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Accurate simulation of transcranial ultrasound propagation for ultrasonic neuromodulation and stimulation

Accurate simulation of transcranial ultrasound propagation for ultrasonic neuromodulation and... 18 April 2024 14:53:13 MARCH 13 2017 Accurate simulation of transcranial ultrasound propagation for ultrasonic neuromodulation and stimulation James L. B. Robertson; Ben T. Cox; J. Jaros; Bradley E. Treeby J. Acoust. Soc. Am. 141, 1726–1738 (2017) https://doi.org/10.1121/1.4976339   View Export Online Citation 18 April 2024 14:53:13 Accurate simulation of transcranial ultrasound propagation for ultrasonic neuromodulation and stimulation 1,a) 1 2 1 James L. B. Robertson, Ben T. Cox, J. Jaros, and Bradley E. Treeby Department of Medical Physics and Biomedical Engineering, University College London, London, United Kingdom Faculty of Information Technology, Brno University of Technology, Brno, Czech Republic (Received 13 June 2016; revised 1 December 2016; accepted 31 January 2017; published online 13 March 2017) Non-invasive, focal neurostimulation with ultrasound is a potentially powerful neuroscientific tool that requires effective transcranial focusing of ultrasound to develop. Time-reversal (TR) focusing using numerical simulations of transcranial ultrasound propagation can correct for the effect of the skull, but relies on accurate simulations. Here, focusing requirements for ultrasonic neurostimula- tion are established through a review of previously employed ultrasonic parameters, and consider- ation of deep brain targets. The specific limitations of finite-difference time domain (FDTD) and k-space corrected pseudospectral time domain (PSTD) schemes are tested numerically to establish the spatial points per wavelength and temporal points per period needed to achieve the desired accuracy while minimizing the computational burden. These criteria are confirmed through conver- gence testing of a fully simulated TR protocol using a virtual skull. The k-space PSTD scheme performed as well as, or better than, the widely used FDTD scheme across all individual error tests and in the convergence of large scale models, recommending it for use in simulated TR. Staircasing was shown to be the most serious source of error. Convergence testing indicated that higher sampling is required to achieve fine control of the pressure amplitude at the target than is needed for accurate spatial targeting. 2017 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4976339] [JFL] Pages: 1726–1738 invasive neural excitation and modulation, with focusing on I. INTRODUCTION the scale of the acoustic wavelength. Table II shows a selec- The use of implanted electrodes for deep brain stimula- tion of UNMS papers published in the last decade, and dem- tion (DBS) is a well-established, invasive treatment for onstrates the variety of acoustic intensities and frequencies multiple neurological conditions and has directly resulted used, target structures sonicated, and neural responses in a greater understanding of functional neuroanatomy observed. The physical mechanism underlying UNMS and deep brain circuitry. Unfortunately, its usefulness is remains unclear, although a non-thermal mechanism is sus- limited by the inherent risks of the required neurosurgery pected, and lower acoustic frequencies have been shown to combined with difficulties in targeting and repositioning 3,6 evoke a response more reliably. Most recently ultrasound the stimulatory focus. Non-invasive alternatives such as has been used to elicit electro-encephalogram (EEG) and transcranial magnetic and direct current stimulation have both sensory responses in human subjects, although this has been met with success in research and clinical settings. However, restricted to superficial cortical brain areas using unfocused they are limited in terms of their ability to achieve tight 7–9 single element transducers. If UNMS is to develop as an spatial focusing, and their penetration deep into tissue. Table effective non-invasive neurostimulation technique, its appli- I demonstrates a selection of existing and planned DBS target cation to human subjects must be extended to deep brain tar- structures alongside their approximate dimensions and devia- gets. Based on the dimensions of DBS targets shown in tion from the approximate center of the brain—the mid- Table I, and the range of effective ultrasonic intensities 4,5 commissural point (MCP). These dimensions demonstrate shown in Table II, the following focusing requirements may the millimeter scale size of the target structures, and their be defined: position close to the center of the brain. Thus, the ability to A spatial targeting error of less than 1.5 mm. non-invasively target these nuclei for modulation and stimula- Control of the intensity at the focus with 10% error will tion would represent a revolutionary neuroscientific tool with ensure that neurostimulation occurs. Greater accuracy both clinical and research applications. may be desirable in studies of the mechanisms and thresh- Ultrasonic neuromodulation and stimulation (UNMS) olds of UNMS. offers a potential solution to these requirements, and has An ultrasonic stimulation focus of no greater than 3 mm recently received a great deal of interest. Transcranial focus- diameter will ensure stimulatory specificity. ing of ultrasound offers the potential for reversible, non- Steering of the ultrasonic focus up to 30 mm from the MCP to allow stimulation of arbitrary deep brain a) Electronic mail: james.robertson.10@ucl.ac.uk targets. 1726 J. Acoust. Soc. Am. 141 (3), March 2017 0001-4966/2017/141(3)/1726/13/$30.00 V 2017 Acoustical Society of America 18 April 2024 14:53:13 TABLE I. Approximate dimensions of DBS targets (Ref. 6). AP/DV/ML— verified by MRI thermometry. Marquet et al. showed that Anteroposterior/dorsoventral/mediolateral, MCP—Mid-commisural point. model-driven TR is capable of restoring 90% of the peak pressure that can be obtained with gold-standard hydrophone AP DV ML MCP deviation based methods when focusing through an ex vivo skull bone. Target [mm] [mm] However, model-driven TR remains subject to systematic Ventral intermediate nucleus 10 15.8 11 17 errors and uncertainties with a resulting loss in focusing Ventral anterior nucleus 7 12.6 10 15 quality or efficiency. Four categories of uncertainty are: Centro-median nucleus 8 4.5 414 Nucleus Accumbens 9.5 10  12 21 (i) The underlying physical model and how the govern- Globus pallidus externus 21.5 10323 ing equations model the physics of propagation Globus pallidus internus 12.5 8620 including phenomena such as absorption, nonlinear- Sub-thalamic nucleus 8 4 6.3 13 ity, and shear wave effects. (ii) Numerical approximations due to the discretization of the simulation domain, including numerical dispersion, The primary obstacle to achieving these ultrasonic the representation of medium heterogeneities, and the focusing criteria within the brain is the presence of the skull, effectiveness of any absorbing boundary conditions. which aberrates and attenuates incoming wavefronts. Time- (iii) The inputs to the model, such as the map of acoustic reversal (TR) focusing, first adapted for transcranial focusing medium properties and the representation of acoustic by Aubry et al., is able to correct for the aberrating effect of transducers. the skull by taking advantage of the time-symmetry of the (iv) How the numerical simulations are used within a lossless acoustic wave equation. In model-driven TR, broader TR protocol, including how the simulated numerical models simulate the propagation of ultrasound source is related to the desired pressure at the target, from a target area to a virtual transducer using acoustic prop- and how phenomena that are not time-invariant, such 9,10 erty maps of the head derived from CT or MRI images. as absorption, are accounted for. The pressure time series at the simulated transducer surface TR simulations for transcranial focusing have typically is then time-reversed, and used to generate drive signals for a multi-element acoustic transducer array. For high-intensity made use of finite-difference time domain (FDTD) numeri- 11,12 thermal applications, model-driven TR may be combined cal models. Recently a k-space corrected, pseudospectral with MRI thermometry for treatment verification. Chauvet time domain (PSTD) numerical scheme was used in model- 11 13 et al. confirmed the potential for model-driven TR-based driven TR and shown to give comparable accuracy. Both focusing inside the human head to millimeter precision, FDTD and PSTD schemes use consistent approximations to TABLE II. Review of selected recent ultrasonic neuromodulation and neurostimulation literature. SPPA—Spatial peak pulse average, SPTA—spatial peak temporal average, SPTP—spatial peak temporal peak, VEP—Visual evoked potential, LGN—lateral geniculate nucleus, FEF—frontal eye field, PET—posi- tron emission tomography, fMRI—functional magnetic resonance imaging, GABA—gamma-aminobutyric acid, S1—primary somatosensory cortex, MC— motor cortex. *—0.5 MHz achieved with 2 MHz carrier. Intensity at focus Target: In-vivo(IV) vs Year Author Freq. [MHz] [W/cm ] Ex-vivo(EV) Neural Response & Observations 2008 Tyler et al. 0.44 and 0.67 2.9 SPPA EV mouse hippocampus Imaging of ion channel opening and synaptic activation 2008 Khraiche et al. 7.75 50–150 SPTP EV rat hippocampus Increased neuronal spike rate 2010 Tufail et al. 0.25 and 0.50 0.228 SPPA IV mouse brain Motor response, cortical spiking, ion channel opening 2011 Yoo et al. 0.69 3.3–12.6 SPPA IV rabbit cortex Motor response VEP suppression and fMRI activity 2011 Min et al. 0.69 2.6 SPPA IV rat epileptic focus Suppression of induced epileptic behavior 2011 Yang et al. 0.65 3.5 SPPA IV rat thalamus Decrease in extracellular GABA levels 2012 King et al. 0.50 1–17 SPTP IV rat brain Motor response above an intensity threshold 2012 Kim et al. 0.35 8.6–20 SPPA IV rat abducens nerve Motor response in abducens muscle 2013 Menz et al. 43 20–60 SPPA EV salamander retina Retinal interneuron stimulation 2013 Deffieux et al. 0.32 4 SPPA IV primate FEF Altered visual antisaccade latency 2013 Younan et al. 0.32 17.5 SPPA IV rat cortex Motor response 2014 Legon et al. 0.50 5.9 SPPA IV human S1 Altered sensory evoked EEG oscillations 2014 Kim et al. 0.35 3.5–4.5 SPTA IV rat thalamus Glucose uptake change, motor response 2014 King et al. 0.5 3 SPTA IV mouse MC Motor response varying with targeting 2014 Juan et al. 1.1 13.6–93.4 SPTA IV rat vagus nerve Reduced vagus compound action potential 2014 Mehic et al. 0.5* 2–8 SPTA IV rat brain Motor response scaling with intensity 2014 Mueller et al. 0.5 5.9 SPPA IV human S1 Altered EEG beta phase dynamics 2015 Lee et al. 0.25 0.5–2.5 SPPA IV human S1 Evoked sensations and EEG changes 2015 Lee et al. 0.25 6.6–14.3 SPPA IV sheep cortex Motor and EEG responses 2016 Ye et al. 0.3–2.9 0.1–127 SPPA IV mouse MC Motor response, more effective at low frequencies 2016 Ai et al. 0.5 and 0.86 6 SPPA IV human brain fMRI activity at stimulation site and deep brain 2016 Darvas et al. 1.05 1.4 SPTA IV rat brain EEG response, focal effects on gamma band activity J. Acoust. Soc. Am. 141 (3), March 2017 Robertson et al. 1727 18 April 2024 14:53:13 the wave equation, and can be made stable by choosing the problems, a Fourier basis is typically used, with the basis discretization parameters appropriately. As the rate of spatial function weights calculated via the fast Fourier transform. and temporal sampling increases, they will converge on the The subsequent gradient calculation is exact, eliminating true solution at a rate dependent on the particular approxima- numerical dispersion due to spatial discretization. However, tions of the numerical model [(ii) above]. However, due to for an explicit time-stepping scheme, temporal gradients the large scale of these simulations, it is desirable to mini- must still be approximated via a finite difference method, mize the grid size and resulting computational burden with- with resulting dispersive error. Fortunately, for a second- out compromising accuracy, so knowledge of the minimum order accurate approximation, this error can be calculated sampling criteria necessary to achieve the required accuracy analytically, and used to introduce a correction factor, j ¼ is valuable. In the present paper, these numerical schemes sincðc kDt=2Þ; in the spatial frequency domain. Here, k ¼ ref are briefly described, and the various factors affecting the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 rate of numerical convergence are examined. This is quanti- k þ k þ k is the magnitude of the wavevector x y z fied in terms of the spatial and temporal sampling required to ðk ; k ; k Þ at each grid point in the spatial frequency domain, x y z obtain acceptable accuracy in the simulation of ultrasound and c is a user defined reference sound speed. This method ref propagation from the scalp to a deep brain target. While the is often called the k-space PSTD method, and in homoge- criteria used are established for the application of transcra- neous media it is unconditionally stable and free from numer- nial UNMS, these results are also applicable to other thera- 15,16 ical dispersion for arbitrarily large Dt: In media with a pies that require accurate transcranial ultrasound simulation, heterogeneous sound speed, the application of j in the spatial such as high intensity focused ultrasound (HIFU) ablation frequency domain means that a single sound speed must be and opening the blood brain barrier with ultrasound. chosen for the correction factor. As a result, numerical dis- persion will arise, the extent of which will depend on the II. NUMERICAL METHODS FOR ULTRASOUND temporal sampling and the difference between the local PROPAGATION sound speed cðxÞ; and the reference sound speed c : As ref A. FDTD with FDTD schemes, simulation-dependent limits on the CFL number must be observed to ensure numerical stability. FDTD methods have seen extensive use in the simula- tion of ultrasound propagation, and have accordingly been C. The BLI 9–12 used for the purpose of model-driven TR with success. FDTD and PSTD methods both use functions to interpo- In finite difference methods, partial derivatives are calcu- lated using a linear combination of function values at neigh- late between the values of the acoustic variables at the grid boring grid points. The finite difference approximations are points. The interpolating functions are used to approximate derived by combining local Taylor series expansions trun- the field gradients at these points, and the values of the field cated to a fixed number of terms. When simulating ultra- variables are updated at the grid points at each time step. sound propagation, this approximation causes an unphysical FDTD methods use polynomials to interpolate between dependence of the simulated sound speed on the number of neighboring points, while PSTD methods use a Fourier grid points per wavelength (PPW ¼ k=Dx) and the number series to interpolate between all points simultaneously. of temporal points per period [PPP ¼ 1=ðfDtÞ] where f and This Fourier series is bandlimited (truncated) to ensure a k are acoustic frequency and wavelength, respectively, and unique Fourier representation and is therefore known as the Dx and Dt are the spatial and temporal discretization, respec- bandlimited interpolant (BLI). This can be considered the tively. This manifests as a cumulative error in the phase representation of the discretely sampled pressure field within of propagating waves, termed numerical dispersion. In addi- PSTD schemes. When a time-varying source is used, the tion, stability conditions must also be met to ensure the resulting pressure signal is formed from a sum of one or numerical scheme is stable. These conditions are contingent more weighted BLIs. As a result of this, a discrepancy can on the exact scheme used and the number of simulated arise between the BLI and the intuitive expectation of what dimensions. A useful metric when considering stability is the sampled function represents. This is shown in Fig. 1(a) the Courant-Friedreichs-Lewy (CFL) number, defined as for a Kronecker delta represented on a discrete grid. In this case, because the Fourier coefficients of the sampled func- cDt PPW tion do not decay to zero before the Nyquist limit of the grid, CFL ¼ ¼ ; (1) Dx PPP the intended field is replaced with a BLI representation with Gibbs type oscillations. It is important to understand that this where c is the sound speed. Stability criteria are often representation is not erroneous per se, but that there is a dis- 14–16 expressed as limits placed on the CFL number. parity between the desired input to the PSTD scheme (in this case a Kronecker delta), and what the scheme is capable of B. PSTD representing via a bandlimited Fourier series. To reduce the In PSTD methods, spatial derivatives are calculated by size of the disparity, smoothing of the intended field can be decomposing the spatially varying acoustic variables into a used to force the Fourier coefficients to decay. This is finite sum of weighted global basis functions. This decom- shown in Fig. 1(b) for the same Kronecker delta function position allows efficient computation of spatial derivatives when frequency is filtered with a Blackman window. using the derivatives of the basis functions. For wave Although this remains an inexact representation of the 1728 J. Acoust. Soc. Am. 141 (3), March 2017 Robertson et al. 18 April 2024 14:53:13 within the head is considered when necessary, with each reverberation consisting of 2 cm propagation through bone, and 20 cm propagation through brain tissue. The combined effects of these numerical inaccuracies and the validity of the established sampling criteria are then examined through con- vergence testing of fully simulated two-dimensional (2D) and three-dimensional (3D) TR protocols. Numerical simulation of ultrasound was carried out with the open source k-Wave toolbox using PSTD and k-space corrected PSTD (with a user defined c )numerical ref 16,19 schemes. These are henceforth referred to as “PSTD” and “k-space” schemes, respectively. The toolbox also includes a FIG. 1. (Color online) Discrete pressure maps and their BLIs. (a) Unsmoothed delta function and (b) delta function frequency filtered with a second-order accurate in time, fourth-order accurate in space Blackman window. (2–4) FDTD scheme, which was also tested. This scheme is described in detail by Strikwerda, and is widely used to sim- original Kronecker delta function, the non-oscillating BLI ulate acoustic wave propagation, including in simulated 11,12 more closely matches the intended underlying pressure dis- TR. Unless stated otherwise, the CFL number was set to tribution as defined by the values at the discrete grid points. 0.3, one-dimensional (1D) tests were carried out on a spatial grid of 4096 grid points, and 2D tests on a 1024  1024 grid. III. NUMERICAL TESTING OF INDIVIDUAL ERRORS Frequency filtered Kronecker delta functions, like that shown in Fig. 1(b), were used to create broadband pressure sources. A. Overview Homogeneous simulation grids were given the acoustic prop- In this section, the impact of various factors which erties of brain tissue, a density of 1040 kg/m ,and a sound affect the convergence of FDTD and PSTD models for the speed of 1560 m/s (also used to represent superficial soft tis- case of transcranial ultrasound simulation is presented. sues). For heterogeneous simulations, bone tissue was These comprise: the influence of the BLI, changes in the assigned a density of 1990 kg/m and a sound speed of effectiveness of the absorbing perfectly matched layer 3200 m/s. When it was necessary to define a specific ultra- (PML), the impact of numerical dispersion, the representa- sonic frequency of interest to calculate the required sampling tion of discontinuities in medium properties, and staircasing criteria, 500 kHz was used. This frequency has seen extensive of acoustic sources and material geometry. The first two rep- use in studies of UNMS (see Table II), sits within the range of resent fundamental considerations in numerical simulations, ultrasound frequencies demonstrating optimal transcranial 21,22 and are dealt with independently. For the subsequent phe- transmission, and has a theoretical minimum focus size of nomena, the specific inaccuracies occurring when simulating 3 mm diameter in soft tissue. the propagation of ultrasound from a source in the deep brain B. The BLI to an external transducer are established. This is modelled as consisting of 10 cm propagation through cerebral soft tissue, The BLI represents a fundamental component of both 1 cm propagation through bone, and 1 cm additional propaga- k-space and PSTD schemes. As such, it is necessary to tion through superficial soft tissue, shown in Fig. 2. Accuracy examine its impact on simulation accuracy before moving is quantified in terms of the resulting error in the amplitude on to more complex factors that affect the rate of conver- and position (calculated using time of arrival) of the temporal gence. Bandlimited interpolation, as described above, can maximum intensity at the target position and the sampling cri- result in a discrepancy between the intended pressure field teria constraining these errors below 10% and 1.5 mm, respec- and the representation of that field within PSTD schemes tively, are established. Beam steering capabilities are when the Fourier coefficients of the intended field have not determined primarily by hardware, and are not considered decayed sufficiently. Practically, this manifests globally as here. Modeling of the skull as a single homogeneous layer in undesired, oscillating pressure values across the simulation this way was recently validated for low frequency model- grid [see Fig. 1(a)]. Therefore, to examine the impact of BLI driven TR by Miller et al. The influence of reverberations effects, it is necessary to determine the amplitude of these undesired pressures relative to that of an intended input. In practice, the error in the representation of a particular pressure distribution will depend on how well it can be rep- resented by a discrete Fourier transform at a specific spatial discretization. Tonebursts have a well-defined power spec- trum determined by their length and central frequency. Therefore, to approximate the BLI error likely to be gener- ally observed, a series of time-varying 10, 30, and 50 cycle acoustic toneburst sources with central wavenumbers approaching the spatial Nyquist limit were used as input FIG. 2. (Color online) A scaled schematic of the simulation model used to evaluate the impact of numerical errors. signals. These sources have 22.7%, 7.4%, and 4.3% full J. Acoust. Soc. Am. 141 (3), March 2017 Robertson et al. 1729 18 April 2024 14:53:13 schemes. In the FDTD scheme used here, outgoing pressure waves are perfectly reflected from the edge of the grid, while for k-space and PSTD schemes outgoing pressure waves are “wrapped” to the opposite edge of the simulation grid (i.e., the grid is toroidal). To replicate free field conditions, k-Wave employs Berenger’s split field PML, where the pres- sure field is artificially divided into Cartesian components to allow selective absorption of the normally incident compo- 19,23 nent. The response of the PML to different frequencies was established by propagating broadband pressure sources toward a 20 point PML on a homogeneous 1D grid using the 15,19 PML profile given in Tabei et al. Rather than the 2-4 FIG. 3. (Color online) (a) Non-causal pressure amplitude as a function of FDTD scheme used elsewhere, a second order accurate in toneburst central PPW for different toneburst lengths and (b) normalized amplitude spectra of non-causal pressure signals and the corresponding space and time (2-2) FDTD scheme with a CFL of 1 was spectra of their source tonebursts, demonstrating how non-casual pressures used to prevent numerical dispersion. It should be noted that relate to the 2 PPW component of the source signal. this FDTD formulation is not practical outside the homoge- width at half maximum (FWHM) bandwidth as a percentage neous 1D case due to stability constraints. The time-varying of central frequency, respectively. The source was positioned pressure traces of the incident wave, the wave reflected from a quarter of the way along a homogeneous 1D computational the surface of the PML, and the wave transmitted to the edge grid with no PML. The simulations were run for the time of the computational grid were recorded. The power spectra of these signals were used to calculate reflection and trans- taken for waves to travel from the source to the center of the mission amplitudes relative to the incident wave as a func- grid. The pressure was recorded at every grid point of the tion of spatial sampling. No notable difference was observed other half of the grid which, according to causality, should between k-space and PSTD schemes. Results are shown in have remained quiescent if the BLI of the pressure field Fig. 4 for k-space and 2-2 FDTD schemes. In both schemes, matched the intended input of compactly supported tone- the pressure reflection coefficient demonstrates a dependence bursts. Error was quantified as the maximum pressure on spatial sampling, rising steadily from below 120 dB for recorded across the second half of the grid relative to the frequencies sampled at above 4 PPW to total reflection at 2 peak pressure of the source toneburst. The results are shown PPW. Transmission to the edge of the grid remains constant in Fig. 3(a) as a function of the PPW of the central wave- at below 70 dB for both schemes until spatial sampling number of the toneburst. The amplitude of the non-causal drops beneath 3 PPW, below which the k-space scheme pressure drops rapidly as the number of PPW increases from shows an increase in transmission and the FDTD scheme the Nyquist limit. Reducing the error requires a higher num- shows a reduction in transmission. These results indicate ber of PPW for shorter tonebursts due to their wider power that the effectiveness of the PML is greatly reduced for spectra, but for all three toneburst lengths the error drops to wavenumbers sampled at below 3 PPW, and it cannot be below 60 dB by 3 PPW. An additional observation was relied on at these PPW values. However, erroneous reflection that wavenumbers corresponding to less than 2 PPW are not and transmission reduce rapidly as sampling increases. It aliased or otherwise propagated on the grid. should be noted that pressure reaching the edge of the grid To determine what frequencies comprised the observed for both schemes is subject to further attenuation within the non-causal pressure, the results obtained from 10 cycle tone- PML when reflected or wrapped back into the grid. bursts were examined further. Time-varying pressure signals Furthermore, the BLI will have influenced the behavior of were recovered from the grid points closest to the wave these tests for frequencies sampled at close to the Nyquist front, which experienced the peak non-causal pressures. The limit, which may explain why the k-space scheme shows an normalized amplitude spectra of these signals resulting from increase in both reflection and transmission close to 2 PPW. source tonebursts with central wavenumbers sampled at 2 The PML was also tested in 2D to determine its depen- and 2.4 PPW are displayed in Fig. 3(b), alongside the corre- dence on the angle of incidence of incoming waves. A sponding amplitude spectra of the source tonebursts. The broadband point source was placed close to the edge of the recorded spectra demonstrate a sharp peak at 2 PPW regard- PML on the 2D grid and propagated into, and across the less of the central frequency of the source toneburst, and the surface of, the PML. The pressure was recorded at the edge amplitude of the peak scales with the amplitude of the 2 of the simulated domain to examine transmission, with each PPW component of the source toneburst. Practically, these recording position corresponding to a particular angle of results demonstrate that this error reduces very rapidly as the incidence. The peak pressure transmission at each angle was spatial sampling of the pressure distribution increases, and at calculated through comparison with a reference recording 3 PPW BLI errors are reduced to below 60 dB. In higher obtained with PML absorption set to zero. The results are dimensions, BLI errors are less severe than the 1D case. shown in Fig. 4(c). Transmission to the edge of the grid is lowest for normally incident waves, rising with increasing C. The PML angle of incidence crossing to above 60 dB at 40 .No The interaction of outgoing pressure waves with the clear relationship between angle of incidence and reflection edge of the simulation grid presents a problem for numerical from the PML was observed. These results should be 1730 J. Acoust. Soc. Am. 141 (3), March 2017 Robertson et al. 18 April 2024 14:53:13 TABLE III. Temporal sampling required to obtain <1.5 mm targeting error. Direct path Reverb. I Reverb. II Reverb. III Target [PPP] [PPP] [PPP] [PPP] FDTD 17.9 21.9 23.9 25.1 PSTD 11.2 18.9 24.2 28.6 k-space 4.0 4.0 4.6 5.4 Results are given in terms of temporal PPP to allow a comparison between dispersive error for the FDTD scheme, which is dependent on both spatial and temporal sampling, and PSTD and k-space schemes, which are dependent on temporal sampling only. Equation (1) shows how the CFL defines a fixed ratio between spatial and temporal sampling. Different CFL numbers will result in a different combination of spatial and temporal requirements for the FDTD scheme. Values for both PSTD schemes are dependent only on tem- poral sampling, and do not require a particular spatial sam- pling to reduce dispersive error. However, with that in mind, the results shown in Table III do demonstrate a clear reduc- tion in the temporal sampling required to minimize disper- sive positional error below acceptable levels for the k-space FIG. 4. (Color online) (a) Reflection from and (b) transmission through the PML as a function of spatial sampling, and (c) transmission to the edge of scheme compared to FDTD and PSTD schemes. the grid as a function of angle of incidence. Here 0 dB means that the reflected or transmitted wave has the same amplitude as the incident wave, E. 1D medium discontinuities i.e., total reflection or transmission. To examine the delay in convergence due to inaccura- considered when designing acoustic sources and considering cies in reflection and transmission from medium discontinu- ities, broadband pressure sources were propagated across a the angles at which pressure waves will impinge on the PML. bone-soft tissue interface (propagation direction makes no difference). The incident, reflected, and transmitted waves were recorded and the power spectra used to calculate inten- D. Numerical dispersion sity reflection and transmission coefficients for each wave- To examine the impact of numerical dispersion, a broad- number. Percentage error in these coefficients was calculated band pressure source was defined on a homogeneous 1D through comparison with the analytical values. To examine grid. Grids with the medium properties of both bone and the dependence of this error on the size of the impedance brain tissue grids were tested, and for the k-space scheme change, these tests were repeated with the impedance of the c was set to the speed of sound in brain tissue. For FDTD ref bone varied up to ten times that of the soft tissue, with sound schemes, the temporal and spatial dispersive errors oppose speed and density varied independently. No difference was each other, with reduced dispersive error at higher CFL num- observed between k-space and PSTD schemes. bers. Therefore the CFL was set to 0.5 for these simula- Figure 5 shows the resulting error in intensity transmis- tions, the highest value at which both schemes are stable in sion and reflection coefficients as a function of PPW, and as 14,15 3D. The time-varying pressure was recorded at a dis- a function of impedance change for a PPW of 6. FDTD and tance of 1 cm, and the phase spectra of the recorded pressure k-space schemes demonstrate a similar error even at high signals were compared to a dispersion-free reference simula- tion obtained with perfect k-space correction. This allowed calculation of phase error per cm propagated in either tissue type as a function of acoustic frequency. Using the model for transcranial propagation of ultrasound to a deep brain tar- get described in Sec. III A, this was used to calculate the sampling criteria required to obtain <1.5 mm positional error for the direct path, and for each reverberation, shown in Table III. To compare, when sampled at 18 PPP, the k-space scheme is exact for soft tissue and gives a 19 lm error in the focal position per cm propagated in bone, PSTD gives a 25 lm error per cm in soft-tissue and a 50 lm error per cm in bone, and FDTD gives 25 lm error per cm in soft-tissue and FIG. 5. (Color online) Error in simulated intensity (a) transmission (Te) and a 130 lm error per cm in bone. (b) reflection (Re) coefficients. J. Acoust. Soc. Am. 141 (3), March 2017 Robertson et al. 1731 18 April 2024 14:53:13 impedance changes. Changes in density result in increased Pythagorean triangles on the grid, specifically: 14.3 (with Pythagorean triple 16, 63, 65), 22.6 (25, 60, 65), 30.5 deviation from the correct coefficient due to the interpolation (33, 56, 65), and 36.9 (39, 52, 65). For these angles, the of the density values on a staggered grid. line-source endpoints are coincident with specific grid point To calculate how the error in 1D reflection and transmis- positions, and any error is only due to the staircased repre- sion will affect transcranial focusing, the error in the inten- sentation of the line, rather than endpoint misregistration sity following transmission across a bone layer (i.e., across (both are aspects of staircasing error). A source defined two bone/soft-tissue interfaces) was computed as parallel to a Cartesian axis was used as a non-staircased ref- f erence. The sources were excited with 10 cycle acoustic Te Te Error ¼ ; (2) 2 tonebursts with a range of central wavenumbers sampled Te at 3–100 PPW. The amplitudes of the source signals were normalized based on any change in the number of distinct where Te is the analytical energy transmission coefficient source points used when defining an angled line source when between bone and soft tissue, and Te is the simulated energy compared to the aligned case. The time-varying field was transmission coefficient as a function of spatial PPW. recorded at 100 points positioned in front of the line source, Reflections inside the skull and skull cavity were not consid- and the sensor map was rotated with the line source to main- ered. To obtain <10% intensity error, the FDTD scheme tain source-sensor geometry. The simulation layout is shown requires 5.9 PPW while the k-space and PSTD schemes in Figs. 6(a) and 6(b). require 4.3 PPW. This result is notable, as the representation Mean errors in the amplitude and position of the peak of discontinuities in medium properties has previously been intensity across the sensor field were calculated indepen- identified as a limitation of PSTD methods. During the dently for each angle tested relative to the aligned, non- update steps of these schemes, the pressure field is multiplied staircased reference case. The maximum mean errors across by the maps of medium density and sound speed, before the range of angles tested for each PPW value are shown in being evaluated by a truncated Fourier series. Step changes Fig. 7. These results demonstrate that staircasing errors in medium properties will therefore introduce Gibbs phe- worsen with lower spatial sampling and are less serious for nomenon into the pressure field, as described in Sec. II C. Pythagorean angles, when endpoints are correctly registered. However, these results indicate that, for the step change in The error in the position of the intensity peak never rises medium properties between bone and soft tissue, the error above 50% of wavelength for any source. Seventeen PPW resulting from the representation of this change within the are required to obtain <10% error in the amplitude of the FDTD scheme tested is greater. intensity peak for all angles tested, while Pythagorean angles require only 7 PPW. Although the error examined here does F. Staircasing not relate directly to the model for transcranial ultrasound Staircasing refers to the spatial approximation that is propagation described above, these results do indicate that necessary when attempting to define continuous geometries staircasing and spatial sampling must be considered when on a discrete, regular Cartesian grid in 2D and 3D. Curved defining acoustic source distributions, and that error can be surfaces and lines at an angle to the Cartesian directions will reduced by ensuring endpoint registration. The testing of be approximated in a stair-stepped manner, and certain multiple angles also allowed examination of how the exact vertex and edge positions do not correspond to points on the mapping of the staircased line relates to the error observed in grid. The impact of staircasing was examined separately the resulting field. No clear relationship between the angle of for acoustic sources and medium distributions. Tests the line source and the level of staircasing error was involved recording the time-varying pressure at a number of observed. However, a staircasing metric was defined as the positions across the field resulting from a staircased repre- average distance between the staircased source points and sentation of a source or medium, and comparison of these their equivalent equispaced points on an ideal angled line. It signals with references obtained from a staircase free simula- was observed that the convergence rate of the error of the tion. Error was then quantified as the percentage error in the staircased source maps, quantified as the average L2 error in amplitude of the temporal peak intensity (calculated using a the recorded pressure signals at the maximum spatial sam- plane wave assumption) and its positional error (derived pling tested, showed a strong dependence on this staircasing from the change in the time of arrival of the intensity peak) metric. Although only a simple metric, this indicates that the as a percentage of wavelength. A positional error of 50% of severity of staircasing error can be predicted through com- wavelength corresponds to 1.5 mm for a source frequency of parison of an ideal or parametric map of the intended geome- 500 kHz in brain tissue. These errors were calculated for try with its staircased representation. each recording position, and then averaged across the field to The impact of staircasing of heterogeneous medium give mean errors in peak intensity amplitude and position. properties was examined in two separate tests. The first was No notable difference in error was observed between FDTD, conceptually similar to the examination of source staircas- PSTD, and k-space schemes across all tests. ing. An acoustic point source excited by ten cycle acoustic The impact of staircasing on acoustic sources was exam- tonebursts with central frequencies ranging from 3 to 80 ined using line-sources with a length of 65  dx, where dx is PPW was propagated across a planar medium boundary (soft the spatial discretization step, at a series of angles to the tissue-to-bone), defined at varying angles to the Cartesian Cartesian grid. These included four angles that form axes. The time varying pressure field was recorded at 100 1732 J. Acoust. Soc. Am. 141 (3), March 2017 Robertson et al. 18 April 2024 14:53:13 FIG. 6. (Color online) Simulation layouts used to test the impact of staircasing (not to scale). Sources are shown in red, and pressure recording positions in black. (a) Non-staircased line source used as reference, and (b) staircased line source used to examine error. (c) Non-staircased medium map used as a refer- ence, and (d) staircased medium used to examine error. (e) High resolution map of a bone-tissue layer used as reference, and (f) downsampled medium. sensor points following the interaction with the medium boundary. The sensor points were rotated with the medium boundary to maintain the simulation geometry. A non- staircased boundary defined along a Cartesian axis was used as a reference map. This simulation layout is shown in Figs. 6(c) and 6(d). The second test was designed as a more accu- rate model of staircasing in transcranial transmission. A 10 cycle, 2520 PPW toneburst was propagated through a medium map comprising a quarter circle bone layer. The medium was then artificially staircased through spatial downsampling, before being remapped to the original grid. A 3780  3780 simulation grid was used due to the large number of integer factors of 3780, which allowed the medium to be successively downsampled while maintaining positioning. The time- varying pressure was recorded across a quarter-circle, and error metrics computed through comparison with the least staircased medium distribution. This simulation layout is shown in Figs. 6(e) and 6(f). An effective PPW value for each level of downsampling was calculated through comparison of the source PPW with the new effective spatial discretization. Mean error measurements across the recorded fields as a func- tion of PPW are shown in Fig. 8 for both medium staircasing tests. For the single interface model, the maximum mean errors across the range of angles tested are shown. In terms of the impact on the model for transcranial FIG. 7. (Color online) Error in staircased line sources as a function of spatial propagation, the results for the bone layer model shown in PPW. (a) Percentage error in peak intensity. (b) Positional error as a percent- Fig. 8 suggest that 20 PPW or above are required to obtain age of wavelength. (c) L2 error at 100 PPW sampling against average devia- less than 10% mean error in intensity transmitted to an exter- tion from ideal line source. Pythagorean sources have both vertices of the parametric line source exactly defined on the grid. nal transducer surface. As might be expected, the errors for J. Acoust. Soc. Am. 141 (3), March 2017 Robertson et al. 1733 18 April 2024 14:53:13 testing of fully simulated TR was carried out in 2D and 3D. The general method is outlined in Fig. 9. It consists of for- ward propagation of a 10 cycle toneburst from a source point inside a virtual skull model to a circular (2D) or hemispheri- cal (3D) virtual transducer-sensor array. As these tests were to examine the specific impact of numerical convergence, the simulated transducer array surface was modelled as a continuous surface made up of point transducers at the resolution of the spatial grid, and no attempt was made to replicate real transducer characteristics. This ensures that convergence is dependent only on the numerical accuracy of the forward simulations and will apply to alternate source FIG. 8. (Color online) Error resulting from propagation through bone-layer and single interface staircased medium boundaries. (a) Percentage error in conditions. The spatial discretization of these simulations peak intensity magnitude. (b) Positional error of peak intensity as a percent- was varied to correspond to a range of spatial PPW values age of wavelength. for the central frequency of the source toneburst. The time- varying pressure signals recorded at the virtual transducer the single interface are lower, with 6 PPW required to obtain position were then time reversed and propagated into the the same error in peak intensity amplitude. The error in peak head to refocus onto the target position. The reversal simula- intensity position is less serious, with mean positional error tions were carried out at the finest discretization feasible, never rising above 50% of wavelength (1.5 mm for 500 kHz and the CFL was 0.3 for all simulations. Due to the change ultrasound in brain tissue) for both tests, as with source stair- in spatial and temporal discretization, it was necessary to casing. This may be due to staircasing introducing a random interpolate the pressure signals recorded in the forward sim- error in acoustic pathlength, leading to a defocusing and ulations onto the spatial and temporal grids used in the rever- change in amplitude rather than a shifting in the peak posi- sal simulations [see Fig. 9(b)] using Cartesian triangulation tion. To place this in context, the voxel size of clinical CT and Fourier interpolation, respectively. In addition, the posi- images is on the order of 0.5 mm at best. This corresponds to tion of the source, defined at a grid point on the high resolu- 6 PPW at 500 kHz, and fewer at higher frequencies. This sug- tion reversal grid, was not conserved due to the varying gests that staircasing may have a significant impact on simula- discretization of the forward simulations, and instead the tions using image derived medium property maps. When nearest neighboring point was used. The peak pressure considered alongside the results for source staircasing, these occurring in a time window of 20 acoustic cycles centered results indicate that staircasing error is likely the most serious on the expected time of refocusing was recorded across the of the numerical errors tested. The single interface medium brain volume. Convergence was established by examining model was also briefly tested using an elastic PSTD model, focusing quality as a function of the discretization used in which indicated that medium staircasing may also have a pro- the forward simulations. The focusing metrics examined nounced impact on simulated mode conversion, although were the spatial and temporal peak pressure across the brain more rigorous testing is necessary. volume, the distance of the peak from the target position, IV. CONVERGENCE TESTING and the FWHM of the focal spot size. Peak pressure and focus FWHM were normalized relative to the results A. Overview obtained for the most highly resolved forward simulation. To examine the combined effects of numerical errors on Simulations in 3D were carried out with toneburst the effectiveness of transcranial TR focusing, convergence sources with central frequencies of 500 kHz, while testing FIG. 9. (Color online) Method for convergence testing in both 2D and 3D. (a) Forward simulation of ultrasound propagation from target point to simulated transducer surface with low-PPW forward discretization. (b) Triangulation is used to extract pressure signals at the transducer positions for the reversal simula- tion. This signal is then Fourier interpolated onto the finer temporal grid, and time-reversed. (c) The time reversed pressure signals are propagated back into the high-PPW head model in a reversal simulation. An example of the pressure field recorded across the brain volume and used to evaluate focusing is shown. 1734 J. Acoust. Soc. Am. 141 (3), March 2017 Robertson et al. 18 April 2024 14:53:13 TABLE IV. Simulation criteria used in convergence testing. 2D 3D Simulation parameter 250 kHz 500 kHz 500 kHz 2 2 2 2 3 3 Simulation grid size 108 –3096 162 –3096 144 –1024 Spatial discretization [mm] 3.1–0.0618 1.6–0.0624 1.6–0.18683 Forward and reverse simulation times [ms] 0.40499 and 0.42499 0.38864 and 0.39864 0.3879 and 0.3979 Temporal discretization [ns] 292.5–5.79 146.25–5.85 146.25–17.515 Transducer radius [mm] 90.9 91.8 95.0 Target Cartesian deviation from transducer focus [mm] [2,3] [2,3] [5,5,5] Simulation runtime [mins] 0.2–83 0.3–78 10–6736 in 2D used both 250 and 500 kHz tonebursts. The simula- key points can be derived from these results. First, for all three tion parameters employed across these tests are shown in metrics, the k-space scheme demonstrates convergence at Table IV. Grid sizes include the absorbing PML layer. The approximately 2 PPW below the FDTD scheme. Second, forward simulations were run for the time taken for an although there is some difference in the position and size of the focus at very low sampling, both 250 and 500 kHz demon- acoustic wave to propagate across the grid three times, strate similar behavior as a function of spatial sampling. This plus the duration of the source toneburst. The reversal sim- indicates that these results can, to some extent, be generalized, ulation was run for the additional time of five acoustic cycles in order to fully capture the reconstructed toneburst. and suggests that the reversal simulations have converged for both frequencies. Finally, of the three refocusing metrics The medium property map used in the convergence tests examined, normalized peak pressure across the brain volume was derived from a T1-weighted MR image obtained from [Fig. 10(a)] requires higher spatial sampling to converge than the Imperial College brain development dataset. Brain either focal volume or the deviation of the focus from the tar- and skull volumes were extracted using the FSL MRI proc- get. This suggests that when fine pressure control is not essing toolbox and converted into a surface mesh using required, coarser sampling criteria may suffice. the iso2mesh toolbox. The reference surface mesh was then sampled onto a 2D or 3D grid with the required spatial C. Convergence testing in 3D discretization steps. Examples of the 2D maps used in for- ward and reversal simulations, and their varying discretiza- 3D convergence testing employed a similar protocol to tions, are shown in Figs. 9(a) and 9(c). In each case, the the 2D convergence testing described above. Ten cycle skull was modelled as a single homogeneous bone layer, and the rest of the simulation domain was assigned brain tissue medium properties. Although fully heterogeneous models of the skull have demonstrated tighter model- driven TR focusing in some cases, homogeneous models remain effective. Furthermore, they allow accurate resampling of the bone map to multiple spatial discretiza- tions without interpolation, and ensure that convergence is dependent on the accuracy of the numerical simulation, rather than the mapping and homogenization of acoustic medium properties. For the k-space scheme c was set to ref the speed of sound in brain tissue. B. Convergence testing in 2D 2D convergence testing was carried out using 10 cycle acoustic toneburst sources corresponding to 250 and 500 kHz frequencies. Reversal simulations were carried out using a 3072  3072 grid, with a spatial discretization corre- sponding to 50 PPW for 500 kHz and 101 PPW for 250 kHz. Forward simulations were carried out using the k-space and FDTD schemes. Reversal simulations were carried out using the k-space scheme only. The results of the 2D convergence testing are shown in FIG. 10. (Color online) Results of 2D convergence testing. (a) Peak pressure Fig. 10, with error in peak pressure position given relative to recorded across the brain volume. (b) The deviation of the peak pressure from the source point in the forward simulations. Refocusing quality the location of the forward simulation source. (c) Normalized FWHM area of in the reversal simulations increases with the spatial PPW of the focal spot size. Normalization is relative to results obtained with the most the simulated frequency in the forward simulations. Several highly resolved forward simulation. J. Acoust. Soc. Am. 141 (3), March 2017 Robertson et al. 1735 18 April 2024 14:53:13 acoustic tonebursts with a central frequency of 500 kHz Accordingly, the reduction in error and lack of dependence were propagated from a source point inside a full 3D model on spatial sampling when calculating positional error relative of the human skull to a simulated hemispheric transducer. to the actual source position from the forward simulation Reversal simulations were carried out using a 1024 1024 suggests that any error in the position of the peak is in 1024 simulation grid with a spatial discretization corre- fact due solely to the misregistration of the source points. sponding to 16.7 PPW. Forward and reversal simulations Generally, these results confirm that when targeting accuracy were carried out using the k-space scheme only. Testing was is the prime concern, spatial sampling requirements are laxer also carried out using homogeneous media for forward and than when a tight focal volume with known peak pressure reverse simulations, to test the accuracy of the spatiotempo- amplitude is required. This is in agreement with previous ral interpolation. The results for heterogeneous 3D conver- studies which have demonstrated that good spatial targeting gence testing are shown in Fig. 11. The results for of HIFU can be obtained via simulated TR using relatively 11,30 normalized peak pressure amplitude in Fig. 11(a) show simi- coarse spatial sampling. Homogeneous testing demon- lar trends to the 2D results. Six PPW are required to obtain strated total convergence across all metrics by 3.5 PPW, 95% reconstruction of pressure at the target (corresponding which is to be expected given the behavior of the PML and to <10% drop in intensity) and 10 PPW to attain conver- BLI, discussed in Sec. III. gence. The convergence of the volume of the focal spot V. SUMMARYAND DISCUSSION [Fig. 11(c)] shows similar behavior, although it only requires 6 PPW to fully converge. Given the known impact of stair- In this paper, a comprehensive assessment of the impact casing in 2D, the faster convergence here is likely due to a of different factors that affect the convergence of numerical reduced staircasing error for 3D geometry. The results in models for the simulation of transcranial ultrasound propaga- Fig. 11(b) show the error in the position of the pressure peak tion was carried out. The spatial and/or temporal sampling relative to both the reversal target and the shifted forward required to reduce inaccuracies below the levels required for source point. This error is reduced compared to the 2D case, targeting of deep brain nuclei for neurostimulation were never rising above 1.1 mm and, when computed relative to determined. the position of the forward source, is stable using sampling Initial simulations examined reduction in the effective- as low as 2 PPW. However it can be seen from the other ness of the PML, and the impact of the BLI when using results that at this sampling the peak pressure amplitude is k-space and PSTD methods. Both the PML and BLI lead to much lower, with a larger focal spot. The apparent periodic- erroneous pressures appearing on the grid when simulating ity in the positional error when calculated relative to the frequencies sampled at close to the spatial Nyquist limit. parametrically defined target point is likely due to the oscil- Although both of these effects have the potential to seriously lating distance of a definable source point from this position. reduce the accuracy of the simulations, they decrease in sever- ity rapidly as the rate of spatial sampling increases. Above 3 PPW, erroneous pressures resulting from both BLI and PML effects were at least 60 dB below the amplitudes of the ultra- sound sources being simulated. Numerical dispersion has a serious effect on the accuracy of FDTD and PSTD schemes, resulting in high temporal sam- pling requirements to reduce positional error. However, this wasnot thecasefor the k-space scheme, where 3 PPW will serve to limit dispersion sufficiently for transcranial transmis- sion for any stable CFL value. Errors in reflection and trans- mission from discontinuous medium properties manifest in the magnitude of reflected and transmitted simulated intensities. Despite the representation of step changes in media previously being identified as a key limitation of PSTD schemes, the error wasshowntobemoresevere for the2-4 FDTD scheme tested. To reduce error in the intensity below 10% following transcra- nial transmission, k-space and PSTD schemes require 4.3 PPW, while FDTD requires 5.9 PPW. Staircasing of source and medium geometries was shown to require the most stringent sampling criteria to obtain required accuracy, affecting FDTD, PSTD, and k-space schemes equally. Both source and medium staircasing were shown to have a greater impact on the intensity amplitude of FIG. 11. (Color online) 3D convergence testing results. (a) Normalized peak the toneburst signal being examined than the position of the pressure amplitude across the brain. (b) Deviation of pressure peak from both intensity peak. The results shown in Figs. 7 and 8 indicate that the parametrical defined target and the source used in the forward simulation. 20 PPW are required to reduce the error in peak intensity fol- (c) Normalized half-maximum focal volume. Normalization is relative to results obtained with the most highly resolved simulation. lowing transcranial transmission below 10%. The preliminary 1736 J. Acoust. Soc. Am. 141 (3), March 2017 Robertson et al. 18 April 2024 14:53:13 examination of a potential staircasing metric also suggests that alternative low-intensity, transcranial ultrasonic therapies the error resulting from a particular staircased geometry is such as opening the blood-brain barrier with ultrasound, as directly related to its deviation from the ideal geometry. well as existing transcranial HIFU ablation therapies. Use of Convergence testing of a fully simulated TR protocol appropriately discretized simulations will ensure accurate using 2D and 3D head models was used to examine the targeting and effective therapy as the field of ultrasonic neu- impact of all numerical errors in concert. Testing in 2D for rostimulation develops. 250 and 500 kHz ultrasound showed a faster rate of conver- gence for all focusing metrics for the k-space scheme when ACKNOWLEDGMENTS compared to FDTD. In addition, the error in the peak pres- sure amplitude at the focus showed slower convergence than The authors would like to thank Charlotte Stagg and both the positional error, and the volume of the focus. This Adam Thomas for helpful discussions, and Nishant Ravikumar is likely due to the most serious source of error, medium and Zeike Taylor for the provision of the skull and brain staircasing, which was shown to have a greater impact on meshes. This work was supported by the Engineering and the peak intensity amplitude of transcranially transmitted Physical Sciences Research Council (ESPRC), UK. J.J. is ultrasound, than the position of the peak. Results in 3D financed from the SoMoPro II programme. This research has showed similar trends to the 2D results for the convergence acquired a financial grant from the People Programme (Marie of the peak pressure amplitude. The focal spot size showed Curie Action) of the Seventh Framework Programme of EU slightly slower convergence in the 3D case, while the posi- according to the REA Grant Agreement No. 291782. The tional error demonstrated almost no dependence on the sam- research is further co-financed by the South-Moravian Region. pling rate of the forward simulation. This indicates that less This work reflects only the author’s view and the European stringent sampling may suffice for applications concerned Union is not liable for any use that may be made of the only with the position of the focus, rather than the size of the information contained therein. Computational resources were focal spot and the exact amplitude at the target. When fine provided by the IT4Innovations Centre of Excellence project control over the pressure amplitude is required, stricter sam- (CZ.1.05/1.1.00/02.0070), funded by the European Regional pling may be necessary. Despite the relatively severe error Development Fund and the national budget of the Czech resulting from staircasing at higher spatial sampling, all Republic via the Research and Development for Innovations three metrics of focusing quality were well converged at Operational Programme, as well as Czech Ministry of below 20 PPW. This discrepancy may be due to the differ- Education, Youth and Sports via the project Large Research, ences between the convergence testing protocol and the spe- Development and Innovations Infrastructures (LM2011033). cific test used to examine staircasing across a bone layer, and suggests that the influence of staircasing is case specific. The work described above is subject to some limita- APPENDIX tions, primarily the degree to which the examination of indi- Simulations were carried out using the open source k- vidual numerical errors can be generalized to different Wave toolbox for MATLAB, Cþþ. The toolbox includes k-space, setups, although trends and qualitative observations remain PSTD, and 2-4 FDTD codes for the time-domain simulation of valid. Many of the tests only examine toneburst sources, and acoustic fields. 1D simulations were carried out in the MATLAB the error is evaluated over a small field, with pressure environment on a Dell Precision T1700 with an Intel Xeon recorded at a limited number of sensor positions (see Fig. 6). E3-1240 3.40 GHz CPU and 16 GB of RAM running A separate 2-2 FDTD scheme was used to examine the effec- Windows 10 64 bit. 2D simulations were carried out in the tiveness of the PML, which may not be exactly relatable to MATLAB environment with CUDA hardware acceleration on a the commonly used 2-4 FDTD scheme. Furthermore, the Dell PowerEdge R730 compute server with 2  6-core Xeon impact of shear wave propagation was not examined. This E5-2620 2.4 GHz CPUs, 64 GB of 1866 MHz memory, on an will not have affected 1D or homogeneous simulations, but a Nvidia Titan X GPU with 3072 CUDA cores and 12 GB of more thorough examination of medium staircasing should include testing of elastic wave propagation. Similarly, no memory. The largest 2D simulations had a domain size of effort was made to examine the manifestation of numerical 3780 including the PML and comprised 258 462 time steps, errors when modeling nonlinear propagation or acoustic with a total runtime of 10.6 h. 3D simulations were carried out absorption, which will become relevant for applications on the IT4I Salomon supercomputing cluster. Each simulation requiring the simulation of high-amplitude ultrasound, such was carried out on Intel Xeon E5-4627v2, 3.3 GHz, 8cores and as HIFU. It should be noted that, in simulated TR, account- 256 GB of RAM per simulation. The largest 3D simulations ing for acoustic absorption occurs in the post-processing had a domain size of 1024 including the PML and comprised stage, when converting recorded pressure signals into driv- 22 718 time steps, with a total runtime of 112.3 h. The skull ing amplitudes, and work examining absorption should mesh used in convergence testing is Copyright Imperial focus on this stage of the simulated TR process. College of Science, Technology and Medicine 2007. All rights The results presented here are primarily relevant to the reserved. www.brain-development.org. simulation of transcranial ultrasound propagation for TR tar- geting of deep brain structures with finely controlled ultra- P. J. Karas, C. B. Mikell, E. Christian, M. A. Liker, and S. A. Sheth, sound for the purposes of neurostimulation. However, the “Deep brain stimulation: A mechanistic and clinical update,” Neurosurg. criteria and simulations presented are also relevant to Focus 35(5), E1–E16 (2013). J. Acoust. 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Phys. 52(7S), 07HF01 (2013). 1738 J. Acoust. Soc. Am. 141 (3), March 2017 Robertson et al. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Journal of the Acoustical Society of America Unpaywall

Accurate simulation of transcranial ultrasound propagation for ultrasonic neuromodulation and stimulation

The Journal of the Acoustical Society of AmericaMar 1, 2017

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0001-4966
DOI
10.1121/1.4976339
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Abstract

18 April 2024 14:53:13 MARCH 13 2017 Accurate simulation of transcranial ultrasound propagation for ultrasonic neuromodulation and stimulation James L. B. Robertson; Ben T. Cox; J. Jaros; Bradley E. Treeby J. Acoust. Soc. Am. 141, 1726–1738 (2017) https://doi.org/10.1121/1.4976339   View Export Online Citation 18 April 2024 14:53:13 Accurate simulation of transcranial ultrasound propagation for ultrasonic neuromodulation and stimulation 1,a) 1 2 1 James L. B. Robertson, Ben T. Cox, J. Jaros, and Bradley E. Treeby Department of Medical Physics and Biomedical Engineering, University College London, London, United Kingdom Faculty of Information Technology, Brno University of Technology, Brno, Czech Republic (Received 13 June 2016; revised 1 December 2016; accepted 31 January 2017; published online 13 March 2017) Non-invasive, focal neurostimulation with ultrasound is a potentially powerful neuroscientific tool that requires effective transcranial focusing of ultrasound to develop. Time-reversal (TR) focusing using numerical simulations of transcranial ultrasound propagation can correct for the effect of the skull, but relies on accurate simulations. Here, focusing requirements for ultrasonic neurostimula- tion are established through a review of previously employed ultrasonic parameters, and consider- ation of deep brain targets. The specific limitations of finite-difference time domain (FDTD) and k-space corrected pseudospectral time domain (PSTD) schemes are tested numerically to establish the spatial points per wavelength and temporal points per period needed to achieve the desired accuracy while minimizing the computational burden. These criteria are confirmed through conver- gence testing of a fully simulated TR protocol using a virtual skull. The k-space PSTD scheme performed as well as, or better than, the widely used FDTD scheme across all individual error tests and in the convergence of large scale models, recommending it for use in simulated TR. Staircasing was shown to be the most serious source of error. Convergence testing indicated that higher sampling is required to achieve fine control of the pressure amplitude at the target than is needed for accurate spatial targeting. 2017 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4976339] [JFL] Pages: 1726–1738 invasive neural excitation and modulation, with focusing on I. INTRODUCTION the scale of the acoustic wavelength. Table II shows a selec- The use of implanted electrodes for deep brain stimula- tion of UNMS papers published in the last decade, and dem- tion (DBS) is a well-established, invasive treatment for onstrates the variety of acoustic intensities and frequencies multiple neurological conditions and has directly resulted used, target structures sonicated, and neural responses in a greater understanding of functional neuroanatomy observed. The physical mechanism underlying UNMS and deep brain circuitry. Unfortunately, its usefulness is remains unclear, although a non-thermal mechanism is sus- limited by the inherent risks of the required neurosurgery pected, and lower acoustic frequencies have been shown to combined with difficulties in targeting and repositioning 3,6 evoke a response more reliably. Most recently ultrasound the stimulatory focus. Non-invasive alternatives such as has been used to elicit electro-encephalogram (EEG) and transcranial magnetic and direct current stimulation have both sensory responses in human subjects, although this has been met with success in research and clinical settings. However, restricted to superficial cortical brain areas using unfocused they are limited in terms of their ability to achieve tight 7–9 single element transducers. If UNMS is to develop as an spatial focusing, and their penetration deep into tissue. Table effective non-invasive neurostimulation technique, its appli- I demonstrates a selection of existing and planned DBS target cation to human subjects must be extended to deep brain tar- structures alongside their approximate dimensions and devia- gets. Based on the dimensions of DBS targets shown in tion from the approximate center of the brain—the mid- Table I, and the range of effective ultrasonic intensities 4,5 commissural point (MCP). These dimensions demonstrate shown in Table II, the following focusing requirements may the millimeter scale size of the target structures, and their be defined: position close to the center of the brain. Thus, the ability to A spatial targeting error of less than 1.5 mm. non-invasively target these nuclei for modulation and stimula- Control of the intensity at the focus with 10% error will tion would represent a revolutionary neuroscientific tool with ensure that neurostimulation occurs. Greater accuracy both clinical and research applications. may be desirable in studies of the mechanisms and thresh- Ultrasonic neuromodulation and stimulation (UNMS) olds of UNMS. offers a potential solution to these requirements, and has An ultrasonic stimulation focus of no greater than 3 mm recently received a great deal of interest. Transcranial focus- diameter will ensure stimulatory specificity. ing of ultrasound offers the potential for reversible, non- Steering of the ultrasonic focus up to 30 mm from the MCP to allow stimulation of arbitrary deep brain a) Electronic mail: james.robertson.10@ucl.ac.uk targets. 1726 J. Acoust. Soc. Am. 141 (3), March 2017 0001-4966/2017/141(3)/1726/13/$30.00 V 2017 Acoustical Society of America 18 April 2024 14:53:13 TABLE I. Approximate dimensions of DBS targets (Ref. 6). AP/DV/ML— verified by MRI thermometry. Marquet et al. showed that Anteroposterior/dorsoventral/mediolateral, MCP—Mid-commisural point. model-driven TR is capable of restoring 90% of the peak pressure that can be obtained with gold-standard hydrophone AP DV ML MCP deviation based methods when focusing through an ex vivo skull bone. Target [mm] [mm] However, model-driven TR remains subject to systematic Ventral intermediate nucleus 10 15.8 11 17 errors and uncertainties with a resulting loss in focusing Ventral anterior nucleus 7 12.6 10 15 quality or efficiency. Four categories of uncertainty are: Centro-median nucleus 8 4.5 414 Nucleus Accumbens 9.5 10  12 21 (i) The underlying physical model and how the govern- Globus pallidus externus 21.5 10323 ing equations model the physics of propagation Globus pallidus internus 12.5 8620 including phenomena such as absorption, nonlinear- Sub-thalamic nucleus 8 4 6.3 13 ity, and shear wave effects. (ii) Numerical approximations due to the discretization of the simulation domain, including numerical dispersion, The primary obstacle to achieving these ultrasonic the representation of medium heterogeneities, and the focusing criteria within the brain is the presence of the skull, effectiveness of any absorbing boundary conditions. which aberrates and attenuates incoming wavefronts. Time- (iii) The inputs to the model, such as the map of acoustic reversal (TR) focusing, first adapted for transcranial focusing medium properties and the representation of acoustic by Aubry et al., is able to correct for the aberrating effect of transducers. the skull by taking advantage of the time-symmetry of the (iv) How the numerical simulations are used within a lossless acoustic wave equation. In model-driven TR, broader TR protocol, including how the simulated numerical models simulate the propagation of ultrasound source is related to the desired pressure at the target, from a target area to a virtual transducer using acoustic prop- and how phenomena that are not time-invariant, such 9,10 erty maps of the head derived from CT or MRI images. as absorption, are accounted for. The pressure time series at the simulated transducer surface TR simulations for transcranial focusing have typically is then time-reversed, and used to generate drive signals for a multi-element acoustic transducer array. For high-intensity made use of finite-difference time domain (FDTD) numeri- 11,12 thermal applications, model-driven TR may be combined cal models. Recently a k-space corrected, pseudospectral with MRI thermometry for treatment verification. Chauvet time domain (PSTD) numerical scheme was used in model- 11 13 et al. confirmed the potential for model-driven TR-based driven TR and shown to give comparable accuracy. Both focusing inside the human head to millimeter precision, FDTD and PSTD schemes use consistent approximations to TABLE II. Review of selected recent ultrasonic neuromodulation and neurostimulation literature. SPPA—Spatial peak pulse average, SPTA—spatial peak temporal average, SPTP—spatial peak temporal peak, VEP—Visual evoked potential, LGN—lateral geniculate nucleus, FEF—frontal eye field, PET—posi- tron emission tomography, fMRI—functional magnetic resonance imaging, GABA—gamma-aminobutyric acid, S1—primary somatosensory cortex, MC— motor cortex. *—0.5 MHz achieved with 2 MHz carrier. Intensity at focus Target: In-vivo(IV) vs Year Author Freq. [MHz] [W/cm ] Ex-vivo(EV) Neural Response & Observations 2008 Tyler et al. 0.44 and 0.67 2.9 SPPA EV mouse hippocampus Imaging of ion channel opening and synaptic activation 2008 Khraiche et al. 7.75 50–150 SPTP EV rat hippocampus Increased neuronal spike rate 2010 Tufail et al. 0.25 and 0.50 0.228 SPPA IV mouse brain Motor response, cortical spiking, ion channel opening 2011 Yoo et al. 0.69 3.3–12.6 SPPA IV rabbit cortex Motor response VEP suppression and fMRI activity 2011 Min et al. 0.69 2.6 SPPA IV rat epileptic focus Suppression of induced epileptic behavior 2011 Yang et al. 0.65 3.5 SPPA IV rat thalamus Decrease in extracellular GABA levels 2012 King et al. 0.50 1–17 SPTP IV rat brain Motor response above an intensity threshold 2012 Kim et al. 0.35 8.6–20 SPPA IV rat abducens nerve Motor response in abducens muscle 2013 Menz et al. 43 20–60 SPPA EV salamander retina Retinal interneuron stimulation 2013 Deffieux et al. 0.32 4 SPPA IV primate FEF Altered visual antisaccade latency 2013 Younan et al. 0.32 17.5 SPPA IV rat cortex Motor response 2014 Legon et al. 0.50 5.9 SPPA IV human S1 Altered sensory evoked EEG oscillations 2014 Kim et al. 0.35 3.5–4.5 SPTA IV rat thalamus Glucose uptake change, motor response 2014 King et al. 0.5 3 SPTA IV mouse MC Motor response varying with targeting 2014 Juan et al. 1.1 13.6–93.4 SPTA IV rat vagus nerve Reduced vagus compound action potential 2014 Mehic et al. 0.5* 2–8 SPTA IV rat brain Motor response scaling with intensity 2014 Mueller et al. 0.5 5.9 SPPA IV human S1 Altered EEG beta phase dynamics 2015 Lee et al. 0.25 0.5–2.5 SPPA IV human S1 Evoked sensations and EEG changes 2015 Lee et al. 0.25 6.6–14.3 SPPA IV sheep cortex Motor and EEG responses 2016 Ye et al. 0.3–2.9 0.1–127 SPPA IV mouse MC Motor response, more effective at low frequencies 2016 Ai et al. 0.5 and 0.86 6 SPPA IV human brain fMRI activity at stimulation site and deep brain 2016 Darvas et al. 1.05 1.4 SPTA IV rat brain EEG response, focal effects on gamma band activity J. Acoust. Soc. Am. 141 (3), March 2017 Robertson et al. 1727 18 April 2024 14:53:13 the wave equation, and can be made stable by choosing the problems, a Fourier basis is typically used, with the basis discretization parameters appropriately. As the rate of spatial function weights calculated via the fast Fourier transform. and temporal sampling increases, they will converge on the The subsequent gradient calculation is exact, eliminating true solution at a rate dependent on the particular approxima- numerical dispersion due to spatial discretization. However, tions of the numerical model [(ii) above]. However, due to for an explicit time-stepping scheme, temporal gradients the large scale of these simulations, it is desirable to mini- must still be approximated via a finite difference method, mize the grid size and resulting computational burden with- with resulting dispersive error. Fortunately, for a second- out compromising accuracy, so knowledge of the minimum order accurate approximation, this error can be calculated sampling criteria necessary to achieve the required accuracy analytically, and used to introduce a correction factor, j ¼ is valuable. In the present paper, these numerical schemes sincðc kDt=2Þ; in the spatial frequency domain. Here, k ¼ ref are briefly described, and the various factors affecting the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 rate of numerical convergence are examined. This is quanti- k þ k þ k is the magnitude of the wavevector x y z fied in terms of the spatial and temporal sampling required to ðk ; k ; k Þ at each grid point in the spatial frequency domain, x y z obtain acceptable accuracy in the simulation of ultrasound and c is a user defined reference sound speed. This method ref propagation from the scalp to a deep brain target. While the is often called the k-space PSTD method, and in homoge- criteria used are established for the application of transcra- neous media it is unconditionally stable and free from numer- nial UNMS, these results are also applicable to other thera- 15,16 ical dispersion for arbitrarily large Dt: In media with a pies that require accurate transcranial ultrasound simulation, heterogeneous sound speed, the application of j in the spatial such as high intensity focused ultrasound (HIFU) ablation frequency domain means that a single sound speed must be and opening the blood brain barrier with ultrasound. chosen for the correction factor. As a result, numerical dis- persion will arise, the extent of which will depend on the II. NUMERICAL METHODS FOR ULTRASOUND temporal sampling and the difference between the local PROPAGATION sound speed cðxÞ; and the reference sound speed c : As ref A. FDTD with FDTD schemes, simulation-dependent limits on the CFL number must be observed to ensure numerical stability. FDTD methods have seen extensive use in the simula- tion of ultrasound propagation, and have accordingly been C. The BLI 9–12 used for the purpose of model-driven TR with success. FDTD and PSTD methods both use functions to interpo- In finite difference methods, partial derivatives are calcu- lated using a linear combination of function values at neigh- late between the values of the acoustic variables at the grid boring grid points. The finite difference approximations are points. The interpolating functions are used to approximate derived by combining local Taylor series expansions trun- the field gradients at these points, and the values of the field cated to a fixed number of terms. When simulating ultra- variables are updated at the grid points at each time step. sound propagation, this approximation causes an unphysical FDTD methods use polynomials to interpolate between dependence of the simulated sound speed on the number of neighboring points, while PSTD methods use a Fourier grid points per wavelength (PPW ¼ k=Dx) and the number series to interpolate between all points simultaneously. of temporal points per period [PPP ¼ 1=ðfDtÞ] where f and This Fourier series is bandlimited (truncated) to ensure a k are acoustic frequency and wavelength, respectively, and unique Fourier representation and is therefore known as the Dx and Dt are the spatial and temporal discretization, respec- bandlimited interpolant (BLI). This can be considered the tively. This manifests as a cumulative error in the phase representation of the discretely sampled pressure field within of propagating waves, termed numerical dispersion. In addi- PSTD schemes. When a time-varying source is used, the tion, stability conditions must also be met to ensure the resulting pressure signal is formed from a sum of one or numerical scheme is stable. These conditions are contingent more weighted BLIs. As a result of this, a discrepancy can on the exact scheme used and the number of simulated arise between the BLI and the intuitive expectation of what dimensions. A useful metric when considering stability is the sampled function represents. This is shown in Fig. 1(a) the Courant-Friedreichs-Lewy (CFL) number, defined as for a Kronecker delta represented on a discrete grid. In this case, because the Fourier coefficients of the sampled func- cDt PPW tion do not decay to zero before the Nyquist limit of the grid, CFL ¼ ¼ ; (1) Dx PPP the intended field is replaced with a BLI representation with Gibbs type oscillations. It is important to understand that this where c is the sound speed. Stability criteria are often representation is not erroneous per se, but that there is a dis- 14–16 expressed as limits placed on the CFL number. parity between the desired input to the PSTD scheme (in this case a Kronecker delta), and what the scheme is capable of B. PSTD representing via a bandlimited Fourier series. To reduce the In PSTD methods, spatial derivatives are calculated by size of the disparity, smoothing of the intended field can be decomposing the spatially varying acoustic variables into a used to force the Fourier coefficients to decay. This is finite sum of weighted global basis functions. This decom- shown in Fig. 1(b) for the same Kronecker delta function position allows efficient computation of spatial derivatives when frequency is filtered with a Blackman window. using the derivatives of the basis functions. For wave Although this remains an inexact representation of the 1728 J. Acoust. Soc. Am. 141 (3), March 2017 Robertson et al. 18 April 2024 14:53:13 within the head is considered when necessary, with each reverberation consisting of 2 cm propagation through bone, and 20 cm propagation through brain tissue. The combined effects of these numerical inaccuracies and the validity of the established sampling criteria are then examined through con- vergence testing of fully simulated two-dimensional (2D) and three-dimensional (3D) TR protocols. Numerical simulation of ultrasound was carried out with the open source k-Wave toolbox using PSTD and k-space corrected PSTD (with a user defined c )numerical ref 16,19 schemes. These are henceforth referred to as “PSTD” and “k-space” schemes, respectively. The toolbox also includes a FIG. 1. (Color online) Discrete pressure maps and their BLIs. (a) Unsmoothed delta function and (b) delta function frequency filtered with a second-order accurate in time, fourth-order accurate in space Blackman window. (2–4) FDTD scheme, which was also tested. This scheme is described in detail by Strikwerda, and is widely used to sim- original Kronecker delta function, the non-oscillating BLI ulate acoustic wave propagation, including in simulated 11,12 more closely matches the intended underlying pressure dis- TR. Unless stated otherwise, the CFL number was set to tribution as defined by the values at the discrete grid points. 0.3, one-dimensional (1D) tests were carried out on a spatial grid of 4096 grid points, and 2D tests on a 1024  1024 grid. III. NUMERICAL TESTING OF INDIVIDUAL ERRORS Frequency filtered Kronecker delta functions, like that shown in Fig. 1(b), were used to create broadband pressure sources. A. Overview Homogeneous simulation grids were given the acoustic prop- In this section, the impact of various factors which erties of brain tissue, a density of 1040 kg/m ,and a sound affect the convergence of FDTD and PSTD models for the speed of 1560 m/s (also used to represent superficial soft tis- case of transcranial ultrasound simulation is presented. sues). For heterogeneous simulations, bone tissue was These comprise: the influence of the BLI, changes in the assigned a density of 1990 kg/m and a sound speed of effectiveness of the absorbing perfectly matched layer 3200 m/s. When it was necessary to define a specific ultra- (PML), the impact of numerical dispersion, the representa- sonic frequency of interest to calculate the required sampling tion of discontinuities in medium properties, and staircasing criteria, 500 kHz was used. This frequency has seen extensive of acoustic sources and material geometry. The first two rep- use in studies of UNMS (see Table II), sits within the range of resent fundamental considerations in numerical simulations, ultrasound frequencies demonstrating optimal transcranial 21,22 and are dealt with independently. For the subsequent phe- transmission, and has a theoretical minimum focus size of nomena, the specific inaccuracies occurring when simulating 3 mm diameter in soft tissue. the propagation of ultrasound from a source in the deep brain B. The BLI to an external transducer are established. This is modelled as consisting of 10 cm propagation through cerebral soft tissue, The BLI represents a fundamental component of both 1 cm propagation through bone, and 1 cm additional propaga- k-space and PSTD schemes. As such, it is necessary to tion through superficial soft tissue, shown in Fig. 2. Accuracy examine its impact on simulation accuracy before moving is quantified in terms of the resulting error in the amplitude on to more complex factors that affect the rate of conver- and position (calculated using time of arrival) of the temporal gence. Bandlimited interpolation, as described above, can maximum intensity at the target position and the sampling cri- result in a discrepancy between the intended pressure field teria constraining these errors below 10% and 1.5 mm, respec- and the representation of that field within PSTD schemes tively, are established. Beam steering capabilities are when the Fourier coefficients of the intended field have not determined primarily by hardware, and are not considered decayed sufficiently. Practically, this manifests globally as here. Modeling of the skull as a single homogeneous layer in undesired, oscillating pressure values across the simulation this way was recently validated for low frequency model- grid [see Fig. 1(a)]. Therefore, to examine the impact of BLI driven TR by Miller et al. The influence of reverberations effects, it is necessary to determine the amplitude of these undesired pressures relative to that of an intended input. In practice, the error in the representation of a particular pressure distribution will depend on how well it can be rep- resented by a discrete Fourier transform at a specific spatial discretization. Tonebursts have a well-defined power spec- trum determined by their length and central frequency. Therefore, to approximate the BLI error likely to be gener- ally observed, a series of time-varying 10, 30, and 50 cycle acoustic toneburst sources with central wavenumbers approaching the spatial Nyquist limit were used as input FIG. 2. (Color online) A scaled schematic of the simulation model used to evaluate the impact of numerical errors. signals. These sources have 22.7%, 7.4%, and 4.3% full J. Acoust. Soc. Am. 141 (3), March 2017 Robertson et al. 1729 18 April 2024 14:53:13 schemes. In the FDTD scheme used here, outgoing pressure waves are perfectly reflected from the edge of the grid, while for k-space and PSTD schemes outgoing pressure waves are “wrapped” to the opposite edge of the simulation grid (i.e., the grid is toroidal). To replicate free field conditions, k-Wave employs Berenger’s split field PML, where the pres- sure field is artificially divided into Cartesian components to allow selective absorption of the normally incident compo- 19,23 nent. The response of the PML to different frequencies was established by propagating broadband pressure sources toward a 20 point PML on a homogeneous 1D grid using the 15,19 PML profile given in Tabei et al. Rather than the 2-4 FIG. 3. (Color online) (a) Non-causal pressure amplitude as a function of FDTD scheme used elsewhere, a second order accurate in toneburst central PPW for different toneburst lengths and (b) normalized amplitude spectra of non-causal pressure signals and the corresponding space and time (2-2) FDTD scheme with a CFL of 1 was spectra of their source tonebursts, demonstrating how non-casual pressures used to prevent numerical dispersion. It should be noted that relate to the 2 PPW component of the source signal. this FDTD formulation is not practical outside the homoge- width at half maximum (FWHM) bandwidth as a percentage neous 1D case due to stability constraints. The time-varying of central frequency, respectively. The source was positioned pressure traces of the incident wave, the wave reflected from a quarter of the way along a homogeneous 1D computational the surface of the PML, and the wave transmitted to the edge grid with no PML. The simulations were run for the time of the computational grid were recorded. The power spectra of these signals were used to calculate reflection and trans- taken for waves to travel from the source to the center of the mission amplitudes relative to the incident wave as a func- grid. The pressure was recorded at every grid point of the tion of spatial sampling. No notable difference was observed other half of the grid which, according to causality, should between k-space and PSTD schemes. Results are shown in have remained quiescent if the BLI of the pressure field Fig. 4 for k-space and 2-2 FDTD schemes. In both schemes, matched the intended input of compactly supported tone- the pressure reflection coefficient demonstrates a dependence bursts. Error was quantified as the maximum pressure on spatial sampling, rising steadily from below 120 dB for recorded across the second half of the grid relative to the frequencies sampled at above 4 PPW to total reflection at 2 peak pressure of the source toneburst. The results are shown PPW. Transmission to the edge of the grid remains constant in Fig. 3(a) as a function of the PPW of the central wave- at below 70 dB for both schemes until spatial sampling number of the toneburst. The amplitude of the non-causal drops beneath 3 PPW, below which the k-space scheme pressure drops rapidly as the number of PPW increases from shows an increase in transmission and the FDTD scheme the Nyquist limit. Reducing the error requires a higher num- shows a reduction in transmission. These results indicate ber of PPW for shorter tonebursts due to their wider power that the effectiveness of the PML is greatly reduced for spectra, but for all three toneburst lengths the error drops to wavenumbers sampled at below 3 PPW, and it cannot be below 60 dB by 3 PPW. An additional observation was relied on at these PPW values. However, erroneous reflection that wavenumbers corresponding to less than 2 PPW are not and transmission reduce rapidly as sampling increases. It aliased or otherwise propagated on the grid. should be noted that pressure reaching the edge of the grid To determine what frequencies comprised the observed for both schemes is subject to further attenuation within the non-causal pressure, the results obtained from 10 cycle tone- PML when reflected or wrapped back into the grid. bursts were examined further. Time-varying pressure signals Furthermore, the BLI will have influenced the behavior of were recovered from the grid points closest to the wave these tests for frequencies sampled at close to the Nyquist front, which experienced the peak non-causal pressures. The limit, which may explain why the k-space scheme shows an normalized amplitude spectra of these signals resulting from increase in both reflection and transmission close to 2 PPW. source tonebursts with central wavenumbers sampled at 2 The PML was also tested in 2D to determine its depen- and 2.4 PPW are displayed in Fig. 3(b), alongside the corre- dence on the angle of incidence of incoming waves. A sponding amplitude spectra of the source tonebursts. The broadband point source was placed close to the edge of the recorded spectra demonstrate a sharp peak at 2 PPW regard- PML on the 2D grid and propagated into, and across the less of the central frequency of the source toneburst, and the surface of, the PML. The pressure was recorded at the edge amplitude of the peak scales with the amplitude of the 2 of the simulated domain to examine transmission, with each PPW component of the source toneburst. Practically, these recording position corresponding to a particular angle of results demonstrate that this error reduces very rapidly as the incidence. The peak pressure transmission at each angle was spatial sampling of the pressure distribution increases, and at calculated through comparison with a reference recording 3 PPW BLI errors are reduced to below 60 dB. In higher obtained with PML absorption set to zero. The results are dimensions, BLI errors are less severe than the 1D case. shown in Fig. 4(c). Transmission to the edge of the grid is lowest for normally incident waves, rising with increasing C. The PML angle of incidence crossing to above 60 dB at 40 .No The interaction of outgoing pressure waves with the clear relationship between angle of incidence and reflection edge of the simulation grid presents a problem for numerical from the PML was observed. These results should be 1730 J. Acoust. Soc. Am. 141 (3), March 2017 Robertson et al. 18 April 2024 14:53:13 TABLE III. Temporal sampling required to obtain <1.5 mm targeting error. Direct path Reverb. I Reverb. II Reverb. III Target [PPP] [PPP] [PPP] [PPP] FDTD 17.9 21.9 23.9 25.1 PSTD 11.2 18.9 24.2 28.6 k-space 4.0 4.0 4.6 5.4 Results are given in terms of temporal PPP to allow a comparison between dispersive error for the FDTD scheme, which is dependent on both spatial and temporal sampling, and PSTD and k-space schemes, which are dependent on temporal sampling only. Equation (1) shows how the CFL defines a fixed ratio between spatial and temporal sampling. Different CFL numbers will result in a different combination of spatial and temporal requirements for the FDTD scheme. Values for both PSTD schemes are dependent only on tem- poral sampling, and do not require a particular spatial sam- pling to reduce dispersive error. However, with that in mind, the results shown in Table III do demonstrate a clear reduc- tion in the temporal sampling required to minimize disper- sive positional error below acceptable levels for the k-space FIG. 4. (Color online) (a) Reflection from and (b) transmission through the PML as a function of spatial sampling, and (c) transmission to the edge of scheme compared to FDTD and PSTD schemes. the grid as a function of angle of incidence. Here 0 dB means that the reflected or transmitted wave has the same amplitude as the incident wave, E. 1D medium discontinuities i.e., total reflection or transmission. To examine the delay in convergence due to inaccura- considered when designing acoustic sources and considering cies in reflection and transmission from medium discontinu- ities, broadband pressure sources were propagated across a the angles at which pressure waves will impinge on the PML. bone-soft tissue interface (propagation direction makes no difference). The incident, reflected, and transmitted waves were recorded and the power spectra used to calculate inten- D. Numerical dispersion sity reflection and transmission coefficients for each wave- To examine the impact of numerical dispersion, a broad- number. Percentage error in these coefficients was calculated band pressure source was defined on a homogeneous 1D through comparison with the analytical values. To examine grid. Grids with the medium properties of both bone and the dependence of this error on the size of the impedance brain tissue grids were tested, and for the k-space scheme change, these tests were repeated with the impedance of the c was set to the speed of sound in brain tissue. For FDTD ref bone varied up to ten times that of the soft tissue, with sound schemes, the temporal and spatial dispersive errors oppose speed and density varied independently. No difference was each other, with reduced dispersive error at higher CFL num- observed between k-space and PSTD schemes. bers. Therefore the CFL was set to 0.5 for these simula- Figure 5 shows the resulting error in intensity transmis- tions, the highest value at which both schemes are stable in sion and reflection coefficients as a function of PPW, and as 14,15 3D. The time-varying pressure was recorded at a dis- a function of impedance change for a PPW of 6. FDTD and tance of 1 cm, and the phase spectra of the recorded pressure k-space schemes demonstrate a similar error even at high signals were compared to a dispersion-free reference simula- tion obtained with perfect k-space correction. This allowed calculation of phase error per cm propagated in either tissue type as a function of acoustic frequency. Using the model for transcranial propagation of ultrasound to a deep brain tar- get described in Sec. III A, this was used to calculate the sampling criteria required to obtain <1.5 mm positional error for the direct path, and for each reverberation, shown in Table III. To compare, when sampled at 18 PPP, the k-space scheme is exact for soft tissue and gives a 19 lm error in the focal position per cm propagated in bone, PSTD gives a 25 lm error per cm in soft-tissue and a 50 lm error per cm in bone, and FDTD gives 25 lm error per cm in soft-tissue and FIG. 5. (Color online) Error in simulated intensity (a) transmission (Te) and a 130 lm error per cm in bone. (b) reflection (Re) coefficients. J. Acoust. Soc. Am. 141 (3), March 2017 Robertson et al. 1731 18 April 2024 14:53:13 impedance changes. Changes in density result in increased Pythagorean triangles on the grid, specifically: 14.3 (with Pythagorean triple 16, 63, 65), 22.6 (25, 60, 65), 30.5 deviation from the correct coefficient due to the interpolation (33, 56, 65), and 36.9 (39, 52, 65). For these angles, the of the density values on a staggered grid. line-source endpoints are coincident with specific grid point To calculate how the error in 1D reflection and transmis- positions, and any error is only due to the staircased repre- sion will affect transcranial focusing, the error in the inten- sentation of the line, rather than endpoint misregistration sity following transmission across a bone layer (i.e., across (both are aspects of staircasing error). A source defined two bone/soft-tissue interfaces) was computed as parallel to a Cartesian axis was used as a non-staircased ref- f erence. The sources were excited with 10 cycle acoustic Te Te Error ¼ ; (2) 2 tonebursts with a range of central wavenumbers sampled Te at 3–100 PPW. The amplitudes of the source signals were normalized based on any change in the number of distinct where Te is the analytical energy transmission coefficient source points used when defining an angled line source when between bone and soft tissue, and Te is the simulated energy compared to the aligned case. The time-varying field was transmission coefficient as a function of spatial PPW. recorded at 100 points positioned in front of the line source, Reflections inside the skull and skull cavity were not consid- and the sensor map was rotated with the line source to main- ered. To obtain <10% intensity error, the FDTD scheme tain source-sensor geometry. The simulation layout is shown requires 5.9 PPW while the k-space and PSTD schemes in Figs. 6(a) and 6(b). require 4.3 PPW. This result is notable, as the representation Mean errors in the amplitude and position of the peak of discontinuities in medium properties has previously been intensity across the sensor field were calculated indepen- identified as a limitation of PSTD methods. During the dently for each angle tested relative to the aligned, non- update steps of these schemes, the pressure field is multiplied staircased reference case. The maximum mean errors across by the maps of medium density and sound speed, before the range of angles tested for each PPW value are shown in being evaluated by a truncated Fourier series. Step changes Fig. 7. These results demonstrate that staircasing errors in medium properties will therefore introduce Gibbs phe- worsen with lower spatial sampling and are less serious for nomenon into the pressure field, as described in Sec. II C. Pythagorean angles, when endpoints are correctly registered. However, these results indicate that, for the step change in The error in the position of the intensity peak never rises medium properties between bone and soft tissue, the error above 50% of wavelength for any source. Seventeen PPW resulting from the representation of this change within the are required to obtain <10% error in the amplitude of the FDTD scheme tested is greater. intensity peak for all angles tested, while Pythagorean angles require only 7 PPW. Although the error examined here does F. Staircasing not relate directly to the model for transcranial ultrasound Staircasing refers to the spatial approximation that is propagation described above, these results do indicate that necessary when attempting to define continuous geometries staircasing and spatial sampling must be considered when on a discrete, regular Cartesian grid in 2D and 3D. Curved defining acoustic source distributions, and that error can be surfaces and lines at an angle to the Cartesian directions will reduced by ensuring endpoint registration. The testing of be approximated in a stair-stepped manner, and certain multiple angles also allowed examination of how the exact vertex and edge positions do not correspond to points on the mapping of the staircased line relates to the error observed in grid. The impact of staircasing was examined separately the resulting field. No clear relationship between the angle of for acoustic sources and medium distributions. Tests the line source and the level of staircasing error was involved recording the time-varying pressure at a number of observed. However, a staircasing metric was defined as the positions across the field resulting from a staircased repre- average distance between the staircased source points and sentation of a source or medium, and comparison of these their equivalent equispaced points on an ideal angled line. It signals with references obtained from a staircase free simula- was observed that the convergence rate of the error of the tion. Error was then quantified as the percentage error in the staircased source maps, quantified as the average L2 error in amplitude of the temporal peak intensity (calculated using a the recorded pressure signals at the maximum spatial sam- plane wave assumption) and its positional error (derived pling tested, showed a strong dependence on this staircasing from the change in the time of arrival of the intensity peak) metric. Although only a simple metric, this indicates that the as a percentage of wavelength. A positional error of 50% of severity of staircasing error can be predicted through com- wavelength corresponds to 1.5 mm for a source frequency of parison of an ideal or parametric map of the intended geome- 500 kHz in brain tissue. These errors were calculated for try with its staircased representation. each recording position, and then averaged across the field to The impact of staircasing of heterogeneous medium give mean errors in peak intensity amplitude and position. properties was examined in two separate tests. The first was No notable difference in error was observed between FDTD, conceptually similar to the examination of source staircas- PSTD, and k-space schemes across all tests. ing. An acoustic point source excited by ten cycle acoustic The impact of staircasing on acoustic sources was exam- tonebursts with central frequencies ranging from 3 to 80 ined using line-sources with a length of 65  dx, where dx is PPW was propagated across a planar medium boundary (soft the spatial discretization step, at a series of angles to the tissue-to-bone), defined at varying angles to the Cartesian Cartesian grid. These included four angles that form axes. The time varying pressure field was recorded at 100 1732 J. Acoust. Soc. Am. 141 (3), March 2017 Robertson et al. 18 April 2024 14:53:13 FIG. 6. (Color online) Simulation layouts used to test the impact of staircasing (not to scale). Sources are shown in red, and pressure recording positions in black. (a) Non-staircased line source used as reference, and (b) staircased line source used to examine error. (c) Non-staircased medium map used as a refer- ence, and (d) staircased medium used to examine error. (e) High resolution map of a bone-tissue layer used as reference, and (f) downsampled medium. sensor points following the interaction with the medium boundary. The sensor points were rotated with the medium boundary to maintain the simulation geometry. A non- staircased boundary defined along a Cartesian axis was used as a reference map. This simulation layout is shown in Figs. 6(c) and 6(d). The second test was designed as a more accu- rate model of staircasing in transcranial transmission. A 10 cycle, 2520 PPW toneburst was propagated through a medium map comprising a quarter circle bone layer. The medium was then artificially staircased through spatial downsampling, before being remapped to the original grid. A 3780  3780 simulation grid was used due to the large number of integer factors of 3780, which allowed the medium to be successively downsampled while maintaining positioning. The time- varying pressure was recorded across a quarter-circle, and error metrics computed through comparison with the least staircased medium distribution. This simulation layout is shown in Figs. 6(e) and 6(f). An effective PPW value for each level of downsampling was calculated through comparison of the source PPW with the new effective spatial discretization. Mean error measurements across the recorded fields as a func- tion of PPW are shown in Fig. 8 for both medium staircasing tests. For the single interface model, the maximum mean errors across the range of angles tested are shown. In terms of the impact on the model for transcranial FIG. 7. (Color online) Error in staircased line sources as a function of spatial propagation, the results for the bone layer model shown in PPW. (a) Percentage error in peak intensity. (b) Positional error as a percent- Fig. 8 suggest that 20 PPW or above are required to obtain age of wavelength. (c) L2 error at 100 PPW sampling against average devia- less than 10% mean error in intensity transmitted to an exter- tion from ideal line source. Pythagorean sources have both vertices of the parametric line source exactly defined on the grid. nal transducer surface. As might be expected, the errors for J. Acoust. Soc. Am. 141 (3), March 2017 Robertson et al. 1733 18 April 2024 14:53:13 testing of fully simulated TR was carried out in 2D and 3D. The general method is outlined in Fig. 9. It consists of for- ward propagation of a 10 cycle toneburst from a source point inside a virtual skull model to a circular (2D) or hemispheri- cal (3D) virtual transducer-sensor array. As these tests were to examine the specific impact of numerical convergence, the simulated transducer array surface was modelled as a continuous surface made up of point transducers at the resolution of the spatial grid, and no attempt was made to replicate real transducer characteristics. This ensures that convergence is dependent only on the numerical accuracy of the forward simulations and will apply to alternate source FIG. 8. (Color online) Error resulting from propagation through bone-layer and single interface staircased medium boundaries. (a) Percentage error in conditions. The spatial discretization of these simulations peak intensity magnitude. (b) Positional error of peak intensity as a percent- was varied to correspond to a range of spatial PPW values age of wavelength. for the central frequency of the source toneburst. The time- varying pressure signals recorded at the virtual transducer the single interface are lower, with 6 PPW required to obtain position were then time reversed and propagated into the the same error in peak intensity amplitude. The error in peak head to refocus onto the target position. The reversal simula- intensity position is less serious, with mean positional error tions were carried out at the finest discretization feasible, never rising above 50% of wavelength (1.5 mm for 500 kHz and the CFL was 0.3 for all simulations. Due to the change ultrasound in brain tissue) for both tests, as with source stair- in spatial and temporal discretization, it was necessary to casing. This may be due to staircasing introducing a random interpolate the pressure signals recorded in the forward sim- error in acoustic pathlength, leading to a defocusing and ulations onto the spatial and temporal grids used in the rever- change in amplitude rather than a shifting in the peak posi- sal simulations [see Fig. 9(b)] using Cartesian triangulation tion. To place this in context, the voxel size of clinical CT and Fourier interpolation, respectively. In addition, the posi- images is on the order of 0.5 mm at best. This corresponds to tion of the source, defined at a grid point on the high resolu- 6 PPW at 500 kHz, and fewer at higher frequencies. This sug- tion reversal grid, was not conserved due to the varying gests that staircasing may have a significant impact on simula- discretization of the forward simulations, and instead the tions using image derived medium property maps. When nearest neighboring point was used. The peak pressure considered alongside the results for source staircasing, these occurring in a time window of 20 acoustic cycles centered results indicate that staircasing error is likely the most serious on the expected time of refocusing was recorded across the of the numerical errors tested. The single interface medium brain volume. Convergence was established by examining model was also briefly tested using an elastic PSTD model, focusing quality as a function of the discretization used in which indicated that medium staircasing may also have a pro- the forward simulations. The focusing metrics examined nounced impact on simulated mode conversion, although were the spatial and temporal peak pressure across the brain more rigorous testing is necessary. volume, the distance of the peak from the target position, IV. CONVERGENCE TESTING and the FWHM of the focal spot size. Peak pressure and focus FWHM were normalized relative to the results A. Overview obtained for the most highly resolved forward simulation. To examine the combined effects of numerical errors on Simulations in 3D were carried out with toneburst the effectiveness of transcranial TR focusing, convergence sources with central frequencies of 500 kHz, while testing FIG. 9. (Color online) Method for convergence testing in both 2D and 3D. (a) Forward simulation of ultrasound propagation from target point to simulated transducer surface with low-PPW forward discretization. (b) Triangulation is used to extract pressure signals at the transducer positions for the reversal simula- tion. This signal is then Fourier interpolated onto the finer temporal grid, and time-reversed. (c) The time reversed pressure signals are propagated back into the high-PPW head model in a reversal simulation. An example of the pressure field recorded across the brain volume and used to evaluate focusing is shown. 1734 J. Acoust. Soc. Am. 141 (3), March 2017 Robertson et al. 18 April 2024 14:53:13 TABLE IV. Simulation criteria used in convergence testing. 2D 3D Simulation parameter 250 kHz 500 kHz 500 kHz 2 2 2 2 3 3 Simulation grid size 108 –3096 162 –3096 144 –1024 Spatial discretization [mm] 3.1–0.0618 1.6–0.0624 1.6–0.18683 Forward and reverse simulation times [ms] 0.40499 and 0.42499 0.38864 and 0.39864 0.3879 and 0.3979 Temporal discretization [ns] 292.5–5.79 146.25–5.85 146.25–17.515 Transducer radius [mm] 90.9 91.8 95.0 Target Cartesian deviation from transducer focus [mm] [2,3] [2,3] [5,5,5] Simulation runtime [mins] 0.2–83 0.3–78 10–6736 in 2D used both 250 and 500 kHz tonebursts. The simula- key points can be derived from these results. First, for all three tion parameters employed across these tests are shown in metrics, the k-space scheme demonstrates convergence at Table IV. Grid sizes include the absorbing PML layer. The approximately 2 PPW below the FDTD scheme. Second, forward simulations were run for the time taken for an although there is some difference in the position and size of the focus at very low sampling, both 250 and 500 kHz demon- acoustic wave to propagate across the grid three times, strate similar behavior as a function of spatial sampling. This plus the duration of the source toneburst. The reversal sim- indicates that these results can, to some extent, be generalized, ulation was run for the additional time of five acoustic cycles in order to fully capture the reconstructed toneburst. and suggests that the reversal simulations have converged for both frequencies. Finally, of the three refocusing metrics The medium property map used in the convergence tests examined, normalized peak pressure across the brain volume was derived from a T1-weighted MR image obtained from [Fig. 10(a)] requires higher spatial sampling to converge than the Imperial College brain development dataset. Brain either focal volume or the deviation of the focus from the tar- and skull volumes were extracted using the FSL MRI proc- get. This suggests that when fine pressure control is not essing toolbox and converted into a surface mesh using required, coarser sampling criteria may suffice. the iso2mesh toolbox. The reference surface mesh was then sampled onto a 2D or 3D grid with the required spatial C. Convergence testing in 3D discretization steps. Examples of the 2D maps used in for- ward and reversal simulations, and their varying discretiza- 3D convergence testing employed a similar protocol to tions, are shown in Figs. 9(a) and 9(c). In each case, the the 2D convergence testing described above. Ten cycle skull was modelled as a single homogeneous bone layer, and the rest of the simulation domain was assigned brain tissue medium properties. Although fully heterogeneous models of the skull have demonstrated tighter model- driven TR focusing in some cases, homogeneous models remain effective. Furthermore, they allow accurate resampling of the bone map to multiple spatial discretiza- tions without interpolation, and ensure that convergence is dependent on the accuracy of the numerical simulation, rather than the mapping and homogenization of acoustic medium properties. For the k-space scheme c was set to ref the speed of sound in brain tissue. B. Convergence testing in 2D 2D convergence testing was carried out using 10 cycle acoustic toneburst sources corresponding to 250 and 500 kHz frequencies. Reversal simulations were carried out using a 3072  3072 grid, with a spatial discretization corre- sponding to 50 PPW for 500 kHz and 101 PPW for 250 kHz. Forward simulations were carried out using the k-space and FDTD schemes. Reversal simulations were carried out using the k-space scheme only. The results of the 2D convergence testing are shown in FIG. 10. (Color online) Results of 2D convergence testing. (a) Peak pressure Fig. 10, with error in peak pressure position given relative to recorded across the brain volume. (b) The deviation of the peak pressure from the source point in the forward simulations. Refocusing quality the location of the forward simulation source. (c) Normalized FWHM area of in the reversal simulations increases with the spatial PPW of the focal spot size. Normalization is relative to results obtained with the most the simulated frequency in the forward simulations. Several highly resolved forward simulation. J. Acoust. Soc. Am. 141 (3), March 2017 Robertson et al. 1735 18 April 2024 14:53:13 acoustic tonebursts with a central frequency of 500 kHz Accordingly, the reduction in error and lack of dependence were propagated from a source point inside a full 3D model on spatial sampling when calculating positional error relative of the human skull to a simulated hemispheric transducer. to the actual source position from the forward simulation Reversal simulations were carried out using a 1024 1024 suggests that any error in the position of the peak is in 1024 simulation grid with a spatial discretization corre- fact due solely to the misregistration of the source points. sponding to 16.7 PPW. Forward and reversal simulations Generally, these results confirm that when targeting accuracy were carried out using the k-space scheme only. Testing was is the prime concern, spatial sampling requirements are laxer also carried out using homogeneous media for forward and than when a tight focal volume with known peak pressure reverse simulations, to test the accuracy of the spatiotempo- amplitude is required. This is in agreement with previous ral interpolation. The results for heterogeneous 3D conver- studies which have demonstrated that good spatial targeting gence testing are shown in Fig. 11. The results for of HIFU can be obtained via simulated TR using relatively 11,30 normalized peak pressure amplitude in Fig. 11(a) show simi- coarse spatial sampling. Homogeneous testing demon- lar trends to the 2D results. Six PPW are required to obtain strated total convergence across all metrics by 3.5 PPW, 95% reconstruction of pressure at the target (corresponding which is to be expected given the behavior of the PML and to <10% drop in intensity) and 10 PPW to attain conver- BLI, discussed in Sec. III. gence. The convergence of the volume of the focal spot V. SUMMARYAND DISCUSSION [Fig. 11(c)] shows similar behavior, although it only requires 6 PPW to fully converge. Given the known impact of stair- In this paper, a comprehensive assessment of the impact casing in 2D, the faster convergence here is likely due to a of different factors that affect the convergence of numerical reduced staircasing error for 3D geometry. The results in models for the simulation of transcranial ultrasound propaga- Fig. 11(b) show the error in the position of the pressure peak tion was carried out. The spatial and/or temporal sampling relative to both the reversal target and the shifted forward required to reduce inaccuracies below the levels required for source point. This error is reduced compared to the 2D case, targeting of deep brain nuclei for neurostimulation were never rising above 1.1 mm and, when computed relative to determined. the position of the forward source, is stable using sampling Initial simulations examined reduction in the effective- as low as 2 PPW. However it can be seen from the other ness of the PML, and the impact of the BLI when using results that at this sampling the peak pressure amplitude is k-space and PSTD methods. Both the PML and BLI lead to much lower, with a larger focal spot. The apparent periodic- erroneous pressures appearing on the grid when simulating ity in the positional error when calculated relative to the frequencies sampled at close to the spatial Nyquist limit. parametrically defined target point is likely due to the oscil- Although both of these effects have the potential to seriously lating distance of a definable source point from this position. reduce the accuracy of the simulations, they decrease in sever- ity rapidly as the rate of spatial sampling increases. Above 3 PPW, erroneous pressures resulting from both BLI and PML effects were at least 60 dB below the amplitudes of the ultra- sound sources being simulated. Numerical dispersion has a serious effect on the accuracy of FDTD and PSTD schemes, resulting in high temporal sam- pling requirements to reduce positional error. However, this wasnot thecasefor the k-space scheme, where 3 PPW will serve to limit dispersion sufficiently for transcranial transmis- sion for any stable CFL value. Errors in reflection and trans- mission from discontinuous medium properties manifest in the magnitude of reflected and transmitted simulated intensities. Despite the representation of step changes in media previously being identified as a key limitation of PSTD schemes, the error wasshowntobemoresevere for the2-4 FDTD scheme tested. To reduce error in the intensity below 10% following transcra- nial transmission, k-space and PSTD schemes require 4.3 PPW, while FDTD requires 5.9 PPW. Staircasing of source and medium geometries was shown to require the most stringent sampling criteria to obtain required accuracy, affecting FDTD, PSTD, and k-space schemes equally. Both source and medium staircasing were shown to have a greater impact on the intensity amplitude of FIG. 11. (Color online) 3D convergence testing results. (a) Normalized peak the toneburst signal being examined than the position of the pressure amplitude across the brain. (b) Deviation of pressure peak from both intensity peak. The results shown in Figs. 7 and 8 indicate that the parametrical defined target and the source used in the forward simulation. 20 PPW are required to reduce the error in peak intensity fol- (c) Normalized half-maximum focal volume. Normalization is relative to results obtained with the most highly resolved simulation. lowing transcranial transmission below 10%. The preliminary 1736 J. Acoust. Soc. Am. 141 (3), March 2017 Robertson et al. 18 April 2024 14:53:13 examination of a potential staircasing metric also suggests that alternative low-intensity, transcranial ultrasonic therapies the error resulting from a particular staircased geometry is such as opening the blood-brain barrier with ultrasound, as directly related to its deviation from the ideal geometry. well as existing transcranial HIFU ablation therapies. Use of Convergence testing of a fully simulated TR protocol appropriately discretized simulations will ensure accurate using 2D and 3D head models was used to examine the targeting and effective therapy as the field of ultrasonic neu- impact of all numerical errors in concert. Testing in 2D for rostimulation develops. 250 and 500 kHz ultrasound showed a faster rate of conver- gence for all focusing metrics for the k-space scheme when ACKNOWLEDGMENTS compared to FDTD. In addition, the error in the peak pres- sure amplitude at the focus showed slower convergence than The authors would like to thank Charlotte Stagg and both the positional error, and the volume of the focus. This Adam Thomas for helpful discussions, and Nishant Ravikumar is likely due to the most serious source of error, medium and Zeike Taylor for the provision of the skull and brain staircasing, which was shown to have a greater impact on meshes. This work was supported by the Engineering and the peak intensity amplitude of transcranially transmitted Physical Sciences Research Council (ESPRC), UK. J.J. is ultrasound, than the position of the peak. Results in 3D financed from the SoMoPro II programme. This research has showed similar trends to the 2D results for the convergence acquired a financial grant from the People Programme (Marie of the peak pressure amplitude. The focal spot size showed Curie Action) of the Seventh Framework Programme of EU slightly slower convergence in the 3D case, while the posi- according to the REA Grant Agreement No. 291782. The tional error demonstrated almost no dependence on the sam- research is further co-financed by the South-Moravian Region. pling rate of the forward simulation. This indicates that less This work reflects only the author’s view and the European stringent sampling may suffice for applications concerned Union is not liable for any use that may be made of the only with the position of the focus, rather than the size of the information contained therein. Computational resources were focal spot and the exact amplitude at the target. When fine provided by the IT4Innovations Centre of Excellence project control over the pressure amplitude is required, stricter sam- (CZ.1.05/1.1.00/02.0070), funded by the European Regional pling may be necessary. Despite the relatively severe error Development Fund and the national budget of the Czech resulting from staircasing at higher spatial sampling, all Republic via the Research and Development for Innovations three metrics of focusing quality were well converged at Operational Programme, as well as Czech Ministry of below 20 PPW. This discrepancy may be due to the differ- Education, Youth and Sports via the project Large Research, ences between the convergence testing protocol and the spe- Development and Innovations Infrastructures (LM2011033). cific test used to examine staircasing across a bone layer, and suggests that the influence of staircasing is case specific. The work described above is subject to some limita- APPENDIX tions, primarily the degree to which the examination of indi- Simulations were carried out using the open source k- vidual numerical errors can be generalized to different Wave toolbox for MATLAB, Cþþ. The toolbox includes k-space, setups, although trends and qualitative observations remain PSTD, and 2-4 FDTD codes for the time-domain simulation of valid. Many of the tests only examine toneburst sources, and acoustic fields. 1D simulations were carried out in the MATLAB the error is evaluated over a small field, with pressure environment on a Dell Precision T1700 with an Intel Xeon recorded at a limited number of sensor positions (see Fig. 6). E3-1240 3.40 GHz CPU and 16 GB of RAM running A separate 2-2 FDTD scheme was used to examine the effec- Windows 10 64 bit. 2D simulations were carried out in the tiveness of the PML, which may not be exactly relatable to MATLAB environment with CUDA hardware acceleration on a the commonly used 2-4 FDTD scheme. Furthermore, the Dell PowerEdge R730 compute server with 2  6-core Xeon impact of shear wave propagation was not examined. This E5-2620 2.4 GHz CPUs, 64 GB of 1866 MHz memory, on an will not have affected 1D or homogeneous simulations, but a Nvidia Titan X GPU with 3072 CUDA cores and 12 GB of more thorough examination of medium staircasing should include testing of elastic wave propagation. Similarly, no memory. The largest 2D simulations had a domain size of effort was made to examine the manifestation of numerical 3780 including the PML and comprised 258 462 time steps, errors when modeling nonlinear propagation or acoustic with a total runtime of 10.6 h. 3D simulations were carried out absorption, which will become relevant for applications on the IT4I Salomon supercomputing cluster. Each simulation requiring the simulation of high-amplitude ultrasound, such was carried out on Intel Xeon E5-4627v2, 3.3 GHz, 8cores and as HIFU. It should be noted that, in simulated TR, account- 256 GB of RAM per simulation. The largest 3D simulations ing for acoustic absorption occurs in the post-processing had a domain size of 1024 including the PML and comprised stage, when converting recorded pressure signals into driv- 22 718 time steps, with a total runtime of 112.3 h. The skull ing amplitudes, and work examining absorption should mesh used in convergence testing is Copyright Imperial focus on this stage of the simulated TR process. College of Science, Technology and Medicine 2007. All rights The results presented here are primarily relevant to the reserved. www.brain-development.org. simulation of transcranial ultrasound propagation for TR tar- geting of deep brain structures with finely controlled ultra- P. J. Karas, C. B. Mikell, E. Christian, M. A. Liker, and S. A. Sheth, sound for the purposes of neurostimulation. However, the “Deep brain stimulation: A mechanistic and clinical update,” Neurosurg. criteria and simulations presented are also relevant to Focus 35(5), E1–E16 (2013). J. Acoust. 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