Bubble Cloud Dynamics Research Articles

Rayleigh–Plesset-based Eulerian mixture model for cavitating flows

The homogeneous mixture model (HMM) is widely in use for simulation of cavitating flows. The mass transfer is typically ruled by simplified models whose efficiency is strictly dependent on the empirical choice of vaporization/condensation constants. In the present paper, we formulate a physically based mass-transfer model relying on the solution of the complete Rayleigh–Plesset (RP) equation. The latter can model the elasticity of the bubbles and non-linear interaction with the external pressure field. The model is tested in different configurations, also considering comparisons with the Schnerr–Sauer model (SSm) and the linearized version of the RP equation. The preliminary simplified tests show that the SS model responds statically to pressure variations and thus in not able to reproduce the actual dynamics of cavitation, under certain circumstances. On the other hand, the linearized RP model (RPl), although dynamically responsive to pressure variations, produces unrealistic small-amplitude bubble fluctuations, whereas the complete RP model (RPc) gives more realistic results. Tests on the performance of the SSm and RP models were carried out considering the turbulent flow in a convergent–divergent Venturi channel, already tested in numerical and experimental reference research. Here, we use the incompressible HMM. The study highlights various crucial aspects of the RPc model, emphasizing its own ability in replicating the shedding cycle as a three-dimensional, and non-stationary phenomenon. On the other hand, the SSm model results as a valid approximation for initial growth stages but fails to capture complex dynamics during the collapse phase. The results are consistent with recent literature findings, and refinements in grid resolution enhance accuracy in capturing the non-stationary sheet-to-cloud vapor dynamics. Neglecting compressibility may account for disparities between numerical and experimental outcomes, especially concerning shock waves generation and propagation. The RPc model emerges as a good candidate in reproducing bubble cloud dynamics and, in the next future, can be implemented in compressible HMM.

Bubble cloud characteristics and ablation efficiency in dual-frequency intrinsic threshold histotripsy

Histotripsy is a non-thermal focused ultrasound ablation method that destroys tissue through the generation and activity of acoustic cavitation bubble clouds. Intrinsic threshold histotripsy uses single-cycle pulses to generate bubble clouds when the dominant negative pressure phase exceeds an intrinsic threshold of ∼25–30 MPa. The ablation efficiency is dependent upon the size and density of bubbles within the bubble cloud. This work investigates the effects of dual-frequency pulsing schemes on the bubble cloud behavior and ablation efficiency in intrinsic threshold histotripsy. A modular 500 kHz:3 MHz histotripsy transducer treated agarose phantoms using dual-frequency histotripsy pulses with a 1:1 pressure ratio from 500 kHz and 3 MHz frequency elements and varying arrival times for the 3 MHz pulse relative to the arrival of the 500 kHz pulse (−100 ns, 0 ns, and +100 ns). High-speed optical imaging captured cavitation effects to characterize bubble cloud and individual bubble dynamics. The effects of dual-frequency pulsing on lesion formation and ablation efficiency were also investigated in red blood cell (RBC) phantoms. Results showed that the single bubble and bubble cloud size for dual-frequency cases were intermediate to published results for the component single-frequencies of 500 kHz and 3 MHz. Additionally, bubble cloud size and dynamics were shown to be altered by the arrival time of the 3 MHz pulse with respect to the 500 kHz pulse, with more uniform cloud expansion and collapse observed for early (−100 ns) arrival. Finally, RBC phantom experiments showed that dual-frequency exposures were capable of generating precise lesions with smaller areas and higher ablation efficiencies than previously published results for 500 kHz or 3 MHz. Overall, results demonstrate dual-frequency histotripsy’s ability to modulate bubble cloud size and dynamics can be leveraged to produce precise lesions at higher ablation efficiencies than previously observed for single-frequency pulsing.

Breaking wave bubble measurements around ship model by optical probe

The bubble sweep-down phenomenon around oceanographic research vessels generates acoustic disturbances. The phenomenon can be reproduced on a 1/30 scale ship model in a wave and current circulating flume tank taking into account ship motions. Optical probe can then be used to measure locally the size and the velocity of the bubbles generated by breaking waves. A comparison of the obtained results with more classical results obtained from bubble detection by image tracking is discussed. The two methods provide complementary results for bubble characteristics. However, bubble detection by optical probe has the advantage of knowing the bubble position in the three dimensions of space. The overall results of this paper provide new elements for the study and the understanding of the bubble sweep-down phenomenon in addition to previous 2D and 3D studies, where the dynamics of bubble clouds have been characterized.

Cavitation dynamics of histotripsy bubble cloud nucleated from echogenic liposomes

Deep vein thrombosis is a serious health concern worldwide. Administration of catheter-directed thrombolytics is the primary approach to achieve vessel patency for critical obstructions. Adjuvant techniques that localize thrombolytic delivery will improve treatment outcomes and reduce the potential caustic bleeding associated with systemic lytic administration. Histotripsy is a bubble-based focused ultrasound therapy that has been shown to act synergistically with lytic to promote clot degradation. Combining histotripsy with a targeted drug delivery agent may further improve treatment efficacy. Echogenic liposomes are vesicles formulated to entrain the thrombolytic, and can be triggered with ultrasound to release drugs locally. The objective of this study was to assess histotripsy bubble cloud dynamics nucleated from echogenic liposomes in an in vitro venous flow model. The change in liposome echogenicity following histotripsy exposure was assessed with high frame rate imaging. Hypoechogenicity indicative of liposome destruction extended beyond the focal region, with a strong decrease in pixel intensity immediately after the applied histotripsy pulse. Interestingly, echogenicity was restored to baseline levels over the course of ∼4 s, which was attributed to vigorous fluid mixing from bubble cloud oscillations. These results indicate an interesting range of bubble cloud behaviors when histotripsy is combined with echogenic liposomes.

Cavitation Bubble Cloud Break-Off Mechanisms at Micro-Channels

This paper provides new physical insight into the coupling between flow dynamics and cavitation bubble cloud behaviour at conditions relevant to both cavitation inception and the more complex phenomenon of flow “choking” using a multiphase compressible framework. Understanding the cavitation bubble cloud process and the parameters that determine its break-off frequency is important for control of phenomena such as structure vibration and erosion. Initially, the role of the pressure waves in the flow development is investigated. We highlight the differences between “physical” and “artificial” numerical waves by comparing cases with different boundary and differencing schemes. We analyse in detail the prediction of the coupling of flow and cavitation dynamics in a micro-channel 20 μm high containing Diesel at pressure differences 7 MPa and 8.5 MPa, corresponding to cavitation inception and "choking" conditions respectively. The results have a very good agreement with experimental data and demonstrate that pressure wave dynamics, rather than the “re-entrant jet dynamics” suggested by previous studies, determine the characteristics of the bubble cloud dynamics under “choking” conditions.

Bubble cloud configuration effect on the added mass

Understanding the mechanisms that control the dynamics of bubble clouds is essential to many industrial processes in the energy and chemical realms. Due to the complexity of the two-phase flow configurations, modeling the physics of the phenomena driving the mixing of bubbles within a liquid matrix is still a major challenge. One of the weaknesses of most existing two-phase flow models is due to the incomplete handling of bubble dispersion. This difficulty comes from the fact that dispersion can be driven by numerous complex phenomena, such as turbulence, local pressure distribution, bubble to bubble interaction, etc. In this study, we introduce the effect of added mass fluctuations on the dispersion of small bubbles. Existing models based on the Euler–Euler approach do not take into account local flow variations due to bubble distributions. Therefore, these models do not correctly describe fine dispersion features. Solving the potential flow around N bubbles allows to take into account the effect of the added mass on bubble cloud distributions. To this aim, a complete added mass model, which includes local bubble configurations via the void fraction gradient, is developed. The void fraction gradient allows us to account for the asymmetry of the bubble cloud around a single central bubble. Consequently, the proposed model can only represent regular and irregular bubble cloud distributions. This methodology results in a more consistent consideration of the added mass effects as well as Meshchersky's force, which should be included in hydrodynamic two-phase flow models. The proposed approach can be implemented in Euler–Euler models intended to consider the dispersion of bubbles caused by the effect of added mass.

Numerical simulation of the interaction of wave phase conjugation with bubble clouds

The interaction of the Wave Phase Conjugation (WPC) process with bubble cloud dynamics is investigated via numerical modeling in application to bubble size measurements. The WPC simulator by Modarreszadeh and Timofeev (Computers and Fluids 197, 104353) is used as a starting point. In the simulator, a modified version of the high-order Nodal Discontinuous Galerkin method is used for modeling of acoustic wave propagation in the active WPC conjugator and the surrounding linear or non-linear fluid in 2D/axisymmetric domains. The effects of bubble dynamics are incorporated with the help of induced volume fractions. The modified Keller–Miksis model is employed to simulate vibrations of bubbles subjected to acoustic waves. After thorough verification and validation of the numerical scheme and the physical model, the interaction of various bubble clouds, including poly-disperse ones, with the WPC process is examined with the help of some existing and newly suggested descriptive indicators, e.g., the amplification factor and the absolute and relative signal intensities. The results show that signals scattered from bubble-plane clouds, in which the bubbles are non-physically synced, are strong enough to affect the modulation process if the stimulation frequency is close to the natural frequency of bubbles. For more realistic bubble clouds, the modulation process is almost intact. However, conjugate waves do carry some signatures of bubble-cloud dynamics, e.g., signals with frequencies higher than the natural frequency of bubbles are significantly diminished. The study demonstrates that WPC-based techniques have a promising potential to be used for measuring bubble dimensions.

Controlling the Dynamics of Cloud Cavitation Bubbles through Acoustic Feedback

Cloud cavitation causes nontrivial energy concentration and acoustic shielding in liquid and its control is a long-standing challenge due to complex dynamics of bubble clouds. We present a framework to study closed-loop control of cavitation through acoustic feedback. While previous approaches have used empirical thresholding, we employ model-based state estimation of coherent bubble dynamics based on theory and high-performance computing. Using a pulsed ultrasound setup, we demonstrate set-point control of the pulse-repetition frequency (PRF) to modulate acoustic cavitation near a solid target over $O(100)$ s. We identify a quasiequilibrium correlation between the PRF and the bubble dynamics and an optimal PRF to minimize acoustic shielding of the target. This framework can be readily scaled up by enhanced acoustic sensing and computational power.

On the thermodynamic behaviors and interactions between bubble pairs: A numerical approach

Thermodynamic behaviors and interactions between bubble pairs are important to better understand the cavitation phenomena. In this study, a compressible two-phase model, accounting for thermal effects to investigate the thermodynamic behaviors and interactions between bubble pairs, is developed in OpenFOAM. The volume of fluid (VOF) method is adopted to capture the interface. Validations are performed by comparing the simulation results of a single bubble and bubble pairs with corresponding experimental data. The dynamical behaviors of bubble pairs and their thermodynamic effect at different relative distances γ are investigated and discussed, which help reveal the bubble cloud dynamics. The quantitative analysis of γ effects on the maximum temperature during bubble collapse is performed with three distinct stages identified. For a single bubble collapsing near the rigid surface, the thermodynamic characteristics at different relative distances are similar to that of the bubble pairs, but the maximum temperature is higher since the single bubble can collapse to a smaller volume.

A simple model of bubble cluster dynamics in an acoustic field.

The dynamics of bubble clouds induced by ultrasound field are investigated in a regime where the cloud size is much smaller than the ultrasound wavelength. Two frequently used models describing the dynamics of individual bubbles inside a bubble cluster in an acoustic field are studied, one based on the homogeneity assumption, and the other based on the simultaneous motion assumption. A modified formula of the homogenization-based model is presented, and an inherent distinction in bubble-bubble interaction term is found in comparison to the simultaneous motion model. To gain insight into the mechanisms of such distinction, a reduced model unifying these two models is presented, and such distinction is explained by the spatial dependence of the bubble-bubble interaction in a bubble cluster accordingly. To validate the reduced model, the normalized distance γbb and the cloud interaction parameter B0 are used as two scaling parameters, and the comparison between the present model and the coupled Rayleigh-Plesset type equations is made. A conclusion is that, in the weak bubble-bubble interaction case (γbb>10), the reduced model can well reproduce the radial motion of bubbles in the cluster during the growth stage and the collapse stage in each acoustic cycle; in the strong bubble-bubble interaction case (γbb<10), the growth phase of bubbles in the cluster can be accurately predicted by the reduced model only if B0 or the amplitude of driving field is small.

In vitro assessment of stiffness-dependent histotripsy bubble cloud activity in gel phantoms and blood clots

As a bubble-based ablative therapy, the efficacy of histotripsy has been demonstrated in healthy or acutely diseased models. Chronic conditions associated with stiff tissues may require additional bubble activity prior to histotripsy liquefaction. In this study, histotripsy pulses were generated in agarose phantoms of Young’s moduli ranging from 12.3 to 142 kPa, and in vitro clot models with mild and strong platelet-activated retraction. Bubble cloud emissions were tracked with passive cavitation imaging, and the threshold acoustic power associated with phantom liquefaction was extracted with receiver operator characteristic analysis. The power of histotripsy-generated emissions and the degree of liquefaction were tabulated for both clot models. For the agarose phantoms, the acoustic power associated with liquefaction increased with Young’s modulus. When grouped based on agarose concentration, only two arms displayed a significant difference in the liquefaction threshold acoustic power (22.1 kPa versus 142 kPa Young’s modulus). The bubble cloud dynamics tracked with passive cavitation imaging indicated no strong changes in the bubble dynamics based on the phantom stiffness. For identical histotripsy exposure, the power of acoustic emissions and degree of clot lysis did not vary based on the clot model. Overall, these results indicate that a fixed threshold acoustic power mapped with passive cavitation imaging can be utilized for predicting histotripsy liquefaction over a wide range of tissue stiffness.

Bubble cloud dynamics in an ultrasound field.

The dynamics of bubble clouds induced by high-intensity focused ultrasound are investigated in a regime where the cloud size is similar to the ultrasound wavelength. High-speed images show that the cloud is asymmetrical; the bubbles nearest the source grow to a larger radius than the distal ones. Similar structures of bubble clouds are observed in numerical simulations that mimic the laboratory experiment. To elucidate the structure, a parametric study is conducted for plane ultrasound waves with various amplitudes and diffuse clouds with different initial void fractions. Based on an analysis of the kinetic energy of liquid induced by bubble oscillations, a new scaling parameter is introduced to characterize the dynamics. The new parameter generalizes the cloud interaction parameter originally introduced by d'Agostino & Brennen (1989). The dynamic interaction parameter controls the energy localization and consequent anisotropy of the cloud. Moreover, the amplitude of the far-field, bubble-scattered acoustics is likewise correlated with the proposed parameter. Findings of the present study not only shed light on the physics of cloud cavitation, but may also be of use to quantification of the effects of cavitation on outcomes of ultrasound therapies including HIFU-based lithotripsy.

Modeling and numerical simulation of the bubble cloud dynamics in an ultrasound field for burst wave lithotripsy

Modeling and numerical simulation of bubble clouds induced by intense ultrasound waves are conducted to quantify the effect of cloud cavitation on burst wave lithotripsy, a proposed non-invasive alternative to shock wave lithotripsy that uses pulses of ultrasound with an amplitude of O(1) MPa and a frequency of O(100) kHz. A unidirectional acoustic source model and an Eulerian-Lagrangian method are developed for simulation of ultrasound generation from a multi-element array transducer and cavitation bubbles, respectively. Parametric simulations of the spherical bubble cloud dynamics reveal a new scaling parameter that dictates both the structure of the bubble cloud and the amplitude of the far-field, bubble-scattered acoustics. The simulation further shows that a thin layer of bubble clouds nucleated near a kidney stone model can shield up to 90% of the incoming wave energy, indicating a potential loss of efficacy during the treatment due to cavitation. Strong correlations are identified between the far-field, bubble-scattered acoustics and the magnitude of the shielding, which could be used for ultrasound monitoring of cavitation during treatments. The simulations are validated by companion experiments in vitro. [The work was supported by NIH P01-DK043881 and K01-DK104854.]

Modeling and numerical simulation of the bubble cloud dynamics in an ultrasound field for burst wave lithotripsy.

Modeling and numerical simulation of bubble clouds induced by intense ultrasound waves are conducted to quantify the effect of cloud cavitation on burst wave lithotripsy, a proposed non-invasive alternative to shock wave lithotripsy that uses pulses of ultrasound with an amplitude of O(1) MPa and a frequency of O(100) kHz. A unidirectional acoustic source model and an Eulerian-Lagrangian method are developed for simulation of ultrasound generation from a multi-element array transducer and cavitation bubbles, respectively. Parametric simulations of the spherical bubble cloud dynamics reveal a new scaling parameter that dictates both the structure of the bubble cloud and the amplitude of the far-field, bubble-scattered acoustics. The simulation further shows that a thin layer of bubble clouds nucleated near a kidney stone model can shield up to 90% of the incoming wave energy, indicating a potential loss of efficacy during the treatment due to cavitation. Strong correlations are identified between the far-field, bubble-scattered acoustics and the magnitude of the shielding, which could be used for ultrasound monitoring of cavitation during treatments. The simulations are validated by companion experiments in vitro.

Experimental study of the bubble sweep-down phenomenon on three bow designs

The bubble sweep-down phenomenon around the oceanographic research vessels generates acoustic perturbations. A specific experimental protocol has been developed in a wave and circulating tank to study this phenomenon. This protocol is used to carry out trials on three different ship models in order to study the influence of the bow geometry on the bubble generation. For different test configurations, bubble clouds are described and compared in terms of area, maximal depth and vertical velocity to highlight bubble cloud dynamics surrounding the three ship models. The relation between the hydrodynamic flow field and the bubble generation is studied by means of Particle Image Velocimetry (PIV) measurements to study the phenomenon by the use of phase averaged velocity fields. The overall results enable us to characterize the bubble sweep-down phenomenon from the air bubble generation and propagation to the frequency of occurrence and the clouds behaviour.

Numerical study of acoustically driven bubble cloud dynamics near a rigid wall

The dynamics of a bubble cloud excited by a sinusoidal pressure field near a rigid wall is studied using a novel Eulerian/Lagrangian two-phase flow model. The effects of key parameters such as the amplitude and frequency of the excitation pressure, the cloud and bubble sizes, the void fraction, and the initial standoff distance on the bubbles’ collective behavior and the resulting pressure loads on the nearby wall are investigated. The study shows that nonlinear bubble cloud dynamics becomes more pronounced and results in higher pressure loading at the wall as the excitation pressure amplitude increases. The strongest collective bubble behavior occurs at a preferred resonance frequency. At this resonance frequency, pressure peaks orders of magnitudes higher than the excitation pressure result from the bubble interaction when the amplitude of the pressure excitation is high. The numerically obtained resonance frequency is significantly different from the reported natural frequency of a spherical cloud derived from linear theory, which assumes small amplitude oscillations in an unbounded medium. At high amplitudes of the excitation, the resonance frequency decreases almost linearly with the ratio of excitation pressure amplitude to ambient pressure until the ratio is larger than one.

Cavitation-enhanced diagnostics and therapy in HIFU treatment

Medical applications of ultrasound such as High Intensity Focused Ultrasound, Extracorporeal Ultrasound Lithotripsy and Ultrasound-Mediated Gene Transfection have recently been the subject of much interest. In these applications, acoustic cavitation facilitates the medical treatment by improving the image quality and enhancing the therapeutic effects through acoustic emission, localized heating, and erosion. During the ultrasound therapy, it is important to control the focal region on the targeting area, which is moving by the respiratory motion, for the reliable treatment. Ultrasound guide is one of the effective way to track the targeting area compensating the respiratory motion. Cavitation-enhanced diagnostics is useful to obtain the clear image of targeting area utilizing the acoustic emission from the existing micro bubbles during the cavitation-enhanced HIFU where cavitation bubbles are generated at the targeting area to enhance tissue heating. In all these applications, it is essential to understand the cavitation phenomena, including bubble and bubble cloud dynamics. The bubble motion and bubble cloud behavior are strongly influenced by the thermal phenomena inside and outside bubbles and the applied ultrasound wave forms which accelerate therapeutic effects by the cavitation bubbles. In this paper, some fundamental aspects of medical application of microbubbles and acoustic cavitation are highlighted.

Modeling and experimental analysis of acoustic cavitation bubble clouds for burst-wave lithotripsy

Understanding the dynamics of cavitation bubble clouds formed inside a human body is critical for the design of burst-wave lithotripsy (BWL), a newly proposed method that uses focused ultrasound pulses with amplitude of O(10) MPa and frequency of O(0.1) MHz to fragment kidney stones. We present modeling and three-dimensional direct numerical simulations of interactions between bubble clouds and ultrasound pulses in water. We study two configurations: isolated clouds in a free field, and clouds near a rigid surface. In the modeling, we solve for the bubble radius evolution and continuous flow field using a WENO-based compressible flow solver. In the solver, Lagrangian bubbles are coupled with the continuous phase, defined on an Eulerian grid, at the sub-grid scale using volume averaging techniques. Correlations between the initial void fraction and the maximum collapse pressure in the cloud are discussed. We demonstrate acoustic imaging of the bubbles by post-processing simulated pressure signals at particular sensor locations indicating waves scattered by the clouds. Finally, we compare the simulation results with experimental results including high-speed imaging and hydrophone measurements. The time evolution of the cloud void fraction and the scattered acoustic field in the simulation agree with the experimental results. [Funding supported by NIH 2P01-DK043881.]

An Eulerian-Lagrangian study of cloud dynamics near a wall

The dynamics of bubble clouds is studied using an Eulerian-Lagrangian model treating the two-phase medium as a continuum and modeling the microbubbles as discrete sources and dipoles tracked in a Lagrangian fashion. These two are coupled through the local void fractions associated with the instantaneous bubble volumes and locations. Resonance of the bubble cloud, resulting in the highest pressure at the nearby rigid wall, is deduced from simulations with the same initial bubble distribution while varying the excitation frequency and amplitude. This resonance frequency deviates significantly from the classical linearized solution as the relative amplitude of driving pressure increases. It gradually drops as the excitation amplitude increases until reaching a limit value. In the high amplitude driving pressure regime, the peak pressure generated at the wall has a maximum for an optimum value of the initial bubble radius for the same cloud radius. This occurs when the ratio of maximum over initial bubble size is maximum. Too weak or too strong bubble interactions in the cloud inhibit strong collective effects. For the same initial void fraction and for different initial bubble sizes (changing the bubble number), an optimum bubble size also exists, resulting in the highest collective pressure.

Focused Ultrasound and Lithotripsy.

Shock wave lithotripsy has generally been a first choice for kidney stone removal. The shock wave lithotripter uses an order of microsecond pulse durations and up to a 100 MPa pressure spike triggered at approximately 0.5-2 Hz to fragment kidney stones through mechanical mechanisms. One important mechanism is cavitation. We proposed an alternative type of lithotripsy method that maximizes cavitation activity to disintegrate kidney stones using high-intensity focused ultrasound (HIFU). Here we outline the method according to the previously published literature (Matsumoto et al., Dynamics of bubble cloud in focused ultrasound. Proceedings of the second international symposium on therapeutic ultrasound, pp 290-299, 2002; Ikeda et al., Ultrasound Med Biol 32:1383-1397, 2006; Yoshizawa et al., Med Biol Eng Comput 47:851-860, 2009; Koizumi et al., A control framework for the non-invasive ultrasound the ragnostic system. Proceedings of 2009 IEEE/RSJ International Conference on Intelligent Robotics and Systems (IROS), pp 4511-4516, 2009; Koizumi et al., IEEE Trans Robot 25:522-538, 2009). Cavitation activity is highly unpredictable; thus, a precise control system is needed. The proposed method comprises three steps of control in kidney stone treatment. The first step is control of localized high pressure fluctuation on the stone. The second step is monitoring of cavitation activity and giving feedback on the optimized ultrasound conditions. The third step is stone tracking and precise ultrasound focusing on the stone. For the high pressure control we designed a two-frequency wave (cavitation control (C-C) waveform); a high frequency ultrasound pulse (1-4 MHz) to create a cavitation cloud, and a low frequency trailing pulse (0.5 MHz) following the high frequency pulse to force the cloud into collapse. High speed photography showed cavitation collapse on a kidney stone and shock wave emission from the cloud. We also conducted in-vitro erosion tests of model and natural kidney stones. For the model stones, the erosion rate of the C-C waveform showed a distinct advantage with the combined high and low frequency waves over either wave alone. For optimization of the high frequency ultrasound intensity, we investigated the relationship between subharmonic emission from cavitation bubbles and stone erosion volume. For stone tracking we have also developed a non-invasive ultrasound theragnostic system (NIUTS) that compensates for kidney motion. Natural stones were eroded and most of the resulting fragments were less than 1 mm in diameter. The small fragments were small enough to pass through the urethra. The results demonstrate that, with the precise control of cavitation activity, focused ultrasound has the potential to be used to develop a less invasive and more controllable lithotripsy system.