Oxygen Transport in Bioreactors for Engineered Vascular Tissues - Computational Modeling in Tissue Engineering - page 284

v r ¼ L p DP ; ð 3 : 1 Þ

where L p is the hydraulic conductivity of the tissue (measured in separate
experiments [ 19 ]) and DP is the measured lumenal pressure.
These bioreactor configurations were implemented in COMSOL using steady-
state, Navier-Stokes flow in the culture medium subdomain, as shown in Eq. 3.2 ,
and Darcy flow from Eq. 3.1 above in the tissue subdomain.

q v r v ¼r p þ l r 2 v ð 3 : 2 Þ

q is the fluid density, p is pressure, and l is the fluid viscosity. The complete
domain was simplified from 3D to 2D by assuming axisymmetry. Additional
boundary conditions for the flow problem consisted of Poiseuille flow at the inlet
and zero pressure at the outlet(s), as shown in Eqs. 3.3 and 3.4 :

z ¼ 0

dp
dz

¼ 8 l h v z i
R i

ð 3 : 3 Þ

P j z ¼ L
r ¼ R o

¼ 0 ð 3 : 4 Þ

The DO transport problem was solved similarly to Eq. 2.2 . The boundary condi-
tions consisted of constant concentration, C 0 , in the fluid inlet and the tissue outlet
had a zero-flux boundary condition. These are shown in Eqs. 3.5 and 3.6 :

C O 2 j z ¼ 0 ¼ C 0 ð 3 : 5 Þ

Þ r ¼ R o

dC O 2
dr

n ð D O 2 ¼ 0 ð 3 : 6 Þ

where n is the unit vector normal to the ablumenal tissue surface. The model was validated
by integrating the flux over the outlet of the tissue surface for the transmural flow cases
and comparing the calculated bulk-averaged DO value to measurements obtained in the
outlet stream. A summary of COMSOL input values is provided in Table 1 .
Outputs included the lumenal pressure field, velocity fields, and DO concentration
and flux profiles along the ablumenal edge. v r was used to assess the dominant mode
of radial oxygen transport via the Peclet number and to determine the average shear
stress realized by cells within the tissue according to Eq. 3.7 [ 26 ]:

s avg ¼ B l v r j 2

ð 3 : 7 Þ

where B is a shape factor (approximately equal to 1) and j is the Darcy specific
hydraulic permeability (j = L p l (R o -R i )). An average interstitial shear stress of
1 dyne/cm 2 was taken as a physiological reference value [ 26 ].
Model predictions for different bioreactor configurations are summarized in
Fig. 4 with plots of the DO profiles along the ablumenal surface of the tubular
tissue construct (the region of lowest DO). As shown in the figure, higher flow