research article experimental verification of modeled

Research ArticleExperimental Verification of Modeled Thermal DistributionProduced by a Piston Source in Physiotherapy Ultrasound

M I Gutierrez1 S A Lopez-Haro2 A Vera2 and L Leija2

1CONACYT Instituto Nacional de Rehabilitacion Subdireccion de Investigacion Tecnologica Calz Mexico XochimilcoNo 289 Col Arenal de Guadalupe Mexico City Mexico2Centro de Investigacion y de Estudios Avanzados del IPN (Cinvestav-IPN) Electrical Engineering Department Bioelectronics SectionAv Instituto Politecnico Nacional 2508 Col San Pedro Zacatenco Del Gustavo A Madero Mexico City Mexico

Correspondence should be addressed to M I Gutierrez mibrahingutierrezgmailcom

Received 10 August 2016 Accepted 23 October 2016

Academic Editor Ayache Bouakaz

Copyright copy 2016 M I Gutierrez et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Objectives To present a quantitative comparison of thermal patterns produced by the piston-in-a-baffle approach with thosegenerated by a physiotherapy ultrasonic device and to show the dependency among thermal patterns and acoustic intensitydistributions Methods The finite element (FE) method was used to model an ideal acoustic field and the produced thermalpattern to be compared with the experimental acoustic and temperature distributions produced by a real ultrasonic applicatorA thermal model using the measured acoustic profile as input is also presented for comparison Temperature measurements werecarried out with thermocouples inserted in muscle phantomThe insertion place of thermocouples was monitored with ultrasoundimaging Results Modeled and measured thermal profiles were compared within the first 10 cm of depth The ideal acoustic fielddid not adequately represent the measured field having different temperature profiles (errors 10 to 20) Experimental field wasconcentrated near the transducer producing a region with higher temperatures while the modeled ideal temperature was linearlydistributed along the depthThe error was reduced to 7 when introducing the measured acoustic field as the input variable in theFE temperature modeling Conclusions Temperature distributions are strongly related to the acoustic field distributions

1 Introduction

In ultrasonic physiotherapy it is common to employ differentsizes of transducers for treating diverse areas of the bodyEach transducer produces a unique acoustic field patterndue to the particular vibration conditions of the piezoelectricdisks related to the variability of transducers manufacturingprocess [1ndash3]These variations in the radiation pattern shouldinfluence the temperature distributions and could be thecause of unexpected acoustic radiations [1] Even thoughmany efforts have been made in order to establish standardsto increase the effectiveness of the treatment [4ndash7] otherimportant aspects have not yet been considered such asthe effects of intrinsic focusing related to the transducermanufacturing [3] the effect of the transducer front-faceheating due to ineffective ultrasound emission [6 8] and therelation between radiation patterns and temperature distri-bution (exposed in this paper) Although isolated evidence

points out the potential of using ultrasound in the treatmentof many illnesses [9ndash12] in practice conclusive physiologicalimprovement when using ultrasound in controlled trialshas not been accomplished [13ndash17] This situation could bethe consequence of the subjective variables involved in thecommon practice of this therapy and the need to introducemore parameters in order to standardize this application [18]Wemust find better parameters (probably device dependent)that could potentially affect the therapy outcome in orderto completely standardize the technique and to achievereproducible clinical results [18]

In clinical practice the therapeutic ultrasound is com-monly applied following a dynamic treatment protocol inwhich the transducer does not remain static but continu-ously moves around the region where the lesion is locatedAlthough dynamic protocols are often recommended toavoid hot spots this strategy incorporates certain subjectiv-ities to the therapy that could influence the overall efficacy

Hindawi Publishing CorporationBioMed Research InternationalVolume 2016 Article ID 5484735 16 pageshttpdxdoiorg10115520165484735

2 BioMed Research International

For instance the definitive protocol for applying the therapyis often chosen by therapists based on their own experiencefrom which they decide the size of the treatment area (egtwice the transducer emitting surface) [19] and the treatmentdose [20] The scanning speed also depends on the therapistbut it has been reported by Weaver et al [21] that thechanges in average heating using different scanning speedwithin a specific treatment area were not significant Greyin 2003 [19] found significant variations in the scanningspeed and important differences in the dynamic strategiesof therapy application when analyzing the transducer move-ment patterns produced by different therapists Demchak andStone [22] reported a difference in deep heating in humansubjects when using clinically observed treatment parametersand the recommended treatment parameters that researchsuggested The local recorded doses were inconsistent withthose theoretically expected which make us hypothesize thatthese differences could influence the gotten temperature intissues In this work we are interested in reducing the sourceof subjectivities by proposing a different method for therapyapplication in which the therapist does not interfere directlyAlthough studies of dynamic protocols are not discarded forfuture work these would be carried out after determining thefeasibility of improving the static application outcomes

In literature measurements of temperature distributionsproduced by physiotherapy ultrasound have been reported inboth real tissues and tissue-mimickingmaterials (phantoms)From these publications a small number were related toproduce accurate models of acoustic field for the planarradiators used in this application [1 23] Furthermore whenthe acoustic field was modeled it usually referred to a pistonsource [24 25] and few particular efforts were made todemonstrate the validity of this assumption For instanceGelat et al [24] presented the aperture method to determinethe effective radiating area of physiotherapy transducersformulated by assuming the transducer as a plane-pistonAlthough the validity of the method is not questionedsimplifying the transducer as a piston source is not congruentwith the measured fields of this kind of radiators [1 2 23] Luet al [25] developed amethod for numerically approximatingthe acoustic field of hyperthermia devices in multilayermedia They also assumed the radiator as a piston sourceand they validated their numerical results with the Rayleighintegral On the basis of the ideal vibrating piston Grey [19]calculated the distribution of superficial energy that resultedfrom therapy applications by 22 therapists however Greydid not present the validation of either the resulting heatingmodel or the simplified acoustic model Recently Miller etal [26] measured in vivo temperature distributions generatedby therapeutic ultrasound in human triceps surae musclegroup during a dynamical treatment protocol on a regionof two times the size of the transducer head Their resultsindicated that the produced thermal increments were notuniform in the treatment area and among individuals andthey concluded that these differences could be related tosome variables they could not control in the experiments asthe thermocouple final insertion the size of treatment areaand so forth However they did not provide any theoreticalmodel to validate the results or demonstrate their hypothesis

Following a thorough analysis of the state-of-the-art weconcluded that few efforts have been made to evaluate thecapability of current acoustic fieldmodels to produce thermaldistributions congruent with measurements

Studies on hyperthermia therapies specifically in physio-therapy applications seldom consider realistic acoustic fieldpatterns and their effects in thermal distributions Whenmodeling the acoustic field the most used approach inultrasonic physiotherapy literature is the piston-in-a-baffle(uniform vibrating aperture into an infinite rigid and no-vibrating baffle) which creates a characteristic diffractionpattern in the near-field zone Moreover when temper-ature profiles of physiotherapy applicators are presentedthey commonly referred to the ideal acoustic field and noanalysis of the importance of correctly representing thereal acoustic field is made In this paper the thermal pat-tern produced by the piston-in-a-baffle model is comparedto experimental measurements to verify if this ideal fieldaccurately allows the representation of realistic temperaturedistributions additionally anothermodeling approach is alsoused for comparison in order to determine if using the realacoustic profile permits an improvement of the modeledthermal distribution For this purpose a finite element(FE) model of ultrasonic heating produced by a theoreticalpiston is compared with the heating produced by a realphysiotherapy ultrasound applicator the second approachusing the measured acoustic field to feed the FE model anddetermine the heating is also shown For the experimentsand models a static therapy protocol is used since inthis modality the dosage is directly correlated to treatmenttime while presenting few effects of thermal dissipation byconduction [19] Additionally this approach eliminates theneed to consider therapist-dependent variables such as thescanning speed of the applicator and the irregular size ofthe treated area The results of this work will allow a betterunderstanding of these acoustic fields and the thermal effectsthey produce to eventually improve the efficacy of ultrasoundbased physiotherapy treatment

2 Materials and Methods

The models were developed in a workstation with 64GB ofRAM and 8-core processor of 300GHz The experimentswere performed using a physiotherapy ultrasound deviceradiating to a muscle-mimicking material (phantom) Thephantom with a cubic shape of 10 cm per side emulatedthe acoustic properties of muscular tissue and was madeof a water-based mixture of agarose graphite and glycerinThe physiotherapy ultrasonic device comprised a special-designed homemade RF amplifier and a commercial trans-ducer of 1MHz and 10 cm2 of nominal radiating area (Model7310 Mettler Electronics Corp USA)

21 Determination of PhantomAcoustic Properties Thephan-tom acoustic properties used in the models were determinedby measurements The acoustic attenuation was obtainedwith the Ping He formulation [27] The phantom was splitinto 10 equal sections of 1 cm of thickness and the attenuation

BioMed Research International 3

Transducer 1 Transducer 2L

SampleDistilled degassed water at 37∘C

P1

P2

Pw

Ps

(a)

Window

Drilled walls

Transducerholder

Hole for transducer front-face

Phantom

Thermocouples

y

x

z

(b)

Phantom

1

2 3

4

5

z

r0

10cm

5 cm(c)

Figure 1 (a) Setup tomeasure the phantom acoustic attenuation and wave velocity (b) Container used for themeasurementsThe transducerholder is dismountable to fasten other transducers of different sizes (c) Geometry for the computationalmodeling the zoomof thermocoupletip just referred to the simulations for determining the effect of thermocouple insertion in acoustic and thermal distributions

of each section was measured We used two widebandtransducers of 35MHz (Panametrics V383-SU OlympusUSA) separated at fixed certain distance (not required forcalculation) in which a transducer was used as transceiverand the other as receiver The echo signals were ampli-fied by a homemade wideband amplifier and the signalswere recorded by an oscilloscope (Wave Runner 6000ALeCroy USA) at 50MHz of bandwidthThe transducers wereimmersed in distilled degassed water at 37∘C controlled by athermostatic bath (RCTB 3000 OMEGA Engineering IncUSA) the water has a negligible attenuation coefficient at theused frequencies [28] Two ultrasound pulses were requiredfor this technique specifically a first reference pulse for thewave traveling inwater and a second pulse of thewave passingthrough the phantom

Phantom thickness of each section was determined withultrasound using the reference pulse 119875119908 the sample pulse

119875119904 and the reflected echoes produced by the interfacesphantom-water (shown in Figure 1(a) as 1198751 and 1198752) assuming119888119908 = 152361ms at 37∘C [29] Finally the attenuation wascalculated using the phantom thickness and the amplitudeand phase spectra of the received pulses [27] The speed ofsound in the phantomwas determinedwith the same setup byusing the pulses produced at the phantom-water interfaces1198751and 1198752

22 Characterization of Physiotherapy Transducer Theacoustical characterization of the physiotherapy transducerwas carried out by using a 3D scanning automated system(Onda Corp USA) and a wideband needle hydrophone(SEA mod PVDF-Z44-0400 USA) with effective apertureof 04mm and sensitivity of minus260 dB that referred to 1 V120583Paat 1MHz Both the transducer and the hydrophone wereimmersed in distilled degassed water to avoid bubble


formation and to neglect attenuation The measurementswere made point-by-point spaced 1mm The transducer wasexcited with tone bursts of 1MHz by using an RF amplifiercard TB-1000 (Matec Instruments USA) The hydrophonewas centered before the measurements [30] to have anequally distributed radiating energy of the emission at 3mmfrom the transducer This center location was confirmedat the last maximum peak of the beam (255mm from thetransducer) with a deviation smaller than 2mm in the 119909-axisand 1mm in the 119910-axis The acquired data were recorded bya computer during the process The electrical excitation wasmade with 20 sinusoidal cycles and 13ms repetition rate forsimulating nearly a continuous emission of the transducerand to let the system dissipate the energy before the nextmeasurement

119909119910 planes were recorded at different 119911 positions todetermine the transducer properties with reference to IEC-61689 finally an 119909119911 plane (at 119910 = 0) was recorded toobtain the complete acoustic field of the transducer Thetransducer effective radiating area (119860ER) was determined inaccordance with the standard IEC-61689 [4] The ultrasonicradiated power 119875119877 emitted by the device was measured with aradiation force balance (UPM-DT-100N Ohmic InstrumentsCompany USA) and this power was related to the effectiveacoustic intensity 119868 by [7]

119868 = 119875119877119860ER

(1)

23 Experiment Setup for Ultrasound Heating Experimentswere carried out by using a homemade container that heldthe transducer and the phantom and allowed guiding thethermocouples during insertion (shown in Figure 1(b))The container had two lateral windows to monitor thethermocouple insertion with an ultrasound imaging deviceThe phantom and the container were immersed in distilleddegassed water at 37∘C regulated with a thermostatic bath(RCTB 3000 OMEGA Engineering Inc USA) to emulatebody temperatureThe center of the transducer emission waschosen as the origin of the experimental Cartesian coordinatesystem then the transducerrsquos emitting surface was placed onthe 119909119910-plane at 119911 = 0 and the direction of wave propagationwas along the 119911-direction

Thermal maps were determined in transient state using11 thermocouples inserted at the same depth The thermo-couples were displaced (by insertion) into the phantom torecord transient temperature at nine different depths Theinsertion started at 119911 = 9 cm and finished at 119911 = 1 cm withsteps of 1 cmThe initial separation among the thermocoupleswas 05 cm determined with an ultrasound imaging deviceat 119911 = 9 cm for 119910 = 0 this means that at 119911 = 9 cmthe 11 thermocouples were positioned respectively at 1199091 =minus25 cm 1199092 = minus20 cm 1199093 = minus15 cm 11990910 = 20 cm and11990911 = 25 cm The thermocouples were type K of 076mm(003 in) of diameter about 12 of therapy-ultrasound wave-length in the phantom (156mm)This thermocouple size waschosen to permit the insertion with little spatial deviationsand to facilitate the thermocouple detection by an ultrasoundimaging device of 7MHzThermocouples extraction was not

chosen because the thermocouples leave routes of damageduring extraction that can modify the thermal distributionat final depths due to direct destruction of the phantomBecause the thermocouples during insertion did not followa real straight direction the insertion route was monitoredwith an ultrasound scan (Prosound 6 Aloka Japan) throughthe lateral window and the top of the container for postpro-cessing as shown in Figure 1(b) the route was verified at theend of the experiment by splitting the phantom

The experiment comprised a heating-cooling sequencefor each of the nine 119911-depths before the next heating-coolingsequence the 11 thermocouples had been displaced to thenext 119911-depthThephysiotherapy ultrasound devicewas set upat 12W of ultrasonic radiated power in continuous emissionthis power was verified at the end of the experiments withno variation The heating lasted 300 s and the phantomwas cooled to 37∘C before next measurement cooling wasrecorded during 100 s Temperature data were taken every 2 susing NI 9219 (National Instruments USA) and a personalcomputer The thermocouples were calibrated by immersionbefore the experiment from 20∘C to 60∘C at steps of 5∘Cwith an overall uncertainty of plusmn006∘C the thermocouplespresented a linear behavior in the temperature range ofexperimentsThermocouple calibration was verified after theexperiments with no changes

Thermocouple artifacts due to viscoelastic heatingappeared in the temperature measurements [31 32] Thisartifact occurs by the combination of media shear viscosityand vibration [31] Viscoelastic heating produces a rapidtemperature increase at the beginning of the heating which isfollowed by the exponential rise due to ultrasound absorption(effect we are analyzing) and then a fast decrease when theradiation is turned off also due to viscosity The correction ofthermocouple artifact was made by means of the detectionof the stable-state temperature due to shear viscosity (1198790) byback-extrapolating the exponential part of the temperatureincrease (due to absorption) to the start of heating (at 119905 = 0)[31] Then 1198790 was subtracted just from the heating sectionImmediately after ultrasound was turned off the measuredtemperature was affected by both the viscous heatingcomponent and the heat dissipation by thermal conductionwhich makes it complicated to know the behavior oftemperature at that specific moment therefore in Figure 3the fast decay immediately after ultrasound was turned offwas just eliminated For instance an Nth order functioncould be used to determine that sectionrsquos temperature byinterpolation of the cooling curve and the temperature pointat 119905 = 300 s but the accuracy of the result in that part ofthe curve would be unknown A back-extrapolation of thecooling exponential part to the stop-of-heating (119905 = 300 s)produced a stepped variation in temperature profile atthat time for innermost thermocouples mainly thoselocated in large temperature gradients This variation wasproduced because the heat moves from regions with highertemperature towards cooler regions and if the thermocouplewas located in the pathway the measured temperature aftercessation of ultrasound can be higher than the previousThis effect produces an upward displacement of the entirecooling curve and the back-extrapolation does not match the


temperature reached at 119905 = 300 s Another possible methodto correct the measurements used by Nell and Myers [32]was to measure the ldquothermocouple artifactrdquo in nonabsorbingmedia and subtracting it from the measurements Howeverthis procedure was inadequate for this work because theartifact is pressure dependent which means that it shouldbe measured at each spatial point of the acoustic field in anonabsorbing phantom and that same spatial point shouldbe registered in the muscle phantom in our experiments itwas not possible for the thermocouples to follow the sametrajectories in both phantoms

24 Acoustic Field Model A first FE model for axisymmetricacoustic field during uniform acceleration distribution onthe transducer surface was developed The main objectiveof this approach was to obtain a computational model forradiation pattern by using the approach of the piston-in-a-baffle in stationary state during harmonic operation Themodel operation conditionswere based on the experiments inwhich the device emitted a continuous stationary harmonicradiation with a negligible start up time of 100ms A secondapproach was developed to improve the results with respectto the measured data For this the measured acoustic fieldwas used as the input data of the bioheat FE simulationHowever because the transducer acoustic field wasmeasuredat low power excitation in nonattenuating medium datapreprocessing was required for this the measured intensityfield was normalized using the average intensity into theeffective radiating area then the field was attenuated basedon the measured phantom attenuation and was multipliedby a factor corresponding to the acoustic power used forexperiments To clarify we are going to call the model usingthe piston source model A and themodel using themeasuredfield as acoustic data model B

The problem was considered axisymmetric with thegeometry shown in Figure 1(c) (zoom in presented is relatedto other analyses that will be shown later) The FE modelwas developed based on the cylindrical coordinate system(119903 119911) where the transducer was located at 119911 = 0 and theradiation propagated in the 119911-direction The variable 119903 in themodel is related to the experimental coordinates 119909 and 119910 by119903 = radic1199092 + 1199102 The boundaries were defined in accordancewith the experimental setup Boundary 3 was simulated asan acoustically rigid wall by setting the normal velocity atzero Boundary 1 was the propagation axis of the problemBoundaries 4 and 5 had the same acoustic impedance asthe media (119911 = 120588119888) to avoid wave reflections mainly atboundary 5 The FE mesh for acoustic model consisted of80000 squared elements that is about six per wavelengthThe transducer uniform vibrationwas set up at boundary 2 byusing 1MHz harmonic acceleration with constant amplitudeThis accelerationwas related to the acoustic intensity by usingsome basic relations for a planewave that could be consideredtrue just on the transducer surface [33] At first the pressureamplitude on the emitting surface was determined based onthe acoustic intensity by using [34]

10038161003816100381610038161199011003816100381610038161003816 = radic2120588119888119868 (2)

where |119901| is the average pressure amplitude on the radiatorsurface (Pa) 120588 is the average medium density (kgmminus3) 119888is the speed of sound in the medium (m sminus1) and 119868 is theeffective acoustic intensity (Wmminus2) obtained using (1)Thenconsidering a plane wave behavior on the transducer surfacethe piston particle velocity V0 and the pressure on the surface1199010 can be related by 1199010 = 120588119888V0 [33] Moreover if we alsoconsider the harmonic operation the transducer emissioncan be set up as either the uniform normal velocity or theuniform normal acceleration on the surfaceThe acceleration1198860 on the radiator surface can be obtained by deriving theconstant amplitude harmonic normal velocity V0119890119895120596119905 withrespect to the time as

1198860 = V0120596119890119895(120596119905+1205872) = 120596120588119888 1199010119890119895(120596119905+1205872) (3)

The term 1205872 in (3) is the phase shift between the velocityand the acceleration and this does not affect their uniformityalong the transducer surface The acceleration of (3) can beestablished in terms of the intensity by substituting (2) into(3)The considerations used to derive (3) are valid just on thetransducer surface where the wave can be considered planar

The experiments were carried out in continuous-moderadiation during stationary harmonic condition thereforethe acoustic field was modeled in this regime The equationof harmonic wave propagation can be given as [1 35]

nabla2119901 + 1198961198882119901 = 0 (4)

where 119901 is the spatial pressure and 119896119888 is the complexwavenumber (119896119888 = 119896 minus 119894120583 where 120583 is the attenuationcoefficient 119896 = 2120587120582 and 120582 is the wavelength) Basedon the transducer geometry the problem was simplifiedby considering axisymmetric radiation situation in which(4) is still valid The attenuation was assumed constantand homogeneous The attenuated acoustic field computedwith our FE model was compared with the attenuated fieldcalculated with the Rayleigh integral with attenuation

25 Ultrasonic Heating Model Concerning the relation ofultrasound and temperature the time-average heat 119876 pro-duced in the media depends on both the intensity 119868 and thepressure amplitude absorption coefficient 120572 as 119876 = 2120572119868 [36]here the coefficient 120572 is related to the pressure amplitudeattenuation coefficient 120583 by 120583 = 120572 + 120573 where 120573 is thescattering For this case it was considered that all the atten-uation was produced by absorption and the scattering effectwas negligible Subsequently the temperature distribution 119879in the medium was obtained by the Pennes bioheat transferequation with no contribution of metabolism and withoutblood flow given by

120588119862 120597119879120597119905 minus 119896nabla2119879 = 119876ext (5)

where 120588 is the medium density (kgmminus3) 119862 is the mediumheat capacity (J kgminus1 Kminus1) 119896 is the medium thermal conduc-tivity (Wmminus1 Kminus1) and 119876ext is the heat per volume and time


(Wmminus3) produced by an external source This external heatis related to the pressure [34] by

119876ext (119903 119911) = 2120572119868 (119903 119911) = 120572 1003816100381610038161003816119901 (119903 119911)10038161003816100381610038162

120588119888 [Wm3] (6)

where 119888 is the speed of sound in the medium (m sminus1) and | sdot |denotes the amplitude of the spatial pressure

The stable-state harmonic acoustic field in media withhomogeneous attenuationwas used as the input of the heatingproblem employing 119876ext of (6) The coordinate system wasthe same as that of the FE model for the acoustic fieldThe boundary conditions were chosen in accordance withthe experimental setup Boundary 1 was considered as thesymmetry axis similar to that of the acoustic field modelBoundary 2 was set as thermally insulated and then n sdot(119896nabla119879) = 0 where n is a normal unit vector Boundaries3 4 and 5 were configured to have 37∘C which was thetemperature of the water in the experiments (controlled bythe thermostatic bath) The time steps for modeling thetransient temperature were 10 s and the heating finished at300 s as in the experiments in addition cooling of 100 s wasalso included in the model

26 Analysis of Thermocouple Acoustic Field PerturbationThe effect in the acoustic field and thermal distributions ofthermocouples was analyzed with FE simulations under thesame operation conditions of the models and for differentthermocouples sizes that ranged from 01mm to 16mm ofdiameter (Oslash) This effect was analyzed in contrast to thenonperturbed situationwith no thermocouple inserted in themedia Regarding the zoom in Figure 1(c) a thermocouplewas included along the propagation axis as it was insertedin the opposite direction of the field propagation startingfrom 119911 = 10 cm at 119903 = 0 with the tip final location at119911 = 1 cm and 119903 = 0 the tip had a round shape as withreal thermocouples and the tip boundary acoustic conditionwas configured as a rigid wall (zero particle velocity) Theacoustic properties of the thermocouple body were thosefor the thermocouples cover material PFA (PerfluoroalkoxyCopolymer) namely the PFA density 120588PFA = 2150 [kgm3]obtained from DuPont PFA datasheet and the speed ofsound of PFA 119888PFA = 920 [ms] calculated from [37]119888 = radic119910(1 minus ])(120588(1 + ])(1 minus 2])) where 119910 and ] areYoungrsquos modulus and Poissonrsquos ratio respectively whosevalues were 119910 = 480MPa and ] = 045 The mesh sizewas improved to 6 elements per millimeter to get about 9elements per wavelength The results of simulations wereanalyzed around the thermocouple tip to determine howthe thermocouple perturbs the acoustic field and how thisperturbation modifies the measured temperature Variationsin PFA acoustic properties of even 50 did not importantlymodify the results this is because we are not analyzing thefield or temperature perturbation at the thermocouple bodyThe results of simulations were confirmed experimentally byusing two different size thermocouples of Oslash = 076mm andOslash = 026mm inserted in the graphite phantom at 1 cm fromthe transducer

27 Measuring Transducer Front-Face Heating The trans-ducer front-face heatingwasmeasured and analyzed to deter-mine its contribution on the obtained thermal distributionFor this it used a cube shape agarose phantom with 6 cmper side The transducer and the phantom were immersedin distilled degassed water at 37∘C and were fixed togetherto avoid any relative movement A type J wire thermocoupleof 026mm (001 in) of diameter was set between them atthe center (119909 = 119910 = 119911 = 0) to measure the temperatureproduced by the transducer front-face heating Temperaturewas recorded each second using NI 9219 (National Instru-mentsUSA)Theultrasounddevicewas configured to deliver10Wcm2 during 5min to emulate the main experimentsThree independent heating-cooling sequences were acquiredfor this measurement

The measured temperature rise was originated by twodifferent heat sources the transducer front-face heating andthe ultrasound absorption in these experiments viscousheating was not observed The agarose phantom does nothave a negligible attenuation coefficient so the heat producedby ultrasound should be accounted To determine this con-tribution at this specific position but without the effect oftransducer front-face heating the temperature elevation byabsorptionwas calculated with FE simulations If we considerthe heat sources affecting themeasurements independent andseparable the transducer front-face heating can be obtainedby subtracting the simulated temperature due to absorptionfrom the measured temperature Validity of our simulationswas verified with the same setup using a gap between thetransducer and the phantom to eliminate the effect of front-face heating (data not shown in this paper)

3 Results

31 Phantoms Themuscle phantomused in experiments wasprepared with 10 of glycerin (to change speed of sound)7 of graphite (to change absorption) 15 of agarose (tosolidify the mixture) and 815 of distilled degassed water(as base of the phantom) The reported speed of soundand pressure attenuation coefficient for muscular tissue were1550ndash1620ms and 40Npm respectively [36 38] theseproperties were measured for the phantom at 37∘C withrespective values of 1553 plusmn 12ms and 598 plusmn 046Npm Bothvalues were considered constant in simulations

The agarose phantom used for measuring the transducerfront-face heating was prepared with 15 of agarose andwater This phantom was planned to have low attenua-tion coefficient (and then low absorption to ultrasound) tomake the heating by ultrasound negligible However realattenuation coefficient was important and its inclusion indetermining the transducer front-face heating was requiredFor this phantom the measured speed of sound and atten-uation coefficient were 1530 plusmn 15ms and 047 plusmn 008Npmrespectively

32 Acoustic Field Characterization The acoustic field ofthe transducer was measured in distilled degassed water(attenuation-free media) The measured acoustic field was


0 2 4 6 8 1000

02

04

06

08

10

12

MeasuredPiston

z (cm)

Nor

mal

ized

acou

stic p

ress

ure

(a)

0 2x (cm)

z (c

m)

minus2

10

5

0

(b)

0 2x (cm)

minus2

z (c

m)

10

5

0

(c)

Figure 2 Acoustic fields of transducer and ideal piston without attenuation (a) Normalized pressure on the propagation axis (b) and(c) Complete acoustic field of piston and transducer respectively (linear grayscale of normalized acoustic pressure with respect to the lastmaximum peak at Rayleigh distance) Comparisons between FE acoustic field and Rayleigh integral gave insignificant variations (less than004 of error in certain points)

compared with the field produced by a piston called modelA (see Figure 2) We started with this ideal field becausethis approximation is widely used to model the radiationof this kind of transducer As can be seen in Figure 2 thereal acoustic field differs enormously from the ideal situationin the region where the therapeutic effects are produced atthe very near-field zone Then we decided to determine ifthese differences in the acoustic field patterns had an effecton the thermal distribution Moreover to neglect effects inthe thermal profile for calculating the acoustic field withthe FE method the FE acoustic field was compared to theone obtained with the Rayleigh integral We found negligiblevariations with amajor difference in amplitude of 423 closeto the radiator (before 119911 = 1 cm) the overall average relativeerror was under 004 These results suggest that we can

considered valid both situations because the field calculatedwith the FEmethod is equivalent to the field obtainedwith theRayleigh integralThe field with the Rayleigh integral was notincluded in Figure 2 because the differences are not visuallyevident

The measured acoustic field was used to calculate thetransducer characteristic parametersThe ultrasonic radiatedpower of the physiotherapy device was measured to deter-mine the acoustic intensity to be used in the models Themeasured 119860ER was 819 cm2 (opposed to 10 cm2 indicated bythe transducer manufacturer) and the measured ultrasonicpower was 12W Using (1) the effective acoustic intensityemitted to the phantom was 146Wcmminus2 and this value wasused to calculate the amplitude of the acceleration for theacoustic field of model A using (2) and (3) the calculated


0 100 200 300 40037

39

41

43

45

47

49

Calculation for a pistonExperimental after correctionExperimental without correction

Time (s)

Tem

pera

ture

(∘C)

Figure 3 Temperature at 1 cm from the transducer radiating surfacealong the 119911-axis (119903 = 0) for themodel and corrected and uncorrectedmeasurements for thermocouple artifact

acceleration amplitude of the radiating surface was 842 times105msminus2 The 119860ER provided the effective radiating radius tosimulate the uniform emission of model A (piston radius)which was 161 cm Temperature increase was obtained byusing (5) and (6) considering the modeled acoustic fieldas the input for the heating model For the model B theprocedure was quite different to that used formodel A Beforecalculating the heat the acoustic field measured at low powerexcitation was normalized to permit us to adjust the intensityto that used in experiments normalization was carried outwith the average intensity close to the transducer into the119860ERThe normalized intensity wasmultiplied by the intensityused in experiments 146W cmminus2 and then it was artificiallyattenuated along 119911-axis with the exponential factor 119890minus2120583119911Temperature increase was then obtained with FE using (5)and (6)

33 Postprocessing Temperature Measurements Temperaturemeasurements were acquired in a personal computer and thepostprocessing was carried out in MATLAB Thermocoupleartifact was corrected as explained above The uncorrectedtemperature measurement was compared with both thecorrected one and the modeled increment at the sameposition in Figure 3 The overall corrected measurement wasfurther approximated to the real temperature and it wasalso congruent with the modelThemaximum absolute errorbetween the corrected measurement and the model resultedin 036∘C meanwhile the difference between the uncorrectedmeasurement and the model was 17∘C

After the thermocouple artifact was compensated thethermal map inside the phantom was determined by cubicinterpolating the measured temperatures the thermocouplesspatial positions used for the interpolation were obtainedby following the thermocouple trajectories during insertionwith the ultrasound scan an ultrasound scan image obtainedat a depth of 7 cm from the transducer radiating surface is

++5 cm

Figure 4 Ultrasonic image of the thermocouples into the phantomat 7 cm from the transducer the thermocouple is shown as a clearzone followed by a dark shadowThe position of each thermocouplewas determined by using these images along the insertion route andit was verified by splitting the phantom at the end of the experiment

shown in Figure 4 The complete thermal map was obtainedfor the 119909119911-plane perpendicular to the transducer surfacefor 119910 = 0 During insertion thermocouples deviated fromtheir initial position and this deviation was accounted Theoverall average on measured 119910-position and the standarddeviation from the start position 119910 = 0 were 19mm andplusmn40mm respectively In this experiment the ratio of ther-mocouple diameter and the imaging-ultrasound wavelength(022mm at 7MHz) was 345 which was adequate to detectthe thermocouples The ratio of the thermocouple diameter(076mm) and therapy-ultrasound wavelength (156mm)was 049 which means that the thermocouple diameterrepresents about 12 of the wavelength This thermocouplesize should not appreciably affect the acoustic field at regionsbetween the transducer surface and the thermocouple andafter that the thermocouple would modify the acoustic fielddue to a shadowing effect [39] these regions were not of ourparticular interest because theywere located beyond thewavetraveling path

34 Thermocouple Acoustic Field Perturbation As it can beobserved in Figures 5(a) and 5(b) the radial and axial pres-sure profiles when using a 01mm diameter thermocoupledo not importantly differ from the unperturbed field sinceboth lines are approximately superposedThemain differencewhen using this thermocouple was observed at 119903 = 0for Figure 5(a) and on the little delay in 119911-direction forFigure 5(b) Almost the same behavior can be observedin 08mm diameter thermocouple with light differences inamplitude in Figure 5(a) (maximum local difference of 124at 119903 = 2mm) but keeping the shape of unperturbed fieldand some distortions at 2mm in front of the thermocouplein Figure 5(b) but with a negligible delay in 119911-directionAs positive control the field when using a 16mm diameterthermocouple was analyzed showing evident signs of pertur-bation in Figures 5(a) and 5(b) The modeled temperatureelevations on the symmetry axis for the studied cases wereplotted in Figure 5(c) (notice the expanded temperaturescale) The most transcendental variation of these resultswas found when using a 16mm diameter thermocouple for


r (cm)

Acou

stic p

ress

ure (

MPa

)

Free space

0 10 2000

01

02

03

Oslash= 01 mmOslash= 08 mmOslash= 16 mm

(a)

Acou

stic p

ress

ure (

MPa

)

0005 06 07 08 09

01

02

03

04

05

10z (cm)

Free spaceOslash= 01 mm

Oslash= 08 mmOslash= 16 mm

(b)

00 02 04 06 08 10477

479

481

483

485

z (cm)

Tem

pera

ture

(∘C)



(c)

Time (s)0 100 200 300 400

37

39

41

43

45

47

49


Tem

pera

ture

(∘C)

(d)

Figure 5 Simulations under the same experimental conditions with a thermocouple imbedded along the symmetry axis with the tip locatedat 1 cm from the radiator surface (a) Radial pressure profile 1mm in front of the thermocouple (119911 = 9mm) for the different thermocouplediameters (Oslash) and for the unperturbed acoustic field (b) Acoustic pressure on the propagation axis between the thermocouple and thetransducer regions immediately in front of the transducer were omitted because important variations were not observed (c) Simulatedtemperature profile on the 119911-axis between the transducer and the thermocouple (notice the scale) temperature profile of the thermocouplewith Oslash = 16mm (one wavelength) presented larger variations than other diameters with respect to the unperturbed temperature profile(d) Experimental temperature measured with different diameter thermocouples at 1 cm from the transducer in a graphite phantom The reddashed-lines indicate the standard deviation (SD) of measurement using 026mm diameter thermocouple the measurements with the otherthermocouples do not present important deviations and SD lines (that should appear transposing the respective dot line temperature graph)were omitted for clarity The result of using any of two thermocouples does not appreciably differ

which the temperature increment was 085 larger than theincrement for the unperturbed situation possibly because ofacoustic field distortion Other temperature variations shownin Figure 5 smaller than 025 can be considered negligibleas they did not follow a predictable tendency when usingdifferent thermocouple sizes between diameters of 01mmto 08mm we have concluded that this variability could beproduced by the accuracy of the FE method but it was notimportantly reducedwhen increasing themesh to 12 elementsper wavelength

Experiments using two different size thermocouples areshown in Figure 5(d)The curves follow the same exponential

tendency and they reach a similar maximum temperatureat the end of heating The red dashed-lines indicate thestandard deviation (SD) of measurements with thinner ther-mocouple with an averaged SD of plusmn085∘C and a maximumSD of plusmn125∘C these large values of SD possibly occurredbecause the thermocouple positioning into the phantomat a specific location was difficult The SD lines of themeasurements with the thermocouple of 076mm diameterare not shown because the SD value was small (averageSD of plusmn008∘C and a maximum SD of plusmn017∘C) and therespective lines should appear transposing the dot line ThisSD was particularly small because the rigid body of these


thermocouples permitted following a more controllableinsertion route

35 Analysis of Temperature Distributions The acoustic fieldand temperature distributions were determined with FEcalculations FE models were compared with experimentaldata to analyze their correspondence for this particularsituation Before making conclusions the transducer front-face heating was determined to evaluate its effects in themeasurements After knowing all the variables involved inthe problem (most of them well controlled) we are able toanalyze the acoustic field and temperature distributions anddetermine their correspondence

The transducer front-face heating was determined asexplained above and the effect of this heating on the temper-ature measurements at different distances is now discussedThe measured temperature at 119905 = 300 s between thetransducer and the phantom was 3940∘C while the modeledtemperature without transducer front-face heating at thesame position considering the agarose phantom acousticproperties was 3814∘C Then the temperature increase at119905 = 300 s due to transducer front-face heating can beapproximately considered as 126∘C with an exponentialtendency stating at 37∘CThis temperature elevationwas usedto simulate the heat propagation in the graphite phantomwithout ultrasound (simulated temperature curves are notshown) The temperature was set in the boundary 2 ofFigure 1(c) and the rest of the boundaries remained withoutchanges as used in the other models With this simulationthe temperature elevation at different depths produced bythe transducer front-face heating can be determined Theresults indicated that the temperature increase by conductiondue to transducer front-face heating was 039∘C at 119911 =1 cm that is 357 of the temperature increase in the mainexperiment at 1 cm At 2 cm the effect reduces to 008∘C thatis 076 of the temperature in the main experiment Afterthese results the effect of transducer front-face heating wasconsidered negligible for our experiments This effect couldbe considered important at 1 cm from the transducer but ourmain conclusions are more related to other deeper distanceswhere the effect is small

The average acoustic intensity at each depth using allmea-sured andmodeled points was determined in order to analyzeif the produced acoustic field is composed of zones withhigher concentration of energy than others Although this isan undesired behavior for this kind of transducer these irreg-ularities should be considered in our analysis For instanceif this procedure were carried out for a focused transducerthe result would indicate a higher energy concentration at thefocus compared to the energy concentration at other depthsThe averaged intensities at each depth are shown in Figure 6normalized using the overall average intensity of each fieldthe measured field was attenuated using the exponentialfactor 119890minus2120583119911 Because acoustic measurements were made inattenuation-free media but thermal experiments andmodelswere made in a phantom with specific attenuation themeasured intensities were attenuated to analyze the graph ofFigure 6 in the conditions of temperature experiments that is

1 2 3 4 5 6 7 8 9 10 1100

02

04

06

08

10

12

MeasuredModeled

Attenuation

z (cm)

Nor

mal

ized

inte

nsity

Figure 6 Averaged acoustic intensity at each depth of themeasured(with artificial attenuation) and the modeled acoustic fields Thenormalization was carried out using the averaged intensity usedin experiments 146Wcm2 The two curves fittingly follow thetheoretical curve of attenuation

in the mode the intensity linearly acts to produce heat (119876 =2120572119868) [36] Figure 7 shows the thermal distributions obtainedat 300 s for the experiment and the model in contrapositionto their respective acoustic fields Although the measuredand modeled acoustic fields exposed spatial discrepanciesin the intensity diffraction pattern shown in left parts ofFigures 7(a) and 7(b) their average intensities of Figure 6at each depth appropriately followed the theoretical curveof attenuation In accordance with this similar behavior ofdepth-averaged acoustic intensities the thermal patterns ofmodel and experiments should be theoretically congruentconversely these thermal patterns differ when 119911-coordinateincreases (right parts of Figures 7(a) and 7(b))This is becausetemperature depends on heat produced spatially as expressedwith the nabla-operator in (5) and the analysis with respectto depth after simplifying the field distribution which is oftenpresented in the literature is not sufficient to obtain validthermal profiles

The analysis of the spatial heating effect produced byboth the theoretical piston source and the real acoustic fielddistributions was carried out to determine the validity of thiswidely used theoretical approach Returning to the graphsin Figure 3 these presented good agreement between modeland measured temperatures for a specific spatial point closeto the applicator with maximum absolute errors of 036∘CAlso the maximum temperature reached at that point wasof 4783∘C in the model and 4760∘C in the measurementsthe average heating rates were 0036∘Cs and 0035∘Cs forthe model and measurements respectively However in spiteof these similitudes close to the radiator in Figures 7 and8 the discrepancies in the thermal distributions are evidentbefore 6 cm of depth While the modeled acoustic field byuniform vibrating distribution shown in Figure 7(b) wasldquoin averagerdquo uniform the real acoustic field produced by aphysiotherapy applicator shown in Figure 7(a) had regionswith clearly higher intensities at the first 6 cm of depth


z (c

m)

r (mm)

10

9

8

7

6

5

4

3

2

115 10 5 0 15105

(a)

r (mm)15 10 5 0 15105

10

09

08

07

06

05

04

03

02

01

0

(b)

r (mm)

40

30

20

10

0

()

minus10

minus20

minus30

0 5 10 15

(c)

Figure 7 Temperature distribution (right side of each distribution) at 300 s versus stationary acoustic field (left side of each distribution) (a)experiment (b) model A and (c) relative errors between the measured and the modeled temperature increments The acoustic intensitiesattenuate with the increase in 119911 Temperatures and intensities are normalized to use the same visualization scale

The modeled acoustic field comprised both very high andvery low intensities too close to each other which owingto added thermal conduction resulted in an average heatingand soft thermal increments along the depthThe region withthe highest temperature increments in the experiments (theregion within 06 of the normalized temperature plot) wasobtained where the highest intensities were located in themeasured acoustic field (Figure 7(a)) On the basis of theerror plot shown in Figure 7(c) it can be concluded that themodeled thermal distribution is not congruent because themeasured and the modeled acoustic fields differ The relativeerrors of temperature increments presented in Figure 7(c)indicate discrepancies of about 15 in the first 5 cm into thevery near-field zone where the strongest therapeutic effectsare expected Differences of 30 beyond 8 cm were not ofparticular interest because the temperature increments weresignificantly reduced at this depth and the relative error wasmagnified

Although these results could be considered good forexperiments carried out with uncontrolled influential vari-ables (eg blow flow therapist related variability irregularmultilayer tissues and deficiencies in device calibration)the found variations should not be considered acceptablebecause our experiments were carried out under very con-trolled conditions Therefore the model (piston) is not anappropriate representation of the reality and these differencesshould be related to another not-yet-accounted variable asthe acoustic field distribution To test this we used themeasured acoustic field to feed the input variable (119876ext of(5)) and get a more realistic temperature profile For this theacoustic field was artificially attenuated with the exponentialfactor 119890minus2120583119911 and we used the same acoustic intensity as in

the experiments These two models are compared using thethermal evolution at two specific times shown in Figure 8for the piston produced heating (model A) and the measuredfield produced heating (model B) In general the outermostcontours of model A in Figures 8(a) and 8(b) accuratelycorrespond to each other while the inner ones do notThe central contours are strongly dependent on the acousticfield distribution as explained earlier The contours of 47∘Cand 46∘C in the measured part of Figure 8(b) are about1 cm deeper than those of the model part the same can beobserved with the 43∘C contour presented in Figure 8(a)This condition also appears for deeper contours beyond6 cm depth However these discordances are reduced formodel B (Figures 8(c) and 8(d)) The innermost contoursare closer between experiment and model B while theoutermost contours remain at the same position as formodel A

The disagreements are quantitatively noticeable in Fig-ure 9 in which the thermal profile on the propagation axisand the radial temperature at 04 cm are shown for both themodels At the first 6 cm of depth the measured thermalprofile differs evidently from model A while it is closer tomodel B with a maximum relative error of 15 at 4 cmfor model A and 7 for model B (Figures 9(b) and 9(d))Moreover beyond the first 6 cm both themodels gave similarresults specifically at 8 cm from the transducer the modeledtemperature was up to 10 lower than the measured oneFor Figure 9(d) it should be noted that we are comparingthe results by using relative errors which were logicallymagnified when 119903 gt 2 cm due to the small temperatureincrements at those points since these temperature incre-ments were caused mainly by the thermal conduction effect


z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(a)z

(cm

)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(b)

z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(c)

z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C38

∘ C

(d)

Figure 8 Contour plots for the experiment (left side of each distribution) and models (right side of each distribution) at two different heatingtimes (a) and (c) 150 s (b) and (d) 300 s (a) and (b) Experiment is compared with temperature produced by the piston (model A) (c) and (d)Experiment is compared with the results when modeling the heat produced by the measured acoustic field (model B) Contours from borderto center 38∘C 39∘C and 47∘C

and just a little contribution by the ultrasound absorption(11 of measured contribution of acoustic absorption when119903 = 2 cm at 4 cm) these parts were not considered in ouranalysisThedifferences of 15of relative error indicate about14∘C of temperature variation among the model and theexperiment which means in terms of therapeutic effects animportant variation that could imply either not having anytherapeutic effect [40 41] (temperature less than 40∘C [40])or crossing the maximum temperature of 45∘C suggested for

physiotherapy [40] As seen in the average intensities shownin Figure 6 these temperature differences are not directlyrelated to the energy deposition dependent on depth becausethe energy along the 119911-coordinate is similarly distributedHowever these temperature differences can be related to theenergy deposition that depends on the radiation patterns Amore accurate model for acoustic field could help match theresults to efficiently predict the thermal distributions in morereal situations


z (cm)

Tem

pera

ture

incr

emen

ts (∘

C)12

10

8

6

4

2

0

1086420

ExperimentModel A

Model B

(a)

Relat

ive e

rror

()

minus30

minus20

minus10

0

10

20

30

z (cm)1086420

Experiment versus AExperiment versus B

(b)

ExperimentModel A

Model B

Tem

pera

ture

incr

emen

ts (∘

C)

10

8

6

4

2

0

r (cm)0 1 2 3

(c)

Relat

ive e

rror

()

minus40

minus20

0

20

40

60


80

r (cm)0 1 2 3

(d)

Figure 9 (a) Thermal profiles along 119911-direction on the acoustic propagation axis for the experiment and the models Temperature of modelA is produced by the ideal piston and temperature of model B is produced when the measured acoustic field was used as the input excitationof the FE heating model (b) Relative error between the measured data and both of the modeled temperatures of part (a) (c) Radial thermalprofiles at 4 cm of depth for the experiment versus both of the models (d) Relative error between the measured data and both of the modeledtemperature increments of graph (c) Using the same radiation parameters the piston (model A) produced less temperature increments thanthe experiment Relative errors indicate large variations (from 10 to 20 in some regions) due to differences in the acoustic field patternsThe results are better when using the measured acoustic field as the input variable of FE simulations (model B)

4 Discussion

The necessity of a correct validation of the effects producedduring a physiotherapeutic ultrasound treatment has beenpointed out by many authors in recent years [7 14ndash16] Milleret al [15] indicated that this therapy modality has presentedlow level of efficacy and at this time the clinical benefit to thepatient is unclear More research around improving the risk-to-benefit ratio should be carried out for physiotherapeuticultrasound This research should incorporate studies aboutquality assurance and patient safety in order to protect the

patient from injuries and to prevent the use ofmalfunctioningdevices [15 18] Quality assurance studies would permit ana-lyzing the device ultrasonic radiation for determining bettertreatment protocols to produce and standardizemethods thatmake all the physiotherapy ultrasonic devices interchange-able This is a clear necessity because at this moment theproduced thermal effects depend on the device used [1 38 40] Our research group is motivated on identifying themachine and transducer critical parameters to produce anadequate acoustic field that generates the expected thermaldistribution However there are remaining research gaps in


the therapy application that should be filled to validate thetechnique and to explain the specific reasons to any gottenconclusion [18] This paper was written to make researchersget involved in this topic again in order to achieve necessaryimprovements to this technique and make the therapy moreeffective and predictable

With the presented results it has been demonstrated thatthe acoustic field of a piston source does not generate athermal distribution that completely corresponds to thoseobtained using real physiotherapy ultrasonic applicatorsThis is a major problem because the piston theory is oftenapplied to easily model the acoustic energy into the biologicmedia to study the thermal distributions at different depthsAlthough the discordance between this ideal model and realacoustic fields is widely known especially for the narrow-band transducers used in physiotherapy the ideal modelsare continuously used in the study of this radiation withoutconsidering the error source produced by differences in thefield patterns The temperature differences of about 14∘Cfound in this study between the model and the experimentcan impact the practical therapeutic effects because of thesmall temperature range of physiotherapy ultrasound (from40∘C to 45∘C) that determines having or not an effectivetreatment or even provoking injuries to the patient [40]Although the obtained differences are only valid for statictreatment protocols under the detailed experimental condi-tions we demonstrated that the acoustic field distributionsdo have an important effect in thermal distributions andthe piston approximation is not the adequate model for thiskind of transducers The use of a static protocol for theseexperiments is related to the interest of our research group infinding an alternative therapy protocol more predictable (andeffective) than the dynamic protocol currently used in clinicThe effects of the acoustic patterns on the produced thermalprofiles should be deeply and systematically demonstratedfor different real transducers to understand these interactionsand to apply this knowledge for proposing a novel therapyplanning that would account for the transducer particularbeam

In accordance with the results shown in Figures 7ndash9the thermal distribution obtained with the ideal acousticfield differed from the experimental thermal distributionmainly in the first 6 cm which is the region where the majortherapeutic effects are expected Those variations are mainlyproduced by the differences in the acoustic distributionsat the near-field among the model and the experimentwhich can be produced by nonuniform vibrations due totransducer radial resonances and layer defects For themodelthe intensity shown in Figure 7(b) is uniformly distributedalong depth with larger peaks just on the propagation axisThis average distribution produces temperature incrementsalong depth dependent on the media attenuation coefficientHowever for the experimental situation shown in Figure 7(a)there is an energy concentration at the first 6 cm and thenthe most significant thermal increments are produced at thatzone As shown in Figure 6 the average intensity in both casesdecreases with depth in accordance with attenuation Theseaverage intensities were calculated to observe the behaviorof the intensity distribution along depth which could be a

possible explanation of the energy concentration observedin the thermal measurements near the transducer howeverboth the average intensity plots follow the same tendency andfrom this representation we were not able to conclude thatthe differences in thermal distributions when 119911 lt 6 cm wereproduced by a 119911-dependent energy concentration It can beobserved in measurements that heat was concentrated nearthe transducer while in the model it decreased exponen-tially as a function of depth Patient injuries and ineffectivetreatments can be the result of these unexpected thermaldistributions Obtaining more accurate thermal distributionsshould be done by properly modeling the acoustic fieldof real ultrasound radiators Because acoustic field modelsfor this kind of applicators adequate for heat generationhave not been proposed yet temperature predictions underreal working conditions cannot be accurately made Onepossible approach for obtaining more congruent acousticfield was proposed by the present authors in a previous work[1] but this approach has not been applied yet to producethermal distributions congruent with the experiments Someadaptations in this proposed acoustic model are necessarysince the modeled field still differs in specific regions into thetreatment zone mainly close to the radiator

Treatment protocols used in ultrasonic physiotherapyoften consider that the acoustic field is naturally nonuniformand it is required to apply some actions to reduce the effect ofthese nonuniformities for example the transducer circularmovement during the therapy However the effects of thesereal acoustic patterns in the temperature distributions arerarely determined and under the experimental conditionsused for this analysis real thermal profiles can have upto 15 (and even more for some other transducers) ofdeviations from the theory As many physiotherapists use adynamic protocol to apply the therapy these experimentsshould be repeated for this kind of protocols to validate thepiston model for studying the therapy under these otherconditions After analyzing our results it is evident thatmore realistic acoustic field models are required to get morecongruent thermal distributions during ultrasonic physio-therapy These acoustic field models should account morefor other parameters than the beam size and beam acousticpower already included other possible parameters couldbe for example the level of transducer beam convergenceand the kind of intensity distribution at the near-field zonebased onmeasurementsThe latter parameter depends on themeasured acoustic intensity distribution which as analyzedin this work is related to the produced thermal distributionThe former parameter was already studied by Johns et al [3]Better treatment protocols and safer standards could be theresult of a thorough study of the acoustic radiation of theseultrasonic applications

As the ultrasonic energy is applied to real perfused tissuesthe effect of blood flow and in vivo attenuation should beincorporated into the models As a first step in this workthe inclusion of blow flow in the model was omitted sincethe objective was to determine the contribution of the mainacoustic variables into the thermal distribution under verycontrolled and simple conditions For including blow flowin following steps there are some approaches that have


been proposed in the literature using ex vivo animal organshowever the range of control of these approaches shouldbe analyzed to not include uncertainties in the model Thevariability of attenuation among individuals is anothermatterthat should be commented on For modeling the temperaturedistributions dependent on the spatial acoustic pressurethe subdomain representing the phantom was configuredto have the same measured attenuation coefficient of theexperimental phantom which permits the direct comparisonof the model results and the experiments Moreover theconclusions drawn can be valid for another attenuation ifits effects are considered in both the experiments and themodels Accurate methods to determine attenuation in vivoand in real-time could help us in obtaining better modelsto study the effects during real therapy [42] However usinga fixed attenuation is still valid to demonstrate the drawnconclusions including variability of attenuation in the modelis beyond the scope of this paper

5 Conclusions

For the modeled thermal distribution presented in this workthe ideal classical approach of the piston-in-a-baffle was usedintentionally to evaluate its usefulness for this application Itwas evident that this model did not provide an acceptableapproximation to the real acoustic field emitted by thiskind of applicators and the modeled and measured thermaldistributions differed mainly due to the differences in theacoustic radiation patterns This was confirmed when usingthemeasured acoustic field as the input variable in the heatingmodel (model B)

This paper has presented evidences that suggest thatideal acoustic field models using pistons are not sufficientlyaccurate to study the energy deposition of physiotherapyultrasonic applicators As a possible approach the thermalmodels should be determined after modeling real acousticprofile (validation involved) without simplifying the radiatorusing a piston sourceThe thermal profile is tightly connectedto the acoustic field which at the same time depends onthe vibration distribution on the transducer surface [1]simplifying the latter with a piston radiator will produce largeerrors in the determined acoustic field and these errors willcarry important variation in the temperature estimation asdemonstrated

Competing Interests

The authors declare that none of them has any conflict ofinterests regarding the publication of this paper or financialand personal relationship with other people or organizationsthat could inappropriately influence the work

Acknowledgments

Theauthors thank JoseHugoZepeda Peralta andRuben PerezValladares for their invaluable technical support in the tem-perature measurements and their technical assistance duringtransducer characterization M I Gutierrez acknowledges

CONACYT Mexico for the fellowship 203653 This workwas supported by the ProjectCONACYT-ANUIES-ECOSnoM10-S02 and ICyTDF Project no PICCO10-78

References

[1] M I Gutierrez H Calas A Ramos A Vera and L LeijaldquoAcoustic field modeling for physiotherapy ultrasound applica-tors by using approximated functions ofmeasured non-uniformradiation distributionsrdquo Ultrasonics vol 52 no 6 pp 767ndash7772012

[2] A Azbaid A Ramos R Perez-Valladares H Calas and L LeijaldquoPreliminary analysis and simulation of the influence of radia-tions from the rim of therapeutic transducers in the irradiatedacoustic fieldrdquo in Proceedings of the IEEE Pan American HealthCare Exchanges (PAHCE rsquo10) Lima Peru March 2010

[3] L D Johns T J Demchak S J Straub and S M HowardldquoThe role of quantitative schlieren assessment of physiotherapyultrasound fields in describing variations between tissue heat-ing rates of different transducersrdquo Ultrasound in Medicine andBiology vol 33 no 12 pp 1911ndash1917 2007

[4] IEC-61689 IEC-61689 UltrasonicsmdashPhysiotherapy systemsmdashPerformance requirements andmethods ofmeasurement in thefrequency range 05MHz to 5MHz International Electrotech-nical Commission 1998

[5] FDA ldquoPerformance standards for sonic infrasonic and ultra-sonic radiation-emitting productsrdquo Tech Rep 21(8)-1050 USFood and Drug Administration 2008

[6] F A Duck ldquoMedical and non-medical protection standardsfor ultrasound and infrasoundrdquo Progress in Biophysics andMolecular Biology vol 93 no 1ndash3 pp 176ndash191 2007

[7] R T Hekkenberg ldquoCharacterising ultrasonic physiotherapysystems by performance and safety now internationally agreedrdquoUltrasonics vol 36 no 1ndash5 pp 713ndash720 1998

[8] C Kollmann G Vacariu O Schuhfried V Fialka-Moser andH Bergmann ldquoVariations in the output power and surface heat-ing effects of transducers in therapeutic ultrasoundrdquo Archivesof Physical Medicine and Rehabilitation vol 86 no 7 pp 1318ndash1324 2005

[9] S Paliwal and S Mitragotri ldquoTherapeutic opportunities inbiological responses of ultrasoundrdquo Ultrasonics vol 48 no 4pp 271ndash278 2008

[10] L B Feril Jr and T Kondo ldquoBiological effects of low intensityultrasound the mechanism involved and its implications ontherapy and on biosafety of ultrasoundrdquo Journal of RadiationResearch vol 45 no 4 pp 479ndash489 2004

[11] L U Signori S T da Costa A F S Neto et al ldquoHaematologicaleffect of pulsed ultrasound in acute muscular inflammation inratsrdquo Physiotherapy vol 97 no 2 pp 163ndash169 2011

[12] J P C Matheus F B Oliveira L B Gomide J G P OMilani J B Volpon and A C Shimano ldquoEffects of therapeuticultrasound on the mechanical properties of skeletal musclesafter contusionrdquo Revista Brasileira de Fisioterapia vol 12 no 3pp 241ndash247 2008

[13] R Marks S Ghanagaraja and M Ghassemi ldquoUltrasound forosteo-arthritis of the knee a systematic reviewrdquo Physiotherapyvol 86 no 9 pp 452ndash463 2000

[14] V J Robertson and K G Baker ldquoA review of therapeuticultrasound effectiveness studiesrdquo Physical Therapy vol 81 no7 pp 1339ndash1350 2001


[15] D L Miller N B Smith M R Bailey G J Czarnota K Hyny-nen and I R S Makin ldquoOverview of therapeutic ultrasoundapplications and safety considerationsrdquo Journal of Ultrasound inMedicine vol 31 no 4 pp 623ndash634 2012

[16] K G Baker V J Robertson and F A Duck ldquoA review oftherapeutic ultrasound biophysical effectsrdquo Physical Therapyvol 81 no 7 pp 1351ndash1358 2001

[17] I Boyraz A Yildiz B Koc and H Sarman ldquoComparison ofhigh-intensity laser therapy and ultrasound treatment in thepatients with lumbar discopathyrdquo BioMed Research Interna-tional vol 2015 Article ID 304328 6 pages 2015

[18] T Watson ldquoUltrasound in contemporary physiotherapy prac-ticerdquo Ultrasonics vol 48 no 4 pp 321ndash329 2008

[19] K Grey ldquoDistribution of treatment time in physiotherapeuticapplication of ultrasoundrdquo Physiotherapy vol 89 no 12 pp696ndash707 2003

[20] S J Warden and J M McMeeken ldquoUltrasound usage anddosage in sports physiotherapyrdquo Ultrasound in Medicine andBiology vol 28 no 8 pp 1075ndash1080 2002

[21] S L Weaver T J Demchak M B Stone J B Brucker andP O Burr ldquoEffect of transducer velocity on intramusculartemperature during a 1-MHz ultrasound treatmentrdquo Journal ofOrthopaedic and Sports PhysicalTherapy vol 36 no 5 pp 320ndash325 2006

[22] T J Demchak and M B Stone ldquoEffectiveness of clinical ultra-sound parameters on changing intramuscular temperaturerdquoJournal of Sport Rehabilitation vol 17 no 3 pp 220ndash229 2008

[23] X B Fan E G Moros and W L Straube ldquoAcoustic fieldprediction for a single planar continuous-wave source usingan equivalent phased array methodrdquo Journal of the AcousticalSociety of America vol 102 no 5 pp 2734ndash2741 1997

[24] P N Gelat B Zeqiri and M Hodnett ldquoA finite-element modelof the aperture method for determining the effective radiatingarea of physiotherapy treatment headsrdquo Ultrasonics vol 43 no5 pp 321ndash330 2005

[25] X-Q Lu E C Burdette and G K Svensson ldquoUltrasound fieldcalculation in a water-soft tissuemediumrdquo International Journalof Hyperthermia vol 14 no 2 pp 169ndash182 1998

[26] M G Miller J R Longoria C C Cheatham R J Baker andT J Michael ldquoIntramuscular temperature differences betweenthe mid-point and peripheral effective radiating area withultrasoundrdquo Journal of Sports Science and Medicine vol 7 no2 pp 286ndash291 2008

[27] P He ldquoDetermination of ultrasonic parameters based onattenuation and dispersion measurementsrdquo Ultrasonic Imagingvol 20 no 4 pp 275ndash287 1998

[28] J M M Pinkerton ldquoA pulse method for the measurement ofultrasonic absorption in liquids results for waterrdquo Nature vol160 no 4056 pp 128ndash129 1947

[29] J Lubbers and R Graaff ldquoA simple and accurate formula for thesound velocity in waterrdquo Ultrasound in Medicine and Biologyvol 24 no 7 pp 1065ndash1068 1998

[30] IEC-61102 611021991 Measurement and characterization ofultrasonic fields using hydrophones in the frequency range05MHz to 15MHz International Electrotechnical Commis-sion 1991

[31] K Hynynen C J Martin D J Watmough and J R MallardldquoErrors in temperature measurement by thermocouple probesduring ultrasound induced hyperthermiardquo British Journal ofRadiology vol 56 no 672 pp 969ndash970 1983

[32] DMNell andM RMyers ldquoThermal effects generated by high-intensity focused ultrasound beams at normal incidence to abone surfacerdquo Journal of the Acoustical Society of America vol127 no 1 pp 549ndash559 2010

[33] T D Mast and F Yu ldquoSimplified expansions for radiation froma baffled circular pistonrdquo Journal of the Acoustical Society ofAmerica vol 118 no 6 pp 3457ndash3464 2005

[34] W L Nyborg ldquoHeat generation by ultrasound in a relaxingmediumrdquo Journal of the Acoustical Society of America vol 70no 2 pp 310ndash312 1981

[35] M-I Gutierrez L Leija and A Vera ldquoModeling the acous-tic field of physiotherapy ultrasound transducers using nonuniform acoustic pressure distributionsrdquo in Proceedings of the2010 7th International Conference on Electrical EngineeringComputing Science and Automatic Control (CCE rsquo10) pp 256ndash260 IEEE Chiapas Mexico September 2010

[36] E G Moros A W Dutton R B Roemer M Burton andK Hynynen ldquoExperimental evaluation of two simple thermalmodels using hyperthermia in muscle in vivordquo InternationalJournal of Hyperthermia vol 9 no 4 pp 581ndash598 1993

[37] L E Kinsler A R Frey A B Coppens and J V SandersFundamentals of Acoustics John Wiley amp Sons New York NYUSA 4th edition 2000

[38] S A Lopez-Haro L Leija L Favari andAVera ldquoMeasurementof ultrasonic properties into biological tissues in the hyperther-mia temperature rangerdquo Physics Procedia vol 3 no 1 pp 551ndash558 2010

[39] K Hynynen and D K Edwards ldquoTemperature measurementsduring ultrasound hyperthermiardquo Medical Physics vol 16 no4 pp 618ndash626 1989

[40] M A Merrick K D Bernard S T Devor and J MWilliams ldquoIdentical 3-MHz ultrasound treatments with dif-ferent devices produce different intramuscular temperaturesrdquoJournal of Orthopaedic and Sports Physical Therapy vol 33 no7 pp 379ndash385 2003

[41] D O Draper S Sunderland D T Kirkendall and M RicardldquoA comparison of temperature rise in human calf muscles fol-lowing applications of underwater and topical gel ultrasoundrdquoJournal of Orthopaedic amp Sports Physical Therapy vol 17 no 5pp 247ndash251 1993

[42] T A Bigelow B L McFarlinW D OrsquoBrien Jr andM L OelzeldquoIn vivo ultrasonic attenuation slope estimates for detectingcervical ripening in rats preliminary resultsrdquo Journal of theAcoustical Society of America vol 123 no 3 pp 1794ndash18002008

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


MEDIATORSINFLAMMATION

of


Behavioural Neurology

EndocrinologyInternational Journal of



Disease Markers


BioMed Research International

OncologyJournal of



Oxidative Medicine and Cellular Longevity


PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of



Computational and Mathematical Methods in Medicine

OphthalmologyJournal of


Diabetes ResearchJournal of



Research and TreatmentAIDS


Gastroenterology Research and Practice


Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom


For instance the definitive protocol for applying the therapyis often chosen by therapists based on their own experiencefrom which they decide the size of the treatment area (egtwice the transducer emitting surface) [19] and the treatmentdose [20] The scanning speed also depends on the therapistbut it has been reported by Weaver et al [21] that thechanges in average heating using different scanning speedwithin a specific treatment area were not significant Greyin 2003 [19] found significant variations in the scanningspeed and important differences in the dynamic strategiesof therapy application when analyzing the transducer move-ment patterns produced by different therapists Demchak andStone [22] reported a difference in deep heating in humansubjects when using clinically observed treatment parametersand the recommended treatment parameters that researchsuggested The local recorded doses were inconsistent withthose theoretically expected which make us hypothesize thatthese differences could influence the gotten temperature intissues In this work we are interested in reducing the sourceof subjectivities by proposing a different method for therapyapplication in which the therapist does not interfere directlyAlthough studies of dynamic protocols are not discarded forfuture work these would be carried out after determining thefeasibility of improving the static application outcomes

In literature measurements of temperature distributionsproduced by physiotherapy ultrasound have been reported inboth real tissues and tissue-mimickingmaterials (phantoms)From these publications a small number were related toproduce accurate models of acoustic field for the planarradiators used in this application [1 23] Furthermore whenthe acoustic field was modeled it usually referred to a pistonsource [24 25] and few particular efforts were made todemonstrate the validity of this assumption For instanceGelat et al [24] presented the aperture method to determinethe effective radiating area of physiotherapy transducersformulated by assuming the transducer as a plane-pistonAlthough the validity of the method is not questionedsimplifying the transducer as a piston source is not congruentwith the measured fields of this kind of radiators [1 2 23] Luet al [25] developed amethod for numerically approximatingthe acoustic field of hyperthermia devices in multilayermedia They also assumed the radiator as a piston sourceand they validated their numerical results with the Rayleighintegral On the basis of the ideal vibrating piston Grey [19]calculated the distribution of superficial energy that resultedfrom therapy applications by 22 therapists however Greydid not present the validation of either the resulting heatingmodel or the simplified acoustic model Recently Miller etal [26] measured in vivo temperature distributions generatedby therapeutic ultrasound in human triceps surae musclegroup during a dynamical treatment protocol on a regionof two times the size of the transducer head Their resultsindicated that the produced thermal increments were notuniform in the treatment area and among individuals andthey concluded that these differences could be related tosome variables they could not control in the experiments asthe thermocouple final insertion the size of treatment areaand so forth However they did not provide any theoreticalmodel to validate the results or demonstrate their hypothesis

Following a thorough analysis of the state-of-the-art weconcluded that few efforts have been made to evaluate thecapability of current acoustic fieldmodels to produce thermaldistributions congruent with measurements

Studies on hyperthermia therapies specifically in physio-therapy applications seldom consider realistic acoustic fieldpatterns and their effects in thermal distributions Whenmodeling the acoustic field the most used approach inultrasonic physiotherapy literature is the piston-in-a-baffle(uniform vibrating aperture into an infinite rigid and no-vibrating baffle) which creates a characteristic diffractionpattern in the near-field zone Moreover when temper-ature profiles of physiotherapy applicators are presentedthey commonly referred to the ideal acoustic field and noanalysis of the importance of correctly representing thereal acoustic field is made In this paper the thermal pat-tern produced by the piston-in-a-baffle model is comparedto experimental measurements to verify if this ideal fieldaccurately allows the representation of realistic temperaturedistributions additionally anothermodeling approach is alsoused for comparison in order to determine if using the realacoustic profile permits an improvement of the modeledthermal distribution For this purpose a finite element(FE) model of ultrasonic heating produced by a theoreticalpiston is compared with the heating produced by a realphysiotherapy ultrasound applicator the second approachusing the measured acoustic field to feed the FE model anddetermine the heating is also shown For the experimentsand models a static therapy protocol is used since inthis modality the dosage is directly correlated to treatmenttime while presenting few effects of thermal dissipation byconduction [19] Additionally this approach eliminates theneed to consider therapist-dependent variables such as thescanning speed of the applicator and the irregular size ofthe treated area The results of this work will allow a betterunderstanding of these acoustic fields and the thermal effectsthey produce to eventually improve the efficacy of ultrasoundbased physiotherapy treatment

2 Materials and Methods

The models were developed in a workstation with 64GB ofRAM and 8-core processor of 300GHz The experimentswere performed using a physiotherapy ultrasound deviceradiating to a muscle-mimicking material (phantom) Thephantom with a cubic shape of 10 cm per side emulatedthe acoustic properties of muscular tissue and was madeof a water-based mixture of agarose graphite and glycerinThe physiotherapy ultrasonic device comprised a special-designed homemade RF amplifier and a commercial trans-ducer of 1MHz and 10 cm2 of nominal radiating area (Model7310 Mettler Electronics Corp USA)

21 Determination of PhantomAcoustic Properties Thephan-tom acoustic properties used in the models were determinedby measurements The acoustic attenuation was obtainedwith the Ping He formulation [27] The phantom was splitinto 10 equal sections of 1 cm of thickness and the attenuation




P1

P2

Pw

Ps

(a)

Window

Drilled walls

Transducerholder


Phantom

Thermocouples

y

x

z

(b)

Phantom

1

2 3

4

5

z

r0

10cm

5 cm(c)









119868 = 119875119877119860ER

(1)










10038161003816100381610038161199011003816100381610038161003816 = radic2120588119888119868 (2)


1198860 = V0120596119890119895(120596119905+1205872) = 120596120588119888 1199010119890119895(120596119905+1205872) (3)



nabla2119901 + 1198961198882119901 = 0 (4)



120588119862 120597119879120597119905 minus 119896nabla2119879 = 119876ext (5)




119876ext (119903 119911) = 2120572119868 (119903 119911) = 120572 1003816100381610038161003816119901 (119903 119911)10038161003816100381610038162

120588119888 [Wm3] (6)






3 Results





0 2 4 6 8 1000

02

04

06

08

10

12

MeasuredPiston

z (cm)

Nor

mal

ized

acou

stic p

ress

ure

(a)

0 2x (cm)

z (c

m)

minus2

10

5

0

(b)

0 2x (cm)

minus2

z (c

m)

10

5

0

(c)






0 100 200 300 40037

39

41

43

45

47

49


Time (s)

Tem

pera

ture

(∘C)





++5 cm





r (cm)

Acou

stic p

ress

ure (

MPa

)

Free space

0 10 2000

01

02

03


(a)

Acou

stic p

ress

ure (

MPa

)

0005 06 07 08 09

01

02

03

04

05

10z (cm)



(b)

00 02 04 06 08 10477

479

481

483

485

z (cm)

Tem

pera

ture

(∘C)



(c)

Time (s)0 100 200 300 400

37

39

41

43

45

47

49


Tem

pera

ture

(∘C)

(d)










1 2 3 4 5 6 7 8 9 10 1100

02

04

06

08

10

12

MeasuredModeled

Attenuation

z (cm)

Nor

mal

ized

inte

nsity





z (c

m)

r (mm)

10

9

8

7

6

5

4

3

2

115 10 5 0 15105

(a)

r (mm)15 10 5 0 15105

10

09

08

07

06

05

04

03

02

01

0

(b)

r (mm)

40

30

20

10

0

()

minus10

minus20

minus30

0 5 10 15

(c)







z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(a)z

(cm

)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(b)

z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(c)

z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C38

∘ C

(d)





z (cm)

Tem

pera

ture

incr

emen

ts (∘

C)12

10

8

6

4

2

0

1086420

ExperimentModel A

Model B

(a)

Relat

ive e

rror

()

minus30

minus20

minus10

0

10

20

30

z (cm)1086420


(b)

ExperimentModel A

Model B

Tem

pera

ture

incr

emen

ts (∘

C)

10

8

6

4

2

0

r (cm)0 1 2 3

(c)

Relat

ive e

rror

()

minus40

minus20

0

20

40

60


80

r (cm)0 1 2 3

(d)


4 Discussion












5 Conclusions



Competing Interests


Acknowledgments



References

















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of



















P1

P2

Pw

Ps

(a)

Window

Drilled walls

Transducerholder


Phantom

Thermocouples

y

x

z

(b)

Phantom

1

2 3

4

5

z

r0

10cm

5 cm(c)









119868 = 119875119877119860ER

(1)










10038161003816100381610038161199011003816100381610038161003816 = radic2120588119888119868 (2)


1198860 = V0120596119890119895(120596119905+1205872) = 120596120588119888 1199010119890119895(120596119905+1205872) (3)



nabla2119901 + 1198961198882119901 = 0 (4)



120588119862 120597119879120597119905 minus 119896nabla2119879 = 119876ext (5)




119876ext (119903 119911) = 2120572119868 (119903 119911) = 120572 1003816100381610038161003816119901 (119903 119911)10038161003816100381610038162

120588119888 [Wm3] (6)






3 Results





0 2 4 6 8 1000

02

04

06

08

10

12

MeasuredPiston

z (cm)

Nor

mal

ized

acou

stic p

ress

ure

(a)

0 2x (cm)

z (c

m)

minus2

10

5

0

(b)

0 2x (cm)

minus2

z (c

m)

10

5

0

(c)






0 100 200 300 40037

39

41

43

45

47

49


Time (s)

Tem

pera

ture

(∘C)





++5 cm





r (cm)

Acou

stic p

ress

ure (

MPa

)

Free space

0 10 2000

01

02

03


(a)

Acou

stic p

ress

ure (

MPa

)

0005 06 07 08 09

01

02

03

04

05

10z (cm)



(b)

00 02 04 06 08 10477

479

481

483

485

z (cm)

Tem

pera

ture

(∘C)



(c)

Time (s)0 100 200 300 400

37

39

41

43

45

47

49


Tem

pera

ture

(∘C)

(d)










1 2 3 4 5 6 7 8 9 10 1100

02

04

06

08

10

12

MeasuredModeled

Attenuation

z (cm)

Nor

mal

ized

inte

nsity





z (c

m)

r (mm)

10

9

8

7

6

5

4

3

2

115 10 5 0 15105

(a)

r (mm)15 10 5 0 15105

10

09

08

07

06

05

04

03

02

01

0

(b)

r (mm)

40

30

20

10

0

()

minus10

minus20

minus30

0 5 10 15

(c)







z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(a)z

(cm

)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(b)

z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(c)

z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C38

∘ C

(d)





z (cm)

Tem

pera

ture

incr

emen

ts (∘

C)12

10

8

6

4

2

0

1086420

ExperimentModel A

Model B

(a)

Relat

ive e

rror

()

minus30

minus20

minus10

0

10

20

30

z (cm)1086420


(b)

ExperimentModel A

Model B

Tem

pera

ture

incr

emen

ts (∘

C)

10

8

6

4

2

0

r (cm)0 1 2 3

(c)

Relat

ive e

rror

()

minus40

minus20

0

20

40

60


80

r (cm)0 1 2 3

(d)


4 Discussion












5 Conclusions



Competing Interests


Acknowledgments



References

















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of



















119868 = 119875119877119860ER

(1)










10038161003816100381610038161199011003816100381610038161003816 = radic2120588119888119868 (2)


1198860 = V0120596119890119895(120596119905+1205872) = 120596120588119888 1199010119890119895(120596119905+1205872) (3)



nabla2119901 + 1198961198882119901 = 0 (4)



120588119862 120597119879120597119905 minus 119896nabla2119879 = 119876ext (5)




119876ext (119903 119911) = 2120572119868 (119903 119911) = 120572 1003816100381610038161003816119901 (119903 119911)10038161003816100381610038162

120588119888 [Wm3] (6)






3 Results





0 2 4 6 8 1000

02

04

06

08

10

12

MeasuredPiston

z (cm)

Nor

mal

ized

acou

stic p

ress

ure

(a)

0 2x (cm)

z (c

m)

minus2

10

5

0

(b)

0 2x (cm)

minus2

z (c

m)

10

5

0

(c)






0 100 200 300 40037

39

41

43

45

47

49


Time (s)

Tem

pera

ture

(∘C)





++5 cm





r (cm)

Acou

stic p

ress

ure (

MPa

)

Free space

0 10 2000

01

02

03


(a)

Acou

stic p

ress

ure (

MPa

)

0005 06 07 08 09

01

02

03

04

05

10z (cm)



(b)

00 02 04 06 08 10477

479

481

483

485

z (cm)

Tem

pera

ture

(∘C)



(c)

Time (s)0 100 200 300 400

37

39

41

43

45

47

49


Tem

pera

ture

(∘C)

(d)










1 2 3 4 5 6 7 8 9 10 1100

02

04

06

08

10

12

MeasuredModeled

Attenuation

z (cm)

Nor

mal

ized

inte

nsity





z (c

m)

r (mm)

10

9

8

7

6

5

4

3

2

115 10 5 0 15105

(a)

r (mm)15 10 5 0 15105

10

09

08

07

06

05

04

03

02

01

0

(b)

r (mm)

40

30

20

10

0

()

minus10

minus20

minus30

0 5 10 15

(c)







z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(a)z

(cm

)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(b)

z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(c)

z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C38

∘ C

(d)





z (cm)

Tem

pera

ture

incr

emen

ts (∘

C)12

10

8

6

4

2

0

1086420

ExperimentModel A

Model B

(a)

Relat

ive e

rror

()

minus30

minus20

minus10

0

10

20

30

z (cm)1086420


(b)

ExperimentModel A

Model B

Tem

pera

ture

incr

emen

ts (∘

C)

10

8

6

4

2

0

r (cm)0 1 2 3

(c)

Relat

ive e

rror

()

minus40

minus20

0

20

40

60


80

r (cm)0 1 2 3

(d)


4 Discussion












5 Conclusions



Competing Interests


Acknowledgments



References

















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of




















10038161003816100381610038161199011003816100381610038161003816 = radic2120588119888119868 (2)


1198860 = V0120596119890119895(120596119905+1205872) = 120596120588119888 1199010119890119895(120596119905+1205872) (3)



nabla2119901 + 1198961198882119901 = 0 (4)



120588119862 120597119879120597119905 minus 119896nabla2119879 = 119876ext (5)




119876ext (119903 119911) = 2120572119868 (119903 119911) = 120572 1003816100381610038161003816119901 (119903 119911)10038161003816100381610038162

120588119888 [Wm3] (6)






3 Results





0 2 4 6 8 1000

02

04

06

08

10

12

MeasuredPiston

z (cm)

Nor

mal

ized

acou

stic p

ress

ure

(a)

0 2x (cm)

z (c

m)

minus2

10

5

0

(b)

0 2x (cm)

minus2

z (c

m)

10

5

0

(c)






0 100 200 300 40037

39

41

43

45

47

49


Time (s)

Tem

pera

ture

(∘C)





++5 cm





r (cm)

Acou

stic p

ress

ure (

MPa

)

Free space

0 10 2000

01

02

03


(a)

Acou

stic p

ress

ure (

MPa

)

0005 06 07 08 09

01

02

03

04

05

10z (cm)



(b)

00 02 04 06 08 10477

479

481

483

485

z (cm)

Tem

pera

ture

(∘C)



(c)

Time (s)0 100 200 300 400

37

39

41

43

45

47

49


Tem

pera

ture

(∘C)

(d)










1 2 3 4 5 6 7 8 9 10 1100

02

04

06

08

10

12

MeasuredModeled

Attenuation

z (cm)

Nor

mal

ized

inte

nsity





z (c

m)

r (mm)

10

9

8

7

6

5

4

3

2

115 10 5 0 15105

(a)

r (mm)15 10 5 0 15105

10

09

08

07

06

05

04

03

02

01

0

(b)

r (mm)

40

30

20

10

0

()

minus10

minus20

minus30

0 5 10 15

(c)







z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(a)z

(cm

)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(b)

z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(c)

z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C38

∘ C

(d)





z (cm)

Tem

pera

ture

incr

emen

ts (∘

C)12

10

8

6

4

2

0

1086420

ExperimentModel A

Model B

(a)

Relat

ive e

rror

()

minus30

minus20

minus10

0

10

20

30

z (cm)1086420


(b)

ExperimentModel A

Model B

Tem

pera

ture

incr

emen

ts (∘

C)

10

8

6

4

2

0

r (cm)0 1 2 3

(c)

Relat

ive e

rror

()

minus40

minus20

0

20

40

60


80

r (cm)0 1 2 3

(d)


4 Discussion












5 Conclusions



Competing Interests


Acknowledgments



References

















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















119876ext (119903 119911) = 2120572119868 (119903 119911) = 120572 1003816100381610038161003816119901 (119903 119911)10038161003816100381610038162

120588119888 [Wm3] (6)






3 Results





0 2 4 6 8 1000

02

04

06

08

10

12

MeasuredPiston

z (cm)

Nor

mal

ized

acou

stic p

ress

ure

(a)

0 2x (cm)

z (c

m)

minus2

10

5

0

(b)

0 2x (cm)

minus2

z (c

m)

10

5

0

(c)






0 100 200 300 40037

39

41

43

45

47

49


Time (s)

Tem

pera

ture

(∘C)





++5 cm





r (cm)

Acou

stic p

ress

ure (

MPa

)

Free space

0 10 2000

01

02

03


(a)

Acou

stic p

ress

ure (

MPa

)

0005 06 07 08 09

01

02

03

04

05

10z (cm)



(b)

00 02 04 06 08 10477

479

481

483

485

z (cm)

Tem

pera

ture

(∘C)



(c)

Time (s)0 100 200 300 400

37

39

41

43

45

47

49


Tem

pera

ture

(∘C)

(d)










1 2 3 4 5 6 7 8 9 10 1100

02

04

06

08

10

12

MeasuredModeled

Attenuation

z (cm)

Nor

mal

ized

inte

nsity





z (c

m)

r (mm)

10

9

8

7

6

5

4

3

2

115 10 5 0 15105

(a)

r (mm)15 10 5 0 15105

10

09

08

07

06

05

04

03

02

01

0

(b)

r (mm)

40

30

20

10

0

()

minus10

minus20

minus30

0 5 10 15

(c)







z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(a)z

(cm

)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(b)

z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(c)

z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C38

∘ C

(d)





z (cm)

Tem

pera

ture

incr

emen

ts (∘

C)12

10

8

6

4

2

0

1086420

ExperimentModel A

Model B

(a)

Relat

ive e

rror

()

minus30

minus20

minus10

0

10

20

30

z (cm)1086420


(b)

ExperimentModel A

Model B

Tem

pera

ture

incr

emen

ts (∘

C)

10

8

6

4

2

0

r (cm)0 1 2 3

(c)

Relat

ive e

rror

()

minus40

minus20

0

20

40

60


80

r (cm)0 1 2 3

(d)


4 Discussion












5 Conclusions



Competing Interests


Acknowledgments



References

















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















0 2 4 6 8 1000

02

04

06

08

10

12

MeasuredPiston

z (cm)

Nor

mal

ized

acou

stic p

ress

ure

(a)

0 2x (cm)

z (c

m)

minus2

10

5

0

(b)

0 2x (cm)

minus2

z (c

m)

10

5

0

(c)






0 100 200 300 40037

39

41

43

45

47

49


Time (s)

Tem

pera

ture

(∘C)





++5 cm





r (cm)

Acou

stic p

ress

ure (

MPa

)

Free space

0 10 2000

01

02

03


(a)

Acou

stic p

ress

ure (

MPa

)

0005 06 07 08 09

01

02

03

04

05

10z (cm)



(b)

00 02 04 06 08 10477

479

481

483

485

z (cm)

Tem

pera

ture

(∘C)



(c)

Time (s)0 100 200 300 400

37

39

41

43

45

47

49


Tem

pera

ture

(∘C)

(d)










1 2 3 4 5 6 7 8 9 10 1100

02

04

06

08

10

12

MeasuredModeled

Attenuation

z (cm)

Nor

mal

ized

inte

nsity





z (c

m)

r (mm)

10

9

8

7

6

5

4

3

2

115 10 5 0 15105

(a)

r (mm)15 10 5 0 15105

10

09

08

07

06

05

04

03

02

01

0

(b)

r (mm)

40

30

20

10

0

()

minus10

minus20

minus30

0 5 10 15

(c)







z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(a)z

(cm

)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(b)

z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(c)

z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C38

∘ C

(d)





z (cm)

Tem

pera

ture

incr

emen

ts (∘

C)12

10

8

6

4

2

0

1086420

ExperimentModel A

Model B

(a)

Relat

ive e

rror

()

minus30

minus20

minus10

0

10

20

30

z (cm)1086420


(b)

ExperimentModel A

Model B

Tem

pera

ture

incr

emen

ts (∘

C)

10

8

6

4

2

0

r (cm)0 1 2 3

(c)

Relat

ive e

rror

()

minus40

minus20

0

20

40

60


80

r (cm)0 1 2 3

(d)


4 Discussion












5 Conclusions



Competing Interests


Acknowledgments



References

















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















0 100 200 300 40037

39

41

43

45

47

49


Time (s)

Tem

pera

ture

(∘C)





++5 cm





r (cm)

Acou

stic p

ress

ure (

MPa

)

Free space

0 10 2000

01

02

03


(a)

Acou

stic p

ress

ure (

MPa

)

0005 06 07 08 09

01

02

03

04

05

10z (cm)



(b)

00 02 04 06 08 10477

479

481

483

485

z (cm)

Tem

pera

ture

(∘C)



(c)

Time (s)0 100 200 300 400

37

39

41

43

45

47

49


Tem

pera

ture

(∘C)

(d)










1 2 3 4 5 6 7 8 9 10 1100

02

04

06

08

10

12

MeasuredModeled

Attenuation

z (cm)

Nor

mal

ized

inte

nsity





z (c

m)

r (mm)

10

9

8

7

6

5

4

3

2

115 10 5 0 15105

(a)

r (mm)15 10 5 0 15105

10

09

08

07

06

05

04

03

02

01

0

(b)

r (mm)

40

30

20

10

0

()

minus10

minus20

minus30

0 5 10 15

(c)







z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(a)z

(cm

)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(b)

z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(c)

z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C38

∘ C

(d)





z (cm)

Tem

pera

ture

incr

emen

ts (∘

C)12

10

8

6

4

2

0

1086420

ExperimentModel A

Model B

(a)

Relat

ive e

rror

()

minus30

minus20

minus10

0

10

20

30

z (cm)1086420


(b)

ExperimentModel A

Model B

Tem

pera

ture

incr

emen

ts (∘

C)

10

8

6

4

2

0

r (cm)0 1 2 3

(c)

Relat

ive e

rror

()

minus40

minus20

0

20

40

60


80

r (cm)0 1 2 3

(d)


4 Discussion












5 Conclusions



Competing Interests


Acknowledgments



References

















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















r (cm)

Acou

stic p

ress

ure (

MPa

)

Free space

0 10 2000

01

02

03


(a)

Acou

stic p

ress

ure (

MPa

)

0005 06 07 08 09

01

02

03

04

05

10z (cm)



(b)

00 02 04 06 08 10477

479

481

483

485

z (cm)

Tem

pera

ture

(∘C)



(c)

Time (s)0 100 200 300 400

37

39

41

43

45

47

49


Tem

pera

ture

(∘C)

(d)










1 2 3 4 5 6 7 8 9 10 1100

02

04

06

08

10

12

MeasuredModeled

Attenuation

z (cm)

Nor

mal

ized

inte

nsity





z (c

m)

r (mm)

10

9

8

7

6

5

4

3

2

115 10 5 0 15105

(a)

r (mm)15 10 5 0 15105

10

09

08

07

06

05

04

03

02

01

0

(b)

r (mm)

40

30

20

10

0

()

minus10

minus20

minus30

0 5 10 15

(c)







z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(a)z

(cm

)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(b)

z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(c)

z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C38

∘ C

(d)





z (cm)

Tem

pera

ture

incr

emen

ts (∘

C)12

10

8

6

4

2

0

1086420

ExperimentModel A

Model B

(a)

Relat

ive e

rror

()

minus30

minus20

minus10

0

10

20

30

z (cm)1086420


(b)

ExperimentModel A

Model B

Tem

pera

ture

incr

emen

ts (∘

C)

10

8

6

4

2

0

r (cm)0 1 2 3

(c)

Relat

ive e

rror

()

minus40

minus20

0

20

40

60


80

r (cm)0 1 2 3

(d)


4 Discussion












5 Conclusions



Competing Interests


Acknowledgments



References

















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of





















1 2 3 4 5 6 7 8 9 10 1100

02

04

06

08

10

12

MeasuredModeled

Attenuation

z (cm)

Nor

mal

ized

inte

nsity





z (c

m)

r (mm)

10

9

8

7

6

5

4

3

2

115 10 5 0 15105

(a)

r (mm)15 10 5 0 15105

10

09

08

07

06

05

04

03

02

01

0

(b)

r (mm)

40

30

20

10

0

()

minus10

minus20

minus30

0 5 10 15

(c)







z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(a)z

(cm

)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(b)

z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(c)

z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C38

∘ C

(d)





z (cm)

Tem

pera

ture

incr

emen

ts (∘

C)12

10

8

6

4

2

0

1086420

ExperimentModel A

Model B

(a)

Relat

ive e

rror

()

minus30

minus20

minus10

0

10

20

30

z (cm)1086420


(b)

ExperimentModel A

Model B

Tem

pera

ture

incr

emen

ts (∘

C)

10

8

6

4

2

0

r (cm)0 1 2 3

(c)

Relat

ive e

rror

()

minus40

minus20

0

20

40

60


80

r (cm)0 1 2 3

(d)


4 Discussion












5 Conclusions



Competing Interests


Acknowledgments



References

















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















z (c

m)

r (mm)

10

9

8

7

6

5

4

3

2

115 10 5 0 15105

(a)

r (mm)15 10 5 0 15105

10

09

08

07

06

05

04

03

02

01

0

(b)

r (mm)

40

30

20

10

0

()

minus10

minus20

minus30

0 5 10 15

(c)







z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(a)z

(cm

)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(b)

z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(c)

z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C38

∘ C

(d)





z (cm)

Tem

pera

ture

incr

emen

ts (∘

C)12

10

8

6

4

2

0

1086420

ExperimentModel A

Model B

(a)

Relat

ive e

rror

()

minus30

minus20

minus10

0

10

20

30

z (cm)1086420


(b)

ExperimentModel A

Model B

Tem

pera

ture

incr

emen

ts (∘

C)

10

8

6

4

2

0

r (cm)0 1 2 3

(c)

Relat

ive e

rror

()

minus40

minus20

0

20

40

60


80

r (cm)0 1 2 3

(d)


4 Discussion












5 Conclusions



Competing Interests


Acknowledgments



References

















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(a)z

(cm

)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(b)

z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C

38∘ C

(c)

z (c

m)

10

9

8

7

6

5

4

3

2

1

r (cm)5 4 3 2 1 0 54321

38∘ C38

∘ C

(d)





z (cm)

Tem

pera

ture

incr

emen

ts (∘

C)12

10

8

6

4

2

0

1086420

ExperimentModel A

Model B

(a)

Relat

ive e

rror

()

minus30

minus20

minus10

0

10

20

30

z (cm)1086420


(b)

ExperimentModel A

Model B

Tem

pera

ture

incr

emen

ts (∘

C)

10

8

6

4

2

0

r (cm)0 1 2 3

(c)

Relat

ive e

rror

()

minus40

minus20

0

20

40

60


80

r (cm)0 1 2 3

(d)


4 Discussion












5 Conclusions



Competing Interests


Acknowledgments



References

















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















z (cm)

Tem

pera

ture

incr

emen

ts (∘

C)12

10

8

6

4

2

0

1086420

ExperimentModel A

Model B

(a)

Relat

ive e

rror

()

minus30

minus20

minus10

0

10

20

30

z (cm)1086420


(b)

ExperimentModel A

Model B

Tem

pera

ture

incr

emen

ts (∘

C)

10

8

6

4

2

0

r (cm)0 1 2 3

(c)

Relat

ive e

rror

()

minus40

minus20

0

20

40

60


80

r (cm)0 1 2 3

(d)


4 Discussion












5 Conclusions



Competing Interests


Acknowledgments



References

















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

























5 Conclusions



Competing Interests


Acknowledgments



References

















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of
















research article experimental verification of modeled

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