Page 328 - Advances in Biomechanics and Tissue Regeneration

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

m 0 ðiÞ m  , and F S ¼ðF 0 ð1Þ ,…,F 0 ðmÞ ) Eq. (16.58)for i ¼ 1, …, m can be expressed in a compact
0
ðiÞ
Defining  S ¼ i¼1  ,  S ¼ i¼1
form as:
_ (16.67)
 S S +  S S ¼ F S
If we explicit the functional dependences, we have
_ (16.68)
 S S +  S ðC,S,P,W,θÞS ¼ F S ðC,S,P,V,θÞ
Besides, using again vectorial notation, the regeneration term may be stated as
_ (16.69)
P ¼ RðC,θÞ
Eqs. (16.63), (16.68) may be combined if we define U ¼ (C, S), H ¼ (W, V, θ),  ¼  C  S ,F ¼ (F C ,F S ), and

2 3
 C  CS
4 0  S 5 (16.70)
 ¼
arriving finally, to the equation:
_
U + ðU,P,HÞU ¼ FðU,P,HÞ
_
P ¼ RðU,HÞ (16.71)
Eq. (16.71) shall be combined with Dirichlet boundary conditions, given in Eqs. (16.46), (16.56). For the former, C i ¼f i (t)
while for the later, S i ¼ f ðtÞ is known in nodes belonging to Γ . The first line of the
0
is known in nodes belonging to Γ D i i D 0
i
compact Eq. (16.71), if U ¼ UðtÞ are the constrained variables, may be split symbolically in free and constrained
variables:
_
_
 f U f +  c U +  f U f +  c U ¼ F (16.72)
So, we finally obtain
∗ _ ∗ ∗ ∗ ∗
 U +  U ¼ F (16.73)
_
∗
∗
∗
where  ¼  f ,  ¼  f , F ¼ F  c U  c U, and U* ¼U f are the unknowns of the problem. Finally, Eq. (16.71) writes
_
∗ ∗ ∗ ∗ ∗ ∗ _
 U +  ðU ,P,HðtÞÞU ¼ FðU ,P,HðtÞ,UðtÞ,UðtÞÞ
_ ∗ (16.74)
P ¼ RðU ,HðtÞ,UðtÞÞ
16.5.1.3 Time Integration
In order to solve Eq. (16.71), that is, to compute U* ¼U*(t) and P ¼ P(t), for t 2 [0;T], it is necessary to define the initial
∗
conditions U ¼ U ðt ¼ 0Þ and P 0 ¼ P(t ¼ 0) and to specify H ¼ H(t) (physical stimulus, including electrical stimulus,
∗
0
thermal stimulus, and flow stimulus) and U ¼ UðtÞ (boundary conditions) for t 2 [0;T].
This problem may be expressed in a standard form using the symbolic notation
∗
(16.75)
x ¼ðU ,PÞ
1 ∗ ∗ ∗ ∗
2 _ 3
 ðF ðU ,P,HðtÞ,UðtÞ,UðtÞÞ  ðU ,P,HðtÞÞÞ
∗
4 5 (16.76)
fðx,tÞ¼
RðU ,HðtÞ,UðtÞÞ
∗
0 (16.77)
x 0 ¼ðU ,P 0 Þ
thus to obtain
dx
dt ¼ fðx,tÞ
xðt ¼ 0Þ¼ x 0 (16.78)
Eq. (16.78) is a nonlinear ordinary differential equation system that may be solved using different numerical schemes,
accounting for the high nonlinearity, coupling between variables, and stiff behavior of the differential equation. Many

II. MECHANOBIOLOGY AND TISSUE REGENERATION