Page 330 - Advances in Biomechanics and Tissue Regeneration

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

Let us define

f i ¼ p i vC i , i ¼ 1,…,n
f n + i ¼ q i vS i , i ¼ 1,…,m (16.86)
T
and f ¼ (f 1 , …, f n , f n+1 , …, f n+m ) .
Finally let us define

n n
∂v X X
s i ¼ C i + C i F i C i F ij + C j F ji , i ¼ 1,…,n
∂x
j ¼ 1 j ¼ 1
j 6¼ i j 6¼ i
n n
∂v X X
s n + i ¼ S i C j G ij + C j G ji , i ¼ 1,…,m (16.87)
∂x
j¼1 j¼1
T
and s ¼ (s 1 , …, s n , s n+1 , …, s n+m ) . With these notations, neglecting the ECM remodeling, it is possible to summarize the
governing equations as:
∂u ∂f
+ s (16.88)
∂t ¼ ∂x
∂u
where f ¼ f x,t,u, and s ¼ s(x, t, u). Eq. (16.88) has sense if and only if we define suitable boundary conditions and
∂x
initial conditions. Boundary conditions are for each variable u i , i ¼ 1, …, n given by Eq. (16.2)or(16.3) and for each
variable u i , i ¼ n +1, …, m by Eq. (16.5) or Eq. (16.6) except for the fact that we replace directional derivatives ∂ by
∂n
∂
partial derivatives .
∂x
Finally, the initial conditions are
0 (16.89)
u ¼ u ðxÞ
0
0
0
where u ¼ C for i ¼ 1, …, n and u 0 ¼ S for i ¼ 1, …, m.
i i n + i i
16.5.2.2 Weak Form
The differential equation (16.88) with boundary conditions (16.2), (16.3), (16.6), (16.6) and initial condition (16.89)is
a nonlinear parabolic differential equation in time and one space variable. We solve numerically this equation using a
method based on a simple piecewise nonlinear Galerkin second-order accurate in space [47], which is compatible with
1
this kind of nonlinear equation and boundary condition. Multiplying the PDE by a test function ϕ 2H ð½α;βŠÞ and inte-
0
grating by parts in [α;β], we arrive to:
Z β ∂ϕ Z β
ϕQ dx (16.90)
ϕðβÞfðβÞ ϕðαÞfðαÞ f dx ¼
α ∂x α
∂u ∂u
where Q ¼ Q x,t,u, sðx,t,uÞ. As a test function we select
∂t ¼ ∂t
β x
α
β α
ϕ ðxÞ¼
x α
β
ϕ ðxÞ¼ β α (16.91)
When using ϕ α , Eq. (16.90) becomes

Z β ∂ϕ Z β
f α dx ¼ fðαÞ + Qϕ dx (16.92)
α
α ∂x α
and when using ϕ β , we get
Z β ∂ϕ Z β
f β dx ¼ fðβÞ + Qϕ β dx (16.93)
α ∂x α

II. MECHANOBIOLOGY AND TISSUE REGENERATION