Modeling Tumor Blood Vessel Dynamics - Mathematical Methods and Models in Biomedicine - page 119

simple approach is to omit the collision step above at all solid nodes and instead
reflect all incoming fluid packets in the reverse outgoing direction. This is referred
to as the mid-plane bounce-back rule:

f out f in f ∗ )

α (

x , t )=

− α (

x , (6)

c − α ,andthe t ∗
indicates the half-time step just prior to collision. The mid-plane bounce-back rule
gets its name from the fact that the wall is actually situated half-way between
adjacent fluid/solid nodes. Note, in practice, the rule works best when

where α , − α correspond to opposite lattice directions, i.e., c α ,

1.
When a lattice node lies on the boundary of the domain, one has a problem
during the streaming step, as certain fluid packets will be undefined. To illustrate,
consider the 2-D example at a north wall interface in Fig. 2 b. After streaming,
the direction-specific densities f 6 ,

τ ≈

f 8 are unknown since the implied off-lattice
nodes above the boundary don't contribute during the streaming step. The simplest
solution is to invoke periodic boundary conditions where outgoing fluid packets
are wrapped around the domain. Additionally, one can specify Dirichlet (pressure)
boundaries at the inlet and outlet of a channel. This requires solving for the three
unknown fluid packets as well as the macroscopic velocity in the direction normal
to the boundary. The solution involves simple algebra. Similarly, one can specify an
incoming velocity and outgoing pressure (flux boundaries). Details of the solution
in both 2/3-dimensions can be found in [ 100 ].

f 7 ,

5.2 Transvascular Convection and Diffusion

Another attractive aspect of the LBM technique is its ability to simulate multi-
component flows including a fluid and gas phase. When a gas species, e.g., oxygen,
has a momentum that is negligible compared to blood plasma, it can be considered a
passive solute , which does not impact the pressure or velocity fields of the advecting
fluid. Rather, the solute is simply advected by the background fluid.
The key artifice is to introduce another distribution function that uses a simplified
equilibrium distribution function:

g eq
i

( x , t )= w i C g ( 1 + 3 c i · u ) (7)

where C g analogously represents the gas concentration rather than a fluid density.
The macroscopic velocity u is taken directly from the underlying fluid. Constant
flux/concentration boundary conditions can be specified in a manner similar to the
Dirichlet boundary condition described above [ 80 ].
An important aspect of LBM solute transport models is that they are known
to generally solve the convection-diffusion/dispersion equation under a variety of
conditions [ 80 ]. For example, in the case where the fluid has no velocity, LBM can