Page 363 - IJB-9-4

International Journal of Bioprinting A computational model of cell viability and proliferation of 3D-bioprinted constructs



∂φ O2 −∇⋅( D ∇ ) +φ ψ (φ , ρ )= 0 in ,Ω ∀ >0 φ ( k 1) − φ k ()  ( k 1) 
+
t
+
∂t O2 O2 O2 O2 ∫ Ω ( Oh2 ∆ t Oh2 v Oh2 + D ∇ φ O h2 ⋅ ∇ v Oh2
O2
(X) φ φ ( k 1)  ρ k ()   φ φ ( k 1)
+
+
k ()
k ()
g


∂φ gl −∇⋅( D ∇ ) + (φρ , in , Ω ∀ >0 + ρ m m φ Oh2 + K m + ρ     1− ρ h       m O2 φ Oh2 + K g ∂ ) Ω
h 
k ()
h
O2
k ()
ψ
φ
t
2
∂t gl gl gl gl ) = 0 (XI) Oh2 O2 max ( k++ ) 1 −φ Oh () k O2
φ
+
OUT
∂+

∂ρ =  ρ 1 − ρ   φ O2  φ gl    + ∫ Γ γ ( φ O h ( k 1) − φ O2 ) Γ ∫ Ω ( glh ∆ t glh v glh

 

O2
2

 
G

g 
 
  
  φ
∂t    ρ max   O2 +K O2   + K    + ) 1  φ glh k ( +11)
g
 gl
g gl 
() k
 φ   +D gl ∇φ glh (k ⋅ ∇v glh +ρ m m () k m
gl

O2
1
−ρ H − φ + K     (XII) φ glh + K gl
d 



k
k
  O2 +ρ ()    1− ρ ()    m g φ glh k ( +11) ∂ ) Ω
h
OUT
φ
−D O2 ∇φ O2 ⋅n = γ O2 ( O2 −φ O2 ) on Γ, ∀>t 0 (XIII) h    ρ max     gl φ glh () k + K gl g
  (k + 1) k ()
−D ∇φ ⋅n = (φ −φ OUT ) on Γ, ∀>t 0 (XIV) (k + 1) OUT ρ h −ρ h h
γ
∂+
gl gl gl gl gl + ∫ Γ γφ ( glh −φ gl ) Γ ∫ Ω ( ∆ t v ρ h
gl
f ( t = 0)= f O2 in Ω (XV) ρ k () φ k () φ k k ()
IN
O2
2
(
f t = )=0 f IN in Ω (XVI) −ρ ( h k+ ) 1 1 ( − ρ h G ) ( k () Oh g ) ( k () glh g )
gl gl max φ Oh + K O2 φ glh + K gl
2
r ( 0 r t = )= IN in Ω (XVII) φ k ()
(
2
∂ , for
−ρ ( k+ ) 1 H −1 Oh )) Ω k = , 0 11,,… N − 1
h φ k () + K d T
2.2.1. Numerical approximation Oh2
The system of equations representing the model (XVIII)
(Equations X, XI, XII, XIII, XIV, XV, XVI, and XVII) is f ( k 0= ) = f IN in Ω (XIX)
solved numerically. For the numerical approximation of Oh2 O2
(
the PDEs, the finite element method (FEM) is used. The f glh k= ) 0 = f gl IN in Ω (XX)
Galerkin FEM relies on the reformulation of the strong ( k= ) 0 IN
[22]
PDE into its weak formulation and on the construction r h = r in Ω (XXI)
of a finite-dimensional subspace where the approximate
solution is looked for. The spatial 3D domain is discretized 2.2.2. Volume-averaged model
with tetrahedra finite elements with piecewise linear A simplified version of the mathematical model (Equations
basis functions. The PDEs are considered in vectorial X, XI, XII, XIII, XIV, XV, XVI, and XVII) was obtained by
form, with the weak formulation of the problem written averaging the PDEs over the spatial domain. Therefore, the
as the inner product of the vector PDE and the vectorial point values of oxygen, glucose, and cell concentration are
test function . Regarding the time domain, a first- approximated by their volume average over the domain
[23]
order accurate semi-implicit numerical scheme based (Equations XXII, XXIII, and XXIV), thus obtaining a
on backward Euler is implemented, so that the problem system of first-order ordinary differential equations,
is linear in the unknown solution at each time instance. described in Equations XXV, XXVI, and XXVII. For
The linear solver used is the Generalized Minimal the time discretization, the semi-implicit discretization
Residual Method (GMRES) with Incomplete Lower-Upper scheme based on backward Euler was used (Equations
factorization (ILU) preconditioner. The computation is XIX, XX, and XXI). The initial conditions are described
implemented in FEniCS, an open-source FEM software in Equations XXVIII, XXIX, and XXX. The numerical
library. The numerical approximation of the problem is model was implemented in Matlab. As in the experiment,
found in Equation XVIII. Time discretization consists in 7 days of culture were simulated. The time step was set
subdividing the time domain into a finite number of time to 0.001 h. The model parameters are summarized in
steps. Given T the time length and N the number of time Table 1.
steps, the time step width is given by T/N. Each iteration of f O2 ∫ f ∂ΩΩ/ , for k = 01…,, , N − 1 (XXII)
k ()
k ()
=
the scheme consists of finding the solution at the current Ω O2 T
time step k, which cycles from 1 to N at each node h of the f k () = f ∂ΩΩ/ , for k = ,,0 1 …, N −1 (XXIII)
k ()
discretized spatial domain. Equations XIX, XX, and XXI gl ∫ Ω gl T
describe the initial conditions of oxygen, glucose, and cell k () k ()
concentration, i.e., at k = 1. r = ∫ Ω r ∂ΩΩ/ , for k = ,,0 1 …, N −1 (XXIV)
T
Volume 9 Issue 4 (2023) 355 https://doi.org/10.18063/ijb.741