Page 289 - IJB-10-3

International Journal of Bioprinting Design and optimization of 3DP bioscaffolds

2.3.2. Oxygen transport process where R is the oxygen consumption rate; C is the
2
0
Oxygen serves as the sole substrate for cell growth. As oxygen concentration in the porous domain; Vo max is the
the flow velocity within the system changes, the oxygen maximum oxygen absorption rate by the cells; K denotes
m
transport transitions from a single diffusion to a coupled the oxygen concentration value at which the absorption
convection–diffusion process. The oxygen diffusion– rate reaches half of its maximum; and ρ is the cell density.
c
convection process in region Ω is governed by Equation Additionally, it is necessary to consider the adaptation
1
12: process, cell spreading, and deformation within the
scaffold. In this model, it is assumed that Vo max increases
∂C from zero at the beginning to a maximum value at a certain
1 =∇⋅DC t()  −v ⋅∇Ct() (12)
∇


∂t 1 1 1 time, as expressed in Equation 18:
where C represents the dissolved oxygen concentration  Vt ⋅ r (0≤<

tt )
1
in Ω , and D indicates the diffusion coefficient of oxygen in V o max  3 1 (18)
1
the nutrient solution. This equation describes the oxygen   V r ( 1 ≤
tt)
concentration changes due to the combined effects of
spontaneous oxygen diffusion (the first term on the right- where Vo max represents the actual maximum oxygen
hand side) and convection caused by fluid flow (the second absorption rate, and t is the nodal time.
1
term on the right-hand side). Under different spatial and 2.3.4. Cell growth kinetics
temporal conditions, C should satisfy the corresponding After oxygen consumption, cell proliferation and division
1
boundary conditions as in Equation 13: occur. The cell growth kinetics described by the first-order
growth process is simulated by utilizing the modified
C = c xy zt(, ,, ) (13) Contois model, as expressed in Equation 19:
1
1
In the porous medium region Ω of the scaffold, the µ ⋅ Ct()
2
2
oxygen transport should satisfy Equation 14: µ t() = K c ρ ⋅ m cmax ⋅ ρ t()+ C t() (19)
c
V ⋅
c
2
c
∂Ct ()
2 =∇⋅Dt () ∇Ct ()  −v ⋅∇C t () − R t() (14) where µ indicates the specific growth rate of cells; µ


cmax
∂t e 2 B 2 0 denotes the maximum specific growth rate of cells during
c
the growth process; C is the oxygen concentration within
where C expresses the oxygen concentration within the 2
2
biological scaffold; D is the effective diffusion coefficient the porous medium domain; K is the Contois parameter;
c
ρ represents the mass density of cells; V is the volume of
e
of oxygen within the scaffold; and R denotes the oxygen an individual cell; and ρ represents the density of cells. The
c
m
0
consumption rate by cells. C should satisfy Equation 15 change in cell density can be expressed using the Contois
c
2
within the region.
equation as follows:
C = c xy zt(, ,, ) (15)
2
2
∂ρ t() = µ t() ⋅ ρ t () (20)
c
The effective diffusion coefficient of oxygen D varies ∂t c c
e
due to changes in porosity caused by cell growth, and this
process satisfies Equation 16. In reality, the growth process of cells in the scaffold
should consider an initial phase lag due to cell adhesion and
spreading. Therefore, a step-wise profile for the maximum
D = ε t () ⋅ D (16) specific growth rate µ is necessary in the model:
e
Γ t () cmax
2.3.3. Oxygen consumption process  0 ( 0 t t )
≤<
Oxygen enters the scaffold (region Ω ) and is then absorbed  µ ⋅− ) 0
(
tt

2
t
t
0
r
and consumed by the cells. The cell oxygen consumption  t − t ( t ≤< ) (21)
1
0
term in Equation 14 is calculated by the Michaelis–Menten  1 µ 0 ( t < t)

1
equation (Equation 17).  r
Vo Ct() where µ represents the actual input of the maximum
r
max
2
Rt() = K + C t() ρ c t () (17) specific growth rate in the model, and t represents time. t 0
0
2
m
is the cell adhesion time, while t indicates the spreading
1
Volume 10 Issue 3 (2024) 281 doi: 10.36922/ijb.1838