Zamboni Venous Pulse Research

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Research ArticleThe Oscillating Component of the Internal Jugular Vein Flow:The Overlooked Element of Cerebral Circulation

Francesco Sisini, 1,2 Eleuterio Toro, 3 Mauro Gambaccini, 1 and Paolo Zamboni 2

Department of Physics and Earth Sciences, University of Ferrara, Via Saragat , Ferrara, Italy Vascular Diseases Center, University of Ferrara, Via Aldo Moro , Cona, Ferrara, Italy Laboratory of Applied Mathematics, DICAM, University of rento, Via Mesiano , rento, Italy

Correspondence should be addressed to Francesco Sisini; ss @uni e.itReceived October ; Accepted November

Academic Editor: John H. Zhang

Copyright © Francesco Sisini et al. Tis is an openaccess article distributed underthe CreativeCommons AttributionLicense,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Te jugular venous pulse (JVP) providesvaluable in ormation about cardiac haemodynamics and lling pressuresand is an indirectestimate o the central venous pressure (CVP). Recently it has been proven that JVP can be obtained by measuring the cross-sectional area (CSA) o the IJV on each sonogram o an ultrasound B-mode sonogram sequence. It has also been proven thatduring its pulsation the IJV is distended and hence that the pressure gradient drives the IJV haemodynamics. I this is true, then itwill imply the ollowing: (i) the blood velocity in the IJV is a periodic unction o the time with period equal to the cardiac periodand (ii) the instantaneous blood velocity is given by a time unction that can be derived rom a ow-dynamics theory that uses the

instantaneous pressuregradient as a parameter. Te aimo the present study is to con rm the hypothesis that JVP regulates the IJVblood ow and that pressure waves are transmitted rom the heart toward the brain through the IJV wall.

1. Background

Te evaluation o the jugular venous pulse (JVP), de ned asthemovemento expansiono the jugular vein due to changesin pressure in the right atrium, provides valuable in orma-tion about cardiac haemodynamics and lling pressures [ ],characteristic wave patterns pathognomic o cardiac diseases[ ], and an indirect estimate o the central venous pressure(CVP). Te JVP evaluation can be use ul in the diagnosisand/or prognosis o many heart diseases [ ]. Such a pulseconsists o three positive waves ( , , and V ) and twodescents,de ned, respectively, as and . Wave corresponds tothe atrial contraction and is synchronized with the waveo the electrocardiogram (ECG). Descent corresponds tothe lowering o the atrioventricular septum, interrupted by asmallpositive wave in relation to theclosure o the tricuspid valve; the third wave V corresponds to the cardiac systole andis ollowedbythe descentwhichcorresponds to theopeningo the tricuspid valve.

In a recent paper [ ] we proved that JVP can be obtainedby a simple ultrasound (US) B-mode investigation that con-sists in measuring the cross-sectional area (CSA) o the IJV

on each sonogram o video clip acquired in the transversalplane. In that paper, we acquired the time-dependent CSAdatasets o three healthy subjects and then calculated theautocorrelation unction o the datasets to show that they were periodic and nally we showed that their wave ormpresented the same , , and V waves as the JVP. Our study gave a quanti cation o the IJV CSA variation during thecardiac cycle. We also have seen that the IJV perimeter,measured on each sonogram o thevideo clip,wascorrelatingwith the IJV CSA ( > 0.9). On this point, we believethat it is desirable to have a con rmation o this ndingalso using a different imaging modality. However, a directexplanation o this nding is that, when in supine position,the pulsation o the IJV is a distension o its wall. Tis resultisvery importantbecause itmeans thatthere is a time varyingtransmural pressure whose effect is clearly visible and cannotbe neglected; in act, large changes in transmural pressureare required to induce CSA de ormation accompanied by astretching o the wall [ , ]. I this is true then the ollowingpoints are also true: (i) the blood velocity in the IJV isa periodic unction o the time with period equal tothe cardiac period and (ii) the instantaneous blood velocity

Hindawi Publishing CorporationBehavioural Neurology Volume 2015, Article ID 170756, 9 pageshttp://dx.doi.org/10.1155/2015/170756

http://dx.doi.org/10.1155/2015/170756http://dx.doi.org/10.1155/2015/170756


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Behavioural Neurology

z

c

w

F : Schematic representation o the heart-head axis. A

pressure wave propagating rom the heart toward the head with velocity is represented together with the direction o the internal jugular vein blood velocity that propagates rom the head towardthe heart.is given by a time unction that can be derived rom a

ow-dynamics theory that uses the instantaneous pressuregradient as a parameter, or example, the linear Womersley solution o the Navier-Stokes equations [ ].

It is worth noting that, in an elastic medium as is the IJVwall, an impulse that gives rise to a wall distension will bepropagated ollowing the wave equation o d’Alembert. Te

time periodic variation o the CSA and hence o pressuremeasured or the IJV is hence related to the propagation o pressure waves propagating rom the heart toward the brain.Tus, in supine position, when the IJVis distended, theblood

ow is governed by an oscillating pressure gradient [ ] andpressure waves generated by the cardiac revolution are thentransmitted rom the heart toward the brain (see Figure ).

Te aim o the present study is to veri y such a claimby comparing the blood velocity assessed by the current USDoppler method with the blood velocity calculated using thelinear Womersley equations. Te correspondence betweenthem will con rm the hypothesis that JVP regulates theIJV blood ow and that theoretically pressure waves aretransmitted rom the heart toward the brain through the IJVwall.

2. Methods

. . Subjects Scanning and Protocol. We have chosen twosubjects, labeled as#1, #2(a27-year-old emaleanda47-year-old male, resp., with no history o cardiovascular, hepatic,gastrointestinal, renal, and cerebral diseases. ECD screening

or CCSVI was completely negative [ – ]), out o ourdatabase o US scan o the neck by using My-Lab x-visionsystem (ESAO E, Genoa, Italy) with a linear array probe . –

MHz and Vivid-q ultrasound system (GE Medical SystemsUltrasound, Horten, Norway) equipped with a linear probe(L - MHz). Tey were chosen or this study becausethey were representative o a small and normal IJV CSA,respectively (with respect to the average diameter value o about . cm reported in [ ]).

Te assessmento the jugular CSA was per ormed using aB-mode scan in the transverse plane o the right IJV at c /clevel. Such region corresponds to the segment close to the junction o the IJV with the subclavian vein. For each subjectwe recorded rst a transversal video clip o the CSA (i.e., asonogram sequence)and immediately afera velocity spectralDoppler trace, both or at least our cardiac cycles. Te tracewas obtained selecting the “mean velocity mode,” whichautomatically produces a weighted average o the velocity

over the whole US re ected spectrum. Tis results in a greentrace (see Figure ) that corresponds to the blood velocity averaged across the sample volume. Te latter is selected toroughly correspond to the IJV lumen.

Tis study was conducted in accordance with the EthicalStandards o the Committee on Human Experimentation o

the Azienda Ospedaliera Universitaria di Ferrara. All the volunteers signed an in ormed consent orm.

. . Pulsed Flow-Dynamics Assumptions. Following the orig-inal paper o Womersley [ ], we denote as ( ) the instanta-neous blood velocity in direction and averaged over the IJVCSA. Te velocity ( ) is calculated, at any time , using theWomersley equations, or which the time-dependent pres-sure gradient ( , )/ is required. Te internal pressure( , )washereassumed to be linearly related to theCSA ( ,[ , , ] as ollows:

( ,) = 0 + 1 × CSA( ,), ( )where 0 is a convenient additive constant and is the IJVcompliance or unit o length de ned as

= CSA . ( )Givena pressurewave velocity , thewave equation statesthat

= ±1 . ( )Substituting (1/)CSA( , ) rom ( ) into ( ), the pres-sure gradient

/ is obtained. Te minus sign on the right

side o ( ) represents a wave moving toward the positivedirection o the -axis while the plus sign represents a wavemoving toward the negative . In order to simpli y theinterpretation o the results, in this paper we chose to assumethe positive direction o the rom the head toward the heart(see Figure ). Te pressure waves are generated rom theheart pulsation and propagate toward the jugular vein in thenegative direction; or this reason they are represented by positive sign in ( ). In this work we neglect the effects o re ected waves propagating rom the brain toward the heart.

Moreover, the Womersley equations only give the instan-taneous oscillating part o the blood velocity. Te net instan-taneous value o the blood velocity is given by summingthe steady velocity with the oscillating velocity. Te steady component o the velocity can be obtained by the Poiseuillelaw once the pressure gradient between the head and theheart is known. In this study we are not interested in a

ull assessment o the mean velocity so only the oscillating velocity is considered and analyzed.

Having assumed a positive rom the head to the heart,the net blood velocity is positive (see Figure ).. . IJV Blood Velocity Assessment: Te Oscillating Compo-

nent. For each subject, the CSA dataset, representing theinstantaneous value o the CSA during a cardiac cycle, wasderived rom theacquired video clip using thesemiautomatic


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87 96 106 ,5 9 ,58 ,57 ,5

Time (s)

0 ,15

0 ,2

0 ,25

0 ,3

0 ,35

C S A

( c m

2 )

IJV CSA

a

c

v

IJV CSA

86 7 95 108 ,55 ,5 9 ,56 ,5 7 ,5

Time (s)

1 ,0001 ,0501,100

1 ,1501 ,2001 ,2501 ,3001 ,3501 ,4001 ,450

C S A

( c m

2 )

F : Instantaneous cross-sectional area (CSA) o the internal jugular vein measured using ultrasound B-mode or about seconds.

algorithm described in [ ]. Te period o the IJV pulsationwas obtained by its discrete Fourier trans orm calculatedusing the Grace sofware [ ]. Ten we write = 2 ( / )−so that

runs rom

− to

during one pulse period. Te

mean CSA, denoted by CSA, over a period is calculated.Te Fourier coefficients o pressure gradient / up to thetenth harmonic are calculated. Tese coefficients are thenconverted to modulus and phase . Te blood velocity is then calculated rom Equation (4) in [ ] by substitutingthe pressure gradient expressed as a Fourier series. Te blood velocity ( ) is given by the summation o ten terms as inEquation (27) in [ ]:

( ,)= −CSA( ,)

2 10

2

=10

=1

sin + + 10 , ( )where we have substituted the term 2 in [ ] by CSA( ),which represents the time-dependent vein CSA. Te param-eter is the blood viscosity ( . Pa), that runs rom to is the order o the Fourier coefficient, and 10 and 10 arederived rom theBessel unction andtabulated in [ ]. Finally,

is the well-known Womersley number given by

= √2 , ( )where

is the blood density ( . g/mL) and

= CSA

/.

. . Spectral Doppler Averaged Velocity. Below we describethe procedure adopted to obtain a dataset sampled romthe US Doppler velocity trace. Common commercial USsystems do not allow the Doppler dataset to be exported;

or this reason, in this work, the mean velocity dataset isobtainedbydigitallyprocessing theimageshot o theDopplertrace. Te mean blood velocity ( ( )) is represented as aline overlapping with the spectral Doppler trace (Figure ).Te line is composed o pixels. An in-house developedprocedure identi es the pixel belonging to the line by itsRGB values. Te coordinates and o each pixel wereautomatically recorded by the procedure, where is an index

: ime period or jugular venous pulse (JVP) and spectralDoppler (SD) periodic diagram is reported or two healthy subjectstogether with the time averaged internal jugular vein (IJV) cross-sectional area (CSA) and the wave propagation velocity

.

Subject JVP period (s) SD period (s) CSA(cm2 ) (m/s) . . . . . . . .

going rom to . Te unction ( )has been obtained rom and values using the ollowing procedure: = ( ) = × 100, ( )

where is the index or the pixels and is the distance inpixels between two time divisions (separated by s) o thetimeaxeswhile isthe distance inpixelsbetween the and

cm/s divisions, measured along the -axis. Combiningthe two expressions in ( ) produces ≡ = × 100. ( )

Te sampled Doppler velocity dataset {}is then plottedtogether with the calculated Womersley averaged velocity ( ( )). Te instant o time corresponding to the maximum value o velocity was detected or both the Doppler and

(and it was used as re erence marker to overlap the two plots.. . . Detailed Calculation. Te detailed algorithm o allthe calculations needed to produce the ollowing results isreported in the appendix.

3. Results

. . IJVBloodVelocity Assessment. Each resultis produced orboth subjects #1 and #2 and presented in the text ollowingsuch order. Each gure starting rom Figure is divided intotop and bottom or subjects #1and #2, respectively. Te mainnumerical results are summarized in able .


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F : Ultrasound spectral Doppler trace o the right internal jugular vein blood ow. Te mean velocity trace is highlighted.

Te US-JVPare plotted orboth subjects in Figure ; theirrepetition periods are . and . s or subjects #1 and #2,respectively; the time averaged cross-sectional areas CSA are

. and . cm2 . Both subjects show clearly visible

and V

waves, while the one is barely detectable.Te time derivatives o the internal pressure and pressuregradient were obtained rom the CSA dataset as described inthe appendix.

Te mean blood velocity was calculated ollowing ( )and is plotted together with the CSA in Figure . It is worthnoting that or both subjects the blood velocity changesinversely with respect to the IJV CSA; that is, when the IJVCSAgoes toward itsmaximumtheblood velocity goestowardits minimum and vice versa.

In Figure , is plotted together with the pressuregradient /.. . Spectral Doppler Averaged Velocity. Figure depicts the

spectral Doppler velocity traces. Te time periods o suchtraces turned out to be . and . s or subjects #1 and#2, respectively (see able ). Such values are very close tothat o the US-JVP period; this is an expected result since thespectral Doppler trace was acquired as soon as the B-modeIJV investigation was completed so as to avoid changes inphysiological conditions.

It is evident that the traces o both subjects #1 and #2show a “two-wave”pro le. Figure depicts theblood velocity sampled rom the spectral Doppler versus the calculatedones or subjects #1 and #2, respectively. For both subjects,sampled and calculated velocity values show the same timedependence or, in other words, the same wave orm. Areasonable agreement o their amplitudes was obtained by requiring a CVP up to mmHg (see Appendix A. ordetails).

4. Discussion

. . IJV Blood Velocity Assessment. Te main result o thepresent study is the well apparent relationship between theblood velocity in the IJV measured by the current Dopplersystem and the same parameter derived rom its pulse. Forboth subjects the presence o the two positive peak waves,“ ” and “V ” waves, respectively, in the US-JVP (the CSA

Blood mean velocity CSAPressure gradient

CSABlood mean velocity Pressure gradient

0 ,1 0 ,2 0 ,3 0 ,4 0 ,5 0 ,6 0 ,7 0 ,8 0 ,90,0

Time (s)

−40,000

−30,000

−20,000

−10,000

0,000

10,000

20,000

30,000

40,000

d p

/ d z

( m m

H g / m

) , v e l o c

i t y

( c m

/ s )

1 ,0001 ,0501 ,1001 ,1501 ,2001 ,2501 ,3001 ,3501 ,4001 ,450

C S A

( c m

2 )

−60

−50

−40−30

−20

−10

0

10

20

30

40

d p

/ d z

( m m

H g / m

) , v e

l o c i

t y ( c

m / s

)

0 ,000

0 ,050

0 ,100

0 ,150

0 ,200

0 ,250

0 ,300

0 ,350

C S A

( c m

2 )

0,0 0 ,2 0 ,3 0 ,4 0 ,5 0 ,6 0 ,7 0 ,8 0 ,90 ,1

Time (s)

F : Te pressuregradient ( /) calculated using the“tube-law” is plotted together with the internal jugular vein blood mean velocity ( ) calculated using Womersley equations and the cross-sectional area (CSA). A single cardiac cycle is reported.

diagram), generates two descents in the assessed velocity time diagram within the same cardiac period. Although therelationship between the JVP and the variation o pressurein the heart has been known since [ ], until now, this

nding has not been taken into consideration or assessingthe brain out ow. Currently, the Doppler ow quanti cationis provided by the product o the time average velocity o ow and the CSA [ – ]. Tis quantitative approach currently providesan accepted objectivecriterion orassessing the ow rate. However, despite the pulsating variation during the car-diac cycle o the IJV CSA, the ow, according to this method,is calculated by using just a single CSA value. Tis raises thequestion as to whether overlooking the pulsatile variation o the IJV CSA may affect the US evaluation o cerebral venousreturn in the clinical setting. In addition, the results o thepresent study urther corroborate the role o the mechan-ical propagation o the cardiac contraction in the upwarddirection. Tis wave propagation is not haemodynamically negligible because the present study clearly demonstrateshow this phenomenon in uences blood ow velocity. Tereare several studies which suggest that ow waves directed


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0

20

40

60

80

100

V e l o c

i t y

( c m

/ s )

0,00 0 ,20 0 ,30 0 ,40 0 ,50 0 ,60 0 ,70 0 ,80 0 ,900 ,10

Time (s)

0 ,100

0 ,150

0 ,200

0 ,250

0 ,300

0 ,350

C S A

( c m

2 )

Blood mean velocity (Womersley)

Blood mean velocity (Doppler trace)CSA (cm2 )

0,00 0 ,20 0 ,30 0 ,40 0 ,50 0 ,60 0 ,70 0 ,80 0 ,900 ,10

Time (s)

0

10

20

30

40

50

60

70

V e l o c i t y

( c m

/ s )

1 ,0001 ,0501 ,1001 ,1501 ,2001 ,2501 ,3001 ,3501 ,4001 ,450

C S A

( c m

2 )

Blood mean velocity (Womersley)

Blood mean velocity (Doppler)CSA

F : Ultrasound spectral Doppler mean velocity ( ) is plottedtogether with the internal jugular vein blood mean velocity ( )calculated using Womersley equations. A single cardiac cycle isreported.

up towards the brain can contribute to neurodegenerativediseases by affecting cerebrospinal uid absorption and/orbrain per usion, such as Alzheimer [ , , ].

Our data clearly demonstrates how the assumption o thePoiseuille theorem in thecase o pulsatile ow mayexpose the

ow assessment to error. We ocused our study on US but o course the MRV methodology used or IJV ow assessmentcould also be affected by the same incorrect assumption. Weknow that the distal segment o the IJV is characterized by signi cant CSA variation over time, an effect linked with thetransmission o pressure waves rom the right atrium. Tisphenomenon, very well known as JVP, is strongly connectedwith important regulators o the cardiovascular unction:degree o lling, heart pulsation, and capacity o emptying o the IJV.

Moreover, the positive waves “ ,” “ ,”and“V ” clearly show a brain directed propagation o pressure waves with signi -cance never investigated, as yet. We know that propagationis greater when the degree o lling o the venous systemis higher and/or when the heart unction is compromised.Acute symptomsin the latterrapidly appear in thepulmonary

unction but we have never investigated i this may have

signi cance or cognitive or neurological decline already described in this particular category o patients.We speculatethat cardiac, carotid, and jugular signals should be synchro-nized and used to assess the circulatory axis between theheart and brain, with signi cant changes in current clinicalpractice, in neuroimaging as well.

Te results reported here can be reproduced using MRVbecause CSA and blood velocity can be measured at eachinstant o time as we did.

. . Pulsed Flow-Dynamics Assumptions. Te Womersley approach presented in [ ] derives the blood velocity rom thepressure gradient assuming a rigid tube as a model. Such amodel accounts or the pressure gradient time dependencebut neglects the movement o the tube wall in the longitu-dinal and lateral direction. In later papers, Womersley madecorrections accounting or such variability [ , ] resultingin a new equation having the same orm as the ormer (seeEquation

(12)in [ ]) and producing a ow about % higher

in magnitude. In this paper we choose to ollow the simplerapproach presented in [ ] or three reasons: (1) the citedpaper is widely known and widely adopted, (2)the presentedanalytical model can be easily implemented making ourresults easily reproducible, and (3) such a model results inan easily treatable linear equation and its results also hold orpulsing ow in an elastic tube [ , ].

However, we encourage the adoption o more sophisti-cated models when the blood ow has to be calculated orclinical reasons (see, e.g., [ ]).

Te assumption o simple linear or quadratic relation-ships between CSA and pressure seems to be correct in thepresent case. In act, considering the well-known “tube-law”

[ ], the CSA/perimeter correlation means that IJV pulsationlies in the curve region (i) where the transmural pressureis greater than the buckling pressure (see Figure 3 in [ ]);hence a linear relationship between pressure and CSA isstraight orward. On the other hand, the elliptical shape o the IJVduring itspulsation could suggest that the transmuralpressure is below the buckling value; hence region (ii) couldalso be affected by the pulsation. O course this nding needsto be urther investigated.

. . Implication for Pressure Wave. Te oscillating nature o the IJV is potentially relevant in the understanding o therelationship between brain drainage and several neurological

disorders [ – ]. Inparticular, theeffect o pulsatile owhasbeen recently hypothesized to be associated with retrogradehypertension transmitted rom the IJV which, presumably,underlies transient global amnesia [ ] and perhaps otherneurological disorders, as recently hypothesized [ – ].

Appendix

A. Velocity Calculation Detailed Procedure

A. . IJV Elasticity Model. In this paragraph we present anexpression or the vein volumetric compliance , that is,the ratio o volume variation

Δ with respect to pressure


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variation Δ, and consequently or the compliance per unito length . We assume = CSA

Δ = ΔCSA (A. )that implies the idea that the vein may change in radius butnot in length. From the de nition o volumetric compliance

it ollows that

= ΔΔΔ = ΔCSA .

(A. )

Te coefficient o unctional elasticity is de ned as ≡ ΔΔ ; (A. )

then, using the de nitions given in (A. ), we have

Δ = ΔCSACSA . (A. )From the two de nitions o Δ given in (A. ) and (A. ) itollows that

1 = CSA. (A. )For cylindrical geometry, the relationship between the YoungModulus and can be assumed to be [ ]

= 2ℎ , (A. )where

ℎ

is the vein wall thickness. Te pressure wave velocity , that is, the velocity o propagation o the JVP in the jugular

vein, is calculated ollowing the Moens-Korteweg equation:

= √ × ℎ2 (A. )which or the assumed relationship between and isdependent only on the parameters and :

= √. (A. )Te central venous pressure (CVP), measured close to the

IJV, is known to vary up to mmHg during the cardiac cycle.Since Δ during a cardiac cycle is given by Δ = 1 (max(CSA()) − min (CSA())) (A. )then

= max(CSA()) − min (CSA())5 . (A. )Equation (A. ) can also be used with (A. ) to determine which is given as

= CSA . (A. )

A. . Parameter Data. Te ollowing parameter value hasbeen assumed [ , ]:

(i) Blood viscosity = 4.5 × 10−3 Pa × s.(ii) Blood density = 1.04 × 103 kg/m 3 .

(iii) Vein wall thickness ℎ = 0.5mm. A. . Input Data. From the US transversal scan o the IJV the

ollowingdatarepresents the input or the velocity calculationprocedure:

(i) Te US-JVP, consisting in a dataset {CSA} with running rom to , where is the number o sono-grams in the US video clip with each element CSA o thedataset representing the IJVarea measurementonthe th sonogram, in pixels.

(ii) Te US video clip duration (

Δ) in seconds.

(iii) Te US sonogram pixel dimension ( ) in cm.

From the longitudinal US Doppler trace, consider theollowing:

(i) Te space averaged velocity time diagram, consistingin a dataset {} in pixels.

(ii) Te distance ( ) between two divisions in the timeaxis o the Doppler trace.

A. . Cross-Sectional Area

( ) {CSA} dataset discrete Fourier trans orm (DF ) isoperated (e.g., using Grace sofware).( ) Te sonogram repetition period ( ) o the {CSA}dataset is obtained by the maximum o the DF

produced at (1).( ) Te sonogram time interval ( Δ) is calculated asΔ =Δ /.( ) Te {CSA} dataset time repetition period is thencalculated as = × Δ.( ) Te

{CSA

} dataset is multiplied by the pixel area

(2

) in order or each element CSA o the datasetto represent the IJV area measurement on the thsonogram, in cm 2 .

( ) Te differential CSA dataset {ΔCSA} is obtained as{ΔCSA} = ΔCSA +1 − ΔCSA .( ) Te discrete partial time derivative {ΔCSA/Δdataset is obtained dividing the {ΔCSA} by Δ.( ) Average CSA is calculated as CSA = ∑ CSA/ .( ) Te average vein diameter is calculated as = 2

CSA

/.


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A. . Pressure. Te ollowing steps ocus on the pressuredataset calculation:

( ) Te compliance orunit o length is calculated as =(Max({CSA})−Min(CSA))/CVPmax , where CVP maxis a convenient parameter representing the maximum value reached by the CVP during the cardiac cycle. Atypical value is mmHg.

( ) Te pulse wave velocity ( , lower case) is calculated as = / × ℎ/ = /.( ) ime derivative pressure dataset, in unit o mmHg, is

given by {Δ/Δ } = (1/){ΔCSA/Δ }.( ) It is then converted into Pa by multiplying the{Δ/Δ } dataset by . Pa/mmHg.( ) Te pressure gradient along the -axis direction iscalculated as {Δ/Δ } = ±(1/ ){Δ/Δ }.( ) Te parameter isde nedas = 2 (( −1)×Δ )/ −.( ) Te pressure gradient is then de ned as DZP ( ) =Δ/Δ.

A. . Pressure Gradient Fourier Series. Te pressure gradientis then decomposed in its Fourier terms as

DZP() = 0 +

=1

cos × + . (A. )Te series coefficients are calculated as ollows:

( ) 0

= (2/2 )∑=1

(Δ/Δ ) × (2 / ).( ) = (2/2 )∑=1(Δ/Δ )cos( × ) × (2 / ).( ) = (2/2 )∑=1(Δ/Δ )sin( × ) × (2 / ).( ) = 2 + 2 .( ) = −arctg( / ).

A. . Blood Velocity Fourier Series. Te blood velocity expres-sion is obtained by the pressure gradient expression as

() = −CSA⋅ 1 0 +

=1

102 sin + + 10 .

(A. )

( ) Te parameter or each harmonic is calculated as = ( /2) (2 / )( / ).( ) Te coefficients 10 / 2 and 10 are tabulatedin [ ] oreach value rom to . For > 10 the asymptoticexpression was used [ ].

Abbreviations

Acronyms

CCSVI: Chronic venous cerebrospinalinsufficiency

CSA: Cross-sectional areaCVP: Central venous pressureDF : Discrete Fourier trans ormDZP: Pressure gradientECD: Echo color DopplerECG: ElectrocardiogramIJV: Internal jugular veinJVP: Jugular venous pulseMRV: Magnetic resonance venography

: Distance in pixels between two timedivisions

: Distance in pixels between the andcm/s divisions

: Pearson coefficientRGB: Red Green Blue

: Sonogram repetition periodUS: Ultrasound.

Symbols

: Womersley number: Pressure waves velocity : Compliance: Compliance or unit o length

Δ: Sonogram time interval

: Coefficient o unctional elasticity

ℎ: Vein wall thickness: Blood viscosity : Young Modulus: Pixel dimension

: Pressure: Blood density : Cardiac period

V : Volume: Blood ow velocity averaged over the IJV CSA.

Conflict of Interests

Te authors declare that they have no con ict o interests.

Authors’ Contribution

Francesco Sisini conceived the study, developed the algo-rithm or the data analysis, analyzed the US studies, per-

ormed the data analysis, and wrote the paper rom thescratch. Eleuterio oro wrote the paper and supervised allthe mathematical aspects o the research. Mauro Gambacciniwrote the paper and critically revised it. Paolo Zamboni ana-lyzed the jugular diagram, wrote thepaper, andsupervised allthe medical aspects o the research.


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Acknowledgments

Tis study was partially supported by the Italian Min-istry o Education, University and Research (MIUR Pro-gramme PRIN - ), Grant no. XE L R, andthanks are due to Jenni er Cowd, Xpress ranslations Ltd.([email protected]) or the proo reading o thepaper.

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