Annals of Biomedical Engineering, Vol. 33, No. 4, April 2005 (© 2005) pp. 415–428DOI: 10.1007/s10439-005-2495-2

Fundamental Study of Ultrasonic-Measurement-Integrated Simulationof Real Blood Flow in the Aorta

KENICHI FUNAMOTO,1 TOSHIYUKI HAYASE,2 ATSUSHI SHIRAI,2 YOSHIFUMI SAIJO,3 and TOMOYUKI YAMBE3

1Graduate School of Engineering, Tohoku University, 6-6-01 Aaramaki-Aoba, Aoba-ku, Sendai 980-8579, Japan; 2Institute of FluidScience, Tohoku University, 2-1-1 Katahira, Aoba-ku, 6-6-01 Sendai 980-8577, Japan; and 3Institute of Development, Aging and Cancer,

Tohoku University, 4-1 Seiryomachi, Aoba-ku, Sendai 980-8575, Japan

(Received 4 June 2004; accepted 6 October 2004)

Abstract—Acquisition of detailed information on the velocityand pressure fields of the blood flow is essential to achieve accu-rate diagnosis or treatment for serious circulatory diseases suchas aortic aneurysms. A possible way to obtain such informationis integration of numerical simulation and color Doppler ultra-sonography in the framework of a flow observer. This method-ology, namely, Ultrasonic-Measurement-Integrated (UMI) Sim-ulation, consists of the following processes. At each time stepof numerical simulation, the difference between the measurableoutput signal and the signal indicated by numerical simulation isevaluated. Feedback signals are generated from the difference, andnumerical simulation is updated applying the feedback signal tocompensate for the difference. This paper deals with a numericalstudy on the fundamental characteristics of UMI simulation usinga simple two-dimensional model problem for the blood flow inan aorta with an aneurysm. The effect of the number of feedbackpoints and the feedback formula are investigated systematically.It is revealed that the result of UMI simulation in the feedbackdomain rapidly converges to the standard solution, even with usu-ally inevitable incorrect upstream boundary conditions. Finally, anexample of UMI simulation with feedback from real color Dopplermeasurement also shows a good agreement with measurement.

Keywords—Bio-fluid mechanics, Computational fluid dynamics,Ultrasonic measurement, Color Doppler imaging, Measurement-integrated simulation, Aneurysm, Boundary condition, Pulsatileflow.

INTRODUCTION

Blood flow, which plays important roles in vital susten-tation and homeostasis, is hindered by disabilities result-ing from circulatory diseases.22 For example, with aging,asymptomatic aneurysms may develop due to arterioscle-rosis, and their rupture may be fatal. It has been empiricallyfound that the size of aneurysms is a reliable anatomi-cal factor predictive of rupture, and statistics have beencompiled regarding the size of ruptured and unruptured

Address correspondence to Dr. T. Hayase, Institute of Fluid Science,Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan.Electronic mail: hayase@ifs.tohoku.ac.jp

aneurysms.27,29 For example, Ujiie et al. suggested that theaspect ratio of aneurysms is a useful parameter in predictingimminent aneurysmal rupture.29 Although hemodynamicforces such as blood pressure and wall shear stress due toblood flow are known to be related to the development,progress, and rupture of aneurysms,2,7,14,25 detailed mech-anisms remain to be elucidated. Clarification of the rela-tionship between pathology and the cause of circulatorydiseases is essential.

Among a number of existing imaging modalities, colorDoppler ultrasonography is widely used for the diagnosis ofblood vessel diseases since it noninvasively provides real-time images of the blood flow structure and vessel config-uration by a relatively compact system. Figure 1 shows anexample of a color Doppler image around an aneurysm ofthe descending aorta. The configuration of the vessel wallis reconstructed as a B-mode image from time delays andmagnitudes of the ultrasonic echo, which are emitted froma probe (located at the center of the sector), and reflected bythe structures in vivo. Furthermore, the blood velocity com-ponent in the direction of the ultrasonic beam or the Dopplervelocity is measured by the Doppler shift frequency. In or-der to visualize the blood flow, the frequency shifts are con-verted to colors of graduated intensity, blue for flow awayfrom the probe and red for flow approaching the probe, asdisplayed on top of the B-mode image. Since the equipmentmeasures only the Doppler velocity, it is difficult to recog-nize the exact three-dimensional blood flow field. Recently,three-dimensional reconstruction of the blood vessel con-figuration, velocity profile, and pressure distribution fromthe ultrasonic measurement have been investigated.1,18,21,28

Capineri et al. developed a technique for dynamic displaysof vector velocity maps, in which the velocity vectors ob-tained by Doppler measurements in two independent di-rections are properly interpolated.1 Tortoli et al. presentedthe real-time two-dimensional velocity profile of descend-ing aortic blood flow using a set-up based on an esophagealprobe connected to a multigate Doppler-processing system,and confirmed extremely complex flow in the proximal

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FIGURE 1. Color Doppler image around a thoracic aneurysm (Center frequency: 4.4 MHz, Pulse repetition frequency: 4 kHz).

portion of the aortic arch or in the case of aortic diseases.28

Ohtsuki et al. developed a new method for nondisturb-ing, quantitative measurement of pressure, in which thevelocity component orthogonal to the Doppler velocity isdeduced after measurement of the velocity field by the pulseDoppler technique, and the pressure is calculated from theacceleration.18 However, in experimental works, there aresome limitations such as the direct effects of noise andthe Doppler transducer position on data acquisition and theassumption of symmetrical flow in the calculation, so thatno method to obtain complete information in real time yetexists.

As a counterpart of measurement, numerical simulationof the blood flow has been studied extensively. Realisticrepresentation of blood flow can be obtained by solvingthe fundamental equations of the flow with realistic ves-sel geometries obtained by medical imaging techniquessuch as magnetic resonance imaging (MRI) or computedtomography (CT).3,4,8,12,15,16,24,26,30 Numerical simulationdealing with flow in an aorta with an aneurysm has alsobeen carried out by such a combination.3,4,26 Taylor andYamaguchi reported the appearance and disappearance ofa primary vortex and regions of high shear stresses both atthe proximal and distal ends of the aneurysm.26 Di Martinoet al. studied the complex mechanical interaction betweenblood flow and wall dynamics in a three-dimensional cus-tom model of an abdominal aortic aneurysm and provideda quantitative local evaluation of the stresses.3 However,numerical simulation has an inherent problem of speci-fication of the boundary conditions,8,12,15,16 and thus thecalculated blood flow is usually similar but not identical tothe real one. In order to minimize the effect of inaccurateinlet velocity profile specification on local wall shear stresscomputations in the vessel segment, Liu et al. extended

their model of the carotid artery in the upstream directionwith a straight tube, proposing that such extension maybe useful until more accurate, noninvasively obtained ve-locity profile data for the inlet become available.15 In thedescending aorta, however, extension by adding the as-cending aorta and the aortic arch does not eliminate the ef-fect of the upstream boundary condition on reproduction ofthe complicated flow structure (e.g., Kilner et al. observedhelical and retrograde secondary flow patterns in the aor-tic arch using magnetic resonance velocity mapping).13,28

By numerical simulation with an integrated model of theleft ventricle and the aorta, Nakamura et al. revealed thatthe upstream boundary condition strongly influences de-velopment of the helical flow.16 Glor et al. performed MRIvelocity measurements and computation in the U bend,confirming the importance of accurate inflow boundaryconditions.8 Since velocity data cannot yet be reliably mea-sured by MRI due to artifacts such as partial volume effect,inevitable errors are introduced to the computation of thefull three-dimensional velocity field and wall shear stress.Therefore, development of new methodology that can ex-actly provide detailed information on blood flow is stronglyrequired.

Integration of computation and measurement in theframework of a flow observer is a possible way to solvethese problems.9,17 Hayase et al. proposed such a flowobserver as an analytic methodology for general flowproblems.9 Conceptually, a flow observer is the state ob-server in control theory applied to flow analysis. For inves-tigation of a real flow, a simulation model is constructed in acomputer using standard numerical analysis methodologybased on fundamental equations with appropriate bound-ary and initial conditions. Some output signals are definedfor real flow measurement and for numerical simulation

Ultrasonic-Measurement-Integrated Simulation 417

in order to evaluate the difference between the two results.Numerical simulation is carried out with an additional bodyforce or a boundary condition as a feedback signal that isderived from the difference between the two output signals.If the feedback law is designed properly, the computationalresult converges to the real flow. In the cited study by Hayaseet al.,9 numerical simulation of the flow observer was per-formed for a turbulent flow through a square duct. Turbu-lent flow structure including fluctuation was successfullyreproduced by the feedback of errors in the axial velocitycomponents estimated at 100 points on a cross section ofthe duct to the pressure boundary condition based on thesimple proportional control law. Nisugi et al. developed ahybrid wind tunnel based on a flow observer and investi-gated the flow with a Karman vortex street, revealing thesubstantial advantage of such methodology over ordinarysimulation, especially in its ability to reconstruct real flowand in its computational efficiency.17 As a flow analysismethod, a flow observer generally has several advantages:(1) simulation is performed using real flow conditions,(2) the simulation is simultaneously validated by measure-ment, and (3) real-time simulation can possibly be carriedout owing to a significant improvement in computationalefficiency.

In this paper, a flow observer is applied for the analysisof blood flow by integrating numerical simulation and mea-surement with color Doppler ultrasonography. We term thismethod Ultrasonic-Measurement-Integrated (UMI) simu-lation. In UMI simulation, feedback signals are added tothe Navier-Stokes equation and the pressure equation inthe form of a source term to compensate for the differ-ence between the computation and the measurement. Ofcourse, UMI simulation is expected not only to reconstructthe color Doppler image but also to provide the velocityvectors and the pressure distribution in detail. This paperconcerns UMI simulation using a simple two-dimensionalmodel problem for blood flow in an aorta with an aneurysm.We focus on the problem of whether the feedback loop inUMI simulation effectively reduces the error in the flowdomain resulting from an incorrect velocity profile settingat the upstream boundary, which usually takes place innumerical simulations of blood flow. A numerical simula-tion with an assumed boundary condition is first definedas a model of real flow since existing ultrasonic mea-surement does not provide complete information on realblood flow and, therefore, is not suitable as a standardto evaluate UMI simulation. UMI simulation is investi-gated as to the convergence of its result to the standardsolution when the real velocity profile at the upstreamboundary is unknown and is incorrectly assumed. Feed-back algorithms of UMI simulation are designed by eval-uating the error from the standard solution. Using thesefeedback algorithms, an example of UMI simulation withthe feedback using real color Doppler measurement is alsopresented.

UMI SIMULATION USING SIMULATED COLORDOPPLER MEASUREMENT

Subject and Computational Method

This paper deals with the blood flow in the vicin-ity of an aneurysm in the descending aorta as shown inFig. 1. A 62-year-old male patient with a chronic aor-tic aneurysm in his descending aorta participated in thestudy. He had no significant cardiac complications. Hiscardiac output was 9.17 × 10−5 m3 s−1 (5.5 l min−1) andhis heart rate was 0.87 Hz (52 bpm) during the measure-ment. Transesophageal echocardiography was performedwith an ultrasound device (SONOS 5500, Philips Medi-cal Systems, Andover, MA, USA) with a transesophagealultrasonic transducer (T6210, Philips Medical Systems,Andover, MA, USA). The central frequency was variable,ranging from 4 to 7 MHz. The images were stored with adigital video recorder (DCR-TRV30, SONY, Tokyo, Japan).In UMI simulation, the B-mode image of the blood vesselobtained by ultrasonic diagnostic equipment is digitized toextract the cross-sectional surface manually, and the pixeldata is allocated to define a two-dimensional computationaldomain. The shape of the blood vessel is extended in boththe upstream and downstream directions in order to performnumerical simulation (see Fig. 2, the right side is upstream).The x-axis is defined in the flow direction at the upstreamboundary (x = 0), and the y-axis is defined against it withthe left-handed system.

Table 1 shows parameters used in this computation. Car-diac cycle T is calculated from the heart rate. The upstreamshape of the blood vessel is assumed to be cylindrical,and the diameter D is calculated from the image. Sincethe upstream boundary is located at some distance fromthe aneurysm, we consider that the blood vessel can beassumed to be cylindrical. Three-dimensional reconstruc-tion from B-mode images in many directions may givethe correct shape of the cross section. We are not, how-ever, so concerned with this since this paper deals with

FIGURE 2. Computational grid and arrangement of monitoringpoints.

418 FUNAMOTO et al.

TABLE 1. Computational conditions.

Heart rate 0.87 Hz (52 bpm)Cardiac cycle T 1.15 sCardiac output 9.17 × 10−5 m3/s (5.5 l/min)Entrance flow 6.42 × 10−5 m3/s (3.85 l/min)Maximum mean velocity u ′

max 0.74 m/sEntrance vessel diameter D 28.25 × 10−3 mKinematic viscosity ν 4.0 × 10−6 m2/s

fundamental two-dimensional analysis. With reference tothe blood flow measurement data, we assume that 30%of the cardiac output flows into the branches and that theremaining 70% (6.42 × 10−5 m3 s−1) flows into the de-scending aorta.6 The variation of the flow rate q is modeledas shown in Fig. 3 according to the MR measurement byOlufsen et al.19 Blood is assumed to be Newtonian fluidwith a density ρ = 1.00 × 103 kg m−3 and a dynamic vis-cosity µ = 4.0 × 10−3 Pa s within normal range.

As a fundamental consideration of blood flow simula-tion integrated with ultrasonic measurement, we assumetwo-dimensional flow, although intravascular blood flow invivo has a complex three-dimensional structure. Governingequations for two-dimensional incompressible and viscousfluid flow are the Navier-Stokes equation,

∂u∂t

= −(u · grad) u + 1

Re∇2u − grad p, (1)

and the equation of continuity,

div u = 0, (2)

where u = (u, v) is the velocity vector and p is the pressure.All the values are nondimensionalized with the diameter ofthe blood vessel D, the cross-sectional average velocity umax

at the upstream boundary for the maximum cardiac output,and the kinematic viscosity ν of the blood. From here on,the same symbols are used for both dimensional and nondi-mensional values since it does not cause confusion.

FIGURE 3. Time variation of flow volume in descending aortaand of cross-sectional average flow velocity at upstreamboundary (nondimensional).

With regard to the upstream boundary conditions forthe standard solution, we assume a Poiseuille flow with aparabolic profile in x-directional velocity. For the conditionof UMI simulation and the ordinary simulation, in contrast,we assume a different flow with a uniform profile in x-directional velocity.

u(Xu, y, t) ={

32 u′(t){1 − (y − Yu)2/(D/2)2} (Standard solution)

u′(t) (UMI simulation and ordinary simulation)

v(Xu, y, t) = 0 , (3)

Yu − D/2 ≤ y ≤ Yu + D/2

where Xu and Yu are the x and y coordinates of the centerof the vessel at the upstream boundary, respectively, andu′(t) is the time-varying upstream cross-sectional averageflow velocity determined by the modeled flow rate19 asshown in Fig. 3. As mentioned above, we define the stan-dard numerical solution as a model of the real flow. Inorder to investigate the effect of the feedback signal onUMI simulation, its upstream boundary velocity profile isassumed to be different from that of the standard solution.At the downstream boundary of the computational domain,the free-flow condition (∂/∂n = 0, n: coordinate normal tothe boundary) is applied, and the no-slip condition is appliedto the blood vessel wall.

The governing equations are discretized by means ofthe finite volume method. These equations are solved withan algorithm similar to the SIMPLER method.10,20 In theSIMPLER method, the x-directional momentum equationin Navier-Stokes equations is expressed as

ui =(∑

B j u j + Si

)/Bi + di (pi − pi−1), (4)

where(∑

B j u j)

means the summation of the values at fouradjacent nodes. By substituting Eq. (4) and similar equa-tions for y-directional momentum to the integrated form ofthe equation of continuity, the pressure equation is obtained.

ai pi =∑

a j p j + spi , (5)

where(∑

a j p j)

represents the summation of the valuesat four adjacent nodes. The concrete notations of the pa-rameters in Eqs. (4) and (5), as well as supplementary pres-sure correction equations and velocity correction procedurein the SIMPLER method, are explained in a reference.20

In descretization of the convective terms in Navier-Stokesequations, a consistently reformulated QUICK scheme11 isapplied in order to assure continuity of the flux at the inter-face of control volumes during the iterations. The QUICKscheme is a three-point upstream-weighted quadratic inter-polation technique within the context of a control-volumeapproach. In the discretization for the time derivative terms,a second order implicit scheme5 is used. The system oflinear algebraic equations is solved with the pentadiago-nal matrix solver, i.e., the MSI method,23 which is a veryefficient iterative solver of linear equations.

Ultrasonic-Measurement-Integrated Simulation 419

In this study, we introduced three types of staggeredgrid systems, in which nodes for velocity components areshifted in half grid size from the pressure node, with differ-ent numbers of grid points in the x and y directions: 32 × 20with grid spacing of 2.973 × 10−3 m (Grid A), 65 × 40with 1.487 × 10−3 m (Grid B), and 130 × 80 with 0.743 ×10−3 m (Grid C). Among these three types of grid systems,Grid A is too coarse to represent the shape of the bloodvessel precisely. On the other hand, Grid C is fine enoughto represent the shape, but it requires huge computationaltime (about 300 h for one case) to converge to the periodicsolution. Hence, in the numerical study for the validationof UMI simulation, we use Grid B as shown in Fig. 2,compromising between reproducibility of the vessel shapeand computational time. The adequate residual at conver-gence and the maximum iteration number are respectivelydetermined as 1 × 10−5 and 300 after test computations.

Feedback Law

In designing UMI simulation, we define monitoringpoints in the relevant flow domain. At the monitoring points,output signals are evaluated both in ultrasonic measure-ment and simulation. Feedback signals, derived from thedifference between these output signals, are applied to thecomputation in order to compensate for the computationalerror and to make the computational result converge to thereal flows. In the present study, the real flow is modeledas the standard numerical solution, and the error of UMIsimulation is due to an inaccurate boundary condition asmentioned above. For the validation of UMI simulation,we define two arrangements of monitoring points: 1) globalarrangement, all 675 grid points in the large rectangle re-gion which covers the blood vessel near the aneurysm inFig. 2, and 2) local arrangement, all 325 points in the smallrectangle above the aneurysm in Fig. 2.

Figure 4 explains how we calculate the feedback signal.The velocity vector obtained by the numerical simulation

FIGURE 4. Schematic diagram of calculation of feedbacksignals.

is defined as uc, and that of the standard solution (modelof the real flow) is us. The purpose of the feedback isto force the velocity uc to converge to us. Doppler ve-locity of the blood flow is obtained with ultrasonic diag-nostic equipment. The Doppler velocities of the numeri-cal simulation and the standard solution, Vc and Vs, arethe projection of uc and us in the ultrasonic beam direc-tion (a chain line in Fig. 4), respectively. Here, it shouldbe noted that Doppler velocities Vc and Vs each representonly one component of the velocity vector, and, therefore,it is not a straightforward task to reconstruct the veloc-ity vectors from the Doppler velocities. In other words,the velocity vector us is generally unknown in real ultra-sonic measurement. In the staggered grid system used inthe SIMPLER method, node points for velocity compo-nents u and v are shifted from the pressure node, “•”, inhalf mesh size. Therefore, the velocity components at thepressure node are evaluated by interpolation in the abovedescription. The origin of the ultrasonic beam, where ultra-sound is emitted from the probe, is set at the same positionas that of the measurement (the center of the sector inFig. 1).

In this study, we deal with two feedback algorithms,feedback to velocity field (feedback A) and feedback tovelocity and pressure fields (feedback B), as follows.

Feedback A: As the feedback signal, the artificial forcefv proportional to the difference between Doppler ve-locities of the standard solution and UMI simulation isapplied to the Navier-Stokes equation in the direction ofthe ultrasonic beam.

In this formula, the force fv is calculated by the followingequation:

fv = −Kvρ(Vc − Vs)u′max�S, (6)

where Kv is the feedback gain (nondimensional), u′max is

the maximum average flow velocity of the blood at theupstream boundary, and �S is an interfacial area of thecontrol volume of pressure. Referring to Fig. 4, if the com-putational result Vc is smaller than Vs, the artificial forcefv has a positive value and accelerates the fluid along theultrasonic beam in UMI simulation to reduce the error. Theforce fv is decomposed to the x-directional component fvx

and the y-directional component fvy , which are added to thecontrol volumes of u(i, j) and v(i, j) in the Navier-Stokesequation, respectively.

Feedback B: In feedback A, artificial forces fvx and fvy notonly accelerate the fluid to reduce the error in velocitybut also increase the pressure of the control volume (i, j)through the pressure equation derived from the equationof continuity in the SIMPLER method.20 Hence, we in-troduce additional feedback to the pressure equation tocounteract the effect of artificial force fv.

420 FUNAMOTO et al.

In this formula, a source term sp, proportional to thedifference between the Doppler velocities, is added to thepressure equation and calculated artificial force fv is addedto the Navier-Stokes equation in feedback A:

fv = −Kvρ(Vc − Vs)u′max�S

sp = −Kpρ(Vc − Vs)�S, (7)

where Kp is the feedback gain for the pressure (nondimen-sional).

UMI simulation with feedback B is specified by thecombination of gains (Kv, Kp). Note that the special casewith Kp = 0 corresponds to feedback A, and Kv = Kp = 0corresponds to the ordinary simulation without feedback.

Evaluation Method

In order to evaluate the accuracy of UMI simulation, wedefine the error norm en for an arbitrary variable a, whichis the velocity component u, v , V, or the pressure p, by thefollowing equation:

en = 1

amaxT

∫T

|acn(t) − asn(t)| dt, (8)

where T is the cardiac cycle, amax is the characteris-tic value for normalization: amax = u′

max for velocity oramax = ρu′

max2 for pressure, where u′

max is the maximumaverage flow velocity at the upstream boundary. Subscriptcn corresponds to UMI simulation or ordinary simulation,and sn corresponds to the standard solution, respectively.In the subscripts “cn” and “sn,” n is the index of the gridpoint.

The average error norm eN is evaluated over the moni-toring points in a domain.

eN = 1

N

∑n

en. (9)

In this section, 675 monitoring points in the global ar-rangement or 325 points in the local arrangement are usedfor the calculation of eN , and they are called e675 or e325,respectively.

The optimum values of the gain Kv for feedback A andthe combination of gains (Kv, Kp) for feedback B are deter-mined so that the average error norms eN of some specifiedvariables take the minimum value over a number of trialcomputations.

The accuracy of UMI simulation is evaluated by compar-ing the average error norms eN of UMI simulation for ve-locity components, Doppler velocity, and pressure, againstthose of the ordinary simulation without feedback.

Results and Discussion

The standard solution is first obtained from the numeri-cal simulation with the upstream parabolic velocity profile.Solid lines in Fig. 5 represent the velocity components and

FIGURE 5. Comparison of convergence with periodic solu-tions regarding (a) x-directional velocity u, (b) y-directionalvelocity v, (c) Doppler velocity V, and (d) pressure p (all resultsare nondimensional).

Ultrasonic-Measurement-Integrated Simulation 421

FIGURE 6. (a) Velocity vectors and pressure distribution and(b) color Doppler image of the standard solution at t = 0.35sin deceleration phase (all results are nondimensional).

the pressure for 32 s (about 28 cardiac cycles) at one mon-itoring point R (see Fig. 2). The results of UMI simulationshown by dotted lines are discussed later in this section. In26 cardiac cycles, the solution converges to almost periodicoscillation (black arrows). We define the standard solutionas the repetition of the result in the 26th cardiac cycle.Figure 6 shows velocity vectors, pressure distribution, andthe color Doppler image of the standard solution at t = 0.35s in the deceleration phase (see Fig. 3). It is noted that thecolor Doppler image representing only one velocity com-ponent does not show the structure of the blood velocityvector field. For example, vortices above the aneurysm aredisplayed by a mosaic pattern consisting of red and blueregions (as can be seen in the white rectangle in Fig. 6).

UMI simulation was performed with the above-mentioned standard solution and the feedback algorithmA in Eq. (6), as well as with the feedback algorithm B inEq. (7). For a given value of Kv in feedback A or (Kv, Kp)in feedback B, time-dependent calculation was carried outuntil periodical oscillations were obtained, and the averageerror norm eN was evaluated using a periodical solution.Figure 7 shows the average error norm e675 with the globalarrangement of the monitoring points (see Fig. 2) for theDoppler velocity, x and y-directional velocity components,and the pressure as a function of the gain Kv. Note that weinvestigated only the limited case of Kv = Kp for feedbackB. Study of the full (Kv, Kp) parameter plane remains as afuture work.

In the figures on the left-hand side of Fig. 7 (a), theaverage error norm e675 of the Doppler velocity for feedbackA monotonically decreases as the feedback gain increasesin the range of 0 ≤ Kv ≤ 7. Over Kv = 8, UMI simulationbegins to diverge, and e675 shows a steep increase. Thefigures on the right-hand side are magnifications of theregion 0 ≤ Kv ≤ 1 in order to show the result of feedback Bclearly. The result for feedback B shows a faster reduction,but it undergoes sudden divergence at some critical value ofKv. The average error norms e675 of the velocity componentsu and v show changes similar to that of the Doppler velocity[Fig. 7(b)]. The error of the pressure, however, takes theminimum value at a relatively small Kv value [Fig. 7(c)].Considering these results, the optimum gains for feedbackA and B are determined with the average error norms eN forthe l1 norm (|u| + |v|) of the velocity vector. The optimumfeedback gain Kv with the global arrangement is determinedas Kv = 7 for feedback A, and Kv = Kp = 0.3 for feedbackB. Note that the optimum gains change if the average errornorm of the pressure is taken into account, especially forfeedback A.

In order to evaluate the validity of the present UMI sim-ulation, the error of UMI simulation against the standardsolution is compared with that of the ordinary numericalsimulation. The color Doppler images around an aneurysmat t = 0.35 s in the deceleration phase of the cardiac cycleare compared in Fig. 8. The mosaic pattern of the aneurysm,which implies a disturbed flow structure such as vortices(see Fig. 6), is different between the ordinary simulation[Fig. 8(b)] and the standard solution [Fig. 8(a)] since the up-stream boundary condition is different. On the other hand,the color Doppler images of UMI simulations with bothfeedback A [Fig. 8(c)] and B [Fig. 8(d)] closely resem-ble that of the standard solution in the aneurysm despitethe upstream boundary condition being different. The samemosaic pattern in the aneurysm implies correct reproductionof the blood flow structure.

Variations of the velocity components and the pressureat the monitoring point R (see Fig. 2) in a cardiac cycle arecompared between the standard solution, UMI simulationsof feedback A and B, and the ordinary simulation in Fig. 9.Due to the incorrect upstream boundary condition, the re-sult of the ordinary simulation is different from that of thestandard solution. In contrast, in spite of the same incor-rect boundary condition, excellent agreement is attained byUMI simulation with feedback A with regard to the veloc-ity components u and v , as well as Doppler velocity. UMIsimulation with feedback B also shows results very close tothose of the standard solution except for u. In the velocitycomponents of UMI simulation with feedback B, Dopplervelocity V agrees with that of the standard solution the bestsince this study, uses it as the output signal for the feedbackartificial force is applied in the ultrasonic beam directionin UMI simulation. The y-directional velocity component vshows better agreement with the standard solution than does

422 FUNAMOTO et al.

FIGURE 7. Variation of average error norm e675 of (a) Doppler velocity, (b) x, y-directional velocity components, and (c) pressureagainst feedback gain Kv in UMI simulation with global arrangement of monitoring points. Figures on right-hand side show0 ≤ Kv ≤ 1.0 magnification.

the x-directional velocity component u. This is because theorigin of the ultrasonic beam is away from the monitoringpoints in the y-direction and, therefore, the Doppler velocityat each monitoring point mainly consists of the y-directionalvelocity component. This means that the computational ac-curacy of the x-directional velocity component could pos-sibly be improved if the origin of the ultrasonic beam weremoved to change the angle of the ultrasonic beam so that theDoppler velocity would contain more of the x-directionalvelocity component. The variation of the pressure clearlyshows the difference between the two feedback formulae.The result of feedback formula A shows poorer agreementthan the ordinary simulation, while that of feedback formula

B shows fairly good agreement with that of the standardsolution. Generally, feedback A can reproduce the velocitycomponents excellently while feedback B can reproduceboth the velocity components and the pressure.

Figure 10 compares the distribution of the error normin the blood vessel around the aneurysm for the velocitycomponents and pressure between the ordinary simulationand UMI simulation with two monitoring point arrange-ments and two feedback formulae. In the result of theordinary simulation, the area of large error shown in redappears in the aneurysm for all velocity components and inthe upper side of the blood vessel for x-directional veloc-ity component u. For the pressure, an area with relatively

Ultrasonic-Measurement-Integrated Simulation 423

FIGURE 8. Comparison of color Doppler images at t = 0.35sin deceleration phase among (a) standard solution, (b) ordi-nary simulation, and (c) and (d) UMI simulations with feed-back A and B, respectively. Feedback A: Kv = 7, feedback B:(Kv, Kp) = (0.3, 0.3). Color bar is the same as that in Fig. 6 (b).

large error exists in the downstream region of the compu-tational domain. In the result of feedback A with globalarrangement, a fairly good result is obtained for all thevelocity components. The error norm, indicated by blue,is reduced to almost zero. In the result for the pressure,however, the error is larger than that of the ordinary sim-ulation in the aneurysm. A better result for the pressure

FIGURE 9. Comparison of periodic solutions of (a) x-directional velocity u, (b) y-directional velocity v, (c) Dopplervelocity V, and (d) pressure p at monitoring point R (all re-sults are nondimensional). Feedback A: Kv = 7, feedback B:(Kv, Kp) = (0.3, 0.3).

is obtained with feedback formula B with global arrange-ment. The errors of the y-directional velocity component vand the Doppler velocity V are almost the same as thosein the former case, although the error of the x-directionalvelocity component u is somewhat larger than that in theformer case but still smaller than that of the ordinarysimulation.

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FIGURE 10. Comparison of error norms between the ordinary simulation and UMI simulations with feedback A and B using globaland local arrangements.

The results of UMI simulation with the local arrange-ment of the monitoring points are also given in Fig. 10.For obtaining the optimum gain for the case of local ar-rangement, systematic computation was performed in away similar to that in Fig. 7. With the local arrangement ofthe monitoring points (Fig. 2), the optimum gains, whichminimalize the average error norms e325 for the l1 norm(|u| + |v|) of the velocity vector, is Kv = 8 for feedback Aand Kv = Kp = 0.3 for feedback B. In all results, reductionof the error norm en is significant in the aneurysm wherethe monitoring points are arranged. As has been mentioned,feedback A gives a better result for x-directional velocityu, while feedback B is better for the pressure p.

Table 2 summarizes the average error norms of velocitycomponents and pressure normalized with the results of theordinary simulation. Note that the smallest error in eachcomparison is shaded. The smallest errors of e675 for thevelocity components are obtained by feedback A using theglobal arrangement of the monitoring points. In this case,UMI simulation reduces the error e675 to 8% for u, 7% for v ,

and 2% for V of those of the ordinary simulation. Compar-ison between the results with feedback formulae A and Breveals that feedback A is superior to feedback B for repro-duction of the velocity components, but is inferior for the re-production of the pressure. UMI simulation with feedback Busing the global arrangement of monitoring points reducesthe error e675 to 32% for p of that of the ordinary simulation.As for the arrangement of the monitoring points, the localarrangement, which seems more realistic in practical ap-plication of UMI simulation using color Doppler measure-ment, yields a poorer result than the global arrangement forthe global error e675, but gives comparatively good resultsif evaluated by the localized error e325. This implies that wemay locally arrange the monitoring points in the region ofconcern.

Finally, the computational load of UMI simulation is dis-cussed in comparison with ordinary simulation. In Fig. 5,the dotted lines represent the velocity components and thepressure of the result of UMI simulation with feedback B ofthe gain (Kv, Kp) = (0.3, 0.3) using the global arrangement

Ultrasonic-Measurement-Integrated Simulation 425

TABLE 2. Comparison of average error norm.

Monitoring pointarrangement Feedback u v V p

e 675

None Nonea 1 1 1 1Global A 0.078 0.069 0.015 0.997

B 0.224 0170 0.102 0.315Local A 0.279 0.152 0.217 0.482

B 0.503 0.216 0.354 0.330e 325

None Nonea 1 1 1 1Global A 0.066 0.036 0.016 1.701

B 0.223 0.137 0.107 0.358Local A 0.093 0.047 0.012 0.935

B 0.229 0.113 0.085 0.312

aOrdinary simulation.

of the monitoring points. The periodic solution is obtainedin eight cycles (see gray arrows in Fig. 5), implying thatUMI simulation requires less than one-third the computa-tional time steps than the ordinary simulation for 26 cyclesto obtain steady oscillation. As for computational time,however, the standard solution requires 4500 s for one car-diac cycle T in comparison with UMI simulation, whichrequires 6300 s. The time required by UMI simulationis 1.4 times longer than that for the ordinary simulationfor 1 cardiac cycle. Consequently, UMI simulation short-ens the computational time to obtain the periodic solutionby a factor of 0.4. It is noted that computational load ofUMI simulation depends on the feedback formula and thenumber of monitoring points. UMI simulation with feed-back B requires more computational time than that withfeedback formula A. For example, UMI simulation usingfeedback formula A with the global arrangement of mon-itoring points takes 5000 s for one cardiac cycle, whichis slightly longer than the time required by the ordinarysimulation.

FIGURE 11. Computational grid used for UMI simulation us-ing real color Doppler measurement. The marked numbers arethe length along the blood vessel wall measured from the up-stream boundary.

UMI SIMULATION USING REAL COLORDOPPLER MEASUREMENT

Computational Method

In this section, we perform UMI simulation using realcolor Doppler measurement. The medical data of the pa-tient, instrument, and equipment are all identical to those ofthe previous section. As a fundamental consideration, theanalysis is limited to a two-dimensional flow problem. Theresult of UMI simulation is compared with those of the ordi-nary simulation and the real measurement. Color Dopplerimages were obtained at a center frequency of 4.4 MHzand a pulse repetition frequency of 4 kHz (see Fig. 1). TheDoppler velocity V of an arbitrary point is determined fromthe intensity of color, red for positive velocity and blue fornegative velocity digitized into 256 grades using the colorbar at upper right of Fig. 1.

For computation, we use grid C, which is twice as finein each direction as grid B of the former section in orderto depict the blood vessel shape in detail. The monitor-ing points are set at the grid points in the white rectanglein Fig. 11. The total number of the monitoring points is864 with 36 × 24 points in each direction. In Fig. 1, therectangle, where the monitoring points are placed, consistsof 141 × 93 pixels. The time step of UMI simulation isdetermined as �t = 0.033 s since the measurement data isobtained at that time interval. The uniform velocity profileis used at the upstream boundary condition. The residual atconvergence is 1 × 10−5 and the maximum iteration num-ber is 500. Two feedback formulae, A and B, are usedin UMI simulation. Based on the result of the previoussection, the gain Kv is determined as the maximum valuewhich does not result in divergence of the computation.We performed UMI simulation using color Doppler im-ages at the diagnosis of 6 cardiac cycles to obtain the finalconvergent solution. The timing when the maximum flowvolume occurs in UMI simulation (t = 0.23 s) is designedto coincide with the measurement for synchronization.

Results and Discussion

The gains are determined as Kv = 1.0 for feedback Aand Kv = Kp = 0.2 for feedback B after a number of trialcomputations. We are concerned here with the results in thefinal 6th cycle of UMI simulation and the correspondingultrasound measurement, as well as that of the ordinarysimulation in the 20th cycle in steady oscillation.

Figure 12 shows a comparison of the color Dopplerimages obtained by the color Doppler measurement, theordinary simulation, and UMI simulations at t = 0.38 s inthe deceleration phase as an example. The color Dopplerimage of the ordinary simulation is different from that ofthe measurement. The complicated mosaic pattern arisingin the aneurysm in Fig. 12 (a) cannot be clearly seen inthe ordinary simulation. In spite of the same condition as

426 FUNAMOTO et al.

FIGURE 12. Comparison of color Doppler images at t = 0.38 sin deceleration phase among (a) real color Doppler image,(b) ordinary simulation, and (c) and (d) UMI simulations withfeedback A and B of the real color Doppler data, respectively.Feedback A: Kv = 1.0, feedback B: (Kv, Kp) = (0.2, 0.2).

TABLE 3. Comparison of average error norm ofDoppler velocity.

Average error norm e864

Ordinary simulation 1UMI simulation

Feedback A 0.483Feedback B 0.511

the ordinary simulation except for the feedback, the colorDoppler images of UMI simulations in Figs. 12(c) and(d) closely resemble that of the measurement includingthe mosaic pattern. We calculated average error norm e864

of Doppler velocity in the monitoring region as shown inTable 3. UMI simulation reduces the difference betweenthe real color Doppler measurement and the ordinary sim-ulation by a factor of 0.48 for feedback A or 0.51 forfeedback B.

UMI simulation with feedback formula A reproducesthe real measurement, including the aliasing [a white arrowin Fig. 12 (a)]. Here, aliasing indicated by a color differentfrom the proper one occurs in the case in which the Dopplershift frequency exceeds one-half the pulse repetition fre-quency, which means that there is an excess of measurablemaximum flow velocity and an incorrect result. Figure 13shows the absolute value of the average instantaneous feed-back intensity | fv| at each monitoring point viewed fromthe direction of the arrow in Fig. 11 for the same conditionas Fig. 12. | fv| is the average absolute value of fv duringthe iteration steps at a time step. In Fig. 13, the artificialforce | fv| is large in the regions where the aliasing occurs(denoted with a white arrow) because large artificial forcesare applied to compensate for the sudden velocity changedue to aliasing. This means a peculiarly large feedbacksignal in UMI simulation can be used as an index of incor-rect measurement such as aliasing. Improvement of UMIsimulation using this information remains for future study.

By the first-order numerical differentiation of the com-putational result of the velocity component, the wall shearstress τ is calculated along the vessel wall with theaneurysm. Since the distribution of the wall shear stressshows an unnatural jagged shape because of the stepped-shape vessel wall defined by the orthogonal grid system, theresult is smoothed by averaging five adjacent data points.Figures 14(a) and (b) show distribution of the time-averagedwall shear stress τ av and the RMS value τ rms of the wallshear stress in the aneurysm along the vessel wall mea-sured from the upstream boundary as shown in Fig. 11.Comparison of those results reveals that a substantial dif-ference exists between the results of time-averaged wallshear stress distributions but not those of the RMS values.The results of UMI simulations are probably better thanthat of the ordinary simulation. The wall shear stress issimilar to that of the simulation with an aortic aneurysmby Finol and Amon.4 However, we cannot say much about

Ultrasonic-Measurement-Integrated Simulation 427

FIGURE 13. Average instantaneous feedback intensity with (a)feedback A and (b) feedback B at t = 0.38 s in decelerationphase (nondimensional).

the validity of the results since there is no measurementdata for the real wall shear stress distribution and the as-sumption of two-dimensionality limits the validity of thepresent model. Further research is needed, including three-dimensional analysis and verification by experiment.

CONCLUSIONS

In this paper we investigated fundamental characteristicsof the Ultrasonic-Measurement-Integrated (UMI) simula-tion of blood flow in the aorta as a key issue to develop ad-vanced medical diagnosis and treatment equipment. A UMIsimulation was performed for a two-dimensional modelproblem using a standard numerical solution instead of theresult of real ultrasonic measurement. Particularly, the ef-fect of an incorrect upstream boundary velocity profile onthe simulation result was investigated.

Results obtained in this study can be summarized asfollows. For feedback formula A, by which artificial forceis applied to the momentum equation to compensate forthe error of the Doppler velocity between the numericalsimulation and the standard solution, the error in velocityis significantly reduced in UMI simulation, but the errorin pressure is intensified more than in the case of ordinarysimulation. For feedback formula B, by which additionalcorrection of the pressure equation is applied in feedback

FIGURE 14. (a) Time-averaged wall shear stress τav and (b)RMS value τrms along the vessel wall (all results are nondi-mensional).

formula A, the error is reduced both in velocity and pressurein comparison with the ordinary simulation. Feedback B ismore sensitive to the change of the feedback gain thanfeedback A.

Placement of the monitoring points in a partial domaineffectively reduces the error in that part. UMI simulationrequires additional computational time for feedback, but en-hancement of convergence to the final solution results in areduction of total computational time. Finally, UMI simula-tions with real color Doppler measurement were performedshowing good agreement with the measurement.

The present study has confirmed the potential of UMIsimulation for the reproduction of real blood flows by com-bining ultrasonic diagnostic equipment and a computer. Itis essential to extend UMI simulation to three-dimensionalproblems and to perform complete verification theoreti-cally, numerically, and experimentally in order to applyUMI simulation to the development of advanced medicaldiagnostic and treatment equipment.

ACKNOWLEDGMENTS

The computations were performed using the supercom-puter system SGI ORIGIN 2000 in the Advanced FluidInformation Research Center, Institute of Fluid Science,Tohoku University. The authors are grateful to the staff of

428 FUNAMOTO et al.

the AFI Research Center for their support in the computa-tional work.

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