Experimental demonstration of noninvasive transskull adaptivefocusing based on prior computed tomography scans
J.-F. Aubry,a) M. Tanter, M. Pernot, J.-L. Thomas, and M. FinkLaboratoire Ondes et Acoustique, ESPCI, Universite´ Paris VII, U.M.R. C.N.R.S. 7587,10 rue Vauquelin, 75005 Paris, France
~Received 30 October 2001; revised 24 August 2002; accepted 28 October 2002!
Extracorporeal ablation of human tumors using HigIntensity Focused Ultrasound~HIFU! has been intensivelyinvestigated during the past decades.1–4 Prostate cancer iscertainly the most suitable application:5,6 the target isreached via an endorectal probe close to the prostate tissis much more difficult to cure tumors located deeply in ttissues: in this case, parts of the emitted wave front wencounter regions with different acoustic velocities, and dferent absorption coefficients, so that the heating beamtern will be degraded. The worst medium to focus throughcertainly the skull, as it was first shown by Whiteet al.7
during investigations into echographic brain imaging, aFry et al.8,9 during investigations into brain hyperthermia:large discrepancy between high acoustic velocity of the s~about 3000 m s21! and the low velocity of brain tissue~about 1540 m s21! combined with a severe attenuationultrasound in the bone strongly degrade the beam shape
Several methods have been developed to precisely fothrough the skull. The most invasive method consists inmoving the piece of the skull in regard of the emitting arrof transducers. Such a method has been experimentallyon living rats and cats in the early 1970s.10 More recently,thermally induced lesions in rabbit brains were also studby using a craniotomy.11 One can also correct the aberratioinduced by the skull either by using a Time ReverMirror12 or by time shifting.13,14In the case of Time ReversaMirrors, prior to the treatment and taking advantage ofbiopsy, which ensures that the tumor is malignant, a hydphone could be inserted in the neighborhood of the tumand used as a beacon. Once the diagnostic would befirmed and the hydrophone removed, the time-reversednals with amplitude compensation could be emitted in or
84 J. Acoust. Soc. Am. 113 (1), January 2003 0001-4966/2003
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to correct both phase and amplitude aberrations inducedthe skull. Then, steering the signals15 would enable one toprecisely heat the whole tumor spot by spot. A refined extsion of this method is the use of a spatiotemporal invefilter:16 taking advantage of the biopsy, one could put a setransducers along the path followed by the physician. Thafter recording the whole set of impulse responses comfrom the set of transducers, one could perform a very acrate focusing: it has been shown experimentally at lpower17 that it was possible to reach the same focusing quity through a human skull as the one obtained in water: phdistortion and attenuation induced by the skull can be crected.
Recently, the possibility to deduce the acoustic propties of the skull from MRI and CT images raised new hopfor noninvasive brain therapy. Hynynenet al.18,19 proposedto use MRI images for extracting the skull profile, withoinformation on the internal heterogeneities. Then, usingthree-layer model~water outside the skull, skull, and braiinside the skull!, they numerically highlighted the necessito perform the phase correction to focus through the skThey proposed to perform this correction by deducingphase aberrations induced by the skull thanks to simulatwith the three-layer model.20 In this work, they highlightedthat the model would be enhanced by taking into accountheterogeneities of the skull. As CT images can give theternal structure of the skull, it is definitely more suited fmodeling the ultrasonic properties of the skull.20–22 In orderto take into account all the information of the CT images,propose here to use a three-dimensional~3-D! finite differ-ences numerical simulation of the complete wave equatBy taking into account the internal heterogeneities in densspeed, and absorption, this finite differences simulation tto model all the defocusing effects highlighted by Whet al.7
It will first be explained how the acoustic velocity, th
density, and the absorption coefficient of the skull candeduced from CT images. In Sec. II, comparisons betwcomputed and experimental wave fronts passing throughsame area of the same skull fully validate this model. Thit will be shown in Sec. III that these results lead to nonvasive focusing through the skull. The focusing patterntained with a conventional time-reversal experiment isdeed close to the one experimentally obtained withmethod based on simulations. Thus, by using CT imagethe skull, it is possible to noninvasively correct both tphase and the amplitude aberrations induced by the sBeyond its interest for noninvasive brain therapy, full 3finite differences simulations are an interesting tool for moeling and understanding the complex acoustic wave proption through bones.
II. ACOUSTIC PROPERTIES OF THE SKULLDEDUCED FROM CT IMAGES
It is known that cortical bones give no signal on Manetic Resonance Imaging~MRI!, whatever their density, buMRI is capable to provide the thickness of the skull. In thcase, one can only model the acoustical properties ofskull with a uniform density, sound speed, and absorptiCT images can measure the internal density of the skenabling us to use more accurate models of the acousproperties of the skull, taking into account the heterogeities of density, sound speed, and absorption inside the s
A. CT images
High-resolution CT images have been performed atInstitut Franc¸ais du Petrole~IFP!, on a General Electric FXmodel with a 0.2 mm in-plane spatial resolution. With tranducers working at a central frequency of 1.5 MHz, it corsponds to al/7 precision in the skull, which is of greaimportance when trying to precisely simulate the wapropagation. The entire CT scan of the skull is given in F1. The scan parameters were set to 120 kV, 60 mA. Slwere 1 mm thick with a 0.5 mm pitch so that each slice woverlapping the previous one.
A dried human skull was midsagitally cut into twhalves. In order to reproducein vivo conditions as closely apossible, the skull is immersed in water and degassed sono air bubble is trapped in porous zones. Instead of befilled with marrow, the porous parts of the skull are thfilled with water. However, it is important that those poro
FIG. 1. CT scan of the entire skull.
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parts are not filled with air bubbles that would introduartifacts in the CT scans. Raw CT values were given iHounsfield Units~H!, defined by
H51000mx2mwater
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with mx , mbone, andmwater, respectively, denoting the photoelectric linear attenuation coefficient of the explored tissbone, and water.
As the skull was placed in water and degassed, iexclusively made of water and bone with a varying densConsequently,F being the bone porosity, we propose herelinear relationship between the photoelectric linear attention coefficient of the explored tissue, and the attenuatcoefficients of bone and water:
mx5Fmwater1~12F!mbone. ~2!
In that case, the porosity map is directly linked to the Housfield map:
F512H
1000. ~3!
Finally, all the acoustic properties~density, speed, andabsorption coefficient! of the skull were deduced from thporosity maps. The porosity map of a slice is given in Fig.Basically, the skull presents a three-layered structure, wiporous zone, called the diploe, stacked between two delayers, respectively, the outer and inner tables. We canthat the diploe is very inhomogeneous and that it is somtimes hard to differentiate it from the tables.
B. Acoustic properties of the skull deduced from CTimaging
All the experiments were made with a plane linear arrof 128 transducers working at a central frequency ofMHz, so that frequency ranges from 0.8 to 2 MHz. Televation of the array~VERMON! is 1 cm, with a 0.5 mmpitch.
FIG. 2. Porosity map of a slice of the 3-D CT data.
85Aubry et al.: Noninvasive transskull focusing
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1. Density
Mass density maps are easily deduced from the bporosity:
d5F3dwater1~12F!3dbone. ~4!
dwater is the mass density of water and has been se1000 kg m23; dbone is the maximum mass density in the cotical bone. One can see that our simple model@Eq. ~2! andEq. ~4!# implies a linear relationship between the bone dsity and the Hounsfield units, as observed experimentallyRho et al.23 and Ploeget al.24
Fry et al.25 statistically studied the acoustical propertiof human skulls. They found an average mass density1900 kg m23 in the inner table. Taking into account the heerogeneities inside the inner table, the maximum mass dsity of the cortical bone (dbone) has thus been set to 210kg m23 in order to obtain the same average mass densitthe inner table as the one given by Fryet al. A slice of the3-D mass density map is given in Fig. 3.
2. Speed of sound
The speed of sound is harder to deduce from the sporosity. The diploe and the inner and outer tables haveferent mechanical properties.25 Acoustic waves propagatinin the fluid and the solid media are coupled, which is crectly described by the Biot theory if the wavelength is neligible compared to the size of the heterogeneities. AtMHz, the wavelength is unfortunately of the order of manitude of the heterogeneities. However, Carter and Hay26
showed that the elastic modulus of bone is proportionathe apparent density cubed, which suggests a linear relaship between velocity and porosity:
c5cmin1~cmax2cmin!3~12F!. ~5!
As porous parts are filled with water,cmin has been set to 1.5mmms21. Fry et al.25 measured the same value in the innand the outer table for the speed of sound: 2.9 mmms21.According to this,cmax has been set to 2.9 mmms21. A sliceof the 3-D velocity map is given in Fig. 4. In order to give aidea of the entire 3-D velocity mesh used in the simulationrepresentation is given in Fig. 5.
FIG. 3. Slice of the 3-D mass density map deduced from CT scans
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3. Absorption
As ultrasonic absorption is known to occur mainly in thdiploe,25 the absorption coefficient is expected to be a groing function of porosity. The following power law model waexperimentally adjusted:
abs5absmin1~absmax2absmin!3~F!b. ~6!
Evaluating the mean absorption of wave fronts propagathrough different parts of the skull showed thatb could varybetween 0.3 and 0.7. The best comparison between simtion and experiments were obtained withb50.5; absmin
50.2 dB mm21; absmax58 dB mm21. This definitely sets theabsorption coefficients for all the experiments performthrough this skull. A slice of the 3-D absorption map is givin Fig. 6.
C. Numerical simulations
Simulations were performed with a finite differencprogram called ACEL developed in our lab. For 3-D simutions through the skull, a 20 h computational time is needto compute a 70 mm310 mm330 mm box at an ultrasonicfrequency of 1.5 MHz with a 500 MHz computer. Basicallthe program is based on a discretization of the linear acouwave equation in heterogeneous absorbing media~7!:
S 11t0~r !]
]t• D Fr0~r !“•S 1
r0~r !“p~r ,t ! D G
21
c0~r !2
]2p~r ,t !
]t2 5S0~r ,t !. ~7!
This equation accurately models fluids with sound spedensity, and absorption heterogeneities. Mode conversand shear waves are not taken into account. Neverthethey can be neglected in a first approximation25 as the wavefront incidence angle on the skull’s interface remains closenormal incidence. As the wavelength in the skull is largthan the inclusions, microstructures in the diploe are avaged by introducing an effective medium. In the bandwidof the transducers, insertion loss increases as the sepower of the frequency.25 A second power frequency dependence of the insertion loss corresponds to a constant reation time,28 as set in the simulation. In order to ensure
FIG. 4. Slice of the 3-D acoustic speed map deduced from CT scan
Aubry et al.: Noninvasive transskull focusing
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FIG. 5. A 3-D spatial representation of the sound speed distribution deduced from CT scans and used as input data in the full 3-D finite differencesation~the box is 60 mm330 mm312.5 mm).
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precise simulation, taking into account density, sound speand absorption heterogeneities, the spatial pitch is set tothan one-tenth of a wavelength and the time step is automcally set to ensure the von Neumann stability criteria29
Strongly absorbing boundary conditions based on a fouorder approximation of Higdon conditions30 are used at thesides and edges of the mesh to avoid unphysical reflectiSets of receivers and emitters can be easily defined thanan intuitive graphic interface, as well as the velocity, densand absorption maps@Eqs. ~4!, ~5!, and ~6!# respectively,c0(r ), r0(r ), andt0(r ). Then the numerical code propagatthe wave front from the set of emitters to the set of receiveThe maps presented in Sec. II B were interpolated to otenth of a wavelength.
The position of the set of transducers on the simulatmesh is critical. It is indeed of great importance to carefuplace the transducers at the same location in the simulatand in the experiment. Basically, the exact positions of thmarkers on the skull were recorded and the experimeconfiguration was reproduced with a one-millimeter presion in each direction. We did not notice significant degration of the experimental focusing patterns based on simtions due to this variance in the estimation of the marklocation. Further investigations could quantitatively estimthe effect of positioning errors.
FIG. 6. Slice of the 3-D absorption map deduced from CT scans.
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III. EXPERIMENTAL VALIDATION: COMPARISONBETWEEN COMPUTED AND EXPERIMENTAL WAVEFRONTS
The midsagitally cut human skull was placed betweour linear array of 128 transducers and a needle hydroph~Fig. 7!. The whole experiment was immersed in water.the general case, out of plane refractions have to be tainto account, explaining why a 3-D simulation code hadbe computed. Nevertheless, we first tried to achieve aexperiment by placing the needle hydrophone close toskull. The surface of the skull has been aligned with tsurface of our active elements in order to minimize refractout of the simulation and experimental plane. Thus, fisimulations could be performed in two dimensions, whichmuch more rapid. The hydrophone has a one millimeterternal diameter housing and 0.5 mm diameter active surf~l/2!, so that one can assume that the experiment takes pin one slice of the CT scan~each slice is indeed 1 mm thick!.A plane wave was emitted by the array of a transducer,the hydrophone was translated along thex axis in order torecord the wave front after passing through the skull. Tplane wave front before passing the skull has also beencorded and is plotted in Fig. 8.
As expected, after propagation through the skull,plane wave is strongly distorted, suffering both phase aamplitude distortions. The experimental wave front is psented in Fig. 9~a! and can be compared to the simulated odisplayed in Fig. 9~b!. At first glance, the simulation remarkably reproduces the influence of the skull. In Fig. 9~c! isreproduced the corresponding porosity map of the skul
FIG. 7. Experimental setup.
87Aubry et al.: Noninvasive transskull focusing
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order to explain the influence of the skull. Two differezones have been highlighted. The zone one correspondsdense part of the skull with a relatively uniform absorptioso that the wave front is mainly distorted by the varyithickness of the skull, which is well reproduced on simutions. At position 40 mm on both experimental and simulawave fronts, a dislocation can be seen: this is due to a n@white arrow No. 2 in Fig. 9~c!# that acts like an acoustilens.
A quantitative comparison between simulated andperimental wave fronts can be done. In Fig. 10~a! is plottedthe relative time shifts between simulation and experimeTime shifts have been evaluated by cross-correlating thenals of Figs. 9~a! and 8~b!. In order to obtain a good focusing, Goodman31 showed that the precision on the phasethe emitted signal should be less than T/8, where T isperiod of the signal. One can see in Fig. 10~a! that this cri-
FIG. 8. Plane wave before passing through the skull~experiment!.
FIG. 9. Wave front after propagating through the skull:~a! Experiment;~b!simulation;~c! corresponding porosity map of the skull.
88 J. Acoust. Soc. Am., Vol. 113, No. 1, January 2003
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teria is well respected, except in one zone around positioncorresponding to very low signal amplitude@see Fig. 9~a!#.Respective normalized amplitude at central frequencycorded or simulated at each location of the hydrophonealso plotted in Fig. 10~b! and are quite similar.
Studying the propagation of a plane wave through aman skull enabled us to validate the whole simulation pcess. In the following section, we will apply this processperform noninvasive focusing through the skull.
IV. APPLICATION TO NONINVASIVE FOCUSINGTHROUGH THE SKULL
A. Experimental setup
The experimental setup, presented in Fig. 11, slighdiffers from the one presented in Sec. II. Our aim is to dtermine the set of signals that have to be emitted on the aof transducers in order to obtain a sharp focusing. The qity of the focusing is experimentally checked by translatithe hydrophone along thex axis and recording the maximumof pressure amplitude. In this section, as out of plane refr
FIG. 10. ~a! Relative phase shift between the experimental and simulaplane wave propagating through the skull.~b! A comparison between experimental~dotted line! and simulated~solid line! normalized amplitude.
Aubry et al.: Noninvasive transskull focusing
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tion effects have to be taken into account, all the simulatiwere performed with our full 3-D finite differences cod~See Fig. 12.!
B. Noninvasive time reversal
As seen in Sec. II, when focusing through the skuparts of the wave front encounter regions with strongly dferent acoustic velocity and different absorption coefficieThis induces a severe spreading in the main lobe andincrease of the side lobes level, as one can see later in14, showing the experimental directivity pattern obtainwhen focusing through the skull by emitting a cylindriclaw with the array of transducers~dash–dotted line!. Thisdirectivity pattern was obtained by plotting in a linear scathe square of the maximum of pressure amplitude recorby the needle hydrophone at each location along thex axis.Such a focal spot is definitely not suited for brain hyperthmia: the maximum of pressure amplitude is not on the tar~target at center! and the focusing quality is so poor thnecrosis would also be induced in surrounding tissues.
Time reversal enables to correct the phase aberratinduced by the skull.12 Basically, a pulse emitted by aacoustic source placed at the desired focus is recorded barray of transducers after propagation through the mediThen the recorded signals are time reversed and reemittethat the wave front propagates back to the source as if
FIG. 11. Experimental setup.
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experiment was played backward. The experimental wfront coming from the central location of the hydrophowas recorded on the array of transducers and is presenteFig. 13~a! after the time reversal operation. These timreversed signals were experimentally reemitted by the arWe obtained the corresponding directivity pattern plottedFig. 14 ~dotted line!.
Thanks to simulations, it is possible to avoid the needan ultrasonic source at focus by using a virtual source.acoustic source was indeed placed on our CT images alocation of the desired focus. A pulse emitted by this virtusource was numerically propagated by the 3-D finite diffences code through the skull to the array of transducers.wave front numerically recorded on the array is presenafter the time reversal operation in Fig. 13~b! and is veryclose to the one experimentally acquired@Fig. 13~a!#. Thetime-reversed simulated wave front was then experimentemitted by the array of transducers. Finally, the experimedirectivity pattern obtained by emitting the simulated setsignals is plotted in Fig. 14~gray solid line!. These experi-ments support the feasibility of totally noninvasive brain hperthermia, since the focusing based on CT images~solidline! is close to the one obtained with a real acoustic souplaced at the desired focus~dotted line!: the focusing pat-terns are very similar, excluding an increase of the secondlobe located at110 mm from the center when using thsimulated wave front. The strong defocusing effect obserwhen using a cylindrical law~dash–dotted line! is now wellcompensated for hyperthermia applications. Ebbiniet al.27
proposed an 8 dB side lobes amplitude as a thresholdorder to avoid burning surrounding tissues. This is the chere.
In previous work, Thomaset al.12 showed that at highfrequencies~1.5 MHz! the focusing pattern could be strongenhanced by combining time reversal with amplitude copensation. We will show in the following section that this calso be performed noninvasively.
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FIG. 12. Full 3-D finite differences simulation: 3-D spatial representation at a given time of the acoustic field passing through the skull~in a linear scale!. Thewave front is coming from a point-like source inside the brain. The calculation box (60 mm330 mm312.5 mm) corresponds to the 3-D volume presentedFig. 5. One can notice some field variations along the elevation axis.
89Aubry et al.: Noninvasive transskull focusing
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C. Noninvasive time reversal with amplitudecompensation
Time reversal corrects phase aberrations induced bypropagating medium but does not correct amplitude abetions. When focusing through a strongly absorbing medilike a human skull, the focusing can be improved by correing the amplitude of the recorded signals.12 This is particu-larly important when working at central frequencies abov
FIG. 13. Time-reversed signals emitted by the array of transducers~a!From experimental signals;~b! from the computed wave front~heteroge-neous model!; and ~c! from a computed wave front~homogeneous model!.
90 J. Acoust. Soc. Am., Vol. 113, No. 1, January 2003
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MHz. The skull absorption is indeed increasing with the frquency, suggesting to work at low frequency. But decreasthe frequency also means decreasing the resolution andcreasing the heating of soft tissues, as their absorption cficient is also increasing with the frequency. A compromhas to be determined. Hynynenet al. decided to work at lowfrequency to optimize the Specific Absorption Rate,19 whichis a good choice in order to avoid wasting energy in tskull. In that case, compensating the amplitude would oslightly improve the focusing, and a correct focusing canachieved without amplitude correction. Another concernthe combination of HIFU and cavitation that may induhemorrhages in the brain. As the cavitation thresholdcreases with the frequency, hemorrhages are more likeloccur at low frequency. Our lab chose to work in a highfrequency range~from 800 kHz to 1.5 MHz! for therapy, butalso imaging. In that case, absorption effects become mimportant and an amplitude correction has been propose
Basically, assuming that the skull is an infinitely thlayer close to the emitting array of transducers, one canply a gain on parts of the wave front that suffered a lowhile passing through the skull. This so-called amplitucompensation is fully described in Ref. 12. The timreversed signals obtained, respectively, by experimentsimulation presented in Fig. 13 were amplitude compensa~Fig. 15! and then were both experimentally emitted by tarray of transducers. The corresponding focusing patternsplotted in Fig. 16.
The focusing pattern obtained with a conventional timreversal experiment with amplitude compensation is slighbetter than the one based on simulations. More preciselyright side has a lower level. Further work could showhether a better positioning system or a more precise adment of the parameters of the inhomogeneous modelsented here could enhance the noninvasive process.
Either experimentally acquired or deduced from simutions, the focusing patterns obtained using time-revecombined with amplitude compensation are much sharsince the effective aperture of the emitting array is enhanc
FIG. 14. Directivity pattern through the skull: energy~pressure squared! isplotted in a linear scale after emission of the time reversal of signals expmentally acquired~dotted line!, of simulated signals~gray solid line!, and ofcylindrical law ~dash–dotted line!.
Aubry et al.: Noninvasive transskull focusing
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Moreover, the side lobe level is lower. Thus, the therapy wbe more precise and tissues located in the near-field wilhomogeneously insonified. Adding the amplitude compention protects the brain tissues to the detriment of an overhing in the most absorbing parts of the skull. This overheatwill be studied in further works.
D. Comparison with a homogeneous model
The full 3-D finite differences codes developed in tharticle take into account all the internal heterogeneitiesdensity, speed, and absorption of the human skull detecteprior CT scans. A homogeneous model of the skull can abe proposed.18,20 In a first approximation, one can indeeconsider that the main effect of the skull results in its varythickness, as the acoustic velocity and the density of the sare about twice the velocity and the density of water andthe brain. In that case a fast simulation code can be ubased on ray tracing theory. In order to determine the advtage of taking into account the internal heterogenieties, ononinvasive focus have been conducted on other parts ohuman skull. Two simulations of the wave front coming froa virtual source placed on the target were conducted:with the previously presented heterogeneous model and
FIG. 15. Time-reversed signals with amplitude compensation emitted byarray of transducers:~a! From experimental acquisition and~b! from a com-puted wave front~heterogeneous model!.
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with a homogeneous model. In the homogeneous model,cording to the value deduced by Hynynen20 from Fry et al.,25
a mean velocity of 2.65 mmms21 has been taken, and a meadensity of 1800 kg m23.
The emitted wave front was either not corrected or crected thanks to the homogeneous or the heterogenmodel. They were emitted with the linear array of 128 tranducers. The corresponding energy distributions are plottea linear scale in Fig. 17. One can clearly see that the homenous model~dotted line! improves the focusing compareto a noncorrected wave front~dash–dotted line!. The sidelobes are significantly reduced, with a 10 dB gain. One cnotice that the directivity pattern obtained with the homogneous model is more symmetric and globally centered ontarget, even if the maximum of pressure amplitude is notthe target: on this area of the skull, destructive interferenoccur at the center. But the heterogeneous model develoin this article reaches a better focusing: the directivity patt
FIG. 16. The directivity pattern through the skull: energy~pressure squared!is plotted in a linear scale after the emission of time reversal with amplitcompensation of signals experimentally acquired~dotted line! and simulatedsignals~gray solid line! and of cylindrical law~dash–dotted line!.
FIG. 17. Directivity pattern through the skull: energy~pressure squared! isplotted in a linear scale after the emission of cylindrical law~dash–dottedline!, time reversal of simulated signals with heterogeneous model~graysolid line!, and with a homogeneous model~dotted line!.
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is quite symmetric, the maximum of pressure amplitudeclose to the target and the side lobes level are very low. O90 experiments conducted with various focusing techniqon various areas of the skull, the focusing capabilities wcompared. Using a noncorrected wave front, a mean 1.5absolute deviation from the target was observed, with amm standard deviation. With a noninvasive correctionduced from a homogeneous medium, the mean deviationcreased to 1.15 mm with a 0.2 mm standard deviation.mean gain on the first side lobe was 6.2 dB, and 3 dB onsecond side lobe. With a noninvasive correction dedufrom a heterogeneous medium, the mean deviation decreto 0.63 mm with a 0.3 mm standard deviation. Comparednoncorrected wave front, the mean gain on the first side lwas 9.4, and 8.7 dB on the second side lobe. Thus, theerogeneous model presented here shows a significantprovement, even if the homogeneous model shows goodsults compared to a noncorrected focusing. One also hanotice that the homogeneous model was solved here by ua finite differences code, taking into account higher refrtion angles than codes derived from ray tracing theory. Moover, the wave front coming from the virtual source has besimulated with the homogeneous model in the same confiration used in Secs. III B and III C. This wave front is prsented in Fig. 13~c!, and can be compared to the experimetal one @Fig. 13~a!# and the one obtained with thheterogeneous model@Fig. 13~b!#. One can see that the amplitude of the wave front is not well reproduced. This meathat such a model only allows performing time shiftingtime reversal. With a more complex model like the heteroneous one presented here, one can try to perform more cplex focusing by changing the phase and the amplitude ofsignals. Even if time shifting is necessary and has to be tainto account to perform brain hyperthermia,19 new develop-ments can be envisioned with a more accurate model, sucamplitude compensation.12
Of course, this first study has to be extended on othuman skulls that the laboratory is currently looking for. Tdifferent models also have to be compared at lower frequcies, where the distortions induced by the skull are lessportant.
V. CONCLUSION
Based on prior CT scans, a 3-D finite differences nmerical simulation of the complete wave equation has bproposed. This 3-D code takes into account the internalerogeneities in density, speed, and absorption of the huskull. Thanks to this model, simulated wave fronts passthrough the modeled skull are very close to the correspoing experimental wave fronts propagating through the sapart of the skull. Moreover, the propagation through the skof a wave front coming from a virtual point-like source located inside the brain can be simulated and recorded byof receivers. The same geometry is experimentally repduced and the simulated wave front is time reversedexperimentally emitted by a real array of transducers. Tenables us to achieve a complete time reversal experimcombined with amplitude compensation without the needa physical source inside the brain, as the finite differen
92 J. Acoust. Soc. Am., Vol. 113, No. 1, January 2003
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code guided by CT scans allows us to create virtual souinside the brain. As both phase and amplitude aberrationduced by the skull can be noninvasively corrected, this tenique could show encouraging promise in brain hyperthmia. A time reversal mirror made of 200 high-powtransducers is under construction to bring into play this ninvasive hyperthermia protocol. Beyond its interest for noinvasive brain therapy, full 3-D finite differences simulatioare an interesting tool for modeling and understandingcomplex acoustic wave propagation through the skull.
ACKNOWLEDGMENTS
CT images were performed at the Laboratoire de Tmographie X of the Institut Franc¸ais du Petrole. We wish tothank Corinne Fichen and Camille Schlitter for both techcal assistance and support.
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