Page 325 - Advances in Biomechanics and Tissue Regeneration

16.5 IMPLEMENTATION 323

16.5.1.2.1 CELL POPULATIONS
Plugging Eq. (16.36) into Eq. (16.29), we obtain
N !
Z r Z Z Z
X ∂C ∂C i
ϕ i ϕ dΩ —ϕ A i dΩ ¼ ϕa i dΩ ϕ dΓ, i ¼ 1,…,n (16.37)
r
Ω r¼1 ∂t Ω Ω Γ R i ∂n
Using linear properties of — operator, Eqs. (16.13), (16.15), (16.17), (16.20), (16.22) may be written as
N
X r
q ðC K D,i Þ—ϕ , i ¼ 1,…,n
D,i ¼ i r
r¼1
0 1
N k
X X r
q @ p K m,i, j —ϕ , i ¼ 1,…,n
m,i ¼ j r
A
r¼1 j¼1
0 1
N n
X X r
q @ S K s,i, j —ϕ , i ¼ 1,…,n
s,i ¼ j r
A
r¼1 j¼1
N
X r
q E,i ¼ ðθ K E,i Þ—ϕ , i ¼ 1,…,n
r
r¼1
N
X r (16.38)
q ðV K T,i Þ—ϕ , i ¼ 1,…,n
T,i ¼ r
r¼1
So, as A i ¼ C i v+q i it is obtained
N !
X r
C ϕ v
A i ¼ i r
r¼1
0 1
N k m
X r X r X r r r
+ @ C K D,i + p K m,i, j + S K s,i, j + θ K E,i + V K T,i —ϕ , i ¼ 1,…,n (16.39)
r
i
j
j
A
r¼1 j¼1 j¼1
And —ϕ A i writes
N
X
r
i
r
i
i
r
—ϕ A i ¼ C ð—ϕ ϕ v —ϕ ðK D,i —ϕ ÞÞ
r¼1
0 1
N k m
r
r
r
X X X
r
+ —ϕ @ p K m,i, j + S K s,i, j + V K i,E + θ K i,T —ϕ , i ¼ 1,…,n (16.40)
A
j
j
r
r¼1 j¼1 j¼1
r
Besides, as C ¼ C ðtÞ, ϕ ∂C r i writes
r
i i ∂t
N r N
_
X ∂C X r
ϕ i ϕ ¼ ϕ C ϕ , i ¼ 1,…,n (16.41)
r
r
∂t i
r¼1 r¼1
Choosing ϕ ¼ ϕ s , s ¼ 1, …, N (Galerkin method) and reorganizing terms in Eq. (16.40), we may write
N m N
X ^ ðiÞ r XX ^ ðijÞ r ^ ðiÞ
 C +  S + F , i ¼ 1,…,n
s (16.42)
—ϕ A i ¼ sr i sr j s
r¼1 j¼1 r¼1
with
^
 ðiÞ ¼ð—ϕ ðϕ vÞ —ϕ ðK D,i —ϕ ÞÞ, i ¼ 1,…,n
r
r
sr
s
s
 ¼ —ϕ K s,i, j —ϕ , i ¼ 1,…,n
^ ðijÞ
r
s
2 0 0 1 1 3
N k
^ ðiÞ X X p r A K m,i, j + V K E,i + θ K T,i —ϕ , i ¼ 1,…,n (16.43)
r
r
F
s
s ¼ —ϕ 4 @ @ j A r 5
r¼1 j¼1
II. MECHANOBIOLOGY AND TISSUE REGENERATION