elementwise approach for simulating transcranial mri

PHYSICAL REVIEW RESEARCH 1, 033205 (2019)

Elementwise approach for simulating transcranial MRI-guided focused ultrasound thermal ablation

Nathan McDannold,1 P. Jason White,1 and Rees Cosgrove2

1Department of Radiology, Brigham and Women’s Hospital and Harvard Medical School, Boston, Massachusetts 02115, USA2Department of Neurosurgery, Brigham and Women’s Hospital and Harvard Medical School, Boston, Massachusetts 02115, USA

(Received 4 September 2019; revised manuscript received 6 November 2019; published 26 December 2019)

This work explored an elementwise approach to model transcranial MRI-guided focused ultrasound (TcMRg-FUS) thermal ablation, a noninvasive approach to neurosurgery. Each element of the phased array transducer wassimulated individually and could be simultaneously loaded into computer memory, allowing for rapid (∼2.5 s)calculation of the pressure field for different phase offsets used for beam steering and aberration correction. Wesimulated the pressure distribution for 431 sonications in 32 patients, applied the phase and magnitude valuesused during treatment, and estimated the resulting temperature rise. We systematically varied the relationshipbetween CT (computerized tomography)-derived skull density and the acoustic attenuation and sound speed toobtain the best agreement between the predictions and MR temperature imaging (MRTI). The optimization wasvalidated with simulations of 396 sonications from 40 additional treatments. After optimization, the predictedand measured heating agreed well (R2: 0.74 patients 1–32; 0.71 patients 33–72). The dimensions and obliquityof the heating in the simulated temperature maps were correlated with the MRTI (R2: 0.62, 0.74, respectively),but the measured heating was more spatially diffuse. The energy needed to achieve ablation varied by an orderof magnitude (3.3–36.1 kJ). While this elementwise approach required more computation time up front (thecombined simulation matrices were approximately 4.6 times higher than a single large simulation), it could beperformed in parallel on a computing cluster. It allows for rapid calculation of the three-dimensional heating atthe focus for different phase and magnitude values on the array. We also show how this approach can be usedto optimize the relationship between CT-derived skull density and acoustic properties. While the relationshipsfound here need further validation in a larger patient population, these results demonstrate the promise of thisapproach to model TcMRgFUS.

DOI: 10.1103/PhysRevResearch.1.033205

I. INTRODUCTION

Transcranial MRI-guided focused ultrasound has emergedas a noninvasive neurosurgical approach for thermal lesion-ing in the brain. This method uses a hemispherical phasedarray transducer to correct for aberrations of the acousticfield caused by the skull, allowing for accurate focusingto central regions in the brain. The method is clinicallyapproved in several countries for thalamotomy for essen-tial tremor [1] and tremor-dominant Parkinson’s disease [2],and is under investigation for a number of other functionalneurosurgery applications, including thalamotomy for neu-ropathic pain [3], pallidotomy for Parkinson’s disease [4],and capsulotomy for obsessive-compulsive disorder [5]. Sim-ilar systems are being tested clinically to disrupt the blood-brain barrier as a treatment for Alzheimer’s disease [6]and to enhance delivery of chemotherapy to patients withglioblastoma [7].

TcMRgFUS thermal ablation has several significant lim-itations. First, the treatment is currently restricted to central

Published by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Furtherdistribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.

locations in the brain. When the transducer is focused at moreperipheral targets, less energy can be transmitted through thebone due to large incidence angles between the transducerelements and the skull, leading to overheating. Furthermore,even at central locations, the shape of the focus can be-come oblique [8], which can put nearby structures at riskfor unwanted thermal damage and side effects [9]. The bonyproperties of the skull vary substantially between patients, andthe energy needed to achieve an ablative thermal dose at thefocus can be excessive in some patients. As a result, somepatients are not candidates for this treatment [10]. Finally, forreasons not fully understood, as sonications are delivered atescalating acoustic energies, it becomes more difficult to heatthe tissue at the focus [11]. Being able to improve the focusingthrough the skull could ameliorate some of the challenges,expand the patient population who are candidates for TcM-RgFUS and, perhaps, prevent side effects due to off-targetheating.

To correct for aberrations induced by the irregular skull,the TcMRgFUS system estimates changes in of the ultrasoundfield induced by the bone for each element of the transducer.This correction is based on an acoustic model that uses thegeometry of the skull and estimates of bone density obtainedfrom CT (computerized tomography) scans [12,13]. The den-sity is used to predict the acoustic sound speed and attenuationbased on calibrations obtained with cadaver skulls [14–16].Currently, a simplified proprietary acoustic model is used

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McDANNOLD, WHITE, AND COSGROVE PHYSICAL REVIEW RESEARCH 1, 033205 (2019)

Skull region

Simulated Pressure field

ROI for individual simulation

Array element

Global ROI

Sum of 993 simulations after adjusting

phases/magnitudes

CT scan

Treatment planning MRI

Registration

Sonication + MRTI

TcMRgFUS

Load simulations

Apply phase/magnitude(aberration correction + steering)

Sum

Scale combined intensitybased on power/duration

Estimate ΔT with bioheat equation

Combine

Simulate acoustic pressure

Define simulationspace from CT

Elementalsimulation

Save to file×993 elements

(in parallel)

(b)(a)

FIG. 1. Methods. (a) Flowchart showing order of operations for the treatment, the simulations of the acoustic field, and the combination ofthe individual simulations. (b) Diagram of simulation space. (Left) Sagittal reformat of a CT scan; a diagram of the 30-cm diameter 650 kHzTcMRgFUS phased array is superimposed. A small volume is selected that includes one array element. Right: The pressure field (logarithmicscale shown) is simulated in this volume using the CT scan to estimate the skull acoustic properties. The simulation is then rotated andinterpolated to a global volume that includes the focal region and saved. After repeating for every element, the combined field can be rapidlyobtained with different phase/magnitude values for the elements.

to generate the phase aberration corrections. Several studieshave evaluated three-dimensional acoustic models to predictthe focal heating [8,16–18]. Those studies have been able toreasonably predict the temperature and shape of the focus andthe acoustic energy needed to reach an ablative thermal doseat the focus.

This work took a different approach to the simulationproblem. Instead of simulating the entire acoustic field in onelarge model, we simulated each transducer element separately(Fig. 1). These simulations were rotated and interpolated intoa global frame of reference and saved [19]. All the elemen-twise simulations could be loaded into computer memorysimultaneously, enabling rapid (∼2.5 s) iteration of phase andmagnitude elements to simulate different correction schemes(or no correction) and perform beam steering to differenttargets. Ultimately, we aim to compile a library of these smallsimulations to produce a look-up table and to provide inputsfor machine learning, which could enable us to rapidly simu-late the heating during a treatment by relating each element toa previously simulated one. Rapid optimization of the phaseand magnitude could maximize the peak intensity at the focusand better define the shape and size of the focal ablativeregion.

II. METHODS

A. TcMRgFUS treatments

The treatments used the ExAblate Neuro transcranial MRI-guided focused ultrasound (TcMRgFUS) system (InSightec,Haifa, Israel), which operated at 660 kHz. This device hasa 1024-element hemispherical phased array (993 active el-ements at our site) integrated with a 3T MRI (GE750, GEHealthcare). The patient was placed in a stereotactic framewhich was attached to the MRI table. A flexible membranewas attached to the patient’s head and the open face of thetransducer. Acoustic coupling was achieved with degassedwater that is chilled and circulated between sonications tominimize skull heating. The transducer was attached to amanually operated positioning system that was steered sothat the focus was within ±1 mm of the initial brain targetidentified by the neurosurgeon. Additional steering from thislocation was achieved electronically using the phased array.The location of the transducer was found in the MRI spacevia MRI tracking coils, and imaging was performed using thebody coil.

Before treatment, a CT scan of the head was obtained witha bone reconstruction kernel and a slice thickness of 1 mm

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ELEMENTWISE APPROACH FOR SIMULATING … PHYSICAL REVIEW RESEARCH 1, 033205 (2019)

or less. On the day of treatment, anatomic MRI scans wereacquired in three orientations using a 3D FIESTA sequence[repetition time (TR)/echo time (TE): 5.1/2.4 ms; flip angle:55◦; receiver bandwidth: ±27.3 kHz; field of view: 22 cm;slice thickness 2 mm; matrix (typical) 224 × 288 × 28]. TheCT and MRI were registered to each other using the softwareof the ExAblate system. Phase aberrations were calculatedusing a proprietary method by the manufacturer. The ampli-tudes of each phased array element were set by the devicesoftware to normalize the acoustic exposure over the head toavoid hotspots. The transformation between the CT and MRIspace, the target locations in MRI space, and the magnitudeand phase of each transducer element were saved for eachtreatment. This information was used to simulate the acousticfield as described below.

During each sonication, magnetic resonance temperatureimaging (MRTI) was obtained in a time series in a single,operator-defined plane using a fast spoiled gradient echosequence. The orientation of the imaging plane and fre-quency/phase encoding directions varied between each soni-cation. Two acquisitions were obtained before the start of eachsonication; the first was ignored due to artifacts. Acquisitioncontinued over the course of the sonication; five additionalacquisitions were obtained to map cooling. In the first 17patients, MRTI parameters were TR/TE: 27.8/12.9 ms; flipangle: 30◦; receiver bandwidth ±5.7 kHz; field of view: 28cm; slice thickness: 3 mm; matrix: 256 × 128. In the oth-ers, a multiecho readout was performed (TR: 28.0 ms; TE:3.1/7.8/12.5/17.2/21.9 ms; flip angle: 30◦; receiver band-width: ±35.71 kHz; field of view: 28 cm; slice thickness:3 mm; matrix: 256 × 128). The higher readout bandwidthof the multiecho sequence reduced spatial distortions in thefrequency-encoding direction with minimal loss of SNR [20].

The MRI reconstructed magnitude, real and imaginaryimages that were converted to phase-difference images.Temperature changes were estimated from the phase-differences [21] using a temperature dependence of−0.00909 ppm/◦C [22]. This value was selected becauseit is what is used in the device software. The raw imageswere stored in DICOM format and independently analyzedoffline using MATLAB software developed in-house. Artifactsresulting from changes in magnetic field (presumably dueto patient motion outside the brain) were removed using aprocedure outlined elsewhere where nonheated regions ineach phase-difference images were fit to a smooth 2D surface[23]. The surface was extrapolated into the heated region andsubtracted off. The phase difference maps obtained with themultiecho sequence were combined via a weighted averagebased on the expected signal-to-noise ratio of the MRTI [24],with the weight for the nth echo given by

Wn = (TEne

− TEnT ∗2

)2. (1)

We used 30 ms for the T ∗2 relaxation time for brain.

B. Simulations

Numerical modelling of the acoustic field was performedusing k-Wave, an open-source MATLAB toolbox [25]. For ev-ery patient, the pressure amplitude was simulated individuallyfor each element of the array. The element was assumedto be a 1 cm diameter flat circular piston centered on the

TABLE I. Acoustic and thermal parameters used in the numeri-cal model.

Parameter Water Brain Skull

Density (mg/kg3)a 1000 1030 from CTSound speed (m/s)a 1500 1560 from densityAttenuation at 660 kHz (Np/m)b 0 4.36 from densityThermal conductivity [W/(m K)]a – 0.51 –Specific heat [J/(kg K)]a – 3640 –Perfusion coefficient (l/s)a – 8.33E-03 –

aReference [26].bReference [27].

location provided by the manufacturer. The dimensions of theelemental simulations were 44 × 44 × 492 elements with aspacing of 0.325 mm, which resulted in a simulation spaceof 14.3 × 14.3 × 159.7 mm3 per element. We used a perfectlymatched layer with a size of 10 grid points and an attenuationof 2 Np per grid point. The total simulation space (64 ×64 × 512 spatial elements, including perfectly matched layer,×993 elements) was approximately 4.6 times larger than whatwould be required for a single 30 × 30 × 16 cm3 simulation(944 × 944 × 512 elements).

After the completion of the individual simulations, wetransformed the resulting pressure field from the elementspace to a global space in the xyz frame of the transducer withdimensions of 14.3 × 14.3 × 23 mm3 via cubic interpolation.For a given sonication, the 44 × 44 × 71 × 993 matrix ofelemental simulations was loaded, and the phase correctionssupplied by the manufacturer were applied along with phasevalues needed to electronically steer the focus to the targetlocation. When all simulations were loaded into memory, ap-plying magnitude and phase and calculating the total pressureamplitude took approximately 2.5 s. Phase corrections for“ideal” focusing were found by simply subtracting the phaseat the target location from each element’s simulated pressure.

Next, we used the bioheat equation to estimate the heating[28]. We averaged the heating to match the voxel dimensionsand temporal resolution of the MRTI with the imaging orien-tation used during treatment.

The parameters used in the numerical modelling of theacoustic pressure and the subsequent temperature rise arelisted in Table I. To estimate the sound speed and attenuationof the skull, we first used the density estimated from the CTscans and the empirical relationships presented by Pichardoet al. [15] interpolated to 660 kHz. We assumed a linearrelationship between Hounsfield units and density, and −1000and 57 Hounsfield units for air and soft tissue, respectively.

Overall, we considered 72 patient treatments. The first 32patients were examined in detail and were used to optimizethe relationships between skull density and the acoustic atten-uation and sound speed. Thirty of these patients were treatedin the ventral intermediate (VIM) nucleus of the thalamus foressential tremor; two were treated in the globus pallidus forParkinson’s disease. The number of sonications per treatmentand the acoustic parameters used (power, duration) variedbetween patients. Details for each patient, their treatments,and the CT scans are listed in Table II. A total of 447sonications were performed in the treatments in patients 1–32;

033205-3


TABLE II. Patient information, treatment parameters, and CT scan settings.

Patient Treatment CT Scan

Acoustic energy (kJ)

No. Age SDR No. son. Power (W) Duration (s) Max. Total To reach 55 ◦C Vendor Kernel keV Voxel (mm)

1 58M 0.61 17 139–797 10–21 12.9 107.6 8.0 SIEMENS H60s 120 0.47 × 0.47 × 1.002 73F 0.51 17 139–694 8–27 16.9 112.1 14.3 SIEMENS H60s 120 0.46 × 0.46 × 1.003 58M 0.41 15 185–937 10–16 13.9 99.7 7.7 TOSHIBA FC30 120 0.43 × 0.43 × 1.004 65F 0.47 16 185–937 10–28 22.4 160.7 8.8 SIEMENS H60s 120 0.52 × 0.52 × 1.005 78M 0.41 17 186–934 10–20 18.6 125.2 9.5 SIEMENS H60s 120 0.48 × 0.48 × 1.006 72F 0.48 15 187–932 8–10 9.2 66.8 4.2 SIEMENS H60s 120 0.43 × 0.43 × 1.007 65M 0.49 13 186–1095 10–10 10.8 65.4 5.3 SIEMENS H60s 120 0.38 × 0.38 × 1.008 69M 0.40 25 188–1197 6–44 44.3 418.0 34.6 SIEMENS H60s 120 0.56 × 0.56 × 1.009 68M 0.46 15 187–840 10–24 18.8 114.4 10.0 TOSHIBA FC30 120 0.50 × 0.50 × 1.0010a 73M 0.46 24 187–1198 10–23 27.4 193.1 25.4 TOSHIBA FC30 120 0.47 × 0.47 × 1.0011a 48M 0.50 13 185–1116 10–13 14.4 90.1 12.2 SIEMENS H60s 120 0.48 × 0.48 × 1.0012 77F 0.44 14 189–473 7–24 9.5 73.0 5.8 SIEMENS H60s 120 0.43 × 0.43 × 1.0013 83F 0.42 12 185–1180 10–13 15.2 77.9 8.3 SIEMENS H60s 120 0.43 × 0.43 × 1.0014 80M 0.38 12 235–1292 10–16 21.1 121.8 12.7 SIEMENS H60s 120 0.52 × 0.52 × 1.0015 70M 0.50 9 230–939 10–13 10.8 51.7 5.4 SIEMENS H60s 120 0.50 × 0.50 × 1.0016 80F 0.38 11 232–1030 10–24 24.3 101.2 15.3 SIEMENS H60s 120 0.45 × 0.45 × 1.0017 91M 0.51 10 230–1026 10–13 13.2 48.1 7.4 SIEMENS H60s 120 0.46 × 0.46 × 1.0018 71M 0.55 13 190–853 10–16 13.0 67.6 5.4 SIEMENS H60s 120 0.47 × 0.47 × 1.0019 75M 0.69 12 142–837 8–13 9.5 51.6 4.4 TOSHIBA FC30 120 0.47 × 0.47 × 1.0020 78M 0.45 20 238–807 9–32 21.4 172.6 13.3 SIEMENS H60s 120 0.50 × 0.50 × 1.0021 79F 0.57 12 189–662 10–21 9.5 56.6 3.7 SIEMENS H60s 100 0.39 × 0.39 × 1.0022 71M 0.44 11 190–1123 9–24 19.0 85.4 10.8 SIEMENS H70h 120 0.49 × 0.49 × 0.5023 69F 0.46 13 191–1148 10–33 36.1 181.6 23.3 SIEMENS H37s 120 0.48 × 0.48 × 1.0024 84M 0.55 10 237–792 11–17 9.1 54.3 4.6 GE BONE+ 120 0.49 × 0.49 × 1.0025 77M 0.59 10 189–707 10–20 13.4 61.0 6.4 GE BONE+ 140 0.53 × 0.53 × 0.6326 86F 0.41 10 189–847 10–19 15.3 87.4 6.4 SIEMENS H60s 120 0.41 × 0.41 × 1.0027 72M 0.61 12 189–903 11–15 12.6 77.8 5.7 GE BONE+ 120 0.54 × 0.54 × 0.6328 84M 0.40 13 191–1302 11–36 31.5 186.8 14.3 SIEMENS H60s 120 0.45 × 0.45 × 1.0029 72F 0.45 11 89–953 12–26 23.6 101.3 7.3 TOSHIBA FC30 120 0.47 × 0.47 × 1.0030 81M 0.48 13 188–848 12–33 22.2 160.9 13.9 GE BONE+ 140 0.49 × 0.49 × 1.0031 79M 0.60 12 183–1054 6–14 13.2 61.3 5.4 SIEMENS H60s 120 0.52 × 0.52 × 1.0032 83M 0.66 12 93–566 12–20 6.9 46.9 4.4 GE BONE+ 120 0.49 × 0.49 × 0.63

aPallidotomy.

16 sonications were excluded due to artifacts in the focalregion in MRTI. After completing the analyses of the first 32patients, we validated our results by simulating 40 additionalessential tremor patients (Table S1).

The simulations were performed in MATLAB using theO2 High Performance Compute Cluster, supported by theResearch Computing Group at Harvard Medical School. Westored the complex steady-state pressure amplitude after eachsimulation. The simulations in the first 32 patients were runfive times: using the density/attenuation relationship describedin Pichardo et al. [15], without attenuation, using the op-timized density/attenuation and finally with the two opti-mized density/sound speed relationships. The optimizationprocedure is described below. We simulated the additional40 patients with the two optimized density/sound speed rela-tionships and without attenuation. The elemental simulationstypically required approximately 20 min each to run.

C. Optimizing skull attenuation

We investigated the feasibility of systematically iterat-ing the elemental simulations to find a relationship between

attenuation and skull density that resulted in heating thatbetter matched the MRTI for the first 32 patients. It wasimpractical to repeatedly run the full numerical model, sowe used a simplified attenuation model that we applied tosimulations performed without attenuation. This simplifiedmodel assumed that the primary attenuation effects occuralong the z direction:

Pk (xi, yi, zi ) ≈ Pk0(xi, yi, zi ) × exp

⎡⎣−

i∑j=0

α(xi, yi, z j )�z

⎤⎦,

(2)where Pk0(xi, yi, zi ) is the pressure distribution simulatedwithout attenuation for element k, α(xi, yi, zi ) is the attenu-ation, and �z is the grid size.

We assumed the relationship between attenuation and skulldensity ρsk (xi, yi, zi ) can be approximated by a series ofpolynomials:

αsk (xi, yi, zi ) ≈∑

m

Amρsk (xi, yi, zi )m, (3)

033205-4


FIG. 2. Optimizing the relationships between skull density and attenuation. We examined 10 000 relationships per sonication (blackcurves). To generate the curves, we used two nodes with densities of 1200 and 3500 kg/m3 (red circles) and a third indicated by one ofthe yellow crosses. These nodes were fit to a polynomial. We generated a set of 100 curves for each yellow cross; an example set is shown inmagenta.

resulting in

Pk (xi, yi, zi ) ≈ Pk0(xi, yi, zi ) exp[−Nb(xi, yi, zi )αb�z]

× exp

⎡⎣−

i∑j=0

∑m

Amρsk (xi, yi, z j )m�z

⎤⎦,

(4)

where Nb(xi, yi, zi ) is the number of soft tissue voxels betweenthe transducer and zi. After exchanging the order of summa-tion this becomes

Pk (xi, yi, zi ) ≈ Pk0(xi, yi, zi ) exp[−Nb(xi, yi, zi )αb�z]

× exp

⎡⎣−

∑m

Am�zi∑

j=0

ρsk (xi, yi, z j )m

⎤⎦.

(5)

The values for Nb(xi, yi, zi ) and the cumulative summationsof the skull density (

∑ij=0 ρsk (xi, yi, z j )m) for orders m = 0 −

4 were calculated for each transducer element, interpolatedto the array space, and saved. These data, along with theindividual pressure estimates for each element could thenbe loaded into memory. The total three-dimensional pressureamplitude for all elements and the resulting temperature risecould then be estimated for different values of Am. We onlyincluded the 23 × 23 × 31 spatial simulation points centeredon the focus location of the simulation to save time.

We attempted to find the Am coefficients that minimizedthe difference between the simulated and measured peaktemperature rise for all patients. It was not practical to load thesimulated pressure fields and skull density summations for the32 patients into computer memory simultaneously. We thussystematically explored different coefficients for each patientseparately and in parallel. We defined a grid of nodes at 12values of density between 1200 and 3500 kg/m3 and 10 valuesof attenuation between 80 and 500 dB/cm (Fig. 2). We thenselected a set of three nodes, which were each then fit to afourth-order polynomial (order chosen ad hoc) to determine

the Am coefficients. Two of these nodes had densities of 1200and 3500 kg/m3 (100 possible sets), and the third was one ofthe 100 nodes between these two. Thus, for each patient weexamined 10 000 sets of Am coefficients.

For each set of coefficients, we used Eq. (5) to estimate theattenuated pressure field for the individual elements. We thenset the phase and magnitude for each sonication and estimatedthe focal temperature rise. After compiling these temperatureestimates, we found the coefficients that minimized the meansquared error between the simulated and measured temper-ature rise. Finally, we repeated the full simulations with theoptimized attenuation/density relationship.

As shown in the results below, in many patients, plots ofthe difference between measured and simulated heating as afunction of previously applied energy was relatively constantfor the initial sonications, after which it began to deviate. Weinterpreted this deviation as an irreversible change in skullproperties. Thus, when we found the optimized Am coeffi-cients, we only included sonications that occurred before thisdeviation.

D. Optimizing skull sound speed

The deviations between simulations and MRTI in plots ofheating as a function of acoustic energy suggest that the skullacoustic properties might be changed during treatment. As-suming ad hoc that this change was primarily in sound speed,we investigated whether we could find optimized relationshipsbetween sound speed and skull density that could be usedbefore and after this change occurred. As described abovefor attenuation, we created a simplified model that allowed usto rapidly explore different density/sound speed relationshipsusing previously obtained simulations. We assumed that therelationship between the inverse of the sound speed and theskull density can be approximated by a series of polynomials:

1

csk (xi, yi, zi )=

∑m

Bmρsk (xi, yi, zi )m. (6)

To estimate the effect of changing the skull sound speed onthe transmitted pressure magnitude, we assumed the biggest

033205-5


loss occurred due to the initial reflections at the interfacesbetween soft tissue, the inner and outer tables, and the diploe(i.e., we ignored multiple reflections and other reflectionswithin the skull). The transmission coefficient was calculatedusing the following relationship [12]:

Tk (xi, yi )

≈(

2ZOTcosθicosθt

ZOTcosθi + ZT cosθt

)(2ZDPcosθ ′

i cosθ ′t

ZDPcosθ ′i + ZOTcosθ ′

t

)

×(

2ZITcosθ ′′i cosθ ′′

t

ZITcosθ ′′i + ZDPcosθ ′′

t

)(2ZT cosθ ′′′

i cosθ ′′′t

ZT cosθ ′′′i + ZITcosθ ′′′

t

),

(7)

where the acoustic impedance of the outer and inner tables,the diploe, and soft tissue, respectively, are given by

ZOT = ρOT(xi, yi )cOT(xi, yi ),

ZIT = ρIT(xi, yi )cIT(xi, yi ),

ZDP = ρDP(xi, yi )cDP(xi, yi ),

ZT = ρT cT . (8)

We used brain values for the soft tissue density (ρT ) andsound speed (cT ), and density values measured at the outerand inner tables of the skull and the diploe for ρOT(xi, yi ),ρIT(xi, yi ), and ρDP(xi, yi ), respectively, as described below.The incident and transmitted angles with respect to thenormal vectors at the outer and inner tables and the diploe(θi, θt , θ

′i , θ

′t , respectively) were found via Snell’s law:

sin θt

sin γO= sin θt

sin θi= cOT(xi, yi )

cT,

sin(θ ′t )

sin(γO − γDP + θt )= cDP(xi, yi )

cOT(xi, yi ),

sin(θ ′′t )

sin(θ ′t )

= cIT(xi, yi )

cDP(xi, yi ),

sin(θ ′′′t )

sin(γI − γDP + θ ′′i )

= cT

cIT(xi, yi ), (9)

where γO, γDP, and γI were the incidence angles of the outerskull surface, the center of the diploe, and the inner skull sur-face measured with respect to the z axis. A diagram showinghow the incidence angles were defined for this procedure isshown in Fig. S1. We measured the density and angles ofincidence for the inner and outer tables with respect to thez axis, for each (xi, yi ) coordinate as described below.

To estimate phase shifts resulting from changing the skullsound speed, we multiplied the delays expected the along thez direction by 2π times the FUS frequency ( f ):

�ϕk (xi, yi ) ≈ 2π fi∑

j=0

�z

c(xi, yi, z j ). (10)

After combining with Eq. (6) and changing the order ofsummation, this becomes

�ϕk (xi, yi ) ≈ 2π f �z∑

m

Bm

i∑j=0

ρsk (xi, yi, z j )m. (11)

The cumulative summations of the skull density[∑i

j=0 ρskull(xi, yi, z j )m] saved for the attenuation estimationwere used.

We applied the magnitude and phase changes to simu-lations performed with the optimized attenuation. We usedEqs. (7) and (11) to remove the effects of the density/soundspeed relationship used in those simulations before testingnew ones.

As was done with the attenuation optimization, we es-timated the temperature rise over a predefined set of den-sity/sound speed curves for every sonication. We defineda grid of nodes covering densities between 1200 and3500 kg/m3 and sound speeds between 1500 and 4000 m/s.We then selected a set of three nodes, and we fit the inverseof the sound speed to a fourth-order polynomial to determinethe Bm coefficients. Two of these nodes had densities of 1200and 3500 kg/m3 (100 possible sets), and the third was oneof the 100 nodes between these two. Thus, for each patient,we examined 10 000 sets of Bm coefficients. The relationshipthat minimized the error between the simulations and mea-surements was found for the sonications that deviated fromthe model after attenuation optimization. We then repeatedthe full simulations with the optimized attenuation and soundspeed relationships.

E. Data analysis

All analysis was performed in MATLAB. We compared thepeak temperature rise at the focus measured with MRTI tothat estimated with the simulations using linear regression.To compare “treatment efficiency”, we estimated the energyneeded in each patient to achieve a focal temperature of 55 ◦C.We noted that plots of measured focal heating as a function ofacoustic energy generally followed a logarithmic curve. Thu,we used nonlinear least squares regression (using the function“nlinfit” in MATLAB) to estimate this energy. To further testour ability to predict the “treatment efficacy”, we comparedthe measured and predicted slopes of plots of the logarithm ofthe acoustic energy and the focal temperature rise.

We also compared the shape and orientation of the heating.The simulated and measured temperature maps at the time ofpeak temperature rise were fit using the Levenberg-Marquardtmethod via the MATLAB function “fit.m” to a two-dimensionalGaussian distribution:

�T (x, y) = A + B exp

[− (x − x0) cos θ + (y − y0) sin θ

σx

]2

× exp

[− (x − x0) sin θ + (y − y0) cos θ

σy

]2

.

(12)

We used linear regression to compare the resulting esti-mates of the dimensions (σx, σy) and obliquity (θ ). For theobliquity, not all the temperature maps had a well-definedangle. Thus values for θ that had 95% confidence intervalsgreater than ±30◦ were excluded. In each patient, we selectedrepresentative sonications that were largely free of artifactsin the MRTI. We selected examples from each imaging ori-entation and frequency encoding direction for each patient.Not all orientations and encoding directions were available

033205-6


(b)

z (m

m)

0-5 5x (mm)

-150

-140

-130

-120

-110

-100

-90

-80

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³)

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2000

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2600

2800

z (mm)-70-75-80-85-90

innersurface(zb)

diploe(zDP)

inner table (zIT)

outer table (zOT)

outersurface(za)

FIG. 3. Feature extraction from the CT scans for each phased array element. (a) For every x and y coordinates of the elemental simulation,threshold analysis of the density along the z direction was used to find the first and last voxel of the inner and outer skull surface. Workingfrom the outer surface, the first peak encountered was used to identify the outer table. The same procedure was used working from the innersurface to find the inner table. The minimum density between the inner and outer table was used to identify the diploe. The coordinates of theinner and outer surface and at the diploe [(b) and (c)] were fit to smooth surfaces (d). The normal vectors for these surfaces were found foreach x and y coordinates (e); the individual normal vectors (black) and the mean vector (red lines) are shown. Note that for clarity, (b), (c), and(e) only plot every fourth x and y coordinates.

in every patient. Overall, we examined 110 sonications whencomparing the focal heating shape.

We examined whether different factors gleaned from theskull CT could predict the acoustic energy needed to achievean effective thermal exposure at the focus (i.e., energy re-quired to heat to 55 ◦C). These factors were determined withthe individual volumes used in the elementwise simulations(Fig. 3). Working along the direction of propagation, weidentified the first and last skull voxel, the voxels at the centerof the inner and outer tables, and the voxel that had theminimum density between the outer and inner table for each

of the 44 × 44 coordinates in the elementwise simulation. Thecoordinates of the first and last skull voxel and the diploewere fit to smooth surfaces. Normal vectors to this surfacewere calculated to obtain the angles of incidence of the skulland the direction of ultrasound propagation. We used thesedata to calculate different skull-derived factors (see Ref. [29]for more details). We then examined the relationship betweeneach factor and the energy needed to reach a peak focal tem-perature of 55 ◦C. The ability of different metrics to predictthis energy were evaluated using linear regression with the“fitlm.m” function in MATLAB. Since there were obvious out-

AxialMRTI

CoronalMRTI

SagittalMRTI

None

xyyz

xz

Tx Ideal No skull

(a) (b)

Simulation Simulation Simulation

FIG. 4. Example simulated pressure distributions (a) and measured and predicted temperature maps (b) from a pallidotomy treatment(patient 11). In (a), three orientations with respect to the transducer are shown for one sonication with no aberration correction, the correctionused during treatment (Tx), and ideal correction; the simulated field without a skull (but with magnitude shading and beam steering) is alsoshown. The location of the geometric focus is indicated. For this sonication, the focus was electronically steered away from the geometric focusto x = 3 mm and y = 1 mm. In (b), MRTI for three sonications are shown; the insets are the simulated heating with the aberration correctionused during treatment for these examples. Scale bars: 1 cm.

033205-7


Cor

onal

A

xial

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ttal

Sim MRTI MRTI MRTI MRTI MRTI MRTI MRTISim Sim Sim Sim Sim Sim

1 2 4 5 6 7 8

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(a)

CoronalSagittal

Axial

100 -50 0 50 100Measurement obliquity (°)

-100-80-60-40-20

020406080

100

Sim

ulat

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obliq

uity

(°)

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Slope: 0.39 ± 0.31Y-int: 0.95 ± 1.28R²: 0.62Sonications: 110

Slope: 0.75 ± 0.42Y-int: 0.25 ± 0.56R²: 0.76Sonications: 110

Slope: 1.10 ± 0.66Y-int: -4.76 ± 21.38R²: 0.74Sonications: 87

Measurement σx,y (mm)1 2 3 4 5 6 7 8 9

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0

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2.5

Sim

ulat

ion

σ y/σ

x

FIG. 5. Comparing predicted and measured focal shape, size, and orientation. (a) Simulated temperature maps (Sim) and correspondingMRTI acquired during TcMRgFUS in 32 patients. One example from each imaging orientation was selected per patient; not every orientationwas used in every patient. The shape, relative size, and obliquity of the simulations matched the measurements reasonably well in most cases.However, the measured focal heating was more diffuse than the simulations predicted. The orange lines indicate the 50% contours for theMRTI; yellow lines indicate 25% contours for the simulations. The field of view of each region shown is 2.3 cm. The patient number and thefrequency-encoding direction in the MRTI are indicated. [(b)–(d)] Simulated and measured heating dimensions and obliquity. The simulatedand measured MRTI were both fit to a two-dimensional Gaussian distribution. (b) A good correlation was observed between the dimensions(σx , σy) of these fits, but the dimensions in MRTI were higher. (c) A good correlation was observed in the ratios of widths. (d) In many cases,this fit detected tilted heating distributions. The simulations detected a tilt in the same direction. (error bars: 95% CI of the fits; dotted line:linear regression; solid line: unity).

033205-8


Magnitude mask

No magnitude mask

G

Lesion(T2WI)

Isotherms(model)

x (mm)-5 0 5

x (mm)-5 0 5

z (m

m)

145150155160

y (m

m)

-505

yx

z

xy

z

4 mm inferior

Focal plane

(b)

(c)

4 mm anterior

Lesion center

(e)

(f)(a)

(g) (h)

(d)

(j)

(i)

Oblique lesion (patient 23) Internal capsule heating (patient 31)

FIG. 6. Risks of an oblique focus. (a)–(h) Tilted lesion in an essential tremor patient. [(a)–(f)] T2-weighted MRI 24h after FUS [(a)–(c)axial and (d)–(f) coronal]. The MRTI imaging planes are indicated by dashed lines, the lesion boundary is segmented in red, and the internalcapsule is indicated in green. Coronal imaging through the lesion center [(d) and (e)] revealed a left/right tilt. Axial imaging inferior to thelesion center (c) and coronal images anterior to it (d) show that it was also tilted in the superior/inferior direction and included a portion of theinternal capsule. This portion of the lesion (arrows) was not included in MRTI in any orientation. (g) 3D rendering of isotherms generated bysimulation (red) along with calvaria (from CT) and the transducer array. The predicted heating is tilted both in the left/right and superior/inferiordirections. (h) Stacks of contours from the lesion segmented in MRI and the simulated isotherms from two different viewpoints. Both the lesionsand the isotherms had similar obliquity. [(i) and (j)] Heating in the internal capsule in a different patient. MRTI and simulated heating (insets)during two consecutive sonications with and without a magnitude mask applied to reduce left/right obliquity of the focus. This reduction isevident in both the MRTI and the simulations. However, the simulations did not predict the lateral heat spread into the internal capsule (arrow).

liers, we used the robust fitting option which uses iterativelyreweighted least squares with a bisquare weighting function.

III. RESULTS

A. Comparing shape, size, and obliquity of simulated andmeasured temperature maps

Figure 4 shows examples of the simulated pressure and thecorresponding measured and simulated focal heating duringTcMRgFUS, here in a pallidotomy procedure. This patientwas selected due to the large left-right tilt in the focal heat-ing. Pressure distributions, displayed in the xyz planes ofthe transducer, are shown with the phase corrections usedduring treatment, ideal corrections where the pressure for eachelement have equal phase at the target, and no correction. Thesimulated temperature maps are shown after spatial averagingto match the planes of the MRTI. The relative sizes and theobliquity of the focal heating are similar to the measurements.

This general agreement was evident overall. Figure 5(a)shows example simulated temperature maps and correspond-ing MRTI for patients 1–32 in different imaging orientations.The relative size and obliquity of the focal heating were sim-ilar in most patients. However, the heating was more diffusein the MRTI than was predicted by the simulations; Fig. 5(a)shows 50% isotherms in the MRTI and 25% isotherms in thesimulations.

To characterize the ability of the simulations to predictthe shape of the heating measured with MRTI, we estimatedthe dimensions and obliquity by fitting the focal heatingin the MRTI and the simulated temperature maps to atwo-dimensional Gaussian distribution [Figs. 5(b)–5(d)].A correlation was observed between both the dimensionsand tilt angles measured in the MRTI correlated and thosepredicted by the simulations [correlation coefficient (R2):0.62 and 0.74, respectively; probability value (P < 0.001)].However, the dimensions of the simulations were smaller thanthe measurements in most cases. The simulations predicteda tilt in the same direction as the MRTI, but the absoluteangle was less in the simulations, particularly in the left/rightdirection in coronal MRTI. The ratios of the dimensions(σx/σy) of these fits for the simulations and measurementswere correlated (R2: 0.76).

The simulated heating shown in Fig. 5 used the opti-mized relationships between density and acoustic proper-ties described below. Similar results were found using therelationships described by Pichardo et al. [15]. There wasno significant difference (P > 0.05) between the optimizedand Pichardo relationships in the resulting dimensions orangularity of the simulated focal region.

Left/right obliquity and elongation in the superior/inferiordirection poses a risk during thalamotomy of damaging theinternal capsule, which is located laterally and inferior to the

033205-9


Optimized αOptimized c

S: 1.10 ± 0.03YI: -2.46 ± 0.50R²: 0.74N: 431/431

(c) Measured ∆T (°C)100 20 30

(a)

Pichardo αPichardo c

Measured ∆T (°C)100 20 30

Sim

ulat

ed ∆

T (°

C)

0

10

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50S: 0.46 ± 0.02YI: -0.49 ± 0.34R²: 0.56N: 312/431

(d)

(e)

Optimized α, optimized c

Pichardo α, c

Sim

.-Mea

s.∆T

(°C

)

104 105103

-20-10

01020

Optimized α, optimized c (non-deviants)

Sim

.-Mea

s.∆T

(°C

)

-20-10

010203040

(f)

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.-Mea

s.∆T

(°C

)

Accumulated acoustic energy (J)

-20-10

01020

(b) Measured ∆T (°C)

Optimized αOptimized c (non-deviants)

100 20 30

S: 1.04 ± 0.03YI: -2.08 ± 0.46R²: 0.78N: 312/431

DeviantsNon-deviants

FIG. 7. Comparison of measured and simulated peak tempera-ture rise for 431 sonications in 32 patients. [(a)–(c)] Comparisonusing the density/attenuation (α) and density/sound speed (c) rela-tionships found in Pichardo et al. [15], using optimized attenuation,and using optimized attenuation and sound speed. [(d)–(f)] Differ-ence between the predicted and measured heating as a function ofthe accumulated acoustic energy. With the literature relationships[(a) and (d)], the simulations generally underpredicted the measuredheating. Using the optimized attenuation relationship [(b) and (e)],the agreement was better. At low energies, the difference betweenthe predictions and measurements was constant. However, in 15/32patients, the simulations began to overpredict the focal heating asthe energy increased. These deviant sonications (red crosses) wereexcluded from the attenuation optimization. Using the optimizeddensity/sound speed relationship resulted in better agreement for thedeviant sonications (c). The lines in (d)–(f) connect data obtained ineach patient. (S, YI: slope, y intercept of linear regression).

location of the thalamic target. An extreme example of sucha case is shown in Figs. 6(a)–6(h). In this patient, there wasobliquity in both the left/right and superior/inferior directions,leading to heating in the internal capsule that could not havebeen detected in the single-plane MRTI in any orientation[Figs. 6(c) and 6(f)]. This double obliquity was predicted bythe simulations [Figs. 6(g)–6(h)]. Another example of this riskis shown in Figs. 6(i) and 6(j). The device manufacturer addedan option to change the magnitude distribution of the phased

array to reduce the left/right obliquity. Figures 6(i) and 6(j)show MRTI acquired with and without this magnitude mask.While a corresponding reduction in obliquity is also evidentin the simulations, the shape of the predicted heating is lessdiffuse and does not include the lateral heat spread into in theinternal capsule [arrow in Fig. 6(i)].

B. Predicting focal heating

The examples presented above demonstrate that the simu-lations generally predicted the relative size and shape of thefocal regions that were observed in MRTI. We also comparedmeasurements and simulations of the peak temperature at thefocus for the 431 sonications delivered without MRTI artifactsin these 32 patients.

Our first simulations used the relationship between densityand attenuation found by Pichardo et al. [15]. After comparingsimulations and measurements (Fig. S2), we made two gen-eral observations. First, while in some cases good agreementwas observed, we found that the simulated temperatures weregenerally less than the measurements, particularly when theexposure level was relatively low. Second, we noted that plotsof acoustic energy versus heating followed parallel trajectoriesduring the initial sonications performed during the treatments.However, in many patients, these trajectories deviated as theacoustic energy increased, and the measured heating increasedless than the simulations predicted as the energy increased.

One interpretation of these two observations is that the rela-tionship between skull attenuation and density was incorrect,and that in some patients the acoustic properties of the skullchanged after a certain level of exposure, leading to defo-cusing. Thus we evaluated the feasibility of optimizing thedensity/attenuation relationship to better match the MRTI forthose sonications delivered before this change. Figures 7(a)and 7(d) compare focal heating in MRTI to simulations usingthe attenuation/density relationship found by Pichardo et al.[15]. Results for the optimized attenuation are shown inFigs. 7(b) and 7(e). The sonications indicative of a possiblechange in skull properties were evident after examining plotsof the difference in heating between the simulations andthe measurements as a function of the accumulated appliedacoustic energy [Figs. 7(d)–7(f)]. Those deviant points, shownas red crosses, were selected automatically for each patientwhen two or more sonications exceeded a cut-off of twostandard deviations above the mean difference (indicated bythe light blue regions in Figs. 7(d)–7(f) for the optimizedattenuation model. With the optimized model, 15/32 patientsshowed an obvious deviation in focal heating from the model.The mean previously applied energy at which this deviationwas 17.6 ± 10.0 kJ.

We next tested whether we could find a relationship be-tween sound speed and density that predicted focal heatingthat better matched the measurements for the sonicationsboth before and after the presumed change to the skull. Theresulting estimated peak temperatures for the deviant sonica-tions are shown with the red crosses in Figs. 7(c) and 7(f). Theagreement with the measurements was improved on average,but the heating in some patients was still not well predicted.

Figure 8(a) shows plots of the measured and simulatedtemperature rise as a function of the acoustic energy for

033205-10


0

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Time (s) Time (s) Time (s) Time (s) Time (s) Time (s) Time (s) Time (s)0 3010 20 0 3010 20 0 3010 20 0 3010 20 0 3010 20 0 3010 20 0 3010 20 0 3010 20

MRTI Simulation(b)

1 10Energy (kJ)

1 10Energy (kJ)

1 10Energy (kJ)

1 10Energy (kJ)

1 10Energy (kJ)

1 10Energy (kJ)

1 10Energy (kJ)

1 10Energy (kJ)

MRTI MRTI (deviants) Simulation Simulation (deviants) Simulation (deviants, + sound speed optimization)(a)

ρ ρρρρρρ ρ

ρ ρρρρρρ ρ

ρ ρρρρρρ ρ

ρ ρρρρρρ ρ

FIG. 8. Comparing predicted and measured focal temperatures. (a) Measured and simulated focal heating as a function of the appliedacoustic energy. For the sonications where the measurements and simulations deviated (red crosses in Fig. 7), results from the two differentdensity/sound speed optimizations are shown. The horizontal dotted lines indicate an absolute temperature of 55 ◦C, a rough estimate forthermal necrosis; vertical lines indicate the energy needed to reach this threshold. The insets show histograms of skull density (x axis: 1290 −3500 kg/m3). (b) Measured and predicted temperature rise vs time plots for two sonications selected from each patient. The two selectedsonications had focal temperatures close to 47 ◦C and 55 ◦C during each treatment. The heating and subsequent cooling measured with MRTIwere each fit to an exponential function. The apparent cooling rate for the simulations was higher than the measurements.

patients 1–32. With the optimized attenuation/density rela-tionship, excluding the deviants, the trend in these plots wassimilar for measurements and simulations, and the simulationscaptured effects such as beam steering, changing sonicationduration, and varying the plane used for the MRTI. Theseeffects are evident in Fig. 8(a) where the simulated temper-ature/energy relationships are not smooth. The simulationsgrossly underpredicted the focal heating in a few patients(such as patients 2, 3, and 16). Furthermore, in most patientsthe measurements followed a logarithmic curve [appears lin-

ear in Fig. 8(a), which has a logarithmic scale on the x axis],while the simulations predicted a more linear relationship[appears curved in Fig. 8(a)]. Patient 6 is an example of thisbehavior. Figure 9 compares the slopes of the measured andpredicted heating versus acoustic energy curves in Fig. 8(a). Agood correlation between measurements and simulations wasobserved (R2: 0.57), but the simulated efficiency was slightlyhigher in most patients.

Using the optimized density/sound speed relationship im-proved the prediction of the focal heating for the deviant

033205-11


151050 20 25 30 35 40Efficiency - Measurement [°C/log(kJ)]

0

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25

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Effic

ienc

y - S

imul

atio

n [°

C/lo

g(kJ

)]

Slope: 1.1 ± 0.1

Excluding deviants

Deviants

y-intercept: 2.9 ± 2.6R²: 0.57

FIG. 9. Treatment efficiency. Comparisons of the measured andsimulated slopes of the plots in Fig. 8 of focal heating as a function ofthe logarithm of the acoustic energy for 32 patients. The blue squaresshow the slopes excluding sonications where the measurement andsimulated trajectories of the heating/acoustic energy plots deviated;red circles included those sonications. The error bars are the standarderror of least-squares regression.

sonications. The predicted focal heating with this optimizedrelationship is shown in Fig. 8(a) by the green “×” symbols.There were some patients (patients 29 and 30, for example)that were still not well-predicted by the simulations after thisoptimization.

As we gained experience with these treatments, we usedlarger steps in acoustic energy between sonications. Thus, intwo cases (patients 27 and 30), all sonications except the firstfew low-energy ones had heating less than the simulationspredicted. In these two patients, there was a large differencebetween the simulation and the measurements even though theprior energy delivered was low.

C. Focal heating versus time

We compared plots of the temperature rise as a functionof time for the measured and simulated MRTI [Fig. 8(b)].For each patient we selected two sonications, one wherethe heating was approximately 10 ◦C, and another wherethe temperature rise was approximately 18 ◦C (correspondingto an absolute temperature of 55 ◦C). The heating rates ofthe simulations and measurements were similar. How-ever, the decay of the heating after the sonication was faster inthe simulations than the measurements. To characterize this,we fit the temperature decay to an exponential function andestimated the time required to reduce the heating by one half.The mean half-life was 14.0 ± 3.7 s in the measurements, sig-nificantly higher (P < 0.001) than the 8.6 ± 1.1 s estimated inthe simulations.

D. Optimizing relationships between skull density,attenuation, and sound speed

In optimizing the attenuation/density relationship, we useda simplified attenuation model that allowed us to rapidlyestimate the focal heating for different scenarios. We esti-mated the temperature rise for every patient and sonication for10 000 different attenuation versus density curves. The bestresult we found [excluding the deviants shown in Figs. 8(b)and 8(d)] is shown in Fig. 10(a). The relationships that werewithin 10% of the best curve followed a similar trajectoryexcept for higher densities that were rarely observed in these32 patients. Using the best relationship, the full simulationswere repeated. Comparison of the focal heating betweenthe simplified and full simulations are shown in Fig. 10(b).Overall the two estimates were correlated (R2: 0.94), withheating estimated by the simplified model slightly higher inmost patients than that predicted by the full simulations.

After the attenuation optimization, we performed a similarprocedure with sound speed. Figure 10(c) shows the den-sity/sound speed relationships that best predicted the mea-sured peak focal heating for the deviant and nondeviant son-ications. Figure 10(d) compares the heating predicted usingthe simplified sound speed model to that predicted by the fullsimulation. The two were correlated (R2: 0.84), but there wasconsiderable mismatch in some patients, and on average theheating was slightly higher in the simplified model.

The coefficients characterizing the optimized density/attenuation and sound density/speed found in Eqs. (3) and (6)that resulted in the best agreement between the models and themeasurement are listed in Table III. The difference in heatingbetween measurement and the simulation varied substantiallyfor the different relationships tested (Fig. S3).

E. Validating the optimization

We simulated an additional 40 ET patients (396 sonica-tions) using the optimized relationships we found betweenskull density and the attenuation and sound speed. Over-all the agreement between simulation and measurement inthe peak temperature rise was similar for these patients forthe nondeviant sonications (Fig. 11). The optimization of thedensity/sound speed relationship for the deviant sonicationsimproved the prediction in many cases, but there were severalcases where the simulations continued to overpredict the focalheating. Additional information about these patients is shownin Table S1 and Fig. S4. When the 72 patients were consideredtogether (825 sonications), regression of the simulated versusmeasured temperature rise at the focus yielded a slope of1.05 ± 0.02, and a y intercept of −1.31 ± 0.39 (R2: 0.71).The root mean square difference between simulation andmeasurement was 3.4 ± 3.1 ◦C.

We also repeated the optimization of the density/attenuation and density/sound speed relationships in these 40patients and for all patients (Fig. 12). The best relationshipsfound were similar to those from patients 1–32.

F. Effects of spatial averaging and aberration correction

We investigated the effects of spatial averaging over theimaging planes used for MRTI [Fig. 13(a)] and aberration

033205-12


% Skull Voxels

0.09

0.08

0.07

0.06

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0.01

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ρ from 32 patients

302010 40 50 60 700

∆T (°C) - Estimated attenuation

Slope: 0.86 ± 0.01Y-int: 0.69 ± 0.2R²: 0.94

Skull Density (kg/m³)1500 2000 2500 3000 3500


Atte

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(a)

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∆T (°C) - Estimated sound speed

Slope: 0.90 ± 0.02Y-int: 1.16 ± 0.21R²: 0.84

∆T (°

C) -

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l sim

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ion

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(d)

FIG. 10. Optimized skull density/attenuation and density/sound speed relationships. [(a) and (b)] For each of the 431 sonications, 10 000different density/attenuation relationships were investigated using simulations performed without attenuation and a simplified attenuationmodel. The relationship that produced the best match to the MRTI and those that were within 10% of the best are shown. The attenuation modeldescribed by Pichardo et al. [15] is also shown. A histogram of the density of all the skulls in the study is shown in green. (b) Plot showingthe heating predicted by the full simulations vs that estimated using the simplified attenuation model in Eq. (5). The simplified attenuationmodel slightly overestimated the focal heating predicted by the full simulations. (dotted line: linear regression; solid line: unity). [(c) and(d)] Optimization of the density/sound speed relationship using a similar methodology. Here, the optimization was performed separately forthe deviant and nondeviant sonications. (d) Focal heating estimated with the full simulation compared to the simplified model outlined inEqs. (6)–(11). Good agreement on average was observed, although the simplified model deviated from the full model in some patients. Thedata points for the individual patients are connected in (b) and (d) by the pale grey lines.

correction [Fig. 13(b)] on the simulated temperature maps.Overall, the predicted temperature rise was 1.5 times higherbefore spatial averaging when the phase corrections usedduring treatment were used. Given that the measured heatingwas more diffuse than the simulations predicted, this may beoverstating the effects of averaging. The simulations predictedthat using the ideal phase corrections would result in improvedheating 1.2 times higher than the corrections used during thetreatments. The simulations also predicted that without the

phase aberration corrections used during the treatments thatthe temperature rise would have been reduced by half onaverage.

G. Predicting ablative energy

There was substantial variability among the different pa-tients in the ultrasound exposure levels needed to achieve atemperature rise sufficient to produce a thermal lesion, pre-

TABLE III. Coefficients for Eqs. (3) and (6) that resulted in the best agreement between MRTI and the simulations.

A0 A1 A2 A3 A4

Before skull change: 5.71 × 103 −9.02 5.40 × 10−3 −1.41 × 10−6 1.36 × 10−10

B0 B1 B2 B3 B4

Before skull change: 3.68 × 10−3 −5.95 × 10−6 4.13 × 10−9 −1.28 × 10−12 1.48 × 10−16

After skull change: 1.24 × 10−3 −7.63 × 10−7 1.69 × 10−10 5.31 × 10−16 −2.79 × 10−18

033205-13


Optimized α, optimized c

Optimized α, optimized c(non-deviants)

Sim

.-Mea

s.∆T

(°C

)

-20-10

010203040

(b)(a) Measured ∆T (°C)

Optimized αOptimized c (non-deviants)

100 20 30

Sim

ulat

ed ∆

T (°

C)

0

10

20

30

40

50S: 0.89 ± 0.03YI: -0.18 ± 0.45R²: 0.78N: 241/394

Optimized αOptimized c

S: 1.04 ± 0.03YI: -0.87 ± 0.61R²: 0.70N: 394/394

Measured ∆T (°C)100 20 30

Sim

.-Mea

s.∆T

(°C

)

Accumulated acoustic energy (J)

-20-10

01020

104 105103(d)

(c)

DeviantsNon-deviants

FIG. 11. Validating the density/attenuation and density/soundspeed optimization in 40 additional patients. [(a) and (b)] Compar-ison of simulated and measured focal heating using the optimizedattenuation (α) and sound speed (c) relationships shown in Fig. 10 forpatients 33–72. [(c) and (d)] Plot of the difference in the simulatedand measured heating as a function of the accumulated energydelivery. The results for these patients are similar to those shownin Fig. 8 for the initial 32 patients used for the optimization. (S, YI:slope, y intercept of linear regression).

sumably due to differences in scattering or other differencesin internal skull properties. To characterize this variability,we estimated the acoustic energy required in each patient toachieve an ablative thermal exposure of 55 ◦C in MRTI [dottedvertical lines in Fig. 8(a)]. This energy range spanned anorder of magnitude; it ranged 3.7–34.6 kJ in patients 1–32 and3.3–36.1 kJ in patients 33–72. We compared the ability of thesimulations to predict this exposure level along with multipleother metrics gleaned from the CT scans in patients 1–32(Fig. S5). Correlation between the measured and simulatedenergy needed to reach 55 ◦C was significant (P < 0.001;R2: 0.78). Several other skull-derived metrics also showeda significant correlation. The three cases that required thehighest energy were outliers in these regressions, and the highacoustic energy required to reach an ablative thermal exposureat the focus (>20 kJ) was only predicted by the simulations.

Non-deviantsPat. 1-32Pat. 33-40All


Atte

nuat

ion

(Np/

m)

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Deviants


Soun

d Sp

eed

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)

1500

2000

2500

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3500

4000

(b)

within 10% of best (All)

FIG. 12. Optimizing density/attenuation and density/soundspeed relationships using results from 72 patients. The relationshipsfound were similar for those for patients 1–32.

Figure 14 shows color-coded maps of the skulls for patients1–32 showing the spatial distribution of three metrics thathad highly significant correlations with the energy needed toreach 55 ◦C: loss in pressure due to skull attenuation, skullthickness, and cortical/trabecular density ratio. The patientsare ordered by the acoustic energy needed to reach 55 ◦C.While trends are evident, there was substantial variability.Note, for example, patient 19, who’s skull was relatively thickand highly attenuating but required very little energy, andpatient 30, who’s skull was thinner and less attenuating thanothers that required less energy.

We also compared these skull-derived metrics for patientsthat did and did not have sonications with heating/energytrajectories that deviated from the predictions. The skullsof the patients that had sonications with deviations werethinner (6.8 ± 1.0 versus 8.0 ± 1.1 mm; P < 0.001) and hada smaller volume (278 ± 47 versus 330 ± 62 cm3; P < 0.001)than those where the trajectories did not deviate. The corre-

033205-14


(a) (b) ∆T (°C): Tx phase corrections Voxel-averaged ∆T (°C)

0

10

20

30

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20 30010 20 30 400

∆T (°

C):

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l or n

o co

rrec

tions

Peak

∆T

(°C

)

NoneIdeal Tx

Slope:None: 1.4 ± 0.02Tx: 1.5 ± 0.01Ideal: 1.7 ± 0.01

Slope:Ideal: 1.2 ± 0.01None: 0.5 ± 0.01

Phase correction

ideal phasecorrection

no phasecorrection

FIG. 13. Impact of spatial averaging and aberration correction onfocal heating. (a) Effects of spatial averaging on the simulated peaktemperature rise at the focus using the phase corrections calculatedby the manufacturer, the ideal phase corrections, and no correction.The models suggest that the peak temperature rise was 1.4–1.7 timesthe value measured with MRTI. (b) Plot of simulated temperaturerise using ideal phase corrections and no correction versus those usedduring treatment (Tx). The models suggest that the correction usedduring the treatment improved the measured heating by a factor of2, and that using the ideal correction predicted could yield a meanimprovement of approximately 20% (dotted lines: linear regressions;solid lines: unity).

sponding loss due to attenuation was also less (0.69 ± 0.06versus 0.79 ± 0.74 ± 0.05; P < 0.001) in the patients withdeviants. The standard deviation of the attenuation along thedirection of ultrasound propagation was also significantlyhigher (67.5 ± 4.7 versus 60.7 ± 4.3 Np/m; P < 0.001).

H. Comparing “ideal” and treatmentphase aberration corrections

The phases used for aberration correction determined bythe device software and used in the treatments were grosslysimilar to the “ideal” phases predicted by the simulations inevery case in patients 1–32 (R2: 0.84, P < 0.001; Fig. S6).We examined whether the different metrics gleaned from theCT scans were correlated with differences in the predictedideal phase corrections and those used during treatment.Factors that had a significant correlation with the differencebetween the ideal and treatment phases were the numberof elements with incidence angles greater than 25◦ (P <

0.001), the standard deviations of the density, sound speed,and impedance, the density and sound speed of the outertable (P < 0.05), and the difference between the internal andexternal incidence angles (P = 0.001). The proprietary modelused by the manufacturer considers shear mode transmissionfor incidence angles above a certain threshold, so it is notsurprising that differences that were correlated with factorsrelated to incidence angles.

Loss

due

to a

ttenu

atio

n

0.45

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4

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14

Skul

l thi

ckne

ss (m

m)

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tical

/trab

ecul

ar ρ

(SD

R)

0.4

0.5

0.6

0.7

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0.9

1.0

3.7 kJ 4.2 kJ 4.4 kJ 4.4 kJ 4.6 kJ 5.3 kJ 5.4 kJ 5.4 kJ


8 kJ 8.3 kJ 8.8 kJ 9.5 kJ 10 kJ 10.8 kJ 12.2 kJ 12.7 kJ






P21 P06 P19 P32 P24 P07 P15 P18

P31 P27 P12 P25 P26 P29 P17 P03

P01 P13 P04 P05 P09 P22 P11 P14

P20 P30 P02 P28 P16 P23 P10 P08









P21 P06 P19 P32 P24 P07 P15 P18

P31 P27 P12 P25 P26 P29 P17 P03

P01 P13 P04 P05 P09 P22 P11 P14

P20 P30 P02 P28 P16 P23 P10 P08









P21 P06 P19 P32 P24 P07 P15 P18

P31 P27 P12 P25 P26 P29 P17 P03

P01 P13 P04 P05 P09 P22 P11 P14

P20 P30 P02 P28 P16 P23 P10 P08

FIG. 14. Skull-derived metrics and treatment efficiency. Mapsof the outer skull surface color-coded based on different metricsderived from the CT scans are shown for 32 patients. The patientsare sorted by the acoustic energy needed to reach a focal temperatureof 55 ◦C, which covered an order of magnitude. A general trendtowards increasing acoustic energies as the loss due to attenuation,skull thickness, and attenuation coefficient of the diploe increasedand as the trabecular/cortical density ratio decreased is evident. Clearoutliers for each metric are also evident.

IV. DISCUSSION

This work describes our experience testing a simulationframework that enables rapid (∼2.5 s) calculation of thepressure field for different transducer magnitude/phase dis-tributions and that could be iterated using simplified modelsto identify relationships between the CT-derived density andthe skull acoustic properties that better matched the MRTI.This approach required more upfront work to generate theindividual elemental simulations, but it could be performedin parallel using a computing cluster and potentially allowsfor more flexibility in exploring how changing different prop-erties affects focal heating.

Ultimately, we aim to assemble a look-up table of elemen-tal skull segments and pressure distributions to enable us toaccurately predict the three-dimensional pressure distribution

033205-15


in real time. This table could be based on similarity compar-isons between skull segments or based on derived factors suchas thickness, angles, etc. Machine learning methods may beuseful in developing these predictive models to relate the skullsegments to the resulting pressure distributions. If one had theability to predict the three-dimensional pressure distribution inreal time, one could rapidly iterate the phase and magnitude ofthe transducer elements (or the position and angulation of thetransducer itself) to optimize not only the peak temperaturerise but also the shape and size of the focal heating. Withan ability to predict the pressure amplitude in real time, onecould also apply holographic methods [30] to shape the focalregion to match the desired anatomy of interest. Such shapingmay be more useful in TcMRgFUS applications other thanthermal ablation, such as blood-brain barrier disruption [6,7],neuromodulation [31], and nonthermal ablation [32] that donot require high acoustic exposure levels, as modifying thespatial pressure field would likely reduce the peak pressureamplitude at the focus.

To achieve this goal, we need to better predict the shapeof the focal heating. The heating measured with MRTI wasmore diffuse than the simulations predicted. This result couldsuggest that acoustic parameters used in the model were notcorrect, or it could reflect factors that were not considered herein the simulations. Transmission after shear mode conversionin elements with oblique incidence angles, for example, wasnot accounted for in these simulations. While the high shearattenuation coefficient will limit this transmission, studies incadaver skulls suggest that transmission of up to 17%−23%of the power can be transmitted via a shear wave for incidenceangles greater than 30◦ [33]. Complex reflections, such asthose that occur from the face of the transducer and thosethat occurred outside of the elementwise simulation volumesmay have also contributed, and probably to a small effect[34], nonlinear acoustic propagation. These components ofthe transmitted acoustic wave were not taken into account inthe aberration correction, perhaps leading to diffuse heating atand around the focal region in the measurements that was notpresent in the simulations. This diffuse heating may explainthe slower temperature decay after the sonications [Fig. 8(b)].A faster decay in the simulation might also suggest that thethermal properties and the perfusion coefficients used forbrain were incorrect.

Other than the diffuse heating, the simulations did a goodjob overall in predicting the relative peak temperature riseand orientation/obliquity of the focal region. However, itis possible that the parameterization of the CT scans andthe acoustic and thermal properties need refinement. Forexample, based on earlier works [14,15], we used a simplelinear extrapolation to estimate skull density from Hounsfieldunits, and the Hounsfield units were estimated directly usinginformation from the DICOM headers. Others have suggestedpotentially more accurate approaches to estimate skull densityfrom CT scans [16,35], and have shown that scans from CTscanners from different vendors and with different reconstruc-tion kernels can yield different results [36]. Here, the CT scanswere from a variety of vendors, but we did not see any obviouseffects on the simulations.

We began this project using the experimentally derivedskull density/sound speed and density/attenuation relation-

ships found by Pichardo et al. [15]. While we did observegood agreement between the measurements and the simula-tions in some patients, in most cases the peak temperaturewas underpredicted by the numerical model for sonications atrelatively low exposure levels. Furthermore, in many patients,the relationship between acoustic energy and temperature risefollowed a similar trajectory in the initial sonications butappeared to deviate after a certain amount of acoustic energywas applied.

Based on these results, as well as clinical experienceshowing that increasing acoustic energy often produces di-minishing increases in heating as the treatment progresses[11] and that has observed skull damage in some patientsafter treatment at high energies [37], we hypothesized that (1)the relationship between attenuation and skull density foundby Pichardo et al. was not optimal, and (2) the deviationcould represent an irreversible change in skull acoustic prop-erties. We attempted to find a density/attenuation relationshipthat better predicted the MRTI measurements for sonicationsdelivered before the skull changes. This process improvedthe predictive ability of the simulations substantially, butthere were still patients where the agreement was poor. It ispossible that there is no single relationship between densityand attenuation, as we do not capture the microstructureof the bone below the resolution of the CT scan, whichlikely affects the scattering of the acoustic field [38,39].More work is needed to understand how well we can useclinical imaging to predict the attenuation for an individualpatient.

We also explored whether we could find a skull den-sity/sound speed relationship that could be used after a (pre-sumed) change in skull properties. While we were able toidentify a relationship that improved the predictive abilityof the model after the simulated heating deviated from themeasurements, there was still substantial variability. Valida-tion of these relationships in 40 additional patients revealedsimilar agreement between predicted and measured heatingoverall. However, the density/sound speed relationship foundin patients 1–32 after the presumed skull changes had evenmore variability in patients 33–72. Finding such variabilityis probably not surprising, since heat-induced changes to theskull will not be spatially uniform or simply binary. It may noteven be correct to assume that the sound speed is the dominantfactor that changes during the treatment.

Furthermore, our interpretation of the results that the skullacoustic properties changed may not be correct for every pa-tient. Note, for example, patients 27 and 30 in Fig. 8(a), wherethe predicted heating with the optimized density/attenuationrelationship was substantially higher than the measurementseven though the previously delivered energy was low. Similarfindings were observed in several treatments of the 40 ad-ditional patients (Fig. S4). These results might suggest thatthe deviations in temperature/energy were not due changesin skull properties from previously applied sonications, butperhaps instead a dynamic change in acoustic propertieswith heating. It could also be that the sonications in thesepatients should not have been included as “deviants” and thatour optimized density/attenuation relationship needs furtherrefinement. Since we do not have heating measurements atintermediate energies in these patients, we cannot answer this

033205-16


question. Future studies with additional patients are needed tounderstand cases like these.

Other factors may contribute as well. For example, theperformance of the transducer elements might change if theyheat at high power levels, or strong reflections from the skullcould change their impedance.

We have demonstrated how one could use elementwisesimulations to improve the relationships between acousticattenuation and sound speed. Here, we used simplified modelsthat could be rapidly applied to previously obtained simula-tions to explore the effects of thousands of different relation-ships on resulting focal heating. While these simplified mod-els did a good job in predicting the temperature at the focusfor the full simulation, a better approach would be to explorethese relationships to match not only the peak temperature butalso the shape and size of the focal heating. Perhaps it wouldalso be better to optimize over multiple acoustic and thermalparameters simultaneously. Individual patients or groups ofpatients may have skulls with different relationships betweendensity and acoustic properties. Repeating this analysis inmore patients may reveal such trends. Finally, we need to eval-uate whether the relationships between density and acousticproperties found here can be used prospectively improve thefocusing in future TcMRgFUS treatments.

A. Predictors of treatment efficacy

We observed an order of magnitude range in energy re-quired to achieve an effective level of heating. It would beuseful to predict this energy before treatment to screen poten-tial TcMRgFUS patients. Currently, the device manufactureruses the “skull density ratio”, which is the ratio in densitybetween the diploic trabecular and dense cortical bone [10,40]to predict which patients will require high acoustic energiesto achieve focal temperatures sufficient to induce a thermallesion. We examined different skull-derived parameters tosee how they compare in this prediction. The manufacturer-derived “skull density ratio” as well as our own calculation ofthis ratio were both predictive of the energy needed to reach55 ◦C (R2: 0.35, 0.40, respectively; P < 0.01), but a number of

other parameters, such as the skull thickness and the expectedloss due to attenuation were also similarly predictive. How-ever, there was significant variability and there were outliersthat were not predicted by simple skull measurements. Thesimulations did a better job in predicting the acoustic energyneeded to reach 55 ◦C (R2: 0.78; P < 0.001) than any singlefactor.

B. Conclusions

This study demonstrates the feasibility of using an ele-mentwise approach to simulate TcMRgFUS thermal ablationand predict the shape and magnitude of the focal heating inmultiple orientations. While there was significant variability,the numeric model predicted the relative shape and focaltemperature rise on average. Deviations between the measuredand simulated heating were observed in many patients overthe course of the treatments, perhaps reflecting a change inskull properties. We also demonstrated how this approachcould be used to optimize the relationship between CT-derived density and the acoustic attenuation and sound speed.We found initial estimates of these relationships based on32 patient treatments, including an estimated density/soundspeed relationship that could be used after the model andthe measurements deviated. The optimization was validatedin 40 additional patients. Future work will expand on thisstudy to refine these optimized relationships. Importantly, theoptimized relationships need to be validated prospectivelyduring clinical TcMRgFUS treatments.

ACKNOWLEDGMENTS

This work was supported by NIH Grant No.R01EB025205. The authors thank Greg Clement, AlexHughes, Samuel Pichardo, Urvi Vyas, and Brad Treeby fortheir helpful advice, and the engineers at InSightec for theirhelp in extracting the transducer and sonication parametersfor these treatments. Portions of this research were conductedon the O2 High Performance Compute Cluster, supported bythe Research Computing Group at Harvard Medical School.

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