Добавил:
Sekretar
kiopkiopkiop18@yandex.ru
t.me/Prokururor I Вовсе не секретарь, но почту проверяю
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Предмет:
Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_3774_Библиотеки_им_академика_М_И_Перельмана
.pdf
152
==
r
vvDD
1000
1000
Shear Rate (s
)
Viscosity (cP)
S. Jansen et al.
https://t.me/medicina_free
H = 45%
H = 0%
100
10
1
0.01 0.1 1 10 100
-1
Fig. 7.9 Blood viscosity as a function of shear rate for 0 and 45% haematocrit [28]
haematocrit is dependent on the diameter of the blood vessel. As the blood vessel
decreases in diameter, the haematocrit also decreases (Fig.7.11). This effect occurs
because the blood cells tend to move away from the vessel walls and travel where
the ow velocity is a maximum in the centre of the blood vessel. This behaviour is
known as the Fahraeus Effect and it has been shown to occur in tubes with a diameter as small as 29μm [13, 14] in the context of a red blood cell diameter of 8μm.
The combination of these two effects counteracts the effects of shear thinning to
maintain constant viscosity of blood throughout the body. It is important to understand how these properties affect shear stress between blood and vessel walls—or
more relevantly between blood and atheroma.
7.7 Reynolds Number
The Reynolds number is an important dimensionless parameter (it has no units)
used in uid mechanics to measure whether the uid ow pattern in a conduit is
laminar, transient or turbulent. The Reynolds number is related to blood density,
viscosity, velocity, and the diameter of the blood vessel. Reynolds number can be
dened as the ratio of inertia force (ρvD) to viscosity or friction force (μ). The
Reynolds number for ow in a pipe is given by:
Re ,
m
v
(7.11)
0
45
H
Tube Diameter (µm)
7
https://t.me/medicina_free
Vascular Haemodynamics
Fig. 7.10 Rouleaux blood
cell formation seen in
capillaries
153
40
35
30
25
20
15
10
050 100 150200 25
Fig. 7.11 Hematocrit as a function of tube diameter. The initial hematocrit value for each line is
shown in the inset box [29]
H (Hinitial = 45%)
H (Hinitial = 20%)
154
https://t.me/medicina_free
where:
ρ = Fluid density (kg/m3).
v = Flow velocity over cross-section area (m/s).
D = Hydraulic diameter (m).
μ = Dynamic viscosity (N.s/m2).
mm
v=
from laminar to turbulent at a Reynolds number of approximately 2000, which is
known as the critical Reynolds number. For less than the critical number, the viscous effects are dominant and ow is laminar. For numbers near to 2000, ow is
called “transient”, which is neither laminar nor turbulent [15]. Laminar or turbulent
ow can be benecial or detrimental. It is possible the properties in the composite
nature of blood work in this transitional range to prevent turbulent ow. This would
be important during high cardiac output especially in children and athletes.
ascending aorta of diameter, D=3 cm, peak blood ow speed u=60 cm/s, blood
density ρ = 1.06 g/cc and blood viscosity μ = 0.0036 Pa.s. These values give
Re= 5300 and turbulent ow would be expected but there is no evidence for this
happening on ultrasound.
Reynolds number by its inertia. When uid has turbulent ow such as in an arterial
stenosis, an audible bruit is heard called Korotkoff sounds [16]. This is the result of
chaotic ow at high velocity which transforms energy to noise. Flow is inefcient
and may be disruptive as in a carotid stenosis, where turbulence can increase the risk
of dislodging material from plaque as embolus. Turbulent ow is less efcient relative to laminar ow. This means that more energy or a greater pressure drop is
required to drive turbulent ow compared to laminar ow.
position may act to discourage the formation of turbulent ow. The bi-concave
shape of the red cell could be crucial in affecting cell-cell interactions so that ow
tends to remain laminar. This shape may also enhance the efciency of blood ow
by vortex shedding in addition to increasing surface area for oxygen delivery.
= Kinematic viscosity (m2/s).
rr
In an artery, if the ow is assumed to be Newtonian then it is predicted to change
For example, let us calculate Re the peak value of Reynolds number for an
Fluid owing in a laminar fashion is dominated by the viscosity and at a high
It is interesting to speculate that the particulate nature of blood and plasma com-
S. Jansen et al.
7.8 Womersley Number
Now we must consider the effect of pulsatile versus constant ow. The Womersley
number(W) is a dimensionless parameter in haemodynamic and biouid mechanics
which typically characterises pulsatile ow within an artery. It denotes the ratio of
inertial forces to viscous forces. It is named after John R.Womersley (1907–1958)
[17], for his work on blood ow behavior in arteries. The Womersley number of
blood ow can be measured by:
pr
convectiveinertia force
transient inertiaforce
7 Vascular Haemodynamics
https://t.me/medicina_free
155
Wr
f=2
m
(7.12)
where r is the radius of the artery, f is the pulsatile frequency (heart bpm=60times
f(Hz)), ρ=1060kg/m3 is blood uid density, and μ=0.00345 Pascal-Second (Pa.s)
is the viscosity.
If 0<W<1 the ow is governed by viscosity effect, and if W is much greater
than 1, the ow is governed by transient effect (unsteady ow with time-dependent
velocity and pressure). When W is low, the velocity proles are parabolic in shape.
For W more than 10, the unsteady inertial force governs the ow motion.
From here one can see that blood ow steadies with change in diameter and ow
divergence. The arterial system uses this to enable ow during diastole to organs
that do not allow perfusion during systole such as the heart and great vessels themselves. This is enabled through the capacitance of the great vessels because of their
elasticity. The Reynolds number (Eq.7.13) and the Womersley number (Eq.7.14)
are nondimensional parameters that investigate the pulsatile ow pattern. They are
useful in solving haemodynamic problems.
Re =
viscous friction force
W
=
viscous friction force
(7.13)
(7.14)
Utilizing Womersley and Reynolds numbers may provide insights into detrimental
to-and-fro and turbulent ow patterns. For example, does connecting a bypass graft
around a diseased native vessel that is still delivering blood cause the two parallel
blood ows to compete, with the potential for thrombosis in one channel? An increase
in Womersley and Reynolds numbers indicate signicant increase in the complexity
of the blood ow pattern and changes in the vortex ow position. Therefore, understanding quality of ow is crucial for good operative planning and choice of conduit.
Some typical values for the Womersley number are shown in Fig.7.12:
7.9 Bernoulli’s Equation
Johann Bernoulli (1667–1748) was a professor in Basel and taught physics, anatomy and physiology. His understanding lies at the heart of vascular physics and
relates pressure to motion and energy. For a uid that has no viscosity, one can write;
2
v
Pgyconstant++=
rr
2
(7.15)
156
15
Main Pulmonar
https://t.me/medicina_free
14
13
12
11
10
9
8
7
6
5
Womerssley Number
4
3
2
1
0
15
13.2
11.5
S. Jansen et al.
8
4.4
3.5
4
0.005 0.035
0.04
y Artery
Carotid Artery
Ascending Aorta
Descending Aorta
Fig. 7.12 Blood vessel type vs. typical Womersley number
Abdominal Aorta
Femoral Artery
Typical Blood vessel
Coronary Artery
Arterioles
Capillaries
where P is the pressure, ρ the mass density of the liquid, v the speed of the uid, g
the gravitational acceleration, and y the height. In other words, the Bernoulli equa-
tion states that the pressure plus the kinetic energy per unit volume
v
r
potential energy per unit volume, ρgy, is a constant at any point along the blood
vessel. The Bernoulli equation takes into account the effect of gravity as well as
resistance, ow and pressure. This is the basis of Buerger’s test, the Trendelenberg
position and Roos test.
It should be understood that Eq.7.15 is an approximation, as it ignores the loss
of energy due to shearing friction between the owing blood and the walls of the
artery. Even so, it does provide us with an intuitive understanding of the physics of
the arterial/venous system. For example, suppose we wish to measure the blood
pressure of a person. Typically one places an external cuff around the upper arm
which is approximately at the same level as the heart and so the pressure will not be
affected by any difference in height. To measure the systolic pressure, the cuff pressure is increased until all blood ow ceases. From Eq.7.15 we know that this “cutoff” pressure is the maximum pressure in the artery. The pressure in the external
cuff is then reduced until the ow is a maximum. This is the diastolic pressure.
In practice, the arterial system has two sources of potential energy to drive the
blood forward. The rst is blood pressure and this is transformed into the kinetic
energy of ow during the period between systole and diastole, and the second is
stored energy in the wall of the artery, called capacitance. Consider what might happen when the kinetic energy meets a resistive obstacle—some energy is dissipated
as heat as in electrical circuit theory and some is stored for use in diastole for
onward ow in the period of heart lling by the elasticity of the large blood vessels
Venules
2
, plus the
2
L
Vascular Haemodynamics
https://t.me/medicina_free
7
157
acting as capacitors. However, some energy is used up due to repetitive alterations
in forward pressure and resistive back-pressure from pulsatile ow. This phenomenon is called water hammer. The injury and healing cycle effect of these water hammers on atherogenesis and aneurysm behaviour at stress points has yet to be fully
determined.
7.10 Young’s Modulus andPulsatile Flow
Blood ows through the arteries in a pulsatile fashion. Arteries are semi-elastic
tubes and expand and contract as the pulse of blood ows along them. The speed, c,
at which blood ows along an artery is determined by the speed that a pulse of uid
can travel along an elastic tube. This speed is given, approximately, by the MoenKorteweg formula;
Eh
c
»
where E is Young’s modulus for the wall of the artery, h is the thickness of the
artery, d is the inner diameter of the artery and ρ is the density of blood. A schematic
depiction of how a pulsatile wave propagates along an artery is given in Fig.7.13.
As can be seen from Eq.7.16, the speed at which blood travels along an artery is
partially dependent on the Young’s Modulus of the arterial wall. To illustrate the
denition of Young’s Modulus it is useful to consider Fig.7.14, where a block of
material is being stretched due to an applied force on one end of the block.
The block has a natural length denoted by L, when a force F is applied to one
side of the block then the length of the block increases by ΔL. This change in length
is known as a strain, ε, and it is dened by the equation:
,
r
d
(7.16)
Fig. 7.13 An exaggerated,
schematic view of blood
ow in an artery
DL
e
=
,
(7.17)
Eh
~
c
~
rd
r
h
d
158
F
A
s
22
AAuA uA==
L
D
https://t.me/medicina_free
Fig. 7.14 A block of
material with a length, L,
and side area A is subject
to a force F. The applied
force stretches the block a
distance ΔL
L
S. Jansen et al.
A
The stress (σ) that the force applies to the block of material has the denition
s
=
,
Young’s Modulus is dened as the stress over the strain, i.e.,
F
(7.18)
E =
e
(7.19)
Young’s modulus is a measure of how easy it is to stretch and compress a material.
Thomas Young (1773–1829) was a medical physician who made signicant contributions to elds of Physics (through his experiments which demonstrated the wavelike nature of light), medicine (with his studies of blood ow), and structural
mechanics (e.g., Young’s Modulus). Surprisingly however, despite being well aware
of the elastic nature of arteries, he does not appear to have used Young’s Modulus to
describe their properties.
One consequence of ageing is increasing stiffness in the arteries. This means that
the Young’s modulus increases and this, as a consequence of Eq.7.16, increases the
speed of pulsatile ow within the arterial system.
7.11 Mass Conservation
In Fig.7.15, we view a schematic depiction of an artery that is changing in shape as
one travels along the artery. The blood ows in at one end with a speed u1. The area
at the inlet of the artery is given by A1. In its simplest form, the mass conservation
equation provides us with the relationship between the quantities at the proximal
and distal ends of the artery:
vv
11 22 11
,
(7.20)
7 Vascular Haemodynamics
https://t.me/medicina_free
Fig. 7.15 A change in the
diameter of an artery leads
to a change in the blood
ow speed
159
u
1
A
1
A
1
u
1
p
1
A
2
p
u
2
2
u
2
A
2
Fig. 7.16 Flow through a blood vessel with a stenosis
Here v2 and A2 are the outlet ow speed and area, respectively. In plain English,
Eq.7.20 is another way of saying “what goes in must come out”.
We can see from Eq.7.20 that if an artery becomes narrower, i.e., A2 becomes
smaller, then the ow speed, v
, increases. This occurs because the mass ow cannot
2
be created or destroyed and so if the tube becomes narrower, then the ow rate has
to increase. Stenosis in a blood vessel wall may be caused by atherosclerosis or
restenosis following an intervention (Fig.7.16).
7.12 Arterial Dissection, Collateral Circulation
andCompeting Flows
Up to this point we have essentially discussed ow in series. Much of the normal
circulation in the human and some pathological ow occurs in parallel. Examples of
parallel ow include the collateral circulation in each segment of the body; the profunda system in the thigh, the geniculate system around the knee and the tibial
system in the leg. Another good example is the carotid and vertebral systems
160
11
RR
totaln
https://t.me/medicina_free
S. Jansen et al.
combining to form the cerebral circulation. In parallel circulation, the pressure at
the separation of the two systems is theoretically the same for each, and the pressure
at the re-union is also the same for each. The proportion of ongoing ow from the
two systems is determined by the resistance of each system. These two therefore
compete for the proportion of on-ow. This works well to direct or redirect the ow
to different target tissues. The body may select priorities for ow, for example, the
brain and heart in shock or the muscles during exercise. The branches of the great
vessels and arteries to the tissues are resistance vessels and they have muscular
walls for this purpose.
The formula for resistors in parallel circuits is:
11
=+ +¼+ ,
RR
12
(7.21)
where n is the number of parallel circuits.
These circuits also provide alternative channels should the dynamics change
due to injury or disease. Not all parallel circuits are benecial. Detrimental competing ows may occur with articially created channels, for example, aortobifemoral bypass, when one iliac system is normal and the other occluded. The
competing ows on the normal side predispose for either that limb of the graft or
part of the iliac system on that side to occlude. Similarly, with femoro-popliteal
bypass after long-standing supercial femoral artery occlusion when the profunda collateral ow has been well developed. If it is desirable to maintain
patency of a vessel is may be better to do this early for longer term patency,
before the collaterals develop. This is obviously debatable in clinical practice, but
worthy of consideration.
In aortic dissection, the outow from the false lumen is met with greater resistance than the outow from the true lumen. The ows compete at fenestrations or
where the intima has been torn off the origin of a branch vessel. The pressure is
higher in the false lumen at any time in the cardiac cycle other than peak systole.
Figure7.17 shows the trace from true and false lumens of a dissected aorta. Note the
systolic pressure is the same in each lumen at 138mmHg. The diastolic pressure is
higher in the false lumen at 93mmHg compared to the diastolic in the true lumen of
82mmHg. The area under the curve is the same and so the pulse wave in the false
lumen is wider. The mean pressure in the false lumen is higher at 109mmHg than
the true lumen, where the mean is 91mmHg.
This means that the false lumen is generally the larger of the two and is more
likely to dilate. Flow of contrast injected into the true lumen is not seen to ow out
to the false lumen through the holes in the membrane unless the pressure of the
injection and the pressure of the lumen together exceed the pressure of the false
lumen. The membrane that is the remnant of the intima oscillates as the pressure
ratio between the true and false lumen changes during the cardiac cycle. This
dynamic also applies for a Type 1 endoleak into the residual sac of an aortic aneurysm treated by an endovascular graft.
Q
r
False Lumen Pressure 138 / 93 (109) mmHg
Tr ue Lumen Pressure 139 / 82 (91) mmHg
7 Vascular Haemodynamics
https://t.me/medicina_free
Fig. 7.17 Pressure readings from the true and false lumens of a dissected abdominal aorta (courtesy of Dr. John Anderson)
161
7.13 Shear Stress andPressure
All vascular clinicians are familiar with the ultimate shearing force injury of high
velocity impact when the mobile arch of the aorta and heart continue to move forward while the descending aorta is held by the intercostal and posterior mediastinum to the vertebral bodies. What of subtle persistent long-term shear stresses and
the relationship with the greatest risk factor for arterial disease—age? There are
known common sites for occlusive atheromatous plaques e.g. the carotid bifurcation, aortic bifurcation, origins of branches of the aorta and coronary arteries and
shear stress points such as the adductor canal.
Atheroma is an arterial lesion. It is only seen in veins subject to long term pulsatile pressure when they are said to be “arterialised”. Pressure and pulsatility are the
forces involved here and the biochemical and biological responses act as accelerators and decelerators.
Shear stress on an arterial wall, τw, due to Poiseuille uid ow is given by the
formula; [18]
m
4
t
=
w
3
p
(7.22)
where r is the radius of the artery and Q is the volume ow rate of blood through the
artery. From this formula, it can be seen that shear stress increases with the increase
Соседние файлы в папке Библиотека им академика М.И. Перельмана