analysis of nonlinear acoustic wave propagation in hifu...

Scientia Iranica B (2018) 25(4), 2087{2097

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringhttp://scientiairanica.sharif.edu

Analysis of nonlinear acoustic wave propagation inHIFU treatment using Westervelt equation

S. Haddadia and M.T. Ahmadianb;�

a. School of Mechanical Engineering, Sharif University of Technology, Azadi Ave., Tehran, Iran.b. Center of Excellence in Design, Robotics and Automation (CEDRA), School of Mechanical Engineering, Sharif University of

Technology, Tehran, P.O. Box 11155-9567, Iran.

Received 18 July 2016; received in revised form 8 March 2017; accepted 10 June 2017

KEYWORDSHIFU;Nonlinear wavepropagation;Pennes's bio-heatequation;KZK equation;Westervelt equation.

Abstract. Currently, the HIFU (High-Intensity Focused Ultrasound) therapy methodis known as one of the most advanced surgical techniques of tumor ablation therapy.Simulation of the non-linear acoustic wave and tissue interaction is essential in HIFUplanning to improve the usefulness and e�ciency of treatment. In this paper, linear,thermoviscous, and nonlinear equations are applied using two di�erent media: liver andwater. Transducer power of 8.3-134 Watts with the frequency of 1.1 MHz is considered asthe range of study to analyze the interaction of wave and tissue. Results indicate that themaximum focal pressure of about 0.5-4.3 MPa can be achieved for transducer power ratesof 8.3 to 134 W. The simultaneous solving of the acoustic pressure and Pennes's bio-heatequations can help determine the amount of temperature rise at the focal point and ablatedarea. Finally, the linear and nonlinear simulations are compared, and the turning point oftransition from linearity to nonlinearity is determined. The simulated results provide uswith the required information about the behavior of the focalized ultrasound in interactionwith liver tissue. The performance of phased array HIFU transducer can be improved fortreatment considering lesion size as well as temperature rise in tissue and for choosing thebest range of operational power.© 2018 Sharif University of Technology. All rights reserved.

1. Introduction

Cancer has been one of the main reasons of deathin the last decades all around the world. Nowadays,many solid cancerous cells in various parts of thebody, including pancreas, prostate, breast, uterine �-broids, and liver, are eroded by High-Intensity FocusedUltrasound (HIFU) radiation as one of the e�cientemerging medical therapies [1,2]. In comparison withother conventional cancer treatment techniques, such

*. Corresponding author. Tel: +98 21 66165503E-mail addresses: s [email protected] (S. Haddadi);[email protected] (M.T. Ahmadian)

doi: 10.24200/sci.2017.4496

as surgery, radio- and chemo-therapy, HIFU has theadvantage of being non-invasive without ionization; be-sides, less post-treatment requirements are reported [3].However, severe side e�ects may occur during thetherapy if the vital blood vessels around the tumorsare subjected to damage by HIFU Hynynen et al. [4].Understanding the HIFU process makes it possible tocontrol and save the desired treatment area from failureand to assure the safety of patients and e�ectiveness ofthe treatment.

With the advent of the computers, which have theability to solve wave equations numerically, studyingthe e�ects of nonlinear wave propagation in di�erentmedia has become possible Bjorno [5]. Wave propa-gation is a nonlinear phenomenon, especially in tissueand human organs. Therefore, assuming a linear wave

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2088 S. Haddadi and M.T. Ahmadian/Scientia Iranica, Transactions B: Mechanical Engineering 25 (2018) 2087{2097

equation, especially with high-amplitude, does not givea correct solution to wave propagation problem. Manye�orts have been made to better predict the e�ects ofnonlinearities on the wave propagation, making it aninteresting �eld of study for many researchers in orderto determine and investigate the nonlinear behaviors ofwaves [3,6-8].

There are many di�erent methods for solvingnonlinear wave equations, each with advantages anddisadvantages. Implementation of di�erent numericalmethods in modeling can predict system behavior withdi�erent accuracies. The most common method forsolving wave equations is the Finite-Di�erence Method(FDM). Okita et al. [9] used the FDM to simulatethe HIFU waves on brain tumors. The Finite-ElementMethod (FEM) is another way to simulate the waveequation with good accuracy. Wong et al. [10] appliedthe FEM to the simulation of CMUTs. An extensiveset of issues in mechanical engineering, especially inthe �eld of biomechanics, can be usually solved by theFEM method. In the HIFU simulation, the FEM cancalculate the ablation area in an e�cient way basedon the transducer power, tissue type, and exposuretime.

To date, many equations for simulating nonlinearwave propagation have been presented. Burgers equa-tion in 1948, Westervelt equation in 1963, and KZKequation in 1971 Hamilton [11] are the most widelyused equations to predict the propagation of nonlinearwaves. The KZK equation usually gives an acceptableanswer in simulated waves for transducers with aper-ture angles of less than 18 degrees [12,13]. The goodchoice of boundary condition in KZK equation mayextend its area of application. However, to simulate thewave propagation for transducers with aperture anglesof more than 18 degrees, the Westervelt equation isgenerally used [14,15].

The subject of how best to model the physicalinteractions between ultrasound waves and biologicaltissue has been widely considered, and investigationsin this area are still making progress. At high in-tensities, nonlinear wave propagation e�ects lead tothe distortion of the waveform. Higher harmonics aregenerated due to the nonlinear distortion. The tissue,while enhancing the local heating, more readily absorbshigher harmonics. The peak temperature at focuspredicted by a nonlinear equation �eld is markedlyhigher than that predicted by a linear equation due tostrong absorption of the higher-frequency harmonics.

At low focal intensities, nonlinear e�ects causeenergy to be generated up to at least the 10th har-monic [16]. At very high focal intensities wherestrongly shocked waves are produced, as many as 600harmonics might be required to model the focal heatingaccurately [17]. Thus, the frequency content of thepropagating ultrasound waves can be very broadband.

Solving large problems involving high focal intensitiesis still a big challenge.

In the present study, a HIFU transducer that isspherically focused with an aperture angle of maximum30 degrees and a focal length of 50 mm is used. Waveinteraction with the liver tissue is simulated in fourpoints in the range of 8.3-134 W as the transducerpower. Any increase in the transducer power and, con-sequently, pressure intensity leads to an intensi�cationof the nonlinear e�ects. Therefore, the nonlinearityis considered in the numerical simulation. Numericalsimulations can improve the prediction accuracy andincrease the e�ciency of the models in di�erent casestudies. Herein, HIFU wave propagation in bothlossless and thermoviscous media is simulated, and thee�ects of the nonlinear factors in di�erent pressures onwave propagation are studied.

2. Methods

2.1. Numerical methodTo study the HIFU treatment, ultrasound wave propa-gation across a heterogeneous medium, such as a liver,should be explained. Two PDEs for acoustic wavepropagation and biological heat transfer are simultane-ously solved in the liver medium. The acoustic equa-tion in the wave propagation problem can be generallydivided into three di�erent types. The �rst one is alinear wave equation for simulating waves with a lowenergy level and short amplitude in a lossless medium.The second type is studying the e�ects of thermalconductivity and viscosity on the equation. This typeof equation leads to a satisfactory answer for mediawith high loss. Finally, to simulate wave propagationby considering the e�ects of high amplitude and energydissipation in heterogeneous environments, the bestanswer comes from solving the nonlinear equations.Herein, in order to evaluate and compare di�erent wavepropagation types, equations should be solved in thetime domain.

2.1.1. Linear acoustic wave equationIn solving the linear acoustic wave equation, themedium is assumed to be lossless. The linearwave equation used in Comsol software (ComsolMultiphysics® version 5.1, Comsol Company, Sweden)is Eq. (1). This equation in the time domain (transitionstate) can be expressed as follows:

1�c2

@2pt@t2

+r:�� 1�c

(rpt � qd)�

= Qm; (1)

pt = p+ pb; (2)

where p is the pressure, c is the speed of sound, t isthe time, and � is the density. In addition, Qm and qdare the monopole and dipole terms, respectively, and

S. Haddadi and M.T. Ahmadian/Scientia Iranica, Transactions B: Mechanical Engineering 25 (2018) 2087{2097 2089

are de�ned in the software separately. In this equation,attenuation can be de�ned in the monopole term as asource. The linear equation is derived from inputtingthe value of Qm = qd = 0.

2.1.2. Thermoviscous wave equationThe thermoviscous equation in Comsol software is asfollows:

1�c2

@2pt@t2

+r:�� 1�c

(rpt � qd)

+1�c2

�4�3

+ �B +( � 1)kCp

�@rpt@t

�= Qm; (3)

where � is the viscosity, Cp is the heat capacity, is the ratio of speci�c heats, and k is the thermalconductivity.

This equation for the media with high-energy losshas acceptable accuracy. The term ( 4�

3 + �B) in theequation is related to the viscosity, showing that thelosses are enhanced in higher viscosity. This means thatas the medium gets nearer to gas phase, the in uenceof the viscosity reduces. Another term is ( ( �1)k

Cp ),which studies the e�ects of the thermal conductivity.By going from the gas phase to the solid phase, thecompressibility factor will decrease and Cp value willbe equal to Cv. Therefore, reducing the compressibilitydecreases the impact of the thermal conductivity. Inaddition, in the thermoviscous simulation, the valuesof Qm and qd are zero.

2.1.3. Acoustic nonlinear equationA nonlinear acoustic �eld can be simulated by Eq. (4);this equation is known as the Westervelt equation andcan be used to simulate acoustic pressure in tissue:

r2p� 1c20

@2p@t2

+�c40

@3p@t3

= � ��0c40

@2p2

@t2; (4)

where p is the acoustic pressure, � = 1 + B=2Ais a nonlinearity coe�cient, and � is the di�usivityof sound, which comes from viscosity and thermalconductivity. The value of � can be calculated fromEq. (5):

� =1�0

�4�3

+ �B�

+k�0

�1Cv� 1Cp

�: (5)

The �rst and second terms of Eq. (4) represent a linearpropagation of waves in a medium without any loss.The third term is related to the losses due to thethermal conductivity and viscosity of the uid. Thelast term of the equation is related to the nonlinearfactors in uencing the simulation of wave propagation

and causing thermal and mechanical changes withinthe tissue.

Nonlinear propagation of sound beam in a ther-movisous uid may also be described by the KZKequation [11]:

@2p@z@tKZK

=c02r2p+

�2c30

@3p@t3KZK

+�

2�0c30

@2p2

@t2KZK:(6)

To validate the proposed algorithm, harmonics gener-ated by the concave transducer in a two-layer mediumusing the Comsol software are compared with thosesimulated using the KZK model (HIFU Simulatorv1.2, Food and Drug Administration, Silver Spring,MD) [18].

To study the impact of nonlinear factors, bothlinear and nonlinear equations are calculated. Anotherimportant parameter in the simulation is the waveintensity. The intensity of the wave can be calculatedby Eq. (7):

I =P 2t

2�c: (7)

2.1.4. Bio-heat transfer equationThe heat generated anywhere in the tissue is a�ectedby two important parameters. The �rst parameteris the acoustic wave adsorption coe�cient as one ofthe intrinsic properties of the tissue, and the secondis the sound wave intensity, calculated by solving thewave equation on the tissue geometry of the study.Therefore, the heat source can be achieved by Eq. (8):

Q = 2�absI: (8)

In the above equation, �abs is the attenuation co-e�cient, which is proportional to the frequency andcalculated by Eq. (9):

� = �0

�ff0

��: (9)

�0 is the attenuation coe�cient at the frequency off0 = 1 MHz and � = 1 in soft tissue.

The Westervelt equation is obtained for thermo-viscous uids. The attenuation in the thermoviscous uids is proportional to Eq. (9). This relation is linearfor a soft tissue [19].

Eq. (8) is considered a heat source of the heattransfer equation. Heat transfer is studied within thetissue in this simulation; therefore, the main mecha-nism of heat transfer and heating the tissue during wavepropagation or afterward is the thermal conductivity.Therefore, the heat transfer equation can be written asin Eq. (10), which is dependent on time:

�Cp�@T@t

+ (u:r)T�

= r:(krT ) +Q+Qbio: (10)


Considering that convection is not a governing heattransfer mechanism in this case, the second term onthe left hand of Eq. (10) can be neglected. Qbio iscalculated from Eq. (11):

Qbio = �bCb!b(Tb � T ) +Qmet: (11)

Tb is the arterial blood temperature.

2.2. Thermal doseThe thermal dose, which is de�ned by Sapareto andDewey [20], indicates the relation between the time andtemperature to heat the tissue in order for necrosis tooccur. In HIFU surgery (temperature is usually above50�C), the thermal dose is de�ned as follows:

TD =Z tfinal

t0R(T�43)dt �

tfinalXt0

R(T�43)�T: (12)

In Eq. (12), the values of R for T > 43�C and 37�C <T < 43�C are equal to 2 and 4, respectively. Accordingto this equation, if the tissue temperature increases to43�C, the required time is approximately 240 minutesto reach cell death. If the temperature increases to56�C, this time is reduced to 1 second.

2.3. GeometryTo solve the problem in a two-dimensional space, atransducer with a 50 mm length of the focal point anda hole in its center with a diameter of 10.22 mm (inorder to reduce the occurrence of wave interference) ismodelled. In this study, as shown in Figure 1, thetissue is immersed in a uid medium (water). Thetissue dimensions are 50�45 mm, and the propagationmedium is assumed 120� 120 mm.

2.4. Boundary conditionsIn order to simulate wave propagation in a two-dimensional space, two sets of boundary conditionsare needed. The �rst boundary condition is used forsolving the acoustic equation and the second boundarycondition for solving the heat transfer equation. Theboundary conditions are de�ned as follows:

1. Boundary condition Pt = P0 sin(!t) is used as thesource pressure;

2. A plane wave radiation boundary condition is de-�ned on the walls as a boundary condition withoutany re ection (transparent). The equation is:

�n:��1�

(rpt � qd)�

+1�

�1c@P@t

�= Qi: (13)

Figure 1. Geometry and boundary conditions used in thesimulation.

3. The tissue wall boundary condition is de�ned asT = T0 = 310 K.

The initial conditions for the acoustic equationsare P = 0 and @P

@t = 0, and the initial condition forsolving the heat transfer equation is T = T0 = 310 K.Figure 1 shows the boundary conditions of the problem.

2.5. Problem descriptionIn order to investigate the e�ects of nonlinearity onwave propagation, transducer surface pressures of100, 200, 300, and 400 kPa are considered, whichare equivalent to operating power rates of 8.3,33.5, 75.4, and 134 W. The time step and totaltransient simulation time are 1 and 300 microseconds,respectively. After 300 microseconds, as a uniformpressure �eld is obtained, a steady-state acousticpressure equation is considered to �nd the temperaturedistribution. Table 1 shows the parameters used inthe simulation for water and liver tissue.

A frequency of 1.1 MHz is used in the simulation.The wave exposure time to determine the ablated areais 60 seconds.

At the focal point, an improved mesh size of �=8is used. Approaching the walls, the mesh size increases.A mesh dependency test is performed by reducing themesh size by 30%, and the results indicate the changesof less than 1%.

Table 1. Parameters used in the simulation.

AttenuationConstant (1/m)

Density(kg/m3)

Heat capacity(J/kg/�C)

Thermal conductivity(W/m/�C)

Speed of sound(m/s)

B/A

Tissue (liver) 6.915 1078.75 3540 0.52 1586 6.8

Water 0.025 998.2071 4180 0.6 1482 5.2


3. Results

3.1. Model veri�cationIn this article, model veri�cation is based on experi-mental data extracted from the study by Karab�oce andDurmu�s [21] for temperature distribution, and the FDAHIFU simulator is used to verify pressure distribution.

HIFU Simulator is a freely distributed, MATLAB-based software package for simulating axisymmetricHigh-Intensity Focused Ultrasound (HIFU) beams [18]by employing KZK equation. The KZK equation hastwo main limitations, which do not exist in Westerveltequation. First, e�ects of re ection and scattering arenot taken into account. Second, it can be applied onlyfor the transducers with small aperture angles (< 18�).

HIFU Simulator is used to calculate the pressureat the focal point for transducer power rates of 33.5,75.4, and 134 W. Results show that the deviationsbetween Westervelt and KZK equations for transducerpower rates of 33.5, 75.4, and 134 W are 10.6, 12.5, and11.9 percent, respectively.

The large aperture angle is the main reason forthe deviation between KZK and Westervelt nonlinearmodels.

3.2. One-dimensional wave simulationIt is essential to study the one-dimensional wavepropagation characteristics to investigate the wavedistribution and attenuation. For this purpose, one-dimensional simulations of the wave propagation alongthe water and liver tissue for nonlinear model areperformed and represented in Figure 2. In thesediagrams, the blue lines represent the wave propagation

in the liver, and the black dashed lines display thepropagation of waves in the water. These waves aregenerated by a monopole wave source with the abilityof generating a sinusoidal wave with 200 kPa as theinitial pressure.

In the magni�ed areas of a, b, and c, a phase lag ofapproximately 4� along the wave propagation directioncan be seen. This is due to di�erent speeds of sound inliver tissue and water. Furthermore, due to the higherviscosity of the liver, the amount of losses in the livermedium is more than water, which can be inferredfrom wave domain reduction along the longitudinaldirection. Energy losses in the liver medium alongthe 50-mm path of wave propagation are more thanapproximately 28% of the water medium. This loss ofenergy mainly occurs due to the liver's higher viscosity.Inherent viscosity of the liver tissue causes shear force,while the ultrasound wave passes through it, convertingsome of the wave energy into the thermal energy.

It should be considered that di�raction and scat-tering properties of nonlinear model cannot be shownin one dimension, and these terms will be explainedand clari�ed in a two-dimensional simulation.

3.3. Two-dimensional simulationIn the two-dimensional simulation, considering nonlin-earity, thermoviscous terms, di�raction and scattering,and proper choice of acoustic pressure equation to ob-tain a precise answer aligned with the real experimentaldata is essential.

3.3.1. Acoustic pressure �eldThe acoustic �eld in water and liver media produced

Figure 2. One-dimensional nonlinear wave propagation in liver and water versus distance: (a) 0-3 mm, (b) 24-26 mm,and (c) 43.5-47.5 mm from the transducer.


by the HIFU transducer with the power of 8.3-134 Wis simulated that includes the transition range fromlinearity to nonlinearity.

3.3.2. The comparison of the linear and nonlinearacoustic pressure �elds

In this paper, Eq. (1) and the Westervelt equationare used for the numerical simulation of the linearand nonlinear wave propagations, respectively. Usingthe Comsol software, the thermoviscous and nonlinearparameters of the Westervelt equation are de�ned asmonopole sources with time steps of 1 microsecond.Figure 3 shows the simulation results of the linearand nonlinear acoustic pressure equations after 50microseconds radiation. The right picture shows theresults of solving the linear pressure equation, whilethe left one shows the results of the nonlinear pressureequation.

Figure 4 shows the pressure variation along thelateral axis at the focal point. The transducer poweris set to 134 W, and the maximum pressure at thefocal point, as shown in Figure 4, for the nonlinearequation is equal to 4.3 MPa. According to this�gure, the nonlinear curve is signi�cantly a�ectedby the excitation of higher harmonics, and the focalpoint shows a maximum pressure, which is 1.9 timesgreater than the maximum pressure in linear mode.Local maxima can also be seen in the nonlinear casethat is due to superposition of waves with di�erentwavelengths representing the higher harmonics.

The higher the pressure is, the more the nonlinearbehavior of the wave increases. Figure 5 represents thenonlinear pressure curve for transducer power rates of33.5, 75.4, and 134 W along the lateral-axis.

Figure 6 represents the acoustic pressure alongthe axial-axis. As shown in this �gure, the nonlinearpressure is 1.9 times greater than the pressure in linearmode. In the thermoviscous case, a large part of the

Figure 4. Pressure function along the lateral axis in theliver at the focal point. The operating power andfrequency are 134 W and 1.1 MHz, respectively.

wave energy is lost because of the factors such asviscosity and speci�c heat rate that a�ect the waveand its absorption by the environment. Finally, thislost energy is absorbed and the tissue temperature isincreased.

Figure 7 shows the acoustic pressure at the focalpoint for di�erent powers. For powers greater than40 W, the in uence of nonlinear factors intensi�es. Inother words, the linear simulation assumption is notcorrect for powers greater than 40 W.

In the liver medium, at pressures higher than4 MPa, the wave behavior is completely nonlinear,and the nonlinear simulation di�ers vastly from thelinear assumption in this case. The ranges coveringtransducer power of 40-80 W and focal pressure of 1.5-2.5 MPa can be considered as the turning point for the

Figure 3. Acoustic pressure �eld results of the linear (a) and nonlinear (b) equations at radiation time of 50 microsecondsin liver medium and 1.1 MHz as the wave frequency.


Figure 5. Nonlinear pressure curve in the liver fordi�erent operational powers along the lateral-axis acrossthe focal point at 1.1 MHz frequency.

Figure 6. Acoustic pressure value for di�erent equationsalong the axial-axis in the liver. Operating power andfrequency are 134 W and 1.1 MHz, respectively.

transition from linear to nonlinear behavior in the livertissue.

3.3.3. The focal point temperature and ablated areaThe pressure at the focal point increases as the trans-ducer power increases. This pressure rise leads tononlinear behavior at the focal point, resulting ina sharp rise in the tissue temperature. Figure 8shows the temperature increase and ablated area fordi�erent transducer powers. It can be concluded thatthe power increase and excitation of higher harmonicssigni�cantly in uence the ablated area. Ablated area iscalculated from thermal dose equation, and the resultsare shown in Figure 9.

Figure 10 shows the areas of the thermal dose

Figure 7. Acoustic pressure at the focal point in the liverversus di�erent powers using linear, thermoviscous, andnon-linear equations at 1.1 MHz frequency.

above the lethal threshold of 240 CEM43. The Cu-mulative Equivalent Minutes at 43�C model (CEM43),introduced by Sapareto and Dewey [20], represent theconcept of thermal isoe�ect dose: a reference temper-ature (43�C) has been chosen to convert all thermalexposures into equivalent minutes at this temperature.

It can be concluded from the results listed inTable 2 that, at higher powers, a smaller part ofthe tissue is a�ected by HIFU therapy; further, therequired insonation time is much lower. Reduction oftime and ablated area is of great bene�t to surgeriesin order to control the scope of damage to surroundingtissue. On the other hand, increasing the power andthe focal pressure will extend the chance of cavitationoccurrence, and the creation of micro-bubbles willspread the damage to neighbor tissues. Therefore, anoptimum amount of power should be applied as the bestsolution, which is equal to the highest power possiblebefore reaching the cavitation during operation ontissue.

In order to validate the temperature simulation,the ablated area related to radiation times of 5, 10,15, ..., 40 seconds is calculated and compared with theexperimental data from Karab�oce [21]. The deviationbetween the predicted ablated area of the simulationand Karab�oce's reported experimental results is cal-culated, implying that the two groups of data agreeclosely and the overall average error of less than 7% interms of ablated area is observed.

4. Discussion

In this paper, Westervelt nonlinear equation is used toanalyze the HIFU waves and tissue interaction. Two-dimensional analysis is performed in order to obtain theprecise result for pressure distribution and attenuation


Figure 8. (a) Increased temperature at the focal point in the liver by the application of di�erent powers and exposuretime of 60 s. (b) Ablated area at the focal point by the application of di�erent powers and exposure time of 60 s.

Figure 9. Thermal dose contour and ablated area in the liver considering the exposure time of 60 s and 1.1 MHzfrequency for (a) P = 134 W, and (b) P = 75:4 W.

Table 2. Ablated area results obtained for di�erent transducer operating conditions in the liver and 1.1 MHz frequency.

Case Power (W) Radiation time (s) Ablated area (cm2)

a 134 8.3 0.12

b 75.4 35 0.30

c 33.5 660 1.88

d 8.3 1400 3.77

along the focal point. Linear, thermoviscous, and non-linear Westervelt equations are applied by consideringtwo di�erent media: liver and water.

Considering the nonlinear e�ects, the acousticpressure at the focal point is approximately 1.9 timeshigher than the linear result. The results show thatthe transient behavior ranging from complete linearityto nonlinearity happens in transducer power ratesof 40-80 W. At powers more than 80 W, using thelinear equation for analysis results in big di�erencesin measured parameters and experimental �ndings;therefore, at high powers, it is necessary to considerthe nonlinear terms in the equation.

Moreover, the application of higher pressures

leads to a meaningful time reduction and reaching adesirable temperature to start cell death; this is bene-�cial to the achievement of a smaller, more controllableablated area to prevent surrounding tissue damage.However, considering that higher power increases theprobability of the cavitation phenomenon that makescontrolling the surgical operating conditions di�cult,an optimum operating pressure should be selected tohave the best therapeutic e�ect.

In this study, it is shown that, using the trans-ducer surface pressure of 400 kPa (equivalent to 134 Wpower), a 0.12 cm2 ablated area can be achieved in8.3 seconds. In this case, the tissue experiences amaximum focal pressure of about 4.3 MPa, which is


Figure 10. Amount of ablated area at the focal point in the liver and frequency of 1.1 MHz for various operationalexposure times and powers of (a) 8.3 s and 134 W, (b) 35 s and 75.4 W, (c) 600 s and 33.5 W and (d) 1400 s and 8.3 W.

far enough from the 5.3 MPa threshold reported forthe start of cavitation [22].

Cavitation is de�ned as the interactions betweenthe ultrasound �eld and small bubbles containing gas.Usually, bubbles or small gas nuclei are naturallypresent in biological tissues; however, using HIFU willaid the creation and encapsulation of gas bubbles intissues. At the average HIFU intensities, a bubblevibrates �rmly in an acoustic �eld while absorbs andradiates energy to the surrounding, called stable cavi-tation.

However, when the acoustic intensity increases,the bubble vibrations become signi�cantly nonlinearand lead to violent collapse, known as inertial cavi-tation. Inertial cavitation stimulates the micro jets,shock wave, and highly elevated local heat that mayreach thousands of degrees Kelvin. The free radicalsproduced in these temperatures can cause mechanicalcell destruction and even melting the solid cells andtissues.

If controlled, highly localized hyperthermia is thedesired outcome, then most investigators may argue

that cavitation is to be avoided because it has typicallyleads to unpredictable thermal results. Several authorshave observed irregular lesions and collateral damageoutside the focal zone when uncontrolled cavitationoccurred during insonation [24-27]. These studiesall contain unambiguous statements recommendingactively avoiding cavitation when the therapeutic goalis the controlled localized heating.

Although cavitation-related enhanced heating isobserved in some of the preceding studies [22,23], theobservation of enhanced heating due to bubble activitydeserves closer attention. A pressure threshold forenhanced heating of about 5.3 MPa at 1 MHz isreported [22].

The goal of this study is the development of acomputational tool for HIFU ablation therapy to assuresafety of the patient and e�ectiveness of the treatment.The proposed model could simulate the non-linearwave propagation from the source and distribution ofexcitation through tissue layers. The performance ofphased array HIFU can be improved in the treatmentplanning. Numerical simulation represents a useful


approach to predicting the pressure and temperaturedistributions and, consequently, gauging the safety ande�ectiveness of HIFU devices.

Nomenclature

B Nonlinearity parameterC Speed of sound (m/s)CP Heat capacity at constant pressure

(J/kg.K)f Frequency (Hz)I Intensity (W/m2)K Thermal conductivity (W/m.K)nk Wave directionP Pressure (Pa)Pb Background pressure (Pa)Pt Total pressure (Pa)qd Dipole source (N/m3)Qm Monopole source (1/s2)Qmet Metabolic heat source (W/m3)T Temperature (K)Tb Arterial blood temperature (K)� Attenuation coe�cient (NP /m) Ratio of speci�c heats� Di�usivity of sound (m2/s)� Wave length (m)� Dynamic viscosity (Pa.s)�B Bulk viscosity (Pa.s)� Density (kg/m3)! Angular frequency (rad/s)

Index

b BloodMax MaximumMin Minimum

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Biographies

Samaneh Haddadi received her BS degree in Me-chanical Engineering in 2005 from University ofTehran, Tehran, Iran and completed the requirementsfor MS degree in Mechanical Engineering in 2008 fromSharif University of Technology, Tehran, Iran. She hasbeen studying to get PhD in Mechanical Engineeringfrom 2011 in Sharif University of Technology. Herresearch interests are Ultrasonics and Bioengineering.

Mohammad Taghi Ahmadian received his BS andMS degrees in Physics in 1972 from Shiraz University,Shiraz, Iran and completed the requirements for BSand MS degrees in Mechanical Engineering in 1980from University of Kansas in Lawrence. At the sametime, while getting his master degree, he completed hisPhD in Physics and PhD in Mechanical Engineering in1981 and 1986, respectively, from University of Kansas.He started working as an Assistant Professor in Sharifuniversity of Technology from 1989; at present, he isa Professor in the School of Mechanical Engineering inSharif University of Technology, Tehran.

analysis of nonlinear acoustic wave propagation in hifu...

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