A model for the dynamics of gas bubbles in soft tissue

Transcription

1 A model for the dynamis of gas bubbles in soft tissue Xinmai Yang a and Charles C. Churh National Center for Physial Aoustis, The University of Mississippi, University, Mississippi Reeived 21 April 2005; revised 16 September 2005; aepted 17 September 2005 Understanding the behavior of avitation bubbles driven by ultrasoni fields is an important problem in biomedial aoustis. Keller-Miksis equation, whih an aount for the large amplitude osillations of bubbles, is rederived in this paper and ombined with a visoelasti model to aount for the strain-stress relation. The visoelasti model used in this study is the Voigt model. It is shown that only the visous damping term in the original equation needs to be modified to aount for the effet of elastiity. With experiment determined visoelasti properties, the effets of elastiity on bubble osillations are studied. Speifially, the inertial avitation thresholds are determined using R max /R 0, and subharmoni signals from the emission of an osillating bubble are estimated. The results show that the presene of the elastiity inreases the threshold pressure for a bubble to osillate inertially, and subharmoni signals may only be detetable in ertain ranges of radius and pressure amplitude. These results should be easy to verify experimentally, and they may also be useful in avitation detetion and bubble-enhaned imaging Aoustial Soiety of Ameria. DOI: / PACS numbers: Wa, Sh, Ei FD Pages: I. INTRODUCTION Cavitation phenomena are very ompliated due to the nonlinear osillations of small bubbles and the interations between these bubbles. In most ases, avitation ours in water, whih is the most familiar fluid to us, and as a result, studies of bubble dynamis in water have been undertaken for over 80 years. Bubble dynami models are well established for bubbles in water or simple Newtonian fluids. With the development of new materials and new tehniques, the study of bubble dynamis in visoelasti media beomes neessary. The inreasing interest in avitation is partly related to the appliation of medial ultrasound. For example, the use of bubble-based ontrast agents in diagnosti ultrasound has signifiantly inreased the quality of imaging. Reently, this issue has beome more important due to the development of the high intensity foused ultrasound HIFU for therapeuti mediine. High intensity ultrasound will indue avitation in soft tissue, and these mirobubbles have a huge impat on the distribution of the ultrasound energy. In these situations, the surrounding media, i.e., biologial tissues, often exhibit non-newtonian behavior. Understanding the behavior of avitation in vivo may provide a powerful tool to improve the quality of medial ultrasound. The study of these mirobubbles involves bubble osillations in visoelasti media. Many researhers have extended the analysis of bubble dynamis in Newtonian fluids to non-newtonian fluids. Fogler and Goddard 1 ombined the linear Maxwell model with the Rayleigh-Plesset equation and examined the ollapse of a spherial avity in a large body of an inompressible visoelasti liquid. The bubble was modeled as a void, and the effet of elastiity was investigated. Their results showed that the elastiity in the liquid an signifiantly retard the ollapse of a bubble. A three a Eletroni mail: xmyang@olemiss.edu parameter linear Oldroyd model was employed by Tanasawa and Yang 2 to study the free osillation of a gas bubble in visoelasti fluids. They investigated the effets of the visous damping in the presene of elastiity, and found that in the presene of elastiity, the effet of visous damping on bubble ollapse is less than that in the pure fluid. Later, Shima, Tsujina, and Nanjo 3 investigated the nonlinear osillations of gas bubbles in visoelasti fluids using the model first derived by Tanasawa and Yang, 2 and the effets of relaxation time and retardation time were larified. A fully numerial sheme was developed by Kim 4 to investigate ollapse of a spherial bubble in a large body of Upper- Convetive Maxwell fluid. He observed that fluid elastiity aelerated the ollapse in the early stage of ollapse while in the later stages it retarded the ollapse. His approah was very omputationally intensive. Alekseev and Rybak 5 presented the resonane frequeny of gas bubbles in elasti media. The dispersion equation in a visoelasti medium was also derived for bubble louds in their study. Allen and Roy 6 hose the linear Maxwell and Jeffreys models as the liquid onstitutive equation to study bubble osillations in linear visoelasti fluids. After linearization of the original nonlinear differential equation a Rayleigh-Plesset type equation, analytial solutions were obtained and ompared with the Newtonian results. In a later study of nonlinear visoelastiity, they 7 employed the Upper-Convetive Maxwell model as the onstitutive equation with the Rayleigh-Plesset equation. A fully numerial study was onduted to solve the governing system of equations. The results of the linear and nonlinear visoelasti approahes were ompared. Their results showed that tissue visoelastiity may be important for the potential avitation bioeffets. For diagnosti ultrasound examinations, the aousti intensity is usually insuffiient to indue inertial avitation in soft tissue diretly. 8 The only soure of mirobubbles in soft tissue would be diret injetion of a bubble-based ontrast J. Aoust. So. Am , Deember /2005/1186/3595/12/$ Aoustial Soiety of Ameria 3595

2 agent, an unommon proedure. However, the use of higher intensity ultrasound, e.g., HIFU therapy, will ause avitation in soft tissue diretly. Different from ontrast agent bubbles, this avitation involves free bubbles osillating nonlinearly beause of the high intensity of the sound field. Previous models for visoelasti media are all based on the Rayleigh- Plesset equation, whih is not very appropriate for large amplitude osillations. In this work, we seek a model that is apable of aounting for the potentially large-amplitude osillations of bubbles exposed to HIFU fields. In addition, the bubble model must inorporate a visoelasti model onsistent with measured tissue properties. Although the data on visoelasti properties of soft tissue at megahertz frequenies are very limited, the linear Voigt model has proven appropriate for the tissues studied; 9,10 none of the above-mentioned models inorporates these experimental results. The Keller- Miksis equation 11 has been shown to be suitable for large amplitude bubble osillations. 12 In this study, we ombine the general form of the Keller-Miksis equation with the linear Voigt model for visoelasti solids to study the dynamis of bubbles in soft tissue. The importane of the inertial avitation threshold has been addressed by many authors. In a medial ontext, severe bioeffets, inluding both thermal and nonthermal effets, may be indued or exaerbated by inertial avitation during high intensity ultrasound insonations. When inertial avitation ours, strong nonlinear aousti emissions an be deteted. A sudden hange in the emission signals from a bubble is often used to monitor the ourrene of inertial avitation in experiments. 13,14 In addition to the familiar seond, third, and higher harmonis, bubbles may also generate subharmoni signals when they osillate nonlinearly. Therefore, monitoring the generation of subharmonis an be used to detet inertial avitation. Higher frequeny emissions are very easily attenuated, and the signal that needs to be deteted may beome very weak. At the same time, the nonlinear propagation of ultrasound will also generate higher harmonis, and this an be a noise soure for higher harmoni detetion. Compared to harmoni detetors, subharmoni detetors have the advantages that the low frequeny signal is less attenuated in soft tissue, and bubbles are the only soures generating subharmonis in soft tissue. Subharmoni signals have been seen in experiments during avitation events, but mehanisms for the generation of subharmonis are still not entirely lear. Possible explanations inlude that a single bubble will emit subharmonis when it breaks up, or interations inside a bubble loud an emit subharmonis, or haoti osillations of a single bubble will generate subharmonis. In this study, we fous on single bubble dynamis, and simply predit subharmoni signals from the haoti osillation of a single bubble. The effet of elastiity on inertial avitation thresholds and subharmoni emissions will be investigated, providing data that may be useful for avitation detetion and bubble imaging. II. THEORY AND METHOD Consider a spherial bubble in an unbounded visoelasti medium. The equation of ontinuity has the following form in a spherial oordinate system, t + v r + 2v r =0, 1 r r where is the density, v r is the radial veloity, t is time, and r is the radial axis. Conservation of radial momentum for a spherially symmetri radial flow yields, 15,16 v r t + v r v r r = p r + rr r + 2 r rr, where p is the pressure in the surrounding medium, and rr and are the stresses in the r and diretions, respetively. The boundary and initial onditions are: p = p g 2 R + rr at r = R, p = p at r =, R = R 0, Ṙ =0 att =0, where p g is the gas pressure inside the bubble, R is the position of the gas-tissue interfae, the dot indiates the time derivative, R 0 is the bubble equilibrium radius, and is the surfae tension. To derive the Keller-Miksis equation, whih an aount for the ompressibility of the surrounding medium to first order, an asymptoti solution is employed in the near field and far field. A. Near field approximation In the near field r=or, the effets of ompression and expansion of the bubble are dominant, and the surrounding medium may be onsidered inompressible. From the Bernoulli integral momentum equation, one an find the solution for the pressure distribution in the internal zone near field, 2 v r = ṘR2 r 2, 3 Ṙ2 p in = p a 0RR r R2 Ṙ 0 R 4 Ṙ 2 r 2 r 4 + rr R r rr +3R r dr, 4 where p a is the pressure at the bubble surfae, R is the bubble wall aeleration, and indiates the time derivative. B. Far field approximation In the far field rr, the pressure flutuations and the density flutuations are small, and the stress omponents beome negligible, as do the nonlinear onvetion terms. Ignoring these terms, the governing equation in the far field is essentially the linear aousti equation. The solutions for the linear aousti equation are ex = 1 r 1t r + 2t + r, J. Aoust. So. Am., Vol. 118, No. 6, Deember 2005 X. Yang and C. C. Churh: Gas bubbles in soft tissue

3 ex p ex = p 0 0, 6 t where p 0 is the stati pressure, 0 is density at equilibrium, is the veloity potential, 1 and 2 haraterize the outgoing and inident aousti waves, respetively, and is the sound speed in the medium. 1 Ṙ RR Ṙ 3Ṙ 2 =1+ Ṙ p a p I + R d dt p a p I, where 2/ 2 = P A gt is the driving pressure, 12 C. Mathing of the solutions To obtain the equation of the radial motion of bubbles for a given driving pressure and to take aount of ompressibility of the surrounding medium, we need to math the asymptoti solutions in the internal and external zones in the intermediate zone. For the internal solution, the intermediate zone is at r, and for the external solution, the intermediate zone is at r 0. The mathing onditions in the intermediate zone are the equality of the volumetri flow and of the pressure, 4r 2 v rin r =4r 2 v rex r 0 p in r = p ex r 0. 7 For the internal solution, shear stresses vanish as r. By mathing the solutions we finally get RR R2 = p a p R f rrr,t rr r dr, where f =R 2 Ṙ, and 2 is the inident wave. Note that the above equation inludes f, whih will ause a third derivative of R. This was first notied by Prosperetti et al. 12 This third derivative an be eliminated by assuming that f/ is small, and evaluating the f from the above-mentioned equation. Then the equation beomes 8 and p a = p g 2 R + rrr,t, p a p I = p g 2 R p rr 2 +3R r dr. D. Evaluating stress omponents Sine stress omponents will vanish in the far field, we then only evaluate rr in the near field, i.e., in an inompressible material. Beause soft tissue is visoelasti, we need to hoose a proper visoelasti model to determine stresses. We hoose the linear Voigt model for this study beause it is a simple linear model and previous studies have shown that it is appropriate in the low megahertz frequeny range. 9,10 More important, some experimental data for soft tissues are also available for this model. 9,10 This also reates the potential for omparing the resulting preditions with the experimental measurements in vivo. Beause the material is inompressible, rr =2G rr + rr, where rr is the strain, rr is the strain rate with rr =u/r, u is the veloity, and G is the shear modulus or rigidity. 17 In the near field near the bubble surfae, u=r 2 /r 2 Ṙ, therefore, rr = 2/3r 3 R 3 R 0 3 and rr = 2R 2 /r 3 Ṙ. Then, we have RR R2 = p a p I, 9 where p I is the pressure at infinity, p I = p 0 2/ 2 + rr R,t 3 R rr /rdr. This is the form of lassi Rayleigh equation, indiating that the evaluation of f from this equation is aurate to leading order. Equation 9 may be rewritten as f R 1 f 2 2 R 4 = p a p I, 10 then f is evaluated as and 3R rr r d dt3r dr = 4G 3R 3R3 R Ṙ R, rr r dr = 4G R 0 3 Ṙ R 4 Expanding p a p I, finally we have p a p I = p g 2 R p 0 + P A gt Ṙ2 R 4 R 2 + R f = R Ṙ2 2 + p a p I, 11 Substitution of Eq. 11 into Eq. 8 results in the final equation, and 4G 3R 3R3 R Ṙ R, 17 J. Aoust. So. Am., Vol. 118, No. 6, Deember 2005 X. Yang and C. C. Churh: Gas bubbles in soft tissue 3597

4 d dt p a p I = d dtp g 2 R p 0 + P A gt + d dt3r = dp g dt + 2Ṙ R 2 rr r dr + P dgt A 4G R 0 3 Ṙ Ṙ2 R dt R 4 4 R 2 + R. 18a Equations 12, 17, and 18a provide the desired formulation desribing the dynamis of gas bubbles in soft i.e., visoelasti tissue. We note that this equation is atually just the Keller-Miksis equation with extra terms to aount for the elastiity of soft tissue. The equation aounts for the ompressibility of the surrounding medium to first order, and thus it is better suited than the Rayleigh-Plesset equation to simulate large amplitude bubble osillations. The validity of this equation is limited to small Mah numbers. 12 For the results presented here, the gas inside the bubble is assumed ideal, allowing the pressure to be estimated by use of a polytropi relation, p g = p g0 R 0 /R 3, where is the polytropi index. In this ase, Eq. 18a may be written as: d dt p a p I = g 2 R 3p Ṙ R + P dgt A 4G R 0 3 Ṙ dt R 4 E. Analytial solutions 4 Ṙ2 R 2 + R R. 18b Although Eq. 12 was speifially to investigate nonlinear bubble ativity in tissue, it is instrutive to onsider the effets of the various physial parameters on bubble dynamis at low pressure amplitudes. An analytial solution to Eq. 12 may be obtained by assuming that the pulsation amplitude R 0 xt, is small, making the usual substitutions of Eq. 19 into Eq. 12: 18 R = R 0 1+x, U = R 0 ẋ, U = R 0 ẍ, R 3 = R , et., 19 and reognizing that the term ir 0 P A e it / is to first order equivalent to the linear expression for the radiated pressure wave: 15,18 P sa = R R 0 1 ir 0, 20 The resulting equation has the form: mẍ + bẋ + kx = P A e it, 21 where the effetive mass, m, total damping, b tot, and stiffness, k, are given by m = R R 0, b tot =3P g0 2 R 0 +4G R R 0 / 1+R 0 / 2R 0 2, k =3p g G + R 0 1+R 0 / 2R 0 2. Notie that eah term is greater than would be found from a purely linear analysis of a gas bubble in water. 18 The effetive mass ontains a small additional inrement due to the effet of visosity. There are two additional damping terms, one arising from variation in the surfae energy of the bubble and direted opposite to the other damping terms, the other from the rigidity of the surrounding tissue, respetively. The rigidity of the tissue also ontributes to the total stiffness of the system, as has been noted previously. 1 6 Comparison of Eq. 21 with that of a damped harmoni osillator: ẍ +2 tot ẋ x = P A m eit, 22 in whih tot is the total damping onstant and 0 is the natural frequeny, allows identifiation of five omponents visous, thermal, aousti, interfaial, and elasti to the total damping, given by vis = b vis /2m =2R R 0, 23a th = b th /2m =3p g0 R 02R R 0 =3p g0 /2R 0 +8, 23b R 0 / a = b a /2m = 1+R 0 / 2 2 R 0 2 R R 0, 23 int = b int /2m = R R 0 = /R R 0, el = b el /2m =2GR 0R R 0 =2G/R 0 +4, and the expression for the natural frequeny: 2 0 = k/m =3p g0 2 +4G R R 0 / 2R 0 2 R R 0 23d 23e. 24 In this last expression, the ontributions of the aousti and visous terms are important only for large bubbles, while the 3598 J. Aoust. So. Am., Vol. 118, No. 6, Deember 2005 X. Yang and C. C. Churh: Gas bubbles in soft tissue

5 elasti term either dominates or is of the same order as the pressure term exept for very small bubbles. Note that if the aousti, visous, and elasti terms are negleted, the wellknown expression for the resonane frequeny of a gas bubble in liquid is reovered. The sattering ross setion may be defined as the ratio of the total aousti power sattered by an objet at a given frequeny to the inoming aousti intensity. 17 For a spherial bubble osillating at low amplitude, the expression for the sattering ross setion is 4R 2 S = tot F. Numerial solutions To obtain preditions for the nonlinear osillation of a bubble in soft tissue, Eq. 12 must be solved numerially. Results for individual Rt urves, inertial avitation thresholds, and subharmoni emissions are presented in the following. Determination of the avitation thresholds requires seletion of a threshold riterion. Several riteria for the inertial avitation threshold have appeared in literature, e.g., R max /R 0 =2, 19,20 T max =5000 K, 21 et. In this study, we use R max /R 0 =2 as the threshold riterion. This hoie is onsistent with the assumption that the air in the bubble expands and ontrats adiabatially sine the amplitude Rt/R 0 is fairly insensitive to the thermodynami proesses within the bubble. 22 For subharmonis, the relative strength of the emission with respet to that of the strongest frequeny omponent emitted by a single bubble is shown. The reason to hoose this relative strength is from onsideration of experimental detetion. After passing a preamp, whether a frequeny omponent is detetable by a spetral analyzer or not depends on its strength relative to the strongest omponent. A frequeny omponent is only detetable when this relative number is within the vertial resolution of the instrument. Otherwise, it will be suppressed as noise. The relative number is obtained by the following method: first the R-t urve is obtained, next the radiation pressure is determined by p sa r,t = R r 2Ṙ2 + RR, 26 FIG. 1. Calulated values of linear resonane frequeny for free air bubbles in water and air bubbles surrounded by tissue having values of G=1.5 upper, 1.0, and 0.5 MPa lower. where r is assumed to be a unit onstant, and then frequeny omponents are determined by FFT. The amplitude of the subharmoni is expressed in deibels relative to the maximum amplitude over all frequeny omponents. The following material properties are used in the urrent simulations: p 0 = Pa, =1060 kg/m 3, =1540 m/s, and =0.056 N/m the value for blood assumed by Apfel and Holland 19. These parameters are hosen to be lose to values appropriate to soft tissue. The atual properties of speifi soft tissues may be slightly different from these values exept for surfae tension, whih is unknown, but the differenes will generally be small. The polytropi index =1.4, a value appropriate for adiabati osillations of air bubbles. The rigidity and the visosity of tissue are assumed to be G=0, 0.5, 1.0, and 1.5 MPa and =0.015 Pa s. These values span the range obtained by previous measurements, 9,10 with the exeption of G=0, whih is inluded for omparison. The effet of visosity has been investigated previously for a Newtonian medium. We will fous on the effets of elastiity on bubble motion in this study, but to allow easy omparison with previous work, results for water G=0, =0.001 and blood G=0, =0.005 are also presented. The driving frequenies studied are 1 and 3.5 MHz, whih are frequently used in HIFU. All the bubble osillations are solved for 30 yles. III. RESULTS A. Analytial results The following results were obtained for the ase of airfilled bubbles in tissues having a modulus of rigidity equal to 0.5, 1.0, or 1.5 MPa as disussed earlier. In addition, results for either water or blood or both are presented for omparison. The effets of the surrounding tissue on resonane frequeny, damping, and the sattering ross setion for individual single bubbles will be illustrated in the following. 1. Resonane frequeny The undamped linear resonane frequeny for bubbles larger than 1 2 m and surrounded by visoelasti tissue is dominated by the shear modulus G. For these bubbles, as shown in Fig. 1, 0, inreases approximately as the square root of G. The bottom urve in Fig. 1, labeled Water, shows the resonane frequeny for a free bubble with G=0. The three urves above it, for tissues with inreasing values of G, demonstrate that the inrease in stiffness provided by the tissue an inrease the resonane frequeny onsiderably. The effet of rigidity is muh greater than the effet of surfae tension, meaning that a larger bubble will exhibit a muh greater stiffness than a free bubble of equivalent size. For example, the value of 0,fora5-m bubble is about 0.63 MHz, while replaing the water with tissue inreases this value by a fator of 2.4, 3.3, and 4.0 times for the three rigidities studied here. Beause these larger bubbles resonate at higher frequenies than free bubbles of equivalent size, they will tend to appear aoustially smaller than they atually are. J. Aoust. So. Am., Vol. 118, No. 6, Deember 2005 X. Yang and C. C. Churh: Gas bubbles in soft tissue 3599

6 FIG. 2. Dimensional linear damping onstants vs radial frequeny for equilibrium bubble radii of a 10 m andb 1 m, surrounded by tissue with G=1.0 MPa and =0.015 Pa s; resonane radius, thermal damping onstant given by Prosperetti Ref. 25. FIG. 3. Dimensional linear damping onstants vs equilibrium bubble radius for radial frequenies of a 1 MHz and b 10 MHz, for air bubbles surrounded by tissue with G=1.0 MPa and =0.015 Pa s; resonane radius, thermal damping onstant given by Prosperetti Ref Linear damping oeffiients Five soures of damping for bubbles surrounding by visoelasti tissue were identified earlier and quantified by Eq. 23. Two of these expressions, for visous and aousti damping, are idential to those given by Prosperetti 18 for a gas bubble in liquid, see Eqs. 23a and 23, respetively. The expression for thermal damping, Eq. 23b, differs markedly from those given by either Prosperetti 18 or Eller, 23 whih is not surprising given the assumption of a polytropi pressure-volume relation for the gas. However, the numerial values obtained for bubbles smaller than the linear resonane radius are remarkably similar, often to within a few perent, of those obtained using more exat theories. 18,23 The two additional damping terms, due to the surfae energy of the bubble and the rigidity of the surrounding tissue, see Eqs. 23d and 23e, respetively, have a form similar to that for thermal damping, Eq. 23b. It is worth noting that a ombination of four of the damping terms, Eqs. 23b 23e, is proportional to the stiffness, and thus also to the resonane frequeny, of the system. The results of alulations for the damping onstants as a funtion of radial frequeny are shown in Fig. 2 for bubble radii of 1 and 10 m, assuming G=1.0 MPa and =0.015 Pa s. Beause the term for interfaial tension is negative, only its magnitude has been plotted here. In any ase, the ontribution of int to tot is quite modest. The figures have been drawn in suh a way as to allow easy omparison with previous results for free 18,23 and enapsulated bubbles. 17 Due to the high value of, the total damping is dominated by vis for frequenies less than 0 and R 0 10 m, while the aousti term a dominates at higher frequenies. The elasti term beomes inreasingly important as R 0 inreases, with el vis at R 0 =30 m. As noted earlier, the expression for th differs from those obtained using more exat approahes. This is refleted in both the shape and the magnitude of the urves labeled Thermal in Fig. 2, as may be seen by omparison with the dotted urves labeled P-Th, obtained using the theory of Prosperetti. 18 Thus, while th el for all values of R 0 when employing the polytropi assumption as is done here, th would ome to dominate other soures of damping for R 0 30 m and 0 in a more rigorous treatment of thermal effets. The results of alulations for damping onstants as a funtion of radius are given in Fig. 3 for frequenies of 1 and 10 MHz, again assuming G=1.0 MPa and =0.015 Pa s. The total damping is dominated by vis for bubbles smaller than the linear resonane radius, while a dominates at larger sizes. The ontribution of el is never more than about 20% of tot, whih ours near the resonane radius at 1 MHz, see Fig. 3a. For frequenies above about 1 MHz and radii less than the resonane size, the values for th 3600 J. Aoust. So. Am., Vol. 118, No. 6, Deember 2005 X. Yang and C. C. Churh: Gas bubbles in soft tissue

7 FIG. 4. Linear sattering ross setions vs equilibrium bubble radius for frequenies of a 1 MHz and b 10 MHz, for air bubbles surrounded by tissue having values of G=1.5 right, 1.0, and 0.5 MPa left with =0.015 Pa s, and for water and blood. alulated using Eq. 23b and the theory of Prosperetti, 18 dotted urves labeled P-Th agree rather losely, although the ontribution of th to tot is not signifiant. The ontribution of int to tot is also trivial. 3. Sattering ross setion Calulated values of the linear sattering ross setions of individual air bubbles, normalized to their respetive geometrial ross setions, are given in Fig. 4 for driving frequenies of 1 and 10 MHz. The strong effet of tissue elastiity i.e., G is apparent in these results, ausing the resonane peaks to shift to bubble radii two to four times larger than for the resonane peak in water. Even though they are larger, the ross setions for bubbles surrounded by tissue are less, and sometimes muh less, than for resonant bubbles in water. It is also seen that the urves broaden and diminish as either the rigidity G dereases or the frequeny f inreases, indiating that bubbles in tissue may be more diffiult to detet aoustially than are bubbles in water at the same frequeny. The ross setions for blood exhibit maxima at about the same radii as for water, but their magnitudes are less due to the higher visosity of that fluid. B. Numerial results In this part of the paper, simulation results will be given for numerial solutions of Eq. 12. The effets of elastiity FIG. 5. A omparison of radial responses for 1-m bubbles driven by a 1-MHz pulse at a 1 MPa and b 3 MPa, for G=0 thin line, and G=1.0 MPa thik line; the visosity was fixed at Pa s. on R-t urves will be examined first. Next, initial avitation thresholds will be presented for different elastiities and ompared to the result in water. Finally, maps of the strength of subharmoni signals emitted by osillating bubbles will highlight a ommon way to detet avitation and will illustrate the ranges of bubble radii that may be easily deteted using this method. 1. The effet of elastiity on R t urves Examples of Rt urves for a 1-m bubble osillating under 1-MHz driving pressures of 1 and 3 MPa are shown in Figs. 5a and 5b, respetively. The visosity is fixed at Pa s and the rigidity is hosen as 0 and 1.0 MPa. The effet of the elastiity is very obvious that it greatly redues the amplitude, and hene the nonlinearity, of the osillation. At 1 MPa, the amplitude of the osillation is muh smaller when elastiity is inluded. When the driving pressure inreases to 3 MPa, the amplitude of the osillation with nonzero elastiity is still smaller than that with zero elastiity, but the differene between the two ases is less. This indiates that the effet of elastiity will be less when the driving pressure is strong. Another feature whih is worthy of omment is that for the zero-elastiity ase, the bubble osillation approahes a steady-state resonane 22,24 of order 2/2, an example of period doubling and an indiation of the start of J. Aoust. So. Am., Vol. 118, No. 6, Deember 2005 X. Yang and C. C. Churh: Gas bubbles in soft tissue 3601

8 FIG. 6. A omparison of radial responses for 1-m bubbles driven by a 3.5-MHz pulse at a 1 MPa and b 3 MPa, for G=0 thin line, and G =1.0 MPa thik line; the visosity was fixed at Pa s. haoti osillation and very strong nonlinearity. For the nonzero-elastiity ase, no period doubling is observed under this driving pressure, although the inertial ollapses are still very strong. Clearly, the presene of the elastiity has redued or eliminated some nonlinear omponents of the bubble osillation. Figure 6 shows examples of Rt urves for a 1-m bubble driven by a 1 MPa and b 3 MPa at 3.5 MHz. Similar onlusions as those at 1 MHz an be drawn from these results. For the same driving pressure and the same size bubble, the nonlinearity appears to be weaker at the higher frequeny beause no period doubling is observed. Figure 7 shows the results for a 5-m bubble at 1 MHz driven by a 1 MPa and b 3 MPa. For zero elastiity, bubbles osillate with larger amplitudes during the first few yles than that for the nonzero-elastiity ase. Interestingly, after the initial state, the presene of the elastiity inreases the amplitude of osillation in both ases. In Fig. 7a, at zero-elastiity, the bubble osillation exhibits strong nonlinear behavior by a derease at the average radius, and the presene of elastiity reovers the linear osillation around the bubble equilibrium radius. In Fig. 7b, although the osillation amplitude is smaller at zero-elastiity ase, the osillation itself beomes haoti, while it is periodi when elastiity is inluded. Compared to bubbles of smaller size, a stronger nonlinearity is FIG. 7. A omparison of radial responses for 5-m bubbles driven by a 3.5-MHz pulse at a 1 MPa and b 3 MPa, for G=0 thin line, andg =1.0 MPa thik line; the visosity was fixed at Pa s. observed, whih is indiated by the approah to haoti osillations. Certainly, the nonlinearity does not always inrease when the bubble size inreases. 2. The effet of elastiity on the inertial avitation threshold In Fig. 8, predited inertial avitation thresholds are shown for a driving frequeny of 1 MHz, G=0, 0.5, 1.0, and 1.5 MPa, and =0.015 Pa s. The thresholds in water and FIG. 8. Predited thresholds for inertial avitation at 1 MHz J. Aoust. So. Am., Vol. 118, No. 6, Deember 2005 X. Yang and C. C. Churh: Gas bubbles in soft tissue

9 FIG. 9. Predited thresholds for inertial avitation at 3.5 MHz. blood are also shown in the plot for referene. The effet of elastiity on the threshold is obvious. Threshold values inrease as the elastiity inreases, as expeted based on the results for the Rt urves given earlier. For nonzero elastiity, the thresholds have signifiant struture that deserves omment. For example, at G = 1.5 MPa, for small bubble sizes, the threshold value is onsistent with the Blake threshold. As the bubble size inreases, the threshold value reahes a minimum value and then slowly rises. At around 4 m, the threshold value suddenly drops and then omes bak. The same shape ours at around 6 m, and another drop in threshold ours around 8 m. The lowest threshold attained following eah drop is less than the preeding minimum at a smaller bubble size. Similar strutures are observed for the lower elastiity ases, the only differene being that these drops our at different positions. This resonant struture appears to be related to the frational-order subharmoni resonane minima desribed previously. 22 The positions of the minima are determined by the elastiity of the surrounding medium, shifting to larger radii as G inreases. Figure 9 shows the same thresholds but at a driving frequeny of 3.5 MHz. At this higher frequeny, the threshold values are greater than at 1 MHz, and the inrease in thresholds for larger bubble sizes is muh faster than that at 1 MHz. The resonant strutures appearing in the nonzeroelastiity ases at 1 MHz are also observed on these threshold urves. The resonane struture differs signifiantly however in that the minimum values attained following the drops in the urves at 3.5 MHz are not less than the immediately preeding minima. The effet of visosity on the threshold an be observed by omparing the threshold for G=0 MPa and those of blood and water the only differene among the three is the value of visosity, =0.015, 0.005, and Pa s, respetively. In omparing, we onlude that thresholds inrease and have less struture as visosity inreases. 3. The map of the strength of subharmonis Figure 10 shows subharmoni emissions at a driving frequeny of 1 MHz as a funtion of driving pressure and bubble equilibrium radius. Results for tissues with G = 0, 0.5, and 1.0 MPa and =0.015 Pa s are shown, with the result for blood also shown for referene G=0 MPa, =0.005 Pa s. We will define a subharmoni signal as strong when it is greater than 30 db beause above this level, the subharmoni signal an be easily deteted by an instrument with an 8-bit dynami range. From Figs. 10a 10, it is seen that as the elastiity inreases, the strong subharmoni signal region moves toward larger bubble sizes and higher driving pressures. The strong subharmoni signal region generally lies above the orresponding inertial avitation threshold. For small values of elastiity, the inertial threshold oinides with the lower boundary of the strong subharmoni emission region better than does that for higher elastiity ases. This result is expeted beause the subharmoni signal investigated here arises from haoti bubble osillations, and haoti osillations usually our after the bubble motion beomes strongly nonlinear. In omparing the map for blood and that for tissue with G =0 MPa, we find that the strong subharmoni signal region beomes smaller when the visosity inreases. The ontour lines on these plots indiate the boundaries between regions in whih the maximum emission ours at a partiular frequeny. The results show that all strong subharmoni emissions our only when the fundamental frequeny omponent has the maximum emission level. Figure 11 shows results for the same ases as in Fig. 10, but at a driving frequeny of 3.5 MHz. As the elastiity inreases, the strong subharmoni emission region shrinks and moves toward higher pressure amplitudes. Signifiantly perhaps, the relative signal also seems stronger. Compared to results at 1 MHz, the strong subharmoni emission region at 3.5 MHz is muh smaller and is limited to the small bubble region. This limitation to the small bubble region probably is related to the resonane struture of these bubble responses. At 3.5 MHz, the linear bubble resonane size is smaller than that at 1 MHz 3.07 vs m. Again, all strongemission regions are above the inertial avitation thresholds for the orresponding elastiity, and strong subharmoni emissions our only when the fundamental frequeny omponent has the maximum emission level. IV. DISCUSSION AND CONCLUSIONS In this study, we developed a theoretial model for the pulsations of gas bubbles in simple linear visoelasti solids and presented some potentially useful results for the ase of soft tissues. As pointed out in the text, although the model is simple, it is onsistent with experimental data taken for some soft tissues. However, at high intensity, bubble osillations are strongly nonlinear. Although strong nonlinear osillations do not automatially imply that a nonlinear visoelasti model is neessary to desribe the bubble motion, the suitability of this linear model also remains unlear. As a matter of fat, there is little evidene that the strain-stress relation in tissue is not linear. Even though the hange in bubble radius is signifiant ompared to its initial dimension, the overall strain in the tissue ould still be onsidered small if this hange were ompared to the dimension of the soft tissue. Use of a different visoelasti model ertainly would result in different preditions, but the sparse measurement data at J. Aoust. So. Am., Vol. 118, No. 6, Deember 2005 X. Yang and C. C. Churh: Gas bubbles in soft tissue 3603

10 FIG. 10. Subharmoni emissions in db relative to the peak emission at a driving frequeny of 1 MHz as a funtion of driving pressure and bubble equilibrium radius for three tissues: a G=0 MPa and =0.015 Pa s; b G=0.5, =0.015; G=1.0, =0.015; and for d blood, G=0, = The ontour lines indiate the boundaries between regions in whih maximum emissions our only at one frequeny omponent: fundamental right-most region and seond harmoni solid line; seond harmoni and third harmoni dashed line; third harmoni and fourth harmoni dotted line; fourth harmoni and fifth harmoni dash-dot line. megahertz frequenies limits the study of verified visoelasti models for soft tissue. Also, the present study provides preditions that an be examined experimentally. Verifiation of this model will be pursued in the future. Linear analysis of this inherently nonlinear formulation provided analytial preditions of bubble responses to insonation at low pressure amplitudes. The result for resonane frequeny inreases as the modulus of rigidity inreases, as was expeted based on previous work. 17 The results for damping onstants showed that the visosity of the tissue tends to dominate either thermal or elasti damping for bubbles smaller than 30 m for frequenies smaller than the linear resonane size, while aousti damping predominates at higher frequenies. It is also expeted that thermal damping would dominate other soures of damping for R 0 30 m and 0 in a more rigorous treatment of thermal effets than is given by the polytropi assumption used here. The peaks in the urves for sattering ross setion shift to larger radii as the rigidity inreases due to the inrease in resonane frequeny, although their magnitudes are less, and sometimes muh less, than is the ase for resonant bubbles in water. This is onsistent with the numerial results used in produing Figs. 10 and 11. The effet of elastiity on bubble dynamis was investigated in some detail. Overall, the presene of the elastiity in a bubble dynamis equation will redue, sometimes greatly, the nonlinearity of bubble osillations. As might be expeted, the inertial avitation threshold was shown to be greater in tissue than in liquids suh as water or blood, in ontrast to the assumptions underlying the mehanial index. 19,25 This result should prove useful for understanding the prevalene of potentially damaging inertial avitation in vivo. This will be the subjet of a later study. Subharmoni emissions from an osillating visoelasti bubble were also studied. Sine soft tissue generally is not transparent, deteting aousti emissions is often the best way to gain information about a bubble. A passive or ative avitation detetor system an easily provide information about bubbles inside soft tissues. When interpreting suh re J. Aoust. So. Am., Vol. 118, No. 6, Deember 2005 X. Yang and C. C. Churh: Gas bubbles in soft tissue

11 FIG. 11. Subharmoni emissions in db relative to the peak emission at a driving frequeny of 3.5 MHz as a funtion of driving pressure and bubble equilibrium radius for three tissues: a G=0 MPa and =0.015 Pa s; b G=0.5, =0.015; G=1.0, =0.015; and for d blood, G=0, = The ontour lines indiate the boundaries between regions in whih maximum emissions our only at one frequeny omponent: fundamental right-most region and seond harmoni solid line; seond harmoni and third harmoni dashed line; third harmoni and fourth harmoni dotted line; fourth harmoni and fifth harmoni dash-dot line. sults however, it will be important to keep in mind that the sizes of the bubbles deteted will be greater than estimated using linear theory for water. Shape osillations and retified diffusion are not onsidered in this study, although they are very important to prediting the stability and equilibrium size of a bubble. 26 Shape osillations ould also generate larger stresses that ould ause severe mehanial damage to soft tissue. Retified diffusion will hange the equilibrium size of a bubble, and the bubble motion an thereby be greatly affeted. These effets will be the fous of future studies. ACKNOWLEDGMENTS The suggestions and support of the Biomedial Aousti group at NCPA are gratefully aknowledged. This work was supported by Award No. DAMD , administered by the US Army Medial Researh Aquisition ativity, Fort Detrik, MD. The information ontained herein does not neessarily reflet the position or poliy of the US government, and no offiial endorsement should be inferred. 1 H. S. Fogler and J. D. Goddard, Collapse of spherial avities in visoelasti fluids, Phys. Fluids 13, I. Tanasawa and W. J. Yang, Dynami behavior of a gas bubble in visoelasti liquids, J. Appl. Phys. 41, A. Shima, T. Tsujino, and H. Nanjo, Nonlinear osillations of gas bubbles in visoelasti fluids, Ultrasonis 24, C. Kim, Collapse of spherial bubbles in Maxwell fluids, J. Non- Newtonian Fluid Meh. 55, V. N. Alekseev and S. A. Rybak, The behavior of gas bubbles in insonated biologial tissues, Aoust. Phys. 44, J. S. Allen and R. A. Roy, Dynamis of gas bubbles in visoelasti fluids. I. Linear visoelastiity, J. Aoust. So. Am. 107, J. S. Allen and R. A. Roy, Dynamis of gas bubbles in visoelasti fluids. II. Nonlinear visoelastiity, J. Aoust. So. Am. 108, C. C. Churh, Spontaneous homogeneous nuleation, inertial avitation and the safety of diagnosti ultrasound, Ultrasound Med. Biol. 28, L. A. Frizzell, E. L. Carstensen, and J. F. Dyro, Shear properties of mammalian tissues at low megahertz frequenies, J. Aoust. So. Am. 60, J. Aoust. So. Am., Vol. 118, No. 6, Deember 2005 X. Yang and C. C. Churh: Gas bubbles in soft tissue 3605

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