Page 323 - Advances in Biomechanics and Tissue Regeneration

16.5 IMPLEMENTATION 321

16.4.2.3 Source Terms and Diffusion for Chemical Species
Source terms in Eq. (16.4) include production and consumption of species. The net production/consumption of the ith
chemical species by the jth cell phenotype can be modeled using an equation of the kind:

F ¼F ðC 1 ,…,C n ,S 1 ,…,S m ,θÞ, i ¼ 1,…,m, j ¼ 1,…,n (16.24)
0
0
ij ij
16.4.2.3.1 DIFFUSION
As for cellular phenotypes, diffusion is stated as
q ¼ K 0 —S i , i ¼ 1,…,m (16.25)
0
i D,i
Here, matrix K 0 D,i is the diffusion matrix that will be expressed as
K 0 0 ðC 1 ,…,C n ,θ,p 1 ,…,p k Þ, i ¼ 1,…,m (16.26)
D,i ¼K D,i

16.4.3 ECM Remodeling Coupling
One last step in this global framework corresponds to regeneration considerations. Mechanical parameters of the sub-
strate p 1 , …, p k can be seen as constant parameters, such as elastic parameters (Young modulus E, Poisson coefficient ν,
etc.), or may be thought of as evolving parameters. In this latter approach, a dynamic approach to the problem is
adopted, being necessary to define an evolution relationship:
_ p ¼R i ðC 1 ,…,C n ,θÞ, i ¼ 1,…,k (16.27)
i

16.5 IMPLEMENTATION

16.5.1 3D Finite Element Implementation

16.5.1.1 Weak Form
1 1 1
0
0
The weak form of Eqs. (16.1), (16.4) is derived. Let ϕ 2H ðΩÞ, i ¼ 1, …, n, j ¼ 1, …, m, being H ðΩÞ the closure in H ðΩÞ
1
2
(H ðΩÞ¼ W 1,2 ðΩÞ), with this latter, the Sobolev space with respect to the L norm of differentiable functions (in the
weak sense) of order 1 of infinitely differentiable functions compactly supported in Ω or equivalently, the space of
1
functions in H ðΩÞ that vanish at the boundary ∂Ω.
1
Eq. (16.1) can be multiplied by a test function ϕ 2H ðΩÞ and integrated in Ω to obtain
0
Z Z
ϕ ∂C i + ϕ— A i dΩ ¼ a i ϕdΩ, i ¼ 1,…,n (16.28)
Ω ∂t Ω
where we have defined
A i ¼ C i v + q
i
n
X
a i ¼ C i F i + ðC j F ji C i F ij Þ
j ¼ 1
j 6¼ i
where, respectively, Dirichlet or Robin-
Integrating by parts Eq. (16.28), splitting the boundary ∂Ω ¼ Γ D i [Γ R i
Neumann boundary conditions are applied, and using that ϕ vanishes at the boundary, results in:
Z Z Z
∂C i ∂C i
ϕ —ϕ A i dΩ ¼ ϕa i dΩ ϕ dΓ, i ¼ 1,…,n (16.29)
Ω ∂t Ω Γ R i ∂n
we arrive to
Multiplying Eq. (16.2) by a test function ϕ and integrating in Γ D i
Z Z
ϕf i dΓ, i ¼ 1,…,n
ϕC i dΓ ¼ (16.30)
Γ D i Γ D i

II. MECHANOBIOLOGY AND TISSUE REGENERATION