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CDa

b

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d

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7

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K

F

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c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 9 8 ( 2 0 1 0 ) 151–160

journa l homepage: www. int l .e lsev ierhea l th .com/ journa ls /cmpb

omersley number-based estimation of flow rate withoppler ultrasound: Sensitivity analysis andrst clinical application

hristian Vergaraa,∗, Raffaele Ponzinib, Alessandro Veneziani c, Alberto Redaelli d,anilo Negliae, Oberdan Parodi e,f

Department of Information Technology and Mathematical Methods, Università degli Studi di Bergamo, ItalyCILEA (Consorzio Interuniversitario Lombardo per l’Elaborazione e l’Automazione), Segrate (MI), ItalyDepartment of Mathematics and Computer Science, Emory University, Atlanta, GA, USADepartment of Bioengineering, Politecnico di Milano, ItalyCNR Clinical Physiology Institute, Pisa, ItalyCardiovascular Department, Niguarda Ca’ Granda Hospital, Milan, Italy

r t i c l e i n f o

rticle history:

eceived 9 December 2008

eceived in revised form

September 2009

ccepted 21 September 2009

a b s t r a c t

In this paper we continue in investigating the approach we have proposed in a paper recently

published, for a reliable estimate of (peak systolic) blood flow rate from velocity Doppler

measurements. Basic features of this approach together with some in silico test cases were

discussed in that work. Here, we provide more insights of this approach by performing a sen-

sitivity analysis of the formulas relating blood flow rate to velocity. In particular we analyze

how our estimates are affected by perturbation or errors in measurements in comparison

eywords:

low rate estimation

oppler measurements

omersley number

oronary flow reserve

with a standard method for catheter based estimates based on the assumption of a parabolic

velocity profile. A first glance to in vivo clinical applications is given as well.

© 2009 Elsevier Ireland Ltd. All rights reserved.

. Introduction

he correct estimation of blood flow rate Q through a vascu-ar surface is a major issue in clinical practice, since it could

ive important informations about the cardiovascular state ofpatient. This can be pursued with a good precision by usingifferent approaches, such as the Electromagnetic flow meter

Abbreviations: PC-MRI, phase-contrast Magnetic Resonance ImagingNR, Consiglio Nazionale delle Ricerche; LAD, left anterior descending (lectrocardiogram.∗ Corresponding author at: Università degli Studi di Bergamo, viale Mar

ax: +39 035 562779.E-mail address: christian.vergara@unibg.it (C. Vergara).

169-2607/$ – see front matter © 2009 Elsevier Ireland Ltd. All rights resoi:10.1016/j.cmpb.2009.09.013

(see, e.g., [27,5]), the transit time thermodilution (see, e.g., [3]),the phase-contrast Magnetic Resonance Imaging (PC-MRI) (see, e.g.,[1,2]) and the Doppler-based technique (see [10,15,35]). The latterapproach estimates quite accurately blood velocity by measur-

; CFR, coronary flow reserve; CFD, Computational Fluid Dynamics;one of the coronary arteries); IVUS, intravascular ultrasound; ECG,

coni 5, 24044 Dalmine (BG), Italy. Tel.: +39 035 2052314;

ing the difference in frequency between a transmitted waveand the reflected signal (Doppler shift effect). In particular inthe continuous Doppler velocity the signal transmission is basedon a continuous wave (see [12]), while in the pulsed Doppler

erved.

s i n

152 c o m p u t e r m e t h o d s a n d p r o g r a m

velocity short pulses are transmitted (see [18]). Moreover, in thesingle-point Doppler method the reflected signal is retrieved ina single point of the vascular section. Other multi-gate devicesacquire information over a large portion of the section. Amongsingle-point Doppler techniques, one is based on the acquisi-tion of the maximum velocity over the section at hand throughthe introduction of an intravascular guide-wire. The underlyingassumption is that the sample volume is small compared tothe vessel. Since only the maximum velocity can be measured,an assumption on the velocity spatial profile is needed in orderto recover a flow rate estimate (see [20]). Within the multigatetechniques no assumption concerning the velocity profile isneeded since the velocity can be partially or totally retrieved.For example, with the uniform insonation method described in[12] an estimate of the mean velocity is obtained by the spec-trum measured by the Doppler. However, sufficient uniformityof the sample volume is difficult to achieve (see, e.g., [17]).Alternatively, other techniques allow to estimate the wholevelocity profile at the section at hand, under the assumptionthat the sample volume is large compared to the vessel, with-out any further hypothesis on the velocity across the section(see, e.g., [52,28]). Finally, Color Doppler technique allows tomeasure the Doppler shifts in a few thousand sample volumeslocated in an image plane (see, e.g., [53,30]). Unfortunately,this technique has a low velocity resolution and therefore itis mainly used for qualitative investigations (see [53]). In [54]a combination of pulsed-wave Doppler and Color Doppler hasbeen proposed to overcome the limitations of both these tech-niques. Nowadays none of such techniques seems to be closerto a standard clinical practice for the estimation of flow rate.Different techniques are well suited for different purposes. Forexample, the single-point Doppler is largely the most usedto estimate the flow rate for small vessels such as coronar-ies, where other Doppler-based methods are hardly applicable.This is based in general on the assumption of a parabolic veloc-ity profile for estimating flow rate from maximum velocity.This method works fairly well for small districts and is muchsimpler than other algorithms based on Doppler velocity (see[20,35]). Available highly accurate velocity measures at differ-ent anatomical sites make the single-point Doppler approachattractive also for other other vascular districts (see [36,25] inthe pulmonary artery, [49,44] in the renal artery, [46] for theaorta and [57] for the great cardiac vein). Also for the deter-mination of the stroke volume in the left ventricular outflowtract the conventional single-point Doppler method has beenclinically widely accepted (see [8]). In this case, the estima-tion of the flow rate is based on the assumption of flat velocityprofile.

There are mainly two estimates of the flow rate obtainedwith the single point Doppler method used in the clinicalpractice. The flow rate at the peak velocity instant, that isthe instant where the velocity at the center-line is maxi-mum over the whole cardiac cycle (referred in the sequelas peak flow rate, see, e.g, [46,11,51]) and the time averagedflow rate, that is the mean value of the flow rate over thecardiac beat (referred in the sequel as mean flow rate, see,

e.g., [44,49]). In this paper we mainly focus on the peak flowrate even if the mathematical approach is readily extendedto other flow rate estimates, as we will point out (Remark2.1).

b i o m e d i c i n e 9 8 ( 2 0 1 0 ) 151–160

It is worth stressing that the a priori assumption of spa-tial velocity profile (usually a parabolic one) required by thesingle-point Doppler method introduces a bias in any flowrate estimate with potentially remarkable consequences. Forwhat concerns the mean flow rate, even if some authorssupported using a parabolic velocity profile (see [6,29]), inmore recent years this has been considered too simplisticand misleading (see [43,21,22]). The limitations obtained byestimating the peak flow rate by using a parabolic velocityprofile have been pointed out in [40] where a mathemat-ically more sophisticated approach has been proposed. Inparticular, a new parametrized formula linking the maximumvelocity and the flow rate at the peak instant has been intro-duced, with an explicit dependence on the heart pulsatilitythat is discarded in the parabolic assumption. The quantita-tive estimation of parameters is based on new methods ofComputational Fluid Dynamics (CFD) (see Section 2). In thispaper we continue the investigation of this approach witha sensitivity analysis of the proposed formula with respectto the measures of the velocity and the diameter. We showthat this formula has stability features comparable with theparabolic one (Section 3). Moreover the two methods, namelythe one based on the parabolic assumption and the new one,are applied to a clinical dataset retrieved from the databaseof the CNR Clinical Physiology Institute of Pisa. In particu-lar, we aim at estimating the coronary flow reserve (CFR) in agroup of patients. Single-Doppler peak flow rate based on theparabolic assumption has become the standard practice forthis application where the knowledge of CFR is needed, suchas during catheterisation and in percutaneous transluminalcoronary angioplasty (see, e.g., [24,23,19,31,58,32,4]). Compar-isons between classical and new estimates suggest that thenew approach introduces a significant improvement with noextra cost in medical devices or clinical procedures (Section4).

2. The parabolic and the Womersley-basedformulas

Let � be a cross-section of the vascular district at hand. Themass flow rate Q(t) through � is defined as

Q(t) =∫

�

�u · nd�, (1)

where � is the blood density (hereafter assumed to be con-stant), u(t, x) the blood velocity, n the normal unit vector andd� the infinitesimal area element. As anticipated in the Intro-duction, we focus on the peak instant t. In principle, the wholevelocity field u(t, x) on � is needed for estimating Q := Q(t).However, Eq. (1) can be rewritten in terms of the mean velocityvalue U := U(t) as

Q = �UA (2)

where A := A(t) and A denotes the area of section � . Unfortu-nately, mean velocity U is not available from measures. Onthe other hand, as pointed out in the Introduction, single-point Doppler velocimetry analysis provides reliable measures

i n b

o�

mp

U

Tt(fiTefvqe

W

HiwtpV

smHi

pIvid

U

wfpe

aiPu

rate. Alternatively, an ad hoc formula still in the form (6) for the

c o m p u t e r m e t h o d s a n d p r o g r a m s

f the maximum velocity at the peak instant VM := VM(t) on. Eq. (2) requires therefore an appropriate formula relatingean velocity U to the maximum one VM. In current clinical

ractice it is usually assumed

ˆ = VM

2. (3)

his equation stems from the hypothesis of a parabolic spa-ial profile for the velocity. For this reason in the sequel Eq.3) will be referred to as parabolic formula. Strictly speaking thisormula assumes that blood flow is steady or quasi-static, lam-nar and Newtonian in a rectilinear cylindrical vessel (see [35]).hese assumptions are far to be fulfilled in real situations (see,.g., [47,39,50,7]). In particular, it has been pointed out by dif-erent authors the relevance of blood flow pulsatility on theelocity profiles ([59,14,35]). In [40] it has been proposed touantify pulsatility with the adimensional index (called Wom-rsley number)

= D

2

√2�f

�. (4)

ere D := D(t), D is the vessel diameter, � the blood viscos-ty and f the main frequency of the heart beat. This choiceas motivated by observing that W can be evaluated be using

he quantities retrieved during standard Doppler acquisitionrocedure. Obviously, a more accurate spectral evaluation of

M(t) would lead to better evaluation of the incidence of pul-atility and eventually to the flow rate, by taking into accountore frequencies of the heart beat together with the main one.owever, this would need specific acquisition procedures and

t would be not of practical use.The higher the value of W the more the assumption of

arabolic velocity profile is incorrect (see, e.g., [59,14,35,40]).n [40] improved blood flow rate estimates from maximumelocity have been devised by exploiting CFD results. The basicdea was to generalize Eq. (3), by introducing an explicit depen-ence on the Womersley number of the mean velocity,

ˆ = g(VM, W), (5)

here g is a suitable function. For this reason, in the sequel,ormula (5) will be referred to as Womersley-based formula. Morerecisely, three ranges of the Womersley number are consid-red and associated with three different formulas:

Q = �Ag1(VM, W) = 12

�AVM(1 + a1Wb1 )

for 0 < W ≤ 2.7, (6a)

Q = �Ag2(VM, W) = 12

�AVMb2 arctan(a2W)

for 3.1 ≤ W ≤ 15, (6b)

Q = �Awg1(VM, W) + �A(1 − w)g2(VM, W)

for 2.7 < W ≤ 3.1. (6c)

Here, a , a , b and b are parameters to be determined by

1 2 1 2

fitting procedure and w = w(W) is a weight function mix-ng the formulas for low and high values of W respectively.arameters quantification was based on 200 numerical sim-lations in cylindrical geometries performed on the entire

i o m e d i c i n e 9 8 ( 2 0 1 0 ) 151–160 153

heart cycle, for different shapes of the flow rate (prescribed asboundary conditions) and values of the Womersley number.Flow rate boundary conditions were prescribed without anybiased arbitrary assumption on the velocity profile, followinga new mathematical technology proposed in [9] and analyzedin [55,56]. A non-linear least squares approach has been thenused for fitting the results of numerical simulations (see also[38,37]). This allows to obtain the following parameters (see[40]){

a1 = 0.00417, b1 = 2.95272a2 = 1.00241, b2 = 0.94973

,

and

w(W) = e

(W−2.7)2

(W−2.7)2−0.42 (7)

where the latter function has been devised in order to guar-antee a mathematically smooth superimposition of the lowand high Womersley numbers formulas. Preliminary valida-tion in [40] has been based on in silico test cases, i.e., onnumerical simulations performed in cases different from theones used for fitting formula (5). These results show that thenew formula improves blood flow rate estimation with respectto the one based on (3). In some cases the improvementswere remarkable. We stress that since formulas (6) establish adependence of the flow rate at the peak instant on the Wom-ersley number W, they are able to describe in a more realisticway different blood flow regimes. For W = 0 (steady condi-tions) from (6a) we recover the parabolic formula (3). However,for the same reason formulas (6) are more delicate in termsof sensitivity from the data, being estimates possibly pol-luted by error on maximum velocity VM, diameter, frequencyand viscosity measures. On the contrary, parabolic formula (3)is independent of frequency and viscosity. This means thaterror in measuring these parameters do not affect the esti-mate. On the other hand, this formula is unable to accountfor flow rate modifications induced by a physical change ofviscosity or pulsatility, as we have pointed out. In the nextsection we investigate with more detail the sensitivity of thedifferent formulas on the parameters that are subject to mea-sures, so to quantify the impact of possible errors on the finalestimates.

Remark 2.1. It is worth pointing out that from the mathemat-ical viewpoint the time when the velocity is measured is notrelevant. In other words, our procedure can be carried out forany given instant t when VM(t) is collected. A formula similarto (6) can be fitted in the same manner. This means that ourapproach is readily extended for evaluating also the minimalflow rate, if VM measures are available when the maximum (inspace) velocity is minimal over the heart beat. If a sequence ofmeasurements VM(ti) for i = 1, 2, . . . , N is available, a time aver-age of the maximum velocity and of the corresponding flowrates can be computed so to have an estimate of the mean flow

mean flow rate could be fitted following the same guidelinesadopted for the peak flow rate. Which is the best approach forevaluating the mean flow rate is still an open question and itcould be subject of future developments of the present work.

s i n

154 c o m p u t e r m e t h o d s a n d p r o g r a m

3. Sensitivity analysis

3.1. Amplification factors

In order to evaluate sensitivity on measurements errors of for-mulas (6) in comparison with (3), we introduce an index �,called amplification factor, as follows. Let us consider a genericfunction y = y(x). Suppose that x could be subject to perturba-tions ı possibly induced by measurements errors. Our goal isto evaluate the ratio between the relative errors, namely

� = (y(x + ı) − y(x))/y(x)ı/x

= y(x + ı) − y(x)y(x)

x

ı.

Let ı tends to 0, so we finally obtain

� = dy

dx

x

y. (8)

This amplification factor states the impact of a perturbationon x on the result y(x). In the context of numerical analysis,� is called condition number of y(x) (see, e.g, [45]). Observe thatfrom the definition we have

y(x + ı) � y(x)(

1 + �ı

x

), (9)

the approximation being good for ı small enough. Whenquantity y depends on more variables xj, j = 1, 2, . . ., partialderivatives are to be taken into account in order to weight thedependence of y on each independent variable xj separately. Inthis way, the sensitivity indeces read �j = (∂y/∂xj)(xj/y). Largevalues of � mean that small perturbations on x (due, for exam-ple, to an error in the measurement) could strongly affect theestimate y. The sensitivity of the estimate has not to be con-fused with its accuracy, that is how this estimate y is “close”to the real value yex. Hereafter, we focus our attention on thedependence of our formulas on the measure of the maximumvelocity and of the diameter, which are those parameters informulas (3) and (6) likely to be most operator-dependent.

3.1.1. Sensitivity to the maximum velocityAll the proposed formulas depend linearly on VM, i.e., are inthe form

Q = c(W)VM

where c(W) is a function of the Womersley number (and inparticular a constant for the parabolic formula (3)). From (8)we have for all the formulas considered here

�VM= ∂Q

∂VM

VM

Q= c(W)

VM

c(W)VM

= 1. (10)

Sensitivity of the formulas to maximum velocity measures istherefore the same. A possible error ı on this measure affectsflow rate estimates, both with the parabolic and with theWomersley-based formula with a perturbation of the sameorder of ı.

3.1.2. Sensitivity to the diameterSensitivity on the diameter D is even more critical with respectto the operator skills and experience. The sensitivity index

b i o m e d i c i n e 9 8 ( 2 0 1 0 ) 151–160

related to D is given by

�D = ∂Q

∂D

D

Q, (11)

where Q is one of the formula for the estimation of the flowrate. By a simple application of (11) and recalling that A =�D2/4, we obtain the following sensitivity indexes:

1. Parabolic formula: in this case we have from (3)

�D,parabolic = ��DVM

4D

(�/2)(�D2/4)VM

= 2. (12)

2. Womersley-based formula for small Womersley numbers: from(6a), by algebraic manipulation we have

�D,g1= 2 + b1

a1Wb1

1 + a1Wb1. (13)

3. Womersley-based formula for large Womersley numbers: from(6b), we obtain

�D,g2= 2 + a2W

(1 + a22W2) arctan(a2W)

. (14)

4. Womersley-based formula for intermediate Womersley numbers(Eq. (6c)): in this case computations are made more diffi-cult by the presence of the weight function w that dependson D through the Womersley number. Let us introduce thefollowing notation:

�12 = wg′1

(1 − w)g2D, �21 = (1 − w)g′

2

wg1D, �w = w′

wD,

�w12 = (g1 − g2)w′

g2D.

Then, it is possible to verify that

�D,g12= 2 +

(1

�−1g1 + �−1

12

+ 1

�−1g2 + �−1

21

+ 1

�−1w + �−1

w12

). (15)

3.2. Forward and backward analysis of perturbations

Sensitivity analysis can be profitably used for evaluating theaccuracy of the flow rate estimates in presence of measure-ment errors.

Computation of a dependent variable y = y(x) based on ameasure of the independent one x is affected by two kind oferrors. The first one is the measurement error on x. The secondone is the approximation introduced by the estimation itself.In the forward sensitivity analysis introduced in the previoussection the effects of perturbations of x on the final resultsare considered. In the backward analysis, errors are evaluatedby quantifying perturbations to be impressed on the originaldata such that, with the approximated estimation procedure,

the final estimation is exact.

With reference to Fig. 1, the solid line corresponds to theexact calculation of yex in x, whilst the dashed line correspondsto the approximated computation which leads to yappr(x).

c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 9 8 ( 2 0 1 0 ) 151–160 155

Fig. 1 – Abstract representation of forward and backwardimpact of data perturbations. Improvement on the resultop

TpplIts

|

L

H

Hs

i

Ts

ı

Nhsgacc

3

3Ib2

Fig. 2 – Amplification factor �D for Womersley-basedformulas as a function of the Womersley number W.Dashed line: index for the original formula featuring theexponential weight function (7) for Womersley numbers in

btained by an approximated process can be the result of aerturbation on the data.

hen, in the spirit of the backward analysis, we look for theerturbation on the data ı such that the approximated com-utation applied to the perturbed data x + ı is equal (or at

east “close”) to the exact value, namely yappr(x + ı) = yex(x).n other words, we look for a perturbation ı that compensateshe approximation of the process. Actually, we look for ı > 0uch that

yex − yapp(x + ı)| < |yex − yapp(x)|. (16)

et us assume the following hypotheses:

1. x > 0, yapp > 0, y′app > 0 (and then � > 0).

2. the approximation process is affected by a constant biasuch that yapp(x) < yex.

By exploiting Eq. (9) and assuming that H1 and H2 hold,nequality (16) becomes

yapp(x) − yex < yex − yapp(x + ı) � yex − yapp(x)(

1 + �ı

x

)⇒ 2(yapp(x) − yex) < −� ıyapp(x)

x< 0.

hen, under the previous assumptions the latter inequality isolved by

<2(yex − yapp)x

� yapp(x). (17)

otice that as a consequence of our hypotheses, the rightand side is positive. This means that positive perturbationmall enough on the independent data does not necessarilyet worse estimates. If ı fulfills (17), measure based on x + ı isctually more accurate then on x. A small overestimation of x

an partially balance the intrinsic error in the approximatedomputation.

.3. Results and discussion

.3.1. Sensitivity to the diametern Fig. 2 we illustrate the stability index �D of the Womersley-ased formulas as a function of W. We observe that for W ≤.7 and W ≥ 3.1, Womersley-based formula is slightly more

the range 2.7 < W < 3.1. Solid line: index for the modifiedformula with the linear weight (18).

sensitive than the parabolic formula, as was to be expectedsince this formula actually depends on the Womersley num-ber, that in turn depends on the diameter. In particular, forW < 2.7 the sensitivity increases with the Womersley num-ber, while for W > 3.1 it decreases with W. In this range, theincrement of �D is in any case less than 13% of the index ofparabolic formula.

On the contrary, for 2.7 < W < 3.1, we observe that theamplification factor increases up to 68% with respect to theindex of parabolic formula. This stems from the fact that inthis range of the Womersley number, our formula is given bya weighted linear combination of g1 and g2. In the superim-position of the effects the amplification factor is affected bythe sum of the two contributions. In order to reduce the sensi-tivity of the “weighted” Womersley-based formula, we modifythe weight function w(W) in (6). In particular, we propose thelinear function

w(W) = 3.1 − W

0.4. (18)

From the mathematical viewpoint, this choice introduces aless regular function. Indeed, the Womersley-based formulaover the entire range of physiological ranges of Womers-ley numbers will be only continuous, with discontinuousderivatives. However, it reduces the sensitivity to D of theWomersley-based formula in the range W = (2.7, 3.1), asshown in Fig. 2, since it features a slope smaller than withthe weight (7). The amplification factor reduces to 38% morethan the one of parabolic formula.

It is important to outline that, while the stability ofthe Womersley-based formula with weight (18) is greatlyimproved, the accuracy is maintained. This is confirmed bynumerical results referring to the same in silico validation testcases considered in [40]. We apply the original and the modi-

fied Womersley-based formula (given by weights (7) and (18),respectively) to the brachial flow wave test case. The resultsin Table 1, show that the accuracy of the Womersley-basedformula is not worsened.

156 c o m p u t e r m e t h o d s a n d p r o g r a m s i n

Table 1 – In silico validation: comparison between therelative errors obtained with formulas (3) (Ep), (6b) withweight (7) (Ew) and (6b) with weight (18) (Ew

mod).

Ep Ew Ewmod

W = 2.868 18.42% 9.52% 8.03%W = 3.049 18.17% 2.77% 3.33%

To be more concrete, we detail some examples of clinicalrelevance (in all the examples we set � = 0.035 Poise).

(1) Coronary vessel: Let us consider the measure of flow rateat the peak instant in a coronary vessel. We assume that thediameter of such a district at that instant is D = 2 mm. Inbasal conditions, frequency f = 1 Hz, consequently W = 1.34.For example if the error in the measurement of the diameterwere equal to 10%, the perturbation induced by the parabolicformula would amount to 20% (see (12)), while from (13) it fol-lows that the perturbation of the Womersley-based formulawould amount to 20.6%.

If an applied stimulation elicits an increase in cardiacchronotropism, such as during pharmacological stress test(dobutamine) or physical stress, like during exercise, the heartrate may increase two–three-fold with respect to baselinevalues. In these circumstances the Wormersley number willincrease. In particular, if we assume f = 3 Hz, corresponding toW = 2.32, the perturbation of the Womersley-based amountsto 22.8%.

(2) Brachial artery: Let us consider the brachial artery: in thiscase, a possible value of the diameter is D = 4.2 mm and thenthe Womersley number in basal conditions (f = 1 Hz) would beW = 2.81. In this case we refer to the perturbation (15) of theweighted formula (6c) with weight (18). With a 10% error in thediameter measure, the perturbation of the Womersley-basedformula given by (15) would be of 37.4% (20% for the parabolicformula).

(3) Femoral artery: Here we can assume D = 10 mm. In basalconditions we have W = 6.69 and the perturbation of theWomersley-based formula given by (14) would amount to21.0%, while under adenosine we have W = 11.60 and the per-turbation would be of 20.5%, versus the 20% of the parabolicformula.

3.3.2. Overestimation of the measures

Referring to Section 3.2, in our application the datum x is thediameter D or the maximum velocity VM at the peak instant,whereas the calculations yex and yapp are the exact and theestimated flow rates at that instant Qex and Qapp, respec-tively.

Let us focus on the parabolic formula, that is Qapp is givenby (3). It is worth noting that clinical evidence (see, e.g., [7])suggests that parabolic formula underestimates the real flowrate, Qex > Qapp. In other words, there is a systematic error witha constant bias (i.e., Qex − Qapp > 0 constantly). Moreover, weremark that D > 0, Qapp > 0 (if we focus on downstream fluxes)and Q ′

app = ∂Qapp/∂D > 0 for construction. This means that H1

and H2 in Section 3.2 hold and we can apply (17).

For example, using one of the in silico test case shown in [40],we have VM = 442.10 mm/s, D = 1.2 mm, Qex = 1000 mm3/s,Qapp = 987.6 mm3/s, W = 1.737. From (10), (12) and (17), it fol-

b i o m e d i c i n e 9 8 ( 2 0 1 0 ) 151–160

lows that an error on the maximum velocity at the peakinstant fulfilling 0 < ı < 11.102 mm/s leads to a better estimateof the flow rate. A small positive perturbation on the mea-sures of VM and D can improve the flow rate estimate basedon (3). Therefore, the application of the results obtained inthe previous section and in particular of the estimate (17) hasan immediate practical consequence: when different measuresof VM or D are available it is worth retaining the largest one,since small overestimations can partially balance the errorsintrinsic to the parabolic formula.

In the case of Womersley-based formulas, there is no avail-able experimental evidence of a constant bias in flow rateevaluation, so it is not possible at the moment to give anypractical suggestions. Numerical in silico results presented in[40,41] suggest however that also this formula features a con-stant underestimation (even if sensibly reduced with respectto the parabolic one as will be illustrated in Section 4). Ifthese results will be confirmed by in vivo validation, then theindication moving towards an overestimated value of the max-imum velocity (and possibly of the diameter) will apply to theWomersley-based formula as well.

4. Some steps to “in vivo” validation

Validation of (6) in [40] was based on CFD results, by perform-ing numerical simulations in geometries and regimes differentfrom the ones used for fitting the parameters in such formula.In [41] Womersley-based formulas have been applied to Y-graftbypass. The advantage of in silico test cases is that prescriptionand comparison of data is completely under control. Resultsobtained in this way show that Womersley-based formula cansignificantly improve flow rate estimates at the peak instantin comparison with parabolic formula.

Our next step is in vivo validation. In what follows weprovide a first clinical application of the Womersley-based for-mula. We point out that this application is just preliminary,even if it is an important step in that direction.

Among all the clinical flow rate applications, we havefocused on catheter-based Doppler ultrasound velocimetryanalysis for the measurement of the CFR. We point out thatif the shape of the velocity profiles at the two conditions werethe same and the area could be assumed constant, the CFRcould be estimated with a good precision by the ratio betweenthe velocities. However, in [43] it has been pointed out thatthis assumption can be a source of error. For this reason, weneed a good estimate of the flow rate to obtain a reliable CFRmeasure. In particular, a 2.9 F, 10 MHz intravascular ultrasound(IVUS) catheter (Eagle Eye Gold, Volcano Corp., San Diego, CA,USA) was passed over the flow wire. This application is one ofthe most relevant in clinical environment (see [13,32,4]). In par-ticular, the CFR has been extensively used to assess coronaryvasomotricity in patients with coronary artery disease (CAD)(see [48]). CFR is known to be defined as the ability of coronaryvessels to increase blood flow adjusting it for the myocardium

demands for oxygen and energy. CFR can be defined as theratio between the flow rate QS measured at the peak instantin a coronary vessel during maximal vasodilation and the flowrate QR measured at the peak instant in resting conditions,

i n b

t

C

Tptm

sopwbb[atrej

idonpca(amaaaEppcaIpcsvaposvf(nwswr¡a

rate in resting conditions and under adenosine obtained withthe parabolic formulas is Q

pR = 1.80 ± 0.44 and Q

pS = 4.27 ±

1.33, respectively, while for the Womersley-based formula weobtain Qw

R = 1.98 ± 0.49. and QwR = 4.89 ± 1.76.

Fig. 3 – Estimation of the coronary flow reserve (CFR)obtained with the parabolic and with the Womersley-based

c o m p u t e r m e t h o d s a n d p r o g r a m s

hat is

FR = QS

QR

. (19)

herefore, CFR could represent a clinical diagnostic andrognostic index concerning the coronary vessel abilityo increase flow proportional to increases in myocardial

etabolic demand.We have applied the Womersley-based formula (6) to 13

ubjects, 8 with idiopathic dilated cardiomyopathy and 5 with-ut cardiac diseases (control group). To guarantee an unbiasedrocess, this has been done in a blind fashion that means thate did not know a priori which of the patients were affected

y an idiopathic dilated cardiomyopathy. The patients haveeen selected from the database of CNR of Pisa, Italy (see

33] for details). The patients signed an informed consentnd the study was approved by the Local Ethical Commit-ee and conformed to the Declaration of Helsinki on humanesearch. (World Medical Association Declaration of Helsinki:thical principles for medical research involving human sub-ects. JAMA 2000;284:3043–3045).

In particular, concerning the data we are going to analyzen this paper, following previous publication [33], we brieflyescribe the acquisition procedure. In the morning, after anvernight fast, all patients were submitted to routine coro-ary angiography and definitively enrolled after excluding theresence of significant coronary stenosis. Measurements oforonary perfusion pressure, flow velocity, and cross-sectionalrea in the proximal third of the left anterior descendingLAD) were performed at completion of diagnostic coronaryngiography. A 6F guiding catheter was placed in the leftain coronary ostium through a 7F femoral artery introducer

fter an intracoronary heparin bolus (100 U/kg) was given and,0.014-in. Doppler flow wire (FloWire, Volcano Corp.) was

dvanced into the LAD. A 2.9 F, 10 MHz IVUS catheter (Eagleye Gold, Volcano Corp.) was passed over the flow wire andositioned in the LAD immediately distal to the first septalerforating branch. The Doppler flow wire was positioned twoentimeters distal to the tip of the IVUS catheter to avoidrtifact caused by the catheter wake. The position of theVUS catheter and flow wire was documented by angiogra-hy and maintained throughout the study by fluoroscopicontrol. After calibration, phase and mean coronary perfu-ion pressure (from the guiding catheter) and coronary flowelocity signals were continuously recorded on paper andcquired, together with ECG (leads I–II–III), on a personal com-uter equipped with dedicated software. IVUS images werebtained for a time greater or equal then 30 seconds at eachtep of the protocol, with temporal synchronization with flowelocity signal, and recorded for off-line analysis (see [33]or further details). After the instrumentation was completedabout 30 min), continuous monitoring of coronary hemody-amic signals was started and the baseline measurementsere registered (baseline). Afterwards, an intravenous adeno-

ine infusion (140 �g/kg/min) was given for 3 min unless it

as not clinically tolerated or significant bradycardia (heart

ate ¡ 50 beats/min) or hypotension (systolic blood pressure90 mmHg) occurred. Coronary parameters were acquired

t the end of the third minute of adenosine administra-

i o m e d i c i n e 9 8 ( 2 0 1 0 ) 151–160 157

tion, or just before suspension, and were used to estimateCFR.

Idiopathic dilated cardiomyopathy is a suitable model forassessing coronary flow rates in a more clean fashion thanin ischemic heart disease. Patients with idiopathic dilatedcardiomyopathy are known to have impaired myocardialblood flow and coronary flow reserve [34]. These patientshave microcirculatory abnormalities reducing the coronaryvasodilating capability, although the epicardial vessels areangiographically normal. Measurements by intracoronaryDoppler flow wire technique are not affected by the idrauliceffect of stenosis, allowing a better steady state situationduring the entire duration of the measurements (basal andvasodilating phase).

Using these data, CFR has been estimated using theparabolic and the Womersley-based formula for the compu-tation of the flow rates, thanks to formula (19).

We observe that no ad hoc data acquisition has been neededin order to evaluate the flow rates (and then the CFR) usingthe Womersley-based formula. An important feature of thisformula is actually that it can be used from data commonlymeasured in the clinical practice.

As shown in Table 2 and in Fig. 3, Womersley-based formulaprovides an higher value of the estimate of the CFR (meanvalue 2.65 ± 0.85), with the respect to the one performed by theparabolic formula (mean value 2.56 ± 0.75), in all the patientsbut one (patient no. 11). We observe that this patient is the onlyone such that the Womersley number is smaller under adeno-sine rather then at rest. This is due to the fact that for thispatient heart rate is slower under adenosine than in restingconditions.

Moreover, in Table 2 the values of the flow rate esti-mated with the parabolic formula (Qp

R and QpS ) and with the

Womersley-based formula (QwR and Qw

S ) are shown. We noticethat Womersley-based formula provides an higher value inall the patients. In particular, the mean value of the flow

formula. (×) Womersley-based formula; (©) parabolicformula. The patients with idiopathic dilatedcardiomyopathy (1–8) and the healthy ones (9–13) areseparated by the dashed line.

158 c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 9 8 ( 2 0 1 0 ) 151–160

Table 2 – Flow rates and coronary flow reserve (CFR) estimated with the parabolic formula (3) and with theWomersley-based formula (6), from the data collected at the Centro Nazionale delle Ricerche (CNR)—Clinical PhysiologyInstitute, Pisa, and relative difference ε = (CFR based on (6)-CFR based on (3))/CFR based on (6). The index p and w refersto parabolic and Womersley-based formulas, respectively.

Patient W basal/adenosine

QpR (cm3/s) Q

pS (cm3/s) Qw

R (cm3/s) QwS CFRp based

on (3)CFRw based

on (6)ε

1 2.88/3.19 1.73 6.50 1.93 7.83 3.75 4.06 7.64%2 3.00/3.15 2.53 5.68 2.95 6.83 2.25 2.32 3.02%3 3.53/4.00 1.79 4.80 2.20 6.05 2.68 2.75 2.55%4 2.05/2.44 2.46 3.78 2.55 3.99 1.53 1.57 2.55%5 2.54/2.49 2.61 4.06 2.79 4.31 1.55 1.55 0.0%6 3.24/3.58 1.44 5.20 1.74 6.41 3.61 3.68 1.90%7 2.49/2.73 1.52 2.65 1.61 2.87 1.75 1.78 1.69%8 2.13/2.34 1.49 3.39 1.55 3.57 2.28 2.30 0.87%

9 2.68/3.32 1.81 3.09 1.95 3.75 1.70 1.92 11.46%.06.79.27.78

10 2.33/2.54 1.96 3.98 211 2.68/2.44 1.66 4.71 112 2.07/2.41 1.23 4.33 113 3.00/3.41 1.53 5.93 1

At the moment, we do not have accurate measures for adeeper error analysis. We limit ourselves to comment the com-parison between our results and the parabolic formula andhow they do fit with clinical expectation. As a matter of fact,experience of medical doctors suggests that parabolic formulatends to underestimate flow rate and CFR as we have pre-viously pointed out. Since our results feature a systematicoverestimation in comparison with parabolic ones, we con-sider them fairly promising. A more accurate comparison byusing a set of accurate in-vivo measures obtained with PC-MRIis found in [42]. Moreover, in Table 2 the relative differences ε

between the two CFR estimates are shown. The mean value ofε related to the first 8 patients in Table 2 (those with idiopathicdilated cardiomyopathy) is 2.53 ± 2.42%, while the mean valuein the healthy patients is 3.55 ± 4.98%. Because of the smallsample size, the two groups are still not well separated.

Data collected so far are just a small sample intended fora preliminary exploration and not for building a statisticallysignificant data set. Starting from these promising results, weplan to enlarge our data base, in particular including caseswith high Womersley number, namely those observed in ves-sels with larger diameter than that of coronary arteries. Infact, in the present study, formula (6) was applied to arte-rial vessels with small Womersley numbers (namely in therange 2.07–4.00). As shown in [40], formula (6) should furtherimprove accuracy of CFR calculation when applied to clinicalconditions characterized by higher values of the Womersleynumber, such as for elevated heart rates, as during atrial pac-ing tachycardia.

5. Conclusions

In this paper we have analyzed the estimation of the bloodflow rate at the peak instant based on available measures ofthe maximum velocity obtained by the single-point Dopplertechnique which is used extensively in clinical practice. In

particular, we have considered the formula proposed in [40](called Womersley-based formula) which is able to incorporatedata on the blood pulsatility (by means of the Womersley num-ber W).

4.25 2.04 2.07 1.45%4.99 2.84 2.79 -1.79%4.58 3.52 3.59 1.95%7.24 3.88 4.07 4.67%

A sensitivity analysis of this formula on the diameter andmaximum velocity measures quantifies the errors affectingthe flow rate estimate due to measurements errors of diame-ter and velocity. Parabolic formula is in general less sensitive tothis kind of errors, being in general less sensitive to the hemo-dynamic conditions of the patient. However, we have shownthat the increment in sensitivity of the Womersley-based for-mula is not significant when the Womersley number is W < 2.7or W > 3.1. On the contrary, for intermediate Womersley num-bers, original formula is remarkably more sensitive to theseerrors. Consequently, we modified the formula proposed pre-viously in a way that reduces the sensitivity to 38% more thanthe parabolic formula versus 68% of the original one.

A first preliminary clinical application of the Womersley-based formula has been provided too. We considered acatheter-based Doppler velocimetry analysis for the measure-ments of the CFR. We have compared flow rate and CFRestimates provided by both parabolic and Womersley-basedformulas. Estimates of flow rates provided by the new for-mula are always greater than the ones given by the parabolicone. The same situation occurs for the CFR apart from onepatient. These results have suggested that the new formulais more accurate in view of the clinical evidence of a bias ofthe parabolic formula, that tends to underestimate CFR andflow rate. Obviously, other Doppler-based techniques could inprinciple improve the estimation of the flow rate as well.

We point out that the Womersley-based formula does notrequire specific ad hoc measurements procedures and it istherefore readily applicable in single-point Doppler measures.A specific in vivo validation comparing the estimates providedby the new formula with accurate flow rate measures obtainedby PC-MRI (referred to as the gold standard technique) is pre-sented in [42].

Conflict of interest statement

All authors disclose any financial and personal relationshipswith other people or organizations that could inappropriatelyinfluence (bias) our work.

i n b

A

TaVH

I

g

(

r

c o m p u t e r m e t h o d s a n d p r o g r a m s

cknowledgments

his study was supported partially by grants fromFP7-ICT-2007 project (grant agreement 224635)

PH2—Virtual Pathological Heart of the Virtual Physiologicaluman.

The research of the first author has been supported by thetalian MURST, through a project COFIN07.

The third author is supported by the project Aneuriskranted by Fondazione Politecnico, Milano, Italy.

The first author wishes to thank Dr. Giuseppe VergaraRovereto Hospital, Italy) for many fruitful discussions.

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