Page 322 - Advances in Biomechanics and Tissue Regeneration

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

16.4.2.2 Migration Terms in Cell Population Equations
Now, migration terms (biological related transport) are defined. The flux vector q i is postulated as a linear
decomposition:
q ¼ q D,i + q m,i + q + q E,i + q , i ¼ 1,…,n (16.12)
s,i
i
T,i
where the flux vectors q D,i , q m,i , q s,i , q E,i , and q T,i are associated with diffusion, mechanotaxis, chemotaxis, electrotaxis,
and thermotaxis phenomena, respectively.
16.4.2.2.1 DIFFUSION
The common framework is used to model diffusion, that is
q ¼ K D,i —C i , i ¼ 1,…,n (16.13)
D,i
Here, matrix K D, i is the diffusion matrix that can be expressed as
K D,i ¼K D,i ðC 1 ,…,C n ,θ,p 1 ,…,p k Þ, i ¼ 1,…,n (16.14)

16.4.2.2.2 MECHANOTAXIS
The mechanotaxis term can be expressed in a similar manner, using the expression
k
X
q K m,i, j —p j , i ¼ 1,…,n (16.15)
m,i ¼
j¼1
where K m, i, j is the mechanotactic motility matrix of species i with respect to parameter p j expressed as:
K m,i, j ¼K m,i, j ðC 1 ,…,C n , p j ,—p j ,θÞ, i ¼ 1,…,n (16.16)

16.4.2.2.3 CHEMOTAXIS
The chemotaxis term is expressed as
m
X
K s,i, j —S j , i ¼ 1,…,n
s,i (16.17)
q ¼
j¼1
Here, K s, i, j is the chemotactic motility matrix with respect to the chemical species j. As before, it is possible to express
the matrix as
K s,i, j ¼K s,i, j ðC 1 ,…,C n ,S 1 ,…,S m ,—S 1 ,…,—S m ,θ,p 1 ,…,p k Þ, i ¼ 1,…,n, j ¼ 1,…,m (16.18)
It is common to use saturation models for chemotaxis [42], getting:
χ C i
i
K s,i, j ðC 1 ,…,C n ,S 1 ,…,S m ,—S 1 ,…,—S m ,θ,p 1 ,…,p k Þ¼ (16.19)
σ j + λ j k —S j k
where σ j and λ j are parameters depending on the chemical species j and χ i is a sensitivity parameter depending on the
cellular phenotype i.

16.4.2.2.4 ELECTROTAXIS
For electrotaxis, a similar expression can be written:
q ¼ K E,i —V, i ¼ 1,…,n (16.20)
E,i
where K E, i is the electrostatic motility matrix and V is the electric potential. As before,
K E,i ¼K E,i ðC 1 ,…,C n ,θ,p 1 ,…,p k ,V,—VÞ, i ¼ 1,…,n (16.21)

16.4.2.2.5 THERMOTAXIS
Thermotaxis can be modeled using analogous equations:
p T,i ¼ K T,i —θ, i ¼ 1,…,n (16.22)

where K T, i is the thermotactic motility matrix and θ is the temperature field, with
K T,i ¼K T,i ðC 1 ,…,C n ,θ,—θ,p 1 ,…,p k ,Þ, i ¼ 1,…,n (16.23)

II. MECHANOBIOLOGY AND TISSUE REGENERATION