Biomedical Engineering Reference
In-Depth Information
of fluid networks. Each network has its own individual porosity
n
a
, density
ρ
a
,
permeability
k
a
and finally fluid velocity relative to the aforementioned solid matrix
w
i
/n
a
.
The complete system presents a highly non-linear set of equations which require
vast computational effort. Fortunately, we are dealing with biological flows and so
acceleration frequencies can be neglected and this can simplify the system to a con-
cise set of
A
+
1 equations, i.e.
+
ρ
b
A
∂
2
u
∂t
2
σ
−
α
a
∇
p
a
∇·
−
=
0
,
(22.1)
a
=
1
k
a
ρ
a
b
p
a
A
∂p
a
∂t
+
∂
2
u
∂t
2
1
Q
a
α
a
∂(
∇·
u
)
∂t
+∇·
k
a
−
−
∇
−
a
˙
s
b
→
a
=
0
.
b
=
1
,b
=
(22.2)
Here,
σ
is the effective stress in the solid matrix,
b
is the local body force,
u
is
the displacement vector of the solid matrix,
ρ
=
a
=
1
n
a
p
a
+
−
ρ
s
(
1
n)
is the
=
a
=
1
n
a
is the total porosity of the combined fluid
networks,
ρ
s
is the density of the solid,
k
is the isotropic permeability,
Q
is the
combined compressibility of the system,
∂(
total density of the system,
n
ε
isthestrainrateinthe
solid matrix,
α
a
is the Biot parameter of the defined network
a
and finally
∇·
u
)/∂t
= ˙
˙
s
b
→
a
is
the rate of fluid exchange from network
b
to
a
(Tully and Ventikos,
2011
).
The MPET framework strives to capture the independent nature of the fluid trans-
fer within the brain. The quadruple MPET model takes into account the arterial
blood network
(a)
, the arteriole/capillary network
(c)
, venous blood network
(v)
and extracellular/CSF network
(e)
. To transform the system in Eqs. (
22.1
) and (
22.2
)
into the quadruple MPET system required, one sets
A
(a,c,v,e)
. In ad-
dition to this, further simplifications are made by assuming: a linear stress-strain re-
lationship, isotropic permeability, no external forces on the system, negligible grav-
itational effects, stationary reference frame, quasi-steady system due to the large
time scales in the development of HCP and physiology-derived constraints of spe-
cific directional transfer of water between networks in order to avoid breaches in
continuity and finally a spherically symmetric geometry. This gives the following
simplified relationships for the new quadruple MPET system, i.e.
=
4 and
a
=
α
a
∂p
a
∂
2
u
∂r
2
α
c
∂p
c
α
ν
∂p
ν
α
e
∂p
e
∂r
2
r
∂u
∂r
−
2
u
r
2
1
−
2
ν
+
−
∂r
+
∂r
+
∂r
+
=
0
,
(22.3)
2
G(
1
−
ν)
k
a
∂
2
p
a
∂r
2
∂p
a
∂r
2
r
−
+
+|˙
s
a
→
c
|=
0
,
(22.4)
k
c
∂
2
p
c
∂r
2
∂p
c
∂r
2
r
−
+
−|˙
s
a
→
c
|+|˙
s
c
→
e
|+|˙
s
c
→
ν
|=
0
,
(22.5)
