351 En gi neer ing Ter race, mail code Amsterdam Avenue New York, NY

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1 ULTRASONIC IMAGING 32, (2010) Sim u la tion Study of Am pli tude-mod u lated (AM) Har monic Mo tion Im ag ing (HMI) for Stiff ness Con trast Quan ti fi ca tion with Ex per i men tal Val i da tion CAROLINE MALEKE, 1 JIANWEN LUO, 1 VIKTOR GAMARNIK, 1 XIN L. LU 1 AND ELISA E. KONOFAGOU 1, 2 Departments of 1 Biomedical Engineering and 2 Radiology 351 En gi neer ing Ter race, mail code Amsterdam Avenue New York, NY ek2191@columbia.edu The ob jec tive of this study is to show that Har monic Mo tion Im ag ing (HMI) can be used as a re li able tu mor-map ping tech nique based on the tu mor s dis tinct stiff ness at the early on set of dis ease. HMI is a ra di a tion-force-based im ag ing method that gen er ates a lo cal ized vi bra tion deep in side the tis sue to es - ti mate the rel a tive tis sue stiff ness based on the re sult ing dis place ment am pli tude. In this pa per, a fi niteel e ment model (FEM) study is pre sented, fol lowed by an ex per i men tal val i da tion in tis sue- mim ick ing polyacrylamide gels and ex cised hu man breast tu mors ex vivo. This study com pares the re sult ing tis - sue mo tion in sim u la tions and ex per i ments at four dif fer ent gel stiffnesses and three dis tinct spher i cal in clu sion di am e ters. The elas tic moduli of the gels were sep a rately mea sured us ing me chan i cal test - ing. Iden ti cal trans ducer pa ram e ters were used in both the FEM and ex per i men tal stud ies, i.e., a 4.5-MHz sin gle-el e ment fo cused ul tra sound (FUS) and a 7.5-MHz di ag nos tic (pulse-echo) trans - ducer. In the sim u la tion, an acous tic pres sure field was used as the in put stim u lus to gen er ate a localized vi bra tion in side the tar get. Radio fre quen cy (rf) sig nals were then sim u lated us ing a 2D con vo lu tion model. A one-di men sional cross-cor re la tion tech nique was per formed on the sim u lated and ex per i - men tal rf sig nals to es ti mate the ax ial dis place ment re sult ing from the har monic ra di a tion force. In or - der to mea sure the re li abil ity of the dis place ment pro files in es ti mat ing the tis sue stiff ness dis tri bu tion, the con trast-trans fer ef fi ciency (CTE) was cal cu lated. For tu mor map ping ex vivo, a har monic ra di a - tion force was ap plied us ing a 2D ras ter-scan tech nique. The 2D HMI im ages of the breast tu mor ex vivo could de tect a ma lig nant tu mor (20 10 mm 2 ) sur rounded by glan du lar and fat tis sues. The FEM and ex per i men tal re sults from both gels and breast tu mors ex vivo dem on strated that HMI was ca pa ble of de tect ing and map ping the tu mor or stiff in clu sion with var i ous di am e ters or stiffnesses. HMI may thus con sti tute a prom is ing tech nique in tu mor de tec tion (>3 mm in di am e ter) and map ping based on its dis tinct stiff ness. KEY WORDS: Breast; dispacement; FEM; har monic-mo tion im ag ing; modulus; ra di a tion force; tu mor. IN TRO DUC TION Pal pa tion is a stan dard screen ing pro ce dure for the de tec tion of tu mors based on their dis - tinct stiff ness in a clin i cal set ting. Sev eral elas tic ity-im ag ing tech niques, such as those de - vel oped in con junc tion with ul tra sound and MRI, have been de vel oped to es ti mate tis sue stiff ness and thus de tect tu mors us ing var i ous forms of tis sue per tur ba tion for the de tec tion of stiffer masses. 1 In the field of ul tra sound, Krouskop et al 2 im posed an ex ter nal vi bra tion and es ti mated the tis sue elas tic modulus by mea sur ing the re sult ing Dopp ler shift. Parker et al 3 mea sured the tis sue re sponse to me chan i cal vi bra tions for sonoelasticity im ag ing and ap plied it on healthy hu man skel e tal mus cle (rectus femoris and bi ceps brachii) in vivo. 4 Ophir et al 5, 6 de - vel oped the method of elastography that ap plied a small ex ter nal static com pres sion (on the /10 $18.00 Copy right 2010 by Dynamedia, Inc. All rights of re pro duc tion in any form re served.

2 AM PLI TUDE-MOD U LATED (AM) HAR MONIC MO TION IM AG ING 155 or der of 1%) and used crosscorrelation tech niques on radio fre quen cy (rf) sig nals in or der to es ti mate tis sue strains re sult ing from the ex ter nal com pres sion. This method has been proven to pro duce good qual ity strain im ages (or, elastograms) in sev eral tis sues, es pe cially in the breast and mus cle in vivo On the other hand, in ter nal per tur ba tion meth ods can pro duce a lo cal ized force deep in side the tis sue that can be used for prob ing and an a lyz ing tis sue prop er ties. 1, Sugimoto et al in - tro duced the use of a fo cused ul tra sound trans ducer to pro duce an im pulse ra di a tion force that caused a lo cal ized static com pres sion deep in side the tis sue to eval u ate lo cal ized tis sue stiff ness. 18 The tis sue dis place ment was es ti mated as a func tion of time by us ing pulse-echo meth ods. Other re search groups have also used the im pulse ra di a tion force to in duce brief me chan i - cal ex ci ta tions lo cally and fol low the re sult ing tis sue re sponse while rf data were col lected dur ing tis sue re lax ation (Acous tic Ra di a tion Force Im ag ing; ARFI) 19, or shear-wave prop a ga tion, e.g., with Shear Wave Elas tic ity Im ag ing (SWEI), 25 ARFI, 26 Su per sonic Shear Im ag ing 27 and Shearwave Dis per sion Ul tra sound Vibrometry (SDUV). 28 These tech niques have been used clin i cally in the di ag no sis of patho log i cal tis sues, such as in ca rotid and popliteal arteries in vivo, 29, 30 hu man liver in vivo, 31 hu man breast in vivo, 32 hu man mus cle in vivo 33 and swine liver in vivo. 28 An other ap proach to mea sure tis sue me chan i cal prop er ties is the use of the ra di a tion force to gen er ate an in ter nal vi bra tion. The SWEI method em ploys a fo cused ul tra sound trans ducer and an ul tra sound im ag ing trans ducer or low-fre quency acous tic de tec tor. An am pli tude-mod u lated fo cused beam is used to gen er ate a time-vary ing acous tic-ra di a tion force. The mod u la tion fre quency is typ i cally on the or der of a few khz. The shear wave re - sult ing from the ra di a tion force is de tected by an ul tra sound im ag ing trans ducer or a sur - face de tec tor that is used to char ac ter ize the viscoelastic prop er ties of the tar geted me dium. 25 In Ul tra sound-stim u lated Vibro-Acoustography (USVA), two con fo cal ul tra sound trans duc ers and a hydrophone are typ i cally used. The in ter fer ence of two con fo cal ul tra - sound trans duc ers at slightly dif fer ent fre quen cies (low khz range) causes a vi bra tion at the fo cus. The am pli tude or the phase of the shear wave is re corded by a hydrophone and used to form an im age (i.e., USVA), or tracked by a sep a rate ul tra sound beam and/or la ser vibrometer (i.e., SDUV). 28 In SDUV, the phase of the shear wave at two dif fer ent lo ca tions is used to cal cu late the shear-wave prop a ga tion speed at dif fer ent fre quen cies. An in verse ap - proach is used for es ti ma tion of tis sue vis cos ity and elas tic ity mea sure ment. Among other ap pli ca tions, USVA has been ap plied on hu man breast in vivo and SDUV has been ap - plied on swine liver in vivo. 28 Har monic mo tion im ag ing (HMI) uses two fo cused ul tra sound (FUS) trans duc ers with an ul tra sound im ag ing trans ducer 39 or one (FUS) trans ducer with an ul tra sound im ag ing trans - ducer. 40 The im ag ing and the FUS trans duc ers are con fo cal and con cen tric. Maleke et al 40 have shown that, in the two-fus-trans ducer con fig u ra tion, the in ter fer ence of the two fo - cused beams pro duced an acous tic ra di a tion force that con tin u ously moved across the fo cal re gion. On the other hand, in the sin gle FUS-trans ducer con fig u ra tion, the am pli tude-mod u - lated FUS beam (on the or der of Hz) gen er ates a time-vary ing radiation force at the fo cal re - gion (i.e., the force did not move spa tially). The re sult ing os cil la tory mo tion at the fo cus is de tected dur ing the force ap pli ca tion us ing an im ag ing trans ducer. The am pli tude of the in - duced mo tion is es ti mated us ing 1D cross-cor re la tion on the ac quired rf sig nals. 39, 40 Since the in duced mo tion is highly lo cal ized, the re sponse of the tis sue is mainly re lated to the un der ly - ing tis sue me chan i cal prop er ties. The o ret i cal and ex per i men tal stud ies of the HMI tech nique us ing two sep a rate FUS trans - duc ers have been pre vi ously re ported. 39 Finite-element and Monte-Carlo simulations were

3 156 MALEKE ET AL used to sim u late the os cil la tory dis place ment within an ap plied force fre quency range of 200 to 800 Hz. The force was ap plied at the spe cific nodes within the fo cal re gion in a fi nite-el e - ment model (FEM). 39 How ever, the ap plied ra di a tion force am pli tude was not re al is tic be - cause the 2D acous tic pres sure field was not taken into ac count. Lo cal ized Har monic Mo tion (LHM) is a tech nique sim i lar to HMI that uses a sequence of quasi-static ex ci ta tions at a spe cific rate. Heikkila et al 41,42 have de vel oped a LHM sim u la tion frame work to test the per for mance of LHM for le sion de tec tion. Two con fig u ra tions were sim u lated in volv ing ei ther a 1D lin ear phased ar ray trans ducer or two con fo cal sin gle-el e - ment trans duc ers, for both sonication and im ag ing. A burst wave form with sev eral rep e ti tion fre quen cies (e.g., 50, 100 and 150 Hz) was then used to in duce dy namic ex ci ta tion in side a me dium. Their sim u la tion find ings in di cated good agree ment with the in vivo LHM ex per i - men tal re sults in the rab bit mus cle. 41,43 In this pa per, we pres ent the sim u la tion study of HMI where the os cil la tory force is gen er - ated by our pre vi ously de vel oped tech nique that uti lizes a sin gle, am pli tude-mod u lated FUS beam. 40 A dif fer ent sim u la tion frame work is con sid ered herein to study the per for mance of the am pli tude-mod u lated HMI method (i.e., us ing one FUS trans ducer) in a re al is tic, acous - tic frame work. The FUS trans ducer is ex cited by an am pli tude-mod u lated wave form to in - duce a sta tion ary vi bra tion deep in side the tis sue. 40, 44 The 2D pres sure field of the FUS trans ducer is sim u lated in Field II. 45, 46 The 2D pres sure field is then used as a load ing con di - tion in the FEM study. The FEM is used to eval u ate the de pend ence of the es ti mated dis - place ment on the acous tic pa ram e ters and me chan i cal tis sue prop er ties, such as the acous tic in ten sity and the tis sue Young s modulus. We ex am ine the po ten tial of HMI in as sess ing dif - fer ent sizes/stiffnesses of an in clu sion em bed ded in a softer me dium. One of the ob jec tives of this study was to show that HMI could be used as a re li able tu mor-map ping tech nique based on the tu mor stiff ness dif fer ence at the early on set of dis ease. The stiffer in clu sions at various different diameters represent the tumor mass at different stages, e.g., early tumor de - vel op ment would en tail a smaller mass. The manu script is or ga nized as fol lows. The sim u la tion study is first de scribed to - gether with the the ory in volv ing the pres sure and in ten sity der i va tion for the am pli - tude-mod u lated (AM) wave form, the acous tic pres sure sim u la tion, the FEM, the im age for ma tion model, and dis place ment es ti ma tion method. Sec ond, the de tails of the setup used for HMI ex per i ments on polyacrylamide gels and postsurgical breast spec i mens, as well as me chan i cal test ing, are pro vided. The re sults of the sim u la tion and ex per i ments are com pared and dis cussed, and, fi nally, con clu sions on the quan ti ta tive ca pa bil i ties of HMI are sum ma rized. MA TE RI ALS AND METH ODS Fig ure 1 shows the sche matic of the sim u la tion pro cess to study the per for mance of the HMI tech nique. The pro cess is as fol lows: (1) The 2D pres sure field of the FUS trans ducer is sim u lated in Field II. 45, 46 (2) The 2D pres sure field is then used as a load ing con di tion in the FEM study. (3) The FEM so lu tion yields the tem po rally-varying dis place ment (os cil la tory dis place ment) that is used to gen er ate a time-de pend ent scat ter ers dis tri bu tion in the sim u - lated gel. (4) The im age for ma tion model is em ployed to sim u late the rf data. (5) One-di men - sional cross-cor re la tion is ap plied on the rf sig nals ob tained to es ti mate the mo tion. (6) The peak-to-peak displacements (D HMI ) from the FEM so lu tion and 1D cross-cor re la tion tech - niques are com pared to val i date the HMI mo tion estimates. The fol low ing sec tions de tails ev ery step used in this sim u la tion study.

4 AM PLI TUDE-MOD U LATED (AM) HAR MONIC MO TION IM AG ING 157 FIG. 1 Flowchart of the sim u la tion pro cess. The di a gram on the left with white ar rows de notes the FEM method while the di a gram on the rightwith black ar rows rep re sents the HMI tech nique. Comparison be tween the two es ti - mates reveals the qual ity of the HMI dis place ment. Theory The ra di a tion force is caused by the change in the mo men tum of the acous tic wave as it prop a gates through a me dium. In the case of a sin gle-el e ment fo cused ul tra sound (FUS) trans ducer, the ra di a tion force is mainly lo cal ized in the fo cal re gion. In an at ten u at ing ho - mog e nous me dium and as sum ing plane wave prop a ga tion, this force can be ex pressed as Ft () 2 It ( ) c (1) where t is time, F(t) is a volumic force [N/m 3 ], is the tis sue ab sorp tion co ef fi cient [1/m], I(t) is the peak-av er age acous tic in ten sity [W/m 2 ] and c is the speed of sound [m/s]. When an AM wave form is used to drive the FUS trans ducer, the ra di a tion force is oscillating at the mod u la tion fre quency ( m ). The acous tic pres sure p(t) of the AM wave gen - er ated at the fo cus can be ex pressed by p( t) p cos( t) cos( t) o m c (2) where p o de notes the max i mum in stan ta neous acous tic pres sure, c is the car rier fre quency and m is due to the mod u la tion fre quency. The spa tial peak-pulse av er age in ten sity (I sppa ),

5 158 MALEKE ET AL I(t) of the AM wave form can be cal cu lated by sim ply in te grat ing p 2 (t) over time, i.e., the time at which the pres sure is ap plied 50 It () t 0 2 p( ) d c (3) or 2 t po 2 I ( t) cos( ) cos( ) d m c c 0 (4) where is the vari able of in te gra tion and t the insonation time. The tis sue pa ram e ters used were: den sity ( ) =1000 kg/m 3 and speed of sound (c) =1540 m/s to sim u late soft tis sues. The re sult ing mo tion, which is re lated to the mul ti pli ca tion and square of the two si nu soi dal func - tions (Eq. 4), os cil lates at fre quen cies equal to twice the fre quency of mod u la tion (2 m ). This re la tion ship was also con firmed through ex per i men tal anal y sis, for in stance, when a 15 Hz mod u la tion fre quency was used, the har monic dis place ment os cil lated at a fre quency equal to 30 Hz. The mo tion re sult ing from this acous tic ra di a tion force is de tected through sig nal pro cess ing of the rf sig nals ac quired on an im ag ing i.e., a pulse-echo trans ducer, and then used to char ac ter ize the me dium be ing stud ied. Acoustic pressure field simulation The acous tic pres sure field was sim u lated in Field II. 45, 46 This frame work em ploys a lin ear acous tic-prop a ga tion model to cal cu late the pres sure field cor re spond ing to spe cific trans - ducer geometry and parameters. We modeled a single-element concave (FUS) transducer (Fig. 2a) with a cir cu lar open ing in its cen ter for the place ment of a pulse-echo trans ducer with iden ti cal pa ram e ters used in the ex per i ments (Fig. 2b). The con cave trans ducer used had a cen ter fre quency of 4.5 MHz, a fo cal length of 40 mm and in ner and outer di am e ters of 30 mm and 70 mm, re spec tively. The ap er ture was di vided into 1256 rect an gu lar el e ments 51 and the area of each el e ment was equal to 1 mm 2. There was a void at the cen ter of the trans ducer, i.e., about 57% of the el e ment was ac tive. The pres sure field sim u la tion showed that the fo cal spot size was about mm 3 (lat eral elevational ax ial), which was in agree ment with the ex per i men tal beam pro file. 52 The apodization tech nique was ap plied for each el e ment to rep re sent ac tive and nonactive re - gions. The cen ter open ing had a di am e ter of 30 mm and the apodization in this re gion was set to zero, i.e., nonactive el e ments, as in di cated by the blue-shaded area in fig ure 2b. The apodization of the ac tive area, an an nu lar re gion, was set equal to one (ac tive el e ments) as de - noted by the pink-shaded area in fig ure 2b. This method was used to sim u late the FUS trans - ducer with an AM wave form, i.e., a com bi na tion of a car rier fre quency ( c ) at 4.5 MHz and an AM fre quency ( m ) at 15 Hz, was used to drive the trans ducer. The 2D pres sure field, p(x, z), was sam pled at 80 MHz and cal cu lated at the fo cal re gion. The fo cal re gion size was equal to 60 mm (ax ial) by 40 mm (lat eral) ob tained at a res o lu tion of 0.1 mm and the at ten u a tion co ef fi cient ( ) of 0.3 db/cm to sim u late soft tis sue. The acoustic intensity levels (I sppa ) were var ied be tween and W/cm 2 to ver ify the lin - ear relationship between the acoustic intensity and the estimated displacement. This particu - lar acoustic intensity range was selected to match the previously-used experimental parameters 40, 54 for im ag ing and ther apy pur poses.

6 AM PLI TUDE-MOD U LATED (AM) HAR MONIC MO TION IM AG ING 159 FIG. 2 Con cave trans ducer with di am e ter of 70 mm and a cen ter open ing with a di am e ter of 30 mm, sim u lated in Field II. Color bar in di cates the apodization value, i.e., zero means nonac tive el e ment (hole) and one de notes ac tive el e ment. (a) side view and (b) top view of the sim u lated FUS trans ducer. Finite-element (FE) model An axisymmetric FE (Comsol MultiphysicsTM, Comsol Inc., Burlington, MA, USA) cy - lin dri cal gel was con structed with a ra dius of 18.5 mm and a height of 20 mm. A hard spher i - cal in clu sion was de fined in side the sim u lated phantoms with di am e ters equal to 3 mm, 5 mm and 10 mm. The back ground Young s modulus (E) was equal to 10 kpa. The tri an gu lar mesh was ap plied to the in clu sion ge om e try. The sec ond-or der, quad ra ture el e ment with six nodes per el e ment was used in this model. The di am e ter and stiff ness of the in clu sions were var ied ac cord ing to ta ble 1. The bound ary be tween the in clu sion and back ground was then mod eled based on the tri an gu lar dicretizations. The mesh re fine ment was per formed in or der to guar an tee the el e ment den sity was ad e quate and not in tro duc ing any nu mer i cal ar ti facts. The model was as sumed to be nearly in com press ible (Pois son s ra tio of ), with zero vis cos ity and a den sity of 1000 kg/m 3. The ul tra sound trans ducer was moved ax i ally and down ward with a step size of 1 mm to cover the en tire phantom depth (Fig. 3). The har - monic ra di a tion force was ap plied se quen tially at n preselected locations (p1, p2, pn) in and around the in clu sion (Fig. 3).

7 160 MALEKE ET AL Ta ble 1 FE model parameters. Back ground Young s modulus (kpa) In clu sion di am e ter (mm) In clu sion Young s modulus (kpa) Type and 50 Type and 50 Type and 50 Displacement p 1 p 2 d1 d 1 Axial depth p 5 d5 Axial depth d 5 pn Lateral direction FIG. 3 (Left) 1D ras ter-scan di a gram along the ax ial di rec tion at dif fer ent lo ca tion, e.g., p1, p2, pn, d1, d5,, dn represents the peak-to-peak displacement amplitude for each location. (Right) Graph representation of displacement amplitude vs. axial depth. d n dn The bot tom bound ary of the cy lin dri cal model was con strained in the ax ial di rec tion, while the re main ing bound aries were free to move in both the ax ial and lat eral di rec tions. In to tal, the num ber of nodes and tri an gu lar el e ments gen er ated was equal to ap prox i mately 7000 and 3000, re spec tively. This in cluded the re fined tri an gu lar el e ments that were spe cif i - cally se lected around and in side the in clu sion, thus al low ing suf fi cient res o lu tion for the spa - tial vari a tion of the cor re spond ing tis sue dis place ment within the trans ducer fo cus. Im age for ma tion The rf sig nals were sim u lated in Matlab 7.2 (Mathworks, Natick, MA, USA) us ing a lin ear convolutional scat ter ing model. 55, 56 The scat terer was ran domly dis trib uted. The lin ear ar ray had 64 el e ments, a cen ter fre quency of 7.5 MHz, a frame rate of 124 frames/s, a beam width of 2 mm and 60% band width. In this sim u la tion, the speckle pat tern was sim u lated as a Ray - leigh distribution 57 and thus, the tis sue mo tion in curred by the ap plied force could be es ti - mated. The rf sig nals were sam pled at 40 MHz. A 1D cross-cor re la tion tech nique was ap plied on con sec u tively-gen er ated rf sig nals at a win dow size equal to 1 mm and an 85% over lap. In this study, a sin gle-el e ment, pulse-echo trans ducer was used to ac quire rf ech oes in the ex per i ments; thus,only the es ti mated dis place ment along the cen ter rf line was con sid - ered in the com par i son with the sim u la tion re sults. Displacement estimation A cu mu la tive ax ial dis place ment was es ti mated by us ing 1D cross-cor re la tion on con sec - u tive rf sig nals with the ref er ence sig nal cho sen to be the rf sig nal when the in put force was

8 AM PLI TUDE-MOD U LATED (AM) HAR MONIC MO TION IM AG ING (a) (b) (c ) FIG. 4 HMI ap plied on an ex-vivo ca nine liver tis sue with AM fre quency of 25 Hz, i.e., dis place ment os cil lates at 50 Hz: (a) Cu mu la tive dis place ment at the fo cus lo ca tion (black line) and os cil la tory dis place ment af ter least-square lowpass fil ter to re move high-fre quency noise (red line). (b) Spec trum of the os cil la tory dis place ment. (c) Peak of the spec tra at twice the AM fre quency. The peak of the spec tra was used to es ti mate the peak-to-peak dis place ment amplitude (D HMI ). first ap plied. Co sine in ter po la tion was ap plied around the peak of the cross-cor re la tion func - tion in or der to im prove the pre ci sion of the dis place ment es ti ma tion. 58 The mag ni tude of cu - mu la tive dis place ment de noted the in curred mo tion. Thus, when the ra di a tion force was ap plied, the tis sue moved away from the trans ducer in di cated by the in creas ing dis place ment am pli tude. When the ra di a tion force de creased, the tis sue re laxed, as in di cated by the decreasing displacement amplitude. The estimated cumulative displacements was estimated through out the en tire tis sue depth and dis played over time. Fig ure 4a shows an ex am ple of the es ti mated os cil la tory dis place ment at the fo cus from an ex vivo ca nine liver to il lus trate the tis sue mo tion dur ing the force ap pli ca tion. The AM fre - quency used in this ex per i ment was equal to 25 Hz, i.e., the op ti mum fre quency for this par - tic u lar case. The mo tion has a 50 Hz os cil la tion fre quency (twice the mod u la tion fre quency; re fer to Eq. (4)). A dig i tal least-square lowpass fil ter (cut off fre quency of 200 Hz) was ap - plied on the es ti mated dis place ment (shown in red) in or der to show the dis place ment with - out the higher fre quency noise. Fig ure 4 shows a tran sient after the ap pli ca tion of the force (at 20 ms) be fore reached steady state. The mo tion con tin u ously oscillated through out the force application. The peak-to-peak dis place ment am pli tude de notes the HMI dis place ment (D HMI ) and was es ti mated at all depths by us ing a Fou rier trans form (FT) method. Based on the FT method, the spec tral peak cen tered at twice the AM fre quency was used to cal cu late the peak-to- peak displacement amplitude (HMI displacement, D HMI ) us ing

9 162 MALEKE ET AL Table 2 Young s moduli of polyacrylamide gels with dif fer ent acrylamide con cen tra tions and load ing con di tions. Young s Modulus (kpa) 20% pre-compression (5% dy namic load ing) 5% pre-compression (2% dy namic load ing) Acrylamide con cen tra tion (%) 0.1 Hz 1 Hz 0.1 Hz 1 Hz D HMI c pft N f s 2 d (5) where c is the sound speed [ m/s], f s is the sam pling fre quency (80 MHz), P FT is the peak of the dis place ment spec tra and N d is the dis place ment data points. 59 The cal cu - lated D HMI (Fig. 4c) at steady state was equal to 12 m in this case. The dis place ment re - gions were av er aged within a 1D re gion, i.e., 1.6 mm (ax i ally), to ap prox i mate the in duced mo tion. Phantom experiments Four uni form and six spher i cal in clu sion-em bed ded polyacrylamide gels with dis tinct stiffnesses were used for both me chan i cal test ing and HMI ex per i ments. Polyacrylamide gels were pre pared us ing the fol low ing guide lines: pre mixed 40% liq uid acrylamide (19:1 acrylamide:bis-acrylamide ra tio) (Thermo Fisher Sci en tific, Waltham, MA) was di luted in deionized wa ter to pro duce a range of acrylamide con cen tra tion from 25% (weight/vol ume) to 40% (weight/vol ume) in 5% in cre ments. The per cent age of acrylamide in the mix ture de - ter mines the stiff ness of the gel af ter it is poly mer ized. The re sult ing so lu tion was dis solved (1.75 ml per to tal ml) in 1M trishydroxymethylaminomethane (TRIS, 1.0 ml per to tal ml) with deionized wa ter (7.16 ml per to tal ml). 10% am mo nium per sul fate (APS, 8.4 l per to tal l) and N,N,N,N -tetramethylethylenediamine (TEMED, Sigma-Aldrich, St. Louis, MO, 0.5 l per to tal ml) sub se quently added. The mix ture was al lowed to poly mer ize at room tem per a ture for ap prox i mately 15 min utes prior to use. In or der to in de pend ently mea sure the Young s modulus of all four ho mo ge neous gels, a dy namic in den ta tion test was per formed. The de scrip tion of the in den ta tion test setup is pro - vided in the next sec tion. Based on the test re sults (Ta ble 2), 25% acrylamide con cen tra tion was used to gen er ate the soft back ground (E = 13 kpa), that is a sim i lar modulus value to the soft back ground in the sim u la tion (E = 10 kpa). The acrylamide con cen tra tions of 30% and 40% were re spec tively used for the hard in clu sions of 25 kpa and 50 kpa. These val ues were cho sen for quan ti ta tive com par i son be tween sim u lated and polyacrylamide gels. The three in clu sions used were 3 mm, 5 mm and 10 mm in di am e ter to sim u late grow ing tu mor di men - sions.

10 AM PLI TUDE-MOD U LATED (AM) HAR MONIC MO TION IM AG ING 163 FIG. 5 Block di a gram of the ex per i men tal setup. From top left, (a) pulse-echo trans ducer, (b) FUS trans ducer, (c) de gassed wa ter and (d) tis sue-mim ick ing gels sub merged in de gassed wa ter. An ab sorber (e) was placed on the bot - tom of a glass beaker at a 45 ( ) in order to reduce possible reflections, i.e., specular reflections. The HMI ex per i ment setup is shown in figure 5. Two fre quency gen er a tors (Agilent (HP) 33120A, Palo Alto, CA, USA) were used to pro duce a car rier fre quency of 4.5 MHz and a low mod u la tion fre quency of 15 Hz, be cause it was de ter mined to be the op ti mized os cil la - tory dis place ment through out the stiff ness range used in the study. The re sult ing AM wave - form was am pli fied by 50 db us ing a rf power am pli fier (E&I, Roch es ter, NY, USA). The AM wave form was used to ex cite a 4.5 MHz FUS trans ducer (Imasonic, Voray sur l Ognon, France) that gen er ated the os cil la tory acous tic ra di a tion force at the fo cal re gion. The acous tic pres sure was mea sured us ing a 0.2-mm nee dle hydrophone (Pre ci sion Acous tics LTD, Dorchester, Dorset, UK). The pos i tive and neg a tive peak pres sures at the fo cus had the same ab so lute am pli tude of 3.15 MPa due to the ap plied si nu soi dal force. The acous tic pres sure at the fo cus in the gel ex per i ments was es ti mated us ing a de rat ing fac tor of 0.3 db/cm/mhz at 4.5 MHz fre quency and 1.5 cm depth to ac count for the at ten u a tion through the gel. The es ti mated acous tic pres sure at the fo cus was equal to 1.98 MPa and the me chan i cal in dex (MI) af ter tak ing into ac count the de rat ing fac tor 0.3 (MI 0.3 ) was equal to 0.93, i.e., be low the FDA limit of 1.9. The peak av er age in ten sity (I sppa ) was cal cu lated nu - mer i cally us ing Eq. (4) with a p o of 1.98 MPa, the gel den sity ( ) was as sumed to be 1000 kg/m 3, c of 1540 m/s and t of 400 ms. The acous tic in ten sity (I sppa ) was equal to 32.5 W/cm 2, which is also be low the FDA I sppa limit of 190 W/cm 2. The potential temperature rise ( T) dur ing the force ap pli ca tion can be es ti mated by solv ing the bioheat trans fer equa tion. Here, the bioheat trans fer equa tion is as sumed to have a lin ear dis tri bu tion of ther mal sources with - out con vec tion or con duc tion ef fects, i.e., I T t v (6) where T is the tem per a ture rise in C, I the acous tic in ten sity (I sppa ), v the vol ume spe cific heat for tis sue (4.2 J/cm 3 / C), the ab sorp tion co ef fi cient of tis sue (0.16 Np/cm, i.e., 0.3 db/cm/mhz at 4.5 MHz) and t the du ra tion of the os cil la tory force (400 ms). For the ap plied

11 164 MALEKE ET AL oscillatory force in a single location, the anticipated peak temperature rise was estimated to be up to 0.96 C. Note that the re ported tem per a ture rise is the max i mum pos si ble value since heat con vec tion and blood per fu sion were not taken into ac count. A 7.5-MHz pulse-echo trans ducer (Panametrics, Waltham, MA, USA) with a di am e ter of 12 mm was placed through the cen ter of the FUS trans ducer, with the beams of the two trans - duc ers, i.e., FUS and pulse-echo, prop erly aligned prior to all ex per i ments. A pulser/re ceiver (Panametrics 5051PR, Waltham, MA, USA) was used to drive the pulse- echo trans ducer at a pulse rep e ti tion fre quency (prf) of 5.4 khz. The rf sig nals were ac quired us ing a stan dard pulse-echo tech nique. An an a log bandpass fil ter, i.e., 7 th or der type II Chebyshev band-pass fil ter, (Reactel, Inc., Gaithersburg, Mary land, USA) with cut off fre quen cies of f c1 = 5.84 MHz and f c2 = 8.66 MHz (at 60 db) was used to fil ter out the fun da men tal fre quency of the FUS beam and its har mon ics from the pulse-echo spec trum. 44 A sil i cone rub ber/ab sorber (McMaster-Carr, Dayton, NJ, USA) was placed at 45 o underneath the gel to fur ther re duce the spec u lar re flec tion from the bot tom of the glass con tainer. The FUS trans ducer was moved ax i ally and down ward by us ing a com puter con trolled positioner with a step size of 1 mm to cover the en tire gel depth (Fig. 3). The fil tered rf sig nals were sam pled at 80 MHz and a 14-bit digitization level (CS14200, Gage Ap plied Tech nol o gies, Lachine, Can ada). A one-di men sional cross-cor re la tion was ap plied on con sec u tive rf sig nals with a ker nel size equal to 1 mm and 85% over lap (as de scribed in the pre vi ous sec tion). Ex vivo experiments The same HMI ex per i men tal setup de scribed above was used to im age breast tu mors ex vivo. The breast spec i mens were col lected im me di ately af ter sur gery from the Irving Pa vil - ion of the Co lum bia Uni ver sity Med i cal Cen ter. The tu mor and its perilesional tis sue were ob tained ap prox i mately 30 min utes af ter ex ci sion and prior to histopathology. These tis sues were im me di ately placed upon ex ci sion in a ster ile con tainer and sub merged in de gassed phos phate buf fered sa line (PBS) so lu tion for the HMI ex per i ments. The spec i mens were then care fully re moved from the con tainer and placed in side a gel a tin ma trix for pro tec tive pur poses but also to con tain them and sim u late the sur round ing breast. All spec i mens were scanned us ing the HMI tech nique and the find ings were com pared to mam mog ra phy or sonography and, fi nally, histopathology find ings. Ori en ta tion of the spec - i men was pro vided by the sur geon. For tis sue map ping, the FUS trans ducer was moved in a 2D ras ter-scanned fash ion with a step size equal to 1 mm us ing a com puter-con trolled positioner (Velmex Inc., Bloomfield, NY, USA). The scan plane had a di men sion of mm 2, an acous tic in ten sity (I sppa ) equal to 32.5 W/cm 2 with a du ra tion of 0.2 s at each point. In or der to choose the op ti mal AM fre quency, a lin ear chirp test con tain ing fre quen cies within the range of 10 to 30 Hz was used to in ves ti gate the op ti mal vi bra tion fre quency for the higher me chan i cal con trast be tween the tu mor and the sur round ing re gions. The op ti mal AM fre quency for the spec i men pre sented in this pa per was 15 Hz. HMI scan ning lasted ap prox i - mately 40 min utes. The spec i mens were re turned to pa thol ogy for di ag no sis. All pro ce dures were ap proved by the In sti tu tional Re view Board (IRB) board of Co lum bia Uni ver sity prior to all stud ies. The le sion re gion was seg mented based on the dis place ment dif fer ence between the stiff in clu sion and the sur round ing re gion. Mechanical testing A dy namic in den ta tion test was used to as sess the Young s modulus of the polyacrylamide gels. The sys tem con sisted of an in den ta tion probe (in denter) that ver ti cally ap plied a small si nu soi dal de for ma tion on the gel sur face. The in denter had a flat and rough sur face to avoid

12 AM PLI TUDE-MOD U LATED (AM) HAR MONIC MO TION IM AG ING 165 FIG. 6 (a) Me chan i cal test ing ma chine, Instron Microtester (Instron, Inc., Norwood, MA) used in the ex per i - ment. (b) closer view of the me chan i cal test ing. The in denter (i) was loaded by a 10 N load cell with a pre ci sion of 0.5% of the to tal ca pac ity (10 N). (ii) tis sue-mim ick ing gels placed be neath the in denter. pos si ble slip page dur ing test ing. The di am e ter of the in denter was equal to 6.33 mm. The load ing site was se lected by plac ing the in denter at the cen ter of the gels. The gels were placed on a rigid cy lin dri cal plate with a thick ness of 14.3 ± 0.56 mm and a di am e ter of 87.1 mm, which was four times larger than that of the in denter so that the dimensionality as sump - tions of the the o ret i cal model (Eq. 7) were sat is fied. 63 A small amount of deionized wa ter was added spar ingly to keep the gels mois tur ized. The tem per a ture was kept within the range of 25.5 C to 25.6 C. The ap plied dy namic dis place ment was con trolled by an Instron Microtester (Instron, Inc., Norwood, MA) by us ing a 10 N load cell (Fig. 6). The pre ci sion of the load cell was 0.5% of its to tal ca pac ity (10 N). Two dif fer ent tests were per formed on each gel. The first test en tailed a 20% precom pres sion (i.e., 2.56 ± 0.12 mm) fol lowed by an os cil la tion at a mag ni tude of 5% ap par ent strain (dis place ment di vided by ini tial thick ness, i.e., ± mm); and the sec ond test en tailed a 5% precom pres sion (i.e., ± mm) fol - lowed by an os cil la tion mag ni tude of 2% ap par ent strain (i.e., ± mm). To test re li abil ity, each in den ta tion test was per formed and com pared at two dif fer ent load ing fre - quen cies of 0.1 Hz and 1 Hz (Fig. 7). The tem po rally vary ing force was mea sured dur ing the en tire in den ta tion pro cess and these data were used to cal cu late the Young s modulus of the gel sam ples us ing the equa - tion E v q a w (7)

13 166 MALEKE ET AL FIG. 7 Graphic rep re sen ta tion of the dy namic me chan i cal test ing. Precompression (i.e., 20% and 5%) was im - posed, fol lowed by an os cil la tion of ap par ent strain, i.e., 5% and 2% (dis place ment di vided by ini tial thick ness). Both load ing pro to cols were tested at the fre quen cies of 0.1 Hz and 1 Hz. where E is the Young s modulus, v is the Pois son s ra tio (as sumed to be nearly in com press - ible material, v = ), q is the load ing pres sure (force per unit area), a is the ra dius of the in denter tip and w is the ap plied dis place ment am pli tude. Con trast-trans fer ef fi ciency (CTE) In HMI, the ap plied force/stress is highly lo cal ized at the fo cus, typ i cally on the or der of 1 to 2 mm, and thus, the re sult ing dis place ment is re lated to the rel a tive tis sue modulus (stiff - ness) within the fo cal re gion. The dif fer ence be tween the dis place ment in the in clu sion and back ground re gions can be used to de tect in clu sions by their stiffnesses dif fer ence. To quantitatively evaluate the performance of HMI, the contrast-transfer efficiency (CTE) (Eq. (8)) pa ram e ter was used. Here, CTE was de fined as the ra tio of the av er age displacement contrast (C d ) to the elas tic ity con trast (C e ). CTE rep re sents the per cent age, at which dis place ment pro files de pict the un der ly ing elas tic ity dis tri bu tion in the me dium given by C dbackground d (8) d inclusion CTE C E E e inclusion background The average displacement contrast (C d ) was cal cu lated based on the ra tio be tween the av - er age dis place ment of the back ground (d background ) to the av er age dis place ment of the in clu - sion (d inclusion ). The elas tic ity con trast (C e ) is the ra tio of the in clu sion Young s modulus (E inclusion ) to the back ground Young s modulus (E back ground ). For elas tic ity con trast (C e ) in the sim u la tion case, the Young s moduli were equal to the in put FE val ues whereas, in the ex per - i ment cases, the Young s moduli were ob tained from me chan i cal test ing. The value of dis place ment con trast is nearly equiv a lent to the value of the elas tic ity con - trast when the CTE is close to 1. Dis place ment con trast can be used to closely rep re sent the rel a tive stiff ness of the in clu sion as long as the CTE re mains rel a tively high. Thus, the change of the elas tic ity at the in clu sion/back ground re gions may be used to ex plain the sim i - lar ity be tween the dis place ment and the elas tic ity con trast. A re gion of in ter est (ROI) was cho sen in the mid dle of the in clu sion and above the in clu - sion for the back ground in or der to cal cu late the in clu sion-to-back ground con trast; both ROIs had the same size. The ROI was al ways smaller than the di am e ter of the in clu sion, and cho sen around the center of the in clu sion for dif fer ent in clu sion sizes, for in stance in fig ure 8a, the ROI for a stiff in clu sion was 1 mm from 10 to 11 mm and the ROI for the sur round ing re gion was 1 mm be tween 5 and 6 mm. The ROIs for the sur round ing re gion were cho sen above the in clu sion be cause the dis place ments were more uni form.

14 AM PLI TUDE-MOD U LATED (AM) HAR MONIC MO TION IM AG ING 167 FIG. 8 Normalized D HMI from sim u la tion along the cen tral axis of the FUS trans ducer for var i ous in clu sion sizes and two dif fer ent Young s moduli ( kpa and kpa). (a, b and c) re sults for 3, 5, and 10-mm in clu sion di - am e ters, re spec tively. The back ground Young s modulus was equal to 10 kpa. The dot ted lines in di cate the di am e ter of the in clu sions. The ROI in the sim u la tions (Fig. 8) for the 5-mm and 10-mm in clu sion di am e ter were of sixe equal to 4 mm (be tween 9 and 12 mm) and 8 mm (be tween 7 and 15 mm), re spec tively. In the phan tom ex per i ments (Fig. 11), the ROIs for 3 mm, 5 mm and 10 mm in clu sion di am e ters were 2 mm (be tween 12 and 14 mm), 4 mm (be tween 17 and 21 mm) and 8 mm (be tween 11 and 19 mm). RESULTS To study the per for mance of HMI, spe cific pa ram e ters are tested in or der to as sess its ca - pa bil ity of es ti mat ing the en tire re sult ing dis place ment range. Com par i son be tween the FEM and HMI-es ti mated dis place ments in a ho mo ge neous me dium with a Young s modulus of 10 kpa is shown in fig ure 9. The high est dis place ment is lo cated at the cen ter of the fo cal zone (depth of 40 mm), ra di ally span ning ap prox i mately ±3 mm from the sym me try axis.

15 168 MALEKE ET AL FIG. 9 Two-dimensional cumulative displacement fields resulting from the acoustic-force distribution. The high - est dis place ment oc curs at a depth of 40 mm (FUS fo cus). (a) FEM so lu tion and (b) es ti mated HMI dis place ment. Figure 8 depicts the relationship between the displacements and inclusion size/stiffness of the sim u lated phan toms. The Young s moduli of harder in clu sions were 25 kpa and 50 kpa and their di am e ters were equal to 3, 5, and 10 mm. The peak-pos i tive acous tic-pres sure am pli tude was equal to 1.98 MPa, the same value as that used in the ex per i men tal cases. The di am e ter of the in clu sion can be es ti mated based on the dis place ment pro files. For ex am ple, in fig ure 8a, the HMI dis place ment am pli tude de creases at the ax ial depth of 9 mm and then in creases at the ax ial depth of 12 mm; thus, the 3-mm-di am e ter in clu sion is cor rectly de picted with a di am e ter of 3 mm. Fig ures 8b and c show the es ti mated in clu sion sizes at 5 mm and 10 mm, re spec tively. As the fo cus of the FUS beam moves to ward the cen ter of the in clu sion, the re sult ing dis place - ment am pli tude steadily de creases, thus cor rectly rep re sent ing the rel a tive stiff ness of the in - clu sion. For in stance, the dis place ment am pli tude close to the cen ter of the hard in clu sion, i.e., be tween 10 and 12 mm, is the low est (Figs. 8b, c). The dis place ment at the depth of 20 mm is ap prox i mately zero, be cause the bot tom sur face of the sim u lated gel was con strained. The me chan i cal test ing re sults show that the mea sured Young s modulus in creases with the con cen tra tion of acrylamide (Ta ble 2). The cal cu lated Young s moduli for 25%, 30%, 35% and 40% are ± 0.32, 25.4 ± 2.46, ± 1.08 and 50.2 ± 1.43 kpa (mean ± SD), re spec - tively. These re sults were used as a guide line to gen er ate a hard in clu sion in the gel with stiff - ness sim i lar to the sim u lated gels. The acrylamide con cen tra tion of 25% was used to model the soft back ground with a Young s modulus of E = 13 kpa (Ta ble 2). We se lected acrylamide con cen tra tions of 30% and 40% to match the in clu sion Young moduli (E) in the sim u lated gel, which are equal to 25 kpa and 50 kpa, re spec tively (Ta ble 2). The es ti mated Young s moduli ob tained from me chan i cal test ing and the in verse of the es ti mated HMI dis place ment from gel experiments exhibit a linear relationship (r 2 = 0.97) (Fig. 10). This fur ther confirms that the HMI dis place ment can be in di rectly re lated to the un der ly ing tis sue stiff ness. To com pare the sim u la tion re sults (Fig. 8) with the gel ex per i men tal find ings, the force was ax i ally moved across the spher i cal in clu sion. Rf frames were ac quired for each gel in four in de pend ent it er a tions to test the reproducibility of the HMI tech nique in re li ably im ag - ing in clu sion sizes. The av er age dis place ment pro files with its stan dard der i va tion among the four it er a tions for each gel are shown in fig ure 11. The small in clu sions, i.e., 3 and 5 mm in di am e ter, are both de tect able (Figs. 11a, b). The as sess ment of the es ti mated dis place ment con trast that is re lated to the un der ly ing elas tic ity dis tri bu tion in the het er o ge neous me dium was eval u ated by cal cu lat ing CTE. The

16 AM PLI TUDE-MOD U LATED (AM) HAR MONIC MO TION IM AG ING 169 FIG. 10 Lin ear re la tion ship be tween the in verse of the HMI dis place ment from ex per i ment and Young s Moduli from mechanical testing. CTE val ues from the sim u lated gels (tri an gle) and polyacrylamide gels (cir cle) are shown in fig ure 12. Here, CTE is cal cu lated for three in clu sion di am e ters (3, 5, and 10 mm) with two dif fer ent Young s moduli (25 kpa and 50 kpa). In this ex per i ment, the dis place ment con trast is close to the elas tic ity con trast for larger and stiffer in clu sions (CTE > 0.7). The CTE is be - low 0.5 for the 3-mm in clu sion, even though the in clu sion/back ground in ter face is clearly de picted on the HMI pro file (Fig. 11a). Fig ure 13a shows the HMI dis place ment im age of ex vivo breast tis sue sam ple where the dis place ment am pli tude is in m. The spec i men was con firmed to be an in va sive lob u lar car ci noma (ILC) by a pa thol o gist and har vested im me di ately af ter sur gery. The spec i men was first im aged us ing mam mog ra phy for pa thol ogy val i da tion and then im aged by HMI. The B re gion (i.e., D HMI was above 40 m) in di cates soft breast tis sue (Fig. 13a, B ). The T re gion (i.e., D HMI was be low 20 m) in di cates a stiffer re gion, i.e. tu mor (Fig. 13a, T ). Fig ure 13b dis plays a mammogram of the same spec i men. The white line (Fig. 13b, N ) de picts the tip of the nee dle at tached to the le sion (tu mor) af ter bi opsy. A bi opsy nee dle was placed prior to sur gery as a guid ance tool for the sur geon to iden tify the le sion and conirm its suc cess ful re moval. Tu mor size was found to be equal to ap prox i mately mm 2. The HMI im age is thus in good agree ment with the mam mog ra phy find ings for both the size and shape of the breast and the can cer tis sue.the spec i men was then sub mit ted to pa thol ogy, where a small tis sue spec i men was dis sected for anal y sis and the re main ing tis - sue was stored in the Tu mor Bank Fa cil ity as re quired by the In sti tu tional Re view Board (IRB) of Co lum bia Uni ver sity. The his tol ogy im age of the spec i men was not avail able due to pa tient confidentiality agreement and the Institutional Review Board (IRB) of Columbia Uni ver sity reg u la tion.

17 170 MALEKE ET AL FIG. 11 D HMI from HMI ex per i ments along the cen tral axis of the FUS trans ducer for var i ous in clu sion sizes and two dif fer ent stiffnesses (- - 30% acrylamide ( 25 kpa) and % acrylamide ( 50 kpa)). The av er age and stan dard de vi a tion (i.e., er ror bars in fig ures) of the es ti mated dis place ment (D HMI ) were cal cu lated based on four it er - a tions com pleted for each in clu sion size. (a, b and c) re sults for 3, 5 and 10-mm in clu sion di am e ters, re spec tively. The back ground stiff ness was 25% acrylamide ( 13 kpa). The dot ted lines in di cate the di am e ter of the in clu sions. DISCUSSION The aim of this study was to in ves ti gate the po ten tial of the am pli tude-mod u lated (AM) har monic-mo tion im ag ing (HMI) tech nique as a noncontact method for map ping and quan - tifying relative tissue stiffness. Owing to the AM application, the oscillatory radiation force was sus tained at the same lo ca tion at a fre quency twice the AM fre quency (Eq. (4)). Since the FUS fo cal spot was small (ap prox i mately mm 3 ), the force was highly lo - cal ized and the probed tis sue within the fo cal re gion could be in di rectly as so ci ated with the un der ly ing tis sue modulus. For im ag ing, the ra di a tion force could be ap plied in a 2D fash ion us ing a ras ter-scan tech nique and a 3D HMI dis place ment im age could be ob tained by com - 40, 67 bin ing mul ti ple 2D planes at vari able depths.

18 AM PLI TUDE-MOD U LATED (AM) HAR MONIC MO TION IM AG ING 171 FIG. 12 Con trast-trans fer ef fi ciency (CTE) for het er o ge neous gel with in clu sion (3, 5, and 10 mm in di am e ter). Triangle denotes result from simulated gels and circle denotes experimental results from tissue-mimicking gels. Solid circle/triangle represents results for 25 kpa-inclusion. Clear circle/triangle represents results for 50 kpa-inclusion. FIG. 13 (a) HMI im age, T blue (stiffer re gion) de notes breast tu mor (In va sive Lob u lar Car ci noma, ILC) and B yel low/red de notes nor mal breast tis sue (softer). (b) Mam mog ra phy of the postsurgical breast spec i men with N was a bi opsy nee dle. In the FE anal y sis, the axisymmetric model al lowed for a 3D anal y sis of the pres sure field applied. In this case, the geometry, tissue-mechanical properties distribution and the acousticra di a tion force were axisymmetrically dis trib uted. This is a lim i ta tion of the axisymmetric model because biological tissues are typically asymmetrically heterogeneous. The acoustic pres sure level of 1.98 MPa was ap plied at the cen ter axis and moved down ward with a step size of 1 mm. Since the model was lin ear, the be hav ior at higher acous tic pres sure lev els could be ex trap o lated in or der to pre dict the re sult ing dis place ments. In this ex per i ment, the ap plied pres sure was ap prox i mately 1.96 MPa for 400 ms with an ex pected peak tem per a - ture rise of 0.9 C. This tem per a ture rise was con sid ered low ( T<1 C) be cause it would not pose a risk to the pa tient dur ing ul tra sound ex po sure. 61, 68 This tem per a ture rise is con sid ered safe and ac cept able for ul tra sound im ag ing. Scan ning and im age gen er a tion lasted ap prox i -

19 172 MALEKE ET AL mately one hour, which can be fur ther im proved through cus tom hard ware im ple men ta tion. Me chan i cal scan ning and cou pling to the pa tient will also be in cluded as part of the new sys - tem design for clinical applications. The ef fects of in clu sion size and stiff ness on the dis place ment pro file were dem on - strated in sim u la tions (Fig. 8) and ex per i ments (Fig. 11). Three dif fer ent hard in clu sion di - am e ters (3, 5, and 10 mm) were used to in ves ti gate the ca pa bil ity of the HMI tech nique in small in clu sion de tec tion. This in di cates that HMI can be used for de tec tion and map ping of small (>3 mm) tu mors with a spe cific stiff ness con trast. In the case of the 10 mm in clu - sion (Fig. 11c), the bound ary of the in clu sion is not dis tinct. If the area of the back ground re gion were twice larger than the in clu sion like in the other cases, it is ex pected that the bound ary of the in clu sion would be more dis tinct. The ef fect of the back ground will be in - ves ti gated in fu ture stud ies. The CTE value rep re sented the qual ity of the HMI dis place ment, which is re lated to the un der ly ing elas tic ity dis tri bu tion in the het er o ge neous me dium. The re sults show that the CTE was higher than 0.5 but might be less than the ideal case (e.g., 1) for most in clu sion cases. Thus, these CTE val ues in di cate that the HMI dis place ment may be use to es ti mate the lo ca tion and size of the le sion based on its dis tinct dis place ment but not nec es sar ily be used to ob tain its ac cu rate stiff ness con trast. Two pos si ble ex pla na tions are pro vided, i.e., that (1) dif fer ent gel ge om e tries were used and (2) the con di tions of the two tech niques used to mea sure dis place ment con trast by HMI and elasticity contrast by mechanical testing were different. Here, the displacement was mea sured by HMI where the lo cal ized force was ap plied deep in side the het er o ge neous gels whereas the elas tic modulus was mea sured by me chan i cal test ing, where the force was ap - plied on the sur face of the uni form gel. The HMI and me chan i cal test ing (i.e., dy namic me - chan i cal test ing) could not be tested within the same fre quency range due to fun da men tal dif fer ences. In HMI, the data stor age ca pa bil ity has been op ti mized for an AM fre quency range above 10 Hz. The in stru men ta tion is cur rently be ing up graded in or der to test a lower fre quency range (<10 Hz). Vappou et al 69 showed that polyacrylamide gels mainly ex hibit an elas tic be hav ior and the es ti mated Young s moduli re mained rel a tively con stant within the fre quency range of 0.1 to 40 Hz. There fore, we as sumed that the stiff ness of the polyacryl - amide gels stayed con stant within the range of 0.1 Hz to 15 Hz. 69 In mechanical testing, when the load ing fre quency ex ceeded 1 Hz, the strain does not im me di ately fol low the ap plied stress, fi nally caus ing a gap (~ 1 mm) be tween the in denter and the gel. The two tech niques thus re quire dif fer ent test ing con di tions and have dif fer ent bound ary con di tions and im - posed stress lev els. Nev er the less, we dem on strated that the HMI dis place ment con trast is in - di rectly re lated to the elas tic ity con trast of a stiffer in clu sion em bed ded in a soft back ground. The dis place ment con trast could be used to rep re sent the rel a tive in clu sion stiff ness, which, in turn, can be used for fu ture tu mor de tec tion. When the CTE value is lower, the con trast of dis place ment un der es ti mates the stiff ness con trast; how ever, as shown in fig ure 11a, HMI could clearly de tect the 3-mm in clu sion de - spite the rel a tively lower CTE. The HMI sen si tiv ity might thus be sufficient to de tect small tu mors (>3 mm in a di am e ter). Pre lim i nary re sults from postsurgical breast spec i mens dem on strated a good stiff ness con trast be tween the tu mor and sur round ing breast tis sue (Fig. 13). The tu mor lo ca tion was also con firmed with mam mog ra phy and bi opsy nee dle lo ca tion. The histopathology re sults in di cate that the tu mor was an in va sive lob u lar car ci - noma (ILC) In this study, the os cil la tory dis place ment was es ti mated with out tak ing into con sid er ation the ef fect of re flec tion or any in ter fer ence oc cur ring. Shear wave and re flec tive waves might af fect the es ti mated dis place ment. At ten u a tion and dif frac tion from these prop a gat ing waves, as well as vis cos ity, can not be ig nored. The ef fect of this in ter fer ence would be min i -