Histological characterization of HIFU lesions.

Background: High intensity focused ultrasound (HIFU) can destroy tissue by thermal ablation which may be accompanied by acoustic cavitation and/or tissue water boiling, but the biological and histological effects of these treatments have not been fully documented. Here, detailed histological analysis over time using well characterized HIFU exposures in in vivo rat livers is described.Methods: Exposures used invoked either (i) thermal, with acoustic cavitation and/or tissue water boiling or (ii) predominantly thermal damage. Cavitation activity was detected using both active and passive methods. Histological assessment involved hematoxylin and eosin (H&E), picrosirius red and immunohistochemical staining.Results: Distinct concentric damage regions were identified after HIFU exposures. The outermost ring showed a red H&E-stained rim that was characterized by hemorrhage. The adjacent inner band appeared white due to increased extracellular spaces. The morphology of the next zone depended on the exposure. Where there was no tissue acoustic cavitation/water boiling, this was the lesion center, in which heat-fixed cells were seen. Where acoustic cavitation/boiling occurred, a centermost zone with irregular holes up to several hundred microns across was seen. Cleaved caspase-3 and Hsp70 staining in the periphery of both types of HIFU exposures was seen within the outermost ring of hemorrhage, where an inflammatory response was also observed. By day 7, a distinct acellular region in the center of the HIFU lesions had been created.Conclusions: These results identify the morphological effects and elucidate the similarities and differences of HIFU-induced thermal lesions in the presence or absence of acoustic cavitation/tissue water boiling.

Degradation of hydroxypropyl methylcellulose (HPMC) by acoustic and hydrodynamic cavitation.

The present study aims to investigate the degradation of HPMC on a laboratory scale by acoustic and hydrodynamic cavitation. The effects of temperature and the addition of an external oxidizing agent on the effectiveness of HPMC degradation were systematically investigated by SEC/MALS-RI, FTIR and 1H NMR. The results of the experiments without cavitation show that an external oxidizing agent alone reduces the weight-average molar mass at 60 °C in 30 min for 45.1 % (from 335 to 184 kg mol-1). However, the weight-average molar mass of HPMC decreased significantly more in the cavitation treatment, for 98.8 % (from 335 to 4 kg mol-1) in 30 min at optimal operating conditions of hydrodynamic cavitation (i.e. addition of external oxidant and 60 °C) with a concomitant narrowing of the molar mass distribution, as shown by the dispersity value, which decreased from 2.24 to 1.31. Compared to acoustic cavitation, hydrodynamic cavitation also proved to be more energy efficient. The FTIR spectra of the cavitated HPMC samples without the addition of H2O2 show negligible oxidation of the hydroxyl groups and the glycosidic bonds, confirming that mechanical effects predominate in HPMC degradation in these cases. In contrast, when H2O2 was added, FTIR and 1H NMR show typical signals for cellulose oxidation products, especially when the experiments were performed at 60 °C, confirming that chemical as well as mechanical effects are responsible for the extensive HPMC degradation in these cases. Since treatment methods that lead to lower molar masses and narrower molar mass distributions of the polymers are lacking or require longer treatment times (e.g. 24 h), mechanochemical treatment methods such as cavitation have great potential, as they enable faster polymer degradation (in our case 30 min) through a combination of mechanical and/or chemical degradation mechanisms.

Acoustic cavitation-induced microstructure evolution in ultrasonically brazed Al/Cu joints using Zn-Al alloy fillers.

Tailoring the phase constitutions of the interfacial reaction layers under the assistance of ultrasonic vibration is a convenient method to fabricate high-strength Al/Cu brazing joints. In this study, 1060-Al and T2-Cu dissimilar metals were ultrasonically brazed with Zn-3Al (wt. %) filler metals. Effects of ultrasonic brazing time on the microstructure and mechanical properties of joints were investigated. Results showed that the CuZn5 intermetallic compound (IMC) layer and Cu-based diffusion layer were created on the Cu substrate surface in the joint ultrasonically brazed at 400 ℃ for 2 s. However, the CuZn5 IMC layer was gradually transformed into a thin Al4.2Cu3.2Zn0.7 IMC layer by increasing the ultrasonic vibration time to 15 s. A well-matched coherent interface was formed between the Al4.2Cu3.2Zn0.7 ternary phase and the Cu-based diffusion layer. The phase transition of the Cu-side interfacial layer correlated closely with the acoustic cavitations induced super-saturation regions near the Cu substrate surface. The measured tensile strength of the Al/Zn-3Al/Cu joint ultrasonically brazed for 15 s was 89.3 MPa, which was approximately 2.5 times higher than that brazed for 2 s, and the tensile failure mainly occurred at the interface between the Al4.2Cu3.2Zn0.7 layer and the Cu-based diffusion layer.

Boiling histotripsy exhibits anti-fibrotic effects in animal models of liver fibrosis.

Liver fibrosis is a hallmark of chronic liver disease which could lead to liver cirrhosis or liver cancer. However, there is currently lack of a direct treatment for liver fibrosis. Boiling histotripsy (BH) is an emerging non-invasive high-intensity focused ultrasound technique that can be employed to mechanically destruct solid tumour at the focus via acoustic cavitation without significant adverse effect on surrounding tissue. Here, we investigated whether BH can mechanically fractionate liver fibrotic tissue thereby exhibiting an anti-fibrotic effect in an animal model of liver fibrosis. BH-treated penumbra and its identical lobe showed reduced liver fibrosis, accompanied by increased hepatocyte specific marker expression, compared to the BH-untreated lobe. Furthermore, BH treatment improved serological liver function markers without notable adverse effects. The ability of BH to reduce fibrosis and promote liver regeneration in liver fibrotic tissue suggests that BH could potentially be an effective and reliable therapeutic approach against liver fibrosis.

Cardiac gene delivery using ultrasound: State of the field.

Over the past two decades, there has been tremendous and exciting progress toward extending the use of medical ultrasound beyond a traditional imaging tool. Ultrasound contrast agents, typically used for improved visualization of blood flow, have been explored as novel non-viral gene delivery vectors for cardiovascular therapy. Given this adaptation to ultrasound contrast-enhancing agents, this presents as an image-guided and site-specific gene delivery technique with potential for multi-gene and repeatable delivery protocols-overcoming some of the limitations of alternative gene therapy approaches. In this review, we provide an overview of the studies to date that employ this technique toward cardiac gene therapy using cardiovascular disease animal models and summarize their key findings.

Intensified degradation of tartrazine dye present in effluent using ultrasound combined with ultraviolet irradiation and oxidants.

Effluent containing tartrazine can affect the environment and human health significantly prompting the current study into degradation using a sonochemical reactor operated individually and combined with advanced oxidation processes. The optimum conditions for ultrasound treatment were established as dye concentration of 10 ppm, pH of 3, temperature as 35 °C, and power as 90 W. The combination approach of H2O2/UV, H2O2/US, and H2O2/UV/US resulted in higher degradation of 25.44%, 57.4%, and 74.36% respectively. Use of ZnO/UV/US approach increased the degradation significantly to 85.31% whereas maximum degradation as 93.11% was obtained for the US/UV/Fenton combination. COD reduction was found maximum as 83.78% for the US/UV/Fenton combination. The kinetic analysis showed that tartrazine dye degradation follows pseudo first-order kinetics for all the studied processes. Combination of Fenton with UV and US was elucidated as the best approach for degradation of tartrazine.

Cavitation-induced pressure saturation: a mechanism governing bubble nucleation density in histotripsy.

Objective.Histotripsy is a noninvasive focused ultrasound therapy that mechanically disintegrates tissue by acoustic cavitation clouds. In this study, we investigate a mechanism limiting the density of bubbles that can nucleate during a histotripsy pulse. In this mechanism, the pressure generated by the initial bubble expansion effectively negates the incident pressure in the vicinity of the bubble. From this effect, the immediately adjacent tissue is prevented from experiencing the transient tension to nucleate bubbles. Approach.A Keller-Miksis-type single-bubble model was employed to evaluate the dependency of this effect on ultrasound pressure amplitude and frequency, viscoelastic medium properties, bubble nucleus size, and transducer geometric focusing. This model was further combined with a spatial propagation model to predict the peak negative pressure field as a function of position from a cavitating bubble.Main results. The single-bubble model showed the peak negative pressure near the bubble surface is limited to the inertial cavitation threshold. The predicted bubble density increased with increasing frequency, tissue viscosity, and transducer focusing angle. The simulated results were consistent with the trends observed experimentally in prior studies, including changes in density with ultrasound frequency and transducerF-number.Significance.The efficacy of the therapy is dependent on several factors, including the density of bubbles nucleated within the cavitation cloud formed at the focus. These results provide insight into controlling the density of nucleated bubbles during histotripsy and the therapeutic efficacy.

Mechanism study of aging oil demulsification and dehydration under ultrasonic irradiation.

With the tertiary oil recovery in the oilfield, the content of aging oil emulsion with high water content and complex components has become more prevalent, so it is crucial for aging oil to break the emulsification. In this paper, the experimental laws of water content are explored under the conditions of different transducer input powers through the ultrasonic reforming of aging oil, and the microscopic topography, particle size, components, etc. of oil samples before and after the irradiation of ultrasound are characterized through the microscopic analysis, particle size analysis and component analysis and other ways. The results show that the oil samples achieve the effect of demulsification and dehydration in the presence of ultrasonic cavitation effect, with a maximum dehydration rate of 98.24 %, and that the dehydration rate follows an "M-type" trend with the increase of power. The results of microscopic and particle size analyses demonstrate that ultrasonic irradiation destabilizes the oil-water interfacial membrane, and causes droplets of different sizes to collide, agglomerate, and settle. It was also observed that the droplets of the emulsion system are more evenly distributed and the intervals are increased. Furthermore, we hypothesize that ultrasound may be less irreversible in demulsification and dehydration of aging oil.

Influence of interactions between bubbles on physico-chemical effects of acoustic cavitation.

Ultrasound technology has been extensively used as one of the efficient and economic methodology to achieve the desired outcomes in many applications by harnessing the physico-chemical effects of acoustic cavitation. However, the cavitation-associated effects, primarily determined by the oscillatory dynamics of cavitation bubbles, are considerably complex and still remain poorly understood. The main objective of this study was to perform a numerical analysis of the acoustic cavitation (i.e., the cavitation dynamics, the resultant temperature, pressure and chemical yields within collapsing bubbles), particularly focusing on the influence of the interactions between bubbles. A comprehensive model was developed to simulate the acoustic cavitation dynamics via combining the influences of mass transfer, heat conduction and chemical reactions as well as the interaction effects between bubbles. The results demonstrated that only the large bubble exerts a greater impact on the small one in a two-bubble system. Specifically, within parameter ranges covered this study, there are noticeable decreases in the expansion ratio of the small bubble, the resultant temperature, pressure and molar yields of free radicals, hence weakening the cavitation intensity and cavitation- associated physico-chemical effects. Moreover, the influences of the interactions between bubbles were further assessed quantitatively under various parameters, such as the ultrasound amplitude PA and frequency f, the distance between bubbles d0, the initial radius of the large bubble R20, as well as the liquid properties (e.g., surface tension σ and viscosity µ). It was found that the suppression effect can be amplified when subjected to ultrasound with an increased PA and/or a decreased f, probably due to a stronger cavitation intensity under this condition. Additionally, the suppression effect is also enhanced with a decrease in d0, σ and µ, but with R20 increasing. This study can contribute to deepening knowledge about acoustic cavitation and the resultant physical and/or chemical effects, potentially further facilitating the ultrasound-assisted various applications involving acoustic cavitation.

Exploring viscosity influence mechanisms on coating removal: Insights from single cavitation bubble behaviours in low-frequency ultrasonic settings.

The role of acoustic cavitation in various surface cleaning disciplines is important. However, the physical mechanisms underlying acoustic cavitation-induced surface cleansing are poorly understood. This is due to the combination of microscopic and ultrashort timescales associated with the dynamics of acoustic cavitation bubbles. Here, we have precisely controlled single-bubble cavitation in both space and time. Ultrasonic excitation leads to the cavitation of generated single bubbles. A synchronous ultrafast photomicrographic system simultaneously records the dynamics of single acoustic cavitation bubbles (SACBs) and the cleaning process of the nearby surface in liquids with varying viscosities. Finally, we analysed the correlation between bubble dynamics and surface cleaning situations. The differences in the typical dynamic characteristics of the bubbles during collapse in liquids with varying viscosities reveal two main mechanisms underlying surface cleaning by acoustic cavitation, which are respective the Laplace pressure during the bubble's movement and liquid jets during bubble collapse. Our study provides a better physical understanding of the ultrasonic cleaning process based on acoustic cavitation, and will help to optimize and facilitate the applications of surface cleaning, especially for the cleaning of substrates with tightly attached dirt.

Numerical investigation of acoustic cavitation and viscoelastic tissue deformation.

Acoustic cavitation and tissue deformation are studied by modifying a level-set method for compressible two-phase flows to consider viscoelastic tissue deformation. The numerical simulations performed using different shear moduli and bubble-tissue distances demonstrate various interactions between bubble and viscoelastic tissue, including inverted cone-shape bubbles, bubble migration, liquid jet formation, compressive and expansive tissue deformation, and tissue perforation. The bubble is observed to grow larger with increasing tissue bulk modulus and density. The maximum tissue deformation generally increases with decreasing initial bubble-tissue distance and with increasing tissue bulk modulus and density. The tissue shear modulus conditions that maximize tissue deformation are in the range of 1-10 MPa, unless the tissue density is very large.

Ultrasound for the degradation of endocrine disrupting compounds in aqueous solution: A review on mechanisms, influence of operating parameters and cost estimation.

Availability of drinking water is one of the basic humanitarian goals but remains as a grand challenge that the world is facing today. Currently, water bodies are contaminated not only with conventional pollutants but also with numerous recalcitrant pollutants, such as PPCPs, endocrine disrupting compounds, etc. These emerging pollutants require special attention because of their toxicity to living organisms, bio-resistant and can sustain even after primary and secondary treatments of wastewater. Among different treatment technologies, sonolysis is found to be an innovative and promising technique for the treatment of emerging pollutants present in aqueous solution. Sonolysis is the use of ultrasound to enhance or alter chemical reactions by the formation of free radicals and shock waves which ultimately helps in degradation of pollutants. This review summarizes several studies in the sonochemical literature, including mechanisms of sonochemical process, physical and chemical effects of ultrasound, and the influence of several process variables such as ultrasound frequency, power density, temperature and pH of the medium on degradation performance for endocrine disrupting compounds. In addition, this review highlighted techno-economic perspectives focusing on the total cost required for translating the ultrasound-based processes on a large scale. Overall, the objective of this study is to exhibit a critical review of information available in the literature to encourage and promote future research on sonolysis for the degradation of Endocrine Disrupting Compounds (EDCs).

Effects of alcohols and dissolved gases on sonochemical generation of hydrogen in a 300 kHz sonoreactor.

The sonochemical generation of hydrogen (H2) was investigated using various water/alcohol solutions under argon (Ar) 100 % in a 300 kHz sonoreactor. Five types of alcohols-methanol, ethanol, isopropanol, n-propanol, and n-butanol-were used at various concentrations (0 - 100 % v/v). The H2 generation rate in water was 0.31 µmol/min in the absence of alcohols. The H2 generation rate increased, peaked, and then decreased as the alcohol concentration increased. The concentrations used for the peak H2 generation were 5 %, 1 %, 0.5 %, 0.5 %, and 0.1 % for methanol, ethanol, isopropanol, n-propanol, and n-butanol, respectively. The highest generation rate (5.46 µmol/min) was obtained for methanol 5 % among all conditions in this study, and no H2 was detected for 100 % alcohol concentrations. The reason for the enhancement of the sonochemical H2 generation by the addition of alcohols might be due to strong scavenging effect of alcohols for sonochemically generated oxidizing radicals and vigorous reactions of alcohol molecules and their derivatives with H radicals. No significant correlations were found between the H2 generation rates and physicochemical properties of the alcohols in any of the data in this study. As alcohol concentration increased, the calorimetric power decreased. This indicates that the calorimetric power does not represent the degree of sonochemical reactions in the water/alcohol mixtures. The effect of oxygen (O2) content in the dissolved gases on the generation of H2O2 (representing sonochemical oxidation activity) and H2 (representing sonochemical reduction activity) was investigated using Ar/O2 mixtures for water, methanol 5 % and n-propanol 0.5 %. In water, the highest H2O2 generation was obtained for Ar/O2 (50:50), which is similar to previous research results. However, the H2O2 generation increased as the O2 content increased. In addition, H2 generation decreased as the O2 content increased under all liquid conditions (water, methanol, and n-propanol).

Mechanisms underlying the influence of skin properties on a single cavitation bubble in low-frequency sonophoresis.

As a safe and effective method for systemic transdermal drug delivery (TDD), sonophoresis has drawn much attention from researchers. Despite numerous studies confirming cavitation as the main reason for sonophoresis, the effect skin has on cavitation bubble dynamics remains elusive due to the difficulty of experimental challenges. For a start, we reveal how single cavitation bubble (SCB) dynamics are affected by skin properties, including elasticity, hydrophilicity and texture. We use polydimethylsiloxane (PDMS) to simulate human skin and record the temporary evolution of SCBs with synchronous ultrafast photography. The influences of skin properties on SCBs are concluded: 1) SCBs collapse later near walls with better elasticities and generate microjets with higher speed; 2) SCBs collapse later near hydrophilic walls with slower microjets; and 3) the existence of a texture structure on walls also delays the time of bubble collapse near them and slows the velocities of microjets (v) during collapses.

An introductory laboratory class in sonochemistry.

This article proposes a substantive scope and scenario of a laboratory class that introduces students to the field of sonochemistry. The class requires only basic laboratory equipment - typical laboratory glassware like graduated pipettes and conical flasks, as well as simple inorganic chemicals. It is designed to acquaint students with fundamental aspects of sonochemistry. In the qualitative aspect, they will conduct and observe some sonochemical reactions like a synthesis of hydrogen peroxide and ultrasound-assisted degradation of toxic chromates(VI) which will demonstrate the indirect consequences of water sonolysis which is the most basic sonochemical reaction, as well as they will illustrate the applications of sonochemistry. In the quantitative aspect, students will learn about how to measure the power of ultrasound and the sonochemical efficiency of the reaction and will conduct experiments allowing for the calculation of these parameters. Finally, an introduction to and demonstration of the sonocatalytic effect is planned. An evaluation system, consisting of a report and test, is also proposed.

Numerical investigation on the influence of dual-frequency coupling parameters on acoustic cavitation and its analysis of the enhancement and attenuation effect.

To understand the effect of coupling parameters between two ultrasonic waves on acoustic cavitation, in this work, Keller-Miksis equation was introduced to built a bubble dynamics model that was used to describe the dynamic evolution of bubble and to discuss the effect of dual-frequency coupling parameters, such as frequency difference f (5 âˆ¼ 280 kHz), phase difference φ (0 âˆ¼ 7π/4 rad), and power allocation ratio ß (0 âˆ¼ 9), on acoustic cavitation in the presence of two ultrasonic waves irradiation. The enhancement and attenuation effect of cavitation have also been analyzed in detail by comparing the different dual-frequency combinations with single-frequency mode. It was found that all coupling parameters have a significant impact on acoustic cavitation, where the smaller values of f and φ were employed when ß = 1, the stronger cavitation intensity was observed. Nevertheless, as the power allocation ratio is increased from 1 to 9 at φ = 0 for different frequency differences, the acoustic cavitation exhibits an attenuation trend. When the total acoustic power is evenly distributed, namely ß = 1, the largest maximum expansion ratio (i.e. 12.96) was obtained at φ = 0 and f = 5 kHz, which represents a strongest cavitation effect. In addition, for different frequency combinations, the enhancement effect is found under the mixture of low and low frequency, whereas attenuation effect is generated easily by the combination of high and low frequency. Moreover, the effect become more pronounced as the proportion of high frequency component increases.

Investigation of interaction effects on dual-frequency driven cavitation dynamics in a two-bubble system.

The cavitation dynamics of a two-bubble system in viscoelastic media excited by dual-frequency ultrasound is studied numerically with a focus on the effects of inter-bubble interactions. Compared to the isolated bubble cases, the enhancement or suppression effects can be exerted on the amplitude and nonlinearity of the bubble oscillations to different degrees. Moreover, the interaction effects are found to be highly sensitive to multiple paramount parameters related to the two-bubble system, the dual-frequency ultrasound and the medium viscoelasticity. Specifically, the larger bubble of a two-bubble system shows a stronger effect on the smaller one, and this effect becomes more pronounced when the larger bubble undergoes harmonic and/or subharmonic resonances as well as the two bubbles get closer (e.g., d0 < 100 µm). For the influences of the dual-frequency excitation, the results show that the bubbles can achieve enhanced harmonic and/or subharmonic oscillations as the frequency combinations with small frequency differences (e.g., Δf < 0.2 MHz) close to the corresponding resonance frequencies of bubbles, and the interaction effects are consequently intensified. Similarly, the bubble oscillations and the interaction effects can also be enhanced as the acoustic pressure amplitude of each frequency component is equal and the pressure amplitude pA increases. Above a pressure threshold (pA = 215 kPa), a larger bubble undergoes period 2 (P2) oscillations, which can force a smaller bubble to change its oscillation pattern from period 1 (P1) into P2 oscillations. In addition, it is found that the medium viscosity dampens the bubble oscillations while the medium elasticity affects the bubble resonances, accordingly exhibiting stronger interaction effects at smaller viscosities (e.g., µ < 4 mPa·s) or certain elasticities (approximately G = 70-120 kPa, G = 160-200 kPa and G = 640-780 kPa) at which the bubble resonances occur. The study can contribute to a better understanding of the complex dynamic behaviors of interacting cavitation bubbles in viscoelastic tissues for high efficient cavitation-mediated biomedical applications using dual-frequency ultrasound.

Ultrasound-Mediated Ocular Drug Delivery: From Physics and Instrumentation to Future Directions.

Drug delivery to the anterior and posterior segments of the eye is impeded by anatomical and physiological barriers. Increasingly, the bioeffects produced by ultrasound are being proven effective for mitigating the impact of these barriers on ocular drug delivery, though there does not appear to be a consensus on the most appropriate system configuration and operating parameters for this application. In this review, the fundamental aspects of ultrasound physics most pertinent to drug delivery are presented; the primary phenomena responsible for increased drug delivery efficacy under ultrasound sonication are discussed; an overview of common ocular drug administration routes and the associated ocular barriers is also given before reviewing the current state of the art of ultrasound-mediated ocular drug delivery and its potential future directions.

Extensive investigation of geometric effects in sonoreactors: Analysis by luminol mapping and comparison with numerical predictions.

This investigation focuses on the influence of geometric factors on cavitational activity within a 20kHz sonoreactor containing water. Three vessels with different shapes were used, and the transducer immersion depth and liquid height were varied, resulting in a total of 126 experiments conducted under constant driving current. For each one, the dissipated power was quantified using calorimetry, while luminol mapping was employed to identify the shape and location of cavitation zones. The raw images of blueish light emission were transformed into false colors and corrected to compensate for refraction by the water-glass and glass-air interfaces. Additionally, all configurations were simulated using a sonoreactor model that incorporates a nonlinear propagation of acoustic waves in cavitating liquids. A systematic visual comparison between luminol maps and color-plots displaying the computed bubble collapse temperature in bubbly regions was conducted. The calorimetric power exhibited a nearly constant yield of approximately 70% across all experiments, thus validating the transducer command strategy. However, the numerical predictions consistently overestimated the electrical and calorimetric powers by a factor of roughly 2, indicating an overestimation of dissipation in the cavitating liquid model. Geometric variations revealed non-monotonic relationships between transducer immersion depth and dissipated power, emphasizing the importance of geometric effects in sonoreactor. Complex features were revealed by luminol maps, exhibiting appearance, disappearance, and merging of different luminol zones. In certain parametric regions, the luminol bright regions are reminiscent of linear eigenmodes of the water/vessel system. In the complementary parametric space, these structures either combine with, or are obliterated by typical elongated axial structures. The latter were found to coincide with an increased calorimetric power, and are conjectured to result from a strong cavitation field beneath the transducer producing acoustic streaming. Similar methods were applied to an additional set of 57 experiments conducted under constant geometry but with varying current, and suggested that the transition to elongated structures occurs above some amplitude threshold. While the model partially reproduced some experimental observations, further refinement is required to accurately account for the intricate acoustic phenomena involved.

Memory-friendly fixed-point iteration method for nonlinear surface mode oscillations of acoustically driven bubbles: from the perspective of high-performance GPU programming.

A fixed-point iteration technique is presented to handle the implicit nature of the governing equations of nonlinear surface mode oscillations of acoustically excited microbubbles. The model is adopted from the theoretical work of Shaw [1], where the dynamics of the mean bubble radius and the surface modes are bi-directionally coupled via nonlinear terms. The model comprises a set of second-order ordinary differential equations. It extends the classic Keller-Miksis equation and the linearized dynamical equations for each surface mode. Only the implicit parts (containing the second derivatives) are reevaluated during the iteration process. The performance of the technique is tested at various parameter combinations. The majority of the test cases needs only a single reevaluation to achieve 10-9 error. Although the arithmetic operation count is higher than the Gauss elimination, due to its memory-friendly matrix-free nature, it is a viable alternative for high-performance GPU computations of massive parameter studies.