ULTRASONIC IMAGING 27, 181-198 (2005) Spatial Angular Compounding for Elastography without the Incompressibility Assumption MIN RAO AND TOMY VARGHESE Department of Medical Physics The University of Wisconsin-Madison 1300 University Avenue, 1530 MSC Madison, WI 53706 tvarghese@wisc.edu Spa tial-an gu lar com pound ing is a new tech nique that en ables the re duc tion of noise ar ti facts in ul - tra sound elastography. Pre vi ous re sults us ing spa tial an gu lar com pound ing, how ever, were based on the use of the tis sue incompressibility as sump tion. Com pounded elastograms were ob tained from a spa - tially-weighted av er age of lo cal strain es ti mated from radio fre quen cy echo sig nals ac quired at dif fer ent insonification an gles. In this pa per, we pres ent a new method for re duc ing the noise ar ti facts in the ax ial strain elastogram uti liz ing a least-squares ap proach on the an gu lar dis place ment es ti mates that does not use the incompressibility as sump tion. This method pro duces ax ial strain elastograms with higher im age qual ity, com pared to noncom pounded ax ial strain elastograms, and is re ferred to as the least-squares an - gu lar-com pound ing ap proach for elastography. To dis tin guish be tween these two an gu lar com pound ing meth ods, the spa tial-an gu lar com pound ing with an gu lar weight ing based on the tis sue incompressibility as sump tion is re ferred to as weighted com pound ing. In this pa per, we com pare the performance of the two an gu lar-com pound ing tech niques for elastography us ing beam steer ing on a lin ear-ar ray trans ducer. Quantitative experimental results demonstrate that least-squares compounding provides comparable but smaller im prove ments in both the elastographic sig nal-to-noise ra tio and the con trast-to-noise ra tio, as com pared to the weighted-com pound ing method. Ul tra sound sim u la tion re sults sug gest that the least-squares com pound ing method performs better and pro vide ac cu rate and ro bust re sults when com - pared to the weighted com pound ing method, in the case where the incompressibility as sump tion does not hold. Key words: An gled beams; an gu lar com pound ing; com pound ing; elasticity; elastogram; elastography; imaging; strain; signal-to-noise; ultrasound. IN TRO DUC TION Elastography is an im ag ing mo dal ity that is ca pa ble of map ping lo cal in ter nal strains that a tissue experiences after a quasistatic compression. 1-14 In this tech nique, lo cal dis place ments are typically computed by applying a cross-correlation analysis to the pre- and postcompression ul tra sonic radio fre quen cy (rf) echo sig nals. Strains are then es ti mated as the gra di ent of the dis place ments along the beam axis and dis played as a lo cal strain im age, re ferred to as an elastogram. Many al go rithms have been de vel oped to im prove the im age qual ity of elastograms, such as tem po ral stretch ing, 15-17 multicompression averaging, 18 wave let denoising 19 20, 21 and other com pound ing ap proaches. Tem po ral stretch ing of the postcompression sig nal by the ap pro pri ate fac tor can com pen sate for ax ial decorrelation ef fects. 15-17 Multi compression av er ag ing re duces decorrelation by av er ag ing elastograms gen er ated us ing suc ces sive small 181 0161-7346/06 $18.00 Copy right 2005/2006 by Dynamedia, Inc. All rights of re pro duc tion in any form re served.
2 RAO AND VARGHESE com pres sions. 18 Wave let denoising smoothes the dis place ment es ti mates in the wave let do - main with out los ing edge in for ma tion. 19 Most of these al go rithms can be used in con junc tion with each other for ar ti fact re duc tion in elastography. In this pa per, we com pare two spa tial an gu lar com pound ing tech niques used to re duce noise ar ti facts in ax ial strain im ages: spa tial-an gu lar com pound ing us ing an gu lar weight - ing, 22-24 re ferred to as weighted com pound ing, and com pound ing based on a least-squares ap - proach pre vi ously used to ob tain the nor mal and shear strain com po nents, 25 re ferred to as least- squares com pound ing. For weighted com pound ing, the com pounded ax ial strain im - age is ob tained by weighted av er ag ing of mul ti ple strain es ti mates around the same re - gion-of-in ter est (ROI) ac quired from dif fer ent insonification an gles. This tech nique uti lizes the same con cept uti lized to re duce speckle noise 26 in con ven tional ul tra sound im ag ing with an gu lar com pound ing of the B-mode sig nals. 27-29 The least-squares com pound ing was pre - vi ously used by Techavipoo et al 25 to es ti mate lat eral and shear strains im ages. How ever, this method also can be used to im prove im age qual ity of es ti mated ax ial strains. In this tech - nique, dis place ments at each spa tial lo ca tion in the com pressed me dium are mea sured along each beam di rec tion us ing time-de lay es ti ma tion tech niques. 30 A lin ear model of the re la - tion ship be tween these di rec tional dis place ments and com po nents of the ac tual dis place - ment vec tor is then con structed. Or thogo nal com po nents of the dis place ment vec tor are then es ti mated us ing a least squares so lu tion. 25 The com pounded ax ial strain im age is then es ti - mated from the gra di ent of the ax ial com po nent of the dis place ment vec tor. The performance of weighted com pound ing and least-squares com pound ing for elasto - graphic im ag ing is com pared us ing phan tom ex per i ments. We pres ent qualititative and quan ti ta tive re sults ob tained us ing a sin gle in clu sion phan tom that dem on strates the im - prove ments in the elastographic sig nal-to-noise ra tio (SNR e ) and con trast-to-noise ra tio (CNR e ), com pared to noncom pounded elastograms. Sim u la tion re sults pre sented in this pa - per also il lus trate the con di tions where the least squares com pound ing method pro vides ac - cu rate re sults when com pared to the weighted com pound ing ap proach. AN GU LAR COM POUND ING AL GO RITHMS FOR ELASTOGRAPHY The two an gu lar com pound ing al go rithms for elastography that are com pared in this pa per are de scribed in the fol low ing sec tion. The weighted com pound ing al go rithm that uses the incompressibility as sump tion can be de scribed in four steps: 23 (1) Af ter data ac qui si tion, each of the an gu lar rf pre- and postcom pres sion frame pairs were analyzed separately to determine angular tissue displacements at the specified beam an gle. These are re ferred to as an gu lar dis place ment frames. (2) The gra di ent of the an gu lar dis place ments was ob tained us ing a least-squares strain es ti ma tor to gen er ate lo cal strain es ti mates or an gu lar elastograms at each beam an gle. (3) Lin ear in ter po la tion was then ap plied for im age reg is tra tion of the an gu lar elasto - grams onto the elastogram ob tained for the zero-an gle grid. (4) Com pound elastograms were ob tained by a weighted av er ag ing of the strain es ti mates 22, 23 over the mul ti ple an gu lar elastograms: n 1 C(, N) wx, ia( i) N in (1) where C(, N) rep re sents a com pound elastogram. N is the num ber of an gu lar elastograms used for com pound ing, which equals 2n + 1, where n is the num ber of an gu lar elastograms in
AN GU LAR COM POUND ING FOR ELASTOGRAPHY WITH OUT INCOMPRESSIBILITY 3 ei ther the pos i tive or neg a tive an gu lar di rec tion. A() and w x (,) are the an gu lar elastogram and the weight ing fac tor for ax ial strains at an gle re spec tively. The weight ing fac tor is given by 22,23 w x 2 2, 1/(cos sin ) (2) where v is the Pois son s ra tio, which is as sumed to be 0.495 for in com press ible soft tis sue. 32 The least-square based spa tial an gu lar com pound ing al go rithm is sum ma rized as fol lows: 25 (1) Af ter data ac qui si tion, each of the an gu lar rf pre- and postcom pres sion frame pairs was analyzed separately to determine tissue displacements at the specified beam angle. These are re ferred to as an gu lar dis place ment frames. (2) Lin ear in ter po la tion was then ap plied for im age reg is tra tion of the an gu lar dis place - ment data to a Car te sian spa tial grid ob tained for the zero-an gle data. (3) The dis place ment vec tor com po nents at each pixel on the zero-an gle grid are then es - ti mated us ing a least-squares ap proach on the an gu lar dis place ment es ti mates at each pixel. (4) The com pounded elastogram is ob tained from the gra di ent of the ax ial com po nent of the dis place ment vec tor us ing the least squares strain es ti ma tor. 31 The relationship between the actual displacement vectors and the measured angular dis - place ment is given by: 25 q Ad n (3) where q 1 q 2 q M. q m cos cos A M. cos 1 1 2 2 m sin sin M. sinm d dz d x n 1 n 2 n M. n m q rep re sents an ob ser va tion of the dis place ment vec tor dat beam an gle i for i = 1,, m, where m is the to tal num ber of beam steer ing an gles; n is the noise in the ob ser va tion at an - gle i. The terms d z and d x de note, re spec tively, the or thogo nal ax ial and lat eral com po nents of the dis place ment vec tor. We can min i mize the squared er ror be tween the mea sure ment q and the lin ear model Ad with re spect to d to es ti mate the value of d. The so lu tion is the leastsquares so lu tion, 33 which is given by: ~ d= % T 1 T d A A A q (4) In this pa per, for both weighted com pound ing and least-squares com pound ing ap - proaches, we con sider the case where the com pound ing of both sym met ri cal pos i tive and neg a tive an gles, as shown in the sum ma tion in dex of Eq (1) is performed. Estimation of SNR e, CNR e and strain con trast The val ues of elastographic SNR e, CNR e, and the strain con trast C 0 were an a lyzed for com - pounded elastograms of the sin gle-in clu sion phan tom. The SNR e is de fined as:
4 RAO AND VARGHESE s SNR e s (5) where s and s are, re spec tively, the mean and stan dard de vi a tion of the strains over pix els in the se lected ROI. The strain con trast C 0 and the elastographic CNR e are de fined as fol lows: 34 C 0 = s 2 /s 1, (6) 2 2( s1 s2 ) CNR e 2 2 ( ) 1 2 (7) where s and 2 de note the mean and vari ance of the strains in the se lected ROI. The sub - scripts 1 and 2 re fer to the re gions in side and out side the in clu sion, re spec tively. In our ex - per i men tal data ac qui si tion, we ac quired eight sets of pre- and postcompression rf data along the same im ag ing plane of the phan tom us ing dif fer ent precompression lev els rang ing from 0 to 4% of the phan tom height, to ob tain sta tis ti cally-in de pend ent re sults for the SNR e, strain con trast and CNR e estimates. Method EXPERIMENTAL RESULTS The per for mance of the two an gu lar-com pound ing tech niques are eval u ated us ing rf data ac quired us ing a TM phan tom. A sin gle-in clu sion TM phan tom of size 9 9 9 cm 3 was used to eval u ate the two com pound ing meth ods. The in clu sion phan tom con tains a 1 cm di am e ter cy lin dri cal in clu sion en cased within a uni form back ground. The in clu sion is three times stiffer than the back ground. The phan tom was scanned us ing an Ultrasonix 500RP (Ultrasonix Med i cal Cor po ra tion, Bothell, WA, USA and Van cou ver, BC, Can ada) realtime scan ner equipped with a 5 MHz lin ear-ar ray trans ducer with an ap prox i mately 60% band width. In or der to ac quire rf data at dif fer ent beam an gles, we de vel oped a cli ent pro - gram, to com mu ni cate with the Ultrasonix Ul tra sound Re search In ter face (URI) and soft - ware server and to con trol the beam steer ing al go rithm. The URI cli ent pro gram en ables the op er a tor to in put the max i mum an gle and the an gu lar in cre ment and the sys tem will au to mat - i cally scan the phan tom along the spec i fied an gu lar sweep. In our ex per i ment, the phan toms were scanned from -15º to 15º with a min i mum an gu lar in cre ment of 0.75º. A sche matic il - lus tra tion of the an gu lar data ac qui si tion is shown in fig ure 1. A com pres sion plate with a rect an gu lar slot for the trans ducer face was mounted on a lin - ear stage driven by a step per mo tor. The com pres sion plate is larger than the phan tom surface, to pro vide a uni form com pres sion of the phan tom as shown in fig ure 1. The com - pres sion was ap plied along the ax ial di rec tion i.e., an gle 0º, the nonsteered case. Echo sig - nals were ac quired over a to tal depth of 4 cm cen tered about the in clu sion im aged, be fore and af ter a com pres sion of 0.5% of the phan tom height. The step per mo tor con trolled com pres - sion pro cess is con trolled by the URI pro gram on the Ultrasonix 500RP sys tem, en abling syn chro nized ac qui si tion of both the pre- and postcom pres sion rf data sets. We im ple - mented an au to mated beam-steer ing and data ac qui si tion al go rithm on the sys tem. The al go - rithm first ac quires the precom pres sion data along the spec i fied an gu lar sweep (-15 º to 15 º in 0.75º in cre ments). The step per mo tor is then ac ti vated to com press the phan tom to a spec i - fied com pres sion in cre ment, fol low ing which the postcom pres sion rf data are ac quired fol - low ing the same an gu lar se quence.
AN GU LAR COM POUND ING FOR ELASTOGRAPHY WITH OUT INCOMPRESSIBILITY 5 z x Compression Direction - 0 + Plate Angular Beams Phantom Plate FIG. 1 A sche matic di a gram de pict ing elastographic im ag ing us ing a lin ear ar ray trans ducer with beam steer ing. The z and x di rec tions are de fined as the ax ial and lat eral di rec tion of the dis place ment vec tor. FIG. 2 An gu lar dis place ments (top) and an gu lar elastograms (bot tom) of the sin gle-in clu sion phan tom. The first, sec ond and third col umns show dis place ment and elastograms ob tained at 0º, 7.5º and 15º insonification an - gles, re spec tively. The grey scale bar for dis place ments im age de notes ax ial dis place ment in mm and the grey scale bar for elastograms de notes strain, where a 1% strain is dis played as 0.01. Re sults Fig ure 2 pres ents a typ i cal ex am ple of the an gu lar dis place ment (a) to (c) and an gu lar elastograms (d) to (f), be fore ap pro pri ate weight ing, ob tained from the sin gle-in clu sion phan tom, for an gles of 0º, 7.5º, and 15º, re spec tively. As ex pected, we ob serve in creased noise ar ti facts in the dis place ment and strain im ages at the larger insonification or beam an - gles. This is due to the fact that an gu lar elastograms ob tained where the beam an gle rel a tive to the di rec tion of com pres sion is large, suf fer from sig nif i cant decorrelation of the pre- and
6 RAO AND VARGHESE FIG. 3 Com pound elastograms ob tained with least-squares com pound ing (a to c) and with weighted com pound - ing (d to f). An an gu lar in cre ment of 0.75º was used over a max i mum an gle of 1.5º (a & d), 7.5º (b & e) and 15º (c & f), respectively. postcom pres sion rf echo sig nals due to the in creased move ment of the scat ter ers. Note also that the an gu lar elastograms be come darker as the insonification an gle in creases, sug gest ing that the strains mea sured at larger an gles are bi ased lower than those mea sured at smaller an - gles. This bias is com pen sated by ap pro pri ate an gu lar weight ing of the an gu lar strain es ti - mates. Fig ure 3 pres ents com pounded elastograms for the sin gle in clu sion phan tom ob tained with the least-squares com pound ing method (top) and weighted com pound ing method (bot - tom). An an gu lar in cre ment of 0.75º and max i mum an gles of 1.5º, 7.5º and 15º were used dur ing com pound ing for the elastograms shown in the first to third col umns, re spec tively. These com pounded elastograms dem on strate the sig nif i cant re duc tion in noise ar ti facts and sub se quent im prove ment in the detectability of the in clu sion. Plots of the SNR e ver sus the max i mum an gle used for com pound ing are shown in fig ure 4. The er ror bars de note the stan dard de vi a tion of the SNR e es ti mates over eight in de pend ent ex per i ments. For each data set, the value of SNR e was calculated using the rectangular region within the in clu sion shown in fig ure 2(d). Ob serve that the im prove ment in the SNR e of the com pounded elastograms ob tained with weighted com pound ing is larger than those ob - tained with least-squares com pound ing. Quantitative variations in the CNR e and strain con trast were cal cu lated us ing the rect an gu - lar ROI within the in clu sion and in the back ground re gion, as shown in fig ure 2(d). The CNR e curves are plot ted ver sus the max i mum an gle used in the com pound ing pro cess, as shown in fig ure 5. The weighted com pound ing method pro vides a higher CNR e in the com - pounded elastograms when com pared to the least-squares com pound ing method. Note also that the CNR e curve for the weighted com pound ing method de creases slightly af ter a max i - mum an gle of around 6º. T his is due to the in creased strain vari ance of elastograms ob tained for the larger beam insonification an gles. The CNR e curve for the least-squares com pound -
AN GU LAR COM POUND ING FOR ELASTOGRAPHY WITH OUT INCOMPRESSIBILITY 7 40 35 SNRe (db) 30 25 20 15-2 0 2 4 6 8 10 12 14 16 Maximum Angle Used ( o ) FIG. 4 Plots of mean SNR e and stan dard de vi a tion shown as the er ror bars over eight in de pend ent re al iza tions of the com pound elastograms vs. max i mum an gle used for com pound ing. 55 50 45 CNRe (db) 40 35 30-2 0 2 4 6 8 10 12 14 16 Maximum Angle Used ( o ) FIG. 5 Plots of mean CNR e with stan dard de vi a tion shown as the er ror bars over eight in de pend ent re al iza tions of the com pound elastograms vs. max i mum an gle used for com pound ing. ing method also sat u rates around a max i mum an gle of 6º, pro vid ing an im prove ment of 18 db over the max i mum an gle from 0 to 6º. The strain con trast plot ted ver sus the max i mum an gle used for com pound ing is shown in fig ure 6. The er ror bars de note the stan dard de vi a tion of the mean con trast ob tained over eight sets of com pounded elastograms. Note that an gu lar com pound ing has lit tle ef fect on the strain con trast.
8 RAO AND VARGHESE Contrast 1.75 1.70 1.65 1.60 1.55 1.50 1.45 1.40-2 0 2 4 6 8 10 12 14 16 Maximum Angle Used ( o ) FIG. 6 Plots of mean strain con trast with stan dard de vi a tion shown as the er ror bars over eight in de pend ent re al - iza tions of the com pounded elastograms vs. max i mum an gle used for com pound ing. Fig ures 7 and 8 show the ef fect of the an gu lar in cre ment on the com pound ing per for mance. The SNR e plot ted ver sus the an gu lar in cre ment is given in fig ure 7, with a max i mum an gle of (a) 6º and (b) 9º used in com pound ing. In our data set, the small est an gu lar in cre ment was 0.75º. There fore, we could use this an gu lar in cre ment value or a mul ti ple of 0.75º. Fig ure 8 pres ents the CNR e curves plot ted against the an gu lar in cre ment with a max i mum an gle of (a) 6º and (b) 9º used in com pound ing. For small an gu lar in cre ments (0.75 to 3º), the SNR e and CNR e is al most con stant with in creases of the an gu lar in cre ment. For an gu lar in cre ment greater than 3º, the SNR e and CNR e de creases faster for weighted com pound ing com pared to least-squares com pound ing. UL TRA SOUND SIM U LA TION RESULTS This sec tion il lus trates con di tions where the least-squares com pound ing method would pro vide more ac cu rate and ro bust es ti ma tions of the com pounded elastograms. Since the weighted com pound ing al go rithm is based on the incompressibility as sump tion where the Pois son s ra tio is 0.495, we will com pare the per for mance of the two com pound ing meth - ods where the un der ly ing ma te rial would have a dif fer ent value of the Pois son s ra tio. For ex am ple lung tis sue has a Pois son s ra tio of 0.3, 36 while car ti lage has a Pois son s ra tio of 0.17. 37 Method A sim u lated sin gle in clu sion phan tom of di men sions 4 4 4 cm 3 was con structed us ing Fi - nite El e ment Anal y sis (FEA) soft ware (ANSYS Inc., Canonsburg, USA), with a Pois son s ra tio of 0.3 as sumed for both the in clu sion and the back ground. The in clu sion was three times stiffer than the back ground. The incompressibility as sump tion is not sat is fied in this
AN GU LAR COM POUND ING FOR ELASTOGRAPHY WITH OUT INCOMPRESSIBILITY 9 (a) 44 40 SNRe (db) 36 32 28 (b) 40 0 1 2 3 4 5 6 7 Angular Increment ( o ) SNRe (db) 36 32 28 0 1 2 3 4 5 6 7 8 9 10 Angular Increment ( o ) FIG. 7 Plots of mean SNR e and stan dard de vi a tion shown as the er ror bars over eight in de pend ent re al iza tions of the com pound elastograms vs. an gu lar in cre ment, with a max i mum an gle of (a) 6º and (b) 9º used for com pound ing. sim u lated phan tom and the abil ity of least-squares an gu lar com pound ing to pro vide ro bust re sults un der these con di tions is eval u ated in this sec tion. Ax ial and lat eral dis place ments gen er ated, us ing FEA of the plane-strain elas tic ity prob - lem de scribed above, were used to dis place the point scat ter ers uti lized in a fre quency-do - main ul tra sound sim u la tion pro gram. 38 A lin ear-ar ray trans ducer was mod eled, which con sist of 0.1 10 mm el e ments with a 0.1 mm cen ter-to-cen ter el e ment sep a ra tion. Each beam line was formed us ing 128 con sec u tive el e ments. The in ci dent pulses were mod eled to be Gaussi an shaped with an 8 MHz cen ter fre quency and a 60% band width. The sim u la tions were per formed as sum ing the sound speed in the phan tom to be con stant at 1,540 m/s and at - ten u a tion to be neg li gi ble. The scat ter ers were ran domly dis trib uted with an av er age con -
10 RAO AND VARGHESE (a) 56 52 CNRe (db) 48 44 (b) 56 0 1 2 3 4 5 6 7 Angular Increment ( o ) CNRe (db) 52 48 44 40 0 1 2 3 4 5 6 7 8 9 10 Angular Increment ( o ) FIG. 8 Plots of mean CNR e and stan dard de vi a tion shown as the er ror bars over eight in de pend ent re al iza tions of the com pound elastograms vs. an gu lar in cre ment, with a max i mum an gle of (a) 6º and (b) 9º used for com pound ing. cen tra tion of 40/mm 3. The sam pling fre quency uti lized was 52 MHz. In our sim u la tion, the phan tom was scanned over an insonification an gle range from -15º to 15 º in 3º in cre ments us ing beam steer ing, which was im ple mented in the ul tra sound sim u la tion pro gram by ap - ply ing ap pro pri ate time de lay to each el e ment in the beam form ing pro ce dure. Re sults Fig ure 9 shows the noncom pounded and com pounded elastograms of the sim u lated in - clu sion phan tom ob tained with least-squares com pound ing and weighted com pound ing re - spec tively. An an gu lar in cre ment of 3º and max i mum an gles of 15º were used dur ing the
AN GU LAR COM POUND ING FOR ELASTOGRAPHY WITH OUT INCOMPRESSIBILITY 11 FIG. 9 Noncompounded elastogram (a) and com pounded elastograms ob tained with least-squares com pound ing (b) and with weighted com pound ing (c) of the sim u lated in clu sion phan tom. An an gu lar in cre ment of 3º was used over a max i mum an gle of 15º. 2.30 2.28 2.26 Contrast 2.24 2.22 2.20 2.18-2 0 2 4 6 8 10 12 14 16 Maximum Angle Used ( o ) FIG. 10 Plots of mean strain con trast with the stan dard de vi a tion shown as the er ror bars over eight in de pend ent sim u la tions of the com pounded elastograms vs. max i mum an gle used for com pound ing. com pound ing. The strain con trast and the CNR e were calculated using the rectangular ROI within the in clu sion and in the back ground re gion, as shown in fig ure 9(a). Plots of the strain con trast and the CNR e ver sus the max i mum an gle used for com pound ing are shown in fig - ures 10 and 11, re spec tively. The er ror bars de note the stan dard de vi a tion of the strain con - trast and the CNR e es ti mates over eight in de pend ent sim u la tions. Note that the least squares com pound ing has lit tle ef fect on the strain con trast but weighted com pound ing leads to an er ro ne ous in crease of the strain con trast due to the use of the incompressibility as sump tion. Al though this in crease in not sig nif i cant at lower max i mum insonification an gles, it sug gests that the weighted com pound ing ap proach may pro vide in cor rect in for ma tion about the strain con trast when the incompressibility as sump tion is not sat is fied. Fig ure 11 shows that the least-square com pound ing pro vides better re sults than weighted com pound ing in this sim u - lated case where the incompressibility as sump tion does not hold. Ob serve that the im prove - ment in the CNR e of the com pounded elastograms ob tained with least-squares com pound ing is sig nif i cantly larger than those ob tained with weighted com pound ing as seen from the in -
12 RAO AND VARGHESE CNRe (db) 60 58 56 54 52 50 48 46 44-2 0 2 4 6 8 10 12 14 16 Maximum Angle Used ( o ) FIG. 11 Plots of mean CNR e with the stan dard de vi a tion shown as the er ror bars over eight in de pend ent sim u la - tions of the com pounded elastograms vs. max i mum an gle used for com pound ing creased sep a ra tion of the er ror bars with the in creased max i mum an gle uti lized for the com - pound ing. DISCUSSION Ex per i men tal re sults dem on strate that weighted com pound ing pro vides an im prove ment in the SNR e and CNR e of around 15 and 17 db, re spec tively, with a max i mum an gle of around 6º, us ing a sin gle-in clu sion phan tom model, when com pared with an elastogram ob tained with out com pound ing. The im prove ment in the SNR e and CNR e ob tained with the leastsquares com pound ing ap proach is around 11 and 15 db, re spec tively, with a max i mum an - gle rang ing from 6 to 7.5º. Both of these com pound ing tech niques do not sig nif i cantly af fect the strain con trast. How ever, the re sults for the sim u lated phan tom with a Pois son ra tio of 0.3, sug gest that the least-squares com pound ing method performs better and pro vides ac cu - rate and ro bust re sults when com pared to the weighted com pound ing method. The dif fer ent re sults ob tained from the ex per i ment and sim u la tion can be ex plained by the as sump tions be hind the two dif fer ent com pound ing meth ods. For the weighted com pound - ing tech nique, tis sue is as sumed to be in com press ible with a Pois son s ra tio of 0.495, as shown in Eqn. (2). The an gu lar elastograms can be av er aged us ing a weight ing fac tor to gen - er ate com pounded elastograms only when the Pois son s ra tio is a con stant over the en tire im - aged ob ject. While in the least-squares com pound ing method, com pounded elastograms are ob tained by es ti mat ing dis place ment vec tors from an gu lar dis place ment data. No as sump - tions are made re gard ing com press ibil ity of the ob ject in this method. Note that the incompressibility as sump tion may not hold in some tis sues, for ex am ple, lung tis sue with a Pois son s ra tio of 0.3 36 and car ti lage with a Pois son s ra tio of 0.17. 37 The sim u la tion re sults dem on strate that the weighted-com pound ing method pro vides er ro ne ous re sults when the incompressibility as sump tion is not sat is fied. In ad di tion, the Pois son s ra tio may not be con stant, such as for poroelastic tis sue when edema is pres ent. 39, 40 Al though weighted com -
AN GU LAR COM POUND ING FOR ELASTOGRAPHY WITH OUT INCOMPRESSIBILITY 13 pound ing pro vides better com pound ing re sults un der cer tain sit u a tion, it can not be ap plied to all types of tis sues to be im aged. It is safer to use the least-squares com pound ing ap proach when the com press ibil ity of the tis sue is un known. Both weighted com pound ing and least-squares com pound ing re quire the ac qui si tion of an gu lar rf frames from dif fer ent beam an gles, which in creases the com pu ta tional time and also re duces the frame rate of the ultrasound sys tem. T he max i mum an gle and an gu lar in - cre ment are two im por tant pa ram e ters that af fect the ef fi ciency of com pound ing. The SNR e and CNR e curves pre sented in fig ures 4 and 5 in di cate sat u ra tion around an insonification an - gle be tween 4º to 6º for both weighted com pound ing and least-squares com pound ing meth - ods. For the weighted-com pound ing method, the CNR e val ues ob tained even de crease slightly af ter a max i mum an gle of 9º, as shown in fig ure 5. For our ex per i men tal setup, the op ti mum value of the max i mum an gle is there fore around 6º be yond which no ap pre cia ble in creases in the SNR e or CNR e are ob tained. This can be ex plained by the sig nif i cant in crease in the decorrelation of the pre- and postcom pres sion rf echo sig nals at larger an gles, as shown in figure 2. An other im por tant fac tor that af fects the ef fi ciency of the com pound ing is the an gu lar in - cre ment used in com pound ing. An op ti mum an gu lar in cre ment whereby fewer an gu lar data are re quired to ob tain sim i lar im prove ments in the SNR e or CNR e can sig nif i cantly im prove the en tire com pound ing pro ce dure. An gu lar in cre ments of 0.75º to 3º pro vide sim i lar im - prove ments in the SNR e and CNR e for a fixed max i mum an gle (see fig ures 7 and 8). This is be cause rf data from a tis sue vol ume viewed with beams sep a rated by a small an gu lar in cre - ment are highly cor re lated. How ever, for a smaller an gu lar in cre ment (0.75º to 3º), the com - pound ing ef fi ciency is low since a larger num ber of an gu lar data are re quired to achieve a given SNR e or CNR e level. For a very large an gu lar in cre ment (greater than 4.5º), on the other hand, the im prove ment in im age qual ity is lower than that ob tained for the smaller an gu lar in cre ment, due to the fewer num ber of an gu lar data avail able for com pound ing. In our ex - per i ment, the op ti mum an gu lar in cre ment lies be tween 3 to 4.5º for both weighted com - pound ing and the least-squares com pound ing meth ods. The use of this op ti mum an gu lar in cre ment could re duce the num ber of an gu lar elastograms that have to be uti lized in the com pound ing pro cess, and hence re duce the com pu ta tion time and im prove the frame rate. CON CLU SION In sum mary, weighted com pound ing pro vides greater im prove ment in both the SNR e and the CNR e, when com pared to least-squares com pound ing. How ever, weighted com pound - ing is based on the incompressibility as sump tion, which makes this method only suit able for tis sues with a Pois son s ra tio of around 0.495. The least-squares com pound ing tech nique can be ap plied to any tis sue with out re gard to whether the incompressibility as sump tion would hold. For both com pound ing meth ods, the max i mum an gle and an gu lar in cre ment used for com pound ing are two im por tant fac tors that af fect the ef fi ciency of the com pound - ing al go rithm. AC KNOWL EDGE MENTS This work is sup ported in part by NIH grants R21 EB003853 and R21-EB002722. The au thors would also like to thank Dr. Laurent Pelissier for the loan of the Ultrasonix 500 RP sys tem used on this re search.
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