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materials

Article

Ultrasound Axicon: Systematic Approach to Optimize Focusing Resolution through Human Skull Bone

Fabián Acquaticci 1,2,*, Sergio E. Lew 1 and Sergio N. Gwirc 3,* 1 Instituto de Ingeniería Biomédica, Universidad de Buenos Aires, Buenos Aires C1063ACV, Argentina; slew@fi.uba.ar 2 Instituto Nacional de Tecnología Industrial, Ministerio de Producción y Trabajo, San Martín, Buenos Aires B1650WAB, Argentina 3 Departamento de Investigaciones Tecnológicas, Universidad Nacional de La Matanza, San Justo, Buenos Aires B1754JEC, Argentina * Correspondence: facquaticci@inti.gob.ar (F.A.); sgwirc@unlam.edu.ar (S.N.G.)

Received: 16 August 2019; Accepted: 17 October 2019; Published: 20 October 2019

Abstract: The use of axicon lenses is useful in many high-resolution-focused ultrasound applications, such as mapping, detection, and have recently been extended to ultrasonic brain therapies. However, in order to achieve high spatial resolution with an axicon lens, it is necessary to adjust the separation, called stand-off (δ), between a conventional transducer and the lens attached to it. Comprehensive ultrasound simulations, using the open-source k-Wave toolbox, were performed for an axicon lens attached to a piezo-disc type transducer with a radius of 14 mm, and a frequency of about 0.5 MHz, that is within the range of optimal frequencies for transcranial transmission. The materials properties were measured, and the lens geometry was modelled. Hydrophone measurements were performed through a human skull phantom. We obtained an initial easygoing design model for the lens angle and optimal stand-off using relatively simple formulas. The skull is not an obstacle for focusing of ultrasound with optimized axicon lenses that achieve an identical resolution to spherical transducers, but with the advantage that the focusing distance is shortened. An adequate stand-off improves the lateral resolution of the acoustic beam by approximately 50%. The approach proposed provides an effective way of designing polydimethylsiloxane (PDMS)-based axicon lenses equipped transducers.

Keywords: ultrasonic lens; axicon lens; focused ultrasound; transcranial ultrasound



1. Introduction

Focused ultrasound (FUS) in pulsed mode is unique among transcranial brain stimulation methods in combining exceptional spatial resolution (on the millimeter scale) [1–3] with the potential to target subcortical structures (deeper than 10 cm) [4] through the intact skull. It also has potential for inducing neuronal excitation or suppression without evidence of tissue damage [5].

Recently, we demonstrated the advantages of focusing ultrasound (US) through polydimethylsiloxane (PDMS)-based axicon lenses to selectively drive brain activity [6]. The ultrasound axicon is shown in Figure 1. It has the shape of a cone. As the cone angle (φ) decreases, the focus moves closer to the lens. Developing low-intensity applications include the opening of the blood–brain barrier and ultrasonic neuromodulation. Both techniques have recently been extended to human subjects and are under active research. Spherically FUS transducer has been, until now, the most commonly used for transcranial focusing ultrasound, but it has a large focal length (several centimeters) that may hinder its coupling with the head, for example, when the focal plane is too close to skull inner ivory layer. With spherical segment ultrasound transducers, FUS is commonly delivered through a big plastic bag containing degassed water placed over the scalp. This is due to the fact that, for cortical stimulation,



Materials 2019, 12, 3433; doi:10.3390/ma12203433



www.mdpi.com/journal/materials



Materials 2019, 12, 3433 Materials 2019, 12, x FOR PEER REVIEW



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FFiigguurree 11.. ((aa)) EExxppllooddeedd mmooddeell ooff tthhee aaxxiiccoonn lleennss.. TThhee ppoollyyddiimmeetthhyyllssiillooxxaannee((PPDDMMSS))ffiillllss tthhee ccoonniiccaall ccaavviittyyooffththeelelnenssanadndthtehestasntadn-do-fof.ffA. nAunltrualstoransiocnleicnsleins sfoirsmfoerdmbeydthbeyPtDheMPSD/pMlaSs/tpiclainsttiecrfiancteesr.faTchees. wThinedwowindisowa itshianthminatmeraiatel rliaayl elarybeertwbeetewneethnethceonciocnailccaal vcaitvyitaynadndthtehefafcaeceofofththeelelennss; ;((bb)) sscchheemmaattiicc uullttrraassoonniiccbbeeaamm ppaatttteerrnn ffoorr ttrraannssdduucceerrss wwiitthh aaxxiiccoonn lleennsseess.. TThheesseeaarreeddeessccrriibbeeddbbyy iittss ttoottaall iinncclluuddeedd ccoonnee aannggllee ((φφ))..DDeeppththooffffooccuuss((DDOOFF))iisstthheeffooccaallrreeggiioonnaannddFF iiss tthhee ddeessiirreedd ffooccaall lleennggtthh.. BBootthh DDOOFF aanndd FF ddeeppeenndd oonn tthhee ssoouunndd vveelloocciittyy iinnttoo tthhee mmaatteerriiaall uusseedd ffoorr tthhee iinntteerrffaaccee..

SSkkuullll iissggeenneerraallllyy ccoonnssttiittuutteedd ooff tthhrreeee rreelatively homogeneous layers: TThhee oouutteerraannddiinnnneerrttaabblleess ooff ssoolliidd iivvoorryy bboonnee,, aanndd tthhee cceennttrraall llaayyeerr ooff dippllooee off canceelllloouuss bboonnee,, wwiitthh aa blloooodd-- aanndd fatt--ffiilllleedd ppoorrous structure. TThheeddiimmeennssiioonnssoofftthheebblloooodd--aannddffaatt--ffiilllleeddiinncclluussiioonnssaarree rraannddoomm,, aanndd aann aavveerraaggee tthhiicckknneessss iinn tthhee ddiirreeccttiioonn nnoorrmmaall ttoo tthhee ssuurrffaaccee iiss aabboouutt 00..66 mmmm.. [[55]],, ssoo the fundaammeennttaall ffrreeqquueennccyy uusseedd was cchhoosseenntotohheelplpalalellveivaitaetethtehceocnocnercnesrnfos rfoarcoaucsotuicsteinceergnyeragbysoarbpstoiorpntoiornreofrarcetfiorancbtiyonthbeysktuhlel. sFkourllf.reFqour efrnecqieusenlocwieesrlothwaenrathbaonuta0b.o5uMt 0H.5zM, reHflze,crteioflnecotfiosnouonf sdoiusntdheisptrhinecpiprianlccipaualsceaoufsienosef rintisoenrtlioosns. lAostsf.reAqtuferenqciueesnbceietws beentwaebeonuat b0o.6uat n0d.60a.n9dM0H.9zM, tHheza, bthseorapbtsiornpltoiossnilnosths eindtihpelodeilpalyoeerlbayegerinbsetgoinlismtoit lsiomuint dsoturanndstmraisnssimonis,ssioonth, esootshcielloasticoilnlastaiorensdarmepdeadmopuetdaonudttahnedinthseeritniosnerltoiossnilnocsrseianscerselainseesarlilnyewariltyh wfreitqhuefrnecqyu. eAntcayb. oAut 0a.b9oMutH0z.,9thMe Hscza,tttehreingsclaotstseriinngthelodsisplionetlhaeyedribpelogeinlsatyoelrimbietgsionusntdo tlriamnistmsiosusinodn, tsroanthsme iinssieornti,osnoltohses ibnesgeirntsiotno lionscsrebaesgeinas toheinfocrueratshepaoswtheer ofofufrethqupeonwcyer[5o]f.frequency [5].

A full-wave nonlinear ultrasound model based on the k-space pseudo-spectral method was developed and released as part of the open-source k-Wave Acoustics Toolbox [9,10]. This model can account for the propagation of nonlinear ultrasound waves in homogeneous or heterogeneous media



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A full-wave nonlinear ultrasound model based on the k-space pseudo-spectral method was developed and released as part of the open-source k-Wave Acoustics Toolbox [9,10]. This model can account for the propagation of nonlinear ultrasound waves in homogeneous or heterogeneous media with power acoustic absorption and without restrictions on the directionality of the waves. The accuracy of the implementation of nonlinear ultrasound model in the k-Wave Toolbox was validated using experimental measurements of the ultrasound made with a linear diagnostic ultrasound probe and a membrane hydrophone [11].

A computational model for elastic wave propagation in heterogeneous media can be constructed



based on the solution of coupled first-order acoustic equations given in Equations (1)–(3) using the Fourier pseudo-spectral method. This uses the Fourier collocation spectral method to compute spatial derivatives, and a leapfrog finite-difference scheme to integrate forwards in time. Using a temporally



and spatially staggered grid, the field variables are updated in a time stepping [12,13]. To simulate free-field conditions, a perfectly matched layer (PML) is also applied to absorb the waves at the edge of the computational domain [14]. Without this boundary layer, the computation of the spatial derivate



via the FFT causes waves leaving one side of the domain to reappear at the opposite side. The use



of the PML thus facilitates infinite domain simulations without the need to increase the size of the



computational grid.



∂u ∂t



=



−1 ρ0



∇p



−



α·u



(1)



∂ρx ∂t



=



−ρ0



∂ux ∂x



− αxρx



(2)



p = c02 ρx,y,z



(3)



where u is the acoustic particle velocity, ρ0 is the medium density, ρ is the acoustic density, c is the thermodynamic sound speed, p is the acoustic pressure, and p0 = p(t = 0) is the initial pressure distribution.

There are two main stages in this work: The definition of general design equations for PDMS-based axicon lenses with optimal stand-off, through comprehensive ultrasound simulations, for an optimal transcranial focusing; and the measurements and simulations performed to determine focusing performance of the proposed lenses, through a human skull phantom.



2. Materials and Methods

In order to develop a design model, axicon lenses with φ angles between 80◦ and 170◦ in steps of 5◦ were simulated with increments of δ in steps of λ/4 with 8 grid points per wavelength (λ) in the stand-off medium. The final pressure field along with the RMS beam pattern was calculated. The normalized cross-section profile area at the focus F was determined, for each δ. The minimum area represents the greatest improvement with the optimum stand-off, which minimizes transmitted energy outside of the main beam and improves lateral spatial resolution with lower possible sidelobes.

The domain was discretized using a grid point spacing of 250 µm (giving a maximum supported frequency of 2.06 MHz), and a grid size of 512 × 512 grid points (corresponding to a domain size of 128 × 128 mm). Simulations were run on an NVIDIA® GTX 950 graphics processing unit (Santa Clara, CA, United States) using the MATLAB® Parallel Computing Toolbox (Natick, MA, United States). The simulation of all angles/stand-off combinations can be completed in approximately 60 h. By default, numbers in MATLAB® are stored in double precision. However, in almost all cases, k-Wave does not require this level of precision. In particular, the performance of the PML generally limits the accuracy to around 4 or 5 decimal places. We use a PML thickness of 20 grid points that gives a transmission coefficient of −100 dB. This corresponds to a reduction in the signal level of 1 × 10−5, which is significantly less than double precision. Further, there will also be uncertainties in the definition of the materials properties. A list of the main simulation inputs is given in Table A1. In this work, a heterogeneous medium was defined as a layered interface on both sides of the conical



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cavity of the axicon lens as shown in Figure A1. The convective nonlinear effects from the convection of



mass was considered. However, at low frequencies and amplitudes, nonlinearity will only have a small



effect on the wave field. At higher frequencies and amplitudes, this effect become more important.



The accuracy of the implementation of ultrasound model with the k-Wave Toolbox was validated



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in our previous work [6] using experimental measurements of FUS made with the same axicon lens



attachedstmraanllsedffueccteronanthde awnaevedfileeldh. yAdtrohipgheornfere(qFuoenrcieesTeacnhdnaomloplgityudMesH, t2h8is) ewffietcht ibnecao6m-eLmanoreechoic test

tank. Ouirmpproertvainot.us study has already shown that there is a good agreement between the simulated and expevraimlideTanhtetedalaincbceouauramrcpypreoavftitoteuhrsenwsim.orpIknle[mt6h]eiunsstawintigoonrekxop, fewruiemltareanlsstooaul cnmhdeaamrsauocrdteeemlreiwznetistdhotfthhFeeUaSkc-mWoauadsveteiwcTipothoreltbshosexusrawemaaesmplitude of the beaaxmicopnaltetnesrnattoafchtehdetaraxniscdouncelrenansdwahneenedFleUhSydwroapshotrnaen(sFmorcietteTdechthnorolougyghMaH2h8u) mwiathninskau6l-lLphantom

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human skull phantom for experimental validation of simulated transcranial ultrasound propagation.

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shown inasTapbarlteo1f .the current work. The reported and measured property values, and estimated uncertainty

To teinstthtohsee emffeeacsutsreomfeanths,uamreashnoswknuilnl ToanblFeU1.S fields, we inserted a 5 mm thick fragment of parietal bone phantoTmo tbeesttwtheeeefnfetchtseoftraahnusmdauncsekruallnodn FtUheS fhieylddsr,owpehinosneret,edasa s5hmomwtnhicink fFraiggmuerent2o.f pTahrieettarlansducer

bone phantom between the transducer and the hydrophone, as shown in Figure 2. The transducer

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Table 1.TCabolme 1p. rCeosmsipornesaslioannadl asnhdesahreasrpsepeeded, ,aatttteennuuaatitoion,na,nadnddendseitnysoiftyCloeafrCMleeda6r1M0 medat6e1ri0alm. aterial.



Comp. Speed (ms−1)



Comp. Speed Shear Speed Absorption Density



S(mhes−a1)r Speed (m(mss−−11))



A(dbBsocmrp−1t)ion ((kdgBmcm−3)−1)



Density (kg m−3)



2495 ± 8



2495 ± 18081 ± 311081 ± 31 3.70 ± 03..170 ± 01.180



1180



Figure 2. Photograph of the ultrasound test tank showing the axicon lens equipped transducer detail, parietal bone phantom, and hydrophone.



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Figure 2. Photograph of the ultrasound test tank showing the axicon lens equipped transducer detail, parietal bone phantom, and hydrophone.



Materials 2019, 12, 3433



Table 2. Specifications of the axicon lens characterized.



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TabPlea2r.amSpeetceifircations of the axicon lens charVacateluriezed.



Transducer frequency (f) Transducer diPamaraemteert(eDr )



0.445 MHz 28 mm



Value



TrTarnasndsudcuercefrrenqueaenrcfyie(lfd) in water (N) TrRaantsidouocferFd/Niam(Fe/tNer)(D) TRrAaatnxiosicdooufncFe/lNrenn(seFaa/rNnfi)gelled(iφn)water (N) AFxoiccoanl lleennsgathng(lFe)(φ)

FoDceapl ltehnogfthfo(Fc)us (DOF)



64.705.4m45mM(H−6z dB) 0.1628 mm 144°06.41.675 mm (−6 dB) 10.514m4m◦

11 m10m.5 (m−3mdB)/22 mm (−6 dB)



DFeopcthusofdfioacmuset(eDrO(dF)F) FoIncusesrdtiioamn elotesrs((dIFL)) SIntSastenardnti-doo-nffol(foδfs)(sδ()IL) EEncnacpaspusluatliaotnioEnchEochRoedRuecdtiuonct(iEonR)(ER)



2.5 m11mmm(−3(−d3Bd)/B3)./522mmmm((−−66ddBB)) 8.4 d2.B5 mm (−3 dB)/3.5 mm (−6 dB) 30 m380.m4mdBm −12−d1B2 dB



33.. RReessuullttss



33..11.. EEssttiimmaattiioonn ooff tthhee AAxxiiccoonn LLeennss AAnnggllee



TThhee rreellaattiioonn bbeettwweeeenn aaxxiiccoonn lleennssaannggleleφφananddthtehevavlauleueofoFf /FN/Nprpordoudcuecded(se(eseAe pAppepnednixdiAx)Ais)



iislluilsltursattreadtegdragprhapichailclyalliyn iFnigFuirgeur3e. T3.heTthreantsradnuscderucneeranr efaierldfieleldnglethngNthisNgiivs egnivbeyn Nby=ND2=f/4Dc.2fT/h4ce.



Tcoheeffcioceieffinctioefndt oetfedrmeteinrmatiinoantiRo2nisR02.i9s70w.9i7thwpit-hvapl-uvealsuigensiifgicnaifincceanlecevelel voefl<o0f.<0000.0010.0T01h.isTrheilsarteiolantiiosnfoisr



fδo=r δδO=ptimδuOmp:timum:



   φ==9.9957.9.0575820.582+∙ +·11l00nl−n2   NF   [d e g  r e  e s  ] .    .



(4(4) )



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TThhee ffoolllloowwininggrerlealtaitoionn, b, absaesdedonoonuorustrusdtyudsuymsummarmizaerdiziendTainbleTaAb2le(sAee2A(spepeenAdpipxeAn)d, iwxaAs )f,ouwnads

efoxupnerdimexepnetarilmlyevnatlaidllyfovratlhide fleonr sthdeeslecnrisbedde.scTrhibeeodp. tTimheuomptriamtiuomofrFa/tNioaopfpFe/aNrsatpopbeearbsettowbeeenb0e.t1waenedn

00..31.aTnhdis0p.3r.oTdhuicsepsrfoodcualcebseafomcadlibameaemtedrsia(mdFetweritsh(dthFewaixthicothneleanxsiceoqnuliepnpseedqturiapnpsedductrearns sadnudcedrNs oanf da cdoNnovfeantcioonnvalentrtaionnsdalutcrearn),sdanudcedr)e,patnhds odfefpotchusso(fDfoOcFu)so(fDsOimFi)laorf rsaimtioilaarccroartdioinagcctoor:ding to:



F = dF = DOFF .



(5)



N dN DOFN



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Materials 2019, 12, 3433



F N



=



d d



=



DOF DOF



.



(5)

6 of 17



For values of F/N > 0.4, some evidence of the original near field still remains. As the value F/N



decrFeoarsevsablueeloswof0F.4/N, al>l e0v.4id, seonmceeoefvtihdeenorciegoinfatlhneeoarrigfiienladl inseraarpfiidellydssutipllprreemssaeidnsin. Athsethleenvs aslyuseteFm/N.



dTehcriseacsoenstrbaesltoswwi0t.h4,thaellbeevhiadveinocreoof fspthheeroicriaglilneanlsenseawrhfieereldsoismreapoirdiglyinsaul pneparresfiseeldd iins atlhwealyesnsprseyssetnetm. .



This cWonetroabstsserwvieththtahtetbheehreavisioarnoifnsvpehrseerliycapl rleonpsoerstiwonhaelrreeslaotmioensohriigpinbaeltwneeaenr fiFe/Nld aisnadlwκδayvsalpureess,eanst.



showWne ionbFseigrvuereth4aftotrhderifefeirseannt vinavlueersseolyf apnrgolpeoφr.tiTohnealrreelalatitvioenpsehricpenbteatgweeleonssF/oNf laantedraκlδrevsaolluuetsio, nas



(sLh(thLLoeLRwRb)n)eaasisntsalaFafiftuguenurnacrcltetiioro4ennsfoooolrffuκdκtiδδioffniiessrtssehhhnaoottwwvcnaanlniuinnebtsethhoeaefcssahaamniemgveleeefdifigφguw. urTiert,hehw,eκwhrδeehlrveaearteiκlvuκiees



ptheercwenatvaegneulmosbseor.f LlaLtReriaslrreelsaotilvuetitoon itshtahtemwinaivme inzuesmebneerr.gLyLtRrainssrmeliattteivde



otoowuuttashttiseedirdeb,eeoasfosttfhltathehteeenravnaralarlorrureweosewoosleftusptδtioopisnnoscistbrhseliaebatlsmeceasam,ninatbhibneeeaabwmcehaa.vimeSevi.lneeScndiengwtcthheiestt-hshwpeκeeδisegpdvheaotelefdudseooatufhvnseaodrtuamignnediPnsDiinmoMuPinzSDedisMsselpSoneweiesredgrloyttwhhtrareaonrnuitsgnhmhawnittahtiteneedr,



mahrsaeettdhtieoieurovomgfaelFwun/eNeitooh.ufssδteminpecddreiisaucsmoensw,titnihtuehiwtsytaeovpfedlveienslcgootchnitstyi-nwdueeiitcgyrheotaefsdevsea,lvowecrhiatiygchedseeocffureencatdsivessep,leywerdheidtchhurcoeeufsfgethchtetivhreealthyieorteeodrfouFgc/eeNns.etohues



33.2.2. .RReeflfeleccttivivitityyEEfffefeccttoonntthheeIInntteerrnnaallWWaallllss ooff tthhee LLeennss HHoouussiinngg TTooilillulustsrtarateteththeeeefffefeccttooffoouutteerrccaasseeaannddiinnnneerr iissoollaattiioonn ((FFiigguurree 55)) oonn tthheellaatteerraallrreessooluluttioionnooffththee

lelenns,s,thtehererlealtaivtievpeeprceerncetangtaegLeLLRLaRs aasfuanfcutinocntioofnκoδf, aκnδd, athnedntohremnaolirzmedalliazteedrallabteeraaml bperaomfileparorefislehoawren inshFoiwgunrein6Ffiogrutrwe o6 dfoiffr etwreontdhifofeurseinntgh. oWuistihnga. rWeflitehctainrgefhleocutisnigngh,owusitihnogu, wt iinthnoeur tslienenveer,stlheeevree,wthilelrbee 1whm3cinwionmioo0otuithnr◦lnpesel,er-tiearnnhbidneaseaegasfntlleeniwmgenrmrccelnnieeotftbiealrhvwelorelecefirwidtnitehinh1vnaflwo3tlieeteul0thisrryct°s.neth,iioisanAoiaftonlgnhss0lnrsasew.aeee8tiofmririnlodtepewhbnsBtctuhli.taiemdinlelFodtlon,sunolw.etewsrmhAnrneeiisisn,tsxshlsttaoaaahytmtlhntreaesheetdtrpreesiae-olumlloelenolarff,nt,p.et,wsisFstitviniohoinmsetlseryhuciLusdeeltLtaamixeetotRahPmenamersDv,haortpaaeMlfoslnlirutuaenehSd,rescs-e-eiwoeboinslalafeigutftbrsnhheteoriisedundofardusnpeIlPinectedoacDecnterfrdorseMeatnaabhaneaSsyetshleb--i4bslhor6te,4auiau2noc5slp5ssonfeitkdmwnasdHtngeorilpdtecazfahonrnareaefsisecnannidIoastedtniteureotcgso4rasno,nyt4nsncic5actoooelrokmnilnarf-mHee6n-prafl2pszaeac5meflenoracdeeguiwnttdcailstdvetinettitiidvhocaoctyeef oof u0t.s8iddeBtdheowmna,inthbeearemlaitsiv1e0%LLhRigvhaelru, ecoismrpeadruecdetdobinynhearlsfleaenvdeernefelregcytivtritaynosmf 4i0tteddB oduotwsind.e the main beam is 10% higher, compared to inner sleeve reflectivity of 40 dB down.



33.3.3. .EEssttimimaattioionnoofftthheeOOppttiimmuummSSttaanndd--OOffff



cimoc(ewoqmpmuhrbaoWibicWtnviihenoeaesantifmiotfosiiouposnpunnasgrntdosifidvvoaafereloansnrarnexusbsaupimyocxma:olieutcneirotariilnlcioecarnnalele)slsrnearoesesnlluela(dasstttiietioo(ehosnnneeA)ssevhpahainApiplpduepnfefptoodhoerrienfxδδdvFAbbai/xNaa)lu.ssAeTeeisddh)o.sefoohTrnnFohe/wlttNeahhnteeriieosissnlnisiammhtFbioiuoeugwtnlluwaantrtbieeioieoen7ntnnw.FrtrTeeihegeshesuuneuorlltlepttishsnt7oieeo.mfaTfo2ru2hp4mr4et2ei20gmlv0irnaaueanelsmnuagsgeriloelvroen/easf/gstleδuatrqaene(unwsdsaod-hitfoo-iioofcnδfffnh



is given by:



  



1100.9.92211 ++ llnn δOptimum== 11.9.966∙·11002



F   N  



[m e te  r  s ] .    



.



(6)(6)



TThheeccooeeffiffcicieiennttooffddeetteerrmmiinnaattiioonnRR22 iiss 00..9966 wwiitthh aa pp--vvaalluuee ssiiggnniiffiiccaanncceelleevveellooff<<00.0.000000011. .



Figure 4. (Top) Value of F/N and (Bottom) relative loss of lateral resolution, both, vs. stand-off given in number of wavelengths (κδ) for 445 kHz PDMS-based axicon lens with different angles. (a) φ = 140◦; (b) φ = 150◦; (c) φ = 160◦; (d) φ = 170◦.



Materials 2019, 12, x FOR PEER REVIEW



7 of 17



Figure 4. (Top) Value of F/N and (Bottom) relative loss of lateral resolution, both, vs. stand-off given Materiainls n20u1m9,b1e2r, 3o4f3w3 avelengths (κδ) for 445 kHz PDMS-based axicon lens with different angles. (a) φ = 140°7; of 17

(b) φ = 150°; (c) φ = 160°; (d) φ = 170°.



FFiigguurree 55.. HHoouussiinngg ooff tthhee aaxxiiccoonn lleennss..



FFiigguurree66. .OOuuteterrcacsaeseeffeefcftefcotrf4o4r54k4H5zkPHDzMPSD-bMasSe-dbaasxeidconaxleicnosnwlietnhscowniethancgolneeφa=n1g3le0◦φ. (=To1p3)0R°.el(aTtoivpe) lRoesslaotifvleatleorsasl roefsolaltuetriaolnrvess.olsutatinodn-ovffs.gsitvaenndi-nofnfugmivbeenr ionf wnuamvebleenr gotfhws (aκvδe)leanngdth(Bso(tκtoδm) a)nndor(mBoatltiozemd)

pnroersmsuarleizaemd pprlietsusduerse oafmtphleitluadteersaol fbtehaemlapterorafilleb.ea(am) Lperonfsilhe.o(uas)inLgenws ihthourseiflnegctwiviitthyroeffl0ec.8tivdiBtydoofw0n.8; (dbB) ldeonws hno; u(bs)inlegnws ihthourseiflnegctwiviitthyroeffl4e0ctdivBitdyoowf n4.0 dB down.



Materials 2019, 12, 3433 Materials 2019, 12, x FOR PEER REVIEW Materials 2019, 12, x FOR PEER REVIEW



8 of 17 8 of 17 8 of 17



TFihgeuvrTdFFeaiiihgfgl7efuuue.evrrrIeeealonll77uuft..esδlIIeltlollnreuufasssstδttatiirronmeaansgtttaiiliooeomtnnsef.adotothffebedtthyhrbeeelyirrlnaeelllteiaaniaotteiirnooarnrnberbbegeeetrgwtterwweseesseeseienoinnonnttthhhiieeess



aaxxiiccoonnlelnens sopotpimtiummumstasntda-nodff-δoffanδdathnedrtahtieoroaftFio/No.f F/N. aiinnxiddciioccnaatlteeedndstotoopotobimtbatiuanminthstethahneidgh-hoiegfsfhtδelaastntedlraattlhererearsalotrliuoetsoioofnlFu/ftNoior. n for



differTehnet vleanluseaonfgδleess.timated by linear regression is indicated to obtain the highest lateral resolution for



Adisffaernenetxlaemnspalneg, Fleisg.ure 8 shows the focusing Adisffaernenetxvamalupelse,oFf iδg(uar:e08msmho, wb:s20th.5emfomcu, cs:in34g



mbbeemhha,avavinioodrrodof: fP4P5D.D2M5MSm-Sbm-abs)ea. dsTeh1de4s41e°4ad4xi◦fifcaeoxrneicnleotnnsseltwetinintshgswfoaiurtehr



four



differienndticvAataseldauneinsexoFaifmgδupr(leae,:9F0sighmuormwei,8nbgs:hto2hw0e.s5retmhlaetmifvo,eccuL:sLi3nR4gambsemaha,fvuainnocrdtioodfn:PDo4f5M.κ2Sδ5.-bWmasimethd).a14Ts4ht°aenasdxei-codofinfffoleefrn3es4nwtmistmeht,ftoitnhuegr s are



indicldaaittfefderarelinsptFavitagialul eresso9oflsuδhti(oaw:n0iimnmgpmrto,hvbee: sr2e0bl.y5atumipvmeto,Lc4L:03R%4,amcsomma,pfauanrndecdtdit:oo4n5th.o2e5fsκmamδm.e)W.leTnihtshewsaeitsdhtioafufnetrdse-tnoatnffsdeo-totfifnf3.g4smarme , the



laterainl dspicaTathtieeadloripnetsiFmoilguuumtrieosn9taisnmhdo-powrfoifnvpgreetshdbeicytreeudlapbtityvoeE4qL0uL%aRt,iaocsnoma(6fp)uawnrcaetsdiocnthoeoctfkhκeeδd.sfaWomritdehiflaefensrsteanwntdiftr-hoeqfofuuoetfnsc3ti4aensm.dFm-iog, ffuth.ree



T1la0hteecraolmpstppimaatriueasml rteshsteoalnuedxti-pooenffriimmprepenrdotaivcleteslodbsybs uyopEf tqolua4tae0tr%iao,lncro(e6ms)opwlaurateisodcnthoe(LtchkLeeRsd)amffooerrldetnirffasenwrseditnuhtcoefurtesqstwuaenitndhc-oieaffcs.o. uFsitgicure 10



compfarerqeusTethnheceioeepsxt(pifm)erouifmm(aes)nt0at.na2ld22l-oo5fsMfspoHrfezdl,ai(ctbet)er0da.l4br4ye5sEMoqluHuatzit,oio(ncn)((0L6.)L89wR0)aMsfocHrhzetr,cakanneddsdf(uodrc) ed4r.i4sff5ewMreitnHhtzfa.rceoquusetnicciefrse. qFuigeunrceies (f) 10 compares the experimental loss of lateral resolution (LLR) for transducers with acoustic

of (a)fr0e.q2u22en5cMiesH(fz),o(fb()a0).04.42522M5 MHHz,z(,c()b0) .08.49405MMHHzz,, a(cn)d0.8(d90) 4M.4H5zM, aHndz.(d) 4.45 MHz.



Figure 8. 445 kHz-Cigar-shaped acoustic focus for different settings of the stand-off. (a) δ = 0 mm; (b)

δ = 20.5 mm; (c) δ = 34 mm; (d) δ = 45.25 mm. (Top) Focusing behavior of the PDMS-based 144° axicon FigurlFeeing8su.rwe4i48th5. 4fk4oH5ukrzH-dCzif-ifgCeraigerna-srt-hvsahapalupeeedsdaoacfcooδu.us(sMttiiccidffodoclcueu)ssNfoforordrmidfaflieiffrzeeendrtepsnerttetssiensutgtrsienogaf msthopelfsitttuahdnedess-otoafffn. td(ha-e)oδffla=.te0(raam)l mδbe;=a(bm0) mm; (b) δ plδ=er=no2s2f0i0wl.e.55.itm(mhBmmofott;;uo((rcmc))d)δiδNf=f=eo3rr4e3mnm4atmlmviz;ame(ldud;e)p(sδdreo=)sf4sδuδ5=..r2e(5M4a5mmi.d2mpd5l.liet(muT) domNpeos.)roF(mTfooatchulpiezs)ieanFdxgoiapcblruebehsesaisanvumgiroeprbraoeomfhftiaplhevlei,ti0PuoDdmr eoMmsfSoit-nhfbdetahiscPeeadDtlea1Mst4et4rhSa°e-lbafbxoaiecscaueomsdn. 144◦ axicopnrolefinles. (wBoitthtofmo)uNr odrimffealrieznedt vparelusseusreoafmδ.pl(iMtuiddeds loef)thNeoarxmiaal bliezaemd pprroefsilseu, 0remammipnldiitcuadteessthoef ftohceusla. teral beam profile. (Bottom) Normalized pressure amplitudes of the axial beam profile, 0 mm indicates

the focus.



Materials 2019, 12, 3433 Materials 2019, 12, x FOR PEER REVIEW

Materials 2019, 12, x FOR PEER REVIEW



9 of 17 9 of 17 9 of 17



FFiigguurree 99.. Stand-offff, givenn in numberr of waveelleennggtthhss,, vvss.. relative loss of lateral resolutioonn for 445 kHzz

PPDFDiMMguSSr--ebba9as.seSedtda1n144d44-◦o°fafx, igciovnenleinnsn..u(am) bδe==r 0o0fmmwmmav;;e((blbe))nδδg=t=h2s20,0.v5.5sm.mremmla;;t(i(cvc)e)δδlo==s3s34o4mf mlammte; r;aaanlndrde(sd(od)l)uδδt=io=4n54f.5o2.5r254m4m5mmk. H. z PDMS-based 144° axicon lens. (a) δ = 0 mm; (b) δ = 20.5 mm; (c) δ = 34 mm; and (d) δ = 45.25 mm.



Figure 10. Stand-off given in number of wavelengths vs. relative loss of lateral resolution for different FfriFegiqugurueerne1c01i.0e.sSS.tatTanhnded-o-voffafflgugieviveoenfniκninδnnufuommr btbheereroooffpwwtiamavvueelmleenngsgtttahhnssdvv-sso..frfe(lraetidvecilrocslse)ooffisllaadtteeirrraealcltrrleeyssooplluruottipiooonnrftfoioorrndadilfifffteeorreetnhntet ffrrfeerqqeuquueennencciycei.se(.sa.)TTfhh=ee0vv.2aa2llu2ue5e MooffHκκzδδ; f(fobor)r ftthh=ee0o.o4pp4tt5iimmMuHmz;s(tca)nfd=-o0fff.f89(r0eMd Hcirzc;l(ed) )isf =d4ir.e4c5tlMyyHprzroo. ppoorrttiioonnaall ttoo tthhee frferqeuqeunecnyc.y(.a()a)f f==00.2.2222255MMHHzz;;((bb))ff== 0.445 MHz; (c) f ==00..889900MMHHzz;;((dd))ff==4.44.545MMHHz.z.



33..344..4. A.AAccoocuuoussttsiitcci--cFF-FiieeilleddldEEExxxpppeeerrriimmimeeennntttaaalllSSScccaaannn



wTbbTwehwbTehiittteehtewhiwhttelwlhTaTeaetlttThehaeethhetenentheheeerernaeearthtshslhahlkthyksydhledeukyduddeiiuslmdslrmrilklskolm,or,lekuoeup,pttneunphllhhthlnsllhsheeolioasiaeoooniannsnsonnnntesedtdnaeatdoosarsrhohfsctrfcthiyfcatiyFaFnniyadFndnUnUgdngUsrsgrSsroSodSwdowpdpbwbipibsheeihseehseteratroaoatraeaoemanmnennmnnpeepccepcc.ec.eeec.reerrEErroErotftfotxfosxoooxsoosspprsprs-rtm-temep-tehmphphrrrreeeireieireommdomdodtfitftrfereiebrlibablaenlanneoneonsonttsttstsatasahsmhhdmlmdldl uwebubbuwweeaececeacaisiieaaeteassttuhrmrmhrhumurooorrwewwueeupupdpdadaatttaaasasstatataaattttnitteneinien−nndr−r−ddrcncnc66n6rrtrssttsedhdedehhpaaparBBpBrrssorsororeeoddeoduouuddddrdrrgdggooouuhhutthptppcoococetteotoeohhd11hd1fdffee00e0ttatahahphprmmprmreeeheheehmmpamapspassnhrnrhrnheetottettotsooooossowoswswmmsmuauanuannrv..vrrv.eieoFeoiFnoiFnaioniaaoidodtFdrttrFrFtitctthgicithcthhgoghoehuoeeeululeerllflirfleforsisioseossecscic1ciccoaiu1oa1ou1aunns1nn1.nnss.. wwwaasas3s.353.5m.5mmmmimn tihnine ftrhteheeesfpfrreaeeceescspopanacdceeiticocoonnnaddniitdtiioo5nnmaamnnddaft55ermmtrammnscaarffatteenrriattlrrtaarnnasnsccsrrmaannisiisaaillonttr.raaWnnsesmmailisssosiiocohnna..rWaWceteeraiazlslesodo tchhcehaararcacotcuetersirtziczeedfidethltdheeianaccotohuuesstaticixcifafilieeldlddirieinnctttihohene,aapxxieiaarllpddeinirrdeeccicttiiuoolnna,,r pptoeerrtppheeennlddeiinccsuullfaaarrcettooantthhdeesllkeenunslsl.ffaaTcceheeaanFndUdSsskkpuurlell.ls.sTTuhhreee hFaFUlUfSSwpprideresthssusourferethhhaealfhlfwawlifdidmtthahxooifmftthuheemhhawallfafmsm2aa2xximimmmuumminwwthaases 22fr22emme smmpaiinncetthhceoenffrrdeeieetissoppnaaaccenedccoo1nn8ddmiittiimoonnaaafntneddr1t1r88amnmsmmcraaafntfetiaerrl ttrrtaarnnasnsmcsrciarsansniioaianl lt.rtUarannnsdsmmerisitsshsioieonsne..UcUonnddeeirtritothhneesss,eetrccaoonnnsddmiittiiisoosnniossn,, ttorraafnn4ss4mm5 iikssHssiiozonnFUooffS44-a44x55ikckoHHnzzlFeFUnUsSSt--haaxrxoiicucoognnhlletehnnesststhkhruroloulugleghdh ttohtheaensksakupullplllreloedxdtimotoaaantnealayppp4pr0ro%oxxilmiomsaastteeilnlyyl4a40t0%e%rallolosrssessiinonllulaatttieeorrnaallorrfeestshooelluuattiicooonnuoostffitcthhbeeeaaaccmoouu, sasttniiccdbboeenaammth,,eaannodtdhooennr ththhaeenodotht,heaernr ahphapnanrdod,x,aianmnaaaptepplpyror1ox8xi%mimaiantetcelryleya11s88e%%iniinntchcrereeaaasxseieaiilnnrtethhseoelauaxxtiiaoalnl rr.eeWssoohlluuettniiooFnnU.. SWWwhheaensntFFrUaUnSSswmwaiatsstettrdraantnhssrmmoiuittgtteheddtththherrosokuuugglhlh,tththheee ascksokuuullsl,lt,icthtphereeasacscououruesstditcircopppprreeesdsssuburyreehdadlrfro.opTpphpeeddinbsbeyyrtihhoaanllffl..osTTshhoeef oiinnussreesrrkttiiuoolnnl pllhooassnsstooomff owouuarrs asspkkupulrllol xppimhhaaannttetoloymm−6wwdaaBss. apappprIornoxtxirmiamcarataetnelyliya−l−6f6odcdBaB.l.characteristics obtained by simulation using the same configuration of Table 2 for diIffnIentrrtaeranccrtartanhniiacialklfnofeocscasalelcschohafarrtahaccetteesrkriisusttliilccsasrooebbitntaadiinniceeaddtebbdyyissniimmTauubllalaettii3oo.nnTuuhsseiinrneggitsthhaeegssaoamomdeecccoooinnnfcfiigigduuerrnaacttieioobnneotowffTeTaeabnblelteh2e2 sfiofmor ruddliafiftfeifoerenrenantnttdhthitcihckkennveesasslsueesessoofoftbththaeeisnskekudulllelaxarpreeriininmddieiccnaatttaeelddlyiinninTTwaabballteer33f..oTTrhhteehrreeesiisksuaalggl oothooiddckccnooeiinsncsciioddfeen5ncmceembbe,etstwweepeeaenrnatthtehede 2sismmimmuulaflartiotoimonnatahnneddltehtnhesev,vaaaslulusehesosowobbnttaaiininneTedadbeelxxepp4ee.rriimmeennttaallllyy iinn wwaatteerr ffoorr tthhee sskkuulllltthhiicckknneessssooff55mmmm,,sseeppaarraatetedd 22mmmmfrforommththeelelennss, ,aasssshhoowwnniinnTTaabbllee 44..



Table 3. Influence of the thickness skull in the focus of axicon lenses.



TTaabblele33..IInnfflluueenncceeoofftthheetthhiicckknneessss sskkuullll iinn tthhee ffooccuuss ooff aaxxiiccoonnlleennsseess..



Thickness 0.75 mm 1.25 mm



1.5 mm



2.5 mm



3 mm



5 mm



6 mm



Skull (ts)



ts < λ/4



FTTh(mhicmikc)knneesssSSkku1ul0l.l7(t5(sts))



00.7.7t5s5m<mλmm/4 tsts<<λ1λ1/4/4



11..22λ55/4mm<mmts < λ/211..55 mmλ/m4 < ts < λ/22.5 mλm/2 < ts < λ33 mmmmλ/2 < ts 5<5mλmmm ts6>6mmλmm ttss<<λλ//4411 λλ//44 << ttss < λ/92.75 λ/4 < ts < λ/28.25λλ//22 <<ttss<<λλ 9λ.λ5//22<<ttss<<λλ 1t0st.s5>>λλ



dFFF(m((mmmmm)) )



3.5



DSdLdFOLF(Fm((d(mmmBm)m) ))



17 −8.2



DDOOFF(m(mmm) )



1100.73.7.555 3−3.5.11508.3 1177



1111 3.5 33..55−1118.5 1188



1111 5



33..5



17.5 −8.4



1188



9.75 4 88..2255



5



17 −8.7



44



17.5



1177



4 99.5.5



17.5 −9.9



44



1177.5.5



41100.5.5 −197.544

117 7



SLSLL(d(dBB) )



−−88.2.2



−−1100..33



−−1111.5



−8.4



−−88..77



−−99.9.9



−−99.5.5



Materials 2019, 12, 3433



10 of 17



Materials 2019, 12, x FOR PEER REVIEW



10 of 17



Table 4. Comparison of simulated and scanned acoustic beam properties obtained using a configuration of 4T4a5bkleHz4.traCnosmdupcaerriswonithoaf 1s4i4m◦uElpatoexdy/aPnDdMsSclaennnse, dthraocuoguhsttihcebpehaamntopmroopfe5rtmiesm othbitcakinneedss,usseipnagraated

configuration of 445 kHz transducer with a 144° Epoxy/PDMS lens, through the phantom of 5 mm 2 mm from the lens.

thickness, separated 2 mm from the lens.



BeamBPeraompePrtrioeps erties SimSiumlautleadted



dF (mdmF )(mm) DOF D(mOmF)(mm) SLL (SdLBL) (dB)



44 17.157.5 −9.−99.9



ScSacnannended

55 1818 −1−41.44.4



FigFuigreur1e11. 1U. Ultrltarsaosuounnddccaannbbeeffoocusedd tthhrroouugghhhhuummanansksuklul lplhpahnatonmto.mEx. pEexrpimereinmtaelnmtaelamsueraemsuernetms oefnts of acouussttiicc pprreessssuurreeffiieeldldeemmititeteddfrforommaa44455kkHHzzaaxxiciocnonlelnesn.s(.a()aF) rFereeespsapcaecwe withitohuotustksukllu; l(lb; )(ba)ftaefrter

tratnrsacnrsacnraianliatrlatnrasnmsmissisiosinonthtrhoruougghhskskuullllbboonneepphhaannttoomm. WWhhiitteelliinneeiinnddicicaatetessththeefofcoucsu.s.

4. D4.iDsciuscsussiosinon

AlAthlothuoguhghthteheanaanlaylsyissiswwaassccaarrrrieiedd oouutt wwiitthh aa ssiinnggllee eelleemmeennttttrraannsdsduucecrerwwithithnonoabaebrerartriaotnion corcroercrteioctnio, na,nadnadsapsepcieficicfiscksuklul lglegoemometertyrydedseisgingnededtotoaapppproroxximimaatteetthheevvaarriieedd sshhaaped of the skull,, itt is expisecetxepdetchteadt ththearteltahteivreeliantflivueenincfeluoefndcieffeorfendtifmfeeredniut mmepdrioupmertpierospaenrdtieasspaencdts aosfpmecetds ioufmmgeedoimumetry wilgleboemmetaryinwtaiilnl ebde.mPahinantationmedg. Peohmanettormy igsetohmeemtrayjoisrtmheatmeraijaolr imnflatueerniacleinofnlutehneceinotnratchreaninitarlaficrealndiaalnd soufinedldsapnededsoisunshdoswpneetdo ibsesthhoewmnotsot ibnefltuheenmtiaolsat cionuflsuteicntpiarlopaceorutystinc fporcoupserptyresinsufroec,upsopsirteisosnu,rea,nd volpuomsieti.oFnr,oamndthveoelxupmeeri.mFeronmtaltbheaemxppeartitmerennstaslhobweanminpFaitgteurrnes1s1h,othwenuinnexFpigeucrted1f1o,ctuhseinugnpexroppecetretdy of thefosckuuslilnign parxoiapleartxyisomf tahye sbkeudlleisncraixbieadl aaxsisamnoaynlbineedaersecrffiebcetdtahsatacnaounsleinsetahreebffeeacmt thtoatrcoatuatseesbtahcekbteoawmard thetoskruoltlaitnesberatcikontopwoainrdt, cthreeatsiknugllaimnsoerretioconmppoainctt,pcrreesastuinrge caigmaro-rsehacpoemdpaaccot upsrteicssfiuerled.ciTgharis-swhaapseadlso obsaecrovuesdticusfiienlgd.sTeghmisewnatesda-lsspohoebrseetrrvaendsduusicnegrsse[1g6m,1e7n]t.eTdh-supsh,etrheetsrkaunslldiuscneorst a[1n6o,1b7s]t.aTchleufso, rthtreasnksuclrlainsial focnuostinagn oofbUstaScalendformtaraynesxcerartniaanl faodcduistiinognaolf aUcSouasntdicmleanysienxgeretffaenctadtodietniohnaanlcaecospuasttiicallernessionlugteioffnecut ntoder ceretnahinanccoensdpiatitoianlsr.esOoluuttioofnfuoncduesr, ctheretasionucnodndpitrieosnssu.rOeudteocfrfeoacsuess, wthiethsoaunvderpyresstseuerpe sdleocpreea. sFesrowmiththe coma pvaerriysosnteienpTsalbolpee.4,Ftrhoemre tihsea cgoomodpacroisionncidinenTcaebbleet4w, etehnerteheissiamguoloadtiocnoianncdidtehnecevableutewseoenbtathineed ienxflpmsouienmmrebinumofclrtaeeohtnimoostfinadttlhhealyesnedolitenfhnthtiwshce.keaRvntseaeekglrsuuasflerolssdtrkihonusbekglrtluaetiihlinnlseettadhihndieefcilxksufpocneoecnernuscitssmeinoooeufffniatt5thayxmleilicyntmohitnnih,ceslkweenanapectsaoseesrursassf,tkotseirucidnlsilckm2ieunpmloletnmhdtheabifncforokcotcenhmu,ewsssitsodihtofehefasdl5xeoiinmfcffosetmn.hreeR,lnessetnekgpvsuaeaerlslrldo,atcsithneiintdegicer2eesthies a doifscsoounntidnupirtoypiangatthioena,caonudstthiceiamvepreadgaenscoeu,nwdistphedediffiesrtehnetavceoluosctiitciepsroopfesrotyutnhdatpmroopstaignaftluioenn,ceasntdhethe aveforacagledsiostuanndces, pweeeodbisserthvee aascoexupseticctepdr,ospmearltlyvtahraiattmionosstinintflheuepnocsietsiotnh,edfioamcaeltderis, taanndcde,ewpteh oobf sfoecruvse, as expfoerctdeidff,esrmenatllthviacrkinaetisosness oinf tthheecproansiitailobno, ndeia, mreesutelrt,inagndindthepetahvoerfafgoecudse,vfioartidoniffseorfenthtethfoicckunselesssessthoafnthe cra1nmialmb.one, resulting in the average deviations of the focus less than 1 mm.

OnOtnhethoetohtehrehrahnadn,dt,rtarnansmsmitittitninggFFUUSStthhrroouugghh hhuummaannccrraanniaiallbboonneecacuausesdedananapapprporxoimxiamtealtyel4y0%40% loslsoisns ilnatleartaelrarlerseosloultuiotinonofofththeeaaccoouussttiiccbbeeaamm,, estimateeddbbyytthheeinintetennsistiytyfuflullwl widitdhthatahtahlfamlfamxiamxiummu. m. HoHwoewvevr,etrh, itshlisoslossosforferseosloultuiotinonisiscocommppeennssaatteeddbbyy adequatee ssttaanndd--ooffff ooffaaxxicicoonnlelnens.sI.nInadaddidtioitnio, nw,ewe



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find that the PDMS provides a smaller focal zone, which is desirable for neurostimulation [6]. All this allows a higher resolution, comparable to spherical transducers (the most commonly used for ultrasonic brain therapy), but with the advantage that the near field is eliminated, and the focus distance is shortened. For other applications, a wider focal zone and a line focus such as that obtained with glycerin or ethylene glycol may be desirable [7]. From Equation (5), the elimination of the near field, by means of an appropriate axicon lens, enables transducers featuring the same wavelength/diameter ratio to produce the same focal spot size. Thus, large diameter, low-frequency transducers may be used. This is useful for brain stimulation where low frequencies are required for penetration of the skull.

One problem of devices with axicon lenses are the relatively high sidelobes [18]. How much this will affect will depend on the proposed applications. With the calculation of the optimum value of δ, by Equation (6), a better lateral resolution is achieved. This relation between axicon lens stand-off and the value of F/N is applicable to high-resolution epoxy resin/PDMS lenses or other similar combination.

5. Conclusions

The numerical approach proposed in this paper provides a complete and effective way of designing axicon lenses for many high-resolution applications, such as mapping or detection. This is also suitable for focused ultrasound through human skull bone. In view of providing better lateral resolution with lower sidelobes, the use of design programs for this task is not as straightforward. The choice of a good starting point is an important factor for successful optimization. It is easier to obtain a starting design using relatively simple formulas and then use it in a lens design program for future analysis and optimization. We believe that this will be an effective way of designing axicon lenses, for example, to build focused windows to the brain for clinically viable transparent cranial implant for chronic ultrasonic therapy and stimulation of the brain.

Author Contributions: Conceptualization, F.A., S.E.L., and S.N.G.; data curation, F.A. and S.N.G.; formal analysis, F.A.; methodology, F.A. and S.N.G.; project administration, S.E.L.; software, F.A. and S.N.G.; supervision, S.E.L.; validation, F.A.; writing—original draft, F.A.; writing—review and editing, S.E.L. and S.N.G.

Funding: This research received no external funding.

Conflicts of Interest: The authors declare no conflict of interest.

Appendix A k-Wave Simulations

The simulations are based on the k-space pseudo-spectral method and implemented in MATLAB® with the open-source k-Wave toolbox. The functions based on the coupled first-order acoustic equations (named kspaceFirstOrder2D) are named with four input structures (kgrid, medium, source, and sensor). These structures define the properties of the computational grid, the distribution of medium properties, stress and velocity source terms, and the locations of the sensor points used to record the evolution of the wave field over time. The propagation of the wave field is computed step by step in a 2-D layered medium, with the pressure values at the sensor elements stored after each iteration. These values are returned when the time loop has completed. A list of the main simulation inputs is given in Table A1. One of the advantages of k-Wave is that the spatial gradients are calculated by using FFTs rather than using a finite-difference stencil. This means that for linear simulations, only two points per wavelength are required (Nyquist). For nonlinear simulations, the number of points per wavelength at the fundamental frequency will depend on the highest frequency of interest. The amplitudes of the harmonics should decay smoothly.



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Table A1. Summary of the simulation inputs for relationships estimation of the axicon lens angle and the optimum stand-off.



Field



Value



Grid



Number of grid points (NX) Size of the domain (x) Grid point spacing (dX = x/NX) Number of grid points for the sensor mask

Perfect Match Layer thickness



512 128 × 10−3 m 2.5 × 10−4 m

512 × 512

20 grid points



Transducer



Sinusoidal transducer frequency (f) Radius of the disc transducer (D/2)



0.515 × 106 Hz 28 × 10−3 m



Target medium (water)



Sound speed (c) Density (ρ)



1480 m/s 1000 kg/m3



Stand-off and filling medium of the Axicon lens cavity (PDMS)



Sound speed (c1) Density (ρ1)



1030 m/s 1030 kg/m3



Axicon lens cavity medium (epoxy resin)



Sound speed (c2) Density (ρ2)



2530 m/s 1170 kg/m3



Inner sleeve



Echo Reduction (ER)



−40 dB



Initially, the computational grid is defined using the function makeGrid. The time steps used in the simulation are defined by the object property kgrid.t\_array. The time array was defined using the function makeTime, which calculate within the simulation functions using the time taken to travel across the longest grid diagonal at the slowest sound speed, and a Courant–Friedrichs–Lewy (CFL) number of 0.1, where CFL = c0∆t/ ∆x. The computational grid is defined by:

% size of the computational grid Nx = 512; % number of grid points in the x (row) direction x = 128e-3; % size of the domain in the x direction [m] dx = x/Nx; % grid point spacing in the x direction [m] % create the computational grid kgrid = makeGrid(Nx, dx, Nx, dx); % create the time array [kgrid.t\_array, dt] = makeTime(kgrid, medium.sound\_speed);

After the computational grid, the properties of the propagation medium shown in Figure A1 are defined by the objects medium.sound\_speed and medium.density. Also, the object medium.BonA represents the nonlinear properties of the medium. With medium.BonA = 0, k-Wave will include convective nonlinear effects in the model, but not include the nonlinear relationship between the acoustic pressure and acoustic density. If medium.BonA is undefined, k-Wave instead solves linearized equations.

The parameters medium.alpha\_coeff (a0) and medium.alpha\_power (y) describe the power law acoustic attenuation in the medium, where the attenuation is of the form a = a0 × fˆy. The power law absorption and acoustic nonlinearity of water is specified by:



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medium.alpha\_power = 2; %[dB/(MHzˆy cm)] medium.alpha\_coeff = 2.17e-3; %[dB/(MHzˆy cm)] medium.BonA = 4.96;

Next, a time varying pressure source is defined by assigning a binary source mask to source.p\_mask (which defines the position of the source points) along with a time varying source input to source.p. Here, a single sinusoidal time series is used to drive the transducer element. The pressure source is defined by:

% define a transducer element source.p\_mask = makeLine(Nx, Ny, startpoint, pi/2, 112); % define a time varying sinusoidal source source\_freq = 0.515e6; % [Hz] source\_mag = 1.0; % [Pa] source.p = source\_mag*sin(2*pi*source\_freq*kgrid.t\_array); % filter the source to remove any high frequencies not supported %by the grid source.p = filterTimeSeries(kgrid, medium, source.p);

Finally, to visualize the acoustic beam produced by the axicon lens, a sensor mask covering the entire computational domain was defined. Only the total beam pattern is required, thus at each time step, k-Wave only updates the maximum and r.m.s. values of the pressure at each sensor point by setting sensor.record to {‘p\_final’, ‘p\_max’, ‘p\_rms’}. The sensor mask is given here:

% create a sensor mask covering the entire computational domain % using the opposing corners of a rectangle sensor.mask = ones(Nx, Ny); % set the record mode to capture the final wave-field and the % statistics at each sensor point sensor.record = {‘p\_final’, ‘p\_max’, ‘p\_rms’};

When the input structures have been defined, the simulation is started by passing them to kspaceFirstOrder2D.

The simulation is invoked by:

% create a display mask to display the transducer display\_mask = source.p\_mask; % assign the input options input\_args = {‘DisplayMask’, display\_mask, ‘PMLInside’, false, . . . ‘DataCast’, ‘gpuArray-single’, ‘PlotPML’, true, ‘PlotLayout’, true}; % run the simulation sensor\_data = kspaceFirstOrder2D(kgrid, medium, source, sensor, . . . input\_args{:});



display\_mask = source.p\_mask;



% assign the input options



input\_args = {‘DisplayMask’, display\_mask, ‘PMLInside’, false,… ‘DataCast’, ‘gpuArray-single’,



‘PlotPML’, true, ‘PlotLayout’, true};



Mat%eriarlus n20t1h9,e1s2i,m34u33lation sensor\_data = kspaceFirstOrder2D(kgrid, medium, source, sensor,… input\_args{:});



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FFigiguurereAA11. .((aa))SSoouunnddssppeeeeddmmaasskkffoorrththeeddiffifefererennttmmeeddiuiummss; ;(b(b))ddeennssitiytymmaasskkfoforrththeeddiffifeferernent tmmedediuiumms.s.

ThTehceomcopmupteudtefdocfoaclallenlegntghth(F()Fa) nadndopoptitmimuummsstatanndd--ooffff((δδ))vvaalluueess ffor difffferent lennss aanngglleess((φφ) )oof f epoexpyoxayndanPdDPMDSMaSxaicxoicno,nw, withitha atrtarnasndsuducecrernneaear-rfi-feieldldddisisttaanncceeiinntthhee water of 6688..22mmmm,,aarreesshhoowwnninin TabTleabAle2.A2.



TabTleabAle2.AV2a.luVeasl,uoebst,aoinbetadinbeydrubnynrinugnnai5n1g2a×551122 s×im51u2lastimonu,loatfifoonc,aol lfefnogctahl (lFe)n,grtahtio(Fo)f, Fr/aNtio, oopftiFm/Num, stanodp-toimffu(δmOpsttiamnudm-o),ffo(cδuOpstimduima)m, feotecrus(ddFi)a,manedterm(adxFi)m, aunmd mimapxrimovuemmeimntporfolvaetmereanl treosfolautteioranl (rMesIoLluRt)i,ofnor epo(xMyI/LPRD)M, foSr-5e1p5okxyH/PzD∅2M8Sm-5m15akxiHcozn∅l2e8nms mwiathxidcoiffnelreennstwanitghledsi.fferent angles.



Angle φ

115◦ 120◦ 125◦ 130◦ 135◦ 140◦ 145◦ 150◦ 155◦ 160◦ 165◦ 170◦ 175◦



F (mm)

0.25 0.50 2.25 3.75 6.25 9.00 11.00 14.50 19.25 22.75 28.75 33.00 42.50



F/N (N = 68.2

mm)

0.0037 0.0073 0.033 0.055 0.092

0.13 0.16 0.21 0.28 0.33 0.42 0.48 0.62



δOptimum (mm)

29.0 32.0 37.0 38.5 44.5 45.0 45.0 45.5 51.0 51.0 52.5 53.0 52.0



κδOptimum (λ = 2 mm)

14.50 16.00 18.50 19.25 22.25 22.50 22.50 22.75 25.50 25.50 26.25 26.50 26.00



dF (mm @-6dB)

1.50 2.00 2.50 2.50 2.50 3.00 3.00 3.50 4.00 4.50 5.50 6.00 8.00



MILR (%)

65 58 50 51 48 54 49 51 49 47 45 44 51



Appendix B Phantom Design

The Skull phantoms were created for the experimental validation using rapid prototyping techniques. The phantom was based on a 3D mesh of the parietal portion of the human skull. This was derived from a human body polygon dataset called “BodyParts3D” [19]. BodyParts3D is maintained by the Database Center for Life Science research located at the University of Tokyo. Polygon data are extracted from full-body MRI images. The MRI image set that BodyParts3D is based on is called “TARO”. TARO is a 2 mm × 2 mm × 2 mm voxel dataset of the human male created by the National Institute of Information and Communications Technology [20]. BodyParts3D polygon data are distributed in the OBJ file format. The 3D mesh was segmented and smoothed with Meshmixer™.

The phantom was 3D printed in Clear Med610, with a one-layered homogeneous structure, using a Connex500 polyjet printer (Stratasys™). Although scattering due to the porous structure of the real skull could be expected to reduce focused transmission (on the other hand, it is possible that the random scattering reduces too existing destructive interference effects), the wavelength corresponding



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to ultrasound frequency used for 445 kHz is 3.37 mm in water, so that for this frequency, the dimensions of the skull inclusions are smaller than one-half the wavelength and, therefore, ultrasound will not be severely scattered by these inclusions.

Appendix C Material Characterization

Clear Med610 material was used to create the skull bone phantom described in Section 2 and Appendix B. The acoustic properties of Clear Med610 are equal to those of the VeroBlack that were reported in [15]; thus, these measurements were not repeated. However, independent measurements of the sound speed and attenuation of epoxy resin and PDMS were conducted and are shown in Table A3. To obtain reliable simulations, it is very important to accurately know the propagation velocity of ultrasound through the materials used for the lens. For materials characterization, both were poured into cylindrical molds and left to set at 25 ◦C for 48 h. The velocity of sound in epoxy resin and PDMS samples was determined by time of flight technique using an ultrasonic echoscope Digital-Echograph 1090 of Karl Deutsch. This measurement was performed in reflection mode with 2 MHz probe Karl Deutsch S6WB2.25. The sound speed in these materials has a weak dependence on frequency, less than 1% measured from 500 kHz to 2 MHz. The Attenuation of ultrasound was determined in reflection at the applied frequency of 2 MHz. The dimensions and weight of the test samples were measured by a micrometer and a digital scale, respectively.



Table A3. Acoustic properties of materials, distilled water, and tissue.



Material

Epoxy (25 ◦C) PDMS (25 ◦C) Water (20 ◦C)



Velocity (mm/µs@2MHz)

2.53 1.03 1.48



Density (g/cm3)

1.17 1.03 1.00



Impedance (MRayl)

2.96 1.06 1.48



Loss (dB/cm@2MHz)

6.8 5.3 0.08



Appendix D Guideline for Axicon Lens Design This guide is based on design Equations (4)–(6) for Epoxy/PDMS materials combination.



(1) Calculate transducer near field, as:



D2f N=

4c



where D is transducer diameter, f is transducer frequency, and c is sound velocity of the material



under inspection



(2) Select the desired lens focus F and check that:



0.1



≤



F N



<



0.4



Values of 0.1 are rarely used because it gives a very near focus. Values or 0.4 give a profile similar to the transducer but remove the N zone. (3) Calculate the angle of the axicon lens, as:



φ



=



9.9708 + ln 5.52·10−2



F N



[degrees].



(4) Calculate the optimum value of stand-off, as:



δOptimum



=



10.921 + ln 1.96·102



F N



[meters].



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(5) Focus diameter is: (6) Depth of focus is:



dF



=



D·F .

2N



DOFF = 2F.



MaterFiailgsu20r1e9A, 122,sxhFoOwRsPEthEeR rReElVaItEivWe LLR as a function of κδ for value of F/N between 0.1 and 0.3.16 of 17



FFigiguurereAA22. .TThheerreelalatitviveeppeerrcceenntataggeelolosssoofflalateteraral lreresosolulutitoionn(L(LLLRR) )aassaafufunnctcitoionnoof fκκδδ. .(a(a))FF/N/N==00.0.09922;; (b(b))FF/N/N==00.1.133;;((cc))FF/N/N==00..1166;;((dd))FF//NN== 0.2211; ((e) FF//NN == 0.28; (f) F/N == 00..3333..

RReeffeerreenncceess

11. . MMeehhic´ić, ,EE.;.;XXuu, ,J.JM.M.;.C; Calaelre,rC, C.J.;J.C; Couolusolsno,nN, N.K..K; M.; Moroitrzi,tzC,.CT..;TM.; oMuoraudra, dP.,DP..DIn.cIrnecarseadseadnaatnomatoicmalicSapleScpifiecciitfyicoitfy noefunroeumrodmuoladtuiolantivoina vmiaodmuoladtueldatfeodcufosecdusueldtruasltoruansodu. nPdL.oPSLOoSNOE N20E142,091,4e, 896, 9e3896.93[C9.rossRef] [PubMed]

22. . RRoobbeertrstosonn, J,.JL.L.;.C; Coxo,xB, B.T..T; .J;aJraorso, sJ,.;JT.;rTeerbeeyb, By.,EB..AE.cAcucrcauterastiemsuimlautiloantiofntorafntrsacrnasnciraalnuialtlruasltoruansodupnrdoppargoaptaiognatfioorn uflotrauslotnraicsonneiucrnomeuordoumlaotdiounlaatinodnsatnimdusltaimtiounl.atJi.oAnc.oJu. sAt.coSuosct..ASmoc.. 2A0m17. ,2104117, 1174216, –11772368–.1[7C3r8o. ssRef] [PubMed]

33. . YYoooo, ,SS.S.S.;.;BByysstrtritistskkyy, ,AA.;.;LLeee, ,J.JH.H.;.;ZZhhaanngg, ,YY.;.;FFisicshcheer,r,KK.;.;MMinin, ,BB.K.K.;.;MMcDcDaannnnooldld, ,NN.J..J;.;PPaassccuuaal-l-LLeeoonnee,,AA.;.; JoJolelesszz,,FF.A.A. .FFooccuusseedduullttrraassoouunnddmmoodduullaatteess rreeggiioonn--ssppeecciifificc bbrraaiinn aaccttiivviittyy.. NNeeuurrooiimmaaggee22001111,,5566,,11226677––11227755. .

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