Nonlinear Wave Propagation - a fast 3D Simulation Method based on Quasi-linear Approximation of the Second Harmonic Field

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1 3.6. H. Torp: Nonlnear Wave ropagaton page Nonlnear Wave ropagaton - a ast 3D Smulaton Method based on Quas-lnear Appromaton o the Seond Harmon Feld Hans Torp ISB NTNU 3.6. rev Abstrat Nonlnear wave propagaton plas an mportant role n medal ultrasound magng. B utlng the seond harmon wave a maor mprovement n mage qualt s aheved. In order to optme mage qualt detaled knowledge o the seond harmon wave-eld as a unton o transduer geometr ousng and pulse shape s needed. Numeral smulaton o nonlnear wave propagaton or pulsed wave ultrasound s tme onsumng espeall when the aperture s not rular smmetral. B usng the square o the lnear propagatng wave-eld as a volume-soure a good appromaton o the seond harmon wave an be aheved. In our smulaton method we started wth the spatal/tme Fourer transorm o the pulsed lnear wave n the oal plane. B usng the angular spetrum method the ontrbuton rom an arbtrarl plane was ound b bak propagaton rom the oal plane squarng the pressure wave ollowed b orward propagaton o the result to the observaton plane. In the spatal/tme Fourer-doman propagaton s perormed b a omple onental ator whereas the square operaton orresponds to a 3D onvoluton. It turned out that the ntegraton along the ultrasound beam ould be solved analtall. The omputaton n the Fourer-doman was then redued to a trppel-ntegral. The resultng seond harmon wave n the observaton plane was ound b 3Dnverse FFT. A smulaton program has been mplemented n Matlab usng a laptop C wth. GH entum III proessor. Appled to a phased arra transduer wth retangular aperture 3*8mm and transmt requen.7 MH the smulaton tme was less than 5 mnutes. Theor The lnear propagatng pulsed pressure wave rom a transduer wth enter n orgo s desrbed b pt where are the amuth elevaton and propagaton dreton. We wll appl the angular spetrum method to propagate the wave along the -as. For an pont along the ultrasound beam we an appl 3D Fourertransorm n the varables and t: dddt p t t Note that the spatal requen varables and are saled b the speed o sound and have unt [H]. Usng the angular spetrum method the relaton between the wave at two planes at along the -as s gven b We wll use vetor notaton or the three requen varables: 3 ; where

2 3.6. H. Torp: Nonlnear Wave ropagaton page In the quas-lnear appromaton to the seond harmon eld the square o the lnear pressure eld pt ats as a volume soure. The ontrbuton to the seond harmon wave n poston rom the lnear wave n poston s obtaned b squarng the lnear wave and propagate the result rom to. In the Fourer-doman ths orresponds to an autoonvoluton n ; 4 The omplete seond harmon eld s obtaned b ntegratng 4 rom = transduer surae to =. d d d 5 The lnear eld n an pont an be obtaned rom the eld n one spe pont e.g. the oal pont usng equaton 3 d d 6 B hangng the order o ntegraton n 6 the seond harmon wave an be ressed as a trppel ntegral smlar to a autoonvoluton o the undamental wave wth a kernel whh s ndependent o. 3 d d 7 Further development o the kernel : d d 8 The ntegraton over s solved sn d 9 and the kernel takes on ts nal orm sn

3 3.6. H. Torp: Nonlnear Wave ropagaton page 3 Summar o equaton -: The seond harmon eld n observaton plane poston an be derved rom the undamental eld n plane poston. 3 d o sn sn ; When the observaton plane s equal to the oal plane; = the kernel takes on a smpler orm sn Lnear wave propagaton to the oal plane The omputaton tme or numeral evaluaton o equaton depends to a large etent on the spatal bandwdth o the undamental wave n plane poston =. Although an poston along the beam ma be used the poston o the oal plane seems to be the preerred hoe. In the oal plane o a oused transduer the spatal Fourer transorm o the wave equals the aperture unton. Ths appromaton s vald or large -number and desrbes the eld on a spheral surae wth enter n the transduer. A 3 The oal dstane s equal n amuth and elevaton and A s the aperture unton. To appl equaton the undamental eld n a plane surae perpendular to the ultrasound beam must be known. I we are lookng at a lmted openng angle the pulsed wave n the oal plane an be obtaned b a tme-sht = o the wave orrespondng to the propagaton rom the spheral surae down to the oal plane t p t p 4 The temporal Fourer transorm o 4 gves 5 Spatal Fourer transorm o 5 gves a onvoluton n dd Hs Hs 6

4 3.6. H. Torp: Nonlnear Wave ropagaton page 4 Summar o equaton 3 6: The spato-temporal Fourer transorm n the oal plane = o an aperture A transmttng a pulse s gven b Hs 7 Hs Fourer A One lmtaton wth ths method s that the oal dstane s equal n amuth and elevaton. For onventonal ld arras wth ed elevaton ous ths wll not n general be the ase. However or.5 D or D arras the oal dstane wll n most ases be equal unless one want to etend the oal depth b usng derent oal dstanes n and. The stuaton wth derent amuth and elevaton ous an be solved b usng a omple and requen dependent aperture unton. The devaton rom a spheral shape o the transduer surae an be ompensated b a requen and spae dependent phase sht n the aperture unton A A 8 Combnng 7 and 8 gves A 9 Eet o requen dependent attenuaton The propagaton equaton an be moded to nlude eets o requen dependent attenuaton provded the attenuaton n db/m s proportonal to the requen. a a Ths gves the ollowng modaton o equaton a The undamental wave n = wth attenuaton s a Combnaton o 9 and gves a 3

5 3.6. H. Torp: Nonlnear Wave ropagaton page 5 Ths shows that requen dependent attenuaton has the same eet to the seond harmon wave as n the lnear ase. Ths observaton was also made b B. Angelsen n [Ultrasound Imagng h.?] Two-wa transmt-reeve response The transmt-reeve response o the ultrasound magng sstem an be ound b ovoluton along the -as o the transmt and reeve responses h ht h 4 The onvoluton n orresponds to multplaton n the -doman. Sne both transmt and reeve responses are alulated n Fourer-doman the most omputer-eent wa to obtan the - wa response s b D Fourertransorm o Ht Hr ollowed b multplaton. The spae/tme response an then be obtaned b nverse t along the - dmenson. The two-d power prole an also be obtaned dretl b the square sum along the - dmenson. In gure eamples o -wa response or a dnam reeve aperture ed ous n elevaton s shown. Implementaton onsderatons A rst prototpe o a smulaton program ropose has been mplemented n Matlab v6.. The spatal and temporal t etent o the smulated beam s gven b the quantsaton step n and respetvel. For smplt the same quantsaton steps was used also n the alulaton o the trppel-ntegral n. The lnear wave as well as the katet unton equ. are tabulated n 3D arra s. For apertures whh are smmetral n and lke the retangular aperture the ntegral an be evaluated or onl one quadrant n the -plane whh redues the pu-tme to 5%.

6 3.6. H. Torp: Nonlnear Wave ropagaton page 6 Fgure. Fundamental and seond harmon eld or =46 and 8 mm. range: 4 db Dnam

7 3.6. H. Torp: Nonlnear Wave ropagaton page 7 Transduer parameters: =.67e6; B=.e6;%pulse bandwdth transmtted pulse =54; lambda=/;d=8.6e-3;%aperure amuth D=3e-3;%aperure elevaton =6e-3;%oal dstane Smulaton range = 55and mm Smulaton tme: 4 se. III. GH Observatons: The sde lobe suppresson n the seond harmon beamprole seems to be arl lose to the double n db o the undamental wave. In a regon beore the oal plane the ar sdelobes seems to be even lower than ths.

8 3.6. H. Torp: Nonlnear Wave ropagaton page 8 Fgure. Two-wa response. Same T aperture as n g. s also used or reeve wth dnam ousng n amuth dreto. Dnam range 8 db