Page 320 - Advances in Biomechanics and Tissue Regeneration

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

velocity, strain, and stiffness of the hydrogel) as well as temperature and electric potential are known or have been
computed previously by the corresponding set of equations. This framework can be easily extended by coupling these
latter variables with the cell number and species concentrations.
This framework is sufficiently general to model almost any possible biological problem. However, it is too complex in
terms of the solution of the equations (highly nonlinear, strongly coupled, and very stiff equations), the number of param-
eters involved (many times unknown or difficult to measure), and the difficulty in validation. This is why this general
framework is particularized and simplified for the problem of interest, keeping the most important influences and dis-
carding those that can be considered comparatively negligible. Here, we shall see how to simplify it by focusing the
research only on the most relevant aspects of our particular experiments. For example, as we will see, the dependence
on external variables is not considered in our experiments, so these equations of the general framework will not be used.

16.4.1 Balance Equations for Cell Populations and Species

16.4.1.1 Cell Populations
The cell concentration (number of cells per unit volume) for each type of cell population is represented as continuous
n
fields C i (x, t), i ¼ 1, …, n where n is the number of cell populations dependent of space x 2  and time t 2  . We note
T
as C(x, t) ¼ (C 1 , …, C n ) . The master equation that regulates cell population evolution is the transport equation with
source terms considering the possible three standard reaction-convection-diffusion phenomena. For the ith cellular
phenotype C i , i ¼ 1, …, n, this equation is
n n
∂C i X X
C i F ij + C j F ji , i ¼ 1,…,n in Ω
i
∂t (16.1)
+ ðv —ÞC i + — q ¼ C i F i
j ¼ 1 j ¼ 1
j 6¼ i j 6¼ i
3
where Ω R represents the domain of study, v is the fluid velocity (convection term), q i is the flux vector (number of
cells at each point per unit surface and per unit time) associated with the migration of phenotype i (diffusion and taxis
terms), and F i is the source term corresponding to the population growth (number of new cells per unit cell and per unit
time), F ij is the source (reaction) term corresponding to phenotype switching, that is, the number of cells that differ-
entiates from phenotype i to phenotype j per unit cell and time. All biological phenomena that influence cell migration
(different types of taxis) will be modeled by expressions affecting the flux vector q i .
corresponds to the part
Boundary conditions are defined in the boundary of the domain Ω, ∂Ω ¼ Γ D i [Γ R i , where Γ D i
of the boundary where the concentration C i is known (Dirichlet boundary part):
(16.2)
C i ¼ f i , i ¼ 1,…,n in Γ D i
corresponds to the Neumann boundary region, where the following general expression is fulfilled:
while Γ R i
∂C i
κ i C i + ¼ g i , i ¼ 1,…,n in Γ R i
∂n (16.3)

16.4.1.2 Species Concentrations
Similarly, the transport equation for the ith chemical species S i , i ¼ 1, …, m, including the reaction-convection-
diffusion phenomena, is
n
∂S i X
0
0 C j F , i ¼ 1,…,m in Ω
i
∂t + ðv —ÞS i + — q ¼ ij
j¼1 (16.4)
Again, v is the fluid velocity, q is the flux vector associated with the chemical species i, and F is the net source term
0
0
i ij
corresponding to the production/consumption of species i per unit cell of phenotype j. Chemical phenomena influenc-
ing species transport are again modeled using flux vectors.
The boundary conditions are defined again in ∂Ω ¼ Γ [Γ 0 with S i satisfying in the Dirichlet part, Γ :
D 0 R D 0
i i i
S i ¼ f , i ¼ 1,…,m in Γ
0
i D 0 (16.5)
i
and in the Neumann part, Γ , S i :
R 0
i
∂S i
κ S i + ¼ g , i ¼ 1,…,m in Γ
0
0
i i R 0 (16.6)
∂n i
II. MECHANOBIOLOGY AND TISSUE REGENERATION