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16.6 SOME APPLICATIONS OF INTEREST 329

16.5.2.3 Spatial Discretization
After numerical quadrature we get, for ξ 2 [α;β]:
∂u α ∂u
Z β ∂ϕ Z β
f ξ,uðξÞ, dx ¼ fðαÞ + Q ξ,t,uðξÞ, ϕ dx (16.94)
α
∂x ðξÞ α ∂x ∂t ðξÞ α
Identifying [α;β]with[x j 1 ;x j ] we get
f j 1=2 ¼ v j 1 + ðξ j 1=2 x j 1 Þð _ u j 1 s j 1=2 Þ (16.95)

where v j 1 ¼f j 1 is considered as a secondary variable of the problem. In a similar way, using ϕ β , we arrive to
f j 1=2 ¼ v j + ðx j ξ j 1=2 Þð _ u j s j 1=2 Þ (16.96)
Adding Eq. (16.95)with j + 1 and Eq. (16.96), we obtain

f j +1=2 f j 1=2 ¼ðξ j +1=2 x j Þð _ u j s j +1=2 Þ + ðx j ξ j 1=2 Þð _ u j s j 1=2 Þ (16.97)

16.5.2.4 Time Integration
Eqs. (16.95)–(16.97), including boundary conditions, form a system of differential-algebraic equations that is integrated
using the MATLAB ODE suite [48]. For the time integrator, an absolute tolerance of 10 6 is fixed with a relative tol-
3
erance of 10 .

16.6 SOME APPLICATIONS OF INTEREST

16.6.1 Reproducing In Silico Measurements of In Vitro Cell Cultures in Microfluidic Devices

We first particularize the general framework presented above to the particular examples we shall simulate. These
correspond to the cell migration processes described above that can be approximated with reasonable accuracy by a 1D
problem. Therefore, we particularize initially the mathematical models and fit all model parameters. In the notation
presented in Section 16.4, this means establishing a proper functional relationship for F i , F ij , K D,i , K m,i, j , K s,i, j , K E,i ,
, and R i . It is common to express these functions through empirical equations with some phenomeno-
0 0
ij
K T,i , F , K D,i
logical meaning. This step, however, includes the definition of some phenomenological parameters that are, in fact,
parameters of the model. The richer and more complex the model, the more parameters will be required. Here, we
present an example of this strategy of modeling and parameter fitting.
16.6.1.1 Model and Parameters
The considered model is a particular case of general equations presented in Section 16.4. Indeed, we consider the
following equations, regulating the evolution of an alive cell population C n , a dead cell population C d , and oxygen
pressure O 2 :
2
∂C n ∂ C n ∂ ∂O 2 1 1
¼ D n 2 χH go ðO 2 ,C n Þ C n + H gr ðO 2 ,C n ÞC n F 12 ðO 2 ÞC n
∂t ∂ x ∂t ∂x τ n τ d
∂C d 1
F 12 ðO 2 ÞC n (16.98)
¼
∂t τ d
2
∂O 2 ∂ O 2
0
αF ðO 2 ÞC n
¼ D O 2 2 11
∂t ∂x
Here, D n is the diffusion coefficient of the normoxic phenotype, χ is the chemotaxis coefficient, τ n and τ d are the growth
and death characteristic times, respectively, and α is the oxygen consumption.
The functions H go (O 2 ), H gr , F 12 , and F 0 are dimensionless activation functions that try to reproduce phenomeno-
11
logical behaviors observed in cell cultures and have the following meaning.
• Go or grow dichotomy: Functions H go and H gr are activation functions that try to reproduce the observed “go or grow”
paradigm in a GBM microenvironment. GBM cells tend to proliferate in oxygenated areas, whereas they become
much more migratory under hypoxic conditions. This behavior is almost exclusive, that is, when a cell invests
energy resources in proliferative activities, it stops investing them in migratory activities. This is determined by a
specific oxygenation threshold, O , the hypoxia threshold. Besides, cell growth and migration should be conditioned
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2

II. MECHANOBIOLOGY AND TISSUE REGENERATION