On the Precision of Time-of-Flight Shear Wave Speed Estimation in Homogeneous Soft Solids: Initial Results Using a Matrix Array Transducer

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1 758 IEEE Tasacios o Ulasoics, Feoelecics, ad Fequecy Cool, vol. 60, o. 4, Apil 03 O he Pecisio of Time-of-Fligh Shea Wave Speed Esimaio i Homogeeous Sof Solids: Iiial Resuls Usig a Maix Aay Tasduce Michael Wag, Sude Membe, IEEE, Be Byam, Membe, IEEE, Mak Palmei, Membe, IEEE, Ned Rouze, Membe, IEEE, ad Kahy Nighigale, Membe, IEEE Absac A sysem capable of ackig adiaio-foceiduced shea wave popagaio i a 3-D volume usig ulasoud is peseed. I coas o exisig sysems, which use -D aay asduces, a -D maix aay is used fo ackig shea wave displacemes. A sepaae sigle-eleme asduce is used fo adiaio foce exciaio. This sysem allows shea wave popagaio i all diecios away fom he push o be obseved. I is show ha fo a limi of 64 ackig beams, by placig he beams a he edges of he measueme egio of iees (ROI) a muliple diecios fom he push, imeof-fligh (TOF) shea wave speed (SWS) measueme uceaiy ca heoeically be educed by 40% compaed wih equally spacig he ackig beams wihi he ROI alog a sigle plae, as is ypical whe usig a -D aay fo ackig. This was veified by simulaio, ad a educio of 30% was expeimeally obseved o a homogeeous phaom. Aalyical expessios ae peseed fo he elaioship bewee TOF SWS measueme uceaiy ad vaious shea wave imagig paamees. I is show ha TOF SWS uceaiy is ivesely popoioal o ROI size, ad ivesely popoioal o he squae oo of he umbe of ackig locaios fo a give disibuio of beam locaios elaive o he push. TOF SWS uceaiy is show o icease wih he squae of he SWS, idicaig ha TOF SWS measuemes ae iisically less pecise fo siffe maeials. Mauscip eceived Novembe 30, 0; acceped Decembe, 0. This wok was suppoed by Naioal Isiues of Healh gas R0 EB-003 ad R0 CA484. The auhos ae wih he Depame of Biomedical Egieeig, Duke Uivesiy, Duham, NC ( mhw@duke.edu). DOI hp://dx.doi.og/0.09/tuffc I. Ioducio Shea wave imagig is a quaiaive mehod of measuig issue siffess oivasively ad i vivo. Shea waves ca be geeaed i issue by muscle aciviy [] [4], exeal mechaical exciaio [5] [7], o by usig acousic adiaio foce [8] []. The shea wave speed (SWS) i issue is diecly elaed o is siffess. By moioig shea wave popagaio usig a eal-ime imagig modaliy such as mageic esoace imagig (MRI) [5], [6], [] o ulasoud [7] [], he udelyig issue siffess ca be esimaed. A commoly used echique fo SWS esimaio fom ulasoically acked issue displaceme is he so-called ime-of-fligh (TOF) mehod [7], [3], [4]. The shea wave aival ime is measued a seveal locaios wihi a spaial egio of iees (ROI), o keel. Ude assumpios of homogeeiy, egligible dispesio, ad a fixed diecio of popagaio wihi he ROI, a liea model ca be fi o he aival imes. The liea elaioship bewee spaial locaio ad aival imes ca he be used o calculae he SWS. This pape peses a ulasoic sysem capable of moioig acousic-adiaio-foce-iduced shea wave displaceme wihi a volume of issue. I coas o pevious sysems which use mechaically swep -D aay asduces o acquie volumeic daa [5], a -D maix aay asduce capable of elecoic beamfomig i boh he laeal ad elevaio dimesios is used fo ackig shea wave displaceme. This eables shea wave aival imes o be measued i muliple diecios fom he adiaio foce exciaio axis wihou he eed fo maual eposiioig of he pobe. The abiliy o moio shea wave popagaio i muliple diecios has seveal advaages. Fis, i allows aisoopic mechaical popeies o be chaaceized [6]. Secod, i iceases he amou of daa ha ca be acquied ad used fo SWS esimaio. Fially, i eables addiioal flexibiliy i he placeme of ackig beam locaios. I his pape, he poeial fo impovig SWS measueme pecisio i a homogeeous maeial by uilizig addiioal ackig beam locaios available fom a -D maix aay is ivesigaed. The fis poio of his pape deives heoeical expessios fo he uceaiy i TOF SWS esimaio. I is he show ha give a fixed umbe of ackig locaios, ad ROI size, he pecisio of TOF SWS esimaio i a homogeeous maeial ca be impoved by iceasig he spead of he ackig beam locaios elaive o he push by usig a -D aay. These heoeical esuls ae he veified by simulaios ad expeimeal daa acquied o phaoms usig he -D maix aay asduce. II. Uceaiy i TOF SWS Esimaio Measueme uceaiy ca be classified io wo goups: adom ad sysemaic [7]. I TOF SWS esimaio usig acousic-adiaio-foce-iduced shea waves, he measued SWS is depede o facos such as he fequecy coe of he shea wave [0], [8] ad he loca /$ IEEE

2 wag e al.: pecisio of ime-of-fligh shea wave speed esimaio i homogeeous sof solids 759 io of he ecosucio egio elaive o he exciaio geomey [9]. These facos ca esul i a sysemaic bias of he measued SWS. TOF SWS measueme is also subjec o adom eos such as jie i ulasoic displaceme ackig [0]. This ype of eo has zeo mea ad gives ise o a age of SWS i epea measuemes. I geeal, sysemaic eos ae difficul o aalyze because he udelyig ue value of he SWS is usually ukow. I coas, adom eo ca be assessed by examiig he spead of epea measuemes. Alhough sysemaic bias amog diffee SWS measueme sysems poses a sigifica challege o he cliical accepace of his echology [], [], he aalysis of souces of bias i SWS esimaio is ouside he scope of his sudy. Isead, he focus of his pape is esiced o adom measueme eo i TOF SWS esimaio. The ems uceaiy ad pecisio houghou his pape efe o he spead, o eo ba size, of epea measuemes, ad do o iclude bias. Whe he spead of epea SWS measuemes is low, he measueme is descibed as havig high pecisio, o low uceaiy. The uceaiy i TOF SWS measueme esulig fom adom eo (ubiased wih zeo mea) is ow deived. Coside leas-squaes fiig of a lie o a se of shea wave aival imes = {,,, } measued a disaces = {,,, } ohogoal o he push axis (he assumed diecio of shea wave popagaio). Noe ha ca epese spaial locaios wihi he eie -D plae ohogoal o he push axis. Muliple pois wihi his plae ca have he same value of (he locus of hese pois is a cicle of adius wih is cee a he push). Le deoe he leas-squaes fi lie, β he coespodig slope, ad β 0 he iecep, such ha = β + β. () 0 The leas-squaes soluio is give by [3]: The covaiace Cov(, ) is a measue of he exe o which vaiaios i ca be pediced by a liea fucio i. Whe he covaiace is zeo, he miimum mea squae eo liea esimao fo i ems of is simply he mea value of (3d). Whe is pefecly coelaed wih, he covaiace is equal o he poduc of he sadad deviaio of he wo vaiables, Cov(, ) = [oe ha is calculaed he same way as i (3b)]. The leas squaes slope (a) i his case becomes he aio / (iuiively, he vaiace i is escaled o have he same vaiace as ). Deviaios i he aival imes abou he leas squaes lie ae give by ε =. (4) I is assumed ha hese deviaios ae due o adom eos i measuig he aival ime of he shea wave. Souces of measueme eo iclude ulasoic displaceme ackig jie esulig fom speckle decoelaio ad fiie ackig keel size [0], [4], as well as udesamplig due o he fiie pulse epeiio fequecy (PRF) used fo ackig. These ype of eos ae omally disibued, ad do o iclude goss oulies, which ca be emoved by a algoihm such as adom sample cosesus (RANSAC) [5]. Assumig ha he vaiace of aival ime measueme eo is ε such ha ε N(0, ε ), (5) he he leas squaes esimaed slope β will also be a omally disibued adom vaiable [3]: β N ( β, β ), (6) whee β is he ue TOF slope ad he sadad deviaio is give by whee β β Cov(,) = (a) = β, (b) 0 Cov(,)= ( i )( i ) (3a) i = i (3b) i = = ( ) = i = = i. i = i (3c) (3d) β ε =. (7) To esimae he TOF SWS (c ), oe calculaes he ivese slope of he above liea model elaig measued aival imes ad disace fom he push locaio: =. We ae ieesed i he saisical disibuio of, amely, he uceaiy i ĉ esulig fom uceaiy i he esimaed slope β. To fid his, we begi by wiig he pobabiliy desiy fucio (PDF) of β : fβ ( β ) = π β exp ( β β) β β. (8) (9)

3 760 IEEE Tasacios o Ulasoics, Feoelecics, ad Fequecy Cool, vol. 60, o. 4, Apil 03 Makig he chage of vaiable β β p =, β such ha (0) β ()= p p + β (a) β dβ ( p) = d β, p (b) ad usig he ule fo asfomaio of adom vaiables [6] dβ () p fp( p)= f ( β( p)), () dp β (9) ca be coveed io he sadad omal disibuio: p fp( p)= π exp. (3) Applyig he same chage of vaiable o (8) gives =. p β β + (4) Eq. (4) ca be expaded as a biomial seies: = p p β β + β β β β β 3 3 p +. (5) The em ( / β ) β i (5) epeses he aio bewee he sadad deviaio of he esimaed slope ad is ue value, ad is equivale o he ivese of he SNR of he aival ime daa. If we assume ha β β, o ha we have high SNR aival ime daa, (5) ca be lieaized by igoig highe ode ems: p β. β β Reaagig (6), oe fids ha (6) β p( ) = ( β ) (7a) β dpc ( ) β =. d β By he ule fo asfomaio of adom vaiables, (7b) Subsiuig (7) io (8) yields f ( c ) = ( β ) exp π. (9) β β β ( ) ( ) Fom (9), i ca be see ha he esimaed SWS ĉ has a omal disibuio of N β β,. β β (0) Because he ivese of he ue TOF slope β is equivale o he ue TOF SWS c, oe fids ha N( c,( β c ) ). () As expeced, he mea value of ĉ is he ue TOF SWS c. The sadad deviaio of he esimaed SWS is give by =. () β c Subsiuig (7) io (), oe fids ĉ ε = c. (3) This esul idicaes ha he uceaiy of TOF SWS esimaio is popoioal o he squae of he SWS. I is liealy popoioal o he aival ime measueme eo, ad ivesely popoioal o he sadad deviaio of he adial disaces sampled by he ackig beams. Fo a fixed disibuio of adii, he SWS eo is also ivesely popoioal o he squae oo of he umbe of ackig beams, as would be expeced fom aveagig muliple measuemes. Thus, give he same aival ime measueme eo ad he same ackig locaios, he uceaiy i TOF SWS measueme iceases wih he squae of he SWS. This is a fudameal limi fo he pefomace of he TOF SWS esimaio mehod. Eq. (3) is valid fo ay wave aival ime esimaio echique. As a example, wo commo appoaches iclude deecig he ime o peak (TTP) ad he ime o peak slope (TTPS) of he shea wave displaceme. I a dispesive medium, hese wo mehods may esul i diffee biases i he measued SWS (a sysemaic eo). Howeve, he disibuio of he SWS abou is mea value fom epea measuemes is sill give by (3). I he ex secio, he poeial fo impovig he pecisio of TOF SWS measuemes by iceasig hough he use of he -D maix aay fo ackig shea waves is exploed. III. Mehods f ( c ) = dpc ( ) d f (( p )). p (8) I ca be see fom (3) ha by iceasig, he sadad deviaio of TOF SWS measuemes ca be e-

4 wag e al.: pecisio of ime-of-fligh shea wave speed esimaio i homogeeous sof solids 76 duced. The em is a measue of he spead of he adial disaces sampled by he ackig beams,, abou is mea value. Iceasig ca be accomplished by iceasig he umbe of ackig beams a he edges of he ROI. Zhai e al. [7] peviously showed ha whe oly 4 ackig beams ae available fom a -D aay, posiioig he beams a he eds of he ROI leads o he lowes SWS esimaio uceaiy. As will be demosaed i he followig subsecios, by akig advaage of he addiioal flexibiliy i he placeme of he ackig beam locaios affoded by he -D maix aay, a lage umbe of ackig beams ca be posiioed a he edges of he ROI o icease. The ex subsecio will pese heoeical pedicios of impoveme i SWS esimaio pecisio fo vaious possible beam locaio cofiguaios usig he -D maix aay. This is followed by descipios of expeimeal ad simulaio sudies pefomed o veify hese esuls. A. Theoy Coside a ROI wih adial age bewee l ad h fom he axis of exciaio such ha l h, as show i Fig.. The size of he ROI, k, is give by k = h l. Geomeically, his coespods o a aulus wihi he plae ohogoal o he push axis ceeed a he push. Le he umbe of uique adial posiios sampled by he ackig beams be. Noe ha moe ha oe beam ca have he same adial posiio. Le he oal umbe of ackig beams be disibued such ha hee ae a equal umbe of beams, θ, a each adius. Tha is, ad ha he TOF SWS uceaiy is ivesely popoioal o he ROI size k. Le us ow coside a ypical ackig beam cofiguaio available fom a coveioal -D aay asduce fo moioig adiaio-foce-iduced shea wave popagaio. The SWS pecisio fom ackig usig a -D maix aay will be compaed wih his efeece cofiguaio houghou his pape. Le he push be locaed a he cee of he imagig field of view (FOV), ad shea wave popagaio acked boh o he lef ad igh of he push. If half he oal umbe of ackig beams ae allocaed o each side of he push (i.e., θ =, = /), ad hey ae equally spaced wihi he aula ROI, he = Φ (, ), (8),D k/ whee he subscip D deoes a -D aay beam cofiguaio. Usig a -D maix aay capable of beamfomig i boh elevaioal ad laeal dimesios, he shea wave aival ime ca be moioed i a abiay umbe of diecios fom he push. This addiioal flexibiliy eables he umbe of aival ime measuemes ake a each adius, θ, o be geae ha wo. To keep he oal umbe of ackig beams cosa, he umbe of uique adii sampled ca be se o = / θ. If hese ae agai evely spaced wihi he adial exe of he ROI, he = Φ ( k, ), (9),D / θ = θ,. (4) I such a cofiguaio, he vaiace i ackig beam adii,, ca be foud usig oly he uique adial posiios sampled (because each oe is weighed equally by he same umbe θ ). If he adii ae evely spaced wihi he exe of he ROI, he he locaio of he ih uique adius, i, is give by i i = k, i. + l (5) Subsiuig (5) io (3b) ad simplifyig, oe fids ha he vaiace of he ackig beam adii of such a cofiguaio is give by he fucio k k = Φ (, ) = ( + ) ( ). Subsiuig (6) io (3), i ca be see ha ĉ ε ( + ) ( ) = c, k (6) (7) Fig.. Geomey of he keel ove which shea wave aival imes ae measued fo ime-of-fligh shea wave speed esimaio i he cooal (laeal-elevaio) plae. The keel (shaded gay) exeds bewee l ad h wih a age k, ad is ceeed o he push locaio (black squae). The push axis is assumed o coicide wih he axial dimesio, which is ohogoal o he page i his view. Example ackig beam locaios measuig shea wave aival imes ae show (black cicles), ad he adial posiio of oe beam, i, is maked. Fo his example cofiguaio, fou uique adial posiios ae sampled by he ackig beams ( = 4) ad hee ae hee beams a each adius ( θ = 3).

5 76 IEEE Tasacios o Ulasoics, Feoelecics, ad Fequecy Cool, vol. 60, o. 4, Apil 03 whee he subscip D deoes a -D aay beam cofiguaio. Le R be he aio of he TOF SWS uceaiy obaied usig he beam cofiguaios fo a -D maix aay as descibed peviously vesus he efeece cofiguaio fo a -D aay asduce:,d R =.,D (30) I is clea fom (3) ha if he umbe of beams, aival ime measueme uceaiy ε, ad udelyig SWS c ae equal fo boh he -D ad -D aay cofiguaios, he R is give by he aio of he spead i beam locaios:,d R =,D (3a) = Φ(, k/ ) Φ(, k/ ). θ (3b) Fially, subsiuig (6) io (3) ad assumig equal ROI age k, oe fids R = ( + )( θ ) ( )( + ). θ (3) Whe θ =, he beam cofiguaio coespodig o a -D aay asduce is obaied, ad R =. As θ iceases, R deceases, ad a educio i TOF SWS uceaiy is obaied. The maximum decease i R occus whe ( θ, ) = (/, ), coespodig o he case whe aival imes ae oly measued a he edges of he ROI, wih half he ackig beams allocaed o each edge. Iuiively, his makes sese, because he spead i he beam disaces o he push is maximized i his cofiguaio, leadig o a lage value of. B. Expeime ) Daa Acquisiio: To expeimeally demosae he educio i TOF SWS uceaiy possible hough muli-diecioal ackig i homogeeous maeials, he Siemes 4ZC maix aay asduce ad SC000 scae (Siemes Healhcae, Ulasoud Busiess Ui, Mouai View, CA) wee used o moio acousic-adiaio-foce-iduced shea waves i phaoms. The 4ZC aay coais (laeal elevaio) 0.4-mm squae elemes. A aula focused high-iesiy focused ulasoud (HIFU) piso asduce (H-0, Soic Coceps, Bohell, WA) was used fo acousic adiaio foce impulse (ARFI) exciaio (. MHz, F/, 63. mm focal deph). The 4ZC was iseed io he ceal opeig of he HIFU piso ad he wo asduces wee igidly coupled usig a specially desiged holde, as show i Fig. (a). The wo asduces wee sychoized usig a cusom iggeig cicui, as show i he sysem block diagam i Fig. (b). Shea waves wee iduced i homogeeous phaoms of vayig Youg s modulus bewee 4.8 ad 7 kpa (CIRS, Nofolk, VA). Fo a liea, isoopic, elasic medium, he elaioship bewee Youg s modulus (E) ad shea wave speed is E c =, ( + νρ ) (33) Fig.. (a) The 4ZC imagig asduce iseed io he ceal opeig of he high-iesiy focused ulasoud (HIFU) piso ad igidly fixed wih he holde. (b) Sysem block diagam of he expeimeal seup used fo ackig adiaio-foce-iduced shea wave popagaio i 3-D. The shea wave is geeaed wih he HIFU piso, ad acked usig he 4ZC maix aay. The wo asduces ae sychoized by he lie syc sigal fom he SC000, which povides pecise imig of asmi eves o he 4ZC. Whe he appopiae umbe of asmis have occued, he iggeig cicui causes he sigal geeao o emi a.-mhz bus of seleced ampliude ad duaio (00 o 400 cycles). This sigal is amplified by 55 db by a RF powe amplifie (E&I A50, Elecoics & Iovaio, Rochese, NY), ad goes hough a impedace machig ewok, befoe fially divig he HIFU piso o geeae he acousic adiaio foce push.

6 wag e al.: pecisio of ime-of-fligh shea wave speed esimaio i homogeeous sof solids 763 whee ν is he Poisso s aio (assumed o be 0.5) ad ρ is he desiy of he maeial (assumed o be 000 kg/ m 3 ). Muliple ieogaios i diffee locaios of he phaoms wee pefomed o obai diffee B-mode speckle ad diffee ealizaios of aival ime measueme oise. A exciaio pulse wih a deaed iesiy of I SPPA.3 = 375 W/cm (i siu I SPPA.7 = 674 W/cm ) was used, ad he same exciaio pulse legh was used fo epeaed ieogaios i he same phaom. This pulse legh was vaied bewee 00 ad 400 cycles fo he diffee phaoms o esue adequae displaceme (> μm) was obaied i all cases. The esulig shea waves wee imaged usig he 4ZC a a cee fequecy of.8 MHz. Because of cue limiaios of he SC000 beamfomig hadwae, abiay placeme of beam locaios was o possible. To ovecome his deficiecy, a fiely spaced ecagula gid of 7 7 (laeal elevaio) beams was used fo moioig he shea wave o povide a lage umbe of possible ackig beam locaios. Desied beam cofiguaios wee he esed by usig he daa fom a subse of he beams fom his gid. A a imagig deph of 60 mm, which is ea he push focal deph whe he wo asduces ae coupled, he aveage beam spacig is 0.54 mm i boh laeal ad elevaio dimesios, ad he FOV is mm. The adiaio foce exciaio was locaed appoximaely a he cee of he FOV, so ha he esula shea wave popagaio ca be visualized i all diecios. 64: paallel eceive was used o beamfom a gid of 8 8 beams o evey asmi so ha oly 8 asmis wee equied o ieogae he eie gid of 7 7 beams. Shea wave displaceme was moioed by epeaig he push, ad sequeially moioig each of he 8 paallel eceive beam goups i u, uil daa fom he eie FOV was acquied. The ime ieval bewee pushes was s, ad was limied by he daa asfe ae o he SC000. Usig his mehod, a high ackig PRF of 7.7 khz is obaied. Alhough his appoach equies a elaively log acquisiio ime, ad 8 pushes o acquie daa fom he eie FOV, hese dawbacks ae o ciical issues whe imagig phaoms. The beefi is ha shea wave displaceme daa wih high samplig aes boh spaially ad empoally is obaied. ) Daa Pocessig: Axial displaceme alog each beamlie was measued usig he zeo-phase displaceme esimaio algoihm descibed by Pesaveo e al. [8] o IQ daa. Acousic-adiaio-foce-iduced displacemes i he laeal ad elevaioal diecios ae o he ode of a magiude smalle ha he axial compoe [9], ad cao be moioed by ulasoic speckle ackig mehods. Theefoe, oly -D displaceme ackig was pefomed. Shea wave aival imes wee measued by fidig he TTPS of he displaceme ime pofile a each locaio. Aival imes wee aalyzed a a imagig deph of 60 mm ea he exciaio focus. The umbe of ackig beams () used fo SWS ecosucio was limied o 64. This is he umbe of paallel eceive chaels available fom he SC000, ad epeses he maximum umbe of locaios a which he shea wave fom a sigle ARFI exciaio ca be moioed wih his scae. Five diffee cofiguaios fo he 64 ackig beams wee esed: ( θ, ) = {(, 3), (4, 6), (8, 8), (6, 4), (3, )}, wih adii equally spaced bewee 4 ad 7 mm fom he push, coespodig o a ROI size of 3 mm. Wih a moe exesively pogammable sysem, o wih a sofwae beamfomig sysem iefaced wih he maix pobe, all he daa fom hese beam cofiguaios could be acquied wih a sigle asmi ad ARFI exciaio. Howeve, as peviously meioed, because of he cue limiaios of he SC000, hese beam cofiguaios wee implemeed by choosig a subse of beams fom a ecagula gid of 7 7 beam locaios. Fo each cofiguaio, beam locaios fom he 7 7 gid closes o he desied se of adii, assumig he push was locaed a he cee of he FOV, wee seleced fo SWS esimaio. Daa a emaiig beam locaios wee o used. Thus, daa fo all five diffee beam cofiguaios wee syhesized fom each full 7 7 gid daa se. The beam locaios seleced fo all cofiguaios ae illusaed i Fig. 3. Noe ha o obai he equied se of disaces bewee he beams ad he push axis fom he limied gid locaios, he beam paes ae o aaged i saigh lies adiaig ou fom he cee of he FOV as oe would expec. Because he locaio of he ARFI push fom he HIFU piso elaive o he 4ZC is o kow pecisely, SWS esimaio usig expeimeal daa was pefomed by leas-squaes fiig a coical suface o he aival imes, as show i Fig. 4(b). The axis of he coe coespods o he axis of he ARFI exciaio elaive o he 4ZC. Whe he HIFU piso ad he 4ZC ae pefecly aliged, he coe axis coicides wih he aival ime axis [he z-axis i Fig. 4(b)] ad goes hough he oigi. The emaiig wo degees of feedom, he coe opeig agle, ad he locaio of is apex alog is axis, diecly coespod o he slope ad iecep paamees of he liea model elaig aival imes o disace fom he push [see ()]. Fo each daa se, he aival imes as a fucio of locaio fom all he beams i he 7 7 gid a 60 mm deph wee fi o a coe o esimae he acual oieaio ad posiio of he push axis. To compesae fo misaligme bewee he HIFU pobe ad he 4ZC, coes wih his same axis wee fi o he aival ime daa fom he seleced beam locaios fo each cofiguaio. I ohe wods, oly he opeig agle ad he apex locaio of he coe wee allowed o vay. Leas-squaes opimizaio of hese wo paamees is equivale o fidig he slope ad iecep i lie-fiig. The SWS is esimaed usig he opeig agle of he leas-squaes coe fo each cofiguaio. C. Simulaio Simulaios wee pefomed i Malab (The Mah- Woks, Naick, MA) o veify he heoeical educio

7 764 IEEE Tasacios o Ulasoics, Feoelecics, ad Fequecy Cool, vol. 60, o. 4, Apil 03 Fig. 3. Beam locaios fom a 7 7 gid a 60 mm imagig deph used fo SWS esimaio: (a) ( θ, ) = (, 3), (b) ( θ, ) = (4, 6), (c) ( θ, ) = (8, 8), (d) ( θ, ) = (6, 4), ad (e) ( θ, ) = (3, ). The sadad deviaio of he beam disaces fom he push a he cee of he FOV,, is show fo each cofiguaio. Alhough he beams fo he ( θ, ) = (, 3) case do o lie o a saigh lie, hey have he same as a ypical cofiguaio whe usig a -D aay fo shea wave imagig. The ohe cases coespod o addiioal cofiguaios possible usig a -D aay. Fig. 4. (a) Shea wave aival imes measued fom he 4.8 kpa homogeeous phaom i he plae ohogoal o he push axis a a imagig deph of 60 mm, which is ea he exciaio focal deph. The push is appoximaely a he cee of he FOV. Aival ime measuemes close o he push ea he cee of he FOV wee o possible, as a esul of evebeaio echoes of he adiaio foce exciaio. (b) Leassquaes coe fi o he aival imes show i (a). The acual push axis locaio ad oieaio, as well as he SWS ca be esimaed fom he paamees of he coe. (c) Suface plos of aival imes a he same imagig deph (60 mm) measued fom hee homogeeous phaoms of diffee siffess. The esimaed SWS fom coe fis o he daa show ae displayed. i TOF SWS uceaiy show i (3). Shea wave aival imes wee geeaed fo he five beam cofiguaios expeimeally esed [( θ, ) = {(, 3), (4, 6), (8, 8), (6, 4), (3, )}] wihi he same adial domai (4 o 7 mm fom he push). The beam locaios used wee exac, ad o quaized by he 7 7 gid as fo he expeime. The aival imes wee geeaed as follows. Fis, he ideal aival ime a each beam locaio was calculaed usig is disace fom he push ad a SWS of.6 ms, which is he same SWS as he 4.8-kPa phaom. The, omally disibued oise wih zeo mea ad a sadad deviaio of ε = 0.6 ms, simila o expeimeally obseved oise levels [5], was added o he ideal aival imes o simulae measueme oise. SWS esimaio was pefomed by calculaig he ivese slope of he leas-squaes fi lie o hese oisy aival imes, as peviously descibed. This was epeaed fo ealizaios of oisy aival ime daa ses fo all five beam cofiguaios. The sadad deviaio (pecisio) of he SWS ove he ealizaios fo he five diffee beam cofiguaios was fially compaed. IV. Resuls Thee-dimesioal displaceme fields measued usig he 4ZC afe ARFI exciaio i he 4.8-kPa phaom wih he HIFU piso ae show i Fig. 5. Alhough a

8 wag e al.: pecisio of ime-of-fligh shea wave speed esimaio i homogeeous sof solids 765 Fig. 5. A 3-D displaceme field measued by he 4ZC afe ARFI exciaio usig he HIFU piso i a 4.8 kpa phaom: (a) = 0.8 ms, (b) = 3.4 ms, ad (c) = 6.0 ms. The displacemes measued alog he beamlie diecio usig -D speckle ackig ae show as a volume edeig a hee diffee ime seps afe he exciaio. Dake pixels coespod o lage displaceme ampliude. The push axis is paallel o he axial diecio ad is ceeed a he oigi i he laeal ad elevaio dimesios. Ohogoal secios of he displaceme field a plaes idicaed by he dashed lies ae also pojeced oo he back walls of he plo. Oe of hese plaes is a cooal view (ohogoal o he push axis) a a axial deph of 60 mm, which is close o he exciaio focus. Thoughou his pape, shea wave daa fom his plae ae aalyzed (i.e., he plae show i Fig. 3). HIFU piso was used fo adiaio foce exciaio, because of he sho duaio of he ARFI pulse (8 o 364 μs), displaceme magiudes o he ode of micos ae obaied axially alog he beamlie diecio. As peviously meioed, displaceme i he laeal ad elevaio diecios ae egligible i compaiso [9], ad wee o measued. I is evide fom Fig. 5 ha as ime afe he exciaio iceases, locaios iceasigly fuhe away fom he push ae peubed by he shea wave. Alhough shea wave daa ae aalyzed i oly oe plae ea he exciaio focus i his pape, ARFI-iduced shea wave displaceme houghou a 3-D volume ca be moioed usig he maix aay. The shea wave aival imes i he FOV a 60 mm fo oe acquisiio o he 4.8-kPa phaom ae show i Fig. 4(a). Because of evebeaio echoes of he adiaio foce exciaio, displaceme esimaes ealy i ime afe he push wee o available. Theefoe, aival ime measuemes close o he push ea he cee of he FOV wee o possible. As expeced, he shea wave akes loge o each locaios fuhe fom he push. The aival ime appeas o be isoopic, as would be expeced fo a homogeeous isoopic maeial. Because of he slow SWS i his phaom, he shea wave did o each he coes of he FOV wihi he ackig duaio. Fig. 4(b) illusaes he coe-fi o he aival imes fo he example i Fig. 4(a). The SWS ca be esimaed fom he opeig agle of he coe. A fase SWS coespods o lowe aival imes, ad cosequely, a bigge opeig agle. Fig. 4(c) shows suface plos of aival imes fom acquisiios o hee homogeeous phaoms of diffee siffess (E = 4.8,, 30 kpa), ad esimaed SWS fom each usig coe fis o he daa. The accuacy of he aligme bewee he HIFU pobe ad he 4ZC was assessed fom he axis of he coes fi o aival imes fom 00 acquisiios colleced o he 4.8-kPa phaom ove muliple days, duig which he HIFU pobe was epeaedly disassembled ad e-aached o he 4ZC. The axis of he HIFU piso ad he 4ZC had a aveage diffeece of 0.5 ± 0.3 i oieaio ad 0.5 ± 0. mm i posiio a a imagig deph of 60 mm. Thus, he adii of he ackig beams i Fig. 3, which wee compued assumig a pefecly aliged push axis ad he 4ZC, should o be sigificaly diffee fom hei ieded values. The educio i SWS uceaiy pediced by (3) fo 64 ackig beams wihi he same size ROI wih diffee disibuios is ploed i Fig. 6. The sadad deviaio of he esimaed SWS fom 0000 ses of simulaed aival imes was calculaed fo he five beam cofiguaios ( θ, ) = {(, 3), (4, 6), (8, 8), (6, 4), (3, )}. These values wee omalized by he sadad deviaio fo he efeece -D aay case [( θ, ) = (, 3)], ad ae show o he same plo. Similaly, he SWS sadad deviaio fom 00 acquisiios o he 4.8-kPa phaom was calculaed fo he five beam cofiguaios, ad he expeimeally achieved educio i SWS sadad deviaio is also show o he same figue. Noe ha he values ploed i Fig. 6 fo he expeimeally implemeed cofiguaios ae fom he acual beam locaios used. These diffe fom he values of he pescibed beam cofiguaios (show by he x-axis locaio of he simulaed daa pois) due o quaizaio eo fom he limied beam locaios available fom he 7 7 gid. Howeve, he diffeeces i wee small (o aveage 0.03 mm fo he five cofiguaios), as ca be see fom he x-axis locaios of he expeimeal ad simulaed daa pois. To calibae he eade, he expeimeal SWS sadad deviaio fo he baselie ( θ, ) = (, 3) case was 0.06 m/s. Because of pacical esicios, he umbe of expeimeal daa ses used o assess TOF SWS pecisio was limied o 00. To illusae he uceaiy i esimaig he sadad

9 766 IEEE Tasacios o Ulasoics, Feoelecics, ad Fequecy Cool, vol. 60, o. 4, Apil 03 deviaio fom 00 samples, he SWS sadad deviaio fom a subse of 00 samples adomly seleced fom he 0000 simulaed daa ses fo each beam cofiguaio was compaed wih he sadad deviaio fo he eie 0000 samples. This was epeaed fo 000 adomly seleced subses. The mea diffeece i sadad deviaio evaluaed ove 00 ad samples ae ploed as eo bas i Fig. 6. I addiio o he 00 acquisiios o he 4.8-kPa phaom, a addiioal 0 daa ses wee acquied o each of five ohe phaoms wih vayig siffess up o 7 kpa. The SWS sadad deviaio was calculaed fo each usig he ( θ, ) = (, 3) cofiguaio. These wee he omalized by he sadad deviaio of he fied slope, β. This effecively omalizes diffeeces i he aival ime measueme eo fo he diffee phaoms. A plo of he omalized SWS sadad deviaio is show i Fig. 7 vesus he mea SWS obaied fom all acquisiios o each phaom. The heoeically pediced cuve fo his elaioship fom () is show o he same figue. Fig. 7. TOF SWS sadad deviaio vesus he mea SWS i six homogeeous phaoms of diffee siffess (squaes), ad afe omalizig fo he aival ime esimaio eo by dividig by he sadad deviaio of he fied slope, (cicles). The elaioship bewee he omalized SWS sadad deviaio ad he SWS pediced usig () is β show by he dashed lie. The ( θ, ) = (, 3) beam cofiguaio was used fo SWS esimaio i all cases. The sample size was 00 fo he phaom wih a mea SWS of.3 m/s, ad 0 fo he ohes. V. Discussio Fig. 6. SWS sadad deviaio fo he five ackig beam cofiguaios esed as a aio of he sadad deviaio fo he efeece -D aay case [( θ, ) = (, 3)]. The pediced values fo his aio usig he values of he beams ad (3) ae show by he dashed cuve. The educio obaied fom simulaed oisy aival ime daa ses is maked by iagles, ad ha fom 00 expeimeally acquied daa ses fom a homogeeous phaom is maked by cicles. Noe ha he values fo he expeimeal daa ae slighly diffee fom he pescibed beam cofiguaios because of he limied beams available fom he 7 7 gid. The eo bas o he expeimeal daa show he expeced uceaiy i esimaig he sadad deviaio fom oly 00 samples. These wee deemied by compaig he sadad deviaio of 000 ses of 00 adomly seleced samples fom he daa ses simulaed fo each cofiguaio wih he sadad deviaio obaied ove all samples. Noe ha hee is o eo ba fo he efeece -D aay case [( θ, ) = (, 3)], because his is used fo omalizaio, ad by defiiio is equal o. As show i Fig. 6, hee is excelle ageeme bewee he educio i SWS uceaiy achieved o simulaed daa ad ha pediced by heoy. The expeimeal esuls, howeve, show deviaios abou he expeced values of R. Neveheless, he oveall ed of deceasig SWS sadad deviaio wih iceasig is maiaied. The lages educio occued fo he ( θ, ) = (3, ) beam cofiguaio, whee a 30% decease i SWS uceaiy compaed wih he baselie ( θ, ) = (, 3) case was achieved expeimeally, ad a 40% decease was pediced by heoy ad simulaio. The dispaiy bewee he expeimeal esuls ad heoy may be due i pa o he elaively small umbe of acquisiios (N = 00) used o assess he SWS sadad deviaio compaed wih he simulaios (N = 0 000). The uceaiy associaed wih esimaig he sadad deviaio fom 00 samples is show by he eo bas i Fig. 6. As he sample size iceases, a moe accuae esimae of he SWS sadad deviaio fo he vaious beam cofiguaios should be obaied. A limiaio of he heoeical deivaio fo he TOF SWS esimaio uceaiy peseed i Secio II is ha cosa aival ime measueme oise is assumed a all beam locaios. This is ulikely o be ue i pacice, ad is aohe faco which could cause he discepacy bewee he expeimeal ad heoeical esuls i SWS uceaiy educio obseved i his sudy. Thee ae wo easos why he aival ime measueme oise may be uequal a diffee beam locaios: ) vaiaio i

10 wag e al.: pecisio of ime-of-fligh shea wave speed esimaio i homogeeous sof solids 767 beam poi spead fucio (PSF) caused by beamfomig, ad ) aeuaio of he shea wave as i popagaes hough he medium. The ackig beam PSF diecly impacs he accuacy of ulasoically acked displacemes. Ideed, he mai souce of oise i ulasoic ackig of adiaio foce iduced displaceme is due o he disoio of he scaee disibuio wihi he ackig beam caused by spaial gadies i he displaceme field [30], [3]. Beams wih highe side lobe levels ae subjec o highe levels of ackig oise because of he geae coibuio of scaee moio fom he side lobes. Thus, whe he PSFs of he ackig beams vay, he accuacy of he measued aival imes fom hose beams will also be diffee. As descibed i Secio III-B, 64: paallel eceive beamfomig was used o acquie gids of 8 8 beams expeimeally i his sudy. A sigle asmi focused a he cee of he gid was used fo isoificaio. Daa a all 64 beam locaios was acquied by shifig he eceive focus away fom he asmi focus o evey locaio i he gid. This has he effec of iceasig he side lobe magiude of he ack beam PSF [3]. As he eceive focus is shifed fuhe away fom he asmi focus, he side lobe level iceases. Thus, he oue beams i he 8 8 gid have highe side lobe levels ha beams locaed ea he cee of he gid. Theefoe, i he expeimes pefomed fo his sudy, a beam cofiguaio i which moe beams ea he edges of he 8 8 paallel eceive beam goups ae seleced fo SWS esimaio will have lowe ha expeced SWS pecisio. To miigae ouifom ackig jie esulig fom paallel eceive beamfomig, a asmi beam wih a highe f-umbe, o a plae wave asmi could be used. The ohe faco which ca cause he accuacy of aival ime esimaes o vay wih locaio is shea wave aeuaio. As a shea wave popagaes hough issue, is ampliude ca be aeuaed by geomeic speadig o viscosiy [0], [8]. This causes he displaceme ampliude a locaios fuhe fom he push o be smalle, leadig o lage aival ime measueme eo a hese locaios. Because his faco was o ake io accou i he deivaio of (3), he expeimeal beam cofiguaios i his sudy which uilize moe beams fa fom he push will have lowe ha expeced SWS pecisio. Leas-squaes fiig wih uequal aival ime measueme accuacy a diffee locaios will sill give a ubiased esimae of he SWS. To exed he aalysis of TOF SWS esimaio uceaiy o accou fo uequal measueme accuacy, he followig fom fo he sadad deviaio of he slope paamee β ca be used [3]: β = i = i = i ( i ) a ( ) i, ε (34) whee a i is he scalig faco fo he oise vaiace a each beam locaio. I ca be show ha if he weighs ae uifomly equal o oe (i.e., a i =, i), he (34) educes o (7). Eq. () elaig he sadad deviaio of he SWS ad he sadad deviaio of he esimaed slope sill applies i his siuaio. Theefoe, he SWS uceaiy ca sill be educed by miimizig (34). I miimizig (34), boh he locaio ( i ) ad measueme oise ( a i ε ) of he beams mus be ake io accou o esue ha deceases i fom speadig he beam locaios is β o offse by beams wih highe measueme oise caused by beamfomig o shea wave aeuaio. The deemiaio of he weighs a i is a difficul poblem, because hey ae depede o may facos, icludig he souce of he shea wave, he beamfomig echique used fo ackig, ad he viscoelasic popeies of he udelyig medium. Oe appoach fo deemiig a i would be o simulae boh he espose of he issue o he adiaio exciaio foce field expeimeally used [33] ad ulasoic imagig of he iduced displacemes [30]. I pacice, he exac viscoelasic popeies of he medium o be imaged ae ulikely o be kow a pioi, so a age of maeial popeies likely o be ecoueed could be simulaed. As show i Fig. 7, he expeimeally measued TOF SWS uceaiy iceased wih mea SWS value i phaoms. Because he udelyig viscoelasic popeies of he phaoms wee diffee, shea waves of vayig badwidhs wee iduced wih impulsive adiaio foce exciaio, leadig o diffeeces i he aival ime measueme accuacy bewee he phaoms. Afe omalizig fo his effec, he SWS uceaiy iceased wih he squae of he SWS, as was pediced by (). The fac ha he uceaiy i TOF SWS measuemes iceases wih he squae of he SWS has implicaios fo he iepeaio of expeimeal esuls usig TOF SWS esimaio. Fis, a geae sample size is eeded fo SWS measuemes i siffe maeials o achieve he same powe i saisical aalysis as i sofe maeials. Secod, i sudies which ivolve he aalysis of vaiabiliy i SWS, fo example, i compaig siffess heeogeeiy i he live as a fucio of fibosis sage, he ihee icease i he vaiace of SWS measuemes i siffe media mus be ake io accou. The aalyical elaioships fo he uceaiy i TOF SWS measueme peseed i his pape ae applicable o issue as well as homogeeous phaoms. Howeve, he ackig beam cofiguaios peseed i his pape ae o ieded fo use i all siuaios. Thei pupose is meely o illusae ha by iceasig he spead i he beam locaios, a moe pecise measueme of he SWS ca be obaied. The specific ackig beam locaios o use i ay applicaio should be chose based o he goal of he expeime, as well as he maeial popeies of he udelyig medium. Addiioal cosideaios o be ake io accou whe selecig ackig beam locaios fo imagig issue iclude shea wave aeuaio caused by viscosiy, as was peviously meioed, poeial vaiaio i siffess wih locaio (heeogeeiy), ad poeial vaiaio i siffess wih diecio (aisoopy).

11 768 IEEE Tasacios o Ulasoics, Feoelecics, ad Fequecy Cool, vol. 60, o. 4, Apil 03 The mai dawback of he appoach used i his pape fo impovig SWS measueme pecisio is he assumpio of issue homogeeiy wihi he ROI. I was show i (7) ha TOF SWS pecisio ad ROI size have a ivese elaioship. Theefoe, hee is a ihee adeoff bewee SWS pecisio ad spaial esoluio. I applicaios i which he issue siffess is elaively homogeeous ad spaial esoluio is o ciical, fo example i assessig live siffess, measueme of he aveage SWS ove a lage ROI is useful. Howeve, i applicaios whee a image of siffess is desied, smalle ROIs mus be employed, which limis he spacig bewee ackig beam locaios ad esuls i lowe TOF SWS measueme pecisio. The ackig beam cofiguaios peseed i his pape also assume he udelyig medium o be isoopic, so ha SWS does o vay wih diecio. Theefoe, beams wee placed a abiay agles elaive o he push i he laeal-elevaio plae. Howeve, i issues which ae aisoopic, whee he SWS is diecioally depede, i is o possible o measue a sigle SWS ove he eie aula ROI. Isead, idepede SWS measuemes mus be made i diffee diecios fom he push. This addiioal esicio equies beams o be placed i a sufficiely wide vaiey of agles o obai he desied agula esoluio i SWS. Neveheless, he geeal piciple of maximizig he spead i he beam locaios ad he heoeical TOF SWS measueme uceaiy [Eq. (3)] ca sill be applied idepedely fo each diecio. Fo ed uses of quaiaive SWS imagig sysems, a useful feaue would be some idicaio of he expeced accuacy of hei measuemes. If hee was a mehod o asceai he aival ime esimaio eo ε, a esimae fo he SWS measueme uceaiy could be calculaed usig (3): s s ε =, (35) whee s c is he esimaed SWS uceaiy ad s ε is a esimae fo he aival ime eo. Fouaely, s ε ca be esimaed by he deviaios of he aival imes abou he leas squaes fi lie [3]: s ε = ε ε, (36) whee ε ε is he sum of he squae eos, ad is he umbe of aival imes used fo he fi. Eq. (35) gives a diec esimae of he uceaiy, o eo ba size, of he shea wave speed measueme i mees pe secod. I is easy o show ha whe Cov(, ) = (pefec coelaio bewee aival imes ad posiio), he uceaiy is 0 m/s, ad whe Cov(, ) = 0 (o coelaio), he uceaiy is ifiie. Thus, i addiio o obaiig a SWS esimae ĉ fom a sigle acquisiio, oe ca also esimae he uceaiy i ha esimae, s c, fom he qualiy of he fi ad (35). I addiio o he abiliy o opimally posiio ackig beam locaios o impove SWS measueme pecisio, aohe beefi of ackig shea wave popagaio i 3-D usig a -D maix aay is he icease i he amou of daa ha ca poeially be acquied. If he umbe of ackig beams is o held fixed, bu allowed o icease by eihe iceasig θ o, he a educio i SWS measueme sadad deviaio would be achieved [see (3)]. This would be feasible usig a maix aay sysem capable of sigle chael acquisiio ad sofwae beamfomig. Alhough he 3-D shea wave imagig sysem descibed i his pape is a useful eseach ool, i has o bee opimized fo daa acquisiio i i vivo expeimes. Alhough capable of acquiig high-qualiy daa wih high samplig aes boh spaially ad empoally, he log acquisiio ime ad lage umbe of pushes equied o sample he eie ecagula FOV is impacical fo cliical sudies. Fo pacical puposes, boh he umbe of ackig beams ad he PRF ca be educed o icease acquisiio speed ad decease he umbe of adiaio foce exciaios eeded. If cue limiaios of he beamfome hadwae ca be ovecome, he abiay beam locaios, such as oe of he cofiguaios esed i his pape, may be ealizable wih a sigle asmi ad 64 paallel eceive. Aleaively, if sigle-chael daa ca be acquied fom he -D aay, beamfomig ca be pefomed offlie i sofwae. Ideally, boh adiaio foce exciaio ad shea wave ackig is pefomed wih a sigle asduce. Howeve, because of he high cos of he 4ZC pobe, his has o ye bee aemped i ou lab. Neveheless, his is a possibiliy ha is ude ivesigaio. VI. Coclusio TOF SWS measueme uceaiy is ivesely popoioal o ROI size, ad ivesely popoioal o he squae oo of he umbe of ackig beam locaios. TOF SWS uceaiy iceases wih he squae of he SWS. This meas ha TOF SWS measuemes ae iisically less pecise fo siffe maeials. Fo a fixed umbe of ackig beams ad ROI size, he TOF SWS uceaiy ca be educed by iceasig he spead of he ackig beam locaios elaive o he push wihi he ROI. This was expeimeally demosaed usig a -D maix aay ulasoud asduce o moio adiaio-foce-iduced shea wave popagaio i a 3-D volume. By akig advaage of he addiioal beam locaios available fom he -D maix aay compaed wih a coveioal -D aay asduce, a icease i he spead of he beam locaios elaive o he push was able o be achieved. Usig a limi of 64 ackig beams, a educio i TOF SWS measueme uceaiy of 40% was show o be heoeically possible by placig he beams a he edges of he ROI a muliple diecios fom he push, isead of spacig he beams equally wihi he

12 wag e al.: pecisio of ime-of-fligh shea wave speed esimaio i homogeeous sof solids 769 ROI alog a sigle plae, as is ypically doe fo shea wave imagig usig a -D aay. This is cooboaed wih simulaed daa, ad a educio of 30% was achieved i pacice usig expeimeal daa acquied wih he -D maix aay o a homogeeous phaom. Alhough he SWS sadad deviaio obaied o a phaom usig he expeimeal seup i his sudy was small (<0.06 m/s), i pacical imagig siuaios whee a lowe ackig PRF, smalle umbe of ackig beams, ad smalle ROI size is used, ad iceased displaceme jie may be pese because of lowe B-mode SNR, he SWS uceaiy will be lage. Fo example, fo epeaed acquisiios i i vivo live usig 3 paallel beams uifomly spaced i a sigle plae, we obseved SWS sadad deviaios of up o 0. m/s [5]. The impoveme i SWS pecisio obaied by ackig shea waves wih he -D maix aay should be mos beeficial i hese siuaios ad whe imagig siff maeials. I a ideal shea wave imagig sysem, he SWS measueme would have boh high accuacy (low bias) ad high pecisio (low spead). This pape has poposed mehods fo chaaceizig SWS measueme pecisio ad a appoach fo iceasig he pecisio of SWS measuemes by iceasig he spead i ackig beam locaios. Howeve, he aalysis ad mehods peseed heei do o exed o sysemaic souces of eos, such as shea wave dispesio, which cause measueme bias. Depedig o he applicaio, sysemaic eos, which coibue o diffeeces i he SWS measued by diffee sysems, may pose a moe sigifica poblem ha measueme pecisio. Aalysis of poeial sysemaic souces of SWS measueme eo i shea wave imagig sysems mus be pefomed i fuue sudies o faciliae wide cliical use of his echology. Ackowledgmes The auhos would hak Siemes Healhcae, Ulasoud Busiess Ui, Mouai View, CA, fo hei sysem suppo. Refeeces [] H. Kaai, Popagaio of spoaeously acuaed pulsive vibaio i huma hea wall ad i vivo viscoelasiciy esimaio, IEEE Tas. Ulaso. Feoelec. Feq. Cool, vol. 5, o., pp , 005. [] M. 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13 770 IEEE Tasacios o Ulasoics, Feoelecics, ad Fequecy Cool, vol. 60, o. 4, Apil 03 [5] M. H. Wag, M. L. Palmei, V. Roembeg, N. C. Rouze, ad K. R. Nighigale, Impovig he obusess of ime-of-fligh based shea wave speed ecosucio mehods usig RANSAC i huma live i vivo, Ulasoud Med. Biol., vol. 36, o. 5, pp , 00. [6] A. Papoulis, Pobabiliy, Radom Vaiables ad Sochasic Pocesses, 3d ed., New Yok, NY: McGaw-Hill, 99. [7] L. Zhai, S. Hsu, R. Bouchad, ad K. Nighigale, A combied ARFI sequece fo D displaceme imagig ad shea wave velociy mappig, i IEEE Ulasoics Symp., 008, pp [8] A. Pesaveo, C. Peey, M. Kuege, ad H. Eme, A ime-efficie ad accuae sai esimaio cocep fo ulasoic elasogaphy usig ieaive phase zeo esimaio, IEEE Tas. Ulaso. Feoelec. Feq. Cool, vol. 46, o. 5, pp , 999. [9] R. Bouchad, M. Palmei, G. Pio, G. Tahey, J. Seee, ad P. Dayo, Opical ackig of acousic adiaio foce impulseiduced dyamics i a issue-mimickig phaom, J. Acous. Soc. Am., vol. 6, o. 5, pp , 009. [30] M. L. Palmei, S. McAleavey, G. Tahey, ad K. Nighigale, Ulasoic ackig of acousic adiaio foce-iduced displacemes i homogeeous media, IEEE Tas. Ulaso. Feoelec. Feq. Cool, vol. 53, o. 7, pp , 006. [3] S. McAleavey, K. Nighigale, ad G. Tahey, Esimaes of echo coelaio ad measueme bias i acousic adiaio foce impulse imagig, IEEE Tas. Ulaso. Feoelec. Feq. Cool, vol. 50, o. 6, pp , 003. [3] J. J. Dahl, M. L. Palmei, V. Agawal, K. N. Nighigale, ad G. E. Tahey, A paallel ackig mehod fo acousic adiaio foce impulse imagig, IEEE Tas. Ulaso. Feoelec. Feq. Cool, vol. 54, o., pp. 30 3, 007. [33] M. L. Palmei, A. C. Shama, R. R. Bouchad, R. W. Nighigale, ad K. R. Nighigale, A fiie-eleme mehod model of sof issue espose o impulsive acousic adiaio foce, IEEE Tas. Ulaso. Feoelec. Feq. Cool, vol. 5, o. 0, pp , 005. Michael H. Wag (S 08) eceived he B.E. (Hos) degee i elecical ad elecoic egieeig fom he Uivesiy of Caebuy, Chischuch, New Zealad, i 003. He eceived he M.A.Sc. degee i elecical ad compue egieeig fom he Uivesiy of Biish Columbia, Vacouve, Caada, i 006. He is cuely pusuig a Ph.D. degee i biomedical egieeig a Duke Uivesiy. His eseach ieess iclude acousic adiaio foce imagig ad chaaceizig he viscoelasic popeies of livig issue. Mak L. Palmei eceived his B.S. degee i biomedical ad elecical egieeig fom Duke Uivesiy, Duham, NC, i 000. He was a James B. Duke gaduae fellow; he eceived his Ph.D. degee i biomedical egieeig fom Duke Uivesiy i 005 ad his M.D. degee fom he Duke Uivesiy School of Medicie i 007. He is cuely a Assisa Reseach Pofesso i biomedical egieeig ad aeshesiology a Duke Uivesiy. His eseach ieess iclude acousic adiaio foce shea wave elasiciy imagig, ulasoic imagig, ad fiie eleme aalysis of sof issue espose o acousic adiaio foce exciaio. Ned C. Rouze eceived bachelo s degees i physics ad mahemaics fom Hasigs College i 977 ad a Ph.D. degee i physics fom he Uivesiy of Washigo i 98. He is cuely a Seio Reseach Scieis i biomedical egieeig a Duke Uivesiy. His eseach ieess iclude adiaio foce imagig, mechaical popeies of sof issue, omogaphic ecosucio, ad pae ecogiio. Kahy Nighigale eceived he B.S. degee i elecical egieeig ad he Ph.D. degee i biomedical egieeig fom Duke Uivesiy i 989 ad 997, especively, havig seved i he Uied Saes Ai Foce fom 989 o 99. Followig he gaduae wok, she pefomed eseach as a Assisa Reseach Pofesso i he Depame of Biomedical Egieeig a Duke Uivesiy fom 997 o 004, ivesigaig acousic-adiaiofoce-based elasiciy imagig mehods. D. Nighigale is cuely eachig ad coducig eseach as a Associae Pofesso i he Depame of Biomedical Egieeig a Duke Uivesiy. He eseach ieess iclude elasiciy imagig, shea wave imagig, he use of acousic adiaio foce i diagosic ad heapeuic ulasoud, ad he accuae chaaceizaio of oliea acousic popagaio i sof issues. Be C. Byam eceived he B.S. degee i biomedical egieeig ad mah fom Vadebil Uivesiy, Nashville, TN, i 004. He eceived he Ph.D. degee i biomedical egieeig i 0 fom Duke Uivesiy, Duham, NC. Bewee 006 ad 007, he spe 0 mohs wokig wih Jøge Jese i he Cee fo Fas Ulasoud Imagig i Lygby, Demak. He is cuely a assisa eseach pofesso i he Biomedical Egieeig Depame a Duke Uivesiy. His ulasoud eseach ieess iclude beamfomig, moio esimaio, ad ohe elaed sigal pocessig asks.