Biomedical Engineering Reference
In-Depth Information
confirm experimental observations made by earlier studies (Srivastava and Srivas-
tava 1983). For instance, the effective viscosity, the frictional resistance and other
flow characteristics are influenced by hematocrit (Srivastava and Srivastava 2009).
Multiphase haemodynamics strengthen understanding of how the effective viscos-
ity and the frictional resistance is influenced by the concentration of the hematocrit.
The review by Freund (2014) discusses some of the modelling challenges to ac-
count for the large deformations in red blood cells. When this is coupled with flow
turbulence, the range of scales from submicron to micron flow structures needs to
be considered. A suggested practice to alleviate this problem is to perform mesh-re-
finement studies to ensure quality in the results. The other challenge Freund (2014)
discusses is the near incompressibility of the membrane, which introduces numeri-
cal stiffness, and thus a significant time-step restriction for explicit time integration
algorithms. Implicit methods, which have no restriction on the time-step, for blood
cells have been peformed by Dimitrakopolos (2007). Another approach is to use im-
mersed boundary methods to employ membrane forces that distribute on the local
mesh. Here the immersed interface enforces membrane jump conditions built into
discrete operators.
A third phase can be introduced to isolate the leukocytes with the red blood cells.
Jung and Hassanein (2008) used a three-phase computational fluid dynamics ap-
proach including plasma, RBCs, and leukocytes to numerically simulate the local
haemodynamics in such a flow regime. The model tracked wall shear stress, phase
distributions, and flow patterns for each phase in a concentrated suspension shear
flow of blood. Higher leukocyte concentration was correlated with relatively low
wall shear stress near a stenosis having a high wall shear stress. Such flow behav-
iour demonstrates the use of the three-phase haemodynamic analysis in to vulner-
able plaque formation in arteries within vivo complex flow conditions.
9.2.2
Direct Numerical Simulations of Blood Cells
By strict definition, a direct numerical simulation of blood flow means to account
for every single blood cell in the domain of interest. In a true physiological flow
the number of red blood cells is in the millions in a small vessel. One microliter of
blood (1 × 10
−6
L) contains about 4 million red blood cells. The surrounding plasma,
which is a viscous fluid, interacts with every red blood cell while there are also
interactions between the cells. This means that accounting for each individual de-
formable cell and its interaction with the surrounding fluid requires an extensive
computational resources.
The first fundamental stage of direct numerical simulation of blood cells is to
account for the individual interactions between blood cells, and platelets in a vessel.
The work by Ii et al. (2012) provides the application of a full Eulerian fluid-mem-
brane coupling method with a smoothed volume-of-fluid approach for spherical
membrane deformation problems, and is applied to a pressure-driven flow with
red blood cells. The advantage of their work compared with traditional immersed
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