Biomedical Engineering Reference
In-Depth Information
4.7.2 Arterial Vessels
Mechanical description of blood vessels has a long and somewhat complicated history.
Much of the advanced mathematics and applied mechanics associated with this work is
beyond the scope of this textbook. This section will therefore give an overview of some of
the main developments and will present a simplified, reduced arterial system model for
use in the following subsection. The reader is referred to the following textbooks for
more in-depth coverage:
Circulatory System Dynamics
[21] by Noordergraaf,
Hemodynamics
[15] by Milnor, and
Biofluid Mechanics
[4] by Chandran and colleagues, and for basic fluid
mechanics,
[30] by White.
Study of the mechanical properties of the heart as a pump requires the computation
of pressures and flows arising from forces and motion of the underlying heart muscle.
Consequently, general equations of motion in the cardiovascular system typically arise
from the conservation of linear momentum. The Reynold's transport theorem, a conserva-
tion equation from fluid mechanics, applied to linear momentum yields the following
general equation of motion for any fluid:
Fluid Mechanics
r d
V
dt
r g
p þ t ij ¼
ð
4
:
59
Þ
where r is fluid density (mass/volume),
p
is pressure,
t ij
are viscous forces, and V is velocity.
is the differential operator
k @
@ z
The general velocity V is a vector function of position and time and is written
V
i @
j @
¼
@ x þ
@ y þ
ð x
,
y
,
z
,
t Þ¼ u ð x
,
y
,
z
,
t Þ
i
þ v ð x
,
y
,
z
,
t Þ
j
þ w ð x
,
y
,
z
,
t Þ
k
where
u, v
,and
w
are the local velocities in the
x, y
, and
z
directions, respectively.
Equation (4.59) comprises four terms:
gravitational
,
pressure
, and
viscous forces
, plus a
time-varying term. Note that this is a
vector
equation and so can be expanded in
x, y
, and
z
components as the set of three equations:
g x @ p
@ x þ @
@ x þ @
t xx
@ y þ @
t yx
t zx
@ z ¼
@ u
@ t þ u @ u
@ x þ v @ u
@ y þ w @ u
r
r
ð
4
:
60
Þ
@ z
@ y þ @
@ x þ @
t xy
@ y þ @
t yy
t zy
@ z ¼
g y @ p
@ v
@ t þ u @ v
@ x þ v @ v
@ y þ w @ v
r
r
ð
4
:
61
Þ
@ z
g z @ p
@ z þ @
@ x þ @
t xz
@ y þ @
t yz
t zz
@ z ¼
r @ w
@ t þ u @ w
@ x þ v @ w
@ y þ w @ w
r
ð
4
:
62
Þ
@ z
This set of nonlinear, partial differential equations is general but not solvable; solution
requires making simplifying assumptions. For example, if the fluid's viscous forces are
neglected, Eq. (4.59) reduces to Euler's equation for inviscid flow. The latter, when integrated
along a streamline, yields the famous Bernoulli equation relating pressure and flow. In appli-
cation, Bernoulli's inviscid, and consequently frictionless, origin is sometimes forgotten.
If flow is steady, the right-hand term of Eq. (4.59) goes to zero. For incompressible
fluids, including liquids, density r is constant, which greatly simplifies integration of the
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