Study on Waveform Superposition and Ultrasonic Gain During Nonlinear Propagation of Ultrasound in Fibrin Clots

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In this research work, the authors propose a new dual-wave-packet amplification strategy to overcome the limitations of viscoelastic dissipation and dispersion that constrain shock wave efficiency for fibrin clot ablation. The method, provide a novel theoretical framework for developing more efficient ultrasonic thrombolysis sequences.

This paper is interesting and it falls with the journal topics. It proposes a new method to overcome the limitations of viscoelastic dissipation and dispersion that constrain shock wave efficiency for fibrin clot ablation. The work is well structured and all results are discussed accurately and justified.

Before suggesting the acceptation of the paper, I have the following comments and questions for the authors:

• The introduction part is divided into section (bold format): I suggest to numerate the section or give paragraphs without those bold titles.
• Improve figures 1, 2, 3 and 7 quality.
• Lines 314 and 333 remove a and b.
• Let a blank line between the figure caption and the text.
• In the caption of figure 7 explain (a), (b),…(e).
• Line 451: why the title frequency is given? The same remark for amplitude and duty cycle lines 471 and 491? Numerate and complete those titles.
• What is the quantitative link between the amplified shock stress (e.g. demonstrated 22.7% gain) and a measurable enhancement of thrombus lysis?
• The method relies on precise tuning of the second wave packet's amplitude, frequency and delay to superpose nonlinearly distorted waves. In a clinical context with thrombi of heterogeneous composition and size, how could a real-time feedback system be implemented to auto-adjust these parameters and ensure optimal superposition for a patient?
• The proposed strategy and its significant gain are supported by numerical simulations, what experimental evidence could directly validate the predicted wave chasing, superposition, and shock amplification phenomena?
• Taking into account the sensitivity of the proposed approach to parameters (e.g. hardening exponent m), how robust are the conclusions across the natural biological variability of blood clots?
• What are the limitations of the prosed approach?

Author Response

Response to Reviewers' Comments

 

We sincerely thank the Reviewers for your valuable time and constructive comments. You have greatly helped us to improve the quality and clarity of our manuscript. In this response letter, the original comments and questions from the reviewer are presented in bold font. Our point-by-point responses follow in regular font. Additionally, for ease of reference, the corresponding revisions made in the revised manuscript are highlighted in blue color.

 

 

Q1: The introduction part is divided into section (bold format): I suggest to numerate the section or give paragraphs without those bold titles.

 

Response: We appreciate the reviewer’s feedback on improving the structure. As suggested, we have removed the bold titles and introduced clear numbered sections (e.g., 1.1, 1.2) in the Introduction to enhance readability.

 

Q2: Improve figures 1, 2, 3 and 7 quality.

Response: Thank you for the suggestion. We have improved the quality of Figures 1, 2, 3, and 7 (Figures 2, 3, 4, and 8 in the revised manuscript) and have updated them accordingly in the revised manuscript.

 

Q3: Lines 314 and 333 remove a and b.

Response: Thank you for the suggestion. We have removed the instances of “a” and “b” from the original Lines 314 and 333. In the revised manuscript, these changes are now reflected in Lines 370 and 389, respectively.

 

Q4: Let a blank line between the figure caption and the text.

Response: Thank you for the suggestion. We have now added a blank line between every figure caption and the following text throughout the manuscript to improve readability.

 

Q5: In the caption of figure 7 explain (a), (b),…(e).

Response: Thank you for the suggestion. We have revised the caption of Figure 7 (Figure 8 in the revised manuscript) to include detailed explanations for each subfigure.“ Diagram of dual-wave-packet chasing: (a) the stress nephograms at different moments during wave propagation; (b) the corresponding line plots at at t=5.6×10^-5 s; (c) the corresponding line plots at at t=1.0×10^-4 s; (b) the corresponding line plots at at t= t=1.516×10^-4 s; (e) the corresponding line plots at at t=2.0×10^-4 s”

Q6: Line 451: why the title frequency is given? The same remark for amplitude and duty cycle lines 471 and 491? Numerate and complete those titles.

Response: Thank you for the suggestion. The original titles were intended to provide a clear structure for the figures. However, as you rightly pointed out, they lacked clarity. We have now renumbered and renamed all relevant figure titles (including those for frequency, amplitude, and duty cycle) to make them more descriptive and complete.

 

Q7: What is the quantitative link between the amplified shock stress (e.g. demonstrated 22.7% gain) and a measurable enhancement of thrombus lysis?

Response:

 

Thank you for this insightful and critical comment. As the reviewer noted, shock wave amplification is of critical practical importance for improving thrombolytic efficacy.

Existing studies have confirmed a clear positive correlation between the mechanical force of ultrasonic shock waves and the thrombolysis rate: at a fixed frequency, higher acoustic intensity leads to stronger mechanical force and a higher thrombolysis rate [R1]. This indicates that enhanced shock wave stress should translate into improved dissolution efficiency. While higher ultrasound intensity indeed pose two main potential risks: mechanical damage to the vessel wall and the generation of harmful embolic fragments due to rapid, uncontrolled thrombus fragmentation. Research on ultra-sound intensity thresholds indicates the existence of a critical intensity (approximately 21.6 W/cm²); below this value, the extent of thrombus damage is minimal (approximately 11-15%); once this threshold is exceeded, the maximum damage degree in-creases rapidly with intensity [R2]. Furthermore, when the intensity (which determines the peak stress) surpasses a specific critical point, the destructive effect is significantly enhanced, exhibiting a strong nonlinear characteristic. Therefore, solely relying on increasing ultrasound intensity is not a viable strategy. Our proposed 'wave-packet chasing and superposition' strategy does not aim to infinitely amplify shock wave energy. Instead, it seeks to more efficiently concentrate energy at specific locations within the thrombus through precise waveform modulation, promoting controllable disruption of its internal structure while maintaining the existing energy input. Compared to traditional methods that simply increase single-pulse amplitude, our strategy offers  intrinsic safety advantages。

 

The content above has been added to Section 4.1 in the revised manuscript.

 

References:

[R1]. Salman-Kesner, N.; Zaltsman, M.-M.; Ertracht, O.; Atar, S. In-Vitro Assessment of the Thrombolytic Efficacy of Therapeutic Ultrasound. Thromb. Res. 2019, 178, 63–68, doi:10.1016/j.thromres.2019.04.002.

[R2] Adzerikho, I.; Kulak, A.; Rachok, S.; Minchenya, V. Dependence of the Rate and Completeness of Fibrin Clot Destruction on the Acoustic Dose and Ultrasound Intensity. Ultrasound Med. Biol. 2022, 48, 846–855, doi:10.1016/j.ultrasmedbio.2022.01.005.

 

 

Q8: The method relies on precise tuning of the second wave packet's amplitude, frequency and delay to superpose nonlinearly distorted waves. In a clinical context with thrombi of heterogeneous composition and size, how could a real-time feedback system be implemented to auto-adjust these parameters and ensure optimal superposition for a patient?

Response: Thank you for raising this profound and highly practical question. Addressing the heterogeneity of thrombi, the core of translating this method into clinical practice lies in developing a fast closed-loop system that integrates real-time monitoring and adaptive control.

 

The system's core would be a rapid "Sense-Decide-Act" closed loop:

1. Sense: The system monitors the mechanical response (e.g., cavitation activity, strain) and structural changes of the thrombus by receiving real-time cavitation acoustic emission signals and ultrasound imaging/elastography data from the treatment area.
2. Decide: An embedded algorithm (e.g., based on pre-trained models or Bayesian optimization) analyzes these feedback signals to determine whether optimal mechanical effects from wave superposition have been achieved. Based on the perceived mechanical properties of the thrombus, it instantly calculates an adjustment plan for the parameters of the second wave packet (amplitude, frequency, delay).
3. Act: An emission module controlled by an FPGA (Field-Programmable Gate Array) regenerates and transmits the optimized, precise wave packet sequence within milliseconds.

 

Implementing such a system undoubtedly faces challenges in signal processing, algorithm generalization, and system integration. However, the clear dependencies on key parameters established in this study precisely provide a clear theoretical framework and control objectives for building such intelligent, individualized therapeutic systems, which will be an important direction for future research.

 

 

 

Q9: The method relies on precise tuning of the second wave packet's amplitude, frequency The proposed strategy and its significant gain are supported by numerical simulations, what experimental evidence could directly validate the predicted wave chasing, superposition, and shock amplification phenomena?

 

 

Response: At present, a significant body of literature has experimentally demonstrated that by tailoring key characteristics of designed wave packet sequences—such as frequency and duty cycle—it is possible to achieve wave chasing, superposition, and ultimately, the amplification of shock responses. such as frequency and duty cycle, enables wave chasing, superposition, and ultimately, the amplification of shock responses. These studies achieve wave amplitude amplification by controlling the propagation velocities of different frequency components, causing them to arrive and superimpose synchronously at a predetermined time and location. For instance, Waters et al. designed a linear frequency-modulated (chirp) excitation waveform to compensate for dispersion in flexural waves, enabling different frequency components to reach the target position simultaneously and achieving significant local shock response amplification in experiments. This work directly validates the principle of "achieving wave front superposition and shock amplification by adjusting the emission delays of different frequency groups (i.e., having subsequent waves 'chase' and catch up with the precursor wave)" [R3]. In phononic crystal waveguides, Kurosu et al. utilized chirped input pulses to achieve temporal focusing of elastic waves at a predetermined time and location, experimentally observing the concentration of wave packet energy [R4]. Wang et al. generated acoustic wave packets via resonant excitation in a toroidal Bose-Einstein condensate and observed that the propagation speed of waves (i.e., density perturbations) depends on the local fluid density. This is analogous to the classical nonlinear acoustics discussed in this paper. Under strong excitation, denser parts within the wave packet propagate faster, leading to wave steepening at the leading edge and broadening at the trailing edge during propagation, with the observation of supersonic shock wave generation under intense excitation [R5]. This aligns with the core physical principle of our work, which leverages the material's nonlinear characteristics by designing key parameters of an ultrasonic dual-wave-packet sequence—such as time delay, frequency, and amplitude—to achieve dual-wave chasing, superposition, and consequent shock wave amplification.

In the context of ultrasonic therapy, Edsall et al. precisely controlled the "time delay" between dual-frequency pulses to achieve coherent superposition of acoustic waves during nonlinear propagation in tissue.[R6] They demonstrated that variations in delay on the order of microseconds or even nanoseconds drastically alter the instantaneous waveform of the superimposed wave and can actively "modulate" the characteristics of bubble clouds, leading to more efficient and precise tissue ablation compared to single-frequency pulses. Experiments in red blood cell phantoms showed that lesions generated by dual-frequency pulses were more precise with clearer boundaries, yet achieved higher ablation efficiency. These experimental findings are consistent with the trend observed in our numerical simulations, where adjusting key parameters (delay time, frequency, amplitude) of the second wave packet leads to shock wave gain.

[R3] Waters, T., et al. “A Chirp Excitation for Focusing Flexural Waves.” Journal of Sound and Vibration 459 (2019): 114863.

[R4] Kurosu, M., et al. “On‑chip Temporal Focusing of Elastic Waves in a Phononic Crystal Waveguide.” Journal of Sound and Vibration 510 (2021): 116288.

[R5] Wang, Y., et al. “Resonant Wavepackets and Shock Waves in an Atomtronic SQUID.” arXiv, arXiv:1510.02968 (2015).

[R6] Edsall, C.; Huynh, L.; Mustafa, W.; Hall, T.L.; Durmaz, Y.Y.; Vlaisavljevich, E. Nanoparticle-Mediated Histotripsy Using Dual-Frequency Pulsing Methods. Ultrasound Med. Biol. 2024, 50, 1214–1223, doi:10.1016/j.ultrasmedbio.2024.04.009.

 

Q10: Taking into account the sensitivity of the proposed approach to parameters (e.g. hardening exponent m), how robust are the conclusions across the natural biological variability of blood clots?

Response:

The issue the reviewer raised is crucial as it directly relates to the generalizability and clinical translation potential of this study. Based on a review of the current literature, although the degree of strain-hardening (manifested by parameter m) varies naturally among different thrombi; since fibrin is an essential, load-bearing structural component within all thrombi (whether red, white, or mixed), the strain-hardening behavior exhibited by its network is a fundamental mechanical property common to all thrombi. Therefore, the core physical mechanism revealed in this study—leveraging this nonlinear property to achieve shock wave amplification—is universally applicable. The observed parameter sensitivity precisely provides a basis for personalized, precise treatment parameter adjustment in the future.

Regardless of the specific magnitude of the hardening parameter, as long as the material exhibits strain-hardening (i.e., m > 0), the propagation of finite-amplitude ultrasound within it will induce nonlinear distortion, dispersion, and shock wave formation. Our study provides the first systematic demonstration that the dual-wave-packet chasing strategy can proactively exploit this inherent property to achieve significant peak shock wave enhancement. This conclusion regarding the feasibility of the physical mechanism is robust.

•  

 

Q11: What are the limitations of the prosed approach?

Response: Thank you for raising this important point. As suggested, we have added a dedicated “Limitations and Future Perspectives” subsection following the Conclusion in the revised manuscript.

 

1.The simulations are based on a simplified constitutive model (the power-law model) of thrombus, which, while capturing the key strain-hardening behavior, may not fully account for the complex frequency-dependent viscoelasticity and the heterogeneous composition (e.g., varying ratios of fibrin, platelets, and red blood cells) of real thrombi. This part has been added in“Limitations and Future Perspectives”.

2. The current model focuses on the mechanical shock wave mechanism and does not incorporate the potential effects of acoustic cavitation, which could interact with the dual-wave packets in complex ways, possibly enhancing or impeding the intended therapeutic effect.This part has been discussed in Line 550-Line 561.

 

We believe that acknowledging these limitations provides a balanced view of our work and clearly outlines important directions for future research.

 

Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

This manuscript investigates nonlinear ultrasonic propagation in fibrin clots and proposes a dual-wave-packet chasing strategy to enhance shock-wave formation for potential thrombolysis applications. The study integrates quasi-static mechanical testing with numerical modelling to examine waveform distortion, overtaking behaviour, and stress amplification. This method tunes the amplitude and timing of a second wave packet to overtake and superimpose upon a precursor wave, theoretically achieving a stress gain of 22.7%. I suggest the publication after the following revision.

1. The manuscript states that the trailing wave B catches up to leading wave A because the material is strain-hardening (higher stress = higher modulus = higher wave speed). However, the paper notes that the trailing wave has a larger amplitude.
2. The authors derive their power-law constitutive model based on quasi-static compression tests performed at a rate of 0.5 mm/min. However, the simulations are conducted at an ultrasonic frequency of 20 kHz. Fibrin clots are viscoelastic materials; their mechanical properties are highly frequency-dependent. The authors should discuss the limitation explicitly.
3. The authors ignore the viscous term when deriving the constitutive equation. However, shock formation in biological soft tissues is strongly counteracted by viscosity. The authors should discuss this influence.
4. The authors could draw a schematic to illustrate all the concepts and application together to make the story clear and quickly to understand.

Author Response

Response to Reviewers' Comments

 

We sincerely thank the Reviewers for your valuable time and constructive comments. You have greatly helped us to improve the quality and clarity of our manuscript. In this response letter, the original comments and questions from the reviewer are presented in bold font. Our point-by-point responses follow in regular font. Additionally, for ease of reference, the corresponding revisions made in the revised manuscript are highlighted in blue color.

 

 

Q1: The manuscript states that the trailing wave B catches up to leading wave A because the material is strain-hardening (higher stress = higher modulus = higher wave speed). However, the paper notes that the trailing wave has a larger amplitude.

 

Response: The reviewer's comment is highly insightful. In fact, assigning trailing wave B a greater amplitude than leading wave A is the critical prerequisite for triggering the 'chasing' behavior. Wave B can only catch up with wave A (rather than maintaining a constant distance or diverging) if its amplitude is larger, because in a strain-hardening material, a higher amplitude directly correlates with a faster propagation speed for the wave. Our work is fundamentally built upon leveraging this intrinsic material property. By actively controlling the frequency, amplitude, and delay time of wave B, we achieve precise spatiotemporal modulation of the shock wave gain at designated locations. Modifying these parameters of wave B directly determines both the specific location of the final shock wave superposition and the magnitude of the gain. Therefore, this strategy provides a novel theoretical foundation and a new dimension of control for achieving controllable and predictable targeted ultrasonic mechanical thrombolysis.

 

 

 

Q2: The authors derive their power-law constitutive model based on quasi-static compression tests performed at a rate of 0.5 mm/min. However, the simulations are conducted at an ultrasonic frequency of 20 kHz. Fibrin clots are viscoelastic materials; their mechanical properties are highly frequency-dependent. The authors should discuss the limitation explicitly.

Response: We thank the reviewer for highlighting this crucial limitation. Our power-law model, based on quasi-static tests (0.5 mm/min), indeed does not capture the frequency-dependent viscoelasticity of fibrin clots at ultrasonic frequencies (~20 kHz). This simplification may cause discrepancies between our simulated shock dynamics and the actual high-frequency material response. We have explicitly acknowledged this significant limitation in the revised Limitations and Future Perspectives Section:

 

“The authors' subsequent research found that by connecting a dashpot with low viscosity in parallel to the current nonlinear elasticity framework based on a power-law model (thereby introducing viscoelasticity), and keeping all other model parameters un-changed, simulations still demonstrate the occurrence of dual-wave superposition and shock wave enhancement. However, the shock wave enhancement value is reduced, and its formation position shifts significantly backward. This indicates that in viscoelastic media, both the magnitude and location of shock wave enhancement depend on the strength of viscosity and the design of the dual-wave sequence. Since the viscoelastic behavior of thrombi may be more complex than that captured by a simple non-linear spring-dashpot model, further research is needed.

Therefore, it must be noted that while the quasi-static power-law model used in this study effectively captures the core strain-hardening nonlinear characteristics of thrombi, it fails to account for their frequency-dependent viscoelasticity under high-frequency ultrasound excitation (e.g., above ~20 kHz). Consequently, the absolute stress enhancement values derived from the simulations may deviate from those observed in real dynamic biological environments.”

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Q3: The authors ignore the viscous term when deriving the constitutive equation. However, shock formation in biological soft tissues is strongly counteracted by viscosity. The authors should discuss this influence.

Response:

We thank the reviewer for their valuable comments. Based on the current strain-hardening model, we have incorporated a viscous coefficient of , a value commonly used for biological materials, as shown in the figure below.

 

 

Figure S1. Nonlinear viscoelastic model.

 

Using this model in conjunction with our existing finite element model and dual-wave packet strategy, simulation results indicate that, when evaluated by the WSF quantitative standard, the shock wave gain is reduced to 12.7%. Additionally, the shock wave formation position is delayed by 2.3×10⁻³ compared to the purely elastic result. Therefore, even with the introduction of a small viscous term, the dual-wave packet chasing strategy remains effective, though it still cannot cover all scenarios，because the viscoelastic properties of thrombi can be more complex and exhibit significant inter-individual variability. Therefore, we have added the limitations in the Limitations and Future Perspectives section.

 

Q4: The authors could draw a schematic to illustrate all the concepts and application together to make the story clear and quickly to understand.

 

Response: Thank you for the constructive suggestion. As suggested, we have added a comprehensive schematic diagram (now included as Figure 1) in the revised manuscript. This figure integrates the key concepts (e.g., material nonlinearity, shock wave formation) and our proposed application (the dual-wave-packet amplification strategy) into a single, clear visual narrative from the core concept to the application, to enhance the clarity of our work.

 

Figure 1. Schematic of the dual-wave-packet chasing strategy for shock wave amplification in a strain-hardening fibrin clot.

 

 

 

 

 

Author Response File: Author Response.docx

Reviewer 3 Report

Comments and Suggestions for Authors

The manuscript presents a numerical study on nonlinear ultrasound propagation in fibrin clots, proposing a dual-waveform sequence to amplify shock wave formation. This is a timely and relevant topic with clear potential to improve ultrasound-mediated thrombolysis. The authors have provided a thorough background and a novel idea. However, significant revisions are needed to address methodological clarity, to validate the findings (even qualitatively) against physical expectations or experiments, and to better position this work in the context of existing ultrasonic thrombolysis techniques. I recommend a Major Revision to resolve the issues detailed below.

 

Commendations:

The authors are commended for investigating an important problem at the intersection of biomechanics and therapeutic ultrasound. The introduction is generally well-researched, covering thrombolysis mechanisms and clot mechanics. The simulation approach is innovative – the idea of a “chasing” second ultrasound wave to reinforce shock formation is novel and scientifically interesting. The manuscript is well structured, with clear figures and definitions (e.g., introduction of the Waveform Superposition Factor, WSF). The effort to derive a constitutive model for fibrin clots from experimental data (Figure 1) is especially noteworthy, as it grounds the simulations in measured material behavior. Overall, the study shows strong potential for advancing noninvasive thrombolysis techniques.

Major Revisions:

1. Experimental Validation and Feasibility

Issue:

The study’s findings (e.g., ~22.7% increase in peak shock stress with dual-wave sequences) are based purely on simulations, with no experimental validation or comparison to physical data. It remains unclear if such waveform superposition would translate effectively to real-world thrombolysis conditions. For instance, no in vitro or in vivo test is presented to confirm that the predicted shock amplification improves clot fragmentation or dissolution. Additionally, practical feasibility (e.g., generating the precisely timed dual-wave packets with clinical ultrasound equipment) is not discussed.

Suggestion:

Please provide some form of validation or at least a stronger argument for real-world feasibility. Ideally, include or cite a simple experiment (even a qualitative bench-top test in a gel or blood mimic) demonstrating enhanced clot disruption with dual pulses. If new experiments are not possible, then compare your simulation trends with existing experimental studies of ultrasound thrombolysis or high-intensity dual-pulse ultrasound, if any. Discuss how the required timing and amplitudes could be achieved with actual transducers and what safety considerations might arise. This addition will reassure readers that the proposed method is not only theoretically sound but also practically implementable.

 

2. Methodological Clarity and Reproducibility

Issue:

Certain aspects of the methods need more clarity to ensure reproducibility. Firstly, the procedure for tuning the second wave packet’s amplitude, frequency, and delay is not sufficiently described. The manuscript should specify how the “chasing” parameters were optimized – was a systematic parametric scan performed, or was an algorithm used to maximize the WSF? Additionally, details about the numerical solver are sparse. The authors mention a “third-order nonlinear wave equation” and presumably solve it with a finite-difference scheme, but important parameters (grid size, time-step, stability criteria) and boundary conditions are not described. It is also not stated whether attenuation and dispersion are modeled frequency-dependently, and how the simulation ensures convergence (especially in shock conditions, which can be challenging for numerical stability).

Suggestion:

Expand the Methods section to better detail the simulation procedure. Explain how the optimal delay and amplitude of the second pulse were determined – e.g., was the 22.7% gain the maximum out of a sweep of time delays and amplitude ratios? If so, provide that range or figure. Include a description of the numerical implementation: the equation solved (e.g., Westervelt or a form of Burgers equation), any assumptions (plane wave, 1D propagation in homogeneous medium, etc.), and how shocks are handled (artificial viscosity, filtering, or shock-capturing schemes). State the spatial and temporal resolution and confirm that results are grid-independent. This information will greatly help readers to trust and potentially reproduce your findings.

 

3. Context of Mechanical Thrombolysis or Mechanotherapy

Issue:

The introduction and discussion do not sufficiently situate the work within the broader landscape of mechanical ultrasound thrombolysis techniques. The authors rightly note that conventional HIFU relies mainly on thermal effects and may not be suitable for clot lysis. However, other non-thermal ultrasound approaches have been developed and should be acknowledged. In particular, the technique of histotripsy uses short, high-amplitude ultrasound pulses to generate shock waves and cavitation that mechanically fractionate clots without drugs. Several studies have demonstrated the ability of pulsed ultrasound (with or without microbubbles) to enhance thrombolysis via mechanical effects. The omission of these references makes the manuscript seem isolated, even though those techniques share a similar goal of shock-wave mediated clot disruption.

Suggestion:

Broaden the discussion to compare your waveform-superposition method with existing mechanical thrombolysis approaches like histotripsy or sonothrombolysis. Emphasize how your approach is different (e.g., using two sequential pulses to amplify an intrinsic shock vs. random bubble cloud shocks in histotripsy) and what advantages it might have. For completeness, cite relevant studies on therapeutic and mechanotherapy ultrasound in clots and other soft biological systems, highlighting mechanical, non-thermal bioeffects. References to consider: Izadifar, Z., et al. ‘Mechanical and biological effects of ultrasound: a review of present knowledge.’ Ultrasound in Medicine and Biology, 43, 1085–1104 (2017).

 

4. Material Model and Viscoelastic Assumptions

Issue:

The fibrin clot material is modeled with a quasi-static power-law constitutive model derived from low-strain-rate compression tests. While this captures strain-hardening, it may not fully represent the high-rate viscoelastic response under ultrasound frequencies. The manuscript currently assumes that the nonlinearity (strain-hardening) is the dominant factor for shock formation. However, real clots also exhibit viscoelastic attenuation and frequency-dependent modulus (as the authors cite in references on shear wave measurements). It’s unclear if the power-law model incorporates any viscous damping or if dispersion is modeled beyond a basic attenuation term. Neglecting important viscoelastic effects could lead to overestimating shock intensity or waveform steepness.

Suggestion:

The authors should discuss the limitations of the constitutive model in the context of high-frequency, dynamic loading. If possible, quantify or describe how viscoelastic damping (e.g., using a standard linear solid model or frequency-dependent attenuation) might alter the results. For example, would a Burgers or Kelvin–Voigt model produce a different shock amplitude due to energy dissipation? If the current model lacks a frequency-dependent loss term, acknowledge that this may over-predict shock sharpness. Ideally, include a brief sensitivity study: e.g., introduce a small damping term or use literature values for clot viscosity to estimate its effect on the WSF. At minimum, add a caution in the Discussion that the power-law fit, while reasonable for static behavior, might not capture all dynamic factors, and that future work should investigate more complex viscoelastic models.

 

5. Significance of Shock Amplification – Link to Thrombolytic Efficacy

Issue:

The results show a 22.7% increase in peak shock stress using the dual-wave strategy, and identify the second-pulse amplitude as the dominant factor for amplification. However, the manuscript stops short of translating what this means for actual clot disruption. It is not discussed whether a ~22% rise in shock pressure is expected to significantly improve thrombolysis (e.g., by exceeding some damage threshold or achieving faster lysis). Without context, it’s hard for the reader to gauge the practical importance of the result – is a 22% increase modest or groundbreaking? Additionally, the potential trade-offs (such as increased exposure risk or energy input) are not addressed.

Suggestion:

Strengthen the discussion by linking the physical metrics to biological impact. For instance, discuss how shock wave amplitude correlates with fibrin network damage or lysis rate, based on prior studies or physical reasoning. If there is a known threshold pressure for causing microstructural damage to clots, mention whether your amplified shock surpasses it. You might speculate that a 22% increase in peak stress could yield disproportionately higher clot fragmentation (if the relationship is non-linear), or conversely, that it might be partially offset by other factors. Also, comment on the possible downsides: does a stronger shock risk damaging healthy tissue or causing emboli due to rapid clot breakage? Framing your results in terms of thrombolytic efficacy and safety will make the findings more meaningful for the reader. This could be just a few sentences connecting your findings to expected improvements in clot dissolution speed or completeness.

 

6. Cavitation Effects and Bubble Nucleation

Issue:

The study focuses on mechanical shock waves in a continuous fibrin medium, but inertial cavitation is not considered. In reality, at the high pressures associated with shock fronts, microscopic cavitation bubbles are likely to form in blood or thrombus (especially if gas nuclei or contrast agents are present). Cavitation can drastically alter wave propagation – for example, bubble collapse can both enhance clot disruption and also absorb energy, potentially limiting further shock transmission. The current simulation neglects cavitation, meaning it might overestimate the sustained shock amplitude in tissue or ignore a key damage mechanism. Not acknowledging this might be seen as a significant gap, since cavitation-based effects are known to contribute to ultrasound thrombolysis.

Suggestion:

Include a discussion of cavitation phenomena as a limitation and/or a potential enhancement mechanism. Acknowledge that at the intensities needed for shock formation, bubbles may form and that your model does not capture this. You could cite studies of cavitation in sonothrombolysis and note how cavitation could either help (by causing additional mechanical damage) or hinder (by soaking up energy or causing shielding) the propagation of your second pulse, along with work in other soft biological systems showing that the temporal pattern of high-frequency ultrasound pulses can significantly alter mechanical or functional responses. If possible, estimate the peak negative pressure in your simulation and whether it exceeds typical cavitation thresholds in blood. This will inform readers about when cavitation might set in. Encouragingly, if cavitation does occur, your dual-pulse method might work synergistically with it – this is worth mentioning. Adding 2–3 sentences on this topic will make the paper more comprehensive. References to consider: Bollen, V., et al. ‘In vitro thrombolytic efficacy of single- and five-cycle histotripsy pulses and rt-PA.’ Ultrasound in Medicine and Biology, 46, 336–349 (2020).

 

7. Parameter Space and Transducer Limitations

Issue:

The parametric analysis in the paper identifies the second wave’s amplitude as the most critical factor for shock amplification, with frequency and time delay presumably also explored. However, the range of parameters tested and their relation to real transducer capabilities are not fully discussed. For example, if extremely high amplitude of the second pulse is needed (beyond what standard diagnostic or therapeutic ultrasound can deliver), that could limit practical use. Similarly, the optimal time delay found in simulation should be contextualized: is this delay achievable and controllable by existing ultrasound sequencing technology? The manuscript does not address how sensitive the WSF gain is to deviations in these parameters – a concern for robustness.

Suggestion:

Provide more insight into the parameter choices and practical constraints. In the Results, briefly note the ranges over which frequency, phase, and delay were varied, and whether the 22.7% gain occurs at an optimal point or a broad plateau. In the Discussion or Conclusion, mention how one might generate such a dual-packet sequence: e.g., using two transducers or programmed bursts from one transducer with controlled inter-pulse delay. If the required second-pulse amplitude is, say, 1.5× the first pulse, discuss if that is within FDA limits or device specs. Conversely, if the second pulse’s frequency is different, can a typical transducer emit both frequencies effectively? By addressing these points, the authors will demonstrate a clear understanding of how the proposed method could be translated into practice, and what the current technical gaps are.

 

8. Presentation: Figures and Terminology

Issue:

A few minor presentation issues should be addressed to improve clarity. For instance, some figures could benefit from more detailed captions. Figure 3 (waveform profiles) is mentioned in the text, but the caption does not explicitly explain all curve labels or the physical scenario depicted (e.g., which line is with vs. without the second wave, etc.). Additionally, terms like “WSF” and “ASF” are introduced; while defined in the text, it would help to also spell them out in figure legends or labels for readers who skim figures. The term “Waveform Superposition Factor” might confuse some readers at first glance – consider briefly clarifying that it quantifies peak amplitude gain from superposition. There are also minor terminology inconsistencies (e.g., sometimes “shock wave” vs “shockwave”) that should be standardized.

Suggestion:

Revise figure captions to be more self-contained. Ensure that each curve or image panel is sufficiently explained (for example, specify “Initial single-pulse waveform vs. dual-pulse waveform at location X” in the caption). Add units to color bars and axes wherever applicable (e.g., stress in MPa, time in µs). Unify terminology in the text: use one form of “shock wave” consistently. Also, double-check that all acronyms (WSF, ASF, etc.) are defined at first use and perhaps remind the reader of their meaning in the Conclusion for emphasis. These tweaks will make the paper more reader-friendly and polished.

 

Conclusion:

In summary, this work addresses an important challenge in ultrasound thrombolysis and presents a novel solution. However, substantial revisions are required to strengthen the validation, clarify methods, and better integrate the study into the existing research context. The eight major points above should be carefully addressed to improve the manuscript’s scientific rigor and clarity. Once these issues are resolved, along with minor language corrections, I believe the paper will meet the criteria for publication. I look forward to seeing a revised version that convincingly demonstrates the feasibility and significance of the proposed dual-wave shock amplification strategy for clot dissolution.

Comments on the Quality of English Language

Overall, the manuscript is written in clear English, but some minor edits will improve readability. Please check for grammar and style consistency. For example, ensure subject–verb agreement in long sentences (e.g., “shock waves is limited” should be “shock waves are limited”). Use consistent tense when describing your results (past tense is often used for methods and results). Make sure to include articles (“the”, “a”) where needed – e.g., “developed a dual-wave-packet strategy”. Also, standardize technical notation and units: write units with a space (e.g., “5 °C”, “10 mm”), and hyphenate compound adjectives (e.g., “strain-hardening material”, “peak-shock stress”). A careful proofread for punctuation (commas, etc.) and phrasing will help. The suggested changes do not alter the science, but they will ensure the paper reads smoothly and meets the journal’s language standards.

Author Response

Response to Reviewers' Comments

 

We sincerely thank the Reviewers for your valuable time and constructive comments. The exceptional professionalism of the reviewer left a profound impression on us. You have greatly helped us to improve the quality and clarity of our manuscript. In this response letter, the original comments and questions from the reviewer are presented in bold font. Our point-by-point responses follow in regular font. Additionally, for ease of reference, the corresponding revisions made in the revised manuscript are highlighted in blue color.

 

Experimental Validation and Feasibility

Issue:

 

The study’s findings (e.g., ~22.7% increase in peak shock stress with dual-wave sequences) are based purely on simulations, with no experimental validation or comparison to physical data. It remains unclear if such waveform superposition would translate effectively to real-world thrombolysis conditions. For instance, no in vitro or in vivo test is presented to confirm that the predicted shock amplification improves clot fragmentation or dissolution. Additionally, practical feasibility (e.g., generating the precisely timed dual-wave packets with clinical ultrasound equipment) is not discussed.

 

Suggestion:

 

Please provide some form of validation or at least a stronger argument for real-world feasibility. Ideally, include or cite a simple experiment (even a qualitative bench-top test in a gel or blood mimic) demonstrating enhanced clot disruption with dual pulses. If new experiments are not possible, then compare your simulation trends with existing experimental studies of ultrasound thrombolysis or high-intensity dual-pulse ultrasound, if any. Discuss how the required timing and amplitudes could be achieved with actual transducers and what safety considerations might arise. This addition will reassure readers that the proposed method is not only theoretically sound but also practically implementable.

 

Response:

We appreciate the reviewer's comment. Our response is structured from two distinct perspectives:

1. Consistency between Physical Principles and Experimental Trends

A significant body of experimental literature has demonstrated that tailoring key characteristics of designed wave packet sequences, such as frequency and duty cycle, enables wave chasing, superposition, and ultimately, the amplification of shock responses. These studies achieve wave amplitude amplification by controlling the propagation velocities of different frequency components, causing them to arrive and superimpose synchronously at a predetermined time and location. For instance, Waters et al. designed a linear frequency-modulated (chirp) excitation waveform to compensate for dispersion in flexural waves, enabling different frequency components to reach the target position simultaneously and achieving significant local shock response amplification in experiments. This work directly validates the principle of "achieving wave front superposition and shock amplification by adjusting the emission delays of different frequency groups (i.e., having subsequent waves 'chase' and catch up with the precursor wave)" [R1]. In phononic crystal waveguides, Kurosu et al. utilized chirped input pulses to achieve temporal focusing of elastic waves at a predetermined time and location, experimentally observing the concentration of wave packet energy [R2]. Wang et al. generated acoustic wave packets via resonant excitation in a toroidal Bose-Einstein condensate and observed that the propagation speed of waves (i.e., density perturbations) depends on the local fluid density. This is analogous to the classical nonlinear acoustics discussed in this paper. Under strong excitation, denser parts within the wave packet propagate faster, leading to wave steepening at the leading edge and broadening at the trailing edge during propagation, with the observation of supersonic shock wave generation under intense excitation [R3]. This aligns with the core physical principle of our work, which leverages the material's nonlinear characteristics by designing key parameters of an ultrasonic dual-wave-packet sequence—such as time delay, frequency, and amplitude—to achieve dual-wave chasing, superposition, and consequent shock wave amplification.

In the context of ultrasonic therapy, Edsall et al. precisely controlled the "time delay" between dual-frequency pulses to achieve coherent superposition of acoustic waves during nonlinear propagation in tissue. They demonstrated that variations in delay on the order of microseconds or even nanoseconds drastically alter the instantaneous waveform of the superimposed wave and can actively "modulate" the characteristics of bubble clouds, leading to more efficient and precise tissue ablation compared to single-frequency pulses. Experiments in red blood cell phantoms showed that lesions generated by dual-frequency pulses were more precise with clearer boundaries, yet achieved higher ablation efficiency. These experimental findings are consistent with the trend observed in our numerical simulations, where adjusting key parameters (delay time, frequency, amplitude) of the second wave packet leads to shock wave gain.

 

2. Feasibility Analysis

The implementation of the proposed dual-wave-packet sequence faces challenges such as precise temporal waveform control and broadband transducer design, placing it at the forefront of current research. However, these challenges are addressable through clear technical pathways, which can be realized at two levels: same-frequency and dual-frequency dual-wave-packet sequences.

(1) For same-frequency dual-wave-packet sequences, the foundation for short-pulse emission can be established by optimizing transducer structure (e.g., adding acoustic backing layers to absorb backward-propagating waves) [R4] and employing broadband materials (e.g., novel composite piezoelectric materials, PMN-PT single crystals) [R5] to enhance damping and broaden the frequency band. Based on this, waveform generation and the precise silent period between the two wave packets can be accurately implemented by counters within an FPGA, achieving nanosecond-level precision.

(2) For dual-frequency dual-wave-packet sequences, two independent transducer elements, each optimized for a specific frequency (f1 and f2), are used. Both elements are controlled by the same FPGA but utilize different transmission channels, driving the transducers to emit in sequence according to the designed protocol.

It must be emphasized that the maturation of the related technology and its engineering application require sustained research efforts and experimental validation for gradual refinement; and that while this technical approach offers therapeutic gains, its potential safety risks require careful assessment and strict control. Firstly, high-intensity, highly focused acoustic energy, if not properly controlled spatially or temporally due to delay errors leading to unintended beam superposition, could damage healthy vessel walls or adjacent tissues surrounding the thrombus. Therefore, future work necessitates extensive in vitro and preclinical experiments to establish precise "therapeutic windows" and safety parameter thresholds for different treatment scenarios (e.g., thrombi in different locations), which is a prerequisite for safe clinical application.

We have condensed the above content, placed it after Section 4.2 (highlighted in blue), and incorporated the relevant citations into the references.

References:

[R1] Waters, T., et al. “A Chirp Excitation for Focusing Flexural Waves.” Journal of Sound and Vibration 459 (2019): 114863.

[R2] Kurosu, M., et al. “On‑chip Temporal Focusing of Elastic Waves in a Phononic Crystal Waveguide.” Journal of Sound and Vibration 510 (2021): 116288.

[R3] Wang, Y., et al. “Resonant Wavepackets and Shock Waves in an Atomtronic SQUID.” arXiv, arXiv:1510.02968 (2015).

[R4]   Zhu, K.; Ma, J.; Qi, X.; Shen, B.; Liu, Y.; Sun, E.; Zhang, R. Enhancement of Ultrasonic Transducer Bandwidth by Acoustic Impedance Gradient Matching Layer. Sensors 2022, 22, 8025. https://doi.org/10.3390/s22208025

 [R5] Tao Wang; Kobayashi, T.; Lee, C. Broadband Piezoelectric Micromachined Ultrasonic Transducer (pMUT) Using Mode-Merged Design. In Proceedings of the 10th IEEE International Conference on Nano/Micro Engineered and Molecular Systems; IEEE: Xi’an, China, April 2015; pp. 238–242.

 

We have condensed the above content, placed it after Section 4.2 (highlighted in blue), and incorporated the relevant citations into the references.

 

 

 

Methodological Clarity and Reproducibility

Issue:

 

Certain aspects of the methods need more clarity to ensure reproducibility. Firstly, the procedure for tuning the second wave packet’s amplitude, frequency, and delay is not sufficiently described. The manuscript should specify how the “chasing” parameters were optimized – was a systematic parametric scan performed, or was an algorithm used to maximize the WSF? Additionally, details about the numerical solver are sparse. The authors mention a “third-order nonlinear wave equation” and presumably solve it with a finite-difference scheme, but important parameters (grid size, time-step, stability criteria) and boundary conditions are not described. It is also not stated whether attenuation and dispersion are modeled frequency-dependently, and how the simulation ensures convergence (especially in shock conditions, which can be challenging for numerical stability).

 

Suggestion:

 

Expand the Methods section to better detail the simulation procedure. Explain how the optimal delay and amplitude of the second pulse were determined – e.g., was the 22.7% gain the maximum out of a sweep of time delays and amplitude ratios? If so, provide that range or figure. Include a description of the numerical implementation: the equation solved (e.g., Westervelt or a form of Burgers equation), any assumptions (plane wave, 1D propagation in homogeneous medium, etc.), and how shocks are handled (artificial viscosity, filtering, or shock-capturing schemes). State the spatial and temporal resolution and confirm that results are grid-independent. This information will greatly help readers to trust and potentially reproduce your findings.

 

Response:

Thank you for the reviewer's valuable comments, which are crucial for ensuring the reproducibility of our research methodology. We will revise and elaborate on the methodology section of the manuscript accordingly.

 

1. As the reviewer pointed out, the reported 22.7% gain was the maximum value obtained from a systematic parameter sweep over both time delays and amplitude ratios. Specifically, the duty cycle was varied within the range of [100%, 85%], and the displacement amplitude range was [9×10⁻⁵ m, 1.3×10⁻⁴ m].
2. Details of the Numerical Simulations:

The simulations are based on the Westervelt equation, which incorporates the material's strain-hardening property. The computational model considers planar longitudinal wave propagation in a two-dimensional homogeneous medium.

To stably capture shock wave formation in the numerical solution, an Artificial Viscosity scheme was employed to handle the solution's discontinuity. This method introduces controlled dissipation at the shock front, preventing numerical oscillations and enabling robust treatment of shocks resulting from finite-amplitude wave propagation.

The computational domain was discretized using a uniform grid of 20,000 points. Time integration was performed using an explicit scheme with a fixed time step of 10⁻⁸ seconds. This time step satisfies the CFL stability condition, ensuring the stability of the numerical solution.

The top and bottom boundaries of the model are free boundaries. The left boundary is a fixed boundary where the excitation is applied. A displacement boundary condition, as specified by the Equation (31) in the manuscript, is imposed at the left end to generate the incident wave packet. The right end is set as a non-reflecting boundary to simulate a semi-infinite medium, thereby preventing wave reflections from interfering with the results.

To ensure the reliability of the numerical results, a rigorous mesh independence study was conducted. Simulations for the same case were performed using three different grid densities: 20,000, 40,000, and 80,000 points. A comparison of the results shows that the relative error in the calculated shock formation position and peak shock stress between the different grids is less than 0.5%. Time integration was performed using an explicit scheme with a fixed time step of 10⁻⁸ seconds. This time step satisfies the CFL stability condition, ensuring the stability of the numerical solution.

The above content has been supplemented in Section 4 and Section 2.2, highlighted in blue.

 

 

Context of Mechanical Thrombolysis or Mechanotherapy

Issue:

 

The introduction and discussion do not sufficiently situate the work within the broader landscape of mechanical ultrasound thrombolysis techniques. The authors rightly note that conventional HIFU relies mainly on thermal effects and may not be suitable for clot lysis. However, other non-thermal ultrasound approaches have been developed and should be acknowledged. In particular, the technique of histotripsy uses short, high-amplitude ultrasound pulses to generate shock waves and cavitation that mechanically fractionate clots without drugs. Several studies have demonstrated the ability of pulsed ultrasound (with or without microbubbles) to enhance thrombolysis via mechanical effects. The omission of these references makes the manuscript seem isolated, even though those techniques share a similar goal of shock-wave mediated clot disruption.

 

Suggestion:

 

Broaden the discussion to compare your waveform-superposition method with existing mechanical thrombolysis approaches like histotripsy or sonothrombolysis. Emphasize how your approach is different (e.g., using two sequential pulses to amplify an intrinsic shock vs. random bubble cloud shocks in histotripsy) and what advantages it might have. For completeness, cite relevant studies on therapeutic and mechanotherapy ultrasound in clots and other soft biological systems, highlighting mechanical, non-thermal bioeffects. References to consider: Izadifar, Z., et al. ‘Mechanical and biological effects of ultrasound: a review of present knowledge.’ Ultrasound in Medicine and Biology, 43, 1085–1104 (2017).

 

Response:

Thank you for the reviewer's comments. We have expanded the introduction by adding a detailed comparative analysis of the dual-wave-packet strategy against mainstream mechanical thrombolysis approaches (such as histotripsy and sonothrombolysis), focusing on their respective mechanisms of action and potential advantages, to better highlight the core innovation and positioning of this study. As shown below.

“1.1 Principles and Current Status of Major Technical Pathways for Ultrasound Thrombolysis

As a therapy combining physical mechanical effects and targeted enhancement capabilities, ultrasound thrombolysis technology has evolved over several decades from an adjuvant method into a promising new treatment paradigm. Currently, its development is characterized by a distinct diversification of technical pathways.

The first is the relatively traditional Sonothrombolysis. Ultrasound, through cavitation effects (generating and oscillating microbubbles near the thrombus) and mechanical stress, alters thrombus structure, promotes drug penetration, and simultaneously activates endothelial function, providing biochemical-level assistance for thrombolysis [R6]. Clinical studies have shown that compared to drug thrombolysis alone, the combination with ultrasound can significantly improve the rate of vascular recanalization. However, it is essentially an auxiliary enhancement method and cannot completely replace drugs. For large-volume or chronic thrombi, treatment relying solely on this technology may require a longer duration. Furthermore, there is the interventional Ultrasound-Enhanced Catheter-Directed Thrombolysis (UTCD), which delivers low-frequency ultrasound energy directly to the interior or surface of the thrombus through an interventional catheter. The ultrasonic cavitation effect directly loosens the thrombus while increasing the permeability of locally perfused thrombo-lytic drugs into the thrombus [R7]. This treatment modality is invasive, requiring vascular interventional surgery and carrying associated risks related to puncture and catheterization.

Building upon classic treatment methods, Histotripsy represents the forefront of high-intensity therapeutic ultrasound as a non-thermal, mechanical ablation technology. It involves emitting extremely short (microsecond-scale), high-intensity pulsed ultrasound towards the target thrombus, generating a dense cloud of microbubbles at the focal point via inertial cavitation effects. The violent expansion and collapse of these bubble clouds produce powerful mechanical forces that physically disintegrate thrombus cells and fibrin networks [R8]. It operates through a purely physical mecha-nism, avoiding drug-related bleeding risks. Moreover, the cavitation effect has a clear pressure threshold, causing minimal damage to normal tissues below this threshold. However, the core action of Histotripsy lies in the dynamics of exogenous bubble clouds. Its destructive efficacy depends on the generation and distribution of cavitation nuclei, and the energy is released primarily through the random, discrete collapse of bubbles.

Therefore, the future development of ultrasound thrombolysis lies in leveraging the inherent mechanical properties of the thrombus itself. By optimizing energy delivery strategies at the mechanical level to achieve more focused and controllable pure mechanical energy (non-thermal effect) action, the ultimate goal is to attain more predictable therapeutic outcomes.“

 

References:

 

[R6]  Petit, B.; Yan, F.; Tranquart, F.; Allémann, E. Microbubbles and Ultrasound-Mediated Thrombolysis: A Review of Recent in Vitro Studies. J. Drug Deliv. Sci. Technol. 2012, 22, 381–392, doi:10.1016/S1773-2247(12)50065-1.

[R7]. Gao, S.; Zhu, Q.; Guo, M.; Gao, Y.; Dong, X.; Chen, Z.; Liu, Z.; Xie, F. Ultrasound and Intra-Clot Microbub-bles Enhanced Catheter-Directed Thrombolysis in Vitro and in Vivo. Ultrasound Med. Biol. 2017, 43, 1671–1678, doi:10.1016/j.ultrasmedbio.2017.03.017.

[R8] Izadifar, Z.; Babyn, P.; Chapman, D. Mechanical and Biological Effects of Ultrasound: A Review of Present Knowledge. Ultrasound Med. Biol. 2017, 43, 1085–1104, doi:10.1016/j.ultrasmedbio.2017.01.023.

 

 

Material Model and Viscoelastic Assumptions

Issue:

 

The fibrin clot material is modeled with a quasi-static power-law constitutive model derived from low-strain-rate compression tests. While this captures strain-hardening, it may not fully represent the high-rate viscoelastic response under ultrasound frequencies. The manuscript currently assumes that the nonlinearity (strain-hardening) is the dominant factor for shock formation. However, real clots also exhibit viscoelastic attenuation and frequency-dependent modulus (as the authors cite in references on shear wave measurements). It’s unclear if the power-law model incorporates any viscous damping or if dispersion is modeled beyond a basic attenuation term. Neglecting important viscoelastic effects could lead to overestimating shock intensity or waveform steepness.

 

Suggestion:

 

The authors should discuss the limitations of the constitutive model in the context of high-frequency, dynamic loading. If possible, quantify or describe how viscoelastic damping (e.g., using a standard linear solid model or frequency-dependent attenuation) might alter the results. For example, would a Burgers or Kelvin–Voigt model produce a different shock amplitude due to energy dissipation? If the current model lacks a frequency-dependent loss term, acknowledge that this may over-predict shock sharpness. Ideally, include a brief sensitivity study: e.g., introduce a small damping term or use literature values for clot viscosity to estimate its effect on the WSF. At minimum, add a caution in the Discussion that the power-law fit, while reasonable for static behavior, might not capture all dynamic factors, and that future work should investigate more complex viscoelastic models.

 

Response:

 

We thank the reviewer for their valuable comments. Based on the current strain-hardening model, we have incorporated a viscous coefficient of , a value commonly used for biological materials, as shown in the figure below.

 

Figure S1. Nonlinear viscoelastic model.

 

Using this model in conjunction with our existing finite element model and dual-wave packet strategy, simulation results indicate that, when evaluated by the WSF quantitative standard, the shock wave gain is reduced to 12.7%. Additionally, the shock wave formation position is delayed by 2.3×10⁻³ compared to the purely elastic result. Therefore, even with the introduction of a small viscous term, the dual-wave packet chasing strategy remains effective, though it still cannot cover all scenarios，because the viscoelastic properties of thrombi can be more complex and exhibit significant inter-individual variability. Accordingly, we have added the following note in the Limitations and Future Perspectives section:

 

"The authors' subsequent research found that by connecting a dashpot with low viscosity in parallel to the current nonlinear elasticity framework based on a power-law model (thereby introducing viscoelasticity), and keeping all other model parameters un-changed, simulations still demonstrate the occurrence of dual-wave superposition and shock wave enhancement. However, the shock wave enhancement value is reduced, and its formation position shifts significantly backward. This indicates that in viscoelastic media, both the magnitude and location of shock wave enhancement depend on the strength of viscosity and the design of the dual-wave sequence. Since the viscoelastic behavior of thrombi may be more complex than that captured by a simple non-linear spring-dashpot model, further research is needed.

Therefore, it must be noted that while the quasi-static power-law model used in this study effectively captures the core strain-hardening nonlinear characteristics of thrombi, it fails to account for their frequency-dependent viscoelasticity under high-frequency ultrasound excitation (e.g., above ~20 kHz). Consequently, the absolute stress enhancement values derived from the simulations may deviate from those observed in real dynamic biological environments."

 

Significance of Shock Amplification – Link to Thrombolytic Efficacy

Issue:

 

The results show a 22.7% increase in peak shock stress using the dual-wave strategy, and identify the second-pulse amplitude as the dominant factor for amplification. However, the manuscript stops short of translating what this means for actual clot disruption. It is not discussed whether a ~22% rise in shock pressure is expected to significantly improve thrombolysis (e.g., by exceeding some damage threshold or achieving faster lysis). Without context, it’s hard for the reader to gauge the practical importance of the result – is a 22% increase modest or groundbreaking? Additionally, the potential trade-offs (such as increased exposure risk or energy input) are not addressed.

 

Suggestion:

 

Strengthen the discussion by linking the physical metrics to biological impact. For instance, discuss how shock wave amplitude correlates with fibrin network damage or lysis rate, based on prior studies or physical reasoning. If there is a known threshold pressure for causing microstructural damage to clots, mention whether your amplified shock surpasses it. You might speculate that a 22% increase in peak stress could yield disproportionately higher clot fragmentation (if the relationship is non-linear), or conversely, that it might be partially offset by other factors. Also, comment on the possible downsides: does a stronger shock risk damaging healthy tissue or causing emboli due to rapid clot breakage? Framing your results in terms of thrombolytic efficacy and safety will make the findings more meaningful for the reader. This could be just a few sentences connecting your findings to expected improvements in clot dissolution speed or completeness.

 

Response:

Thank you for this insightful and critical comment. As the reviewer noted, shock wave amplification is of critical practical importance for improving thrombolytic efficacy.

Existing studies have confirmed a clear positive correlation between the mechanical force of ultrasonic shock waves and the thrombolysis rate: at a fixed frequency, higher acoustic intensity leads to stronger mechanical force and a higher thrombolysis rate [R9]. This indicates that enhanced shock wave stress should translate into improved dissolution efficiency. While higher ultrasound intensity indeed pose two main potential risks: mechanical damage to the vessel wall and the generation of harmful embolic fragments due to rapid, uncontrolled thrombus fragmentation. Research on ultra-sound intensity thresholds indicates the existence of a critical intensity (approximately 21.6 W/cm²); below this value, the extent of thrombus damage is minimal (approximately 11-15%); once this threshold is exceeded, the maximum damage degree in-creases rapidly with intensity [R10]. Furthermore, when the intensity (which determines the peak stress) surpasses a specific critical point, the destructive effect is significantly enhanced, exhibiting a strong nonlinear characteristic. Therefore, solely relying on increasing ultrasound intensity is not a viable strategy. Our proposed 'wave-packet chasing and superposition' strategy does not aim to infinitely amplify shock wave energy. Instead, it seeks to more efficiently concentrate energy at specific locations within the thrombus through precise waveform modulation, promoting controllable disruption of its internal structure while maintaining the existing energy input. Compared to traditional methods that simply increase single-pulse amplitude, our strategy offers  intrinsic safety advantages。

 

The content above has been added to Section 4.1 in the revised manuscript.

 

References:

[R9]. Salman-Kesner, N.; Zaltsman, M.-M.; Ertracht, O.; Atar, S. In-Vitro Assessment of the Thrombolytic Efficacy of Therapeutic Ultrasound. Thromb. Res. 2019, 178, 63–68, doi:10.1016/j.thromres.2019.04.002.

[R10] Adzerikho, I.; Kulak, A.; Rachok, S.; Minchenya, V. Dependence of the Rate and Completeness of Fibrin Clot Destruction on the Acoustic Dose and Ultrasound Intensity. Ultrasound Med. Biol. 2022, 48, 846–855, doi:10.1016/j.ultrasmedbio.2022.01.005.

 

Cavitation Effects and Bubble Nucleation

Issue:

 

The study focuses on mechanical shock waves in a continuous fibrin medium, but inertial cavitation is not considered. In reality, at the high pressures associated with shock fronts, microscopic cavitation bubbles are likely to form in blood or thrombus (especially if gas nuclei or contrast agents are present). Cavitation can drastically alter wave propagation – for example, bubble collapse can both enhance clot disruption and also absorb energy, potentially limiting further shock transmission. The current simulation neglects cavitation, meaning it might overestimate the sustained shock amplitude in tissue or ignore a key damage mechanism. Not acknowledging this might be seen as a significant gap, since cavitation-based effects are known to contribute to ultrasound thrombolysis.

 

Suggestion:

 

Include a discussion of cavitation phenomena as a limitation and/or a potential enhancement mechanism. Acknowledge that at the intensities needed for shock formation, bubbles may form and that your model does not capture this. You could cite studies of cavitation in sonothrombolysis and note how cavitation could either help (by causing additional mechanical damage) or hinder (by soaking up energy or causing shielding) the propagation of your second pulse, along with work in other soft biological systems showing that the temporal pattern of high-frequency ultrasound pulses can significantly alter mechanical or functional responses. If possible, estimate the peak negative pressure in your simulation and whether it exceeds typical cavitation thresholds in blood. This will inform readers about when cavitation might set in. Encouragingly, if cavitation does occur, your dual-pulse method might work synergistically with it – this is worth mentioning. Adding 2–3 sentences on this topic will make the paper more comprehensive. References to consider: Bollen, V., et al. ‘In vitro thrombolytic efficacy of single- and five-cycle histotripsy pulses and rt-PA.’ Ultrasound in Medicine and Biology, 46, 336–349 (2020).

 

Response:

Thank you for this insightful and critical comment. We fully agree that in ultrasound thrombolysis research involving high-pressure shock waves, the inertial cavitation effect is an important and non-negligible physical mechanism. The limitation you pointed out regarding our model is accurate. We have addressed this in the revised manuscript by adding a supplementary discussion in the "Limitations and Future Work" section. The main points are as follows:

 

1. Acknowledgment of the Model Limitation in This Study

Our current continuum mechanics model indeed does not incorporate the inertial cavitation effect. We acknowledge that in a real biological environment, the negative pressure phase of a shock wave front can potentially induce the growth and subsequent inertial collapse of pre-existing microbubbles (e.g., gaseous nuclei in blood/thrombus or exogenous contrast agent microbubbles).

 

2. Preliminary Assessment of Cavitation Potential Under Our Study Conditions

Based on the theory of ultrasound propagation in nonlinear media like fibrin clots and our numerical results, when a finite-amplitude compression wave (positive pressure) propagates, nonlinear waveform distortion does generate higher harmonic components, including rarefactional (negative pressure) waves. Analysis of our simulation data estimates that under the parameters used in this study, the peak negative pressure in the shock formation region is approximately 0.25 MPa. Wang et al. found that under specific conditions (e.g., a 2.4 MHz center frequency), a peak negative pressure of about 1.2 MPa could be sufficient to induce inertial cavitation and collapse of microbubbles, while the cavitation threshold for pure blood and tissue without contrast agents is significantly higher, typically exceeding 10 MPa—a value greater than the peak negative pressure generated by material nonlinearity within the scope of this study. However, in an actual therapeutic process where the dual-wave-packet is delivered as high-frequency pulsed waves to accumulate shock effects, inappropriate temporal patterns could substantially increase the likelihood of cavitation.

 

3. The Dual Nature of Cavitation and Its Potential Synergy with Our Strategy

We concur with the reviewer's perspective on the dual nature of cavitation. On one hand, the violent collapse of bubbles can generate localized micro-jets and secondary shock waves, providing additional mechanical damage that may significantly enhance thrombolytic efficacy (as demonstrated by Bollen et al. [1], where specific pulse sequences optimized cavitation-mediated destruction). On the other hand, the formation of bubble clouds can scatter and absorb acoustic energy, potentially shielding subsequent wave packets and interfering with the precise superposition central to our designed "wave packet chasing" strategy.

 

Promisingly, and as the reviewer encouragingly notes, if cavitation can be controlled, our dual-pulse method holds potential for synergy. By precisely designing the time delay of the second wave packet, it might be possible to couple it spatiotemporally with the collapse of cavitation bubbles induced by the first pulse. This could achieve a combined focusing of energy from both "mechanical shock waves" and "cavitation micro-explosions," potentially yielding a synergistic effect beyond that of either mechanism alone.

 

4. Explicit Addition to the Manuscript

We have added a new paragraph in the Discussion section (Section 4.1) explicitly listing "the exclusion of cavitation effects" as a key limitation of the current model and outlining the analysis above regarding its potential impact and synergy:

 

" However, the current continuum model did not consider inertial cavitation effects. According to nonlinear acoustic theory, the propagation of finite-amplitude compression waves (positive pressure) in media such as fibrin clots leads to waveform distortion, generating higher harmonics that include rarefaction waves (negative pressure). This provides a theoretical possibility for inertial cavitation. While, our numerical simulation results (Fig. 7e) indicated that the peak negative pressure induced by material nonlinearity was approximately 0.075 MPa, theoretically, since both wave packets pro-vide positive pressure. Existing research clearly shows that the cavitation threshold for blood containing ultrasound contrast agents is on the order of megapascals, while the threshold for pure blood or tissue is even higher (typically >9 MPa) [47]. Evidently, the negative pressure level generated in our simulations is significantly below these cavitation thresholds, suggesting a low probability of significant inertial cavitation within the current theoretical framework.

This characteristic represents a potential safety advantage of our strategy: it primarily amplifies mechanical shock by leveraging the material's strain-hardening properties through precise waveform design, rather than relying on uncontrolled cavitation bubbles as the main disruption mechanism. Nonetheless, we fully recognize that in practical high-intensity pulse sequence treatments, improper parameter design (e.g., timing, duty cycle) could substantially increase cavitation risk. Therefore, future engineering explorations could draw on existing studies by optimizing pulse forms (e.g., employing short-pulse, low-duty-cycle sequences) [48]. This approach holds promise for synergistically combining controlled cavitation mechanical forces with shock wave amplification effects while effectively managing thermal effects, thereby achieving ef-ficient and synergistic thrombus ablation."

 

Parameter Space and Transducer Limitations

Issue:

 

The parametric analysis in the paper identifies the second wave’s amplitude as the most critical factor for shock amplification, with frequency and time delay presumably also explored. However, the range of parameters tested and their relation to real transducer capabilities are not fully discussed. For example, if extremely high amplitude of the second pulse is needed (beyond what standard diagnostic or therapeutic ultrasound can deliver), that could limit practical use. Similarly, the optimal time delay found in simulation should be contextualized: is this delay achievable and controllable by existing ultrasound sequencing technology? The manuscript does not address how sensitive the WSF gain is to deviations in these parameters – a concern for robustness.

 

Suggestion:

 

Provide more insight into the parameter choices and practical constraints. In the Results, briefly note the ranges over which frequency, phase, and delay were varied, and whether the 22.7% gain occurs at an optimal point or a broad plateau. In the Discussion or Conclusion, mention how one might generate such a dual-packet sequence: e.g., using two transducers or programmed bursts from one transducer with controlled inter-pulse delay. If the required second-pulse amplitude is, say, 1.5× the first pulse, discuss if that is within FDA limits or device specs. Conversely, if the second pulse’s frequency is different, can a typical transducer emit both frequencies effectively? By addressing these points, the authors will demonstrate a clear understanding of how the proposed method could be translated into practice, and what the current technical gaps are.

 

Response:

Thanks for the valuable comments from the reviewers. We respond to them from the following aspects.

1. In the Results, we provide more insight into the parameter choices：

 “The 22.7% gain was the maximum value obtained from a systematic parameter sweep over both time delays and amplitude ratios. Specifically, the frequency remained 20KHz; the duty cycle was varied within the range of [100%, 85%], and the displacement amplitude range was [9×10⁻⁵ m, 1.3×10⁻⁴ m].”

 

2. The core mechanism of the dual-wave strategy proposed in this paper lies in utilizing the strain-hardening characteristics of the medium to achieve higher shock wave gain. This implies that, under the premise of complying with FDA safety limits, precise design of dual-wave parameters—such as the interval time, amplitude, and frequency—enables the pursuit and superposition of ultrasonic waves within a specific region. Critically, this strategy does not rely on the amplitude of the second wave packet exceeding regulatory limits. In practice, as long as the amplitude of the first wave packet is greater than that of the second within a certain range, the strain-hardening properties of the material can be leveraged within the safe operational window. Through parameter modulation, the wave-chasing and superposition effects can be achieved, thereby optimizing the formation and enhancement of shock waves under compliant conditions.

 

3. The implementation of the dual-wave packet chasing strategy places clear demands on the damping and bandwidth of the transducer. This outlines a clear engineering pathway: achieving ideal waveform transmission through subsequent material optimization and transducer design. Existing theoretical models and experimental trends fully support the feasibility of this pathway in principle. In terms of acoustic implementation, the strategy can be realized on two levels: a single-frequency du-al-wave packet sequence and a dual-frequency dual-wave packet sequence.

(1) For the single-frequency dual-wave packet sequence, damping is increased and bandwidth is broadened by optimizing the transducer structure (e.g., adding an acoustic backing layer to absorb backward-propagating waves) [R11] and using broadband materials (e.g., novel composite piezoelectric materials, PMN-PT single crystals, etc.) [R12]. This provides the foundation for short-pulse emission. On this basis, waveform generation and the precise silence period between the two waveforms are achieved by counters within a Field-Programmable Gate Array (FPGA), with accuracy reaching the nanosecond level.

(2) For the dual-frequency dual-wave packet sequence, two independent trans-ducer elements are used, each optimized for frequencies f1 and f2, respectively. Both elements are controlled by the same FPGA but use different transmission channels. They are driven sequentially according to the designed sequence.

It must be emphasized that while this technical approach offers therapeutic gains, its potential safety risks require careful consideration. High-intensity, highly focused acoustic energy, if not properly controlled spatially or if unexpected beam superposition occurs due to timing errors, may damage the healthy vessel wall or adjacent tis-sues near the thrombus. Therefore, extensive in-vitro and preclinical experiments will be necessary in the future to establish precise "therapeutic windows" and safety parameter thresholds for different treatment scenarios (e.g., thrombi in different locations). This is a prerequisite for achieving safe clinical application.

This refined content has been incorporated into the concluding part of Section 4, highlighted in blue color.

 

References:

[R11] Zhu, K.; Ma, J.; Qi, X.; Shen, B.; Liu, Y.; Sun, E.; Zhang, R. Enhancement of Ultrasonic Transducer Bandwidth by Acoustic Impedance Gradient Matching Layer. Sensors 2022, 22, 8025, doi:10.3390/s22208025.

[R12]. Tao Wang; Kobayashi, T.; Lee, C. Broadband Piezoelectric Micromachined Ultrasonic Transducer (pMUT) Using Mode-Merged Design. In Proceedings of the 10th IEEE International Conference on Nano/Micro Engineered and Molecular Systems; IEEE: Xi’an, China, April 2015; pp. 238–242.

 

3.Regarding safety and device specifications, our parametric study indicated that the optimal amplification often requires the secondary pulse amplitude to be 1.2–1.5 times that of the primary pulse. While this increased local stress is the intended mechanism, the spatial peak, temporal peak intensity (Ispp) of the compounded waveform must be evaluated against regulatory limits (e.g., FDA guidelines for diagnostic ultrasound mechanical index thresholds) and the output limits of clinical HIFU systems. This represents a key translational constraint that necessitates careful system design and dosimetry studies.

 

 

 

Presentation: Figures and Terminology

Issue:

 

A few minor presentation issues should be addressed to improve clarity. For instance, some figures could benefit from more detailed captions. Figure 3 (waveform profiles) is mentioned in the text, but the caption does not explicitly explain all curve labels or the physical scenario depicted (e.g., which line is with vs. without the second wave, etc.). Additionally, terms like “WSF” and “ASF” are introduced; while defined in the text, it would help to also spell them out in figure legends or labels for readers who skim figures. The term “Waveform Superposition Factor” might confuse some readers at first glance – consider briefly clarifying that it quantifies peak amplitude gain from superposition. There are also minor terminology inconsistencies (e.g., sometimes “shock wave” vs “shockwave”) that should be standardized.

 

Suggestion:

 

Revise figure captions to be more self-contained. Ensure that each curve or image panel is sufficiently explained (for example, specify “Initial single-pulse waveform vs. dual-pulse waveform at location X” in the caption). Add units to color bars and axes wherever applicable (e.g., stress in MPa, time in µs). Unify terminology in the text: use one form of “shock wave” consistently. Also, double-check that all acronyms (WSF, ASF, etc.) are defined at first use and perhaps remind the reader of their meaning in the Conclusion for emphasis. These tweaks will make the paper more reader-friendly and polished.

Response:

Thank you for these valuable suggestions to improve the clarity and presentation of our manuscript. We have carefully addressed all points:

1. As suggested, we have revised all figure captions to make them self-explanatory. Please note that due to the insertion of a new schematic figure, the figure numbering has shifted accordingly. For instance, the figure showing Definitiondiagramof wave superposition factor (previously Figure 3) is now Figure 4, and its caption has been revised to explicitly state: “Calculation diagram of  waveform superposition factor at location x=3.927×10^-3m during a dual-wave-packet chasing process”.
2. We have standardized the terminology to consistently use “shock wave” and ensured all acronyms are spelled out in their first use in figure legends (e.g., “Waveform Superposition Factor (WSF)”).
3. All units (e.g., stress in Pa, time in s) are now clearly indicated on axes and color bars.

4.A brief clarification has been added near the first mention of WSF, noting it "WSF represents waveform superposition factor, which is used to characterize the wavefront slope at a given moment.” And it has been confirmed that all abbreviated terms (such as WSF, ASF, PED) have been defined upon their first use.

We hope these comprehensive edits meet the reviewer’s expectations and make the paper more reader-friendly.

 

Conclusion:

 

In summary, this work addresses an important challenge in ultrasound thrombolysis and presents a novel solution. However, substantial revisions are required to strengthen the validation, clarify methods, and better integrate the study into the existing research context. The eight major points above should be carefully addressed to improve the manuscript’s scientific rigor and clarity. Once these issues are resolved, along with minor language corrections, I believe the paper will meet the criteria for publication. I look forward to seeing a revised version that convincingly demonstrates the feasibility and significance of the proposed dual-wave shock amplification strategy for clot dissolution.

 

 

Comments on the Quality of English Language

Overall, the manuscript is written in clear English, but some minor edits will improve readability. Please check for grammar and style consistency. For example, ensure subject–verb agreement in long sentences (e.g., “shock waves is limited” should be “shock waves are limited”). Use consistent tense when describing your results (past tense is often used for methods and results). Make sure to include articles (“the”, “a”) where needed – e.g., “developed a dual-wave-packet strategy”. Also, standardize technical notation and units: write units with a space (e.g., “5 °C”, “10 mm”), and hyphenate compound adjectives (e.g., “strain-hardening material”, “peak-shock stress”). A careful proofread for punctuation (commas, etc.) and phrasing will help. The suggested changes do not alter the science, but they will ensure the paper reads smoothly and meets the journal’s language standards.

 

Response:

 

We sincerely thank the reviewer for their valuable and detailed suggestions to improve the clarity and readability of our manuscript. We have performed a thorough proofreading and language editing of the entire manuscript accordingly.

 

Specifically, we have addressed all the points raised:

 

1. Grammar and Subject-Verb Agreement: We have addressed the noted issues (e.g., 'shock waves is' to 'shock waves are') and conducted a comprehensive proofreading to correct grammatical errors throughout the paper.
2. Tense Consistency: The tense used in describing methods and results has been standardized to the past tense for consistency.
3. Use of Articles: Definite and indefinite articles (e.g., "the", "a") have been checked and added where necessary, for example, ensuring "developed a dual-wave-packet strategy" is used correctly.
4. Standardization of Technical Notation: A space has been inserted between values and units , and compound adjectives have been hyphenated (e.g., "strain-hardening material", "peak-shock stress").
5. Punctuation and Phrasing: The manuscript has been carefully reviewed for punctuation (e.g., commas) and phrasing. Awkward sentences have been rephrased to improve overall flow.

 

We believe these comprehensive revisions have significantly enhanced the linguistic quality of the manuscript, ensuring it reads smoothly and meets the journal's language standards. All changes are incorporated into the revised version.

 

Author Response File: Author Response.docx

Round 2

Reviewer 3 Report

Comments and Suggestions for Authors

In this revised version, the authors have done an excellent job addressing the previous review comments on context, modelling clarity, and practical relevance. The Introduction now situates the work clearly within the broader landscape of ultrasound-based thrombolysis, including mechanical (non-thermal) pathways and related techniques such as sonothrombolysis and histotripsy, and explains where the proposed dual-wave-packet strategy fits conceptually. The derivation and presentation of the constitutive model and nonlinear wave equation are much more transparent, with explicit assumptions and a clear discussion of viscoelastic limitations. The numerical methodology is now reproducible, with governing equations, boundary conditions, discretization details and a brief mesh-independence study described in sufficient detail. The extended discussion of shock amplification, cavitation thresholds, parameter ranges and transducer feasibility brings out the physical meaning of the 22.7% stress gain and appropriately bounds the claims for thrombolytic applications. Overall, the manuscript now reads as a well-structured and self-contained modelling contribution on nonlinear ultrasound propagation and waveform superposition in fibrin clots.

 

Commendations:

The authors are to be commended for the thoroughness and care of their revision. The expanded background on thrombus biomechanics and therapeutic ultrasound provides a solid and up-to-date framework for the study. The clarified constitutive formulation and its relation to the nonlinear wave model make the modelling assumptions explicit and understandable to readers from both mechanics and biomedical backgrounds. The numerical section now gives enough information (equations, initial and boundary conditions, discretization, convergence checks) to allow others to reproduce or extend the simulations. The revised Discussion and Conclusion sections articulate the main contributions—the concept of dual-wave-packet “chasing” to enhance shock formation without increasing total energy, the identification of amplitude and timing as key control parameters, and a frank statement of the model’s limitations (e.g. lack of dynamic viscoelasticity and cavitation modelling)—and link them clearly to potential ultrasound-thrombolysis applications and future experimental work. The authors have clearly engaged constructively with the earlier review and strengthened the paper in all the key areas that were previously flagged.

Conclusion:

The authors have substantially strengthened the manuscript and addressed all of the previously raised major concerns regarding scientific context, constitutive modelling, numerical transparency, physical interpretation and practical feasibility. I have no further scientific or methodological requests, and the remaining issues are at most minor language or stylistic points that can be handled during copy-editing. I therefore recommend acceptance in its present form.