Pulsed magnetohydrodynamic blood flow in a rigid vessel under physiological pressure gradient

(1)

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vessel under physiological pressure gradient

Dima Abi Abdallah, Agnès Drochon, Vincent Robin, Odette Fokapu

To cite this version:

Dima Abi Abdallah, Agnès Drochon, Vincent Robin, Odette Fokapu. Pulsed magnetohydrodynamic

blood flow in a rigid vessel under physiological pressure gradient. Computer Methods in Biomechanics

and Biomedical Engineering, Taylor & Francis, 2009, 12 (4), pp.445-458. �hal-01083975�

(2)

Pulsed Magnetohydrodynami blood ow in a rigid vessel

under physiologial pressure gradient

Dima Abi-Abdallah†^,Âgnès^Drohon†^,^Vinent^Robin‡^,ândÔdette^Fôkapu†

†^Biomehanis^andBioengineering,UniversityofTehnologyofCompiègne,Frane

‡^Laboratory^of^AppliedMathematis,UniversityofTehnologyofCompiègne,Frane (June2007)

Blood ow ina steady magneti eld has been of great interest over the past years. Many re-

searhershaveexaminedtheeetsofmagnetieldsonveloityprolesandarterialpressure,and

majorstudiesfousedon steadyorsinusoidalows.Inthispaper wepresentasolutionforpulsed

magnetohydrodynami blood owwith asomewhat realistiphysiologialpressure waveobtained

usingawindkessellumpedmodel.Apressuregradientisderivedalongarigidvesselplaedatthe

outputofa ompliantmodulewhihreeivestheventrileoutow.Then,veloityproleandow

rateexpressionsarederivedintherigidvesselinthepreseneofasteadytransversemagnetield.

Asexpeted, resultsshowedow retardationand attening. Theadaptability ofour solutionap-

proahallowedaomparisonwithpreviouslyaddressedowasesandalulationspresentedagood

oherenewiththosewellestablishedsolutions.

Keywords:Statimagnetield,magnetohydrodynamiinterations,Halleet,windkessel,lumped

model.

1 Introdution

TheinreaseinexposuretohighmagnetieldsausedbythewideuseofMag-

netiResonane Imaging(MRI)asastandardmedialproedure, hasraiseda

onern in theresearh ommunity and onstituted an inentive for studying

theeets ofmagneti elds onhumanphysiologyand its impaton patients

health. Espeially that, in striving to ahieve higher resolution and greater

spetralseparation,theMRIsannersstatimagneti eldskeepaugmenting.

Studies evaluating theeet of human or animal exposure to magneti elds

haveshownnomajorhanges,exeptforaninreaseofsystolibloodpressure

aswell asalterations of the eletroardiogram (ECG) signalmanifested asel-

evations of theT wave,allof whih aredueto blood ow.

Themovementofaondutinguid,suhastheblood,inanexternallyapplied

(3)

magneti eld is governed by the laws of magnetohydrodynamis. When the

body is subjeted to a magneti eld theharged partilesof theblood ow-

ing transversally to the eld getdeeted by theLorentz fore thus induing

eletrial urrents and voltages, aross the vessel walls and in the surround-

ing tissues, strong enough to be deteted at the surfae of the thorax inthe

ECG. Furthermore, the interations between these indued urrents and the

applied magneti eld an ause a redution of ow rate and thus a reative

ompensatoryinrease inbloodpressure inorder to retain a onstant volume

owrate.

Magneti eld interations withblood ow have been demonstrated by mul-

tiple authors throughout in vitro experiments [1,2℄ where pressure and ow

rateweremeasured, aswellasinvivostudiessuhas[3,4℄whereanimalECG

alterations have been observed,and [5℄where theeetson humanvitalsigns

were foundto onsist essentiallyinanarterial pressure inrease.

Theoretialmagnetohydrodynamibloodowalulationshave,however,been

addressedmuhearlierandgobakasfar astheearlysixties.Korhevskiiand

Marohnik [6℄rst proposedaveloityprole solution for blood ow between

two parallel plates under a onstant pressure gradient with a perpendiular

magneti eld, under the assumption that blood is newtonian. Later other

studies foused on ow in a rigid irular tube with non onduting walls

plaed ina transverse magnetield to oer a more realistimodelfor blood

owinvessels. Inthis ase, the most omplete solution ofthemagnetohydro-

dynami equations of a onduting uid was proposed by Gold [7℄. Setting a

onstant pressure gradient, Gold derived expressions for the veloity prole

as well as indued elds and voltages. Vardanyan [8℄ subsequently published

an approximate steadysolution where veloityprole and ow rate were al-

ulated by negleting the indued elds.More reent studies were essentially

based on these founding works, suh asKeltner et al. [1℄ where a omparison

wasestablishedbetweentheresultsofGoldandVardanyantoassesstheonse-

quenesofnegletingtheindutions.WiththesamehypothesisasVardanyan,

Sudetal. [9℄laterdealtwithasinusoidalpressuregradientthatmodeledabit

loser thepulsed nature of blood ow inarteries. The hypothesis of ondut-

ingwallswasnotintrodued untilKinouhietal.[10℄whoinluded indutions

in the vessel and the surrounding tissues in the steady ow ase in order to

evaluate theindued ECG superimposedvoltages.

In this work, we revisit the ow of blood as a newtonian uid, in a irular

rigidvessel,withnonondutingwalls,inthepreseneofatransverseonstant

magneti eld.Nevertheless, insteadoftakinga onstant pressuregradient or

a sinusoidal one, we apply a realisti pulsed pressure gradient derived using

a windkessel lumped model, where the ompliant module provides the input

owintotherigid vessel.Then, negleting induedelds,wesolve themagne-

tohydrodynamiequationsto obtainveloityprole andowrateexpressions.

(4)

Thefatthatour resolution method isbasedonFourier deompositionmakes

the solutions easily adaptable to steady or sinusoidal ases, thus allowing a

omparison withtheprevious well establishedstudies.

2 General equationsand solution

The ow of a onduting inompressible newtonian uid inthe preseneof a

magnetieldisdenedbyaombinationofMaxwell'sequationsononehand,

andtheNavier-Stokesequationinludingthemagnetiforeontheother,along

withtheonservation equation, aswell asOhm'slaw.

If we neglet the indued elds, the veloity prole an be solely dened by

theNavier-Stokesequation, where the magnetifore term is evaluatedusing

Ohm's law,

ρ ∂~u

∂t +

~u·∇~

~u

=−∇p~ +η∆~u+σ

~u∧B~

∧B~ , ⁽¹⁾

where

B ~

^is ^the ^magneti ^eld,

~u

^,

ρ

^,

η

^,

σ

^are respetively the uid veloity, density,visosityand ondutivityand ∇p

~

^is ^the^pressure ^gradient.

Byassuming thattheowis unidiretional, axisymmetri withno swirl in a

Figure1. Flowmodelgeometry

Thevesselisrepresentedbyaylindrialondutwhereblood owsalongtheOzaxis,inthe

preseneofatransversesteadymagnetieldoriented intheOxdiretion.

irularrigidvessel,itsveloityanbewrittenas

~u = (0, 0, u(˜ r, t))

^(g.^1).^The

uidpressure isthus a funtion of theposition z and time.The external on-

stantmagnetieldisappliedtransversallysuhas

B ~ = (B

₀

cos θ,

−B₀

sin θ, 0)

and the vessel is onsidered to have non onduting walls. The ow would

thereforebe governed bythe

Oz

^projetion ^of ⁽¹⁾^whih^expressedⁱⁿ^ylindri-

aloordinates gives,

a² ν

∂u(˜r, t)

∂t =g(t) +∂²u(˜r, t)

∂r˜² +1

˜ r

∂u(˜r, t)

∂˜r −H_a²u(˜r, t) , ⁽²⁾

(5)

with

r ˜ =

_a^r^, ^where

a

^represents ^the ^vessel ^radius,

H

_a

= B

₀

a

q

σ

η ^is ^the ^Hart-

mannnumber,

ν =

^η_ρ ^is^the ^kinemati ^visosity^,^and g(t) =−a²

η

∂p(t, z)

∂z , ⁽³⁾

withboundary onditionat the walls

u(1, t) = 0

^.

The proposed resolution method onsistsof aFourier deomposition, followed

byaHankeltransform.

Fourier series deomposition. The pulsed ow studied here is periodiwith

period T,inverse ofthe ardia frequeny.

u(˜ r, t)

^and

g(t)

^are^thus ^T^periodi

timefuntions thatan be deomposed inFourier series suhas,

u(˜r, t) =

+∞

X

k=−∞

uk(˜r)e^iω^k^t ^where:uk(˜r) = 1 T

Z T 0

u(˜r, t)e⁻^iω^k^tdt

g(t) =

+∞

X

k=−∞

gke^iω^k^t ^where:gk = 1 T

Z T 0

g(t)e^−iω^k^tdt

with:

ω

_k

= k

^2π_T ^.

By replaingin(2) we get,

a² ν

+∞

X

k=−∞

iωkuk(˜r)e^iω^k^t=

+∞

X

k=−∞

gke^iω^k^t+

+∞

X

k=−∞

∂²uk(˜r)

∂˜r² +1

˜ r

∂uk(˜r)

∂˜r

e^iω^k^t

−H_a²

+∞

X

k=−∞

uk(˜r)e^iω^k^t

⇔ a²

ν iωkuk(˜r) =gk+ ∆uk(˜r)−H_a²uk(˜r) ∀k∈Z . ⁽⁴⁾

Hankel Transform. For a funtion

f (r)

^dened ^over

[0, 1]

^, ^the ^zero ^order

Hankeltransform isdened [11℄ as,

H(f) = (f_n^∗)_n∈_Z ; f_n^∗= Z 1

0

rf(r)J0(rλn)dr ;

withthefollowing properties,

H(∆f(r)) =−λ²_nf_n^∗ (^iff(1) = 0) ^and H(cste) = cste

λn

J1(λn) ,

(6)

where

λ

n ^are ^the^roots^of^Bessel ^funtion

J

0

(x)

^.

Knowing that

u(1, t) = 0

⇔

u

_k

(1) = 0

∀k ∈ Z^,^applying ^the ^Hankel ^trans-

formon equation(4) yields,

u^∗_k,n= J1(λn) λn iωka²

ν +λ²_n+H_a²gk . ⁽⁵⁾

Solution. To derive the solution we must inverse the Hankel transform to

get the veloity's Fourier oeients and then establish the veloity prole

expression.

We know that if

H(f ) = f

_n^∗ ^, ^then ^the ^inverse ^transform ^gives

f (r)

^suh ^as,

f (r) = 2

P∞

n=1 J0(λnr) J1(λn)²

f

_n^∗^.

From (5)we derive,

uk = 2

∞

X

n=1

J0(λn˜r) λnJ1(λn)

gk

iωka²

ν +λ²_n+H_a² , ⁽⁶⁾

and thereforeaveloity prole,

u(˜r, t) = 2

∞

X

k=−∞

∞

X

n=1

J0(λnr)˜ λnJ1(λn)

gk

iωka²

ν +λ²_n+H_a²e^iω^k^t . ⁽⁷⁾

Theowrate will thenbe given as,

q(t) = Z Z

A

u dA= Z 2π

0

Z a 0

u(r, t)rdrdθ

= 4πa²

∞

X

k=−∞

∞

X

n=1

e^iω^k^t λnJ1(λn)

gk

iωka²

ν +λ²_n+H_a² Z 1

0

J0(λnr)˜˜rd˜r .

Using thefatthat

R

x

ⁿ

J

n−1

(x)dx = x

ⁿ

J

n

(x)

⁽ ^[12℄,^p.137),^we ^get,

q(t) = 4πa²

∞

X

k=−∞

∞

X

n=1

e^iω^k^t λ²_n

gk

iωka²

ν +λ²_n+H_a² . ⁽⁸⁾

3 Pressuregradientexpression

Intheliterature,studiesonerningpulsedowsinrigidtubesusuallyonsider

sinusoidalpressuregradients[11℄and[9℄.Inthiswork,inordertosolvetheve-

loityprole,weseektoimposearealistiphysiologialpressuregradient.For

this,we usethe3-element Windkessellumped modelwhere a ompliant mod-

ulerepresentsthelargearteriesandapureresistanerepresentstheperipheral

rigid vessels. We then derive a pressure gradient, governed by the ompliant

module, to be applied on arigid vesselbeyond thelarge arteries.

(7)

(a)2-elementWindkessel (b) 3-elementWindkessel

Figure2. Windkessellumpedmodels

Pv(t):leftventriularpressure,Pa(t):aorti pressure,Q(t):outputowrateoftheleftventrile,

Q1(t)^:inputôw^rate^to^the^peripherâl^vessels,Ra^:resistaneôf^theâortaând^the^largeârteries,

C^:omplianeôf^theâortaând^largeârteries,Rp^:total^peripherâl^resistaneôf^smallârteries,

arteriolesandapillaries

They rely on an analogy with eletri iruits where urrents represent ar-

terial blood ows and voltages represent arterial pressures. In suh models,

resistanes standforresistaneto ow(arterialandperipheral)resulting from

visous dissipation inside the vessels, apaitors represent volume ompliane

of thevessels,and indutorsrepresent blood inertia.

Intheirulatorysystem,thesmall aliberarteriesan beonsideredasrigid.

These arteries get at their input a ow rate imposed by the large ompliant

arteries and an be modeled usingpureresistanes [13℄.

Inthefollowingsetionwewillestablishamodelthatwillallowusto evaluate

this owrate andtherefore deduean expressionof pressuregradient along a

rigid vessel.

The Windkessel model, oneived by Otto Frank in1899 and inspired by air

hamber pumps used in re engines, is a quite simple onguration that de-

sribestheow attheheartoutputand intothesystemi arteries.Themodel

onsistsofanelastiaumulationhamberplaedinarigidondutpreeded

bya valveand followedbyaPoiseuillehydrauliresistane. Whenthevalveis

open(systole)partoftheventriularowaumulatesinthehamber,andthe

restowsintheresistane.Whenthevalveislosed(diastole)thebloodwhih

hadaumulatedinthehamberisforedoutthroughtheresistane.Thisele-

trial modelwasinitiallyoneived withtwo elements omprisinga apaitor

that representstheelastiityof large arteries and aperipheral resistanethat

stands for the resistane of small arteries and arterioles (g.2(a)). The input

pressureoftheiruitisthe leftventriularpressure, assuminganullpressure

atthevenaava.Adiodeplaedattheiruit'sentryplaystheroleoftheaor-

ti valve whihlets theow getthroughonly whentheventriular pressureis

superior tothatoftheaorta. This2-element modelwaslater transformedinto

a 3-element model (g.2(b))where an additional resistanewasintrodued to

takeinto aount theresistaneof theaorta andlargearteries [14℄ .Thelatter

model wasfound to produe quiterealisti pressure and ow rateurvesthat

orretlyreprodue experimental data[15℄and thus remains verywidelyused

(8)

In the following setion we adopt the 3-element windkessel model and om-

pute an expression for pressure along a pure resistane at the output of the

ompliant module. We rst dene a mathematial expression to model real

ventriular pressureinput suhas[15℄,

P v(t) = Pmax

2 (1−cos 2γt) 0≤t≤tp

0 tp≤t≤T ⁽⁹⁾

with

γ =

_t^π

p

.

Thedierential equationdening the3-element iruit an be written as,

dP a dt +P a

τ = Q(t)

C , ⁽¹⁰⁾

where

τ = CR

_p ^.

Solving for

P a(t)

ⁱⁿ êah ârdia ^yle ^phase ^: ^diastole ând isovolumetri ontration phases where

Q(t) = 0

^, âs ^well âs ^the êjetion ^phase ^where

Q(t) =

P v(t)−P a(t)

Ra

,whileensuringurve ontinuity between thephasesyields

theaortipressureexpression,

P a(t) =







Pse⁻^t+T^τ^−ts 0≤t≤t1 (isovolumetriontration)

Ke⁻^t−t1^Zτ +A(t) t1≤t≤ts ^(ejetion)

Pse⁻^t−ts^τ ts≤t≤T ^(diastole)

(11)

with,

A(t) = Rp

Ra+Rp

Pmax

2

1−2γZτsin 2γt+ cos 2γt 1 + 4γ²Z²τ²

;

Ps=P v(ts) ; K=P v(ts)e⁻^t^{1 +}^T−ts^τ −A(t1) ; Z= Ra

Ra+Rp

;

t

₁ ^denotes^the^beginning ^of^the^ejetion^phase^when

P a

^beomes^less^than

P v

^,

t

_s ^denotes ^the ^end ^of ^systole ^when

P a

^beomes ^greater ^than

P v

^, ^and

t

_p ^is

the instant at whih the pressure in theventrile drops to zero, and

T

^is ^the

ardia yleperiod.

The ow rate in the peripheral resistane is given by

Q

₁

(t) =

^{P a(t)}_R

p

and the

pressure drop along a peripheral vessel of radius

a

^and ^length

L

^would ^be

obtained by multiplying the ow rate with a hydrauli Poiseuille resistane

8ηL

πa⁴ ^yielding dP(t)

L

=

−_πa^8η4

Q

1

(t)

^,^and ^thus,

−∂p(t, z)

∂z = 8η πa⁴











Ps

R_pe⁻^t+T−ts^τ 0≤t≤t1 K

Rpe⁻^t−t1^Zτ +^A(t)_R

p t1≤t≤ts Ps

Rpe⁻^t−ts^τ ts≤t≤T

(12)

(9)

Inordertoderivetheveloityproleexpressionby(7)weneedtoomputethe

Fourier oeients

g

_k ^of

g(t)

^dened ⁱⁿ^{(3) .} Âfter^Fourier întegral âlulation

for eah ofthethree phases,

gk = 1 T

Z T 0

−a² η

∂p(t, z)

∂z e^−iω^k^tdt

= 8

T πa²Rp

Z t1

0

Pse⁻^t+T−ts^τ e^−iω^k^tdt+ Z ts

t1

Ke⁻^t−t1^Zτ +A(t)

e^−iω^k^tdt

+ Z T

ts

Pse⁻^t−ts^τ e^−iω^k^tdt

! ,

we get,

gk = 8 T πa²Rp

Ps 1 τ +iωk

e⁻^T−ts^τ

1−e⁻^t¹(τ¹+iωk)−e^−iω^k^T +e⁻^t^s(¹τ+iωk)⁺^Tτ

+ K

1 Zτ +iωk

e^Zτ^t1

e⁻^t¹(Zτ¹ +iωk)−e⁻^t^s(Zτ¹ +iωk)

+ Rp

Ra+Rp

Pmax

2

"

fk− e^−iω^k^t[(−iωk−2γǫ) cos(2γt) + (2γ−iωkǫ) sin(2γt)]

(1 + 4γ²Z²τ²)(4γ²−ω_k²)

ts

t1

#) ,

(13)

where

ǫ = 2γZτ

^and

f

_k

=

(

t

_s−

t

₁

k = 0

e^−iωkt1−e^−iωkts

iωk

k

6= 0

.

4 Results

All numerial omputations were done using the parameter values shown in

table 1.

Figure3showsthepressureandowrateurvesomputedusingthe3-element

Windkessel model (g.2(b)). They agree very well with measured pressure

urvesshownin[16℄.Fromthese omputedresults we analulate otherar-

diayleparameters.Wendameanardiaoutputof

77cm

³

/s

⁽

4.62l/min

^),

a stroke volume of

62cm

³ ând â ^systole ^duration ôf

t

s

= 30%T

^, ^all ^oin-

iding well with the normal range values (Comolet [17℄, for example, gives

C.O.

≈

87cm

³

/s

^,

S.V.

≈

70cm

³ ^and

t

s≈

37%T

^for

P

max

= 140mmHg

^).

Figure 4(a) shows the pressure gradient wave inthe rigid vessel withradius

a

^,âs^well âs^theôw^rates ^for ^dierent ^Hartmann^numbers. Ît ôuld^be^noted

thatforsmall

H

_a^the^vâriation^dynamisôf^theôw^rate^(and^veloity)îsâ^lot

slowerthan thatof thepressure gradient, however for larger

H

_a ^the^ow ^rate

(10)

Windkesseliruitomponents

Ra 0.0334 mmHg.s/cm³

Rp 1 mmHg.s/cm³

C 0.77 cm³/mmHg

Bloodharateristis

η 4.10⁻³ P a.s

ρ 1050 kg/m³

σ 0.5 S/m

Vesselradius

a 0.3 cm

Cardiayleparameters

f req 75 bpm

Pmax 120 mmHg

tp 50%T s

Table1. Numerialvalues

Theresistaneandapaitorvaluesarehosenbased on[15℄andagreeingwith[14℄,blood

harateristisaretakenfrom[10℄andtheardiayleparametersarehosentomaththe

averagetypialvalues.

Figure3. Ventriularpressurewaveandaortipressureandowrateurves

Pressureandowratewavesomputed usingthewindkesselmodelovertwoardiaylesat75

bpm.TheventriularpressureP v(t)îs^setâsⁱⁿêquation ^{(9) ,}^theâorti^pressureP a(t)îs

omputed by(11),andtheaortiowrate(ventrileoutput)isQ(t) = P v(t)−P a(t) R_a

urvestends to follow that of thepressure. The retardation of themovement

an also be learly depited, as theow slows down when the magneti eld

intensity inreases(g.4(b)).

Note that by applying the Poiseuille standard formula for stationary ows

q

_pois

=

^πa_8η⁴^∆p_∆z ^,ⁱⁿâ^vessel ôf^the^sameâliberûnderâ^pressure ^gradient êqual

tothemeanvalueoftheomputedpulsedgradient,wewouldgetapproximatly

77cm

³

/s

^.^Whihîs^the^value^we^get^byômputing^the^mean^valueôf^the^pulsed

owratefor

H

_a

= 0

^.

Given that a Poiseuille prole yields a maximum veloity

U

max

=

^a_4η²^∆p_∆z ≈

544cm/s

^,^gures^5(a)^and ^5(b)^represent ^normalized ^veloity^proles^as^ratios

to this value for

H

_a

= 0

^and

H

_a

= 2

respetively.

(11)

(a) Pressuregradient and owrate inarigid

vessel: The pressure gradient is alulated by

(12),andtheowrateby(8)fordierentHart-

mannnumbersforapulsedowinarigidves-

selwhihharateristivaluesaregiveninta-

ble1

(b)MeanowrateasfuntionoftheHart-

mannumber:Thegraduationsontheleft

representaowratenormalizedasaratio

to a Poiseuilleow rate, whilethe grad-

uationsontherightrepresentaowrate

incm³/s

Figure4. Flowrateforvariouseldintensities

0

T

2T 0

0.5 1 0

0.2 0.4 0.6 0.8 1

Time Normalized velocity profile (Ha=0)

r/a

U /(UmaxPois)

(a) Ha=0

0

T

2T 0

0.5 1 0

0.2 0.4 0.6 0.8 1

Time Normalized velocity profile (Ha=2)

r/a

U /(UmaxPois)

(b) Ha=2

Figure5. Normalizedveloityproles

Theveloityisomputedfrom (7)andthenormalizationisdonerespetivelytothePoiseuille

Umax≈544cm/s

5 Comparison with otherow types

Inthissetionwedisusstheobtainedresultsbyomparingthemtootherwell

establishedases of blood ows,whether stationaryor pulsed,inthepresene