Numerical simulations of heating patterns and ... - Semantic Scholar

Numerical Simulations of Heating Patterns and Tissue Temperature Response due to High-Intensity Focused Ultrasound Francesco F. Curra, Member, B E E , Pierre D. Mourad, Mem,ber, IEEE, Vera A. Kholthlova, R.obin 0 . Clevcland, a,nd Lawrelice A. Crnm Abstract-The results of t h i s papor show-for a n existing high intensity, focuaod ultrasound (IIIFU) transducer-the i m p o r t a n c e of nonlinear oit'ccts on t h e space/time propcrtics of wave propagat,ion a n d heat generation in porfused liver modols when a blood vossol also might b e present. T h e s e simulations a r e based on t h e nonlinoar parabolic oquation for sound propagation a n d t h e bio-heat oqualion for t e m p e r a t u r e generation. T h e use of high initial pressiirc i n HIFU trunsduccrs i n combination w i t h t h e physical characteristics of biological tissue induces shock formation d u r i n g t h e propagation of a t h c r a p c u t i c ultrasound wave. T h e inducod shock directly affects t h c r s t o a t which heat is absorbed by tissuo a t t h e focus without significant influcncc on t h e m a g n i t u d e and spatial distribution of t h e encrgy being dolivcrerl. W h e n shocks f o r m closo t o t h o focus, nonlinear cnhanccrnent of heating is confinod in a small region a r o u n d ttic fociis and generatos a higher localized t h e r m a l impact on t h e tissuc t h a n t h a t predicted by h o a r thoory. T h o presence of a blood vcsscl changes t h e spatial distribotion of b o t h t h o heating r a t a and t c m p e r n t u r o .

I. INITLODUCTION 111: AHILITY of ultrnsonnd to safely penetrate dccp

T.

into tissue makcs its usc appealing for noninvasive treatment of discasc u r d damagc. By using a fociisetl ultrasonic beam, intense rnecliimical encrgy is dclivered to n sniiill region o l tissiic with little or iio cffccl. on the s i r rounding areas. Here absorption proccssrs raise the tissue tcmpcrature to relativcly high values iind caiise thcrrnal ciingidation and ablation o l cells. In ordcr to predict and ne respoiisc to fomsed ultrasound, il. is iniportant t o takc into consiclerntion the spccilic bchavinr (if soinid wavcs and lieat, transfer as ~vcllas thc part,iculav biokigical structiires contained in tlic rcgion of iiiterest. Whcn ultrasonic transdnccrs arc drivcri to producc high focal intcnsitics in tissue, thc sound wavc is, in many cascs, inflncnccd by iioriliiiear propagation [l]-[(I], Thc appcarm c c of higlier harmonics has an effect on thc amount of cnergy absorbed in tissue, especially in thc focal area.

Uiological struci.nrcs also play an iniportant, role and need to bc taken into account loor adeqiiatc predictions. For cxaruplc, tilood is a poor ahsorher o l iiltrasoiind; il; can conslitutc an excdlent sink of hcat, Srnni snrroiindirig tissue and, bct:;iiisc CIS flow, can convect hest away. In most prcvioiis ruodels, thc cflccl. of blood perfusion on hcat, trirnsfer in tlic hiolmit equation is accounted for by ;lilding a rclaxatiori t,crrri that rcmovcs heat nuifi~rrnly throughont 1.11~tissue 171, [8].T h c ariiouiit of heal rcinovcd is dctcrniincd by averaging t,lie dfcct o l hlood perfusion nver all tlic tissiie; that is, tlic lociilized effect of singlc vossels is ncglcctcil. This "homo~ciiization"ol perfiwion is prohal-ily approprink for tissue thtit consists of capillary beds; howcvc N,,,, the sccancl term in (B), arc approximated in (3) and (5) by t,ho a,s,ymptotic values currcsponding to the high-frcqucncy spectrum uf tlic shock with l,lic arnplitndc A, wid pliasc r.?.Tlic values of A, arid rs are calculated at cvcry step in propiigntion distnncc z Sroin t,lii: amplitiidos of tlie h s t t,wo harmonics . ~ ,and C N , , , ~ assiiming ~, that they alresdy belong shock wavc asyrriptolc. An operator-splitting kchniqnc is used for tiiimcriciil intcgration of (3), in mhich thc nonlinear tcrm is solvcd by wing a fourth-urdcr R.ungeKutta method, the nhsorption is calciilatcd using an exact, analytical soliit,iun, and tlic diflractiun is solved iisiiig a mix of irnplicit backward and Criink-Nicolsnn finitc differcnce rnelhods.

'rhe acoiistic pr~:ssiircfield is rcconstructed using ( 5 ) , which can lie rcwritten iis a combination of the first; N,,,, harmonics and tlic siiwtuotli-like:profile:

Tlie first, term on thc right.-hand sidc of (3) is a convolution that iiccoiinls for thc nonlirioar intcraction OF hiirmonics. Thc second tcrm governs absorption and ilispcrsion efftxks [18],which me modcled following thi: powcr law of ahsorption:

and corrcspondirig dispersiun, calcnliitcd using l o d dispersion rclationships [19], [20].Herc N I ) is choscn as ii wilnc of absorption at tlic rcferencc frequency f * = 1 M H n . Diffraction effccts are accounted for by the l.hird term.

The mmi intcnsity f' for thc quiisiplanc wave propaga, tion, ( I ) hiis its main component in z-ilircction, i ~ n dit is the samc as for a plane w i ~ I~ rN I , N (p2)/cupo [11]. Asymptotic cxprcssions for the int,cnsity tliai. Sollow f r m i (6) using i.lic plane wavc form of the intensity can

I079

Tlii: media aajiistic propcrbies mere [XI, [26]: 1;issiic aud hlood dcrisitg, po = 1064 kg/m3; lissuc am1 hlood spccil iif sound, CO = 1684 m j s ; tissue ahsorpl,ion coefficicnt, ao = 0.7 dB/cni at 1

. Area graphed

MHu; tissnc nonlincar paranief.er, R j h = 7.1; powcr law of aljsosptioii 71 = 1.1.

Blood vessel a =aperture radius F = focal distance

Pnr llic pnrpnse of siniiilat,ions and later comparison of rcsnlls, it is convenient t,o inlroiliici: lhc dimcrrsionlcss pararnetcrs N (tiorilinea,rity),C (linear Licilsing gain), i ~ n d A (i~IisorpLiori).They arc deiiried as:

bc obtii,ined in I.hc frcqncricy rloniaiii:

and heill. deposition iliia lo absorption of iiltrasonnd can be obl.ainod from (1):

and r e h t c lhc rcspectivc cllaradxristic longtlis plane wavo shock ~i~rlriationdistance z7, = c ~ p a / 2 ~ . f o P p o , Itaylcigh distance z,j = r~fofne' j q i , and ahsorpt.ion Icnglti z,, r o(f0)-'-to tho focal langth E' of the transduccr. Fix our sccnariii, the focusing gain was eqiial l o G=20 and lhc atisiirplion par;unot,er A = l . The nonlincar ~ial.anieti:r, which is proportional l o l.lm snnrcc amplit,nde, was varied from 0 l o 2, wliich prodiices intcnsiks at lhc Sociis of np to 50011 W J c 1 n ~ .

U.T(m1)erature Modcl The cxprchsiori of the hcal deposit,ion, ( 8 ) , consisk of two tcrnis. The first lcrrri is rela,tcd lo thc ahsnrpt,ion OS tlic 11iwi:r spectral coriqionenls according to tlir alisorplion law. Thc second term accoiiiits for tlie absorpt,ion at the shnck, if it is dcvclopcd, and docs not ilc~ii:nil on lhc a b sorpt,ion propertics of t,lie propagation mcdiiirn [23], Tho asymplolic soliitions for thc wavcforni, iitensily, and lieat dcposition, (6) ( 8 ) , liavc becii compared wi1.h niiriierical soliit,ions obtained hy convcnlional nict,lioils ill tlic Srcyucncy dorriain (using a high niirnbcr of liarnioriics -1000) and in the timc domain (using a very fine tiriie grid) [2]. Tlic resiills wcrc fiiunil t,ii he in very gooil agrecmcnt. Fig. I shows 1 . 1 1 ~cylindrically syitnnctric geoinclry ~ i s a l in thc simulations. In the coinpntal.ions, wc used N,,,,,: = 50 lmrmonics arid initiali~cdthe splicrii:ally curvcd Lransiluccr by a Canssian shailctl pressurc a,niplitudc ilistribii{.ionof tlic fnrin:

p ( z = 0 , r , ~=) pacxp (-(rja,)2)sin(2nfn ( ~ + r ' j l a c o V ) ) .

(9)

IIcrc p o is the soi~rccpressure aniplil.irdc, n is the transiliicer apcrt,iirc radiiis, and F is tlii: geomclrical fni:al lenglh. Tlie transducer was rriodeled aflcr an existing tvansduccr fix acoustic licniostasis with 2-cm apcrtnra diamctcr am1 :3.5-cm radiiis of ciirvature opcratirig tlt il cenler [requcncy ol 3.6 MIIz [24]. l'he acoustic pmpag;~tionwas assumed to be entirely i n liver, and no impedancc tnisnint,ch was introduced by the presence OS a vcsscl (sec Iielow).

Thc tetrrpcral.nrc profile as a fiiriction of t,irni: was cniripiilcil liy two coiipled cxprcssinns ol the hkihcal eynation [27], one fnr the lissiic ilomnin and lho ollicr for tlic hlood dorriairi [ 2 8 ] .Tlic i:oilplcd cquiitiiins can hc cxlxcsscil 3s:

BT

ptCt7

=

dl

kt . V 2 T- 71Ji,