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REVIEW 3 major objections 4 minor 40 references

An Angular Spectrum Method for Nonlinear Propagation in Heterogeneous Tissue with Immersed Sources for Ultrasound

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper claims that a modified angular spectrum method reproduces full-wave nonlinear ultrasound fields through ex vivo skull and from deeply curved bowl transducers, matching focal depth to 2.3 percent and focal intensity to 1.1…

desk verdict The solver is real and the benchmarks are clean, but the headline nonlinear obliquity correction is not actually supported by the bowl benchmark, which sits in the regime the paper admits the correction fails. read the letter →

arxiv 2608.07389 v1 pith:VRIHSONP submitted 2026-08-07 physics.med-ph

Gianmarco Pinton This is my paper · ORCID
classification physics.med-ph
keywords angularspectrummethodnonlinearacousticstranscranialfocusedultrasoundshockcapturingheterogeneoustissuesplit-stepbowltransducerfull-wavevalidation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that an angular spectrum method, which propagates a sound field plane-wave by plane-wave with FFTs, can be extended to handle strongly nonlinear ultrasound in heterogeneous tissue and from deeply curved bowl transducers, a regime previously dominated by full-wave finite-difference solvers. It introduces a consistent obliquity correction so that each plane-wave component travels its true path length $\Delta z/\cos\theta$ through both absorption/dispersion and nonlinear steepening, a shock-capturing Burgers update that resolves sharp waveforms without the CFL-driven temporal refinement of FDTD, and a plane-by-plane injection of curved sources that preserves the aperture-dependent shock-formation distance. If these claims hold, nonlinear transcranial treatment planning and curved-transducer dosimetry become practical on desktop hardware, since the method matches Fullwave 2 focal fields to a few percent using roughly nine times less memory and about a third of the wall time.

What carries the argument

The load-bearing mechanism is the split-step operator sequence $p^{(n+1)} = A^{1/2} H^{1/2} N H^{1/2} A^{1/2} p^{(n)}$, in which diffraction $H$ is exact in wavenumber space, attenuation/dispersion $A$ carries a per-mode $k/k_z$ obliquity factor, and the retarded-time Burgers update $N$ is computed with a Kurganov\u2013Tadmor central-upwind flux. The named objects are the obliquity-corrected attenuation filter $\exp[-(\alpha+i\alpha^*)\,\Delta z\, k/k_z]$ and the beam-averaged nonlinear scaling $N_{\mathrm{eff}} = N\langle k/k_z\rangle$. These filters reuse the same FFT pair as diffraction, so the added cost is small. The plane-by-plane source injection turns a curved bowl into a stack of axial slices added at their own depths, preserving the distance each ray travels before shock formation.

What would settle it

Run the same bowl geometry with a wider rim angle (or a higher-frequency equivalent) so that rim-ray obliquity exceeds about 30 degrees, and compare the near-focal intensity and harmonic spectra against a full-wave solver; if the scalar obliquity correction misfires where wavefronts cross, the focal region will deviate beyond the reported 2.3 percent bound for the R = 80 mm bowl.

Watch

Extended reading notes

Core claim

The central claim is that a split-step angular spectrum solver with three modifications reproduces full-wave-quality nonlinear focal fields in heterogeneous media and curved-source geometries. The attenuation and dispersion filter is applied on the full $(k_x, k_y, \omega)$ grid with a per-mode $k/k_z$ factor, so each plane wave attenuates and disperses over its true path length $\Delta z/\cos\theta$. The nonlinear Burgers step scales its coefficient by the power-weighted mean obliquity $\langle k/k_z\rangle$ of the current spectrum, and is discretized with a second-order Kurganov\u2013Tadmor central-upwind flux with MUSCL reconstruction and SSP-RK2 time stepping, giving stable shock resolution. Curved bowl transducers are injected slice-by-slice at their correct axial depths, and CT-derived phase-and-amplitude screens model transcranial aberration and insertion loss. Against Fullwave 2, the method matches transcranial focal-plane intensity to 1.1 percent RMS with a 5.4 dB predicted skull insertion loss, and matches a clinical bowl's focal depth to 2.3 percent with 9x less memory and 2.9x less wall time.

Load-bearing premise

The method assumes that one power-weighted number, the mean obliquity $\langle k/k_z\rangle$, can stand in for the true path-length correction of every plane-wave component during nonlinear steepening; that approximation loses accuracy where comparable-amplitude plane-wave fronts cross, such as near the focus of a strongly curved bowl.

Editorial extensions

If this is right

  • Nonlinear transcranial treatment plans can be computed with desktop-class memory and GPU-friendly FFTs, making iterative aberration correction feasible.
  • Shock formation at multi-MPa focal pressures is resolved without refining the 3-D spatial grid, since temporal sampling lives on a per-point 1-D axis.
  • Deeply curved bowls like the TIPS annulus can be modeled with their true aperture and focusing delays, preserving shock-formation distances.
  • The per-pixel absorbed-intensity loss map provides a heat-source and radiation-force input that accounts for harmonic content without a single-frequency attenuation assumption.
  • The same flux-conservative machinery extends to the cubic Burgers shear-shock regime in soft tissue, enabling modeling of shear shock waves.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The scalar beam-averaged nonlinear obliquity correction is likely the first approximation to break for very high-numerical-aperture transducers; a per-direction nonlinear discretization would be the natural stress test.
  • Because the loss field is computed from the actual harmonic spectrum at each pixel, thermal-dose and acoustic-radiation-force estimates could be more accurate than single-frequency attenuation models, although the paper does not validate against measured heating.
  • The shear-shock cubic extension suggests the framework could model ultrasound neuromodulation and shear-wave elastography shock regimes, not just longitudinal therapeutic beams.
  • The plane-by-plane injection scheme could be adapted to arbitrary surface sources in range-dependent waveguides, since it only needs a depth-sorted source surface.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The manuscript develops a modified angular spectrum method (ASM) for three-dimensional nonlinear acoustic propagation through heterogeneous tissue, with three claimed contributions: a consistent obliquity correction on both the linear attenuation/dispersion operator and the nonlinear Burgers operator, a flux-conservative shock-capturing discretization of the retarded-time Burgers update using Kurganov–Tadmor/MUSCL with Strang splitting and adaptive CFL sub-cycling, and a plane-by-plane source-injection scheme for deeply curved bowl transducers. It also adds CT-derived phase and amplitude screens, an intensity-loss tracking output, and three absorbing-boundary improvements. Validation includes analytical piston benchmarks, one-dimensional Burgers and Riemann problems, a transcranial comparison against Fullwave 2 through an ex vivo human skull, and a water-bowl comparison against Fullwave 2 for the TIPS annular array. The reported headline agreements are 1.1% RMS focal-plane intensity difference in the transcranial benchmark, a 5.4 dB through-skull insertion loss, and a 2.3% focal-depth match for the bowl with 9x less memory and 2.9x less wall time.

Significance. If the load-bearing concerns are resolved, this would be a practically valuable contribution to transcranial and therapeutic ultrasound planning: the ASM formulation is substantially cheaper than FDTD, the shock-capturing nonlinear update is well tested in one-dimensional benchmark problems, the boundary treatments are quantified, and the source code and validation scripts are openly available, which strengthens reproducibility. The analytical validations of diffraction, attenuation, and the Burgers scheme are clean, and the transcranial focal-plane agreement with Fullwave 2 is strong. However, the amplitude validation of the bowl benchmark relies on a fitted source scaling rather than an independent calibration, and the central novelty for the nonlinear operator, the beam-averaged obliquity correction, is explicitly acknowledged in Section 6 to lose accuracy in the crossing-front regime that a strongly focused bowl produces. These issues must be addressed before the central claims can be accepted.

major comments (3)
  1. [§5.3 (Fullwave 2 comparison)] The statement that both methods produce a focal peak pressure of 13.8 MPa is not an independent validation, because the ASM source amplitude was scaled by an ad hoc factor of 1.31 to align with the Fullwave 2 source convention. Since this factor is fitted to the reference, it calibrates away exactly the amplitude error that the benchmark is supposed to measure. Please specify how the factor was determined from the source-plane normalization, report the unscaled comparison and the sensitivity of the focal-depth and beam-shape metrics to this scaling, or remove the absolute-pressure claim.
  2. [§6 and Eq. (6)] The beam-averaged obliquity correction ⟨k/kz⟩ is the central novel element for the nonlinear operator, but its only direct quantitative validation in Section 3.3 is on single tilted plane waves, which have no crossing fronts. Section 6 explicitly concedes that the approximation loses accuracy where plane-wave fronts of comparable amplitude cross, such as near the geometric focus; the TIPS bowl benchmark focuses electronically at z=50 mm and is exactly such a crossing region. No ablation separates the nonlinear obliquity correction from the per-mode attenuation correction, so the 2.3% focal-depth agreement cannot be attributed to the claimed nonlinear correction. Add a controlled study that toggles Eq. (6) on and off, or an alternative strongly focused configuration with measurable nonlinear effects, and quantify the error in the crossing-front regime.
  3. [§5.3 and §6] The combination of the two headline capabilities, heterogeneous tissue propagation and immersed curved-source injection, is not validated together: the bowl benchmark is in water and the transcranial benchmark uses a flat sparse array. Because the paper's title and conclusions promise a method for heterogeneous tissue with immersed sources, the composition of these two extensions should be demonstrated, or the scope claims should be tempered accordingly.
minor comments (4)
  1. [Figure 10 and §5.3] Because each field in Figure 10 is normalized to its own peak, the absolute level agreement between the ASM and Fullwave 2 is invisible; please also report the peak-intensity ratio or the absolute dB offset before normalization.
  2. [§2.1, Eq. (6)] The normalization of the power-weighted mean obliquity is clear from the displayed formula, but the text should state explicitly that the denominator is Σ|p̃|² and that the sum excludes evanescent modes, as this restriction is mentioned only in passing.
  3. [§5.2 and §5.3] For the bowl benchmark, the paper gives grid sizes and propagation-step counts but does not state the temporal sampling or the acoustic time-window used in the ASM run; these parameters should be reported so the comparison can be reproduced.
  4. [§3.3 and Table 2] The text in Section 3.3 describes 'fixed-step' KT versus adaptive sub-cycling, while Table 2 of the appendix compares Rusanov, Rusanov-TVD, and patched KT; the relation between the two sets of scheme variants should be clarified in the main text.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: analytical benchmarks and disclosed calibrations keep the central claims independently grounded.

full rationale

The paper's derivation chain is self-contained at each load-bearing step. The beam-averaged obliquity correction of Eq. (6) is presented as an explicit approximation, not as a derived consequence of the benchmarks, and its estimator is independently tested on synthetic tilted plane waves (Section 3.3). The transcranial and bowl-transducer benchmarks compare against Fullwave 2, a separate FDTD solver developed by the same author but governed by different equations and discretization; it is not a re-expression of the ASM equations. The 1.31 source-amplitude scaling in Section 5.3 is explicitly disclosed as aligning source conventions, and the paper's headline validation quantities (focal depth to within 2.3%, beam-shape agreement, 1.1% RMS focal-plane intensity, 5.4 dB insertion loss) are not obtained by fitting to those targets. The admitted inaccuracy of the scalar obliquity approximation near crossed wavefronts (Section 6) is a stated limitation and a correctness risk, not a circular step, because no prediction is defined in terms of that approximation's output. No quoted equation reduces to its own input by construction, no fitted parameter is renamed as a prediction, and the self-citations (Fullwave 2 references [1,2] and the companion framework) are not load-bearing in the sense of supplying the central result. Accordingly, the paper warrants a circularity score of 0.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced. The free parameters are the bowl source amplitude scaling factor (fitted to Fullwave 2) and unspecified absorbing-boundary tuning parameters. The key axioms are the forward-only propagation model, the Burgers equation as the nonlinear model, the CT-derived screen approximation, and the scalar obliquity correction.

free parameters (2)
  • Source amplitude scaling factor for bowl benchmark = 1.31
    In Section 5.3, the ASM source amplitude is scaled by a factor of 1.31 to align with the Fullwave 2 source convention. After this scaling both methods produce the same 13.8 MPa focal peak pressure, so this parameter is fitted to the reference for the amplitude comparison.
  • Absorbing boundary layer parameters = not specified
    Section 2.2 introduces the Wendland C2 taper, frequency-weighted damping with an f0/max(f,fmin) exponent, and a super-absorbing correction with strength αs in [0,1]. The numerical values of layer thickness, αs, and fmin are not reported, and Table 1 gives only resulting reflection ratios.
assumptions (5)
  • domain assumption Forward-only propagation; reflections, mode conversion, and multiple scattering are neglected.
    Stated as the principal limitation in Sections 1 and 6. The phase-and-amplitude screen model captures only forward-transmitted wavefronts, and the Fullwave 2 comparison shows the discrepancy is concentrated near the skull where backward effects are strongest.
  • domain assumption The retarded-time Burgers equation (Eq. 2) is an adequate model for cumulative nonlinear distortion in heterogeneous tissue.
    The split-step solver relies on this 1-D pointwise nonlinear update combined with spectral diffraction. This is standard in nonlinear ASM literature, but it neglects full Westervelt cross-terms and mode coupling through heterogeneous media.
  • standard math Power-law attenuation alpha(f) = alpha0 f^n with Kramers-Kronig-consistent dispersion describes tissue absorption.
    Used in Eq. (7) and validated in Section 3.2. The dispersion term alpha* is not explicitly defined in the text, but the validation shows the filter reproduces the expected attenuation and phase velocity.
  • domain assumption Phase-and-amplitude screens derived from CT data, including normal-incidence impedance transmission, capture first-order skull aberration and insertion loss.
    Section 2.3 and Section 4 apply thin screens with phase shifts proportional to sound-speed contrast and amplitude factors for attenuation and impedance mismatch. This neglects refraction and ray bending at non-normal incidence.
  • ad hoc to paper The power-weighted mean obliquity scalar is a sufficient correction for the nonlinear path-length effect.
    Equation (6) scales the Burgers coefficient by the power-weighted average k/kz over the spectrum. The paper acknowledges in Section 6 that this loses accuracy where comparable-amplitude plane-wave fronts cross near the geometric focus.

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Pith. "Pith review of An Angular Spectrum Method for Nonlinear Propagation in Heterogeneous Tissue with Immersed Sources for Ultrasound." pith.science (2026). https://pith.science/paper/VRIHSONP

@misc{pith2026260807389,
  author       = {Pith},
  title        = {Pith review of: An Angular Spectrum Method for Nonlinear Propagation in Heterogeneous Tissue with Immersed Sources for Ultrasound},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VRIHSONP}},
  note         = {Machine review of arXiv:2608.07389}
}
read the original abstract

A modified angular spectrum method (ASM) is developed for three-dimensional nonlinear acoustic propagation through heterogeneous tissue, targeting transcranial and therapeutic ultrasound. First, a consistent obliquity correction is applied to both the linear and nonlinear operators of the split-step update. The attenuation and dispersion filter carries a per-mode k/kz factor on the full wavenumber-frequency grid, so each component accumulates absorption and phase over its true path length dz/cos(theta), while the Burgers coefficient is scaled by the power-weighted mean beam obliquity. Second, the retarded-time Burgers update is discretized with a second-order Kurganov-Tadmor central-upwind flux using MUSCL reconstruction and SSP-RK2 time integration, composed with diffraction and attenuation through Strang splitting with adaptive CFL sub-cycling, resolving fully developed shocks without the temporal refinement that the CFL coupling imposes on FDTD. Third, a plane-by-plane source-injection scheme decomposes deeply curved bowl transducers into axial slices injected at their correct propagation depths, preserving the aperture-dependent shock-formation distance. Phase-and-amplitude screens derived from skull CT data model transcranial aberration and insertion loss, and three enhanced absorbing-boundary treatments reduce boundary reflections by a factor of 2.4. Analytical validation reproduces the baffled-piston far-field pattern to 0.014 RMS and the focused-piston focal pressure to within 2.3 percent. In a transcranial benchmark through an ex vivo human skull, the ASM matches the Fullwave 2 focal-plane intensity to 1.1 percent RMS and predicts a 5.4 dB through-skull insertion loss. For a bowl transducer (R = 80 mm, f0 = 1 MHz), the ASM matches the Fullwave 2 focal depth to within 2.3 percent with 9x less memory and 2.9x less wall time.

Figures

Figures reproduced from arXiv: 2608.07389 by the authors.

Figure 1
Figure 1. Focused piston diffraction (a = 8λ, F = 20 mm, F# ≈ 3.3). The top row shows the on-axis amplitude versus the paraxial solution (left, RMS 0.0645) and the focal-plane lateral profile versus the Airy pattern (right, RMS 0.002). The bottom row shows the x–z and y–z time-integrated intensity. frequency-only filter exactly. At θ = 29◦ , the obliquity-corrected magnitude is 0.885 against 0.898 without correction, a ratio … view at source ↗
Figure 2
Figure 2. Attenuation and Kramers–Kronig dispersion validation (0.5–10 MHz). The top row shows the mea￾sured and theoretical attenuation and their difference. The bottom row shows the phase velocity and a waveform comparison [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Shock-forming Burgers problem, showing the L 2 error versus step size (left), the total variation (center), and the coarsest-step waveforms (right). The adaptive KT scheme is stable and accurate at all step sizes. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Riemann problem at t = 0.1, 0.3, 0.5. Rusanov (circles) and KT (squares) overlaid on the exact solution (solid). 3.4 Boundary treatments The three boundary improvements were evaluated using apertures wide enough to drive significant energy toward the domain edges [PIT…
Figure 5
Figure 5. Figure 5: Combined boundary with nonlinear propagation, showing the final waveform (left) and spectrum (right) for four boundary configurations. The interior waveform is invariant and harmonic levels match to <1 dB. 3.5 Convergence A pairwise convergence study in the strongly no…
Figure 6
Figure 6. Figure 6: Step-size convergence, showing the coarsest waveforms (left) and the pairwise L 2 error (right). Strang+KT achieves rate ≈ 2. 4 Transcranial propagation To validate the phase-and-amplitude-screen extension, focused ultrasound transmission through an ex vivo human skull…
Figure 7
Figure 7. Figure 7: Transcranial simulation setup. The top row shows the whole-skull axial reference slice with the beam axis overlaid (left), the source-plane pressure magnitude from the 1024-element sparse array (center), and the beam-aligned sound-speed map in the y–z plane (right). Th…
Figure 8
Figure 8. Figure 8: Transcranial comparison referenced to the homogeneous ASM field. The top row shows axial (x–z) beam profiles for the homogeneous ASM, the through-skull ASM, and Fullwave 2. The middle row shows the corresponding focal-plane (x–y) intensity maps at each method’s focal d…
Figure 9
Figure 9. Figure 9: TIPS bowl transducer setup. The panels show the bowl cross-section (R = 80 mm) with element boundaries and focal lines (top left), the plane-by-plane source slices spanning the 14.6 mm bowl depth (top right), the 8-element annular aperture (bottom left), and the per-el…
Figure 10
Figure 10. Figure 10: Comparison of the plane-by-plane ASM and Fullwave 2 for the TIPS bowl in water. The top row shows axial (x–z) beam profiles and the middle row shows focal-plane (x–y) intensity maps at each method’s focal depth. The bottom row shows the on-axis intensity (left) and th…
Figure 11
Figure 11. Figure 11: Cubic-Burgers Riemann Test C (uL = +1, uR = −1) at z = 1. The solid line shows the exact entropy solution, with the rarefaction u = p −τ /z on [−z, −z/4] and the sonic shock at τ = −z/4. Markers show standard Rusanov (red +), TVD-Rusanov (orange ×), and KT (blue dots)…

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Works this paper leans on

40 extracted references · 39 canonical work pages

  1. [1]

    A heterogeneous nonlinear attenuating full-wave model of ultrasound,

    G. F. Pinton, J. Dahl, S. Rosenzweig, and G. E. Trahey, “A heterogeneous nonlinear attenuating full-wave model of ultrasound,”IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 56, no. 3, pp. 474–488, 2009

    work page 2009

  2. [2]

    Numerical methods for nonlinear wave propagation in ultrasound,

    G. F. Pinton, “Numerical methods for nonlinear wave propagation in ultrasound,” Ph.D. disserta- tion, Dept. Biomed. Eng., Duke University, Durham, NC, 2007

    work page 2007

  3. [3]

    New approaches to nonlinear diffractive field propagation,

    P. T. Christopher and K. J. Parker, “New approaches to nonlinear diffractive field propagation,” The Journal of the Acoustical Society of America, vol. 90, no. 1, pp. 488–499, 1991

    work page 1991

  4. [4]

    The ultrasonic field of a Gaussian transducer,

    G. Du and M. A. Breazeale, “The ultrasonic field of a Gaussian transducer,”The Journal of the Acoustical Society of America, vol. 78, no. 6, pp. 2083–2086, 1985

    work page 1985

  5. [5]

    Modeling of nonlinear ultrasound propagation in tissue from array transducers,

    R. J. Zemp, J. Tavakkoli, and R. S. C. Cobbold, “Modeling of nonlinear ultrasound propagation in tissue from array transducers,”The Journal of the Acoustical Society of America, vol. 113, no. 1, pp. 139–152, 2003

    work page 2003

  6. [6]

    Acoustic shock wave propagation in a heterogeneous medium: a numerical simulation beyond the parabolic approximation,

    F. Dagrau, M. Rénier, R. Marchiano, and F. Coulouvrat, “Acoustic shock wave propagation in a heterogeneous medium: a numerical simulation beyond the parabolic approximation,”The Journal of the Acoustical Society of America, vol. 130, no. 1, pp. 20–32, 2011

    work page 2011

  7. [7]

    Rapid quantitative imaging of high intensity ultrasonic pressure fields,

    H. Luo, J. Kusunose, G. Pinton, C. F. Caskey, and W. A. Grissom, “Rapid quantitative imaging of high intensity ultrasonic pressure fields,”The Journal of the Acoustical Society of America, vol. 148, no. 2, pp. 660–677, 2020. 17 G F Pinton

    work page 2020

  8. [8]

    Nonlinear reflection of shock shear waves in soft elastic media,

    G. F. Pinton, F. Coulouvrat, J.-L. Gennisson, and M. Tanter, “Nonlinear reflection of shock shear waves in soft elastic media,”The Journal of the Acoustical Society of America, vol. 127, no. 2, pp. 683–691, 2010

    work page 2010

Show all 40 references
  1. [9]

    Numerical simulation of focused shock shear waves in soft solids and a two-dimensional nonlinear homogeneous model of the brain,

    B. Giammarinaro, F. Coulouvrat, and G. Pinton, “Numerical simulation of focused shock shear waves in soft solids and a two-dimensional nonlinear homogeneous model of the brain,”Journal of Biomechanical Engineering, vol. 138, no. 4, p. 041003, 2016

    2016

  2. [10]

    A reduced viscoelastic FDTD formulation for ultrasound-driven shear wave propagation in soft tissue,

    G. Pinton, “A reduced viscoelastic FDTD formulation for ultrasound-driven shear wave propagation in soft tissue,” arXiv:2607.28414 [physics.med-ph], 2026

    2026 arXiv

  3. [11]

    Ultrasound beam simulations in inhomogeneous tissue geometries using the hybrid angular spectrum method,

    U. Vyas and D. A. Christensen, “Ultrasound beam simulations in inhomogeneous tissue geometries using the hybrid angular spectrum method,”IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 59, no. 6, pp. 1093–1100, 2012

    2012

  4. [12]

    Simulation of hemispherical transducers for transcranial HIFU treatments using the hybrid angular spectrum approach,

    S. Almquist, D. L. Parker, and D. A. Christensen, “Simulation of hemispherical transducers for transcranial HIFU treatments using the hybrid angular spectrum approach,”Journal of Therapeutic Ultrasound, vol. 3, no. S1, p. P9, 2015

    2015

  5. [13]

    Evaluation of a wave-vector-frequency-domain method for nonlinear wave propagation,

    Y. Jing, M. Tao, and G. T. Clement, “Evaluation of a wave-vector-frequency-domain method for nonlinear wave propagation,”The Journal of the Acoustical Society of America, vol. 129, no. 1, pp. 32–46, 2011

    2011

  6. [14]

    Numerical modeling of ultrasound propagation in weakly heterogeneous media using a mixed-domain method,

    J. Gu and Y. Jing, “Numerical modeling of ultrasound propagation in weakly heterogeneous media using a mixed-domain method,”IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 65, no. 7, pp. 1258–1267, 2018

    2018

  7. [15]

    Rapid calculation of acoustic fields from arbitrary continuous-wave sources,

    B. E. Treeby, J. Budisky, E. S. Wise, J. Jaros, and B. T. Cox, “Rapid calculation of acoustic fields from arbitrary continuous-wave sources,”The Journal of the Acoustical Society of America, vol. 143, no. 1, pp. 529–537, 2018

    2018

  8. [16]

    Evaluation of the angular spectrum approach for simulations of near-field pressures,

    X. Zeng and R. J. McGough, “Evaluation of the angular spectrum approach for simulations of near-field pressures,”The Journal of the Acoustical Society of America, vol. 123, no. 1, pp. 68–76, 2008

    2008

  9. [17]

    New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations,

    A. Kurganov and E. Tadmor, “New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations,”Journal of Computational Physics, vol. 160, no. 1, pp. 241–282, 2000

    2000

  10. [18]

    Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method,

    B. van Leer, “Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method,”Journal of Computational Physics, vol. 32, no. 1, pp. 101–136, 1979

    1979

  11. [19]

    Strong stability-preserving high-order time discretization methods,

    S. Gottlieb, C.-W. Shu, and E. Tadmor, “Strong stability-preserving high-order time discretization methods,”SIAM Review, vol. 43, no. 1, pp. 89–112, 2001

    2001

  12. [20]

    Ontheconstructionandcomparisonofdifferenceschemes,

    G.Strang, “Ontheconstructionandcomparisonofdifferenceschemes,”SIAM Journal on Numerical Analysis, vol. 5, no. 3, pp. 506–517, 1968

    1968

  13. [21]

    Construction of higher order symplectic integrators,

    H. Yoshida, “Construction of higher order symplectic integrators,”Physics Letters A, vol. 150, no. 5–7, pp. 262–268, 1990

    1990

  14. [22]

    The calculation of the interaction of non-stationary shock waves and obstacles,

    V. V. Rusanov, “The calculation of the interaction of non-stationary shock waves and obstacles,” USSR Computational Mathematics and Mathematical Physics, vol. 1, no. 2, pp. 304–320, 1962

    1962

  15. [23]

    Modeling and simulations of two-dimensional propagation of shear shock waves in relaxing soft solids,

    B. B. Tripathi, D. Espíndola, and G. F. Pinton, “Modeling and simulations of two-dimensional propagation of shear shock waves in relaxing soft solids,”Journal of Computational Physics, vol. 395, pp. 205–222, 2019

    2019

  16. [24]

    R. J. LeVeque,Finite Volume Methods for Hyperbolic Problems. Cambridge: Cambridge University Press, 2002. 18 G F Pinton

    2002

  17. [25]

    Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree,

    H. Wendland, “Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree,”Advances in Computational Mathematics, vol. 4, no. 1, pp. 389–396, 1995

    1995

  18. [26]

    Absorbing boundary conditions for the numerical simulation of waves,

    B. Engquist and A. Majda, “Absorbing boundary conditions for the numerical simulation of waves,” Mathematics of Computation, vol. 31, no. 139, pp. 629–651, 1977

    1977

  19. [27]

    Intensity images and statistics from numerical simulation of wave propagation in 3-D random media,

    J. M. Martin and S. M. Flatté, “Intensity images and statistics from numerical simulation of wave propagation in 3-D random media,”Applied Optics, vol. 27, no. 11, pp. 2111–2126, 1988

    1988

  20. [28]

    Heat generation by ultrasound in a relaxing medium,

    W. L. Nyborg, “Heat generation by ultrasound in a relaxing medium,”The Journal of the Acoustical Society of America, vol. 70, no. 2, pp. 310–312, 1981

    1981

  21. [29]

    JAX: composable transforma- tions of Python+NumPy programs,

    J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, C. Leary, D. Maclaurin, G. Necula, A. Paszke, J. VanderPlas, S. Wanderman-Milne, and Q. Zhang, “JAX: composable transforma- tions of Python+NumPy programs,” http://github.com/google/jax, 2018

    2018

  22. [30]

    L. E. Kinsler, A. R. Frey, A. B. Coppens, and J. V. Sanders,Fundamentals of Acoustics, 4th ed. New York: Wiley, 2000

    2000

  23. [31]

    Modeling nonlinear ultrasound propagation in heterogeneous media with power law absorption using ak-space pseudospectral method,

    B. E. Treeby, J. Jaros, A. P. Rendell, and B. T. Cox, “Modeling nonlinear ultrasound propagation in heterogeneous media with power law absorption using ak-space pseudospectral method,”The Journal of the Acoustical Society of America, vol. 131, no. 6, pp. 4324–4336, 2012

    2012

  24. [32]

    Causality-imposed (Kramers–Kronig) relationships between attenuation and dispersion,

    K. R. Waters, J. Mobley, and J. G. Miller, “Causality-imposed (Kramers–Kronig) relationships between attenuation and dispersion,”IEEE Transactions on Ultrasonics, Ferroelectrics, and Fre- quency Control, vol. 52, no. 5, pp. 822–833, 2005

    2005

  25. [33]

    Causal theories and data for acoustic attenuation obeying a frequency power law,

    T. L. Szabo, “Causal theories and data for acoustic attenuation obeying a frequency power law,” The Journal of the Acoustical Society of America, vol. 97, no. 1, pp. 14–24, 1995

    1995

  26. [34]

    A micro-CT of a human skull,

    T. Kirchner, “A micro-CT of a human skull,” arXiv:2202.11519 [physics.med-ph], 2022, data: doi:10.5281/zenodo.6108435

    2022 arXiv

  27. [35]

    The development of a 1.25MHz 1024-channel sparse array for human transcranial imaging:in vitro characterization,

    J. R. McCall, R. M. Jones, F. Santibanez, K. Latham, J. Zou, P. A. Dayton, and G. F. Pinton, “The development of a 1.25MHz 1024-channel sparse array for human transcranial imaging:in vitro characterization,”Measurement Science and Technology, vol. 35, no. 3, p. 034003, 2024

    2024

  28. [36]

    Attenuation, scattering, and absorption of ultrasound in the skull bone,

    G. Pinton, J.-F. Aubry, E. Bossy, M. Muller, M. Pernot, and M. Tanter, “Attenuation, scattering, and absorption of ultrasound in the skull bone,”Medical Physics, vol. 39, no. 1, pp. 299–307, 2012

    2012

  29. [37]

    Absorbing boundary conditions for difference approximations to the multi- dimensional wave equation,

    R. L. Higdon, “Absorbing boundary conditions for difference approximations to the multi- dimensional wave equation,”Mathematics of Computation, vol. 47, no. 176, pp. 437–459, 1986

    1986

  30. [38]

    Observation of shock transverse waves in elastic media,

    S. Catheline, J.-L. Gennisson, M. Tanter, and M. Fink, “Observation of shock transverse waves in elastic media,”Physical Review Letters, vol. 91, no. 16, p. 164301, 2003

    2003

  31. [39]

    An open-source framework for predicting ultrasound neuromodulation: bridging tissue elastomechanics and neuron firing dynamics,

    G. Pinton, “An open-source framework for predicting ultrasound neuromodulation: bridging tissue elastomechanics and neuron firing dynamics,” arXiv:2608.06321 [physics.med-ph], 2026

    2026 arXiv

  32. [40]

    Acous- toelasticity in soft solids: Assessment of the nonlinear shear modulus with the acoustic radiation force,

    J.-L. Gennisson, M. Rénier, S. Catheline, C. Barrière, J. Bercoff, M. Tanter, and M. Fink, “Acous- toelasticity in soft solids: Assessment of the nonlinear shear modulus with the acoustic radiation force,”The Journal of the Acoustical Society of America, vol. 122, no. 6, pp. 3...

    2007

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.