A frequenc decomposton tme doman model of broadband frequenc-dependent absorpton: Model II W. Chen Smula Research Laborator, P. O. Box. 134, 135 Lsaker, Norwa (1 Aprl ) (Proect collaborators: A. Bounam, X. Ca, H. Sverre, A. Tveto, Å. Ødegård) Kewords: frequenc decomposton, bandpass, Gauss flter, broadband ultrasound, frequenc-dependent attenuaton, tme doman. 1. Summar Ths report s the second n seres about tme-doman modellng of broadband frequencdependent attenuaton va a frequenc decomposton approach. The other two reports are concerned wth the modfed mode superposton model [1] and the fractonal dervatve model []. The medcal ultrasound transducers as well as other real-world acoustc sources mostl produce multple-frequences acoustc wave. Ths stud ams to extend the frequencdependent attenuaton tme-doman model of sngle frequenc exctatons [3] to broadband exctatons va fnte frequenc decomposton. The basc dea behnd ths strateg s to dvde a broadband exctaton nto a bank of narrowband exctatons and then regard each narrowband source as a sngle frequenc source. Fnall, the responses of all narrowband sources are summed up as the sstem response of the correspondng broadband source. In essence, the frequenc decomposton s based on the superposton 1
prncple. A frequenc decomposton expermental smulaton [4] has verfed that the methodolog s accurate, relable and effcent. It s expected that the strateg wll also work wth the FEM tme-doman numercal smulaton. Compared wth the prevous mode superposton model [1], the present broadband model also satsfes the prncple of causalt and s as smple to mplement, more computatonall affordable snce t crcumvents computatonall expansve fractonal power of a matrx. The addtonal computng effort of ths model s to the forward and nverse bandpass flter transforms of broadband exctatons, whch s trval among the whole smulaton. However, the computng effort of broadband exctatons s expected about ten tmes hgher than that of a sngle exctaton. Blessng s that the model s nherentl ver sutable for parallel computng. The drect extenson of the present model to nonlnear meda, however, s blocked snce the superposton prncple ceases to hold. A lnearzaton teraton method [5] ma revtalze the model to nonlnear problems, whch we wll nvestgate n the future. The rest of the report s grouped nto the four sectons. In secton, the tme doman model of sngle frequenc exctaton s brefl dscussed for the completeness of ths selfcontaned report, and then, n secton 3, we outlne the frequenc decomposton procedure to model the ultrasound propagaton under a broadband exctaton n tme doman. Secton 4 s dedcated to the bandpass flter whch decomposes a broadband exctaton nto a bank of narrowband exctatons. Fnall, secton 5 analzes the dsperson propertes of the present broadband tme doman model.. Tme-doman model for sngle frequenc exctaton The frequenc-power attenuaton s gven b p α ( f ) x ( x + x) = p( x) e, (1)
where p represents the ampltude pressure, and f s the frequenc, ( ) α f α =, [,]. () f Here α s the attenuaton constant and the frequenc-power exponent, both of whch are dependent on tssues. Assumng the vscous absorpton lnearl depends on the veloct, the standard tmedoman damped ultrasound wave model can be expressed as 1 = c & p + γ p& p, (3) where γ s the vscous coeffcent, and upper dot denotes the temporal dervatve. The FEM analogzaton of spatal Laplacan elds the sem-dscretzaton equatons () t & p + c γ p& + c Kp = g, (4) where K s the smmetrc FEM nterpolaton matrx of Laplace operator, and vector g(t) s due to the external exctaton source (transducers on boundar surface of our problems). In the case of a sngular frequenc exctaton g(t), [1] derved () t & p + α cf p& + c Kp = g (5) correspondng to the emprcal frequenc-dependent absorpton (1). In the precedng superposton model report [1], t s shown that (5) satsfes the causalt prncple. 3
3. Frequenc decomposton tme doman scheme If the external source s a broadband exctaton s(t) nstead of a sngle frequenc g(t) n (4), the models (4) and (5) can not hold wth the frequenc-dependent attenuaton () except when =. [3] suggested usng the central frequenc of s(t) as a sngle frequenc n determnng the vscous coeffcent, and then the formulaton (5) s appled. In some cases, ths procedure ma ntroduce unpredctable sgnfcant model error. Followng the frequenc decomposton dea presented n [4], a more accurate alternatve scheme should be to decompose the broadband exctaton s(t) nto a sum of narrowband exctatons s (t),.e. s m () t = = 1 s ( t). (6) When the bandwdth of each component s narrow enough, the attenuaton effect can be smulated at the center frequenc of each narrowband component [4]. Then, the model (5) for sngle frequenc exctaton s appled to each narrowband component source s (t) wth ther respectve central frequenc, and we get m sets of sem-dscrete FEM sstem equaton () t & p + α cf p& + c Kp s, (7) = whch f s the central frequenc of s (t). Although the FEM dscretzaton for matrx K s onl done once for all, the repeated soluton of m sets of ordnar dfferental equaton sstems (7) s qute tme-consumng gven that each (7) s a sem-dscrete FEM equaton of large ODEs. The parallel soluton of all temporal ordnar equaton sets (7) could be crucal to mprove the effcenc. In terms of the superposton prncple under the lnear sstem, the response of a broadband exctaton s(t) s a sum of those of all of ts narrowband component exctatons 4
m () t = p = 1 p ( t). (8) (6), (7) and (8) outlne the basc methodolog of the present scheme. Fg. 1 llustrates the schematc soluton procedure va the frequenc decomposton of a broadband pulse. B 1 (f) s 1 (t) p 1 (t) s(t) B (f) s (t) p (t) p(t) B m (f) s m (t) p m (t) Fg. 1. Schematc procedure of frequenc decomposton tme doman model The maor sources of error of the present mathematcal model are 1) the central frequenc of narrowband component source n (7) s approxmatel used as n the sngle frequenc case; ) the formulaton (5) for sngle frequenc exctaton tself s also an approxmate analog of the dampng effect whch gnores the transent soluton. Snce there s no need to calculate the fractonal power of matrx K, the present model s computatonall more effcent compared wth the mode superposton model gven n [1]. However, the present model also faces a new computng challenge: the repeated (about ten tmes) soluton of ODEs (7) for all narrowband responses. 5
4. Bandpass flters for frequenc decomposton Ths secton addresses the bandpass flter ssue, a ke part of the present broadband tme doman model. The flter s to perform frequenc-dependent alteratons of a sgnal,.e. flterng. As shown n Fg. 1, a set of the bandpass flter B (f) dvde the broadband s(t) nto a sum of narrowband s (t). One of the smplest, fastest and convenent approaches to alter the frequenc propertes of a sgnal b flterng s to appl the lnear frequencdoman flter ( f ) B ( f ) S( f ) S =, (9) where f denotes the frequenc parameter, B s the transfer functon as shown n Fg. 1, S the Fourer transform of the nput broadband sgnal s(t), S the Fourer transform of the fltered narrowband sgnal s (t). (9) can be equall expressed as a convoluton operaton s () t b ( τ )( s t τ d )τ =, (1) where b s the nverse Fourer transform of the frequenc doman flter B. Flterng n frequenc doman gves superor performance compared wth the other flters desgn technques. Its man drawback s to requre complete sgnal accessble. Ths s not an ssue n our medcal ultrasound case. Some tpcal bandpass flters are llustrated and brefl commented n Table 1 [6]. 6
Table 1. Tpcal bandpass flters Box flter Ths flter s an deal frame aperture. A frame usng ths flter dstrbutes the tme samples regularl over the specfed amount of tme. Gauss flter The gauss flter smulates the behavor of a tube camera. The tme behavor looks lke a bell where the samples are concentrated n the mddle of the frame. Shutter flter The shutter flter smulates the shutter of a mrror reflex camera. In the frst quarter of the frame the shutter opens. Then the shutter remans open for the half tme and then the shutter closes. Under Blzzard II the proporton s fxed to 1::1. He [4] recommended the use of the Gauss flter to mnmze the reconstructon error. Ths s owng to that f the sstem transfer functon s Gaussan, the Gaussan soluton for an mpulse nput s an excellent approxmaton of the exact causal soluton for most applcaton problems [7]. Each bandpass flter used n [4] has a Gaussan magntude functon B ( f ) = ( 1) f fl B 1 B e π, =1,,,n, (11) 7
where f H f L B =, (1) m 1 f L and f H respectvel denote the lower and upper lmts frequences. Snce the energ at hgher frequenc components are dsspated much more rapdl that those of lower frequences, t ma be more effcent to dvde the frequenc n a non-unform fashon unlke (1). We wll explore ths ssue n the later research. [4] ponted out that the dvson number m s not necessar bg snce the experments show that m=1 or so produces ver accurate reconstructons. As an example, the energ spectrum of a broadband source s llustrated n Fg., where W f s the bandwdth and f C central frequenc. E W f f L f C f H f Fg.. A broadband pulse 8
5. Dsperson analss Wth the Duhamel ntegral and when α f <ω, where ω are the egenvalues of matrx K n (7), we can have the soluton of each modal equaton (7) p m 1 t α cf ( t ) ) cf t ) ) τ α () t = ) s () τ e snc ( t τ ) dτ + e { α sn ct + β cos ct} c (13) where α and β are calculated usng the ntal condtons, and the dstorted phase veloct c ) dependent on the attenuaton s calculated b ) c = c ω α f. (14) (14) shows that the vscous effect causes the dsperson. When α f >ω, the soluton of (7) s p m t () t = s ( ) h( t τ, f, ω ) αcf t ct αω f αcf t+ cωt αω f τ dτ + α e + β e, (15) and under crtcall damped (α f =ω ), the soluton s p m t α cf t () t = s ( τ )( r t τ f, ω ) dτ + e ( α +, β t), (16) where h and r are the mpulse response functon of sstems. It s well known that the ultrasound wave s heavl attenuated whle travelng through soft tssues. The transent solutons, whch are the second rght terms of (13), (15) and (16), are rapdl evanescent and the frequenc spectrum of the propagatng ultrasound wave s close to that of ts 9
transducer exctaton source. Thus, t s reasonable to splt the broadband ultrasound wave n terms of the exctaton frequenc spectrum. References: 1. W. Chen, A drect tme-doman FEM modelng of broadband frequenc-dependent absorpton wth the presence of matrx fractonal power: Model I, Smula Research Report, Aprl,.. W. Chen, Spatal fractonal dervatve modelng of broadband frequenc-dependent absorpton: Model III, Smula Research Report, Aprl,. 3. W. Chen, A smple approach mbeddng frequenc-dependent attenuaton law nto tme-doman lnear acoustc wave equaton, Ultrasound proect report, Smula Research Lab, Nov. 1. 4. P. He, Smulaton of ultrasound pulse propagaton n loss meda obeng a frequenc power law, IEEE Trans. Ultra. Ferro. Freq. Contr., 45(1), 1998. 5. G. Wock, J. Mould, Jr., F. Lzz, N. Abboud, M. Ostromoglsk, and D. Vaughan, Nonlnear modellng of therapeutc ultrasound, 1995 IEEE Ultrasoncs Smposum Proceedngs, 1617-16, 1995. 6. http://ls7-www.cs.un-dortmund.de/~mork/b_dstr.html. 7. T. L. Szabo, Tme doman wave equatons for loss meda obeng a frequenc power law, J. Acoust. Soc. Amer. 96(1), 491-5, 1994. 1