Measurements of bubble-enhanced heating from focused, mhz-frequency ultrasound in a tissue-mimicking material

Ultrasound in Med. & Biol., Vol. 27, No. 10, pp. 1399 –1412, 2001 Copyright © 2001 World Federation for Ultrasound in Medicine & Biology Printed in the USA. All rights reserved 0301-5629/01/$–see front matter

PII: S0301-5629(01)00438-0

● Original Contribution MEASUREMENTS OF BUBBLE-ENHANCED HEATING FROM FOCUSED, MHz-FREQUENCY ULTRASOUND IN A TISSUE-MIMICKING MATERIAL R. GLYNN HOLT and RONALD A. ROY Department of Aerospace and Mechanical Engineering, Boston University, 110 Cummington Street, Boston, MA, USA (Received 27 March 2000; in final form 2 July 2001)

Abstract—Time-resolved measurements of the temperature field in an agar-based tissue-mimicking phantom insonated with a large aperture 1-MHz focused acoustic transducer are reported. The acoustic pressure amplitude and insonation duration were varied. Above a critical threshold acoustic pressure, a large increase in the temperature rise during insonation was observed. Evidence for the hypothesis that cavitation bubble activity in the focal zone is the cause of enhanced heating is presented and discussed. Mechanisms for bubble-assisted heating are presented and modeled, and quantitative estimates for the thermal power generated by viscous dissipation and bubble acoustic radiation are given. (E-mail: [email protected]) © 2001 World Federation for Ultrasound in Medicine & Biology. Key Words: Acoustic, Ultrasound, Hyperthermia, Cavitation, Enhanced heating, Bioeffects, Bubble, Ultrasound contrast agents.

2000), and therapeutic situations in which it is desirable to excite cavitation. For example, it is clear from experimental work (Coleman et al. 1996) in shock-wave lithotripsy that bubbles are unavoidable in the presence of such intense negative pressures. Because the end goal is mechanical destruction of the kidney stone, the muchstudied liquid jet formed by the asymmetric collapse of a bubble near a boundary would seem like a viable candidate for a destruction mechanism. Several research efforts have addressed the role of cavitation in stone comminution (Church 1989; Ding and Gracewski 1994; Bailey 1997; Evan et al. 1998). The same considerations apply in the field of ultrasonic surgery using high-intensity focused US (HIFU), where the goal is to produce irreversible necrosis deep into the tissue with minimal damage in the intervening path (as reviewed in ter Haar 1995, and highlighted in the special issue of IEEE Trans UFFC, Ebbini 1996). The fact that bubbles enhance US imaging via their high backscatter cross-section makes them useful in targeting applications. If gross mechanical damage and tissue ablation is desired, then excitation of cavitation bubbles is the most efficacious (if also unavoidable) method at high insonation pressures and long insonation durations (for example, Fry et al. 1950, 1970, 1996; Lehmann and Herrick 1953; ter Haar et al. 1982; Tavakkoli et al. 1997; Arefiev et al. 1998, Smith and Hynynen 1998).

INTRODUCTION AND LITERATURE Acoustic cavitation (the acoustically induced nucleation and subsequent oscillatory activity of bubbles) is often the result of the application of ultrasound (US) in a biomedical context. There are sound reasons, both experimental and theoretical, for defining therapeutic and diagnostic operating conditions to minimize bubble activity (Williams et al. 1991; Carstensen et al. 1993; Penney et al. 1993; Everbach et al. 1997). Most of the arguments boil down to the fact that bubbles perform a very efficient and highly nonlinear conversion of acoustical energy to mechanical motion. In diagnostic US, the mechanical index (MI) was conceived and implemented because of the perceived risk of mechanical bioeffects from the violent inertial collapses of bubbles (Apfel and Holland 1991). In therapeutic US, bubbles formed in the propagation path inhibit deposition of acoustic energy because they scatter and absorb much of the energy before it can reach the target area. Conversely, there are often diagnostic situations in which the use of bubble-based contrast agents make cavitation activity unavoidable (Fowlkes and Holland

Address correspondence to: R. G. Holt, Department of Aerospace and Mechanical Engineering, Boston University, 110 Cummington Street, Boston, MA 02215 USA. E-mail: [email protected] 1399

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However, if controlled, highly localized hyperthermia is the desired outcome, then most investigators have argued that cavitation is to be avoided because it has typically led to unpredictable thermal results. Several authors observed irregular lesions and collateral damage outside the focal zone when uncontrolled cavitation occurred during insonation (Fry et al. 1970; Lele 1987; Hynynen 1991; Chapelon et al. 1991, 1996, 2000; Sibille et al. 1993; Clarke and ter Haar 1997). These studies all contain unambiguous statements recommending actively avoiding cavitation when the therapeutic goal is controlled localized heating. In the preceding list, three (Lele 1987; Hynynen 1991; Clarke and ter Haar 1997) observed cavitationrelated enhanced heating (the heating rate measured via single thermocouple during insonation increased dramatically above some threshold insonation pressure), with (in the Hynynen study) concomitant dramatic increase in measured acoustic emission. The observation of enhanced heating due to bubble activity deserves closer attention. Lele (1987) (and references therein) simultaneously measured temperature rise and acoustic emission using instrumented calf liver in vitro insonated at 2.7 MHz with 0.2 and 0.3 s CW tone bursts. They observed bubble-enhanced heating (directly from single thermocouple measurements, although these were often reported as an increased effective absorption), and either screening or boiling when cavitation (inferred from subharmonic and anharmonic, i.e., broadband, acoustic emissions) was present. There is some difficulty interpreting their results because, for all insonation intensities greater than ISPTA ⫽ 1 ⫻ 107 W/m2, they increased the ambient pressure to 22 atm (⬃2.2 MPa) to suppress boiling. It is clear from the text that they did not consider the effect that increasing the ambient pressure would have on the bubble activity. Nevertheless it is also clear from their results that they did not entirely suppress cavitation or its thermal effects. They obtained a threshold intensity for enhanced heat generation of ISPTA ⫽ 1.5 ⫻ 107 W/m2 at 2.7 MHz, corresponding to a peak threshold pressure of roughly 7.5 MPa. Hynynen (1991) observed an enhanced heating effect in vivo using a thermocouple embedded in dog thigh muscle that was insonated at frequencies ranging from 0.75 to 5 MHz using 1-s CW tone bursts and a range of pressures. A pressure threshold for enhanced heating of roughly 5.3 MPa at 1 MHz was observed. This was followed by a saturation region at higher pressures, with temperature vs. time results deviating markedly from linear theory. Regions of enhanced echogenicity in US images of the focal region were observed. The onset of enhanced temperatures and echogenicity were correlated with the onset of subharmonic noise detected by a pas-

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sive hydrophone. Hynynen concluded that the enhanced temperatures were the result of acoustically induced cavitation in the focal region. He also concluded that such effects should, perhaps, be avoided in clinical therapy because they would lead to unpredictable thermal and mechanical damage. Watmough et al. (1993) observed enhanced heating in the presence of bubbles artificially introduced via a porous solid inclusion in a gel that was insonated by a 0.75-MHz plane piston transducer operating at ISPTA ⫽ 1 ⫻ 104 W/m2. They noted peak temperatures at the end of a 1-s CW tone burst to be roughly 6 times higher than when they performed the same experiment with a degassed inclusion. Although their pressures were not high enough to induce cavitation, they hypothesized that nearresonant-sized bubbles were introduced by the inclusion and were responsible for the enhanced heating. Clarke and ter Haar (1997) observed the enhanced heating effect ex vivo in dog and cow liver tissue using 1-MHz, 10-s CW tone bursts with varying pressures. They also observed a drop in temperature on occasion at high insonation intensities. They argued that cavitation and/or boiling could explain both types of result. In a study of the effects of cavitation on cell lysis, Daniels et al. (1995) observed a temperature rise in a large reservoir only during 0.75-MHz CW insonations of sufficient amplitude to induce continuous cavitation activity as measured by acoustic emissions. Miller and Gies (1998) observed indirect evidence for bubble-enhanced heating in a study of 400-kHz insonated mouse intestine, in which parametric studies of petechiae (which the authors took as a surrogate for thermal damage) and hemorrhages (surrogate for cavitation damage) were conducted. They found that, above a threshold pressure value, the number of petechiae was greater for CW than for pulsed insonation for the same temporal average intensities. However, they concluded that heating and cavitation “...appeared to have largely independent roles in vascular bioeffects....” Sanghvi et al. (1996) compiled dual-mode acoustical evidence (imaging and noise emission) of in vitro cavitation during clinical treatment of benign prostate hyperplasia using US. They reported successful treatment of the hyperplasia via lesion formation and ablation of the insonated tissue, and also observed enhanced heating when noise from bubble activity (termed “popcorn”) was present. At times, the US images indicated temporally evolving “cloud-like” regions of echogenicity that started at the focus and gradually migrated toward the therapy transducer. They were unable to effectively propagate the US beyond the cloud, a clear indication of bubble activity. Fujishiro et al. (1998) reported a doubling of the temperature elevation when beef was injected with the

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contrast agent Albunex威 (concentration unreported) for a 1.5-MHz, 9 ⫻ 103 W/m2 (SPTA), 3-min CW insonation measured by an inserted thermocouple. For their experiment, the equivalent temperature elevation was achieved without contrast agent by increasing the acoustic intensity to 1.8 ⫻ 104 W/m2. Wu (1998) modeled the temperature rise from US insonation of a suspension of contrast agents by treating the contrast agent as a effective medium with an attenuation coefficient of 40 dB/cm at 1 MHz. He only considers diagnostic insonations, but the temperature rise is, nevertheless, on the order of a few degrees for a 1-s insonation where ISPTA ⬃ 3.5 ⫻ 104 W/m2. Thus, we see that a bubble-mediated heating effect has been demonstrated in the literature in vivo. It is one aim of the current work to provide a basis for using bubbles to enhance controlled localized heating. It is precisely the fact that bubbles are so efficient at transducing acoustic energy into mechanical energy that motivates the present discussion. The best available largeaperture focused transducers exploit geometric gain to achieve energy concentrations on the order of 103 times greater than the unfocused plane wave acoustic energy density; see, for example, Rivens et al. (1996). A sufficiently energetic bubble oscillation can produce an order of 1011 gain in instantaneous acoustic energy density (Barber and Putterman 1991), and that is in addition to any focusing gain from the transducer because the bubble senses the local acoustic field. In this fashion, bubbles can cause temporal and spatial energy transduction and focusing with efficiency unparalleled by any manufactured device. If such energy focusing could be harnessed specifically for controlled hyperthermia, then therapeutic ultrasonically induced hyperthermia could be achieved with much shorter treatment durations and correspondingly reduced time-averaged intensities, albeit at the price of increased peak pressures and some focal mechanical damage. The advantages of such an approach are straightforward. Small and well-controlled heating zones and lesions are achievable at reduced input energies, minimizing secondary scattering and absorption outside the desired target region. There will also be a reduction in unwanted heating in the propagation path, with resultant media changes in the acoustic path minimized; see Hynynen and Jolesz (1998) for a proposed example application. This will again yield more control, and also allow better comparison with theoretical modeling of nonlinear acoustic propagation. We have undertaken a series of laboratory experiments on tissue-mimicking phantoms insonated by 1-MHz focused pulsed US at MPa pressure amplitudes. The need for independently and simultaneously measured pressures and temperatures (as opposed to using a

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Fig. 1. Schematic of the experimental setup.

derating process to infer the pressure), as well as the ease of instrumentation afforded by cast test samples, motivated the use of a phantom rather than animal tissue. Equally important in our choice of phantoms was the desire to quantitatively model (albeit with Newtonian assumptions) both the acoustic absorption and the bubble dynamics to compare to our experiments without employing adjustable parameters. Our phantoms allowed us to independently measure both acoustic and thermal material properties to facilitate the partial achievement of this aim. MATERIALS AND METHODS Apparatus A side cut-away view of the apparatus is shown in Fig. 1. A single-element focused bowl piezoceramic transducer (f ⫽ 7.0 cm, aperture ⫽ 7.0 cm, Sonic Concepts, Seattle, WA) operating at 1 MHz was used for all experiments. The input from a function synthesizer (33120A, Hewlett-Packard Corp., Palo Alto, CA) was attenuated using a variable attenuator, amplified (A-500, Electronic Navigation Industries, Rochester, NY) at a fixed gain of 60 dB, then coupled to the transducer via an

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impedance matching network. The transducer was mounted in a recirculating water tank containing filtered, deionized and degassed water. The transducer position was controlled in 3-D via computer-driven stepper motors coupled to fine-pitch (5 mm/rev) gears; positioncontrol resolution was better than 100 ␮m (Isel Automation, Eiterfeld, Germany). Depending on the experiment, either a PVA-based or an agar-based tissue-mimicking material phantom was cast using a cylindrical form and mounted in a fixed location in the tank. The axially symmetric phantoms were instrumented with an array of eight embedded thermocouples (0.13 mm, type-E, bare wire, Omega Engineering, Stamford, CT) aligned along a radial line normal to the acoustic axis and in the focal plane, with the closest thermocouple positioned in the periphery of the acoustic focus. The uncertainty in the position of a given thermocouple bead was ⫾ 0.5 mm in any direction. For each run, the nearest thermocouple was placed no closer than 0.5 mm from the acoustic axis, determined experimentally each run. This ensured that the nearest thermocouple was near (see acoustic beam data below) the first radial minimum in the acoustic field intensity. The thermocouples were themselves oriented parallel to the acoustic axis. The thermocouples were terminated in an isothermal terminal block, and electronically compensated. Thermocouple voltages were fed through a bank of low pass filters (⬍ 10 kHz), amplified (60-dB gain) and sampled at a rate of 20 sample/s (SCXI-1000 w/1120 module, PCI-MIO-16-E1, National Instruments, Austin, TX). The uncertainty in the measured temperature was approximately ⫾ 0.1°C, and the rise-time of the combined thermocouple/signal-conditioning subsystem was less than 40 ms. A calibrated PZT needle hydrophone (DAPCO Industries, Branford, CT) was embedded in the phantom for pressure measurements and position calibration. The hydrophone voltage was sampled with a bandwidth of 100 MHz using a digital oscilloscope (9304 AM, 8-bit resolution, Lecroy Corp., Chestnut Ridge, NY). The sensitivity of the hydrophone (see below) was 0.54 ⫾ 0.03 V/MPa. For all runs, the outgoing acoustic wave propagated through 2.0 ⫾ 0.1 cm of degassed water (measured from the front face of the transducer), followed by 3.1 ⫾ 0.1 cm of the phantom until reaching the focus center along the acoustic axis in the thermocouple plane. The focal spot size was 1.5 mm in the radial direction and 9.5 mm in the axial direction, defined as full-width half-maximum points of the intensity profile measured in degassed water with a spatial-peak temporal-average peak positive pressure of approximately 0.1 MPa. For the experiments reported here, the phantom was exposed to pulses of CW US lasting from 0.25 to 10 s,

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and time histories of the thermocouple array readings were recorded as a function of acoustic pressure. At each pressure setting, five acoustic pulses were applied, with the phantom allowed to cool for 100 s in between pulses. The temporal-peak temperature for each thermocouple was extracted, and averaged over the five pulses. The spatial-peak temporal-average peak positive pressure for these experiments ranged from 0.05 MPa to nearly 4 MPa. Preparation and calibration Phantom physical properties. Two tissue-mimicking phantom mixtures were used. The first, which we refer to as “PVA,” consists of a mixture of 600 mL of water, 750 mg of methyl paraben as a preservative (methyl 4-hydroxybenzoate, Aldrich, Milwaukee, WI), 20 g of polyvinyl alcohol (PVA powder, 99 to 99.8% fully hydrolized, JT Baker, Phillipsburg, NJ), 65 g of graphite powder as a scatterer (325 mesh, JT Baker), and 48 mL of 1-propanol for adjusting the sound speed (lab-grade reagent, JT Baker). The second, which we refer to as “agar,” consists of 600 mL of water, 750 mg of methyl paraben, 18 g of agar (powder, EM Science, Gibbstown, NJ), 65 g of graphite powder, and 48 mL of 1-propanol. Both were mixed by heating to near boiling, and allowed to cool (in addition, the PVA phantoms had to be frozen to gel properly). In one case study, we degassed the hot liquid agar mix in a mild vacuum for 20 min before allowing it to solidify. In all cases, the hot mixes were poured into PVC plastic molds that supported the thermocouple array and needle hydrophone. By casting the sensors directly into the phantom, we minimized problems associated with postsolidification insertion of the probes. The instrumented phantoms thus obtained were cylindrical, with a typical diameter and length of 10 cm and 6 cm, respectively. The precise dimensions varied slightly, depending on the mold employed for fabrication. For both phantom mixes, we performed routine material characterization measurements to independently obtain relevant thermal and acoustic properties. The results are summarized in Table 1. Some comments on the material parameters are in order. The attenuation values were obtained via an insertion loss technique (Parker 1983), and are corrected for impedance loss (less than 0.1 dB in all cases) at the water/sample interface. However, no attempt has been made to differentiate the relative contributions of absorption and scattering. All losses are assumed to be due to absorption alone; thus, the values in Table 1 represent an upper bound to the true material absorption coefficient. The thermal conductivity of the agar phantom is similar to that of water and, thus, of most soft tissue. The attenuation values for the agar

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Table 1. The measured material properties of the tissue-mimicking phantom mixtures Measured material properties (customary units)

Phantom PVA Agar Water

Density ␳ (g/mL)

Sound speed c (cm/s ⫻ 105)

Attenuation, 1 MHz ␣ (Np/m/MHz)

Thermal conductivity ␬ (W/cm °C)

Specific heat Cv (J/g °C)

1.10 ⫾ 0.05 1.10 ⫾ 0.05 1.00

1.50 ⫾ 0.025 1.60 ⫾ 0.025 1.50

7.3 ⫾ 0.6 18.4 ⫾ 1.2 0.025

– 0.0065 ⫾ 0.001 0.006

– 3.3 ⫾ 0.5 4.18

All values are for a room temperature of roughly 21°C. The water values are accepted nominal values for room temperature water and are included for comparison.

phantoms used are slightly higher than those of most soft tissues at 1 MHz (Goss et al. 1978, 1980). Although this fact will not qualitatively affect the main result of bubble-enhanced heating in the present work, it will cause our peak temperatures to be larger than, and our heated volume (if we were to calculate a necrotic region based on the equivalent thermal dose guideline) to be slightly smaller than what we would have measured in a medium of lower thermoviscous absorption for the same duration of insonation. In situ acoustic pressure. The sound source was calibrated in water using the substitution method. A calibrated (NPL-traceable) bilaminar PVDF membrane hydrophone (Marconi, Chelmsford, UK) served as the reference standard. The source transmitting response was found to be linear for spatial-peak temporal-peak positive pressures up to 2 MPa. (Note: 2 MPa is not the limit of linearity; we terminated our calibration to avoid damage to the Marconi probe.) The 10.25-cm long DAPCO needle hydrophone, which possessed a 1.0-mm diameter active element, was also calibrated via substitution. Its response was also accurately linear, and its measured sensitivity was 0.54 ⫾ 0.03 V/MPa. When the needle hydrophone was embedded in the phantom, short pulse measurements of the pressure waveform in the phantom were obtained for up to 3 MPa spatial-peak temporal-peak positive pressure. The ability to measure the pressure field inside the phantom proved valuable for several reasons. Because the locations of the thermocouples were known relative to the needle hydrophone, one could use the hydrophone to align the array relative to the acoustic focus. In addition, the needle hydrophone calibration admitted direct in situ monitoring of the incident pressure field, thereby eliminating uncertainties associated with the traditional derating procedure. The in situ calibration and thermocouple alignment procedure was repeated before each experiment, and run variations (in both pressure in positioning) were less than 10%.

MODELING I: PRIMARY ACOUSTIC HEATING It will prove useful to compare our results with a well-established analytical model for tissue heating and cooling due to a focused, monochromatic, Gaussian acoustic beam (we refer to this scenario as the primary acoustic heating). Assuming the total thermal energy deposited is proportional to the time-averaged, peak acoustic intensity, and that heat conduction occurs solely via diffusion in a homogeneous medium, the following simplified expressions are obtained for the time-dependence of the temperature T for a thermocouple in the insonatedd region (Parker 1983; Nyborg 1988; Clarke and ter Haar 1997): during insonation T共r, t兲 ⫽

冉

2 ␣ I 0 ⫺r2/a2 a 2 4␬t e ln 1 ⫹ 2 ␳Cv 4␬ a

冊

(1)

following insonation T共t兲 ⫽

冉

T peak , 4␬t 1⫹ 2 a

冊

(2)

where ␣ is the amplitude absorption coefficient, I0 is the spatial-peak temporal-average acoustic intensity, a ⫽ 0.95 mm is the effective Gaussian radius of the radial spatial intensity profile obtained from fitting the measured profile, ␬ ⫽ k/␳Cv is the phantom thermal diffusivity, ␳ is the measured phantom density, Cv is the measured phantom specific heat, and Tpeak is the temporal-peak temperature reached before cooling at the location of the thermocouple. Table 1 gives the measured physical values used in these equations for comparison. It follows from eqn (1) that the maximum temperature Tpeak depends linearly on the intensity and, thus, is proportional to P2, where P is the spatial-peak temporalaverage peak positive pressure.

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Fig. 2. Temperature vs. time for the thermocouple array in the agar phantom resulting from a 1.2-s CW tone-burst insonation at 1.36-MPa peak positive pressure and 1-MHz frequency. Distances given for each thermocouple (labeled TCn) are relative to the acoustic axis. The theoretical curve was generated using eqns (1)–(5), and the measured physical property values listed in Table 1.

RESULTS Temperature vs. time Figure 2 plots the temperature vs. time for the thermocouple array embedded in the agar phantom resulting from a 1.2-s exposure at 1.36 MPa and 1 MHz. (Unless otherwise noted, all acoustic pressures cited in this study correspond to spatial-peak, temporal-average, peak positive pressures.) The indicated distances for each thermocouple are relative to the acoustic axis. Note that the heating rate nearest the focus (TCA, 0.5 mm) is still large up to the time the acoustic field is turned off, illustrating the fact that such short (with respect to thermal relaxation times) insonations do not result in steadystate temperature fields. At this modest pressure, the temperature increase is only about 3°C for the thermocouple nearest the focus. Despite the inherent idealizations (such as the assumption that the measured attenuation is due solely to absorption), the linear analytic theory provides a good estimate of the temperature rise. Figure 3 depicts the temperature for the same phantom and exposure duration, but for a peak positive pressure of 1.62 MPa. In this case, the temperature rise was 15°C (a 400% increase) in the nearest thermocouple, even though the pressure squared increased by only 40%. The time history of the thermocouple nearest the focus deviates dramatically from the theoretical heating curve. We repeated these experiments for exposure durations of 0.35, 0.7, 1.2 and 10.2 s and, in each case, we observed that, above some threshold insonation pressure, the heat-

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Fig. 3. Temperature vs. time for the conditions of Fig. 2, with Pa ⫽ 1.62 MPa.

ing rates increased dramatically from the theoretical predictions. To illustrate the sudden onset of enhanced heating, Fig. 4 plots the temperature field for a 10-s insonation at two pressures: Pa ⫽ 1.39 MPa (just below the threshold pressure) and Pa ⫽ 1.42 MPa (just above the threshold pressure). Note that the heating remains spatially localized (see Fig. 4b). The deviation from the logarithmic heating rate for the 1.42-MPa curve is an indication of bubble activity, as argued below.

Fig. 4. Temperature field for insonation pressures just below and just above the threshold for enhanced heating for a 10.2-s insonation of an agar phantom: (a) temperature vs. time; (b) temperature vs. radial distance from the acoustic axis.

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Fig. 6. Threshold peak-positive pressure for the onset of enhanced heating (squares) and for saturation (triangles) as a function of the insonation duration ␶CW. Results for normal (■, Œ) and degassed (䊐, ‚) agar phantoms are plotted. The lines and shading are guides for the eye. Note the logarithmic time scale.

Fig. 5. Peak temperature increase from ambient vs. insonation pressure for agar phantoms at three increasing CW insonation durations: (a) 0.7 s; (b) 1.2 s; and (c) 10.2 s. The theoretical curves were generated using eqns (1)–(5), and the measured physical property values listed in Table 1. Note the thermocouple locations are slightly different for each run. The error bars represent the 5-point sample SD at each pressure value.

Temperature vs. acoustic pressure To investigate the effect of increasing the acoustic pressure, we monitored and recorded the maximum temperature increase attained (regardless of time of occurrence) at each thermocouple position for a given exposure cycle. Figure 5a– c plots the peak temperature increase, relative to ambient, for the two thermocouples nearest to the focus as a function of acoustic pressure at the focus for the agar phantoms insonated at 1 MHz with 0.7-, 1.2- and 10.2-s pulse durations, respectively. For all these cases, three regions can be delineated. The first region, dominated by the absorption of the incident sound, exhibits peak temperatures that are roughly proportional to the square of the acoustic pressure and are, thus, consistent with predictions from linear analytic theory calculated using the independently measured material and exposure parameters. The second region is a transitional zone character-

ized by a sudden jump in temperature increase at some critical threshold pressure. In this region, the peak temperature increases at a much greater rate with increasing pressure than linear absorption theory predicts. No simple functional dependence is derivable from our data, but, on average, the rate of heating is 5 to 10 times higher than observed for the lower (i.e., subthreshold) pressures. This increase is erratic, with a large run-to-run variation as evidenced by the large error bars associated with runs for these pressure values. The third region is one of apparent saturation, where the peak temperature may actually decrease with increasing pressure. We observed the same behavior for PVAbased phantoms (Holt et al. 1998). The pressure ranges delineating these three behaviors vary with phantom preparation and with insonation duration, but appear to be generic features of relatively short-time, high-pressure insonation conditions. For some of the degassed phantoms, a region of still greater enhancement followed by another saturation can be observed to follow the initial saturation (see Fig. 6). Threshold pressure vs. exposure duration The value of the threshold pressure at the onset of enhanced heating depends on the insonation duration. In Fig. 7, the peak positive pressure at the onset of enhanced heating, or “threshold pressure” (Pth), is plotted as a function of the exposure duration ␶CW for a degassed and a nondegassed phantom. The threshold pressure decreases as the insonation time increases, until it apparently saturates at near 1.5 MPa for exposure times greater than 1 s. It is significant that the onset threshold appears

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Fig. 7. Peak temperature increase from ambient vs. insonation pressure for degassed agar phantoms at two CW insonation durations: (a) 0.7 s; and (b) 1.2 s. The theoretical curves were generated using eqns (1)–(5), and the measured physical property values listed in Table 1. Note the thermocouple locations are slightly different for each run.

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Fig. 8. Temperature vs. time in a degassed agar phantom resulting from a 1.2-s CW insonation at 4.0 MPa peak-positive pressure and 1-MHz frequency. Distances given for each thermocouple (labeled TCn) are relative to the acoustic axis. The theoretical curve was generated using eqns (1)–(5), and the measured physical property values listed in Table 1.

from these data to be independent of the dissolved gas concentration.

temperature “inversion,” in which the temperature nearest the focus was lower than the temperature further away, as is seen in Fig. 6a at Pa ⫽ 3.4 MPa and in Fig. 6b for Pa ⫽ 2.25 to 2.5 MPa. Often, temperature vs. time curves for the degassed phantoms exhibited much more erratic behavior, such as that observed in Fig. 8.

Threshold pressure vs. dissolved gas content Although the threshold pressure for enhanced heating is apparently independent of the dissolved gas concentration, the enhanced heating rate and resultant peak temperature rise appear to be greater for the degassed phantoms. Figure 6a and b plot temperature rise vs. pressure for 0.7- and 1.2-s durations, respectively, in degassed agar phantoms. Figure 6a and b may be compared to Fig. 5a and b, which plot the results for the same insonation times for phantoms that were not degassed in preparation. Within experimental error, the onset pressures were the same for both preparations. For the degassed phantom preparations in Fig. 6, the transition region (region one) exhibits a more rapid heating rate, and the saturation region (region two) occurs earlier. This early saturation is followed by an actual decrease in the peak temperature as the insonation pressure continues to increase, a feature not seen in the more gassy phantoms. Finally, in the 1.2-s case, the temperature dramatically rose again followed by a second saturation region at a peak temperature almost double the temperature reached in the gassy phantom at the same pressures. Another feature observable with the degassed phantoms at pressures beyond the onset of enhanced heating is a

Evidence for bubble activity Figure 9a– c shows the acoustic pressure vs. time and its power spectrum measured with the needle hydrophone positioned at the focus in the agar phantoms. The peak-positive pressure from the source transducer was increased from 0.25 MPa in Fig. 9a to 1.27 MPa in Fig. 9b to 2.5 MPa in Fig. 9c. For Fig. 9a and b, we see gradually increasing nonlinearity due to propagation effects. However, because the phantom was highly absorbing, (especially at higher frequencies) and because, for this phantom, the absorption increased linearly with frequency) even for 2-MPa peak positive pressure, the wave was only mildly nonlinear. Figure 9c shows markedly different behavior, characterized by a sharp transition threshold pressure. The prominent appearance of transient harmonics in the time series and the increase in the broadband subharmonic and harmonic spectrum are both common signatures of cavitation bubble activity. The onset of apparent bubble activity, as measured by the hydrophone, was always an abrupt phenomenon, and exhibited hysteresis. As we increased the pressure, the onset of irregular recorded traces occurred at nearly 0.3 MPa above the pressure at which the phenomenon would cease as we decreased the input power.

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scale of hundreds to thousands of acoustic cycles. Bubble growth will be interrupted by: 1. the onset of shape instabilities leading to breakup of the bubble (Eller and Crum 1970; Brenner et al. 1995; Gaitan and Holt 1999; Hao and Prosperetti 1999) or, 2. the diminution of bubble response as the mean bubble size increases well past the regime of resonant behavior so that bubble growth by rectified diffusion is halted. In either case, an (unknown) asymptotic bubble size distribution is expected for sufficiently long insonation times, and we take an arbitrary upper limit of R0 ⫽ 10⫺4 m for our calculations in what follows.

Fig. 9. Acoustic pressure vs. time and its power spectrum. The pressures were measured with the DAPCO needle hydrophone imbedded in the phantom and positioned at the focus of the transducer for peak-positive pressures of (a) 0.25; (b) 1.27; and (c) 2.5 MPa.

The existence of a sharp threshold pressure for the onset of enhanced heating as seen in Figs. 5 and 6 is a characteristic of acoustic cavitation and effects induced by cavitation bubble activity. The large increase in scatter of the measured temperature data for acoustic pressures above this threshold is further indirect evidence of bubble activity. The saturation effect at higher pressures is probably the result of shielding due to the presence of multiple bubbles in the focal region, which scatter the acoustic wave and defocus the beam, resulting in a more diffuse energy deposition and, thus, locally lower temperatures. MODELING II: BUBBLE-ENHANCED HEATING Bubble size distribution Our working hypothesis is that sufficient nuclei (hydrophobic inclusions, or possibly even stabilized microbubbles) exist in our phantoms so that, above a threshold pressure amplitude, acoustic cavitation occurs. The distribution of bubble sizes is unknown. The nuclei most susceptible to inertial cavitation at 1 MHz will possess an equilibrium radius R0 on the order of 0.1 ␮m (Apfel and Holland 1991; Allen et al. 1997). Growth by rectified diffusion (Eller and Flynn 1965; Crum and Hansen 1982; Church 1988) will shift the mean of the bubble size distribution to larger radii over a long time

Energy loss mechanisms in bubble oscillations There exists a wealth of literature concerned with energy dissipation mechanisms for acoustically forced bubble oscillations. Almost all treatments assume spherical symmetry, and all ignore gaseous mass transfer. A few works deal with energy dissipation due to the effects of vapor and its associated phase transition. The available evidence from our experiments points to inception and growth of cavitation bubbles at relatively low temperatures and, thus, we will consider primarily gas bubbles in this preliminary assessment of mechanisms. Damping of bubble oscillations can be conceptually attributed to thermal, viscous and acoustic dissipation. Thermal damping arises from consideration of the thermodynamics of a bubble oscillation. If we acoustically excite a bubble that is initially in thermal equilibrium with an equilibrium radius R0 (ignoring mass diffusion), thermal energy flows into the bubble during its expansion phase and out of the bubble during its collapse phase. The net amount of such heat flow for a particular bubble oscillation depends on the oscillation amplitude, which, in turn, is a strong function of insonation pressure, frequency, and equilibrium bubble size. For example, a thermally “large” bubble (R0 ⬎⬎ thermal diffusion length in the gas) driven at low amplitudes will undergo primarily adiabatic oscillations. Conversely, a thermally “small” bubble (R0 ⬇ thermal diffusion length in the gas) will undergo primarily isothermal oscillations. This mechanism is very subtle and can, in fact, lead to cycleaveraged heat flow into the bubble for large-amplitude nonlinear oscillations; thus, making the bubble a thermal conduction sink for some combinations of parameters (Kamath et al. 1992). However, it has been shown for air bubbles in water that this effect is small, even for violently nonlinear oscillations (Kondı´c et al. 1995). Although this mechanism will become increasingly important as the bubble heats up (i.e., as the vapor content increases), the heat flow will still be into, not out of the bubble; thus, any full accounting of bubble-mediated heating must deal with this thermal sink term for hot bubbles because otherwise the heat deposition from bub-

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bles will be slightly overestimated. However, to first order, we may ignore this effect, and we will not discuss it further in the present work. Viscous damping arises primarily from viscous dissipation in the relatively thin layer of host medium surrounding the bubble that moves during the bubble oscillations. The dissipation is directly proportional to the bubble interfacial velocity and the shear viscosity of the host medium. As is the case for thermal damping, its relative contribution depends on acoustic parameters and bubble size. Prosperetti (1977; 1991) has shown that, for a large range of linearized bubble dynamics, the viscous contribution for spherical oscillations is small compared to the thermal and acoustic terms for an air bubble in water. But, for ␮m-sized bubbles at MHz frequencies, MPa forcing pressures and a host with biologically relevant rheological properties (notably shear viscosities greater to much-greater than water), we expect the viscous term to be at least as important as the acoustic and thermal contributions and, in some cases, it may well be the dominant damping mechanism and, thus, also be the dominant heat deposition mechanism. Acoustic damping arises because the bubble is a sound source and radiates acoustic energy. This term is an indication of the effects of finite compressibility of the host medium. It becomes important primarily when the collapse phase of a bubble’s oscillation becomes violent, because very large values of acceleration of the bubble wall may be attained during such inertial collapses. Indeed, for very violent collapses such as are typically associated with transient cavitation, the bubble emits a shock wave upon collapse. We now turn to a consideration of the resultant heating rates from these bubble oscillation dissipation mechanisms. We consider estimates from linearized bubble dynamics, then turn to numerical calculations to capture the effects of nonlinearity. To begin to quantify these considerations, we must also establish certain key parameters. Both the agar and PVA phantoms we use (as well as most biologic fluids and tissues) exhibit viscoelastic properties, and we should properly be using a model of the host medium that possesses such features. We can reasonably assume that, at the extremely high strain rates typical of nonlinear bubble oscillations at MHz frequencies (peak values can exceed 109 Hz), we will reside in the high-frequency asymptote for the shear viscosity number. This is important because the asymptotic value of the shear viscosity number is determined by shear thinning and will, in general, be one or two orders of magnitude below the static or low-strain-rate value. This still leaves us with some uncertainty regarding the shear viscosity of our agar phantoms. We can invert the well-known expression relating the mea-

Volume 27, Number 10, 2001

sured acoustic absorption (18 Np/m/MHz) to the shear dynamic viscosity to obtain a value of 3.15 Pa䡠s (water is 10⫺3 Pa䡠s, and whole blood at body temperature has a measured value of 4 to 5 ⫻ 10⫺3 Pa䡠s.). In light of the above discussion of the viscoelastic nature of our phantom, 3.15 Pa䡠s should be considered an upper bound. Viscous heating. The basic hydrodynamic equation for spherical bubble dynamics in a Newtonian host fluid is the Rayleigh–Plesset equation, and the particular version we use here incorporates a correction for the compressibility of the liquid (Keller and Kolodner 1956; Keller and Miksis 1980):

冉 冊 1⫺

冉 冉 冊

R˙ 3 R˙ RR¨ ⫹ R˙ 2 1 ⫺ c 2 3c ⫽ 1⫹

R˙ P共R, R˙ , t兲 R ⭸P共R, R˙ , t兲 ⫹ c ␳ ␳c ⭸t

冉冊

P共R, R˙ , t兲 ⫽ P l

冊

R0 R

3␬

⫹ Pv ⫺ P0 ⫺ ⫺

2␴ R

4 ␮ R˙ ⫺ P a sin共 ␻ t兲 R

Pl ⫽ P0 ⫺ Pv ⫺

(3)

2␴ , R0

(4)

(5)

where R is the instantaneous radius of the bubble, c is the speed of sound in the liquid, ␳ is the density of the liquid, P is the pressure inside the bubble, Pl is the pressure in the liquid, Pa is the acoustic pressure amplitude with frequency ␻, Pv is the vapor pressure, P0 is the ambient pressure, ␬ is the polytropic exponent of the gas in the bubble, ␴ is the surface tension, ␮ is the shear (dynamic) liquid viscosity, R0 is the equilibrium bubble radius, and overdots denote time derivatives. The viscous contribution is typically incorporated into the bubble dynamics equation as a boundary condition on the normal component of the interfacial stress for a spherical bubble in a Newtonian host fluid; it appears as the next-to-last term on the right-hand-side of eqn (4). The power dissipated by this term is then (Prosperetti 1977): W vis ⫽ 16 ␲␮ RR˙ 2.

(6)

This energy is dissipated solely as heat in the medium, and so this term is identically the viscous heating contribution. Equation (6) must be time-averaged for R(t)

Bubble-enhanced heating ● R. G. HOLT and R. A. ROY

obtained via numerical integration of eqns (3)–(5). For linearized bubble dynamics, eqn (6) reduces to: L W vis ⫽ I inc

冉

16 ␲␮ c ␻2 ␳ R 0 共 ␻ 02 ⫺ ␻ 2兲 2 ⫹ 共4 ␤ 2␻ 2兲

冉

␻ 02 ⫽ 3 ␬ P 0

冊

冊

1 ⫹ 2 ␴ /P 0R 0 2␴ ⫺ 3, 2 ␳R0 ␳R0

(7)

(8)

where Iinc is the incident (or insonation) acoustic intensity, ␤ ⫽ 2␮/␳R02 is the viscous damping ratio, and ␻0 is the linear, undamped resonant frequency of the bubble. Heating due to bubble acoustic radiation. The energy converted into heat due to radiated sound from a bubble may be evaluated by assuming the far-field pressure radiated by the oscillating bubble is given by: P b共r, t兲 ⫽

␳R 共2R˙ 2 ⫹ RR¨ 兲, r

(9)

where r is the distance from the bubble center and R(t) must, in general, be obtained from the numerical solution of eqns (3)–(5). In the far field, the acoustic pressure and velocity are in phase; thus, by employing the plane-wave expression for the acoustic intensity radiated from the bubble, Ib ⫽ Pb2/2␳c, we may estimate the thermal power dissipated in the host: W rad ⫽ 共4/3 ␲ R 03兲2 ␣ I b.

(10)

Several assumptions are involved in the estimate given by eqn (10). We choose to evaluate Ib at r ⫽ R0, although we use the far-field expression for the radiated pressure, eqn (9). Likewise, we choose the bubble equilibrium volume as the effective volume for most of the thermal absorption. Finally, we use the measured absorption coefficient for 1 MHz, even though, in general, the radiated pressure Pb(t) will be nonlinear, containing higher harmonic components that will possess higher absorption coefficients. Nevertheless eqn (10) will give correct order-of-magnitude nonlinear estimates for wide ranges of parameters, and we defer a more rigorous treatment of Wrad to a future work. We may also obtain a linearized estimate by treating the bubble as a linear oscillator and calculating the (active part) of the scattering cross-section: L W rad ⫽ 共4/3 ␲ R 03兲2 ␣ I scatt

(11)

where I scatt ⫽

冉

active I inc⌰ scatt ␻4 2 ⫽ I inc 2 2 2 4␲R0 共 ␻ 0 ⫺ ␻ 兲 ⫹ 共4 ␤ 2␻ 2兲

冊

(12)

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is the active (and dominant for these size ranges) part of the scattered acoustic intensity from the bubble, and ⌰activescatt is the cross-section (Leighton 1994). Estimates for bubble-induced heat transfer We attempted to obtain bounding-value estimates of heat transfer from oscillating bubbles to compare with the primary acoustic absorption heating incorporated into eqns (1), (2). We must first present an estimate of the energy per acoustic cycle converted into heat, assuming classical acoustic absorption in a thermoviscous medium. We assumed as a baseline 1-MHz insonation with Pa ⫽ 2.0 MPa at the focus. Assuming plane-wave propagation, the spatial-peak temporal-average intensity is I0 ⫽ 114 W/cm2, (compared to typical HIFU insonations of 1000 W/cm2), and the thermal energy deposition rate per unit volume (thus, the thermal power density) is q ⫽ 2␣I ⫽ 42 W/cm3, where ␣ is the measured absorption coefficient of 18 Np/m/MHz for our agar phantom. Using the measured acoustic intensity profile, we calculated the baseline thermal power for acoustic absorption to be 713 mW. To obtain the enhanced heating rates we observe, we require a power on the order of 1 to 5 W, which is roughly 1 to 5 times greater than the baseline available from the incident acoustic wave alone. We performed our estimates for a single bubble oscillating in a uniform external acoustic field. We considered a range of bubble equilibrium radii 0.01 ␮m ⱕ R0 ⱕ 100 ␮m, and a range of shear dynamic viscosities 0.001 Pa䡠s ⱕ ␮. ⱕ 3.15 Pa䡠s. Table 2 summarizes several test cases for comparison. Although not an exhaustive parameterization of bubble-mediated heating, we may notice several trends. First, for all the cases studied, the viscous heating from the bubble was at least 2 orders of magnitude greater than acoustic emission heating. Second, nonlinearity only matters when inertial-type cavitation characterized by large values of the expansion ratio Rmax/R0 are attained, as in cases 3 and 4. However, the largest heating power Wvis occurs for a super-resonant bubble, 30 ␮m in size, and a viscosity of 0.5 kg/m-s. This represents a local maximum in the viscous heating due to a single bubble, and its value of 32 mW is already 5% that due to HIFU alone. Because this is from a bubble as large as 30 ␮m, it is not hard to imagine 20 such bubbles in any radial cross-section of the focal region being present—this itself would double the heating due to HIFU alone. We cannot on this basis dismiss the absorption of the bubble’s acoustic radiation because, as noted above, our calculation is clearly an underestimate, especially for small bubbles undergoing inertial cavitation and radiating weak shock waves. For example, in case 3 (the 1-␮m bubbles, which are subresonant and undergoing inertial motion), we could easily conclude that both viscous and

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Volume 27, Number 10, 2001

Table 2. Comparison of linear and nonlinear bubble heating powers

R0 (m) ␮(Pa䡠s) L Wvis (W) Wvis (W) L Wrad (W) Wrad (W) Rmax/R0

Case 1

Case 2

Case 3

Case 4

Case 5

3.58E ⫺ 06 3.15E ⫹ 00 9.16E ⫺ 05 9.75E ⫺ 05 3.98E ⫺ 13 0.60E ⫺ 13 1.1E ⫹ 00

1.80E ⫺ 06 3.15E ⫹ 00 1.16E ⫺ 05 1.24E ⫺ 05 3.23E ⫺ 15 3.75E ⫺ 15 1.1E ⫹ 00

1.00E ⫺ 06 1.00E ⫺ 03 5.53E ⫺ 06 3.68E ⫺ 04 4.46E ⫺ 13 1.21E ⫺ 06 18E ⫹ 00

3.58E ⫺ 06 1.10E ⫺ 01 2.62E ⫺ 03 7.50E ⫺ 03 3.26E ⫺ 10 1.33E ⫺ 08 2.8E ⫹ 00

3.00E ⫺ 05 4.95E ⫺ 01 3.22E ⫺ 02 3.14E ⫺ 02 4.38E ⫺ 06 4.76E ⫺ 06 1.2E ⫹ 00

All cases are for 2 MPa and 1 MHz insonation, with host parameters the same as the agar phantom preparation. The ambient pressure was taken to be 0.1 MPa, the surface tension 0.072 Nt/m, and the polytropic constant taken to be 1.4, representing adiabatic bubble dynamics. For comparison, the thermal power due to acoustic absorption at 2 MPa, 1 MHz is 713 mW. Case 1 is for resonance-size bubble at 1 MHz, with a shear viscosity inferred from attenuation measurements. Case 2 is for a bubble that is half resonance size, but with the same viscosity as Case 1. Case 3 is for a 1-␮m bubble in water. Case 4 is a resonance-size bubble with an intermediate value for the shear viscosity. Case 5 is a bubble that is larger than resonance with an intermediate viscosity. The value of the expansion ratio R max/R 0 determined from the numerical solutions is also given, as an indicator of nonlinearity.

acoustic radiation mechanisms are equally important, given the uncertainties in the modeling. Nevertheless, Table 2 establishes that bubbles can easily deposit enough heat to explain our measurements. Alternative explanations Nonlinear absorption. Several authors have noted the possibility of enhanced absorption of nonlinearly propagating acoustic waves (Bacon and Carstensen 1990; Dalecki et al. 1991; Christopher and Carstensen 1996). The argument is that, since the absorption of US varies as a power-law of the frequency, a nonlinear sound field with higher-harmonic components would be preferentially absorbed relative to a linear sound field of the same peak pressure. Also, the heating pattern produced by a nonlinear beam will be narrower and peak closer to the transducer than the comparable linear beam. Although, in principle, this may be a contributing mechanism, it is unlikely to be the source of our observed enhanced heating rates. First, the threshold dependence observed in our data cannot be explained by a nonlinear propagation model. Second, the pressure at the focus, measured in situ and displayed in Fig. 9, exhibits, at most, very weak nonlinearity as evidenced by the slightly narrowed positive peak and the slightly broadened rarefaction peak, and the low level of 2nd harmonic energy in the measured frequency spectrum. Temperature-dependent material absorption. Several experimental groups have observed an increase of the material absorption coefficient as a function of temperature (Bush et al. 1993; Damianou et al. 1997). However, this appears to be a function of real tissue necrosis, and exhibits a temperature threshold of 50°C. In addition, the effect is irreversible. None of these characteristics applies to our measurements, nor do they explain

the earlier measurements of Hynynen (1991) or Watmough et al. (1993). Other bubble-related mechanisms. The simple fact that there could be bubbles in the sound field could lead to enhanced heating by increasing the effective path length in the insonified region due to multiple scattering. This effect could be modeled by utilizing an effective absorption coefficient that increases with the number density of the bubbles; see Wu (1998). However, this explanation would fail to capture the threshold dependence and erratic heating that were observed in our experiments. Several high-amplitude effects could also contribute to bubble-mediated heating. If any liquefaction of local areas of the phantom due to thermal or mechanical sources occurs, then enhanced viscous heating may occur via the streaming fields associated with the high-amplitude bubble motion. If bubbles do not remain spherical due to instability or proximity to other bubbles, then the viscous dissipation would be increased relative to the spherical case, with or without streaming flow around the bubble. Although any of these effects can and may occur, considered alone they fail to explain our results and are, thus, of secondary importance. CONCLUSIONS We have measured bubble-enhanced heating in a phantom as a function of several parameters. Our observations are similar to the earlier work of Lele (1987), Hynynen (1991) and Clarke and ter Haar (1997). To summarize our findings and their significance: 1. There appears to be a generic scenario for temperature rise vs. insonation pressure. An initial phase in which the temperature rise is modest and well-described by linear acoustic absorption theory is followed by the abrupt onset of dramatically increased temperature rise at some critical threshold pressure. Upon further

Bubble-enhanced heating ● R. G. HOLT and R. A. ROY

increasing the insonation pressure, no further gain in temperature rise is achieved. 2. The threshold pressure for the onset of bubble-enhanced heating decreases as the insonation duration increases and, apparently, remains constant at 1.5 MPA for durations greater than 1 s. 3. The radial temperature field measurements show that the enhanced heating effect remains localized. 4. Lowering the dissolved gas concentration appears to increase the enhanced heating effect beyond onset, leaving the onset threshold pressure unchanged. The saturation regime is poorly understood for the degassed measurements. 5. Modeling estimates indicate that the heat deposition from bubbles can, indeed, quantitatively explain our measured temperature rises. The leading order mechanisms are viscous heating and absorption of bubble acoustic emission. Our measurements would require on the order of 100 bubbles active in the focal zone to account for the enhanced heating. 6. For decreasing insonation durations, it becomes easier to pick appropriate combinations of insonation duration and pressure to achieve bubble-enhanced heating. We believe our work here, combined with the observations of earlier researchers (particularly their in vivo work) suggests a scenario for exploiting and controlling nonlinear bubble effects to obtain enhanced local heat deposition in tissues. To maximize controlled local heat deposition, one should operate at relatively short insonation times with pressures greater than (but not too much greater than) some critical threshold pressure for bubble nucleation. The challenge is prediction of the proper acoustic parameters to achieve the elevated heating rate and avoiding saturation and shielding effects. An alternative to absolute prediction would be to employ a diagnostic measurement, such as noise emitted by cavitation; see Hynynen (1991), to determine the onset of the phenomenon, and using modeling to predict the parameters given the measured onset. POSTSCRIPT Since the submission of this work, two papers have theoretically addressed some aspects of bubble behavior in medium and high-intensity US and should be noted here. Hilgenfeldt et al. (2000) consider the nonlinear sound scattered by diagnostic pulse-driven, ␮m-sized bubbles and calculate exactly the same heat transfer mechanisms we consider above, using a version of the Rayleigh–Plesset equation. Chavrier et al. (2000) consider only the bubble acoustic radiation absorption mechanism calculated from a Rayleigh–Plesset equation, but go on to incorporate such results into the heat conduction equation with the use of a nonlinear “global attenuation

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coefficient” for an arbitrary number of bubbles in the focal zone. Acknowledgements—The authors thank R. Cleveland, P. Edson, J. Huang, and the other members of the Boston University Physical Acoustics Group for support and helpful discussions. They gratefully acknowledge the support and collaboration of L. Crum, M. Bailey, P. Kasckowski, P Mourad, and the entire HIFU team at the University of Washington Applied Physics Laboratories. This work was supported by the Defense Advanced Research Projects Agency.

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