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A numerical model of pulsatile blood flow in compliant arteries of a

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A numerical model of pulsatile blood
ﬂ
ow in compliant arteries of atruncated vascular system
☆
Kamil Kahveci
a
, Bryan R. Becker
b,
⁎
a
Faculty of Engineering, Trakya University, Edirne, 22180 Turkey
b
University of Missouri
—
Kansas City, 350C Flarsheim Hall, 5100 Rockhill Road, Kansas City, MO, 64110-2499 USA
a b s t r a c ta r t i c l e i n f o
Available online 26 June 2015
Keywords:
Blood
ﬂ
owBifurcationArterial complianceTaperingArterial deformationBoundary model
Three dimensional blood
ﬂ
ow in compliant, tapering vessels of a truncated vascular system model containingtwo levels of bifurcation was investigated numerically using a commercially available
ﬁ
nite element analysisand simulation software. Although the branching pattern and the geometry of the human vascular systemare complex,they canbespeci
ﬁ
edforsmall arteriesusing Murray'shypothesis that thestructureof the vascularsystem obeys the principle of minimum work. Accordingly, in the current vascular system model, the parent/daughter diameter ratios and the angles of bifurcation were speci
ﬁ
ed according to Murray's law. Another geo-metrical parameter, the ratio of blood vessel length to diameter, was determined according to data found inthe literature. The vascular system model also includes a 5 mm thick layer of tissue surrounding the vessels.This tissue layer helps to resist artery deformation during the cardiac cycle. Experimentally measured time de-pendent blood velocity data, available in the literature, were used as the inlet boundary condition to representthe cardiac cycle. An out
ﬂ
ow boundary model, consisting of an elastic tube followed by a contraction tube, wasused at the four outlets to represent both the compliance and the pressure drop of the small arteries, arterioles,and capillaries that would follow the truncated vascular system. The results show that, at each bifurcation, theblood
ﬂ
ow velocity decreases signi
ﬁ
cantly in the transition from the parent vessel to the daughter vessels dueto the higher total cross-sectional area of the daughter vessels as compared to the parent vessel. This decreaseinvelocityispartiallyrecoveredalongthearteriesduetothetaperingofthebloodvessels.Itcanalsobeobservedfrom the results that thepressuredistributionsand pressure drops along thevascular system are ingood agree-mentwiththephysiologicaldatafoundintheliterature.Theresultsalsoshowthatthevelocitypro
ﬁ
lesimmedi-ately following a bifurcation are not initially symmetric, with their maxima shifted toward the inner part of thebifurcation in the daughter vessels. Finally, the results show that the maximum deformation is about 2% of theaverage vessel radius, which is relatively small and typical for small arteries.© 2015 Elsevier Ltd. All rights reserved.
1. Introduction
The cardiovascular system is one of the most vital systems in thehuman body. It transports nutrients, oxygen and vital hormones andneurotransmitters to all parts of the body. The cardiovascular systemconsistsoftwo
ﬂ
uidpumpsinseriesandacomplicated,branchedpipe-line network made up of large arteries, small arteries, arterioles, capil-laries and veins [1]. The diseases related to the cardiovascular systemsuch as atherosclerosis, aneurysms, heart attack, heart failure, highblood pressure, and stroke are some of the leading causes of death inthe world. The dynamics of blood
ﬂ
ow are an underlying factor inmany of these diseases and a better understanding of blood
ﬂ
owdynamics plays an especially important role in the diagnosis andtreatment of these diseases. A better understanding of blood
ﬂ
ow dy-namicsisalsoessentialinthedesignandperformanceevaluationofcar-diovasculardevicessuchasleftventricularassistdevices,arti
ﬁ
cialheartvalves,stents,andgrafts[2].Therearetwomethodsthatcanbeusedtoinvestigateblood
ﬂ
owinthe cardiovascularsystem:invivostudiesandcomputational studies.Invivo studies are importantbecausethey yieldactualmeasured data thatgroundtheoreticalsolutions,but at thesametime,invivostudiesaredif
ﬁ
cult,timeconsumingandexpensive.There-fore,computationalstudieshavebeenanecessityandhavegainedpop-ularity in the investigation of blood
ﬂ
ow in the cardiovascular system.Althoughmanyattemptshavebeenmadetomodelblood
ﬂ
owinthecardiovascular system during the past decades, three dimensional sim-ulationofthewholevascularsystemisstillanunfeasibletaskbecauseof the complexity of the cardiovascular system that consists of millions of vessels spanning several orders of magnitude in size. Therefore, previ-ousstudiesreportedintheliteratureincludesimpli
ﬁ
edlumpedparam-eter and one dimensional
ﬂ
ow models [3
–
6]. The lumped parametermodels are based on an analogy between the current
ﬂ
ow in an
International Communications in Heat and Mass Transfer 67 (2015) 51
–
58
☆
Communicated by W.J. Minkowycz
⁎
Corresponding author.
E-mail addresses:
kamilk@trakya.edu.tr (K. Kahveci), beckerb@umkc.edu(B.R. Becker).http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.06.0100735-1933/© 2015 Elsevier Ltd. All rights reserved.
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer
journal homepage: www.elsevier.com/locate/ichmt
electrical circuit and the blood
ﬂ
ow in the human circulatory system.The lumped parameter and one dimensional models are generallyusedtosimulatetheperipheralvascularnetworkinordertoobtainare-lationshipbetweentheblood
ﬂ
owrateandpressure.Ontheotherhand,threedimensional
ﬂ
owmodelingisessentialtocaptureallthedetailsof the blood
ﬂ
ow in the vascular system [7].In three dimensional blood
ﬂ
ow modeling, only a small truncatedsystemofbifurcatedvesselsisconsideredandtheremainderofthevas-cularsystemisusuallyrepresentedbyboundaryconditions.Thereareanumber of studies reported in the literature on three dimensional nu-merical simulations of blood
ﬂ
ow [8
–
13]. However, nearly all of thesesimulationscontainonlyasinglelevelofbifurcationinthemodeledvas-cular system. In addition, many of the studies focus only on blood
ﬂ
owin large diameter arterial level vessels and ignore the smaller vessels[12
–
14].Blood
ﬂ
ow in the cardiovascular system is pulsatile in nature. Thepulsatility is generated by the intermittent ejection of blood from theheart during the systolic and diastolic phases. Therefore, the blood
ﬂ
ow is both unsteady and quasi-periodic. However, the pulsatile
ﬂ
owfrom the heart is gradually damped as the blood
ﬂ
ows from large tosmall vessels and eventually becomes steady
ﬂ
owing at the capillarylevel by a combination of the compliance of the large vessels and thefrictional resistance in the small vessels. Therefore, the
ﬂ
ow throughthe smaller arterioles and capillaries can be analyzed based on the as-sumptionofsteadystate
ﬂ
ow.However,thereisstillasigni
ﬁ
cantpulsa-tile effect present in the blood
ﬂ
ow through small arteries and largerarterioles. Even so, this pulsatile effect has been ignored in a numberof studies found in the literature wherein the steady state assumptionhas been used for the sake of simplicity [15
–
17].Mostofthebloodvesselsinthevascularsystemareelasticandtheirdiameter changes depending on the pressure during a heartbeat.However, the elasticity of blood vessels varies with the diameter of the vessels, that is, blood vessels become gradually more rigid as theirdiameter becomes smaller. The capillaries are essentially rigid and canbe modeled with a rigid wall assumption. On the other hand, the rigidwall assumption fails to predict some essential characteristics of theblood
ﬂ
ow through vessels that are larger than capillaries. Even so, therigidwallassumptionhasbeenusedinanumberofstudies[1,18,19]be-causemodelingthe
ﬂ
uid
–
solidinteractionofanelasticwalldramatical-ly increases the computational complexity and cost.Thevesselsinthecardiovascularsystemareinterconnectedtubesof different diameters. Therefore, the diameter of a blood vessel tapersfrom beginning to end. In vivo studies show that the taper angle canbeaslarge as 2°[20]and this conical nature of the blood vessels causessigni
ﬁ
cant changes in blood
ﬂ
ow [21
–
24]. The taper of the tube is alsoan important factor in the pressure development along the vessel [24].Therefore, in order to obtain results consistent with physiologicalblood
ﬂ
ow, the tapering effect of the blood vessels has to be taken intoconsideration. However, many of the studies found in the literature ig-nore the tapering effect and assume that the vascular system is com-posed of straight constant diameter vessels.Therefore, the study presented in this paper is unique in that threedimensional pulsatile blood
ﬂ
ow along small elastic arteries is simulat-ednumericallyinatruncatedvascularsystemconsistingoftwolevelsof bifurcation.Furthermore,thetaperingeffectofthebloodvesselsastheyapproach the next bifurcation point is also taken into consideration inthis study.
2. Description of the vascular system model
2.1. Geometry of the vascular system model
The vascular system model considered in this study is composed of small arteries and has two levels of symmetric bifurcation as shown inFig. 1. Although the branching pattern and the geometry of the humanvascular system are complex, they can be speci
ﬁ
ed for small arteriesusing Murray's hypothesis that the structure of the vascular systemobeys the principle of minimum work. Accordingly, in the current vas-cularsystemmodel,theparent/daughterdiameterratiosandtheanglesof bifurcation were speci
ﬁ
ed accordingto Murray'slaw[25]. The taper-ing effect of the blood vessels as they approach the next bifurcationpointisalsoaccountedforinthisstudy.Thisstructuredtreeassumptionis valid only for small arterial vessels and, accordingly, the diameter of the root vessel in the tree is taken as 0.6 mm.Thediameters of thedaughtervessels after eachbifurcationand theangles between the main vessel and the daughter vessels, as shown inFig.2,arespeci
ﬁ
edaccordingtoamodi
ﬁ
edversionofMurray'slaw[25]:
d
n
0
¼
d
n
1
þ
d
n
2
;
cos
θ
1
¼
d
40
þ
d
41
−
d
42
2
d
0
d
1
ð Þ
2
;
and
;
cos
θ
2
¼
d
40
−
d
41
þ
d
42
2
d
0
d
1
ð Þ
2
;
ð
1
Þ
wherein subscript 0 refers to the parent vessel, and subscripts 1 and 2refertothetwodaughtervesselsatthebifurcation,
θ
1
and
θ
2
arethean-gles between the parent and the daughter vessels, and
n
is the bifurca-tion exponent. The total branch angle (
θ
1
+
θ
2
) between the twodaughter vessels is known as the bifurcation angle.Murray proposed that physiological vascular systems, through evo-lution by natural selection, must have achieved an optimum arrange-ment such that in every segment of the vascular structure,
ﬂ
ow is
Fig. 1.
The geometry and coordinate system for the vascular system model.
Fig. 2.
The geometric parameters of the main and daughter vessels.52
K. Kahveci, B.R. Becker / International Communications in Heat and Mass Transfer 67 (2015) 51
–
58
achieved with the least possible biological work [26]. Murray alsoassumed a value of three for the bifurcation exponent,
n
= 3 [25].The results obtained by Murray's law with
n
= 3 have been comparedextensively with experimental measurements of the vascular system.The measurements give overall support to Murray`s law but thereis considerable scatter in the data. Through the application of variousoptimization procedures, a value of 2.7 for the bifurcation exponent(
n
= 2.7) was found to be optimal for the analysis of the
ﬂ
ow in smallarteries [27]. Therefore, in this study the bifurcation exponent is taken
as
n
= 2.7.A second geometrical parameter required in the construction of thevascular system model considered in this study is the ratio of vessellength, L, to vessel diameter, d, which was determined according todata found in the literature as follows:L
¼
Kd
ð
2
Þ
whereinKisaconstantthatdependsonthevesselsize.Basedonstudiesby Iberall [28], Olufsen et al. [29] report a value of
K
= 25 for small ar-teries. Therefore, in this study, a value of
K
= 25 is used to constructthe vascular tree.The taper angle in this study was determined so as to be in accor-dance with the vessel length and the change in the vessel diameterfrom parent vessel to daughter vessel. The diameters of the parent anddaughtervesselswerecalculatedusingthemodi
ﬁ
edversionofMurray'sLaw given in Eq. (1), while the vessel lengths were calculated usingEq. (2). Accordingly the taper angle used in this study corresponds to0.54°, which is within the physiological range as determined byin vivo studies.
2.2. Elasticity of the vascular system model
In this study, the vascular system model is composed of vesselswhosewallsareassumedtobehomogeneous,impermeableandlinear-ly elastic with low elasticity. Hence, the displacements and strains forthe small vessels in the vascular system model are quite small, and inthis case, it is suf
ﬁ
cient to use a linear elastic material model given bythe following set of equations:
ρ
s
∂
2
u
s
∂
t
2
−∇
σ
s
¼
0
;
σ
s
¼
4
C
:
ε
s
;
ε
s
¼
12
∇
u
s
þ
∇
u
s
ð Þ
T
ð
3
Þ
wherein
u
s
=(u
s
,v
s
,w
s
)isthestructuraldisplacement,
σ
s
isthestructur-al stress tensor.
4
C
represents the fourth-order Hookean elasticity tensorand
ε
s
istheelasticstraintensor.ThematerialpropertiesofthestructurearefullydeterminedbytheYoung'smodulus,E,andthePoisson'sratio,
ν
.Thedensity,Young'smodulusandthePoisson'sratioofthebloodvesselsaretakenas
ρ
=1000kg/m
3
,
E
=4.55×10
5
Paand
ν
=0.28,respective-ly [31,32]. The model also includes the tissue surrounding the vascularsystem because it presents a stiffness that resists artery deformationdue to the blood pressure. The tissue surrounding around the vascularsystemmodelisassumedtohaveathicknessof6mm.Thematerialprop-erties of this tissue were taken as
ρ
= 1000 kg/m
3
,
E
= 7 × 10
6
Pa and
ν
= 0.28.
2.3. Boundary conditions for the vascular system model
The continuity condition was assumed on the interfaces betweenthe arteries and the tissue. Furthermore, the deformation normal totheoutersurfaceofthetissuewasassumedtobezeroasthetissuesur-rounding the arteries has high stiffness. Under these assumptions, theboundary conditions can be expressed as follows:
u
s
;
a
¼
u
s
;
t
ð
4
Þ
u
s
n
¼
0
ð
5
Þ
3. Description of the blood
ﬂ
ow model
3.1. Governing equations for the blood
ﬂ
ow model
As the velocities in the present vascular system are relatively lowand the characteristic lengths of the small vessels considered in thestudy are relatively small, the
ﬂ
ow is laminar in the truncated vascularsystem. Furthermore, blood is assumed to be an incompressible
ﬂ
uidwith a constant viscosity in the present study. The density and the vis-cosityofthebloodaretakenas
μ
=0.005Pa-sand
ρ
=1060kg/m
3
,re-spectively [30].With these assumptions, the governing equations take on the fol-lowing form:
ρ
f
∂
u
f
∂
t
þ
u
f
∇
u
f
¼
−∇
−
p
I
þ
μ
∇
u
f
þ
∇
u
f
T
h i
ð
6
Þ
∇
u
f
¼
0
ð
7
Þ
wherein
u
f
=(u
f
,v
f
,w
f
)isthevelocity,
p
isthepressure,
ρ
isthedensity,and
μ
is the dynamic viscosity of the blood.
3.2. Boundary conditions for the blood
ﬂ
ow model
Boundary conditions are necessary to close the set of governingequations.Atthewallsofthebloodvessels,theno-slipboundarycondi-tion is imposed.Thevelocitypro
ﬁ
leatthesingleinletboundaryofthecurrentvascu-lar system model is assumed to be uniform across the diameter of theblood vessel. However, the magnitude of this pro
ﬁ
le varies with timein accordance with a cardiac cycle with a frequency of 1 cycle per sec-ond. The shape of this time dependence, as shown in Fig. 3, is basedupon the experimental data for small arteries, obtained by Kanariset al. [1].There are four identical outlet boundaries in the current truncatedvascular system model. An out
ﬂ
ow boundary model, proposed byPahlevan et al. [2], is used to specify the boundary condition at thefour outlets. This out
ﬂ
ow boundary model extends the computationaldomain with a straight elastic tube and contraction tube, as shown inFig.4,torepresentthecompliance,thewavere
ﬂ
ection,andthepressuredrop due to thesmall arteries, arterioles, andcapillaries that wouldfol-low the truncated vascular system.
Fig. 3.
Variation of the inlet velocity during a heartbeat.53
K. Kahveci, B.R. Becker / International Communications in Heat and Mass Transfer 67 (2015) 51
–
58
The Young's modulus and the length of the elastic tube are selectedbythefollowing relationstomatch thecompliance, C,and wavearrivaltime, t
arr
[2]:
E
¼
π
r
3
t
arr
C
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2
ρ
f
rh
1
−
v
2
ð Þ
q 0B@1CA
2
;
L
¼
π
r
2
t
2
arr
4
C
ρ
f
1
−
v
2
ð Þ ð
8
Þ
wherein
r
and
h
are the radius and the wall thickness, respectively, of theelastictubeoftheboundarymodel.Theradius,
r
,andthewallthick-ness,
h
,oftheelastictubeoftheboundarymodelwerechosentobethesameastheradiusandthewallthicknessofthelastvesselofthevascu-larsystem model sothat thetransitionfromthevascularsystem modelto the boundary model would be geometrically continuous.The terminal compliances of the different segments in the humanvascular system are given in the literature and ranging from10
−
14
m
3
/Pa to 10
-11
m
3
/Pa [33]. In the current study, the truncatedvascular system contains only small arteries that are expected to havea low value of terminal compliance. Therefore, 10
−
13
was selected asthe terminal compliance for this study. The wave arrival time wastaken as
t
arr
= 0.01 s as the vessel segment considered in this study iscloseto theend of thearterial system.Theinternal diameterand thick-ness of various types of the vessels in the vascular system are given inthe literature [34]. The vessel diameters in the current vascular system
model are in the range for vessels called small rami. Accordingly, thethickness for the vessels in this study was taken as 125
μ
m, typical of small rami vessels.The contraction ratio,
γ
=
d
2
/
d
1
, is the ratio of the diameter of thecontraction tube,
d
2
divided by thediameter of themodeled blood ves-sel,
d
1
(see Fig. 4). The contraction ratio,
γ
, is adjusted to represent ef-fects of the truncated vasculature by the following relation [2]:
γ
¼
γ
R
ð Þ ¼
RQ
−
k
2
k
1
!
−
1
=
4
;
ð
9
Þ
wherein
R
is the peripheral resistance, and
k
1
and
k
2
are de
ﬁ
ned as:
k
1
¼
8
ρ
Q
2
π
2
d
41
;
k
2
¼
E
ϑ
Q
−
8
ρ
Q
2
π
2
d
41
;
ð
10
Þ
whereinQ isthemeanvolumetric
ﬂ
owrateand
Ė
ϑ
istheviscousenergyloss within the boundary model. The viscous energy loss within theboundary model,
Ė
ϑ
, has little effect upon the contraction ratio calcula-tion and is therefore neglected.The range of the contraction ratio,
γ
, used in the present study wasobtained based upon the peripheral resistance of the human vascularsystem,whichvaries between 5.2and 474.2 mmHg-s/ml [35]. Calcula-tions made with Eqs. (7) and (8) indicate that the contraction ratio,
γ
,ranges from 0.3 to 0.8 as the peripheral resistance,
R
, varies from 5.2to 474.2 mmHg-s/ml.As the blood
ﬂ
ows through the large arteries, the blood pressureis not signi
ﬁ
cantly reduced; however, as the blood
ﬂ
ows throughthe small arteries and arterioles, the blood pressure drops 50% to 70%[36]. The resistance value for the present model should therefore havehigh resistance values as the truncated vascular system in this studycontains only the small arteries. In accordance with this fact, the simu-lations showed that a contraction value equal to 0.43 yields a pressuredistribution consistent with the physiological data. Therefore, thecontraction ratio in this study was taken as 0.43 with an end pressureof 1400 Pa [2].
4. Coupling of the blood
ﬂ
ow model and the vascular system model
The governing equations developed in this study constitute acoupledmodelthatconsistsofthe3DNavier
–
Stokesequationsdescrib-ingtheblood
ﬂ
owandthemodelthatdescribesthedisplacementofthevesselwalls.Thecouplingappearsontheboundariesbetweenthebloodandthevesselwalls.ThenumericalsolutionincludestheusageofanAr-bitrary Lagrangian
–
Eulerian (ALE) method to combine the
ﬂ
uid
ﬂ
owmodel formulated using an Eulerian description and a spatial frame,withthesolidmechanicsmodelformulatedusingaLagrangiandescrip-tion and a materialframe. Inthis formulation,thegeometry of the
ﬂ
uidandstructuralelementsarecontinuouslyupdatedateachtimestepdur-ingthecalculations.Thismethodeffectivelycombinesthetwodifferentformulations for solid mechanics and
ﬂ
uid mechanics and involves acontinuous adaptation of the mesh without modifying the meshtopology [37].
Fig. 5.
Velocity magnitude distributions in the mid-plane through the vascular system model.
Fig. 4.
Model for out
ﬂ
ow boundary condition [2].54
K. Kahveci, B.R. Becker / International Communications in Heat and Mass Transfer 67 (2015) 51
–
58
5. Discussion of results
The solution of thegoverningequationsis obtained byusinga com-mercially available
ﬁ
nite element analysis and simulation softwarepackage. The computational domain is meshed by using a built-inmeshing algorithm within the
ﬁ
nite element simulation software. Thecomputational results are obtained by using an iterative solver basedon the generalized minimum residual (GMRES) method with a multi-grid pre-conditioner. The GMRES method developed by Saad andSchultz [38] approximates the solution of discretized equations by thevector in a Krylow subspace with minimal residual. The convergencecriterionischosenastherelativeerror,er
≤
10
−
5
foralldependentvar-iables. Therelative error,er, isbased on theEuclideannorm de
ﬁ
ned as:
er
¼
1
N
X
N i
¼
1
e
i
j j
W
i
2
" #
1
=
2
;
W
i
¼
max U
i
j j
;
S
i
ð Þ ð
11
Þ
where
N
isthenumberofdegreesoffreedom,e
i
istheerrorandU
i
isthedependent variable and S
i
is the scale factor.The velocity magnitude distributions in the mid-plane through thevascularsystemmodelareshowninFig.5.Themodelcardiaccycleisas-sumedtohaveaperiodof1s.Thus,the
ﬁ
gurefor
t
=0.2sinthecardiaccycle shows the velocity near its maximum value. On the other hand,the
ﬁ
gure for
t
= 0.8 s in the cardiac cycle shows the velocity ap-proaching its lowest value.
Fig.9.
Centerlinevelocitiesat.1sintervalsalongthedaughtervesselsfollowingthesecondbifurcation.
Fig. 8.
Centerline velocities at .1 s intervals along the daughter vessels following the
ﬁ
rstbifurcation.
Vessel Centerline
Fig. 7.
Velocity pro
ﬁ
les in a cross section of a daughter vessel immediately following the
ﬁ
rst bifurcation.
Fig. 6.
Pressure magnitude distributions in the mid-plane through the vascular system model.55
K. Kahveci, B.R. Becker / International Communications in Heat and Mass Transfer 67 (2015) 51
–
58

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