{"id":"e3c82758-c0ec-4366-af0e-13e57e4a5955","arxiv_id":"2608.07389","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A modified angular spectrum solver with obliquity corrections, shock-capturing nonlinearity, and plane-by-plane curved-source injection matches full-wave simulations of transcranial and bowl-transducer ultrasound to within a few percent.","lead":"This paper presents a faster way to simulate focused ultrasound waves traveling through the skull and other tissue, including shock-forming nonlinear effects. The new angular spectrum solver claims to match a full-wave reference in focus position and intensity while using much less memory and time.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The scalar ⟨k/kz⟩ nonlinearity correction is admitted-inaccurate near crossed wavefronts, and the absence of an ablation leaves the bowl-benchmark agreement unable to establish the method's central nonlinearity claim.","rationale":"The paper has real independent support: open-source JAX code, analytical diffraction/attenuation checks, and two Fullwave 2 benchmarks. My concern is focused: the beam-averaged nonlinearity obliquity correction is the paper's novel ingredient, but it is unablated in exactly the geometry (a deeply curved bowl near the focus) where Section 6 says it may fail. The TIPS comparison uses normalized beam metrics and a fitted 1.31 source amplitude scale, so it cannot isolate the correction's contribution. The proposed ablation would settle whether the 2.3% focal-depth agreement is robust to the approximation. This is the same weakest assumption the reader identified, and I agree with keeping the verdict conditional pending that check.","tokens_in":17282,"tokens_out":9378,"duration_ms":90212,"concrete_test":"Re-run the TIPS bowl benchmark (Sec. 5.3) twice more: (i) with the nonlinear obliquity correction disabled (N_eff = N, i.e., ⟨k/kz⟩ ≡ 1) and (ii) with ⟨k/kz⟩ computed from the analytic aperture geometry rather than the instantaneous spectrum; compare focal depth, focal-plane RMS intensity, and on-axis gain to the Fullwave 2 reference. If disabling or replacing the correction shifts focal depth by less than ~0.5% and focal-plane RMS by less than ~1%, the scalar approximation is not controlling the claimed agreement; if it shifts more, the match is contingent on an admitted-inaccurate approximation and a per-mode or multi-directional nonlinearity treatment is required before the nonlinear focal claim can be accepted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central novelty is the consistent obliquity treatment of the nonlinear operator: Eq. (6) replaces the per-mode path-length factor k/kz (which is exact for attenuation, Eq. (7)) with a single power-weighted mean ⟨k/kz⟩ multiplying the entire Burgers term (Eq. (2)). This is an uncontrolled approximation for a focused bowl: at the electronic focus (z = 50 mm) of the TIPS transducer, aperture components with widely different kz values cross with comparable amplitudes, and the pointwise retarded-time Burgers update cannot represent their cross-mode nonlinear interactions. Section 6 explicitly concedes the scalar approximation 'loses accuracy where plane-wave fronts of comparable amplitude cross, such as near the geometric focus.' Section 5.3 reports a 2.3% focal-depth match to Fullwave 2 but provides no ablation of the ⟨k/kz⟩ scaling, and the paper says such a study is 'left to future work.' The benchmark therefore cannot distinguish a robust match from a regime where the correction is known to be inaccurate. Because the method's generality claim for deeply curved sources rests on this nonlinearity correction, this is the load-bearing weak point.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":17528,"tokens_out":8439,"duration_ms":88189,"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":[{"comment":"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.","section":"§5.3 (Fullwave 2 comparison)"},{"comment":"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.","section":"§6 and Eq. (6)"},{"comment":"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.","section":"§5.3 and §6"}],"minor_comments":[{"comment":"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.","section":"Figure 10 and §5.3"},{"comment":"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.","section":"§2.1, Eq. (6)"},{"comment":"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.","section":"§5.2 and §5.3"},{"comment":"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.","section":"§3.3 and Table 2"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the Fullwave 2 reference is from the same research group as the ASM solver, so the benchmarks are not fully independent; I do not regard this as disqualifying, but it raises the burden on the 1.31 source scaling in Section 5.3. A further technical concern that I did not make load-bearing for my verdict: in Section 2.4, Eq. (14), the claim that per-pixel intensity-loss differences are guaranteed non-negative by Parseval is not implied by Parseval because the attenuation filter is applied in (kx,ky,ω) and the inverse FFT mixes spatial points; the max(0,·) guard may therefore bias the loss map. If the loss map is used for quantitative thermal-dose or radiation-force calculations, this should be corrected or reformulated as a total-energy or region-integrated quantity."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. The paper is a genuine engineering contribution: the Kurganov–Tadmor flux for the retarded-time Burgers operator is carefully implemented and tested, the plane-by-plane source injection is a sensible fix for deeply curved bowls, and the per-mode k/kz attenuation filter is exact and verified. The transcranial comparison against Fullwave 2 (1.1% focal-plane RMS, 5.4 dB insertion loss) is a good result, and the code is open. But the central novelty—a power-weighted scalar obliquity correction applied to the nonlinear term—is a heuristic that the paper itself concedes loses accuracy where plane-wave fronts of comparable amplitude cross, and that is precisely the regime of the strongly curved bowl benchmark. The focal-depth match (2.3%) is a linear property and does not test the nonlinear correction. The amplitude match (13.8 MPa) is obtained after a 1.31 source scaling fitted to the reference, so it cannot confirm the nonlinear accumulation. No ablation of the correction is given; Section 6 defers it to future work. So the headline claim of consistent obliquity treatment on the nonlinear operator is not backed by the evidence in this paper.\n\nWhat's solid: the KT/MUSCL scheme and the adaptive CFL sub-cycling are validated against Riemann problems and show second-order convergence; the sonic-detector patch for the cubic extension is a nice touch. The plane-by-plane injection for the TIPS bowl is a real improvement over equivalent-source-plane collapse. The boundary treatment results are reproducible and show a clear 2.4x gain. The paper ships code and data, which should count in its favor.\n\nWhat I'd want before accepting: an ablation study that turns off the <k/kz> scaling on the nonlinear term in the bowl case, or a benchmark at an off-focus steep nonlinear regime where the approximation is supposed to work. As written, the bowl benchmark can't distinguish the heuristic from a null effect.\n\nThis paper is for people building treatment-planning pipelines for focused ultrasound. It deserves a serious referee, but the reviewer should push on the nonlinearity claim. It has one significant unvalidated component, not a fatal flaw.","headline":"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.","tokens_in":18012,"tokens_out":3077,"would_cite":true,"duration_ms":28610,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["angular spectrum method","nonlinear acoustics","transcranial focused ultrasound","shock capturing","heterogeneous tissue","split-step method","bowl transducer","full-wave validation"],"falsifier":"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.","tokens_in":1757,"feed_emoji":"🎯","tokens_out":2001,"duration_ms":65532,"temperature":0.7,"pith_summary":"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.","feed_headline":"Ultrasound solver matches full-wave skull field to 1.1%","feed_subtitle":"A split-step method captures shocks and skull aberration while using 9x less memory than full-wave simulation.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the heterogeneous nonlinear full-wave FDTD model that the Fullwave 2 reference solver is based on.","marker":"[1]"},{"why":"Defines the Fullwave 2 solver used as the reference for transcranial and curved-bowl benchmarks.","marker":"[2]"},{"why":"Provides the prior obliquity-corrected absorption treatment in the nonlinear ASM that the present per-mode filter extends.","marker":"[6]"},{"why":"Establishes the hybrid angular-spectrum screen approach for heterogeneous tissue that the CT-derived screens build on.","marker":"[11]"},{"why":"Shows the equivalent-source-plane projection for hemispherical HIFU transducers, which the plane-by-plane injection scheme improves upon.","marker":"[12]"},{"why":"Supplies the Kurganov\\u2013Tadmor central-upwind flux that forms the shock-capturing core of the nonlinear update.","marker":"[17]"},{"why":"Provides the MUSCL reconstruction used to achieve second-order spatial accuracy in the flux scheme.","marker":"[18]"},{"why":"Supplies the strong-stability-preserving Runge\\u2013Kutta time integration used for the nonlinear step.","marker":"[19]"},{"why":"Gives the measured 5\\u201310 dB skull insertion loss range that validates the predicted 5.4 dB value.","marker":"[36]"}],"fun_headline_variants":["Skull ultrasound solver: 9x less memory, 2.9x faster, matches full-wave to 1.1%","New ultrasound ASM resolves skull shocks with 9x less memory, 2.9x less time","Fast ultrasound solver captures skull shocks at 1.1% error","Shock-capturing ASM matches skull simulations, cuts memory 9x","Nonlinear ultrasound through skull: 1.1% error, 9x less memory"],"cache_read_input_tokens":20224,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Skull ultrasound solver: 9x less memory, 2.9x faster, matches full-wave to 1.1%","New ultrasound ASM resolves skull shocks with 9x less memory, 2.9x less time","Fast ultrasound solver captures skull shocks at 1.1% error","Shock-capturing ASM matches skull simulations, cuts memory 9x","Nonlinear ultrasound through skull: 1.1% error, 9x less memory"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00103,"raw_usage":{"total_tokens":4438,"prompt_tokens":1145,"completion_tokens":3293,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":761,"completion_tokens_details":{"reasoning_tokens":3172}},"tokens_in":761,"tokens_out":3293,"duration_ms":20877,"temperature":1.0,"reasoning_tokens":3172,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T05:14:21.441341+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"A heterogeneous nonlinear attenuating full-wave model of ultrasound,","cited_arxiv_id":null,"evidence_quote":"Supplies the heterogeneous nonlinear full-wave FDTD model that the Fullwave 2 reference solver is based on."},{"cited_title":"Numerical methods for nonlinear wave propagation in ultrasound,","cited_arxiv_id":null,"evidence_quote":"Defines the Fullwave 2 solver used as the reference for transcranial and curved-bowl benchmarks."},{"cited_title":"Acoustic shock wave propagation in a heterogeneous medium: a numerical simulation beyond the parabolic approximation,","cited_arxiv_id":null,"evidence_quote":"Provides the prior obliquity-corrected absorption treatment in the nonlinear ASM that the present per-mode filter extends."},{"cited_title":"Ultrasound beam simulations in inhomogeneous tissue geometries using the hybrid angular spectrum method,","cited_arxiv_id":null,"evidence_quote":"Establishes the hybrid angular-spectrum screen approach for heterogeneous tissue that the CT-derived screens build on."},{"cited_title":"Simulation of hemispherical transducers for transcranial HIFU treatments using the hybrid angular spectrum approach,","cited_arxiv_id":null,"evidence_quote":"Shows the equivalent-source-plane projection for hemispherical HIFU transducers, which the plane-by-plane injection scheme improves upon."},{"cited_title":"New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations,","cited_arxiv_id":null,"evidence_quote":"Supplies the Kurganov\\u2013Tadmor central-upwind flux that forms the shock-capturing core of the nonlinear update."},{"cited_title":"Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method,","cited_arxiv_id":null,"evidence_quote":"Provides the MUSCL reconstruction used to achieve second-order spatial accuracy in the flux scheme."},{"cited_title":"Strong stability-preserving high-order time discretization methods,","cited_arxiv_id":null,"evidence_quote":"Supplies the strong-stability-preserving Runge\\u2013Kutta time integration used for the nonlinear step."},{"cited_title":"Attenuation, scattering, and absorption of ultrasound in the skull bone,","cited_arxiv_id":null,"evidence_quote":"Gives the measured 5\\u201310 dB skull insertion loss range that validates the predicted 5.4 dB value."}],"review_version":1}