The Lattice Boltzmann Method as a General Framework for Blood Flow Modelling and Simulations - Bioanalysis Advanced Materials Methods and Devices - page 154

external viscous dissipation (R), vessel compliance (C), and fluid inertia (L) and
compensates for the missing components (lumped parameter model).
In the framework of the LB method, boundary conditions at the inlet and multiple
outlets can be imposed as follows. A constant velocity (with plug or parabolic profile)
is enforced at the entrance of the main artery, as a way to control the amplitude of the
flow. Even if the inlet profiles are not the real ones for irregular geometries, they fulfill
the purpose of imposing the total flow rate in the chosen region. The fluid spontane-
ously and rapidly develops the consistent profile already at a short distance down-
stream. A constant pressure is imposed on the several outlets of themain artery, as well
as on the outlet of all secondary branches (of the order of 10 in typical coronary
systems). This leaves the simulation with the freedom of creating an appropriate
velocity profile in the outlet regions, and building up a pressure drop between the
inlet and the several outlets. The Zou-He method [ 46 ] is used to implement both the
velocity inlet and the pressure outlets. This method exploits information streamed
from fluid bulk nodes onto boundary cells and imposes a completion scheme for
particle populations which are unknown because their neighboring nodes are not part
of the fluid domain. The boundary cells are treated as normal fluid cells where to
execute the conventional LB scheme. Thanks to this natural integration of the bound-
ary scheme, the method is second-order accurate in space, compatible with the overall
accuracy of the LB method (see [ 22 ]). The method handles in a natural way time-
dependent inflow conditions for pulsatile flows. The algorithm requires that all nodes
of a given inlet or outlet are aligned on a plane which is perpendicular to one of the
three main axes, although the injected flow profile and direction can be arbitrary.
However, since the inlet section is typically a critical region of simulation in terms of
numerical stability due to the high fluid velocities, it is preferable to have an incoming
flow direction aligned with one of the Cartesian axes. This requirement can be fulfilled
by rotating the artery in such a way as to secure that alignment, the inlet axis with one
of the Cartesian axis, which guarantees an exact control of the flow imposed at the
inlet. Conversely, the outlet planes are not in general normal to the orientation of
the blood vessels. However, this does not lead to noticeable problems, because the
pressure drop along typical arterial systems is mild, and the error due to imposing a
constant pressure along an inclined plane is negligible.
10.3 Red Blood Cells
RBCs or erythrocytes are globules that present a biconcave discoidal form, and a soft
membrane that encloses a high-viscosity liquid made of hemoglobin: they exhibit
both rotational and orientational responses that deeply modulate blood rheology.
While blood rheology is quasi-Newtonian away from the endothelial region, the
presence of RBCs strongly affects blood flow in proximity of the endothelium,
where the interplay of RBC crowding for hematocrit levels up to 50 % , depletion
due to hydrodynamic forces, and RBC arrangement in rouleaux takes place.