Biological Systems: Multiscale Modeling Based on Mixture Theory - Multiscale Modeling in Biomechanics and Mechanobiology - page 263

T

ʶ ʱ The vectors are:

= ʶ 1 , ··· ,ʶ ndim

T

ʶ ʱ (11.69)

and

ʶ r N p h r d = n (ʱ = 1 ,..., n p )( r = 1 ,..., n dim ) (11.70)

n

Assumptions and Boundary Conditions

Due to the complex interactions of all the physical and chemical processes taking
place, it is necessary to consider some simplifications and assumptions in order to
computationally solve the problem. The model simplifications and assumptions are
as follows:

1. the wall is considered rigid,
2. chemical interactions and the influence of the electric charge of the cells are
neglected,
3. the cells are considered solid spheres,
4. Magnus effect are considered negligible (does not consider particle rotational
effects),
5. the domains are saturated by the moving fluid, so that there are no capillarity
effects,
6. and assume the model relies purely on fluid-particle interactions, so any particle-
particle interaction is currently neglected.

Figure 11.1 depicts the geometry that consists of an in vitro cell culture system
(hydrogel) with the objective of quantifying the major factors that affect the cell
migration process at the macro-scale. This three-dimensional geometry emulates a
simplified vascularized tumor system with the hydrogel acting as the tumor tissue
and a microchannel acting as a vessel. This simple system was chosen for conducting
the simulations because it will be easily developed during experimental tests in future
work. The hydrogel was considered as a homogeneous porous media for this case.
The objective of this study case is to understand the pressure effects, distribution of
nutrients, drag interactions, and viscous shear stress exertions on cell motion at the
macro-scale. A finite element method was implemented to solve the mathematical
model. Tetrahedral elements were used to mesh the three-dimensional computational
domain and mesh sensitivity analysis was carried out by varying the number of mesh
elements in the domain.
The boundary conditions specify that there is no slip at the wall, upstream flow
varies in a parabolic fashion, and there is free flow at the outlet. The tube is long
enough to generate developed axial velocity profiles.
The parameters used in the simulation, in Table 11.1 , consist of both estimated
and experimental values. The velocity inlet was 5

. 45 mm/s, which was calculated

based on the volume flow rate of Q = 0 . 130 mL/min.