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of blood ow through the artery and tends to increase as the artery becomes smaller
in diameter—provided that the volume ow rate and the viscosity are approximately
constant.
Atherosclerotic lesions form at specic areas where low and oscillatory endothelial shear stress occur. High risk plaques have a large lipid core, thin and inamed
brous cap and excessive expansive remodeling [19]. Wall shear stress may rupture
the established plaque. Plaque rupture and intraplaque haemorrhage are recognized
causes of cardiac events. Computational models from CT scans of carotid bifurcations with atherosclerotic plaques showed that stresses in the brous cap and around
the plaque shoulders affect plaque rupture risk, with higher stress and plaque rupture risk for thinner caps [20–23].
S. Jansen et al.
7.14 Forces onGraft Systems
The performance of stent grafts was found to be different to open repair with a
sutured replacement of the artery because of unsuspected inuences, as mentioned
above, that relate to sustained physical forces [1]. The openly-sutured prosthesis
binds the wall of the artery to the prosthesis with a transmural suture. The artery
may expand above or below the prosthesis. However, at the point of attachment the
artery wall is held to the xed diameter by the through-wall suture for as long as the
suture holds. Endoluminal grafts (ELG) to date do not bind the adventitia to the
prosthesis—they merely attach. The ELG must continue to act to bridge the gap
between the normal artery above and below until, if ever, the aneurysm’s cavity
shrinks right down. The diameters of the grafts used for the same Abdominal Aortic
Aneurysm (AAA) differ markedly between the open and ELG methods. The common diameters used for tube replacement surgically of infrarenal AAA is 18 or
20mm. The commonest diameter for a stent graft is 26 or 28mm and 30+mm is not
uncommon. Why is there such a discrepancy when the surgeon judges the diameter
for suitable t? This discrepancy is due to the different types of attachment of an
open graft and a stent graft. With the former, the aortic diameter is permanently
xed to the diameter of the graft in its pressurised state. The diameter of a crimped
vascular graft is, by denition, the minimum internal distance between the crimps
in the non-pressurised state. It is increased by approximately 10% when pressurized. With the ELG, a residual radial force is required for seal and the oversize
allowance must accommodate elasticity and compliance while maintaining the seal
between pulsations for the whole of the length of the sealing zone. With a stent graft
the long-term function and durability demands are different and greater [1].
Understanding the forces involved is basic to the design and use of new technology,
and the weaknesses that lead to aneurysmal disease provide an ongoing challenge
because it is progressive [1, 24].
A mistaken clinical impression is that the forces on a thoracic ELG should be
greater than those on an abdominal ELG.The ow and diameter of the thoracic
aorta are greater and the haemodynamic forces potentially much larger. However,
A
2
7 Vascular Haemodynamics
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163
because the diameter of the graft changes little, if at all, the downward displacement
force in the thoracic ELG is small as the resistance in the graft is low—except on the
curve of the aortic arch. The resistance of any graft that extends into the iliac vessels
is much greater because of the signicant change in diameter and high resistance
with the graft acting like a windsock or sea anchor [24]. An aorto-uni-iliac device
affords greater resistance than a bifurcated graft and detachment at the neck and
migration is a common problem due to high displacement forces. The force applied
to the thoracic graft is on the curve and centrifugal forces apply. Since every action
has an equal and opposite reaction (Newton’s third law), one must ask where is the
reaction? The reaction is to pull the graft out from the top and the bottom almost
equally. When stent grafts were rst used in the thorax, unexpected upward migration of the distal end emerged as the problem, especially when there was a signicant curve on the graft. For the same reason, this ‘lift out’ may also be seen from the
iliac arteries when the graft xation is weak because of ectasia and/or short length
of distal attachment. Type 1B endoleak can be more dangerous than Type 1A if this
factor is ignored and it is important to have an understanding of the possible forces
that may be exerted on a graft.
To illustrate the steps used in determining the forces on a graft system, via analytic equations, we consider the steady ow of blood through a bent pipe (Fig.7.18).
In this gure, the proximal inlet entrance is labelled (1) and the distal exit by (2). D1,
A1 and D2, A2 are the diameters and cross-sectional areas, respectively, of the graft
at the points 1 and 2. The blood ow forces (A1 and A2) on the grafted system are at
angles of θ1 and θ 2 to the vertical, so can only be calculated when the inlet and
outlet force components are in the same plane. Similarly p and v refer to the pressures and velocities at these points. Rx and Ry are the x and y components of the
restoring force. The external pressure on the graft system is denoted by Pex.
Fig. 7.18 The
characteristic velocity,
pressure, area and force
vectors required to
compute the restraining
forces on a bent, singletube graft system
1
p
1
1
q
1
u
1
D
1
R
y
R
x
p
ex
p
2
D
2
u
2
A
2
q
2
164
vA vA
22
=
()
()
+-
11
ns
rq
Av
()
()
--
sc
rq
Zh
22
ZZ
12
è
ø
+-
()
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S. Jansen et al.
In our analysis, we assume a steady-state, i.e., non-pulsatile, ow. We do this as
it gives us a basic idea of how the system is behaving. The rst equation is the
steady-state mass conservation equation, which we rewrite in the form:
11
(7.23)
One should note that v1 and v2 are average ow speeds, where the average is taken
over the areas of A1 and A2 respectively.
The next analysis tool at our disposal is the momentum conservation equation,
which can be expressed in the form;
RPPA PPA vA vA
=-
xexex
2221 11 22221
--
sinsin si
qqrq
2
in
(7.24)
and
R PPA PPAv
=- -
yexex
1112 221212222
--
coscos co
qqrq
A
os
(7.25)
22
where in these formulae, we have ignored the weight of the graft and the weight of
blood in the graft. These terms are easily included into the equations, if required.
Energy is the nal conserved quantity that we can use in our analysis. The energy
conservation equation has the form:
Pv
111
++=+ ++a,
g
g
Pv
Z
1
g
2
a
222
g
2
L
2
(7.26)
where g is the gravitational acceleration, γ=ρg is the weight density of blood, z1 and
z2 are the vertical heights of the proximal and distal ends of the graft, respectively,
and hL is the ‘head loss’ in the pipe, i.e., the amount of pressure or energy that is lost
due to frictional viscous effects as the uid travels through the pipe. Head loss is
usually given by the equation;
where KL is a constant, the value of which is usually dependent on the shape,
length and diameter of the pipe. The coefcients α1 and α2 are kinetic energy
correction factors that have different values depending on the type of ow. For
example, for uniform ow α=1, turbulent ow has α ≈ 1, and laminar ow
gives α=2.
By combining Eqs.7.23, 7.26 and 7.27, one obtains
PP
=+ -+
hK
=
LL
2
æ
g
v
1
ç
aa g
12
ç
2
g
K
()
2
v
2
,
2
g
2
ö
æ
ö
A
1
÷
ç
÷
L21
÷
A
2
è
ø
(7.27)
(7.28)
u
1
7
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Vascular Haemodynamics
165
So, by using Eqs.7.23 and 7.28, we can express p2 and v2 in terms of quantities at
the entrance of the graft. This then allows us to compute the restraining forces on the
graft system by then using Eqs.7.24 and 7.25.
7.14.1 Case 1: TheCylindrical Graft
For this case (Fig.7.19), the inlet and the outlet areas are the same, so, by Eq.7.23,
the inlet and outlet ow speeds are also equal. The angles θ1 and θ2 are equal and
have a value of 90°. The inlet and outlet pressures are not equal due to the frictional,
shear interaction between the blood and the graft (i.e., the head loss as given by
Eq.7.27). This frictional interaction causes the outlet pressure, p2, to be less than the
inlet pressure,P1. This is called the pressure drop.
In considering the restraint forces on the graft, we know there are no vertical
forces generated by blood owing through a horizontal graft in this case. The horizontal force on the graft is quite small, therefore one can conclude that straight,
cylindrical grafts only feel a relatively small drag force in the direction of the ow.
7.14.2 Case 2: TheWindsock Graft
Suppose now we consider a graft in the shape of a wind-sock, such as in Fig.7.20.
For this case, the inlet area is now larger than and the outlet area, so, by Eq.7.23,
the outlet ow speed is greater than the inlet ow speed as given by;
Fig. 7.19 Cylindrical graft
R
1
A
1
p
y
R
x
Fig. 7.20 A stent graft in
the shape of a wind-sock
R
y
u
1
A
1
p
1
R
x
u
2
A
2
p
2
u
2
A
2
p
2
166
p
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æ
ö
A
v
1
=
2
v
ç
÷
1
A
2
è
ø
S. Jansen et al.
(7.29)
As in the previous case, the angles θ1 and θ2 are equal and have a value of 90°
and the inlet and outlet pressures are not equal due to the frictional, shear interaction between the blood and the graft. The restraint forces on this graft act to
create a windsock which has a much larger drag force than a cylindrical graft.
7.14.3 Case 3: TheCurved Graft (Fig.7.21)
As with the cylindrical graft, the inlet and the outlet areas are the same, so, by
Eq.7.23, the inlet and outlet ow speeds are also equal. Due to the symmetry of the
situation, the vertical restraint force is zero, the horizontal restraint force however is
much greater. A tortuous stent graft which also narrows is subjected to both axial
and radial forces.
7.14.4 Case 4: TheSymmetrical Bifurcated Graft
Suppose that we consider a symmetric bifurcated graft, such as shown in Fig.7.22,
where the two outlet distal legs of the graft are at an angle α to the horizontal, the
two distal ends are equal and gravity is ignored. The proximal end of the graft is
labelled by the number 1, the symmetric distal ends by 2 and 3. We also know to
satisfy mass conservation that the ow in has to equal the sum of the outows.
By applying Bernoulli’s equation and the mass conservation equations we can
calculate that the horizontal restraint force is strongly dependent on inlet area, pressure and on the bifurcation angle (especially >15°). However, the blood inlet
Fig. 7.21 Curved graft
1
u
1
p
2
u
2
A
1
A
2
R
y
R
x
A
3
7
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Vascular Haemodynamics
167
Fig. 7.22 Symmetric
bifurcated graft
p
1
υ
1
A
1
R
y
R
x
α
2
υ
2
p
2
p
3
υ
3
A
velocity or ow rate has negligible effect on the horizontal restraint force [25–27].
Naturally, a steady-state assumption is questionable, since pulsatile ow occurs in
the human body. However, it was shown experimentally that a steady-state analytical model can be used, with variable pressure and ow rate inputs, to predict forces
on a symmetric, bifurcated graft in pulsatile ow with reasonable approximation
within design limits [26, 27].
7.15 Conclusion
Understanding the physics of the vascular system in health and disease inuences
vascular management. This is a rich eld for research. Further clues to atherogenesis may lie in the differences in the uid dynamics and stresses applied to the arterial
system particular to branch points. Computational modelling will be of increasing
importance to our understanding of vascular haemodynamics, the behaviour of aortic dissection, the impact of devices on arterial disease and prediction of injury and
rupture risk.
References
1. Lawrence-Brown MM, Semmens JB, Hartley DE, Mun RP, van Schie G, Goodman MA, etal.
How is durability related to patient selection and graft design with endoluminal grafting for
abdominal aortic aneurysm. In: Durability of Vascular and Endovascular Surgery. London:
WB Saunders; 1999. p.375–85.
2. Harris PL, Buth J, Mialhe C, Myhre HO, Norgren L.The need for clinical trials of endovascu-
lar abdominal aortic aneurysm stent-graft repair: the EUROSTAR project. Los Angeles: SAGE
Publications; 1997.
3. Gaze DC. The cardiovascular system—physiology, diagnostic and clinical implications.
London: InTech; 2012.
4. Ku DN.Blood ow in arteries. Ann Rev Fluid Mech. 1997;29:399–434.
5. Greenwald S.Ageing of the conduit arteries. J Pathol. 2007;211:157–72.
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6. Buja LM, Butany J.Cardiovascular pathology. 4th ed. Amsterdam: Elsevier; 2016.
7. Yunus AC, Cimbala JM. Fluid mechanics fundamentals and applications, vol. 185201.
International Edition. NewYork: McGraw Hill Publication; 2006.
8. Taylor M.An introduction to some recent developments in arterial haemodynamics. Aust Ann
Med. 1966;15:71–86.
9. Greenhalgh R, Brady A, Brown L, Forbes J, Fowkes F.Mortality results for randomised con-
trolled trial of early elective surgery or ultrasonographic surveillance for small abdominal
aortic aneurysms. The UK Small Aneurysm Trial Participants. Lancet. 1998;352:1649–55.
10. Lawrence-Brown M, Norman P, Jamrozik K, Semmens J, Donnelly N, Spencer C, et al.
Initial results of the Western Australian ultrasound screening project for aneurysm of the
abdominal aorta: relevance for endoluminal treatment of aneurysm disease. Cardiovasc Surg.
2001;9:234–40.
11. Morris PD, Narracott A, von Tengg-Kobligk H, Soto DAS, Hsiao S, Lungu A, et al.
Computational uid dynamics modelling in cardiovascular medicine. Heart. 2016;102:18–28.
12. Caro CG, Pedley T, Schroter R. The mechanics of the circulation. Cambridge: Cambridge
University Press; 2012.
13. Popel AS, Johnson PC. Microcirculation and hemorheology. Annu Rev Fluid Mech.
2005;37:43–69.
14. Pries A, Secomb T, Gaehtgens P.Biophysical aspects of blood ow in the microvasculature.
Cardiovas Res. 1996;32:654–67.
15. Westerhof N, Stergiopulos N, Noble MI. Snapshots of hemodynamics: an aid for clinical
research and graduate education. Berlin: Springer Science & Business Media; 2010.
16. Barrett KE, Barman SM, Boitano S, Brooks H.Ganong’s review of medical physiology, vol.
23. NewYork: McGraw-Hill Medical; 2009.
17. Womersley JR.Method for the calculation of velocity, rate of ow and viscous drag in arteries
when the pressure gradient is known. J Physiol. 1955;127:553–63.
18. Glagov S, Zarins C, Giddens DP, Ku DN. Hemodynamics and atherosclerosis. Arch Pathol
Lab Med. 1988;112:1018–31.
19. Chatzizisis YS, Coskun AU, Jonas M, Edelman ER, Feldman CL, Stone PH.Role of endo-
thelial shear stress in the natural history of coronary atherosclerosis and vascular remodeling:
molecular, cellular, and vascular behavior. J Am Coll Cardiol. 2007;49:2379–93.
20. Gao H, Long Q.Effects of varied lipid core volume and brous cap thickness on stress distri-
bution in carotid arterial plaques. J Biomech. 2008;41:3053–9.
21. Kock SA, Nygaard JV, Eldrup N, Fründ E-T, Klærke A, Paaske WP, et al. Mechanical
stresses in carotid plaques using MRI-based uid–structure interaction models. J Biomech.
2008;41:1651–8.
22. Tang D, Yang C, Mondal S, Liu F, Canton G, Hatsukami TS, etal. A negative correlation
between human carotid atherosclerotic plaque progression and plaque wall stress: in vivo
MRI-based 2D/3D FSI models. J Biomech. 2008;41:727–36.
23. Gao H, Long Q, Graves M, Gillard JH, Li Z-Y. Carotid arterial plaque stress analysis using
uid–structure interactive simulation based on in-vivo magnetic resonance images of four
patients. J Biomech. 2009;42:1416–23.
24. Liffman K, Lawrence-Brown MM, Semmens JB, Bui A, Rudman M, Hartley DE.Analytical
modeling and numerical simulation of forces in an endoluminal graft. J Endovasc Ther.
2001;8:358–71.
25. Šutalo ID, Liffman K, Lawrence-Brown MM, Semmens JB. Experimental force measure-
ments on a bifurcated endoluminal stent graft model: comparison with theory. Vascular.
2005;13:98–106.
26. Liffman K, Šutalo ID, Bui A, Lawrence-Brown MM, Semmens JB.Experimental measure-
ment and mathematical modeling of pulsatile forces on a symmetric, Bifurcated Endoluminal
Stent Graft Model. Vascular. 2009;17:201–9.
S. Jansen et al.
Vascular Haemodynamics
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27. Zhou S, How T, Black R, Vallabhaneni S, McWilliams R, Brennan J.Measurement of pulsatile
haemodynamic forces in a model of a bifurcated stent graft for abdominal aortic aneurysm
repair. Proc Inst Mech Eng H. 2008;222:543–9.
28. Chien S, Usami S, Dellenback RJ, Gregersen MI.Shear-dependent deformation of erythro-
cytes in rheology of human blood. Am J Physiol. 1970;219:136–42.
29. Barbee JH, Cokelet GR.The fahraeus effect. Microvasc Res. 1971;3:6–16.
169
Further Reading
Barbee JH, Cokelet GR.The Fahraeus effect. Microvasc Res. 1971;3:6–16.
Liffman K, Šutalo ID, Bui A, Lawrence-Brown MM, Semmens JB.Experimental measurement
and mathematical modeling of pulsatile forces on a symmetric, bifurcated endoluminal stent
graft model. Vascular. 2009;17:201–9.
Morris PD, Narracott A, von Tengg-Kobligk H, Soto DAS, Hsiao S, Lungu A, etal. Computational
uid dynamics modelling in cardiovascular medicine. Heart. 2016;102:18–28.
Yunus AC, Cimbala JM.Fluid mechanics fundamentals and applications, vol. 185201. International
Edition. NewYork: McGraw Hill Publication; 2006.
Chapter 8
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Computational Fluid Dynamics
intheArterial System: Implications
forVascular Disease andTreatment
SiamakMishani, ShirleyJansen, MichaelLawrence-Brown,
ChristopherLagat, andBrianEvans
Key Learning Points
•
Pulsatile blood haemodynamics and irregular geometry of arterial wall play an
important role in the formation of vascular disease.
• Due to nonhomogeneous and anisotropic structural behaviour of the blood vessel
and non-Newtonian shear thinning properties of blood ow, the response of the
arterial wall to blood pressure is complex.
• Multidisciplinary numerical modelling (CFD, FEA and FSI methods) can simu-
late pulsatile blood ow in an artery to calculate force transferred from blood
ow into blood vessel and determine the resultant stresses in the arterial layers
(intima, media, and adventitia).
• The accuracy of the numerical model is inuenced by geometries and meshing
resolution, initial and boundary conditions, and uid-solid coupling method.
8.1 Introduction
The arterial tree is a branching tube composed of three nonhomogeneous layers
subject to pulsatile blood pressure. Biomechanical analyses of the strain and stress
in, and between, the layers can provide a good understanding of the relationships
between blood ow and the arterial wall and therefore, of arterial disease. Pulsatile
S. Mishani · C. Lagat · B. Evans
Faculty of Science and Engineering, WA School of Mines: MECE, Curtin University,
Perth, WA, Australia
S. Jansen (
Heart and Vascular Research Institute, Harry Perkins Institute of Medical Research, Medical
School, Curtin University, Perth, WA, Australia
M. Lawrence-Brown
Faculty of Health Sciences, Curtin University, Perth, WA, Australia
R. Fitridge (ed.), Mechanisms of Vascular Disease,
https://doi.org/10.1007/978-3-030-43683-4_8
*)
171© Springer Nature Switzerland AG 2020
172
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blood haemodynamics and arterial wall geometry, which is irregular, play an important role in the formation of vascular disease especially in areas of complex blood
ow [1, 2]. Disturbance of blood ow occurs in regions where arteries branch or
vary in cross-sectional area [3, 4]. The response of the arterial wall to blood pressure
is also complex, due to there being a nonlinear, time-dependent structural response
to blood pressure, and nonhomogeneous and anisotropic structural behaviour of the
blood vessel wall.
Blood pressure affects the arterial wall, and in turn, the elastic deformation of the
arterial wall inuences blood ow. The intima can transduce displacements and
pressure from the blood ow to the arterial wall and vice versa. This phenomenon
can be analysed using three numerical approaches based on coupling uid ow
dynamics with structural analysis:
1. CFD (Computational Fluid Dynamics) uses numerical methods to simulate and
analyse blood ow behaviour.
2. FEA (Finite Element Analysis) uses numerical methods to predict stress and
strain in the arterial wall.
3. FSI (Fluid-Solid Interaction) is modelling the interaction of the elastic blood
vessel with blood ow and pressure.
These three approaches used together can help us understand the behaviour of
blood ow and deformation of the arterial wall to investigate arterial disease and the
behaviour of prosthetic devices. This chapter explains the modelling techniques for
characterizing stress and strain within different layers of the arterial wall and blood
ow patterns in an artery.
S. Mishani et al.
8.2 Terminology
We include the following glossary to explain terms used in this chapter;
• Analytical method: Solving the problem using ordinary analysis which can be
obtained with pencil and paper.
• Numerical method: Using mathematical methods to solve complicated prob-
lems using computer programming.
• Governing equation: The mathematical statements of the three fundamental
equations (continuity, momentum and energy) which describe the physical prin-
ciples of uid dynamics.
• Anisotropic material: A substance with different mechanical properties in dif-
ferent directions.
• Validation in FEA: Using equations to quantify the uncertainty of the results
(get the right physics).
• Verication in FEA: Using equations to quantify the errors of the results (get
the right mathematics).
• Transient ow: An unsteady ow with position and time dependent velocity and
pressure.
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