Page 332 - Advances in Biomechanics and Tissue Regeneration

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

by the nutrient and spatial needs of the cell, so it is reasonable to use a logistic growth model including a
maximum cell capacity C sat and to use the same capacity for avoiding cell migration. This suggests defining
H go and H gr as:

∗
2
H go ðO 2 ,C n Þ¼ ϕ ðO 2 ;O Þϕ ðC n ;C sat Þ (16.99)

∗
+ (16.100)
2
H gr ðO 2 ,C n Þ¼ ϕ ðO 2 ;O ÞρðC n ;C sat Þ
Here, we have used the following notation
8
1 if x 0
>
< x if 0 x β
β
1
ϕ ðx;βÞ¼

>
0 if (16.101)
: x > β
8
0 if x 0
>
< x if 0 x β
+
ϕ ðx;βÞ¼ β
>
1if (16.102)
: x > β
x
(16.103)
α
ρðx,αÞ¼ 1
• Cell death: Cell death is a natural process depending on many factors and agents and has an inherent stochastic
nature. Anoxia is the fundamental cause of cell death in the problem analyzed here, but due to the stochastic nature,
the switch function should be a smooth function that can incorporate death cell variability depending on
oxygenation conditions. Here, a two-parameter sigmoid model is used that is able to capture necrosis and apoptosis
phenomena.
d (16.104)
2
F 12 ðO 2 Þ¼ σ ðO 2 ;O ,δO 2 Þ
where σ is the function:
1 x β
1 tanh (16.105)
2 δ
σ ðx;β,δÞ¼
Here, β is a threshold parameter and δ is a sensitivity parameter. They can be seen as a pair of location-spread param-
d
eters summarizing the stochastic behavior of the considered phenomenon. The parameters O and δO 2 , associated with
2
the oxygen level, define the limits for cell death, capturing the stochastic nature.
• Oxygen consumption: Oxygen consumption is a complex phenomenon related to the oxidative phosphorylation that
occurs in the membrane of cellular mitochondria. Many authors have considered a zero-order consumption
function, that is, a constant consumption rate independent of oxygen concentration O 2 [49–52]. A more realistic
assumption is that the consumption function is described by the Michaelis-Menten model for enzyme kinetics [53,
54]. With this consideration we can define
K
2
F 11 ðO 2 Þ¼ rðO 2 ;O Þ (16.106)
with
x
(16.107)
rðx;KÞ¼
x + K
This type of equation was observed for the oxygen consumption rate in the late 1920s and early 1930s [55]. The
Michaelis-Menten equation has a sigmoid shape that can be interpreted as an almost constant consumption rate
for a high concentration of oxygen, followed by a rapid decrease when the oxygen concentration decreases. This equa-
tion describes more accurately the consumption at low oxygen concentrations and is compatible with previous con-
K
stant consumption rate models, thus allowing the possibility of comparison with previous studies. The parameter O is
2
the oxygen concentration at which the reaction rate is half the rate in a fully oxygenated medium and therefore is
related with oxidative phosphorylation kinetics, cell structure and morphology (size and number of mitochondria),
and the diffusion process at the cytoplasm.

II. MECHANOBIOLOGY AND TISSUE REGENERATION