Biomedical Engineering Reference
In-Depth Information
uum; however, simple models have been developed to incorporate the effect of this
discreteness on the apparent viscosity [2].
Between these extremes, and the focus of this chapter, are individual arteries, usu-
ally branches, bends, sacs and constrictions, which are the cause or consequence of
focal vascular diseases like atherosclerosis and aneurysms. Owing to cycle-averaged
Reynolds numbers
2
on the order of a few hundreds, inviscid or creeping flow approx-
imations are not possible. Owing to the heartbeat, pulsatile (unsteady) flow effects
can often not be neglected, expect perhaps for cases having low Womersley num-
bers
3
or low-amplitude flow rate dynamics.
Decades of research have highlighted the important role of vascular geometry in
giving rise to distinct and surprisingly complicated fluid mechanics in these vessels.
As a result, the three-dimensional nature of the artery shape is critical to capture.
Such models are usually considered in isolation from the rest of the vascular net-
work, or at least those effects are incorporated into the inlet and outlet boundary con-
ditions. Rigid walls and Newtonian rheology are almost always assumed, although
the appropriateness of these assumptions remains the subject of ongoing debate.
4
This chapter will briefly review the rationale and evidence behind these two key
assumptions, particularly as they pertain to studies of large artery hemodynamics
that have been the subject of the author's investigations over the past two decades.
Along the way, important links between these assumptions and the prescription of
flow boundary conditions or the nature of turbulent blood flow will also be consid-
ered. It must be stated from the outset that this chapter does not aim for comprehen-
siveness, but rather seeks to inspire the reader to think critically about assumptions
in hemodynamic modelling for their own applications, especially those employing
image-based or “patient-specific” modelling. It also aims to highlight the challenges,
many ongoing, about when and how to relax these assumptions, and some of the sur-
prising opportunities that arise, or have arisen, from their consideration.
1.2 Rigid vs. compliant walls
As anyone who has taken his or her pulse can plainly feel, arteries are distensible
or compliant.
5
In other words, they deform in response to the pressure pulse pro-
duced by the heart. For a typical large artery, pulse wave velocities (PWV) are on
the order of several metres per second. For a typical resting heart rate, this implies a
2
Re is defined as VD/
ν
, where V is the lumen-averaged velocity, D is the lumen diameter, and
ν
is the blood viscosity, usually assumed to be 0.03-0.04 cm
2
/s.
3
1
/
2
, where R is the lumen radius and
is the frequency of the heart
beat in rad/s. For a typically blood viscosity of 0.035 cm
2
/s and a heart rate of 60 bpm, large artery
Womersley numbers range from around 2 (coronaries) to 20 (aorta).
4
The questions this author is most often asked after presentations are: “How can you assume the
wall is rigid?” and “How can you assume blood is a Newtonian fluid”. The former tends to be asked
by clinicians, the latter by engineers; I'll not hazard a guess why.
5
These descriptive terms, as well as “stiffness” are often used interchangeably, but have specific
meanings for quantifying arterial wall properties [3].
Wo is defined as R
(
ω
/
ν
)
ω
