Biomedical Engineering Reference
In-Depth Information
Table 11.16.
Values
of
coefficients
of
cutaneous
wound
healing
and
angiogenesis
model
(Source: [
1507
];
ρ
l
: density
of fibroblasts and leukocytes). Approximative densities of endothelial cells, macrophages, and
fibroblasts are equal to 10
ρ
t
,
ρ
s
: endothelial cell density of tip and stalk of capillary sprout;
ρ
f
,
m (107 cells/cm
3
), 10
m (65 cells/0.07 mm
3
), and
×
10
×
1
×
10
×
10
m (104 cells/cm
3
).
100
×
100
×
10
10
3
g/cm
3
ρ
t
0
10
4
g/cm
3
ρ
s
0
10
3
g/cm
3
ρ
ic
0
10
3
g/cm
3
ρ
f
0
Injury leads to quick healing associated with angiogenesis. Wound environment
characterized by moderate hypoxia, lactate accumulation, slight pH lowering, and
hypercapnia primes cell migration and proliferation. A system of non-linear partial
differential equations describing space and time interactions between coagula-
tion proteins, endothelial cells of capillary tips and sprouts (density
ρ
t
and
ρ
s
;
Tab le
11.16
), other recruited cells such as fibroblasts (density
ρ
f
) and inflammatory
leukocytes (neutrophils and macrophages; density
ρ
l
), chemoattractants (concentra-
tion
c
gf
), oxygen (concentration
c
O
2
), and the extracellular matrix (density
ρ
ecm
)
has been used to model cutaneous wound healing and angiogenesis as a function of
oxygen availability [
1507
]. Capillary tip flux is assumed to be
J
t
=
−D
t
∇ρ
t
−C
t
(
ρ
t
,
ρ
ecm
)
∇
c
gf
(11.10)
(
t
chemotactic coefficient). Capillary sprouts bear diffu-
sion and drag caused by tip flux (tip velocity
v
t
D
t
: diffusion coefficient;
C
=
J
t
/
ρ
t
) and move with velocity
v
s
=
g
s
(
ρ
)
v
t
(assumption:
g
s
(
ρ
)=
A
ρ
/
(
ρ
t
0
). Fibroblast and leukocyte fluxes are
t
t
t
supposed to be given by
J
f
=
−D
f
∇ρ
f
−C
f
(
ρ
f
)
∇
c
O
2
(11.11)
and
J
l
=
−D
l
∇ρ
l
+
C
l
(
ρ
l
)
∇
c
O
2
.
(11.12)
An effective growth factor is assumed to combine all involved growth factors.
When tissues remodel, zero-stress state of the vessel is modified. Wall
remodeling implies changes in rheological properties, and consequently, if the
elastic behavior is expressed in terms of a strain-energy function (SEF), the material
constants in the function must be updated. Moreover, the constitutive equation must
include not only stress history but also material history due to wall restructuring.
The vessel lumen subjected to sustained increments in blood flow enlarges.
The arterial wall thickens in response to sustained increases in blood pressure. Walls
remodel, mainly to restore the stress field toward their homeostatic values. Flow-
induced changes involved in long-term vascular tissue growth and remodeling have
been studied using the continuum approach and motion decomposition [
1508
,
1509
].
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