligible.
Proliferation of stem cells is considered to be proportional to the mechanical
stimulus (
ψ
), which is identified here with the second invariant of the deviatoric
strain tensor:
,
t
)
ψ
(
x
,
t
)
+
ψ
proliferation
α
·
ψ
(
x
pr
s
proliferation
β
=
(4.2)
α
ψ
where
proliferation
are constants that define the stem cell proliferation.
They assumed that the differentiation process (
f
differentiation
(
proliferation
and
,
t
)) is dependent on
the mechanical stimulus and the time that cells need to mature.
Finally, they considered that callus growth is mainly due to mesenchymal cell
proliferation (
f
proliferation
(
c
s
,
ψ
ψ
)) and chondrocyte hypertrophy during endochondral
ossification (
g
endochondral
(
ψ
,
t
)):
f
proliferation
(
c
s
,
g
endochondral
(
div(
v
)
=
ψ
)
+
ψ
,
t
)
(4.3)
It was assumed that the concentration of mesenchymal cells can vary between zero
and a maximum or saturation cell density. When the saturation concentration of
mesenchymal stem cells is reached, the only way cells can proliferate further is by
increasing the callus size at a constant level of cell concentration.
4.8
Examples of Application of Bone Fracture Healing Models to Implant Design
Different factors affecting bone healing have been studied in computational sim-
ulations using fracture healing models: the size [95, 109] and type of the fracture
[85], the amount of interfragmentary motion [95, 111], the type of load [96], and
the stiffness of the external fixator [90, 110, 113]. All these factors could help in the
design of new fracture treatments and implants.
Loboa
et al
. [85] analyzed hydrostatic stress and maximum principal tensile strain
patterns in regenerating tissue around the site of an oblique fracture and compared
the results with the histomorphology of a typical oblique pseudarthrosis. Tissue
differentiation predictions were consistent with the characteristic histomorphology
of oblique pseudarthrosis. For example, they found that regions of high hydrostatic
pressure correlated with locations of periosteal bone resorption. G omez-Benito
et al
. [109] and Lacroix and Prendergast [95] studied the influence of the gap size
on the fracture healing process. In both works, the delay in the healing process
with the increase of the gap size was predicted. In the work of Gomez-Benito
et al
.
[109], large gap size and intrerfragmentary motions were also analyzed resulting in
a nonunion or pseudarthrosis as in experimental works [37].
