Page 272 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

FINITE-ELEMENT ANALYSIS 249

They used an anisotropic stress loading on the material and achieved a more anisotropic material
response. While an anisotropic, nonlinear elastic-plastic model would be best to model skin, the pre-
ceding may be used as an intermediate step in FEA.
Vascular flow models (i.e., arterial, venous, and cardiac) and airways models have to be con-
cerned with vessel wall elasticity. As a first step, a rigid wall may be useful when investigating simply
the flow patterns. As one progresses toward a more realistic model, elastic properties (e.g., radial,
circumferential, and axial) may be required. Texts such as that by Fung (1996) present some of these
properties. Another issue is that of tissue tethering. Most mechanical property evaluations have used
material removed from its surrounding tissue. Vascular tissue is intimately connected with its sur-
rounding tissue, which may reduce the elastic component slightly. Commercial silicone tubing is a
reasonable material on which to model vessel walls if one is evaluating the material as isotropic. If
one varies vessel compliance (elasticity) from rigid to that of thin-walled silicone tubing (and beyond),
one can achieve a reasonable cross section of possible flow patterns. This is of particular impor-
tance when evaluating pulsatile flows. The downside to adding vessel compliance is the dramatic
increase in computational time. Steinman and Ethier (1994) evaluated an end-to-side anastomosis
pulsatile flow model in two dimensions with either a rigid or elastic wall. The computational time
on a workstation was 3 to 4 hours for the rigid model versus 78 hours for the same model when elastic
walls were included.
The next topic when discussing material behavior in vascular flows is the nonlinear flow material
itself (blood). As a first approximation, a Newtonian fluid (water) may be used to model blood. This
is generally the first material used when evaluating flows by bench testing. Flows in the larger arteries
and veins, such as the carotid artery, may be successfully evaluated in this manner. Whole-blood vis-
cosity is 0.035 cP, with a density just slightly above that of water (Fung, 1996). This model is rea-
sonable for larger vessels because the solid components of blood do not interact enough to greatly
vary the results. The flow in the boundary layer near the vessel walls tends to have fewer red cells in
the large vessels because the particles tend to shift toward the higher-velocity core flow. Smaller-
diameter vessels, such as the coronaries, will demonstrate greater deviation from Newtonian flow as
the red blood cells become a dominant factor in the flow. These flows are best modeled as a combi-
nation of Bingham material and power-law flow. The flow is often modeled as a Casson’s flow,
including a pluglike flow in the vessel core and a combination of the preceding two flows nearer the
wall. A finite shear stress is required for flow to exist, and then the viscosity changes as a power-law
fit to the shear rate. At higher shear rates, this tends to become a straight line and would be well
modeled as Newtonian. Capillary flows should not be modeled as Newtonian or power-law flows
because they are much more complex. The normal red blood cell is larger than the diameter of a
capillary and must be “forced” through the vessel by deforming the cell in either a balloonlike form
or a so-called slipper-zipper form that forces the cell along one side of the vessel.
The decision to “accurately” model blood flow becomes difficult when evaluating the red blood
cell component. Although the red blood cell begins as the biconcave form seen in most anatomy and
physiology texts, it deforms under the shears seen in flow. The cell becomes elongated (looking
much like a football), and the cell membrane slides in a tank-treading motion (Sutera et al., 1989).
The interior of the cell is also a viscoelastic fluid, making the modeling of this system difficult.
While the preceding is correct for normal red blood cells, disease states, such as sickle cell anemia,
modify the mechanical structure of the cell membrane, which in turn alters the flow characteristics
of the blood.

10.4 BOUNDARY CONDITIONS

The final important area of FEA is that of boundary conditions. What are the loadings and constraints
of the systems? For hard and soft tissues, these include loading sites and multiple attachments. For
biofluids, these include pressures, velocities, and pulsatile conditions.
The primary concern when evaluating hard tissues is to determine how many ligament and tendon
attachments are required. For example, the wrist is made up of a vast network of ligaments but only