F speckles are randomly located. Therefore, as a general. Ultrasonic Texture Motion Analysis: Theory and Simulation

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1 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 4. NO. 2, JUNE Ultasoc Textue Moto Aalyss: Theoy ad Smulato Jea Meue, Membe, IEEE ad Mchel Betad Abstuct- A theoetcal model was pevously developed to evaluate the elatoshp betwee the dyamcs of ultasoc speckle ad ts udelyg tssue. The model s dvded to a stumetal pat epeseted by the pot spead fucto ( the fu feld) of the ultasoc appaatus ad a movg tssue compoet descbed by a collecto of scattees. By computg the covoluto of these tems ad the the evelope, oe obtas a smulated ultasoc speckle patte sequece whch shows speckle motos closely lked to the tssue dyamcs whe small moto ampltudes ae volved. I ths pape, a theoetcal study of the coelato betwee vaous lea tasfomatos of the tssue ad the coespodg ultasoc speckle motos s pefomed, based o a 2-D exteso of the evelope coss-coelato aalyss of a aow-bad Gaussa ose. I the lea sca case, obvously, tssue taslato geeates a detcal speckle taslato. Howeve, tssudspeckle moto coelato deceases wth ceasg otato ad/o baxal defomato, lateal defomato (pepedcula to the beam popagato axs) beg much less sestve. Wth espect to the tasduce fequecy, the otato ad the axal defomato of the tssue show a bette elatoshp wth the espectve speckle moto at lowe fequeces whle lateal defomato coelato s depedet of the pulse fequecy. Wth espect to beam (pulse) sze paametes, tssudspeckle coelato deceases wth otato whe a wde ultasoc beam s used whle the axal defomato coelato deceases wth the axal duato of the pulse. Ths study sets the goud fo the developmet of a ultasoc sta gauge patculaly useful fo the assessmet of bomechacal soft tssue ad flud flow popetes based o speckle tackg. I. INTRODUCTION ROM AN obseve pot of vew, soft tssue ultasoc F speckles ae adomly located. Theefoe, as a geeal ule, may studes o ultasoc mage textue aalyss addess a tssue chaactezato poblem usg a statstcal appoach,.e., by tyg to elate fst- ad secod-ode statstcal popetes of the tssue textue to the tssue stuctue [l]. Howeve, small chages tssue posto ae expected to lead to coespodg obsevable chages the speckles. Thus, t makes sese to study speckle tackg as a meas to fe tssue dyamcs. Mauscpt eceved August, 993; evsed Febuay 25, 995. Ths wok was suppoted by the Natoal Scece ad Egeeg Reseach Coucl of Caada, the QuCbec Msty of Educato (FCAR), ad the Fod de Recheche de Isttut de Cadologe de MotCal. The Assocate Edto esposble fo coodatg the evew of ths pape ad ecommedg ts publcato was R. Mat. J. Meue s wth the Dkpatemet d hfoatque et de Recheche OpCatoelle, Uvestt de MotCal, Motkal, H3C 357, Caada; e-mal: meue@ o.umoteal.ca. M. Betad s wth the Isttut de GCe BomCdcal, Uveste de MotCal, Motkal, H3C 357, Caada. IEEE Log Numbe I ode to get a bette udestadg of the elatoshp betwee speckle ad tssue posto chages, we fst developed a compute model of the B-sca magg pocess assocated wth a cotactg myocadum [2]. Usg the model, we epoted speckle tackg feasblty (ad lmtatos) fo the myocadum [3]-[5]. Ths wok allowed us to establsh a clea elatoshp betwee small tssue defomatos ad speckle patte moto ad demostated the potetal of speckle patte tackg as a dagostc tool to study tssue dyamcs. Meawhle, Tahey et al. [6] showed expemetally that the moto of speckle pattes poduced by blood flow ca be used to estmate blood flow velocty. They used a 2-D coelato seach algothm to tack the taslato of speckle pattes. Ou optcal flow (velocty feld) algothm appoach allows the computato of paametes elated to the tssue taslato ad also to otato ad defomato [3]-[5]. Betad et al. [7], Che et al. [8], ad Tahey et al. [9] also vestgated speckle tackg usg a phatom ad skeletal muscle [7]. Ths pape dscusses the 2-D aalyss of the evelope of etued ultasoud sgal ove subsequet mages to extact soft tssue motos. Based o smulatos ad a theoetcal aalyss, the fudametal lmtatos of speckle tackg to assess soft tssue moto ae peseted. I patcula, t dscusses a model to study coelato betwee speckle patte moto ad tssue moto whe a tssue s subjected to a lea geometcal tasfomato (taslato, otato, ad defomato).. MODEL OF THE ECHOGRAPHIC TEXTURE Seveal models to smulate the speckle pattes ecouteed typcal echogaphc mages wee poposed the lteatue. Bascally, f oe assumes leaty ad posto depedet pot spead fucto (PSF) the fa feld, the 3-D RF echogaphc sgal I(z,y,z) ca be descbed by a 3-D covoluto poduct (8) betwee the system PSF H(z, y. z) ad the mpulse espose of the tssue T(z, y. z) [2], [ll], [2] x H(z - p, y - U, z - U)& dv dw (2) whee z s the decto whch the beam popagates, y the lateal decto the magg plae, ad z the decto pepedcula to the magg plae,.e., the decto of elevato. If oe assumes a sepaable PSF (at least fo the z compoet: H(z, y, z) = H(z, y)hz(z)) ad a slce though As dscussed [28], such PSF ca be obtaed wth a ectagula tasduce /95$ IEEE

2 294 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 4. NO. 2, JUNE 995 I(x,y,z), say at z =, to smulate the RF mage poduced dug the B-mode scag, oe gets wth qx, U) = H(z, Y) 8 N(z,!/I (3) N(x, y) =.I T(z, y, z)h,(-z) dz. Ths last pa of equatos foms the bass of ou model alog wth a evelope detecto pocedue to poduce the esultg B-sca mage IB (x, y). We chose a cose modulated by a 3-D Gaussa evelope to model a symmetc plaa ad sepaable PSF (appoxmato of a fa feld PSF) The tssue s assumed to be a collecto of homogeetes (cells o othes) that behave as scattees. Thee ae seveal models [3] to defe the T(z,y, z) tem; to smplfy the tssue model, we assume a lage umbe of vey small homogeetes wth espect to the wavelegth of the PSF (4) qz, y, 2) = - z, y - y 2-27%) (6) whee S(x, y, 2) s the 3-D Dac fucto o mpulse fucto, (z, y, z,) the adomly dstbuted cetes of each homogeety, ad a, the echogecty of each scattee. Cosdeg (4), oe obtas That s, f the beam testy pofle H,( ) s costat wth the thckess of the beam ad f cells (scattees) have equal echogecty a, the N(x,y) s popotoal to the umbe of cells m ovelayg the pxel aea wth the beam thckess. Ths umbe of cells at pxel (2, yo) ca be modeled as a adom vaable wth a Posso dstbuto, ad f t aveages say moe tha fve o so, the dscete dstbuto wll be close to a sampled Gaussa; howeve, wth ( l), the dstbuto wll be cotuous by the fact that cells may have oufom echogecty a, ad wll be set at dffeet postos a beam that does ot have a costat testy pofle 2. I ou smulatos, the beam thckess ad pxel aea ae lage eough to cout much moe tha fve cells pe pxel, theefoe N(z, y) could be modeled as a omal pocess.* Thus, to obta N(s, y), oe smply eeds to geeate a 2-D omally dstbuted adom feld (mage). B. The B-Sca Image All those assumptos allow a teestg smplfcato of the eal 3-D poblem to a much moe ease 2-D poblem that copoates aalytcally the thd dmeso. I ode to get a typcal B-sca mage IB (2, y), a evelope detecto pocedue s eeded. Fst, the complex pe-evelope s computed. Ths s easly doe by usg the Hlbet tasfom to get the magay pat ad the ogal RF mage as the eal pat of the pe-evelope. The evelope s the smply computed as the magtude of the pe-evelope [4]. N(z, Y) = an,(&,)s(z - z, Y - Y). (8) Hece, to model the tssue tem (3), we use a osy patte N(z, y) that wll be dscussed ow. A. The Pocess Fucto N(z, y) I pactce, the fucto I(x, y) s badlmted by the 2-D PSF, H(z, y). If we assume that H(z, y) s aowbad, the fo a suffcetly aow gate fucto ect(az, Ay), (3) could be ewtte as wth I(s, Y) = H(x, w) 8 N(z, Y) R(z, y) = N(z, y) 8 ect(az, Ay) (9) () that s, the spectum of the ect( ) fucto s cosdeed to be ufom wth the badwdth of I(z, 9). Fo a pxel sze (Ax, Ay) at locato (2, yo) the mage plae Examples of the esultg B-sca mages ae peseted Fgs. ad 3. The mage sze s 256 x 256 pxels coespodg to 5.2 x 5.2 mm. The stumetato paametes ae gve Table I. Fo moe detals o ths pat of the smulato, the eade s efeed to oe of the followg papes [2], [5]. C. Modelg Tssue Dyamcs We stat wth a 3-D lea moto descbed by the followg tssue posto tasfomato: (:) =?+ME:). (3) Ths tasfomato defes the moto of a tssue td posto (zo, yo, 2) to ts fal posto (zl, y,zl) whee T epesets a 3-D tssue taslato vecto ad M the 3-D defomato ad otato of the tssue. A costat s added to the model to take to accout the compessblty of muscles [5] ad soft tssue geeal. Ths costat fo the 3-D lea defomato studed hee s smply det(m) =. =l 2 ou smulatos, the pxel aea s ZOpm x 2pm ad the PSF thckess s the ode of 2. mm. Fo example, a typcal cadac cell (scattee) s 5 to 2 pm, ad theefoe oe obtas 4 pm t 2 pm/3 pm cz 53 fbes/pxel. Ths umbe must be educed by appoxmately 3% to take to accout the extacellula medum, whch esults a mea clealy lage tha fve fo the adom vaable ~JI.

3 MEUNIER AND BERTRAND. JLTRASONlC TEXTURE MOTION ANALYSIS 295 TABLE I INSTRUMENTAL PARAMETERS USED FOR THE SIMULATION OF H(..!I), FWHM: FULL WIDTH HALF MAXIMUM % 2.3.;~ STANDARD DEVIATION (a). THESE PARAMETE:RS DEFINE A NARROW-BAND SYSTEM; THEREFORE, I(x. y) IN (9) Is THE REALIZATION OF A NARROW-BAND GAUSSIAN PROCESS. THE PSF SPATIAL FREQUENCY IS DEFINED ALONG THE AXIAL DIRECTION; IT IS EQUAL TO TWICE THE TRANSDUCER FREQUENCY DIVIDED BY THE SPEED OF SOUND Ultasoud Veloc PSF axal dmeso PSF lateal dmeso (m) PSF spatal fequecy PSF axal badwdth 54 &sec.. mm 6.5 cycles/mm Fg. I. Smulato of speckle patte fo a tssue otato of (a) degee, (b) 2 degees. ad (c) degees. (d) Lea tasfomato computed fom (a) to (b) usg a optcal flow algothm. The optcal flow uses a lea velocty feld costat ove the whole j.2 mm x 5.2 m mage. The og s at the cete of the mage ad the.i ad axes u pwtve fom top tu bottom ad left to ght, espectvely. Note the speckle patte decoelato whe the tefame moto s lage ((a) vesus (c)).. SIMULATION RESULTS To smplfy as much as possble the tssue moto aalyss ad smulato, the moto s assumed to take place the x-y plae (mage plae) oly. I ths patcula case, we get the followg esult afte pola decomposto: M=RD (4) T= - (:) (L: b.r= :;:I9!).D= (I :) wth det(d) =. (5) The fst tem T s a vecto that epesets the axal ( x compoet) ad lateal ( y compoet) taslato of the tssue. R ad D ae matces descbg a,-y plae (mage plae) otato ad baxal (that s, a defomato alog two pepedcula ma axes) defomato of the tssue, espectvely. The egevectos of D show the defomato ma axes whle the coespodg egevalues eflect the extet of the defomato. I the followg smulatos, we wll study axalhatea defomatos whch mea that y =. The egevectos ae theefoe potg the axal ad lateal dectos ad the egevalues ae, espectvely, N ad [I. The compessblty costat fo the 2-D moto studed hee s smply det(d) = a/) =. To set ths tasfomato to the pevous secto model, we ca apply ths tasfomato to the 2-D tssue compoet N(.. y) by meas of a chage of vaables (x. g ) to obta the esultg tasfomed tssue pojecto N(.. y ) wth PSF lateal badwdth Howeve, ou mplemetato scheme, we pefeed to tasfom the stumetal tem (PSF) stead of the tssue tem whch, elatvely speakg, s the same: If oe otates the tssue clockwse, ths s elatvely equvalet to otatg the PSF couteclockwse. Ths pocedue s qute teestg fo the followg easo: The tasfomato ca be doe easly sce the PSF s kow aalytcally. Ths also meas that peodcty ad cotuty at the edge ca be easly peseved afte the tasfomato sce the PSF s cofed to the cete of the mage [5, [6. A. Taslato Smulato Taslato s cetaly the most smple tssue moto. I the patcula case of a lea scae, oe ca easly show that fo a mage plae taslato of the tssue, a detcal taslato of the speckle patte wll occu. I fact, fo a tssue taslato (a. b. ), the tssue compoet becomes T( -.y - b.2) ad thus fom ts mage plae pojecto N( x -a.!/ - b), oe obtas the obseved RF mage taslato fom (3) I(s - a,!/ - 6) = H(..?/) C% N(. - a.!/ - b). (7) Note that ths taslato behavo s at the bass of the speckle tackg algothm developed by Tahey et al. [6], [9]. The pot of teest hee s that, fo a lea scae, the taslato of the tssue fes a detcal taslato of the esultg speckle patte but t s teestg to otce that a secto sca magg ths would ot be the case. Hee, the PSF wll ot be soplaetc a Catesa coodate system, but athe pod coodate (. 9). Theefoe, a (.I-. y) taslato of a small ego of tssue would also volve a otato tasfomato wth espect to the PSF ad, as show below, ths would duce speckle decoelato. B. Rotato Smulato Fg. shows smulated echogaphc textues followg a -, 2-, ad -degee tssue otato aoud the mage cete. By

4 296 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 4, NO. 2, JUNE 995 C.8 e.6 a t.4.2 I 5 otato (degees) (a) I 5 otato (degees) (b) C.8 e.6 a t otato (degees) (C) Fg. 2. Coelato betwee the tssue otato ad the speckle moto. Smulatos fo a 5-MHz (a) ad 3-MHz (b) tasduce wth.5 mmx. mm (FWHM) PSF shape ad a 5-MHz tasduce wth a. mm x.5 mm PSF (c) ae show. The eo bas epeset two stadad devatos fo 2 smulatos. The sold cuves ae obtaed theoetcally wth (25) ad (9) (Secto IV). Note the lage eo bas the last fgue (wth a lage PSF) due to the ucetaty toduced by lage speckles the mages, ad theefoe a smalle umbe of them to estmate the coelato. obsevg caefully these mages (usg a amato pocedue, a maual assessmet, o a appopate algothm to tack the speckles [3]-[9], [3]), oe ca measue the udelyg tssue otato fo small tefame moto. Fg. l(d) shows the lea tasfomato that best fts the optcal flow (velocty feld) [3]-[5], [7], [3] computed fom the two mages Fg. l(a) ad (b). As oe ca obseve, thee s a clea otato compoet fo ths 2-degee speckle moto. Howeve, t becomes moe ad moe dffcult to deteme the tssue moto fom the speckle moto as the tefame moto gets lage. Oe ca see, fo example, that the speckle pattem Fg. l(a) s ecogzable (b) (2-degee otato) but ot (c) (-degee otato), theefoe s ot tackable fom (a) to (c). It s theefoe vtually mpossble to deteme the -degee otato that occus betwee mages (a) ad (c) Fg., as f the speckle pattes had bee "coupted" by a moto-duced ose. I ths pape, the wod decoelato [28], [3] s used to efe to ths pheomeo because t s moe dffcult to coelate the obseved textue moto ad the udelyg tssue moto.

5 MEUNIER AND BERTRAND. ULTRASONIC TEXTURE MOTION ANALYSIS 29 IV. RESULT ANALYSIS To expla these smulated esults, a theoetcal study of the coelato betwee these vaous tssue lea tasfomatos (LT) ad the coespodg ultasoc speckle motos s pefomed based o a 2-D exteso of the evelope cosscoelato aalyss of a aow-bad Gaussa ose [ 6, [7]. Ideed, the RF mages geeated wth the model ae aow-bad Gaussa ose sce they ae poduced wth a 2- D aow-bad PSF ad a Gaussa ose tssue. Wth the symmetcal PSF used hee, oe ca elate the coelato p~ to the ease-to-compute RF coelato p.f. wth a sees expaso of a hypegeometc fucto [5, [7], [8] Fg. 3. Smulato of speckle patte fo a tssue defomato of (a) % (o = l), (b) thckeg of % (a =.) wth a coespodg cotacto obeyg o 9 =, ad (c) c) =.3. (d) Lea tasfomato computed fom (a) to (h) usg a optcal flow algothm. The optcal flou uqes a lea velocty feld costat ove the whole 5.2 mm x j.2 mm mage. The og s at the cete of the mage ad the.( ad y axes I-u postve fom top to bottom ad left to ght, espectvely. Note the speckle patte decoelato whe the tefame moto s lage ((a) vesus (c)). I ode to quatfy ths pocess, we compute a measue of coelato pt that establshes the coespodece betwee the tssue &otato ad the ultasoc textue moto. Ths measue ca be defed as the coelato betwee mages ID (.. y ) ad : (.. y), espectvely the -otated efeece mage ad the mage poduced wth a 6 tssue otato Fg. 2(a) shows the esults obtaed fo the PSF descbed Table I. Twety depedet smulatos fo each agle of otato have bee executed to obta the cofdece teval Fg. 2. As oted above, a global coelato decease wth otato s obtaed. Futhemoe, ths decease s educed wth a lowe fequecy (Fg. 2(b)) ad amplfed wth a wde PSF (Fg. 2(c)) as wll be dscussed moe detals Secto IV. C. Defomato Smulato Fg. 3 llustates the esults of axal expasos of,, ad 3 pecet (a =..., ad.3, espectvely) wth the coespodg lateal cotacto (p wth costat o[l = ). Smlaly to otato, by caefully obsevg these mages, oe ca measue the udelyg tssue defomato fo small tefame moto, but as the tefame defomato ceases, the task s moe dffcult due to decoelato. Fg. 3(d) shows the optcal flow computed fom the two mages (a) ad (b). Aga, oe ca compute the coelato betwee tssue defomato ad speckle moto fom the measue of coelato p$ betwee mages IB(.. y ) ad Ig(.. y). Fg. 4 shows the esults: a lage decease the coelato wth hghe fequeces ad/o lage PSF. whee p~ ca ow be smply calculated fom RF mages stead of B-mode mages (8). A. Theoetcal Aalyss I ths subsecto, we wll develop the p.f. tem stead of p~ to smplfy the calculato wthout loss of geealty kowg the elatoshp betwee the two fom (9) LT Covaace[I(.:. y ). I ~~(:x. y)] P.f. = Jvaace [I(.. y ace [ILT (.. y )] (2) whee I[d. y ) ad ILT(x. y) epeset, espectvely, the lea tasfomato (otato o defomato) of the efeece mage ad the mage poduced wth a lea tasfomato of the tssue. Sce the mea value (DC value) of RF mages s zeo, oe obtas JJ I[.d. <y )ILT(,. g)d dy. (2) p% = J-JJJ lilt(.. y)2d. dy Usg the Paseval detty, oe ca easly tasfom ths equato the fequecy doma wth *, the complex cojugate, ad ( d. 7) ) the lea tasfomato the fequecy doma poduced by the spatal doma lea tasfomato (. y ). Usg (3) the Foue space, we ow show that the RF mage coelato coeffcet p::, educes to the coelato coeffcet of the coespodg PSF, that s, H(:. y) ad ts lea tasfomato H(:c. y ), as show (23) at the bottom of the ext page. ln(u.,tl )I* s a whte Gaussa ose powe spectum equal to the N(:. y) vaace ad elated to the costat cell desty [9]; thus, oe obtas LT JJ H(v. V )H*( U. 7l)fl7l d ) P.f. =. (24) IH(u.! ~ ) 2 d u d t J ~ J,[f A stuctve mathematcal model of H(u. U) ca thus be obtaed usg a 2-D Gaussa ceteed o the PSF spatal fequecy f wth stadad devato U?, ad o,, vesely popotoal to the PSF spatal oes, c,~ ad uy, espectvely; these beg elated to the axal ad lateal speckle sze. Fo otato ad defomato (assumg a aow-bad PSF ad

6 298 IEEE TRANSACTIONS ON MEDICAL. IMAGING, VOL. 4, NO. 2, JUNE e.6 :.4 O.2.8 e.6 :.4 O axal defomato (a) (a)..2.3 axal defomato (a) (b) C e a t axal defomato (a) (C) Fg. 4. Coelato betwee the tssue defomato ad the speckle moto, The smulated PSF paametes ae the same as fo Fg. 2. The sold cuves ae obtaed theoetcally wth (26) ad (9) (Secto IV). Note, aga, the lage eo bas the last fgue (wth a lage PSF) due to the ucetaty toduced by lage speckles. small otato ad defomato), e e-a(&) (sze) P.f. (25) PEf. These esults ca be exteded to B-sca mages usg (9) ad agee wth the smulated oe descbed above (sold cuves Fgs. 2 ad 4). Note that, wth the famewok of ths model, otato ad defomato coelatos ae sestve to 2J;YP I 2- e- 5 (&,2 +, (26) the lateal ad axal badwdth, espectvely (a, ad a,) ad the PSF fequecy f whch appea the expoetal tem. J(a2 + l)(p + )

7 MEUNIER AND BERTRAND: ULTRASONIC TEXTURE MOTION ANALYSIS 299 The lateal defomato does ot appea the expoetal tem; theefoe,,o:~, s much less affected by lateal defomato tha by axal defomato. The obseved,o:,~, sestvty to otato s also explaed by the s2 6 tem the expoetal. Remembe that a weak coelato meas that speckle tackg of the tssue moto s moe dffcult. Ths dcates that, wth the famewok of ou model, addto to the obvous small tefame moto, a pulse wth a low fequecylbadwdth s desable fo a speckle tackg methodology. FOURIER FOURIER DOMAIN (&O) B. Coelato wth Lateal Taslato (Secto Sca Imagg) I [9], Tahey et al. vestgated coelato wth taslato usg a secto-sca system. I ou pape, we vestgate a soplaatc PSF whch s vaat wth espect to taslato. I ode to exted ou esult to a secto sca stuato ad epoduce ts speckle moto fo a lateal taslato t,, we eed to taslate the ego of teest (ROI) lateally ad the otate t. Fo small taslato, the otato agle s s-l(t,/d) E t,/d, whee d s the tasduce to ROI dstace. Gve ths, (25) ca be ewtte tems of lateal taslato (fo a secto sca), athe tha otato agle. It s developed as follows: 6 % e-a(&)2(s2 ) P.f. %e-4(j(% L *. (27) Ths last equato ca be futhe smplfed fo a ectagula tasduce usg a dffacto fomula. I the fa feld, o fo that matte eve focus, the beam-wdth s popotoal to /( fd), whee D s the tasduce lateal dmeso; theefoe, the lateal badwdth CT~ s popotoal to the ( f D) poduct. Substtutg ths the above equato, oe fds That s to say, fo a gve (ectagula) tasduce-roi dstace d, the coelato s oly a fucto of the taslato expessed as a facto of tasduce wdth. Ths s deed oe of the coclusos of [9]. C. Gaphcal Aalyss A vey teestg qualtatve sght to the computato of p.f. s obtaed usg a gaphcal epesetato of (3), a appoach somewhat smla to [2]. I Fg. 5, the RF mage obtaed fom the lea tasfomato (LT) of the tssue (otato ad/o defomato), ILT(x, y), ad the lea tasfomato of the efeece RF speckle pattem, I(d, y ), ae show as ovelappg egos the fequecy doma. Oe ccula aea epesets the 2-D spectum of ILT(x,y) ad esults fom a spectal samplg of the lea tasfomed tssue by the umodfed PSF ad s theefoe ceteed o the PSF cetal fequecy f, wth a adus coespodg to ts badwdth A f. The 2-D spectum of I(d, y ) s the ccula (PSF-sampled) aea modfed by the lea tasfomato that affects both the tssue ad the PSF sce t s ow the speckle pattem that s subjected to lea tasfomato. The elatve sze of the ovelappg ego s a good dcato U Fg. 5. Gaphcal tepetato of the coelato betwee speckle ad tssue moto fo (left) otato ad (ght) defomato. ILT ad I sketch the Foue tasfom of ILT (2, y) ad I( x, y ), espectvely. U ad U ae the fequecy doma axes coespodg to the spatal doma axes x ad y,> f ad Af ae the PSF cete fequecy ad 2-D badwdth, espectvely. Remembe that, ude LT, expaso spatal doma becomes compesso the fequecy doma. of the coelato p.f. fo a gve moto sce the spatal coelato s elated to the poduct of oe spectum by the cojugate of the othe. The left sketch Fg. 5 epesets otato ad the ght oe axaylatea defomato. Oe ca ow easly see the deceasg coelato descbed befoe as a coespodg elatve decease the ovelappg ego ad udestad how the fequecylbadwdth ( f /A f ) ato s a ctcal facto fo the decoelato duced by otato ad axal defomato. Fo example, doublg the pulse fequecy o halvg the badwdth Fg. 5 cleas out ay ovelap. A moe caeful vestgato eveals that lateal defomato (@-elated) s much less sestve to decoelato tha the axal defomato (a-elated). Wth espect to the tasduce fequecy, the otato ad the axal defomato of the tssue show a bette elatoshp wth the espectve speckle moto at lowe fequeces whle lateal defomato coelato s depedet of the pulse fequecy. Wth espect to beam (pulse) sze paametes, tssue/speckle coelato deceases wth otato whe a wde ultasoc beam s used whle the axal defomato coelato deceases wth the axal duato of the pulse.3 These gaphcal tepetatos ae pefect ageemet wth the theoetcal oes. V. CONCLUSION Ths study sets the goud fo the developmet of a tool patculaly useful fo the assessmet of bomechacal soft tssue popetes based o speckle tackg. Fom smulatos ad a theoetcal aalyss, the fudametal lmtatos of speckle tackg to assess soft tssue moto wee peseted. I patcula, we have vestgated a model to study coelato betwee speckle patte moto ad tssue moto whe a tssue s subjected to a lea geometcal tasfomato (taslato, otato, ad defomato). Ideed, these theoetcal esults wee successfully appled to the detemato of myocadal ad skeletal muscle dyamcs ad wee also expemetally valdated fom a phatom study eale eseach [2]-[9], [3]. May othe goups [2]-[26] ae wokg elated Remembe that, ude LT, expaso spatal doma becomes compesso the fequecy doma. U

8 3 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 4, NO. 2, JUNE 995 felds ad the appoach s cetaly pomsg. Moe complex (cuved) PSF wee vestgated ecetly [27], [28] to study speckle moto atfacts [29] that must be emoved ode to eveal the tssue moto. These atfacts ae due to phase dstoto effects the badwdth of the PSF. Tssue fuctos T(z, y, z) wth dffeet desty dstbutos N(z, y) usg dffeet mxtues of small ad lage scattees havg the ow desty dstbutos would also be a teestg aea of eseach to smulate, fo example, omal ad schemc tssue, ad to evaluate the sgfcace of specula ad esolved scattee compoets tssue moto estmato [26]. Fally, moe complex 3-D tasfomatos ae ude vestgato to take to accout moe ealstc olea ad out of plae (mage) motos. ACKNOWLEDGMENT The authos wsh to thak J. Raymod fo evewg ths mauscpt. REFERENCES [l] J. F. Geeleaf, Tssue Chaactezato wth Ultasoud. Boca Rato, FL: CRC, 986. [2] J. Meue, M. Betad, ad G. Malloux, A model fo dyamc textue aalyss two-dmesoal echocadogams of the myocadum, SPIE Patte Recogto Acoustcal Imag. Pmc., vol. 768, pp. 93-2, 987. [3] J. Meue, M. Betad, G. Malloux, ad R. Pettclec, Local myocadal defomato computed fom speckle moto, IEEE Comput. Cadol., pp , 988. [4] -, Assessg local myocadal defomato fom speckle tackg echogaphy, SPIE Med. Imag., vol. 94, pp. 2-29, 988. [5] J. Meue, Aalyse dyamque des textues dkchogaphes bdmesoelles du myocade, Ph.D. dssetato, hole Polytechque de Motkal, 989. [6] G. E. Tahey, J. W. Allso, ad O.T. vo Ramm, Agle depedet ultasoc detecto of blood flow, IEEE Tas. Bomed. Eg., vol. BME-34, o. 2, pp , 987. [7] M. Betad, J. Meue, M. Doucet, ad G. Malloux, Ultasoc bomechacal sta gauge based o speckle tackg, IEEE 989 Ultaso. Sympos., 989. [8] E. J. Che, I. A. He, J. B. Fowlkes, R. S. Adle, P. L. Caso, ad W. D. O Be, J., A compaso of the moto tackg of 2-D ultasoc B-mode tssue mages wth a calbated phatom, IEEE Ultaso. Sympos., vol. 2, pp. 2-24, 99. [9] G. E. Tahey, S. W. Smth, ad T. vo Ramm, Speckle patte coelato wth lateal apetue taslato: Expemetal esults ad mplcatos fo spatal compoudg, IEEE Tas. Ultaso. Feoelec. Fequecy Cot., vol. UFFC-33, o. 3, pp , 986. [IO] R. F. Wage, M. F. Isaa, ad S. W. Smth, Fudametal coelato legths of coheet speckle medcal ultasoc mages, IEEE Tas. Ulaso. Feoelec. Fequecy Cot., vol. 35, pp , 988. [I ) J. C. Bambe ad R. J. Dckso, Ultasoc B-scag: A compute smulato, Phys. Med. Bol., vol. 25, pp , 98. [I2 D. A. Segge, S. Leema, ad R. E. Buge, Realstc smulato of B-sca mages, IEEE Ultaso. Sympos., pp , 983. [I3 M. F. Isaa ad D. G. Bow, Acoustc scatteg theoy appled to soft bologcal tssues, Ultasoc Scatteg Bologcal Tssue, K. K. Shug ad G. A. Theme, Eds. Boca Rato, FL: CRC, 993, ch. 4, pp [4] M. S. Rode, Aalog ad Dgtal Commucato Systems, 3d ed. Eglewood Clffs, NJ: Petce-Hall, 99, pp [I5 R. 3. Bask ad P. J. Paol, Muscle volume chages, J. Geeal Physol., vol. 49, pp , 966. [ 6 J. Meue, Ultasoc textue moto aalyss: Theoy ad smulato, SPIE Med. Imag., vol. 896, 993. [I7 R. Pce, A ote o the evelope ad phased-modulated compoets of aow-bad Gaussa ose, IRE Tas. Ifom. Theoy, vol. IT-I, pp. 9-3, 955. [ 8 D. Mddleto, A Itoducto to Statstcal Commucato Theoy. New Yok: McGaw-Hll, 96, ch. 9. [ 9 A. Papouls, Pobablty, Radom Vaables, ad Stochastc Pocesses, 3d ed. New Yok: McGaw-Hll, 99. [2] J. F. Geeleaf, A gaphcal descpto of scatteg, Ultasoud Med. Bol., vol. 2, pp , 986. [2] P. G. M. De Jog, T. Ats, A. P. G. Hoeks, ad R. S. Reema, Detemato of tssue moto velocty by coelato tepolato of pulsed ultasoc echo sgals, Ultaso. Imug., vol. 2, pp , 99. [22] J. Oph, I. Cspedes, H. Poekat, Y. Yazd, ad X. L, Elastogaphy: A quattatve method fo magg the elastcty of bologcal tssues, Ultaso. Imag., vol. 3, pp. -34, 99. [23] I. Akyama, N. Nakajma, ad S. Yuta, Movemet aalyss usg B-mode mages, Acoustcal Imag., vol. 7, pp , 989. [24] R. Shehada, R. S. C. Cobbold, ad L. Y. L. MO, Black holes whole blood: Velocty pofles ad shea ates usg coelato, 4th Dexel Uv. Sympos. Ultasoud [mag., 992. [25] S. G. Foste, P. M. Embee, ad W. D. O Be, Flow velocty va tme-doma coelato: Eo aalyss ad compute smulato, IEEE Tas. Ultaso. Feoelec. Fequecy Cot., vol. 37, o. 2, pp , 99. [26] P. M. Embee ad W. D. O Be, Volumetc blood flow va tmedoma coelato: Expemetal vefcato, IEEE Tas. Ultaso. Feoelec. Fequecy Cot., vol. 37, o. 2, pp , 99. [27] F. Kallel, M. Betad, ad J. Meue, Image segmetato fom moto aalyss of speckle patte, Caadu Co$ Elec. Comput. Eg., pp , 99. [28] -, Speckle moto atfact ude tssue otato, IEEE Tas. Ultaso. Feoelec. Fequecy Cot., vol. 4, pp. 5-22, 994. [29] D. C. Moso, W. N. McDcke, ad D. S. A. Smth, A moto atefact eal-tme ultasoud scaes, Ultasoud Med. Bol., vol. 9, o. 2, pp. 2-23, 983. [3] J. Meue ad M. Betad, Tssue chaactezato fom echogaphc speckle moto, Iov. Techol. Bol. Med., vol. 5, pp , 994.