Biomedical Engineering Reference
In-Depth Information
Substituting equation (11.10) into (11.7) and (11.8), respectively, we obtain
∇• φ f v s
p = 0
∂φ f
∂t
κ
+
(11.11)
( v s
∇•
κ
p ) = 0
(11.12)
Equation (11.12) is a governing equation that links the cartilage matrix
deformation to the interstitial fluid motion. Equation (11.11) is an interme-
diate result that can simplify the (soon to be introduced) solute transport
governing equation.
In case of no mass sink, conservation of a solute phase (e.g., IGF) in a fluid
phase can be expressed as
φ f c I
∂t
J I = 0
+
∇•
(11.13)
where
c I + φ f v f c I
J I =
φ f D I
(11.14)
Here J I is the mass flux of solute in the fluid phase and D I is the effective
diffusion coecient of the solute in the cartilage including the tortuosity factor
for the cartilage matrix.
The cartilage solid matrix volumetric strain, ε v , is given by the divergence
of the solid phase displacement
u s
ε v =
∇•
(11.15)
For small strains,
φ f = φ f
0 + ε v
(11.16)
where φ f
0 represents the initial fluid volume faction.
Thus, the solute transport equation (11.13) can be simplified by using
equation (11.11)
φ f ∂c I
2 c I + φ f v s
−∇ φ f D •∇
φ f D I
c I = 0
∂t
κ
p
(11.17)
Equation (11.17) is the governing equation for the transport of solute in a
deformable porous medium (in this case cartilage).
11.2.2.2
Conservation of Linear Momentum
The incremental total stress tensor σ inside the tissue is the sum of the incre-
mental interstitial fluid pressure p, and the incremental elastic stress σ e , result-
ing from deformation of the solid matrix [52]
p I + σ e
σ =
(11.18)
where I is the identity tensor.
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