Computational Methods in the Modeling of Scaffolds for Tissue Engineering - Computational Modeling in Tissue Engineering - page 110

Fig. 5 Elastic modulus and permeability versus porosity on two pore cell units; spherical pore
(top, in plot with dashed line) and cylindrical pore (bottom, in plot with solid line), modified from
Hollister [ 23 ]

The homogenization theory is used to generate the multiscale equilibrium equa-
tions to compute effective properties of these unit cell designs [ 21 , 22 ]. The study
presented by Hollister [ 23 ] has been selected to illustrate to the reader the
homogeneity method. The mechanical properties such as the macroscopic effective
stiffness E macro (Eq. 4 ) can be computed from the stiffness at a microscopic E micro
level depending of the strain tensor M and the volume of unit cell V. The mass
transport (fluid inside the pores) is also homogenized focusing in the macroscopic
permeability K (Eq. 5 ) computed through the average Stokes flow velocity v and
pressure gradients [ 23 ].

1
V unit cell

Z

E macro ¼

V E micro MdV unit cell

ð 4 Þ

j j

1
V unit cell

Z

K macro ¼

V vdV unit cell

ð 5 Þ

j j

Hollister [ 23 ] have found that an increment of the amount and disposition of
material in a pore induces an increment of elastic properties, but with a reduction
of the permeability (Fig. 5 ). Results demonstrate that the modulus increases as
expected with volume fraction, and that the spherical pore leads to a stiffer
scaffold. The permeability decreases as expected with volume fraction and
cylindrical pore design allows fabricating a more permeable scaffold. Others