Page 324 - Advances in Biomechanics and Tissue Regeneration

322 16. ON THE SIMULATION OF ORGAN-ON-CHIP CELL PROCESSES

we obtain
and repeating it with Eq. (16.3) and integrating in Γ R i
Z Z ∂C i Z
κ i ϕC i dΓ + ϕ dΓ ¼ ϕg i dΓ, i ¼ 1,…,n (16.31)
∂n
Γ R i Γ R i Γ R i
The same strategy is followed with Eq. (16.4) obtaining
Z Z
∂S i
ϕ + ϕ— B i dΩ ¼ b i ϕdΩ, i ¼ 1,…,m (16.32)
Ω ∂t Ω
where we have defined
B i ¼ S i v + q 0 i
m
X
0 0
ij
ji
b i ¼ C j ðF F Þ
j¼1
A new integration by parts of Eq. (16.32) leads to
Z Z Z
ϕ ∂S i —ϕ B i dΩ ¼ ϕb i dΩ ϕ ∂S i dΓ, i ¼ 1,…,m (16.33)
Ω ∂t Ω ∂n
R
Γ 0
i
We do analogously with boundary conditions (16.5), (16.6) in order to obtain
Z Z
ϕf dΓ, i ¼ 1,…,m
0
ϕS i dΓ ¼ i (16.34)
D D
Γ 0 Γ 0
i i
and repeating it with Eq. (16.3) and integrating in Γ 0 we get
R
i
Z Z Z
∂S i
κ ϕS i dΓ + ϕ ϕg dΓ, i ¼ 1,…,m
0
0
i dΓ ¼ i (16.35)
∂n
Γ 0 Γ 0 Γ 0
R R R
i i i
16.5.1.2 Spatial Discretization
All scalar and vectorial fields involved in the problem are discretized using a finite basis of dimension
N, B¼ fϕ ,r ¼ 1,…,Ng, that is,
r
N
X r
C ðtÞϕ ðxÞ, i ¼ 1,…,n
C i ðx,tÞ¼ i r
r¼1
N
X r
S ðtÞϕ ðxÞ, i ¼ 1,…,m
S i ðx,tÞ¼ i r
r¼1
N
r
X
r
θðx,tÞ¼ θ ðtÞϕ ðxÞ
r¼1
N
r
X
r
Vðx,tÞ¼ V ðtÞϕ ðxÞ
r¼1
N
X r
p ðtÞϕ ðxÞ, i ¼ 1,…,k
p k ðx,tÞ¼ k r
r¼1
(16.36)
N
r
X
r
v ¼ v ðtÞϕ ðxÞ
r¼1
Note that all fields are approximated using the same basis and a separation representation in space time is postulated.
Even if it is not the most general approach, the fact that Eqs. (16.29), (16.33) involve up to first-order derivatives is
consistent with this FE approximation.
II. MECHANOBIOLOGY AND TISSUE REGENERATION