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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_3774_Библиотеки_им_академика_М_И_Перельмана
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142
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S. Jansen et al.
7.1 Introduction
Complex parameters such as non-Newtonian uids, turbulent ow, shear stresses
within the blood and between the blood ow and the intima, as well as within the
wall itself, are important for understanding the ow of blood, which is a particulate
mixture of cells and plasma proteins. We will not try to be denitive for some of
these parameters, especially when they are combined, but we do want to impart
some understanding of how they may inuence the uid dynamics.
Accompanying the major advances in computing and imaging capabilities is the
improvement in computational uid dynamics (CFD) modelling which now includes
patient-specic geometry, uid-structure interactions, pulsatile ow and nonNewtonian ow. CFD and Finite Element Analysis (FEA) are now used to assist in
the design of implantable medical devices and enhance understanding of the physics
of vascular systems. They will inuence future vascular disease management and
are presented in the next chapter.
This chapter will summarise and discuss the following laws, equations and phenomena to give a basic understanding of the haemodynamic principles of the conduits and uids with which we work:
• Darcy’s law
• Poiseuille ow
• Laplace’s law of wall tension
• Newtonian uid
• Non-Newtonian uid
• Reynolds number
• Womersley number
• Bernoulli’s equation
• Young’s modulus and pulsatile ow
• Mass conservation
• Shear stress and pressure
• Forces on graft systems
• Venous muscle pump mechanism
7.2 Pump Mechanisms intheCirculation
As Fig.7.1 shows, the rst pump is the heart and it works as a positive displacement pump to supply blood to the organs. For every cardiac cycle, the ventricles,
which are pumps in series, push (displace) a xed amount of blood into the aorta
and pulmonary artery. The compulsory relaxation phase of cardiac muscle and its
control of rate by regular physiological feedback systems ensures a regular
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Vascular Haemodynamics
Fig. 7.1 Schematic diagram of cardiac
pumps in series feeding parallel circuits
143
supply, and this rhythm and control give the heart the qualities of a servo pump.
The aorta acts not only as a vessel to transfer blood but also as a modulator and
dampener (Windkessel effect, as described by Otto Frank, a German physiologist)
(see Glossary) to convert this pulsatile blood ow into a more uniform ow to
supply blood to the tissues whilst ensuring diastolic ow to supply the heart and
walls of the great vessels themselves.
Propulsion in the venous system is independent of the heart and relies on several mechanisms; the most important and powerful of which is the skeletal muscle
pump where every muscle in the body is acting as a ‘heart’. Contraction and relaxation of the skeletal muscles form a venous muscle pump to return blood from the
144
Muscles relaxed,
Muscles contracted,
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S. Jansen et al.
valves closed
valve above muscle opens
Fig. 7.2 Schematic diagram of the venous muscle pumping mechanism
deep veins to the heart—another form of displacement pump. Contraction of the
muscles (increasing during exercise) push the blood to the heart from valve to
valve (Fig.7.2). The pumping mechanism in the muscles is similar to the suckerrod lift pump in oil wells. The sucker-rod pump is based on the positive displacement progressive lift method. When the muscle relaxes, the uid (blood) rushes
into the vein relling the deep veins prior to blood being pumped cranially with
surface veins acting as llers.
The synchronous movements of the chest wall and diaphragm are another displacement pump (Fig.7.3). This affects the working of the other two pumps; heart
and skeletal muscles. During inspiration, venous return to the right side of the heart
DVIR= ,
DPQR= ,
s
moves up
Vascular Haemodynamics
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7
Breath out Breath in
145
Chest moves
inward
Diaphragm
Fig. 7.3 Schematic diagram of respiratory pumping mechanism
Diaphragm
moves down
Chest move
outward
is enhanced and expiration enhances pulmonary venous return. Note that the respiratory muscle and cardiac pumps have different rates; the interaction between the
three pumps is complex but when they coincide the venous return is enhanced. For
example, when we are startled, we contract our muscles and gasp ensuring cardiac
lling for ight or ght response.
7.3 Darcy’s Law
For those who understand electrical circuit theory, there is considerable similarity
between electrical circuit theory and haemodynamics. The physics of blood ow is
aided by the understanding of Ohm’s law (Eq.7.1). When considering uid dynamics instead of:
where ∆V=(V2−V1) is the potential difference (voltage) between two points, I is
the electric current and R is the electrical resistance, we substitute this formula with
Darcy’s law:
with ∆P=(P
−P1) the pressure difference, Q the volume ow rate and R the ow
2
resistance.
(7.1)
(7.2)
146
r
()
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S. Jansen et al.
Pressure is dened as the force exerted perpendicularly on the surface of an
object (expressed as force per area). However, blood pressure readings are not
reported in this way but expressed as millimetres of mercury (mmHg). Since the
pressure is changing over the course of the blood vessel, the pressure parameter
used is pressure difference (∆P), also called pressure gradient, which is the difference between the pressure at the beginning of the blood vessel (P1) and the pressure
at the end of the blood vessel (P2). Because the pressure at capillary level is practically zero, ∆P is effectively the blood pressure (P) in the arterial system. This means
that there is no pressure left from the heart to drive the venous system.
The resistance equation is [3]:
m
L
8
R
=
4
p
(7.3)
where μ is the blood viscosity, L is the blood vessel length and r is the inside radius
of the blood vessel.
Resistance is the main cause for maintenance of pressure in the blood vessel [4].
As seen in Darcy’s law, the greater the resistance the lower the ow rate. Resistance
has an inverse relation with the fourth power of r (inside radius of the blood vessel),
so that with small changes in r, the overall resistance will change dramatically.
Viscosity of a uid measures the interaction (tensile and shear stresses) between
owing particles, which corresponds to the inter-molecular friction of uid.
The organs and limbs can be thought of as resistors in parallel rather than in
series and this is important when it comes to the ability to regulate organ ow and
to cope with ischaemia and the contribution of collaterals [3]. The great vessels, like
the aorta, are without muscle and their walls are composed of collagen and elastin
bres. This allows them to behave as capacitors and store some of the energy in
systole to be released to power ow in diastole; this is important for vessels such as
the coronary arteries. Elastic arteries stiffen with age [5] which explains the loss of
phasicity of ow with aging. The ow to the heart and the vessel walls themselves
is dependent on diastolic ow which is dependent on capacitance. Capacitance
reduces when arteries stiffen, thus affecting blood ow to these organs.
In Eq.7.2, ∆P=P
−P2, where P1 is the pressure out of the heart and P2 the pres-
1
sure in the target organ or peripherally, an increase in P2 results in a decrease of ∆P
and blood ow. For example, if the peripheral arterial resistance increases, blood
ow will decrease. With constant resistance (R), if pressure goes up, ow goes up.
Flow can be regulated by varying resistance rather than by varying pressures.
Resistance is the sum of xed resistance and variable resistance. Clinically this
might equate to a xed stenosis plus variable peripheral resistance. For example, in healthy young people, the pressure may be constant during exercise
because the peripheral resistance falls to allow an increase in ow. In older
people, or in diseased and therefore stiffened arteries, the resistance variability
PPQ
ÞÞ
D Ris constant
1
(7.4)
L
d
d
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Vascular Haemodynamics
147
is reduced and exercise results in increased blood pressure as ow demand
increases. This can have pathological consequences. Once the resistance is xed
or at a steady state, any increase in ow is restricted and the pressure P2 (beyond
the stenosis) falls. This is the basis of the ankle/brachial index (ABI) and the
effect of exercise on ABI.
7.4 Poiseuille Flow
Suppose that you have a Newtonian uid owing in a steady, non-pulsatile manner
down a cylindrical, non-elastic pipe of length L and radius r. If the pipe (Fig.7.4) is
long enough (more than Le, as shown in Eqs.7.5 and 7.6) the ow will develop a
parabolic velocity prole (Fig.7.5), which is generally called a Poiseuille ow prole [6]. The ow takes its name from Jean Louis Poiseuille, a physician with training in physics and mathematics, who rst described this ow structure in 1846.
e
L
e
RFor laminar flow
» 006.,
» 44
e
1
6
RFor turbulent flow
.,
e
(7.5)
(7.6)
where Le is the entrance length, which is the length of conduit that the ow should
travel until the ow velocity prole becomes fully developed, d is the pipe inside
diameter and Re is Reynolds number (Eq.7.11).
If in Eq. 7.2, we substitute R with Eq. 7.3, the volumetric ow rate (Q) for
Poiseuille ow, i.e., the volume of uid owing along the tube per unit time, is given
by the formula (7.7), where P1–P2 is the pressure difference between the two ends
of the tube and μ is the viscosity of the uid.
L
e
Entrance Length
Developing Flow
Fully Developed Flow
Fig. 7.4 Flow development in a pipe
148
()
Pipe
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Fig. 7.5 Parabolic velocity
prole for fully developed
Poiseuille ow
S. Jansen et al.
a
P
1
L
4
PP r
Q
=
p
-
12
8
m
L
u
P
2
(7.7)
The four main characteristics of the Poiseuille (known as the Hagen-Poiseuille)
equation [7] are;
1. Constant uid viscosity;
2. Rigid and cylindrical pipe;
3. Length of the pipe greatly exceeds its diameter;
4. Laminar, steady and non-pulsatile uid ow.
The physics of the ow is nicely described by this equation. That is, ow is
driven by the pressure gradient in the tube or conversely, when there is ow in a
tube, then you must have a pressure gradient to drive the ow. Note that increasing
length will increase friction and consequently reduce blood ow. Patency, therefore,
such as in femoro-popliteal synthetic conduits, is related to the length of the conduit, as well as changes in cross-sectional area, kinking, bending and change in
diameter. Therefore, below knee bypass is more prone to occlusion than above knee.
This explains better patency in shorter bypass grafts. The internal surface and wall
properties of the prosthetic graft should also be considered.
7.5 Laplace’s Law ofWall Tension
Laplace’s law relates the tension in an arterial or venous wall to the pressure that the
elastic tube can apply to the material inside the tube [8]. To assist in understanding
this law we consider Fig.7.6. In this gure, w represents the thickness of the arterial
wall, r is the inner radius of the artery, P the inward pressure force due to the elastic
nature of the artery and T is tensional stress within the wall of the vessel: the tensional stress therefore points in a direction that is tangential to the vessel wall. Due
to mass conservation (see Sect. 7.12), the wall thins as the vessel expands.
w
r
rr
~,
TT
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Fig. 7.6 Cross section of
an artery showing the
various physical
components that make up
Laplace’s law
w
The formula for Laplace’s law is given by Eq.7.8;
P
T= ,
149
P
r
(7.8)
where it is usually assumed that the wall thickness, w is small relative to r. This law
tells us that the inward pressure that is exerted by the vessel wall on the blood is
directly proportional to the tensional stress in the wall and inversely proportional to
the radius of the wall. Thus the smaller the vessel, the larger the pressure it can
apply on the blood.
Large thin-walled vessels are low pressure vessels. Increasing the pressure distends the vessel and increases the vessel volume which is a characteristic property
of veins. For arteries to maintain pressure, the width of the wall must obviously be
greater, so large veins are thin-walled and arteries are thick-walled.
One consequence of this behaviour is that, to a certain extent, an artery acts like
a long cylindrical party balloon. When one attempts to blow up such a balloon, it is
quite difcult to do at the rst blow, however once the balloon reaches a particular
radius, it usually becomes much easier to expand the balloon. That is, you require
less pressure to increase the size of the balloon. This phenomenon is known as instability. If this happens to an artery, then we are dealing with an aneurysm and the
relatively constant blood pressure will keep on increasing the size of the aneurysm.
The radius of the artery at which this instability occurs is difcult to compute
accurately, but some fairly general arguments suggest that the following formula is
a good guide;
2
c
0
(7.9)
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TT
b
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T
P
T
a
Fig. 7.7 Cross sections of a small artery (a) and a very large artery (b) showing the stress distribution within the artery wall
where rc is the critical radius for the onset of the instability and r0 is the initial radius of
the artery. The median diameter of the aorta is 23mm and the thus aortic rupture is very
rare when less than 50mm in diameter, which is consistent with clinical data [9, 10].
This guide also directs us to consider that the ratio of the diameters is probably more
important than the absolute diameter and this should be taken into account when
assessing aneurysms in the smaller diameter vessels of women. How arterial wall instability arises is illustrated in Fig.7.7, where in Fig. 7.7a, we show the stress structure
within a small artery. Here the tensile stresses have a component in the radial direction,
where the letter T labels this component. In Fig.7.7b the aneurysm/balloon has become
very large, such that over a small segment of the wall the artery has hardly any curvature. This is an extreme case, but it does show that there is now no radial component to
the tensile stresses. In such a case, the aneurysm can expand freely for just about any
internal arterial pressure, hence rupture risk is increased at larger diameter.
7.6 Viscosity Behaviour
When we wish to describe the behaviour of a uid it is necessary to know something
about the frictional properties of the uid. Consider the schematic depiction of a
uid shown in Fig.7.8. In this gure, uid is owing from left to right along the x
direction. For the purposes of illustration, we assume that the speed of the uid, v,
is increasing with increasing height (i.e., increasing y). This means that elements of
uid are sliding past each other and so generating frictional shear stress 𝜏. In a
Newtonian uid, the frictional shear stress is proportional to the rate at which the
speed changes as a function of distance, where 𝜇 is the viscosity. Therefore, dv/dy
in Eq.7.10 corresponds to the shear rate [11, 12].
d
y
x
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Fig. 7.8 Elements of uid
slide past each other and
generate a frictional
shear stress
tm
=
dy
u
t
v
,
151
(7.10)
Non-Newtonian uids like blood do not obey Newton’s law of viscosity; so the
viscosity of blood varies with shear rate. As the blood ow increases (during exercise and peak systole) the ow shear rate increases and consequently the viscosity
of blood decreases. This is called “shear thinning”. This behaviour usually occurs
because at rest, a shear thinning uid typically has a tangled molecular structure,
which makes the uid relatively viscous. When force is applied, the molecules
become ordered, the uid viscosity decreases and the uid begins to ow more easily.
In Fig.7.9, we show the experimentally determined shear thinning behaviour of
blood, where the haematocrit value for the blood is 45%. These data show that for
high shear rates, which may occur in the large arteries of the body, the viscosity of
blood is about four times that of water (where the viscosity of water is approximately one centipoise (cP)). However, for lower shear rates, the viscosity of blood
can be over 100times that of water.
This change in viscosity is mostly due to the collective behaviour of red blood
cells. At low shear rates, red blood cells form aggregates where they stack one upon
another, somewhat like a cylindrical pile of coins or “rouleaux” (Fig.7.10). When
the shear rate increases, the rouleaux are broken down and tend to line up with the
ow of uid and blood viscosity decreases.
However, the viscosity of blood does not increase as blood travels from the arteries through the arterioles and into the capillaries but is approximately constant
throughout much of the body. The explanation for this is as follows;
Firstly, the viscosity of blood is dependent on the haematocrit. If the haematocrit
decreases, then the blood viscosity decreases. For example, in Fig.7.9, we show the
viscosity of blood as a function of shear rate for 45 and 0% haematocrit. Reducing
haematocrit essentially changes blood to a more Newtonian uid. Secondly,
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