Three-dimensional transient and harmonic shear-wave scattering by a soft cylinder for dynamic vascular elastography

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1 Three-dimesioal trasiet ad harmoic shear-wave scatterig by a soft cylider for dyamic vascular elastography Ais Hadj Hei a,b Laboratory of Biorheology ad Medical Ultrasoics, Uiversity of Motreal Hospital Research Ceter, Motreal, QC H2L 2W5, Caada Cédric Schmitt Laboratory of Biorheology ad Medical Ultrasoics, Uiversity of Motreal Hospital Research Ceter, Motreal, QC H2L 2W5, Caada ad Istitute of Biomedical Egieerig, Uiversity of Motreal, Motreal, QC H3C 3J7, Caada Guy Cloutier a,c Laboratory of Biorheology ad Medical Ultrasoics, Uiversity of Motreal Hospital Research Ceter, Motreal, QC H2L 2W5, Caada; Istitute of Biomedical Egieerig, Uiversity of Motreal, Motreal, QC H3C 3J7, Caada; ad Departmet of Radiology, Radio-Ocology ad Nuclear Medicie, Uiversity of Motreal, Motreal, QC H3T 1J4, Caada Received 17 April 28; revised 18 July 28; accepted 19 July 28 With the objective of characterizig biological soft tissues with dyamic elastography, a three-dimesioal 3D aalytical model is proposed to simulate the scatterig of plae shear waves by a soft cylider embedded i a ifiite soft medium. The 3D problem of harmoic plae shear-wave scatterig is first formulated ad solved, ad the moochromatic solutio is employed to simulate trasiet wave scatterig. Both harmoic ad trasiet simulatios are compared with experimetal 3D acquisitios. The good agreemets obtaied betwee measured ad calculated displacemet fields allowed to coclude o the validity of the proposed 3D harmoic ad trasiet models. The spatial distributio of displacemets diffractio lobes, displacemet oscillatios, wave diffractio agles, etc. ad their relative amplitudes i both iclusio ad surroudig materials depeded o the cotrast betwee the viscoelastic properties of the differet media. The possibility of solvig a iverse problem to assess soft heterogeeous medium viscoelasticity is discussed ad some future theoretical ad experimetal developmets are proposed. 28 Acoustical Society of America. DOI: / PACS umber s : 43.8.Qf, 43.8.Vj, 43.8.Ev, Mr FD Pages: I. INTRODUCTION Detectio ad characterizatio of vascular pathologies are of great importace sice they are resposible for more tha oe-third of global deaths caused by cardiovascular diseases i the world 1 22 World Health Orgaizatio statistics. I this cotext, the preset work is dedicated to study, theoretically ad experimetally, shear-wave SW scatterig by mimicked veous thrombi 2 i order to erich existig diagostic methods, such as ultrasoud imagig, x-ray agiography, ad magetic resoace imagig MRI, byaew oe based o dyamic elastography. Ideed, it has bee prove that viscoelastic properties of coagulated blood deped o the clot age ad compositio. 3 5 This depedece is curretly exploited to perform qualitative static elastography imagig usig a static or quasistatic loadig to image the strai distributio i the medium 6,7 but this techique is sesitive to mechaical boudary coditios ad motio a Authors to whom correspodece should be addressed. b Electroic mail: ais.hadjhei@crchum.qc.ca c Electroic mail: guy.cloutier@umotreal.ca artifacts ad does ot allow to measure tissue viscosity. However, quatitative elasticity ad viscosity imagig could become a iterestig cliical idicator for pathology diagosis ad therapy plaig. Cotrary to static methods, dyamic elastography 8,9 permits to perform viscoelastic characterizatio of livig tissues by studyig travelig of elastic SWs ito the probed medium. Sice SW speeds i soft tissues are very low few meters per secod compared to compressio waves, it is possible to track them with a ultrafast ultrasoic imagig system that ca typically produce more tha 5 images per secod. Compared to static methods, SW trackig allows to determie their velocity ad atteuatio idepedetly of the mechaical boudary coditios. Moreover, the fast imagig system is ot sesitive to typical motio artifacts sice the propagatio occurs durig few millisecods. Dyamic elastography allows to obtai the spatiotemporal displacemet evolutio ad, cosequetly, the wave-medium iteractio to perform tissue mechaical characterizatio. Here, oe has to otice that cotrary to elastic waves i solids, the total displacemet field i tissues ca be experimetally measured ad imaged J. Acoust. Soc. Am , October /28/124 4 /2394/12/$ Acoustical Society of America

2 FIG. 1. Color olie Left: 3D view of the recostructed B-mode ultrasoic image of the experimetally tested heterogeeous medium. There is o echogeicity cotrast betwee the agar-gelati cylidrical iclusio represeted i the image ad its surroudig soft medium. Right: 3D represetatio of the plae SW scatterig problem. The icidet plae wave makes a agle of with its propagatio directio ad is iclied i the plae o,e x,e y from the y axis by a agle. A. Dyamic elastography i medicie The first applicatios of SW imagig i bioacoustics were performed usig M-mode ultrasoography 1 ad Doppler for sooelasticity, 11,12 MRI for magetic resoace elastography, 13 or by meas of real-time ultrafast ultrasoic imagig for both harmoic ad trasiet elastographies. 14,15 Cocerig trasiet elastography, most of modelig works cocered aalytical or umerical simulatio of SW geeratio obtaied by impact methods 15 or by ultrasoud radiatio forces. 16,17 The iteractio of SWs with cofied heterogeeities i ihomogeeous tissues has ot yet bee precisely modeled despite the fact that to characterize mechaically may pathologies such as clotted vessels, tumors, etc., it is ecessary to simulate ad uderstad SW scatterig by such ihomogeeities. B. Objectives A applicatio of dyamic elastography to veous ad arterial pathologies is preseted here. The aim was to model aalytically ad to study, theoretically ad experimetally, SW scatterig by a veous clot, modeled by a cylidrical soft iclusio surrouded by a ifiite soft medium. From a mechaical poit of view, both iclusio ad surroudig media were assumed to be made of homogeeous, isotropic, ad liear viscoelastic materials. A three-dimesioal 3D aalytical model was developed to simulate the scatterig of a harmoic plae SW by a cylidrical iclusio for a arbitrary icidet agle. Followig theoretical works of Fara, 18 White, 19 Fa et al., 2 ad Hoarvar ad Siclair 21 o scatterig of plae waves by cylidrical solids, the solutio was formulated usig a modal decompositio techique. The superpositio priciple of harmoic solutios statioary displacemet fields served to obtai the 3D scattered field for a arbitrary icidet trasiet plae SW. It is importat to ote that the model proposed i the preset work ca be coupled with magetic resoace elastography or sooelastography techiques. Before validatig experimetally the 3D model, the image acquisitio process ad the experimetal setup that were used to produce ultrafast scaig are first described. The experimetal material was composed of a agar-gelati phatom tissue mimickig material cotaiig a soft cylidrical iclusio, which simulated a veous clot. Both theoretical ad experimetal results are compared ad discussed to ivestigate the validity of simulatios. I the light of this discussio, a set of future possible applicatios ad perspectives is preseted, i particular, to solve the iverse problem allowig to characterize blood clot mechaical properties. II. THEORY To itroduce the model, Fig. 1 shows a B-mode ultrasoic image of the 3D volume experimetally probed. The imaged soft agar-gelati volume cotaied a circular cylidrical iclusio made of a mechaically differet agargelati material. Oe ca observe that there is o echogeicity cotrast betwee the iclusio ad its surroudig medium despite their viscoelasticity differeces. It will be show i Sec. IV that a strog mechaical cotrast appears betwee these two media whe a SW propagates ad iteracts with the iclusio. A. Problem formulatio The iclusio blood clot phatom was assumed to be a circular cylider, of radius R, made of a soft material medium 1 ad surrouded by a ifiite soft tissue medium 2. Both media were assumed to be homogeeous, isotropic, liear, ad viscoelastic. It is kow that, for a certai frequecy rage amely, for a few hudred hertz, the mechaical behavior of the agar-gelati material is govered by the Hooke Voigt viscoelastic behavior law. 22 This is i agree- J. Acoust. Soc. Am., Vol. 124, No. 4, October 28 Hei et al.: 3D harmoic ad trasiet elastography 2395

3 met with a prelimiary experimetal work 23 that also suggested that coagulated blood viscoelasticity follows the same Hooke Voigt model. This latter relies o the frequecy domai, the stress,, ad the strai,, tesors i the followig way: = C + i C, where C ad C are the stiffess ad viscosity tesors, respectively. For a isotropic ad viscoelastic material, the complex stiffess tesor, C=C +i C, depeds o the complex Lamé coefficiets ad. Accordig to this, ad i order to easily take ito accout the viscosity, the elastodyamic problem was expressed i the frequecy domai. Sice studied materials are soft, the bulk moduli of both media are ot sigificatly differet ad were chose to esure a compressio wave speed equal to 154 m/s ultrasoud speed i biological tissues for tissue desities of 11 kg/m 3. Cosequetly, scatterig of SWs i such materials depeds strogly o the complex shear moduli. The problem geometry, give i Fig. 1, was described i a Cartesia system of coordiates o,e x,e y,e z but regardig the cylider circular geometry ad to simplify the expressio of mechaical boudary coditios, the elastodyamic problem was expressed ad solved i a cylidrical system of coordiates o,e r,e,e z. The iclied icidet plae SW, represeted i Fig. 1, propagates with a icidece agle, formed by the icidet shear wavevector k Tic ad the polarizatio plae o,e x,e y. Fially, ad for more geerality, the icidet wave polarizatio, U ic cotaied i the plae o,e x,e y, was doig a agle with the y axis, see Fig. 1. From these geometrical cosideratios, oe ca coclude that the scatterig problem is 3D ad that mode coversios of SWs ca occur. Cosequetly, diffracted waves were expressed as a combiatio of compressio ad SWs i both media. The icidet plae SW was first cosidered to be harmoic with a agular frequecy. The, the harmoic solutio was derived to model the scatterig of arbitrary trasiet plae SWs. B. The statioary problem I the followig, the subscript 1 is assiged to the cylider costitutive material, whereas the subscript 2 desigates the surroudig medium. I additio, the logitudial compressio ad trasverse shear wavevectors i the medium j j= 1,2 are labeled by K j ad k j, respectively. Both of these wavevectors are decomposed ito compoets parallel to the z axis, called K jz =K j si ad k jz =k j si, ad other oes cotaied i the plae o,e r,e perpedicular to the cylider axis give by K j =K j cos ad k j =k j cos. The displacemet field i both media satisfies the Navier elastodyamic wave equatio. 24 Applyig a Fourier trasform to this equatio, oe obtais 1 j 2 U j + j +2 j U j j U j = with j = 1,2, 2 where U j = U rj U j U zj T is the displacemet field i a phase j= 1,2, j is the desity, ad j ad j are the complex viscoelastic Lamé coefficiets. The Helmholtz decompositio techique permits to express the displacemet field i both media ito scalar ad vector potetials: 25 U j = j + j e z + R j e z with j = 1,2. 3 Mechaically, j r, is the displacemet scalar potetial associated with compressio waves i phase j ad j e z ad j e z are the vector potetials of displacemets polarized followig the o,e r,e plae ad aother parallel to the z axis, respectively. Each of these potetials satisfies the well kow Helmholtz wave equatio. Followig the classical modal decompositio method, these potetials ca be expressed as ifiite series of Bessel ad agular fuctios cotaiig ukow coefficiets, except for the kow icidet purely SW cotaied i the plae o,e r,e. This latter depeds oly o the vector potetial ic U ic = ic e z. Cosequetly, we ca choose to express the shear icidet field by its potetial, which is writte i a cylidrical system of coordiates as ic = amp i J k 2 r = cos e ik 2 z z i t, where is the Neuma factor, i.e., =1 ad =2 for 1. Oe has to ote that the icidet wave s icliatio agle, i the plae o,e r,e, acts like a rotatio agle applied to the wave correspodig to = i.e., a icidet wave propagatig followig the x axis. The icidet wave amplitude, amp, is fixed to 1 for harmoic icidet waves ad is equal to the excitatio amplitude complex amplitude i the case of a trasiet icidet wave see Sec. II C. The ormalizatio coefficiet,, esures that the maximum amplitude of the icidet harmoic wave is equal to uity regardless of the frequecy. I a similar way, trasmitted ad reflected displacemet fields ca be expressed via their potetials. Cosiderig the fact that the displacemet at the cylider ceter is fiite, oly the first kid Bessel fuctios J serve to express the refracted potetials. The three displacemet potetials withi the iclusio are the give by 1 = A J K 1 r si e ik 2 z z, = 1 = B J k 1 r cos e ik 2 z z, = 1 = C J k 1 r si e ik 2 z z. = I these expressios ad i the followig, the harmoic time depedece term is omitted for simplificatio. The displacemet potetials i the surroudig medium are composed of those of the icidet plae wave Eq. 4 ad of the scattered oes. Kowig that scattered waves are out J. Acoust. Soc. Am., Vol. 124, No. 4, October 28 Hei et al.: 3D harmoic ad trasiet elastography

4 1 1 Normalized displacemet (a.u.) Normalized displacemet (a.u.) Time (ms) Frequecy (Hz) FIG. 2. Left: icidet trasiet wave temporal form used i experimets ad simulatios. The wave is composed of six modulated pulses with 35 Hz cetral frequecy. Right: the discretized icidet wave spectrum represetig the calculatio poits. goig, they are expressed by meas of first kid Hakel fuctios H 1. Oe fially obtais the followig scattered potetial expressios: 2 = D H 1 K 2 r si e ik 2 z z, = 2 = E H 1 k 2 r cos e ik 2 z z, = 2 = F H 1 k 2 r si e ik 2 z z. = Oe ca ote that the axial wave umber k 2z is the same i both media. It has to be equal to the axial compoet of the icidet SW sice stress ad displacemets are cotiuous at the cylider iterface. Moreover, it is importat to ote that the agular depedecies i Eqs. 5 ad 6 were chose to be i agreemet with the symmetry ad atisymmetry properties of the differet elastic waves i.e., the icidet ad the scattered oes. 19 At the cylider boudary, the cotiuity of displacemets ad ormal stress give by, where is the stress matrix ad is the uit vector ormal to the cylider permits to determie the ukow coefficiets A, B, C, D, E, ad F i Eqs. 5 ad 6. These coditios are summarized as follows: 6 U r1 U 1 U z1 T = U r2 U 2 U z2 T rr1 r 1 rz1 T = rr2 r 2 rz2 T at r = R ad,2. 7 Usig Eqs. 1 ad 3, displacemet ad stress fields expressed at r=r i Eq. 7 ca be writte with respect to potetials give i Eqs. 5 ad 6. The orthogoality property of agular fuctios over the iterval,2 allows to separate the boudary coditios i Eq. 7 ito a ifiite set of equatios with respect to the ifiite series orders =,...,. For a give mode of order, it appears, after calculatios, that the ukow coefficiets A, B, C, D, E, ad F are related to a set of kow terms, which deped o the icidet SW, by meas of a liear system of equatios: T A B C D E F T = b with r = R ad =,...,. 8 I Eq. 8, the 6 6 T matrix depeds o the geometry ad the mechaical properties of media 1 ad 2, while the secod member b represets the icidet wave cotributio to the th displacemet ad stress terms, expressed at the cylider boudary. Both T ad b elemets are detailed i the Appedix. Solvig the system of Eq. 8, by usig a classical matrix iversio algorithm 26 to avoid fastidious calculatios usig the Cramer rule which formulates solutios explicitly, provides for each iteger order the searched coefficiets. These latter permit the to calculate the total statioary displacemet field by meas of expressios 4 6 ad 3. Sice oe caot calculate all terms of the ifiite series appearig i the displacemet field expressio, the maximum calculatio order is limited to a fiite order. The trucatio order, which depeds o the iclusio size ad o the icidet wave frequecy, was fixed for the simulatios preseted here to 5 terms. C. Trasiet-wave scatterig Followig the superpositio priciple i the spectral domai, 24 trasiet SW scatterig by a homogeeous soft circular cylider was modeled usig the previous harmoic wave scatterig model. Ideed, ay icidet wave, havig a arbitrary temporal profile, ca be expressed i the frequecy domai by meas of a temporal Fourier trasform. I practice, the relevat frequecies are frequetly limited to a certai iterval with o sigificat iformatio outside it see the example i Fig. 2. Cosequetly, by choosig to express the trasiet wave scatterig problem i the frequecy domai, oe ca discretize the frequecy rage of iterest ito a fiite umber of calculatio poits Fig. 2 ad solve, for each frequecy-wave amplitude couple,amp i Eq. 4, the associated harmoic problem usig the model previously preseted. This set of statioary solutios provides the diffracted trasiet wave spectrum. The temporal diffracted field was fially obtaied from the spectral solutio by J. Acoust. Soc. Am., Vol. 124, No. 4, October 28 Hei et al.: 3D harmoic ad trasiet elastography 2397

5 y z Fuctio geerator Amplifier Charge amplifier Temperature-cotrolled chamber x Rigid plate Excitatio sigal Oscilloscope Accelerometer Vibrator Sychroizatio US scaer & computer Mimickig blood iclusio Agar-gelati phatom meas of a iverse Fourier trasform. Compared to a temporal solvig strategy, this spectral method is faster ad permits to model more easily viscoelastic effects sice temporal covolutio products appearig whe viscosity is modeled are trasformed ito simple products i the frequecy domai. III. EXPERIMENTAL SETUP A. Geeratio of shear waves ad ultrafast acquisitio system I the experimetal setup schematized i Fig. 3, SWs were geerated by a large rigid plate mm 2 attached to a vibrator type 481, Brüel&Kjær, Nærum, Demark ad applied o the phatom surface. The low frequecy harmoic or trasiet vibratio was produced with a fuctio geerator model 3325A, Agilet, Palo Alto, CA, amplified type 276, low frequecy amplifier, Brüel&Kjær before supplyig the miishaker. I this cofiguratio, the propagatio directio of the plae SWs was orthogoal to the plate motio. A cliical array trasducer model L14-5/38, 38 mm width, 128 elemets, Ultrasoix coected to the Soix RP scaer Ultrasoix Medical Corporatio, Buraby, BC, Caada was positioed parallel to the tissue motio to acquire ad recostruct radio-frequecy RF sequeces at a high frame rate 385 Hz. The ultrafast imagig method used here was ispired from a electrocardiogram-gated image acquisitio strategy to reach high frame rates. 27,28 I the preset case, sychroizatio was achieved usig SW gatig by cosiderig the startig of the mechaical excitatio. The probe excitatio frequecy, the samplig frequecy, ad the bit depth were 1 MHz, 4 MHz, ad 16 bits, respectively. To avoid reverberatio artifacts, a acoustical absorber was placed o the frot of the probe. The true plate motio was acquired with a accelerometer type 4375, Brüel&Kjær coected to a charge amplifier type 2692, US probe US absorber FIG. 3. Color olie A schematic represetatio of the experimetal setup: the SW geeratio ad the ultrasoic acquisitio devices. Nexus Amplifier, Brüel&Kjær. The part of the experimetal setup icludig the phatom was eclosed i a temperaturecotrolled chamber regulated at 2 C. The agar-gelati phatom was made followig a protocol described i a previous work. 29 The surroudig material was made i proportio of the water weight of 4.% porcie ski gelati ad 3.% agar powder product No. G-189 for gelati ad No. A-9799 for agar, Sigma Chemical, St. Louis, MO ad cotaied a agar-gelati cylidrical iclusio composed of 2.5% porcie ski gelati ad 1.% agar. The whole phatom had a parallelepipedal geometry cm 3 ad the iclusio was a 9.8 mm diameter cylider. This latter formed a agle of =46 i.e., the icidece agle with the movig plate positioed vertically ad had a legth equal to 15.5 cm. To image the 3D displacemet field, RF ultrasoic sigals at 14 cosecutive plaes mm 2 were acquired every 2.5 mm alog the z axis for a 35. mm scaig distace by adjustig automatically the probe positio with a positioig step motor. A ormalized cross-correlatio algorithm applied to the acquired RF sigals allowed to obtai the y displacemet field compoet ad its temporal evolutio for the differet slices. Fially, the 3D displacemet field was recostructed by superposig, alog the z axis, the two-dimesioal 2D cosecutive measured fields. I what follows, oe has to ote that both experimetal ad simulated displacemet fields i 3D, 2D ad oedimesioal 1D represetatios have bee ormalized by their respective maximum displacemet value. B. Viscoelasticity of phatom compoets Sice mechaical properties have to be etered ito the model to calculate displacemet fields ad i order to validate simulatios, i.e., to solve the forward problem, it was ecessary to assess viscoelastic properties of the gel materials that were used. The viscoelastic properties of phatom materials were determied by usig 2D dyamic elastography 22 based o our ultrafast imagig system. Usig this approach, the viscoelasticity was obtaied by solvig a iverse problem 22 based o the phase velocity ad atteuatio of plae SWs propagatig i a homogeeous gel sample. A multifrequecy study usig harmoic plae waves at differet frequecies allowed to assess the Hooke Voigt viscoelastic parameters. The two agar-gelati materials surroudig medium ad iclusio were characterized by usig a separate homogeeous ad parallelepipedal phatom for each of them. Viscoelastic properties were assessed betwee 5 ad 44 Hz for the surroudig material ad from 5 to 36 Hz for the iclusio material. Sice this latter was very soft ad viscous, high frequecy plae SWs did ot propagate sufficietly to characterize the iclusio material at frequecies above 36 Hz as it ca be show by simulatio, such a problem is avoided whe the soft material is cofied, for example, i a cylidrical form ito a harder oe. The experimetal viscoelastic mea values ad relative errors i.e., the ratio betwee the stadard deviatio ad the mea value for the two samples were obtaied by assessig plae SW veloc J. Acoust. Soc. Am., Vol. 124, No. 4, October 28 Hei et al.: 3D harmoic ad trasiet elastography

6 Displacemet (a.u.) Displacemet (a.u.) FIG. 4. Color olie 3D represetatio of experimetal left ad simulated right ormalized statioary displacemet fields with =46. The top ad bottom images correspod to the scatterig of 35 ad 45 Hz harmoic icidet waves, respectively. -1. ity ad atteuatio alog 8 lies of 38 mm width at differet depths. Viscoelasticity mea values were = i Pa ad = i Pa for the surroudig medium ad iclusio, respectively. Relative experimetal errors were equal to 2.4% ad 14.% for elasticity ad viscosity, respectively. IV. EXPERIMENTAL VALIDATION OF THE 3D MODELS A. Moochromatic case The first experimetal validatio cosisted i comparig experimetal ad theoretical statioary displacemet fields due to the scatterig of a moochromatic icidet SW. This validatio is ecessary sice the harmoic simulatio also served to model a trasiet icidet wave scatterig. I additio, moochromatic excitatio of soft tissues is commoly used i magetic resoace elastography ad i Doppler sooelastography. Two harmoic excitatios, 35 ad 45 Hz, have bee chose to verify the validity of the harmoic model. This choice has bee made because a large badwidth was covered by the trasiet wave experimetally geerated see Fig. 2. Figure 4 presets a 3D view of experimetal ad theoretical ormalized displacemets for the two harmoic excitatios. A quarter volume was cut out from the etire scaed or calculated space i order to better visualize the displacemet field ito the iclusio. As oe ca see, a great similarity is oted betwee displacemets withi the iclusio ad those withi the surroudig medium for the two icidet waves. Oe ca also otice that the correspodece betwee simulatios ad measuremets is ot satisfied everywhere i the 3D space. I particular, it appears that the experimetal frotwaves are ot always strictly plaar ad that phase ad amplitude shifts exist i certai regios. I order to compare more rigorously simulatios with experimets ad study the differeces, the statioary fields were compared, first, followig a plae amely, the plae z= 17.5 mm, the alog a lie 1D parallel to the x axis ad cotaied i the plae, ad fially alog the z axis i.e., the cylider axis. Figure 5 shows the 2D ormalized statioary displacemet fields i the plae for the two harmoic icidet waves. The horizotal striatios jitter errors appearig i the experimetal displacemets correspod to the piezoelectric elemets of the ultrasoic liear array. It is oticeable that measuremets cotai experimetal oise due to the relatively small amplitude of displacemets at high frequecies at lower frequecies, strog displacemet modulatios do ot appear withi the iclusio. Despite the presece of oise, oe ca see the good agreemet betwee measured ad theoretically predicted fields. For this example, the ex- J. Acoust. Soc. Am., Vol. 124, No. 4, October 28 Hei et al.: 3D harmoic ad trasiet elastography 2399

7 xaxis x (mm) (mm) xaxis(mm) x Displacemet (a.u.) Oe ca also observe i Fig. 5 that the presece of oscillatios i the iclusio is well simulated by the model. This is cofirmed i Fig. 6 that represets, for both selected frequecies, the superpositio of the experimetal ad simu yaxis(mm) y y y axis (mm) xaxis(mm) x (mm) xaxis(mm) x Displacemet (a.u.) y axis y (mm) y y axis (mm) (mm) FIG. 5. Color olie 2D represetatio of experimetal left ad simulated right ormalized statioary displacemet fields i the plae correspodig to z= 17.5 mm. The top ad bottom images correspod to the scatterig of 35 ad 45 Hz harmoic icidet waves, respectively. The measured icliatio, i the plae o,e x,e y, equals perimetal icidet wave was iclied, i the plae o,e x,e y, by a agle = 3 relative to the x axis. Figure 5 shows that this icliatio, which has bee measured ad etered ito the model, is correctly simulated. 1 (a) 1 (b) Displacemet (a.u.).5 Displacemet (mm) (a.u.) xaxis(mm) x xaxis(mm) x FIG. 6. Compariso of the experimetal ad simulated statioary displacemets alog the x axis i the plae for a a 35 Hz harmoic icidet wave ad b for a 45 Hz harmoic icidet wave experimetal; simulatio. The double arrow, i the top of figures, desigates the iclusio spatial locatio. 24 J. Acoust. Soc. Am., Vol. 124, No. 4, October 28 Hei et al.: 3D harmoic ad trasiet elastography

8 zaxis(mm) (a) Displacemet (a.u) Displacemet (a.u.) FIG. 7. Compariso of experimetal ad simulated statioary displacemets alog the z axis ito the iclusio for a the 35 Hz harmoic icidet wave ad b for the 45 Hz harmoic icidet wave experimetal; simulatio. lated displacemet profiles alog a lie y= parallel to the x axis, crossig the iclusio ad cotaied i the plae.a good correspodece betwee calculatios ad measuremets is observable, except for the regio immediately after the iclusio i the positive x directio. Oe ca ote the effect of atteuatio icreasig with frequecy betwee the two plots, particularly i the regio ragig betwee 5. ad 2. mm. Oe may also otice differeces i amplitude betwee measured ad calculated displacemets i Fig. 6 that are also observable i the statioary fields of Fig. 5. This is due to three pricipal reasos. The first oe is the ucertaity i mechaical properties of materials composig iclusio ad surroudig medium give i Sec. III B. Sice theoretical displacemets are calculated for mea values of the measured viscoelastic properties which are global measuremets, locally, slight differeces betwee simulated ad measured displacemets particularly for wave atteuatio ca exist. The secod source of differeces is attributed to the relatively low spatial resolutio i the lateral x directio because of the limited umber of elemets i the array trasducer. The measured displacemets were cosequetly subsampled particularly ito the iclusio where wavelegths are smallest ad a error o the iclusio localizatio occurred. Fially, the whole phatom dimesios were fiite ad the surroudig medium viscosity was relatively low. Experimetally, i the harmoic regime, this caused SW reflectios at boudaries that perturbed the measured displacemets ito the iclusio ad i its eighborig regio. The abovemetioed sources of errors were simultaeously preset but they did ot dramatically affect the quality of measuremets. This is also verified i Fig. 7, where experimetal ad theoretical statioary displacemets alog the z axis are compared for each tested frequecy. As oe ca expect, the wavelegth followig the z directio, as i the other directios, is iversely proportioal to the icidet wave frequecy which is 14.5 ad 11.3 mm at 35 ad 45 Hz, respectively. The best agreemet is thus oticed at the highest frequecy of 45 Hz i Fig. 7. Existig differeces could be explaied by the spatial discretizatio error z z axis (mm) (b) =2.5 mm ad by elastic wave reflectios at phatom boudaries that are strogest at low frequecies sice SWs are less atteuated at low frequecies. Before cocludig for the moochromatic case, it is importat to poit out that the presece of oscillatios i the iclusio, observed i the horizotal plaes see, for example, Fig. 6 ad alog the vertical directio as i Fig. 7, is directly related to the iclusio costitutive material viscoelasticity. Ideed, o oe had oscillatio wavelegths are a fuctio of elasticity, ad o the other had their amplitudes deped o both elasticity ad viscosity. Aother scattered wave characteristic is its spatial distributio i a give horizotal plae, i.e., the SW slowig dow after crossig the iclusio, the orietatio of diffractio lobes i the surroudig medium see Fig. 5, the 2D shape of oscillatios ito the iclusio, the wave diffractio agle ito the iclusio with respect to its axis, etc. All these behaviors are related to the iteral ad exteral viscoelastic properties ad also to the cotrast betwee them. Cosequetly, as it is developed i Sec. IV B, oe could exploit this rich iformatio to characterize mechaical properties of heterogeeous media. B. Trasiet case The harmoic 3D model is ow used to simulate scatterig of a trasiet SW with the previously described strategy. Both simulated ad experimetal icidet waves were costituted by six oscillatios, modulated by a Blackma temporal widow to miimize Gibbs effects, with 35 Hz cetral frequecy see Fig. 2. Theoretical displacemets are compared as a fuctio of time with measuremets obtaied o the same phatom as the oe used for the moochromatic excitatio. Figure 8 shows the trasiet wave propagatio at differet momets i half of the total 3D probed volume. This 3D view has bee chose to visualize more precisely the iteral iteractio of the icidet SW with the cylidrical iclusio ad the wave diffractio agle withi the cylider relative to the z axis. Time-varyig experimetal ad simulated displacemets are i good agreemet i both iclusio ad surroudig media. However, some commets ca be made o the scatterig behavior. First, it is oticeable that the experimetal icidet wave is ot perfectly plae this is observed at t=21.8 ms i the lower left part of the image. This explais slight differeces betwee measured ad predicted displacemet fields. A secod remark cocers oscillatios observable i simulatios i both iclusio ad surroudig media before the icidet wave arrival see the right pael of Fig. 8 at t=11.4 ms. These artifacts are due to the Gibbs pheomeo, which appears whe the scatterig spectral solutio is iversely Fourier trasformed to the time domai. Ideed, sice the scatterig problem is solved for a fiite frequecy rage, we applied the zero-paddig techique outside this rage to get the total spectral respose. This umerical techique itroduced a small discotiuity i the spectra ad, i cosequece, small ophysical oscillatios i the time respose. Despite these sources of errors, i additio to those refereced for the harmoic case except reflectios which are J. Acoust. Soc. Am., Vol. 124, No. 4, October 28 Hei et al.: 3D harmoic ad trasiet elastography 241

9 t=11.4 ms Displacemet (a.u.) t=34.8 ms t=27.3 ms t=21.8 ms t=16.6 ms FIG. 8. Color olie Time-varyig 3D represetatio of the trasiet wave excitatio scattered by the soft cylidrical iclusio. The view represets the half of the volume to visualize the behavior of the wave ito the iclusio durig the propagatio. Icliatio ad icidece agles, ad, were equal to 3 ad 46, respectively. abset i the trasiet case, we ca observe that the 3D trasiet SW scatterig was well simulated by the model. It is importat to otice that, as for the harmoic scatterig problem, the wave distributio ad amplitude are related to the medium viscoelasticity alog the propagatio path. Typically, the wavelegth ad the wave diffractio agle ito the cylider are related to the iclusio mechaical properties ad to the cotrast i viscoelasticity betwee both media, respectively. 242 J. Acoust. Soc. Am., Vol. 124, No. 4, October 28 Hei et al.: 3D harmoic ad trasiet elastography

10 V. CONCLUSION AND PERSPECTIVES A aalytical model was developed to simulate the scatterig of a arbitrary moochromatic or trasiet plae SW by a circular cylidrical heterogeeity. Both iclusio ad surroudig media costitutive materials were soft eough to allow the slow propagatio of SWs which could, cosequetly, be imaged by a ultrafast ultrasoic system. The model validatio was achieved experimetally o a heterogeeous phatom cotaiig a very soft cylidrical iclusio. The compariso of measured ad calculated 3D, 2D, ad 1D statioary displacemet fields has show the model validity to simulate SW scatterig by a cylidrical heterogeeity. The validated model the served to simulate the scatterig of a trasiet plae SW usig the superpositio priciple. I this cofiguratio, 3D theoretical temporal displacemets were also i good agreemet with measuremets. Whe the geometrical ad mechaical cofiguratios of a scatterig problem allow a aalytical solvig, aalytical approaches are geerally more precise, more stable, ad faster tha umerical methods such as fiite-elemet ad fiite-differece methods, well adapted to irregular scatterig cofiguratios. Moreover, aalytical models allow the calculatio of mechaical fields, such as displacemet, strai, ad stress, at ay spatial positio a part of the plae, a lie, or a set of poits, without calculatig i the whole volume like i umerical approaches. These advatages could be iterestig to solve iverse problems usig simulatios. I the cotext of dyamic elastography of soft structures, ad more geerally of livig tissues, the approach developed i the preset work could likely be applied to assess viscoelastic properties of veous blood clots, as itroduced earlier. Ideed, these latter have a cylidrical shape almost circular ad are costituted by coagulated blood with elasticity that does ot exceed few kilopascals ad a viscosity that is geerally high. 5 Clot mechaical properties deped o several factors such as hematocrit level, the time elapsed sice its formatio i a blood vessel, chemical ad metabolic coditios, etc. Oe ca the cosider to formulate ad solve a iverse problem, ito a strategy usig both i vivo ad simulated data, to perform mechaical property assessmets of clots. The correlatio of the assessed viscoelasticity with the clot age, its compactess, ad biological costitutio would provide to cliicias a complemetary tool for veous thrombosis diagosis ad therapy plaig. Moreover, the geeral 3D cofiguratio cosidered here is compatible with real cliical coditios where mechaical excitatio ad ultrasoud imagig agles ca be arbitrary due, for example, to aatomical restrictios. The formulatio ad resolutio of such iverse problem should deserve attetio. Parallel to this direct applicatio of the preset work, the model could be developed to simulate scatterig of SWs by a multilayer cylidrical iclusio. This extesio would permit to treat the more geeral ad realistic case of a ihomogeeous iclusio. To be more realistic, aother aveue of developmet would be the simulatio of SW iteractio with geometrically ad mechaically more complex heterogeeities e.g., hollowed heterogeeities, elliptical cyliders, fiite heterogeeities, aisotropic materials, etc.. To coclude, this study showed that it is possible to aalytically model SW iteractios with soft bodies. The impact of this model is ot limited to ultrasoud elastography i both harmoic ad trasiet regimes but also to MRI based elastography moochromatic regime that ca be formulated i a similar framework. It is fially coceivable to exted this modelig strategy to other pathologies such as breast, brai, prostate ad liver tumors, atherosclerosis, etc. ACKNOWLEDGMENTS This work was supported by a grat from the Caadia Istitutes of Health Research No. MOP-84358, byanatioal Scietist award of the Fods de la Recherche e Saté du Québec G.C., ad by postdoctoral A.H.H. ad doctoral C.S. partial scholarships of the Groupe de Recherche e Scieces et Techologies Biomédicales of the Istitute of Biomedical Egieerig of the École Polytechique ad Uiversité de Motréal. Appedix The 6 6 T matrix elemets i Eq. 8 are give by T 1,1 = 1 2 K 1 J 1 K 1 R J +1 K 1 R, = R 2 k 1ik 2z J 1 k 1 R J +1 k 1 R, T 1,3 = R J k 1 R, H 1 +1 K 2 R, T 1,5 = R 2 k 2 ik 2z H 1 1 k 2 R H 1 +1 k 2 R, T 2,1 = R J K 1 R, T 1,2 T 1,4 = 1 2 K 2 H 1 1 K 2 R T 2,2 T 1,6 = R H 1 k 2 R, = ik 2z J k 1 R, T 2,3 = 1 2 k 1 J 1 k 1 R J +1 k 1 R, T 2,4 = R H 1 K 2 R, H 1 k 2 R, T 2,5 = ik 2z T 2,6 = 1 2 k 2 H 1 1 k 2 R H 1 +1 k 2 R, T 3,1 = ik 2z J K 1 R, J. Acoust. Soc. Am., Vol. 124, No. 4, October 28 Hei et al.: 3D harmoic ad trasiet elastography 243

11 T 3,2 = R k 1 2 T 3,3 =, 4 J 2 k 1 R 2J k 1 R + J +2 k 1 R k 1 2R J 1 k 1 R J +1 k 1 R + 2 R 2J k 1 R, T 3,5 = R + k 2 2 T 3,6 =, T 3,4 = ik 2z H 1 K 2 R, 4 H 1 2 k 2 R 2H 1 k 2 R + H 1 +2 k 2 R + k 2 2R H 1 1 k 2 R H 1 +1 k 2 R 2 R 2H 1 k 2 R, T 4,1 = K J 2 K 1 R 2J K 1 R + J +2 K 1 R 1 2 R 2 + k 2 z 2 J K 1 R + 1 K 1 2R J 1 K 1 R J +1 K 1 R, T 4,2 =2 1 R 4 i k 1 2 k 2z J 2 k 1 R 2J k 1 R + J +2 k 1 R R ik 2 z 2 J k 1 R, T 4,3 =+2 1 R 2J k 1 R 1 R k 1 J 1 k 1 R J +1 k 1 R, T 4,4 = K H 1 2 K 2 R 2H 1 K 2 R + H 1 +2 K 2 R R 2 + k 2z 2 H K 1 2 K 2 R 2 2R H 1 1 K 2 R H 1 +1 K 2 R, R T 4,5 = i k 2 2 k 2z H 1 2 k 2 R 2H 1 k 2 R + H k 2 R 2 2 R ik 2 z 2 H 1 k 2 R, T 4,6 = 2 2 R 2H 1 k 2 R + 2 R k 2 H 1 1 k 2 R H 1 +1 k 2 R, T 5,1 = 2 1 R 2J K 1 R + 1 R K 1 J 1 K 1 R J +1 K 1 R, T 5,2 = 2 1 R ik 2 z J k 1 R + 1 ik 2z k 1 J 1 k 1 R J +1 k 1 R, T 5,3 = 1 k J 2 k 1 R 2J k 1 R + J +2 k 1 R 2 k 1 1 R 2J k 1 R + 1 2R J 1 k 1 R J +1 k 1 R, T 5,4 =+2 2 R 2H 1 K 2 R 2 R K 2 H 1 1 K 2 R H 1 +1 K 2 R, T 5,5 =+2 2 R ik 2 z H 1 k 2 R 2 ik 2z k 2 H 1 1 k 2 R H 1 +1 k 2 R, T 5,6 = 2 k H 1 2 k 2 R 2H 1 k 2 R + H 1 2 k +2 k 2 R + 2 R 2H 1 2 k 2 R 2 2R H 1 1 k 2 R H 1 +1 k 2 R, T 6,1 =+2 1 ik 2z K 1 J 1 K 1 R J +1 K 1 R, T 6,2 = 1 Rk 1 k 2z 2 J 1 k 1 R J +1 k 1 R 1 R k J 3 k 1 R 3J 1 k 1 R +3J +1 k 1 R J +3 k 1 R + 1 k 1 R J 1 k 1 R J +1 k 1 R 1 k J +2 k 1 R J 2 k 1 R 2J k 1 R R 2J k 1 R R k 1 J 1 k 1 R J +1 k 1 R, T 6,3 = 2 1 R ik 2 z J k 1 R, T 6,4 = 2 2 ik 2z K 2 H 1 1 K 2 R H 1 +1 K 2 R, 244 J. Acoust. Soc. Am., Vol. 124, No. 4, October 28 Hei et al.: 3D harmoic ad trasiet elastography

12 T 6,5 = 2 Rk 2 k 2z 2 H 1 1 k 2 R H 1 +1 k 2 R + 2 R k H 1 3 k 2 R 3H 1 1 k 2 R k +3H 1 +1 k 2 R H k 2 R 2 R k H 1 1 k 2 R H k 2 R H 1 2 k 2 R 2H 1 k 2 R + H 1 +2 k 2 R R 2H 1 2 k 2 R 2 R k 2 H 1 1 k 2 R H 1 +1 k 2 R, T 6,6 =+2 2 R ik 2 z H 1 k 2 R. The secod member vector i Eq. 8 cotais the th compoets of displacemet ad stress due to the icidet wave at the cylider boudary. The b vector elemets are as follows: b 1 = amp R i J k 2 R, b 2 = amp 1 2 k 2 i J 1 k 2 R J +1 k 2 R, b 3 =, b 4 = amp 2 2 R 2 i J k 2 R amp k 2 2 R i J 1 k 2 r J +1 k 2 r, b 5 = amp k i J 2 k 2 R 2J k 2 R 4 + J +2 k 2 R + amp k 2 2 2R i J 1 k 2 R J +1 k 2 R Amp 2 2 R 2 i J k 2 R, b 6 = amp 2 2 R i +1 k 2z J k 2 R. 1 World Health Orgaizatio, Last viewed December 12, J. Hirsh ad J. Hoak, Maagemet of deep vei thrombosis ad pulmoary embolism: A statemet for healthcare professioals from the Coucil o thrombosis i cosultatio with the coucil o cardiovascular radiology, Circulatio 93, M. Kaibara, Rheology of blood coagulatio, Biorheology 33, J. V. Shah ad P. A. Jamey, Strai hardeig of fibri gels ad plasma clots, Rheol. Acta 36, J.-L. Geisso, S. Lerouge, ad G. Cloutier, Assessmet by trasiet elastography of the viscoelastic properties of blood durig clottig, Ultrasoud Med. Biol. 32, S. Y. Emeliaov, X. Che, M. O Doell, B. Kipp, D. Myers, T. W. Wakefield, ad J. M. Rubi, Triplex ultrasoud: Elasticity imagig to age deep veous thrombosis, Ultrasoud Med. Biol. 28, J. M. Rubi, S. R. Aglyamov, T. W. Wakefield, M. O Doell, ad S. Y. Emeliaov, Cliical applicatio of soographic elasticity imagig for agig of deep veous thrombosis: Prelimiary fidigs, J. Ultrasoud Med. 22, R. M. Lerer, K. J. Parker, J. Hole, R. Gramiak, ad R. C. Waag, Sooelasticity: Medical elasticity images derived from ultrasoud sigals i mechaically vibrated targets, Acoust. Imagig 16, Y. Yamakoshi, J. Sato, ad T. Sato, Ultrasoic imagig of iteral vibratio of soft tissue uder forced vibratio, IEEE Tras. Ultraso. Ferroelectr. Freq. Cotrol 37, A. Eisecher, E. Schweg-Toffler, G. Pelletier, ad P. Jacquemard, La palpatio échographique rythmée: Echosismographie, J. Radiol. 64, T. A. Krouskop, D. R. Dougherty, ad F. S. Viso, A pulsed Doppler ultrasoic system for makig oivasive measuremets of the mechaical properties of soft tissue, J. Rehabil. Res. Dev. 24, K. J. Parker ad R. M. Lerer, Sooelasticity of orgas: Shear waves rig a bell, J. Ultrasoud Med. 11, R. Muthupillai, D. J. Lomas, P. J. Rossma, J. F. Greeleaf, A. Maduca, ad R. L. Ehma, Magetic resoace elastography by direct visualizatio of propagatig acoustic strai waves, Sciece 269, S. Cathelie, F. Wu, ad M. Fik, A solutio to diffractio biases i sooelasticity: The acoustic impulse techique, J. Acoust. Soc. Am. 15, L. Sadri, M. Tater, S. Cathelie, ad M. Fik, Shear modulus imagig with 2D trasiet elastography, IEEE Tras. Ultraso. Ferroelectr. Freq. Cotrol 49, J. Bercoff, M. Tater, ad M. Fik, Supersoic shear imagig: A ew techique for soft tissue elasticity mappig, IEEE Tras. Ultraso. Ferroelectr. Freq. Cotrol 51, M. L. Palmeri, A. C. Sharma, R. R. Bouchard, R. W. Nightigale, ad K. R. Nightigale, A fiite-elemet method model of soft tissue respose to impulsive acoustic radiatio force, IEEE Tras. Ultraso. Ferroelectr. Freq. Cotrol 52, J. J. Fara, Soud scatterig by solid cyliders ad spheres, J. Acoust. Soc. Am. 23, R. M. White, Elastic wave scatterig at a cylidrical discotiuity i a solid, J. Acoust. Soc. Am. 3, Y. Fa, A. N. Siclair, ad F. Hoarvar, Scatterig of a plae acoustic wave from a trasversely isotropic cylider ecased i a solid elastic medium, J. Acoust. Soc. Am. 16, F. Hoarvar ad A. N. Siclair, Acoustic wave scatterig from trasversely isotropic cyliders, J. Acoust. Soc. Am. 1, S. Cathelie, J.-L. Geisso, G. Delo, M. Fik, R. Sikus, S. Abouelkaram, ad J. Culioli, Measuremet of viscoelastic properties of homogeeous soft solid usig trasiet elastography: A iverse problem approach, J. Acoust. Soc. Am. 116, C. Schmitt, A. Hadj Hei, ad G. Cloutier, Characterizatio of timevaryig mechaical viscoelastic parameters of mimickig deep vei thrombi with 2D dyamic elastography, i IEEE Iteratioal Ultrasoics Symposium, New York 27, pp J. D. Achebach, Wave Propagatio i Elastic Solids North-Hollad, Amsterdam, P. M. Morse ad H. Feshbach, Methods of Theoretical Physics McGraw- Hill, New York, Matlabs User s Guide, Mathworks, Ic., Natick, MA, E. Chéri, R. Williams, A. Needles, G. Liu, C. White, A. S. Brow, Y.-Q. Zhou, ad F. S. Foster, Ultrafast frame rate retrospective ultrasoud microimagig ad blood flow visualizatio i mice i vivo, Ultrasoud Med. Biol. 32, M. Perot, K. Fujikura, S. D. Fug-Kee-Fug, ad E. Koofagou, ECGgated, mechaical ad electromechaical wave imagig of cardiovascular tissues i vivo, Ultrasoud Med. Biol. 33, J.-L. Geisso ad G. Cloutier, Sol-Gel trasitio i agar-gelati mixtures studied with trasiet elastography, IEEE Tras. Ultraso. Ferroelectr. Freq. Cotrol 53, J. Acoust. Soc. Am., Vol. 124, No. 4, October 28 Hei et al.: 3D harmoic ad trasiet elastography 245

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