Noninvasive Measurement of Local Thermal Diffusivity Using Backscattered Ultrasound and Focused Ultrasound Heating

Ultrasound in Med. & Biol., Vol. 34, No. 9, pp. 1449–1464, 2008Copyright © 2008 World Federation for Ultrasound in Medicine & Biology

Printed in the USA. All rights reserved0301-5629/08/$–see front matter

doi:10.1016/j.ultrasmedbio.2008.02.004

● Original Contribution

NONINVASIVE MEASUREMENT OF LOCAL THERMAL DIFFUSIVITYUSING BACKSCATTERED ULTRASOUND AND FOCUSED ULTRASOUND

HEATING

AJAY ANAND* and PETER J. KACZKOWSKI

Center for Industrial and Medical Ultrasound, Applied Physics Laboratory, University of Washington, Seattle, WA, USA.

(Received 22 April 2007; revised 19 October 2007; in final form 4 February 2008)

Abstract—Previously, noninvasive methods of estimating local tissue thermal and acoustic properties usingbackscattered ultrasound have been proposed in the literature. In this article, a noninvasive method of estimatinglocal thermal diffusivity in situ during focused ultrasound heating using beamformed acoustic backscatter dataand applying novel signal processing techniques is developed. A high intensity focused ultrasound (HIFU)transducer operating at subablative intensities is employed to create a brief local temperature rise of no morethan 10°C. Beamformed radio-frequency (RF) data are collected during heating and cooling using a clinicalultrasound scanner. Measurements of the time-varying “acoustic strain”, that is, spatiotemporal variations in theRF echo shifts induced by the temperature related sound speed changes, are related to a solution of the heattransfer equation to estimate the thermal diffusivity in the heated zone. Numerical simulations and experimentsperformed in vitro in tissue mimicking phantoms and excised turkey breast muscle tissue demonstrate agreementbetween the ultrasound derived thermal diffusivity estimates and independent estimates made by a traditionalhot-wire technique. The new noninvasive ultrasonic method has potential applications in thermal therapyplanning and monitoring, physiological monitoring and as a means of noninvasive tissue characterization.(E-mail: [email protected]) © 2008 World Federation for Ultrasound in Medicine & Biology.

Key Words: Thermal ablation, Ultrasound treatment monitoring, Tissue characterization, Ultrasonic signal
processing, Ultrasound thermometry.
INTRODUCTION

The delivery of an accurate thermal dose is essential forthe success of ablative thermal therapies such as highintensity focused ultrasound (HIFU) (Vaezy et al. 1997;Sanghvi et al. 1999; ter Haar 2001; Wu et al. 2002) andradio-frequency ablation (Mahnken et al. 2004; Frerickset al. 2005; Nour and Lewin 2005) in the clinic. Thethermal parameters at the ablation site, namely, thermaldiffusivity and perfusion loss, play important roles in thefinal therapeutic outcome as they influence the evolutionof temperature distributions in space and time achievedin the tissue. The thermal diffusivity is an intrinsic tissuethermal property that combines the thermal conductivityand the heat capacity to express the rate at which heatdissipates and, thus, directly influences the time depen-dence of the temperature distribution in tissue. Perfusion

� Present address: Philips Research North America, BriarcliffManor, NY, USA.

Address correspondence to: Ajay Anand, PhD, Philips Research
North America, 345 Scarborough Road, Briarcliff Manor, NY 10510USA. E-mail: [email protected]
1449

acts as a local sink for excess heat energy by convectivetransport and depends on the degree of vascularity at theablation site and proximity to larger blood vessels. Inreality, these tissue specific thermal parameters arehighly variable between tissue types and also acrossindividuals. Hence, noninvasive methods of measuringthese thermal parameters at the treatment location canreduce the uncertainty in therapeutic dose planning anddelivery. Development of such technologies could alsoprovide a means of tissue characterization to monitorphysiological response to stimulus, or to detect abnormaltissue pathologies, for example (Lagendijk et al. 1988;Cheng and Plewes 2002).

Recently, the use of numerical simulation tools topredict the temperature distribution and thermal dose fortherapy dosimetry planning applications has been widelyreported in the literature (Meaney et al. 1998; Kolios etal. 1999; Curra 2001). The Pennes bio-heat transferequation (BHTE) (Pennes 1948) is used to compute thetemperature distribution and the thermal dose is com-
puted using the formalism proposed by Sapareto and
mailto:[email protected]

mailto:[email protected]

1450 Ultrasound in Medicine and Biology Volume 34, Number 9, 2008

Dewey (1984). These simulation tools typically use apriori knowledge or assume standard values for tissuethermal parameters, namely, thermal diffusivity and per-fusion loss. Uncertainties in the knowledge of theseparameters can result in significant errors in the temper-ature distribution computed using the BHTE and conse-quently the thermal dose, which is a clinically importantindicator of the endpoint of thermal therapy. Providing insitu estimates of the thermal parameters in the treatmentregion as inputs to the simulation tools would reduceerrors in the predicted temperature distributions duringplanning phases and improve the ability of these tools toeffectively adapt to varying local conditions for planningor during treatment.

Imaging techniques based on magnetic resonanceimaging (MRI) (Cheng and Plewes 2002) and infrared(IR) imaging (Telenkov et al. 2001) to noninvasivelyestimate tissue thermal diffusivity and/or perfusion havebeen previously reported. However, no ultrasound basednoninvasive method to estimate these parameters hasbeen developed to the best of the knowledge of theauthors. An ultrasound-based estimation technique is amore attractive option for use with therapy deliverysystems because MRI requires specialized therapy equip-ment compatible with high magnetic fields and has lowertemporal resolution (frame rate) compared with ultra-sound imaging. IR imaging methods are limited toestimation of surface thermal measurements and maynot be suitable for deeper ablation sites due to poorpenetration.

Ultrasonic methods of estimating tissue thermal andacoustic properties have been recently proposed. Yaoand Ebbini (Hui et al. 2004) demonstrated the feasibilityof reliably estimating the initial heating rate at a local-ized heating spot by inducing a temperature change onthe order of 1°C and proposed that the ultrasonicallyestimated initial heating rate can be used to compute thelocal tissue absorption. They also demonstrated that theultrasonically determined initial temperature decay rateafter turning off the heating demonstrates excellentagreement with the decay rate obtained from invasivethermocouple readings and conclude that the local per-fusion can be estimated from these decay rate measure-ments. However, no quantitative estimate of the localtissue properties, absorption and perfusion, is obtained inthat study. Recently, Sumi and Yanagimura (2005, 2007)proposed a technique for reconstructing the thermalproperties such as thermal conductivity and diffusivityprovided that quantitative spatial and temporal tempera-ture information of the region-of-interest and referencevalues of the thermal properties are independently mea-sured a priori. The temperature rise in the sample wasinduced using a heated waterbath. Simulations and ex-
perimental results obtained in a tissue mimicking phan-
tom were presented. Practical application of the methodsin noninvasive therapy modalities such as HIFU was notpresented in that study.

In this article, a noninvasive ultrasound-based tech-nique for estimating the local thermal diffusivity in situusing focused ultrasound or HIFU heating and applica-tion of novel backscattered radio-frequency (RF) signalprocessing techniques is presented. The technique isdesigned such that the thermal diffusivity estimation canbe performed in the treatment region as part of a therapyplanning calibration procedure conducted prior to thetherapy delivery session. The theoretical framework un-derlying the estimation procedure is developed for the invivo situation using the Green’s function solution for theBHTE including the effect of bulk perfusion as a heatsink. It is shown from the theoretical analysis that thethermal diffusivity can be obtained by tracking the rateof change of the size of the heated region without knowl-edge of the absolute temperature. The estimated thermaldiffusivity can then used as a known quantity to estimatethe unknown perfusion term if information about the heatsource is available. The effects of heat transport in thepresence of a large blood vessel are not considered in thistheoretical formulation. Since the current experimentalwork is limited to in vitro scenarios, the estimation ofperfusion is not included in the parameter estimationstep. Discussions on extending the parameter estimationstep to the in vivo case for measuring perfusion areprovided. The parameter estimation technique is basedon visualizing the spatiotemporal variation of tempera-ture induced strain (Miller et al. 2002, 2004) estimatedfrom the raw ultrasound backscatter data after a shortfocused ultrasound heating pulse is applied at subabla-tive intensities. The temperature induced strain is causedby changes in the local sound speed of the medium andthermal expansion (Maass-Moreno et al. 1996; Simon etal. 1998; Varghese et al. 2002). The maximum inducedtemperature rise is less than 10°C to avoid any perma-nent changes in tissue. A mathematical analysis of thesolution to the heat transfer equation illustrates that thespatiotemporal rate of change of the temperature inducedstrain profile transverse to the HIFU beam axis of prop-agation is directly related to the thermal diffusivity. Thetechnique is applied to ultrasound backscatter data col-lected during in vitro experiments performed in a tissuemimicking phantom and excised turkey breast muscletissue.

THEORY

The technique adopted in this work for the estima-tion of the thermal diffusivity is based on determiningunknown thermal parameters from the BHTE. For the in
vitro case, in the absence of perfusion, the BHTE reduces

�

Noninvasive method of estimating local thermal diffusivity ● A. ANAND and P. J. KACZKOWSKI 1451

to the heat transfer equation (HTE). In this analysis, it isassumed that the medium is locally homogenous andisotropic in the local region where the thermal parame-ters are estimated.

Bio heat transfer equationThe differential equation describing the transient

BHTE can be written in axisymmetric cylindrical coor-dinates as,

�T

�t� K�2T � bT � Q(t) · I(r, z) (1)

where T(r,z,t) is the temperature change in °C, K is thethermal diffusivity (m2/s), Q(t) (°C/s) represents the localin situ heating rate due to ultrasound energy absorption,I(r,z) is the normalized spatial acoustic intensity distri-bution profile (unitless), r represents the axis perpendic-ular to beam propagation (transverse) and z representsthe beam propagation axis (longitudinal). b is given bywb�bCb/�C, where wb is the blood perfusion rate (ml/s/ml), �b and � represent the density of blood and tissuerespectively in kg/m3, Cb and C represent the heat ca-pacity of blood and tissue respectively in J/kg/°C. It maybe noted that Q is a scalar number and I(r,z) is a two-dimensional (2D) matrix of values between 0 and 1 withthe rows representing r and the columns representing z.

The solution of transient BHTE can be expressed interms of the Green’s function G and the source heatingterm S(r,z,t) in cylindrical coordinates as (Carslaw andJaeger 1959; Arfken and Weber 2005),

T(r, z, t)

�0

�

��

�

��

�

S(r � , z � , t � )G(r, z, t;r � , z � , t � ) � r � dr � dz � dt�

(2)

The Green’s function of the transient BHTE is given by(Vyas and Rustgi 1992),

G �r � ut�(t)

�2�C�2D�t � t � ��3 2exp��b�t � t�� exp��

�z � z � �2

4D�t � t � ��exp��

�r2 � r�2�4D�t � t��I0� rr�

2D(t � t � )�(3)

where ut��t� is the unit step function,

ut��t��1 for t�t�

� 0 for t�t�(4)

I0(x) is the modified Bessel function of order zero.The focused ultrasound source is modeled with a

Gaussian spatial beam profile along the axial (z) and

radial (r) directions as (Parker 1983,1985; Cheng andRoemer 2005),

S(r, z, t) � Qacoustic(t) � exp(�r2

r) � exp(

�z2

z) (5)

where r and z represent the ultrasound beam widthalong r and z, respectively and Qacoustic�t� is the acousticpower deposited per unit volume in W/m3.

Substituting eqns 3 and 5 in eqn 2 and rearrangingthe terms we get,

T(r, z, t) �1

�2�C�0

t1

�2K�t � t��1 2Qacoustic(t � )exp��b�t � t��dt�

��

�

exp��z�2

z�

(z � z � )2

4K(t � t � )�dz�

��0

�

r�2

�2K�t � t��exp��r�2

r�

(r2 � r�2)

4K(t � t � )�I(rr�

2K(t � t � ))dr�

(6)

Solving the integrals in eqn 6 with respect to r’ and z’and rearranging the terms,

T(r, z, t) � r� z

�C �0

t

dt � Qacoustic(t � )exp��b�t � t��

�

exp�� z2

4K(t � t � ) � z�

�4K(t � t � ) � z�1⁄2

�

exp�� r2

4K(t � t � ) � r�

�4K(t � t � ) � r�(7)

The above equation requires the time integral to besolved numerically. However, by limiting the focusedultrasound heat source to a short burst of energy ofduration t0 applied at t � 0 (Newman and Lele 1985;Parker 1985), the impulse approximation can be used torepresent the heat source Qacoustic(t) as,

Qacoustic(t) � 2�I0t0�(t) (8)

where, I0 is the peak acoustic intensity, � is the acousticabsorption coefficient.

Substituting eqn 8 in 7, we get the temperature
profiles during the decay phase as,


T(r, z, t) �2�I0t0 r� z

�Cexp��bt�

exp�� z2

4Kt � z�

�4Kt � z�1⁄2�4Kt � r�m

Ap(t)

exp�� r2

4Kt � r�

(9)

If the temperature profiles are observed along the radialdirection r for a fixed value of z, the above equation canbe expressed as,

T(r, z, t) � Ap(t)exp�� r2

R(t)� (10)

Equation (10) is the product of Ap(t) and a Gaussiancurve with width related parameter R�t� � 4Kt � r attime t. The equation demonstrates that it is possible toestimate the thermal diffusivity and the perfusion sepa-rately in a sequential manner provided (i) A short dura-tion heating pulse (eqn 8) relative to the times for ther-mal dissipation is applied to induce the temperature rise,and (ii) Flow is not in large vessels which would result ina directional heat loss but rather through perfusion mod-eled as a nondirectional heat sink term in the BHTE.

Focused ultrasound beams, such as those employedthis article, are particularly well suited to induce temper-ature rises rapidly and produce a sharp thermal gradientperpendicular to the direction of propagation of theacoustic beam. The focal field of a uniformly excitedspherical transducer is not Gaussian but the focal shapecan be well approximated by an equivalent Gaussian thatis fit by matching average power over the peak. Inaddition, since the heating effect produced by focusedbeams is local and the proposed thermal diffusivity es-timation technique is also designed to produce a localestimate around that heated location, the use of a nondi-rectional perfusion term as implemented in the BHTEshould be valid. If large blood vessels are present withinthe parameter estimation region-of-interest that result ina directional heat sink, the parameter estimation methoddeveloped based on eqn 1 and the approximate formsthat follow would not be valid.

From eqn 10, K is estimated by tracking the rate ofchange of R(t) i.e.,

d�R(t)]

dt�

d�4Kt � r]

dt�4K (11)

The estimated thermal diffusivity can then be substitutedin Ap(t) to predict the local perfusion b assuming theterms r, z are known from the characteristics of theHIFU beam profile. As part of the estimation procedurefor b, the heat source term 2�I0t0/�C would also have tobe determined. This can be performed using an iterativeapproach such as that proposed by Cheng and Plewes
(2002) where the heat source and perfusion terms are
alternately computed until the parameter values con-verge.

The temperature variation at the center of the focalregion (r � 0,z � 0) is,

T�r�0,z�0,t� �2�I0t0 r� z

�C�4Kt � z�1⁄2�4Kt � r�exp��bt�

(12)

The temperature at the center of the focus will decay asa function of t and is affected by the perfusion constant.

The scope of the experimental results presented inthis article was limited to in vitro conditions for whichthere is no perfusion. Substituting b � 0 in eqn 9, theBHTE reduces to the HTE and we get,

T(r, z, t) �2�I0t0 r� z

�C

exp�� z2

4Kt � z�

�4Kt � z�1⁄2�4Kt � r�m

A(t)

exp�� r2

4Kt � r�

(13)

or equivalent to eqn 10,

T(r, z, t) � A(t)exp�� r2

R(t)� (14)

where A(t) is equal to Ap(t) with the perfusion constantb set to zero, and R�t� � 4Kt � r as before. Thus, theapproach to estimating K relies on measuring the rate ofchange in transverse width of the temperature profilenear the HIFU focus.

Estimation of thermal diffusivity using backscattered ul-trasound

The estimation of thermal diffusivity using back-scattered ultrasound is based on tracking the spatiotem-poral evolution of the temperature induced strain pro-files. In this approach, a short heating pulse is appliedand the resulting temperature induced strain is measuredultrasonically over a period of time sufficient to detectthe spatial diffusion of the heat. This section describeshow quantitative estimates of the thermal diffusivity canbe obtained from time series of the strain profiles.

The experimental arrangement, including orienta-tion of the HIFU transducer (used to induce the localtemperature rise) and the position of the ultrasound im-aging probe and the sample volume being interrogated, isillustrated in Fig. 1. For a short HIFU-induced heatingpulse applied at time t � 0, the resulting temperaturedistribution along the transverse direction r after theHIFU pulse has been turned off is modeled by eqn 14.Previously, it has been demonstrated (Simon et al. 1998;
Varghese et al. 2002; Pernot et al. 2004) that the tem-


perature-induced strain (�) measured from the ultrasoundradio-frequency (RF) backscatter signal, for small tem-perature rises above ambient temperature, is directlyproportional to the induced temperature change T(r,t)

T(r, t) � k�(r, t). (15)

where k is a scalar constant.The maximum temperature rise that can be induced

depends on the range of temperatures for which k isconstant to minimize the estimation error resulting fromthe temperature dependence of this parameter. For ex-ample, based on experiments in turkey muscle tissue, itwas reported that the strain versus temperature relation islinear for temperature rises up to 10°C (Maass-Moreno etal. 1996; Simon et al. 1998). Miller et. al. (Miller et al.2002) have reported on the temperature dependence of kand its variability with tissue composition. For example,in normal liver (low fat content) and fatty liver, k remainsnearly constant (from 37°C to around 50 °C), whereas forliver with intermediate fat content k is small and mayeven change sign. For such conditions, the method hasfundamental limitations and may not work unless thetissue is heated or cooled to a substantially differentoperating point.

From eqns 14 and 15, we get,

�(r, t) � �max(t)e�r2⁄R(t) (16)

where �max � k A(t) (no perfusion) or k Ap(t) (withperfusion).

In this technique, as long as the parameter k isindependent of temperature over the range of interest, itis not necessary to know the value of the constant k. Thedata processing result indicates the degree to which k isconstant with temperature. Integrating eqn 16 along r incylindrical coordinates, the cumulative shift in the ultra-sound RF echo locations s�r,t� along the imaging beam

Fig. 1. Schematic illustration of the orientation of HIFU ther-apy transducer, ultrasound imaging transducer and location of

the geometric focal spot of the therapy transducer.

has the form of the error function erf(x),

s(r, t) � �u�0

u�r

�(u, t)du�smax(t) �u�0

u�r

e�u2⁄R(t)du, (17)

where smax(t) is the peak displacement. From eqn 17, itcan be seen that the parameter R(t) can be obtained fromthe s(r,t) profile for measurement time t. It is important tonote from eqn 17 that since the integration is performedwith respect to r, the perfusion term contained in smax(t)is separable and allows the thermal diffusivity to beindependently estimated.

The above equation does not compensate for thethermal lens effect (decorrelation in the RF signal due torefraction and refocusing of the US beam during heating)(Simon et al. 1998). The thermal “lens” can lead toaberration of the acoustic imaging beam when lateraltemperature gradients are large compared with the imag-ing beam width. As a consequence of this effect, theshape of s(r,t) in the above equation would deviate fromthe ideal error function erf(x), especially distal to theheated zone. Additional sound propagation modeling,signal processing and filtering would be required to min-imize the errors introduced by this effect in the parameterestimation process. Simon and Ebbini (Simon et al. 1998,Fig. 12) have experimentally demonstrated that the effectis dominant at the ends of the HIFU beam and is notpresent in the central region. Based on this observation,care was taken in the experimental design in this work toensure that only scanlines towards the center of the focalregion were used for analysis to estimate the thermaldiffusivity. Placing the HIFU beam in the imaging planereduces the lateral gradient in the plane at the ends of theHIFU focal zone. Additionally, centering the imagingplane on the HIFU focus minimizes the role of lateral(out of plane) distortion by imposing symmetry.

As indicated in eqn 11, differentiating R(t) withrespect to t gives,

d�R(t)]

dt� 4K. (18)

Equation (18) shows that by computing the rate ofchange of R(t) versus t, the thermal diffusivity K can beestimated. The measurement procedure is noninvasiveand can use several ultrasound scanlines over a sequenceof frames to improve the signal to noise ratio. It may benoted from eqn 17 that errors introduced by the signalprocessing process in the estimation of s(r,t) will affectthe estimated R(t) and consequently the estimated valueof thermal diffusivity K. These signal processing relatederrors are in addition to the underlying model shortcom-ings such as neglecting thermal lensing, assuming spatialhomogeneity in K and neglecting temperature depen-
dence of K (and k). In many cases, we expect these


assumptions to hold and we present two experimentalexamples for which the method is successful.

MATERIALS AND METHODS

SimulationsTo validate the ultrasound RF-based estimation

technique for the noninvasive estimation of thermal dif-fusivity, simulations were performed by generating back-scattered RF scanlines during a simulated HIFU expo-sure and applying the estimation algorithm. The proce-dure outlined by Miller and Bamber (Miller et al. 2002)was followed to simulate the effect of temperature onbackscattered RF data. A random scatterer distribution iscreated with Gaussian distributed amplitudes while thespacing between the scatterers was derived from a uni-form distribution. Twenty scatterers per wavelength aregenerated along each scanline to guarantee fully devel-oped speckle. RF waveforms are synthesized by convo-lution of a system impulse response with the point scat-terer distribution. In this study, the evolution of thetemperature field over a 3 cm � 2 cm 2D region aroundthe HIFU focus was computed using an axisymmetricfinite element solution of the heat transfer equation im-plemented in FEMLAB™ (now COMSOL Multiphysics,by COMSOL AB, Stockholm, Sweden). The heat sourcefield was defined using the linear acoustic intensity fieldfor a single element transducer with a geometric focallength of 35 mm and active diameter of 16 mm, operat-ing at 5 MHz; the field of the transducer used in theexperiments was mapped using a hydrophone (GoldenLipstick, 200 �m aperture, SEA, Soquel, CA, USA) toconfirm that the simulated beam pattern matched themeasured transducer field. The HIFU ON-time was 5 sand the cool down phase (OFF-time) was observed for15 s. The simulation parameters are listed in Table 1. TheHIFU exposure intensity was chosen such that the max-

Table 1. Ultrasound and thermal parameter values used insimulation to generate RF data for noninvasive estimation of

thermal diffusivity

Center frequency (MHz) 8 MHz, 60% bandwidthScatterer density 20 per wavelengthSound speed (m/s) 1540Ultrasonic SNR (dB) 20Sampling rate (MHz) 32HIFU ON time (s) 5HIFU frequency (MHz) 5Initial temperature (°C) 25Thermal conductivity (W/m/°C) 0.7Heat capacity (J/kg/°C) 4180Density (kg/m3) 1000

SNR � signal-to-noise ratio; HIFU � high intensity focused ultra-sound; RF � radio-frequency.

imum temperature reached at the focus was no greater

than 10°C above ambient temperature (37°C). The tem-perature maps were converted to corresponding soundspeed using the mapping reported by Bloch et al. (Blochet al. 1998) for liver tissue, which is a primary tissue ofinterest for HIFU treatment applications (in simulations,the acoustic strain coefficient k must be specified but itsvalue is not used in the estimation of diffusivity K). Foreach scanline, the sound speed depth profile was mappedto apparent displacements of the scatterers and new RFbackscattered waveforms were computed for these newscatterer locations. The center frequency of the transmitpulse used in the ultrasound imaging system was 8 MHzwith a fractional bandwidth of 60%. The signal-to-noiseratio (SNR) of the simulated RF signals was set to 20dB by the addition of white noise. Imaging RF wave-forms were generated at a frame rate of 1 frame persecond for the warming and cooling phases. Fiftyrealizations of the simulations using different scattererarrangements were performed to compute first andsecond order statistics. For the estimation of K, signalanalysis was performed along scanlines passingthrough the HIFU focal zone at z � 0 (Fig. 1). The RFwaveform shifts s(r,t) and function R(t) were com-puted by processing each frame acquired during thecool down phase relative to a reference frame acquiredbefore heating commenced.

RF data analysisEquation (17) illustrates that the backscatter wave-

form shifts s(r,t) can be represented by a closed func-tional form and parameterized in terms of smax (maxi-mum amplitude) and R (width parameter). In this article,a novel parametric estimation approach is employed toestimate smax and R in eqn 17 from the RF backscatterdata collected during the HIFU experiment. A schematicrepresentation of the estimation technique is presented inFig. 2. Techniques for estimation of ultrasound echolocation shifts s(r,t) between RF data frames acquiredfrom a spatial region-of-interest have been previouslyused for ultrasound-based temperature estimation (Si-mon et al. 1998; Miller et al. 2002; Miller et al. 2004).However, such methods first estimate strain and thenapply those estimates to temperature estimation. They donot incorporate knowledge of the underlying physicalheat diffusion mechanism in the estimation algorithmand result in noisy and often biased estimates for strainsthat occur over distances that comprise few imagingwavelengths. The approach presented here estimatesthermal parameters smax and R by iterative analysis of theRF data and makes use of thermal diffusion modeling toguide the solution. The processing is cast as a minimi-zation problem and relies on comparing a reference RFwaveform (acquired before heating commenced, at time
t � t0) to a post-heating waveform acquired during cool

e to es


down (t�tcool). As indicated in eqn 17, the heat transfermodel is used to predict the displacement (shift) for aparticular choice of parameters (smax, R) and the pre-dicted shift is applied to the heated waveform. Thedifference between the shift-corrected and referencewaveforms is minimized by searching over the values of(smax, R). This procedure provides an estimate of strainby relating the measured RF shifts to the underlying heattransfer model for the HIFU heating experiment and isaccurate under the assumptions listed above. Further-more, final degree of correlation, goodness of fit andapparent degree of linearity in the various results ofprocessing provide clear assessment of the validity ofseveral key assumptions.

The steps in the algorithm to find the optimal R(t-

cool) for a single scanline passing through the heatedregion are outlined below:

1. s(r,t) is mathematically modeled using eqn 17 withparameters (smax, R) set to reasonable initial guessvalues for a particular time t � tcool.

2. The RF line acquired at t � tcool is shifted byfunction s(r, tcool). Each data sample of the RF lineis shifted by the corresponding value of s(r, tcool).The shifted data points are then re-sampled onto thesame uniform grid as the reference signal acquiredat t � t0. No additional global shift, scaling or anylocalized corrections other than that specified bys(r,t) is applied to the waveform. The shifting op-

Fig. 2. Flowchart of techniqu

eration is similar in principle to temporal stretching

(Alam and Ophir 1997; Varghese and Ophir 1997)and companding (Chaturvedi et al. 1998) earlierproposed in elasticity imaging. However, it must benoted that the phenomenon of thermal lens effect isnot an issue in elasticity imaging as in estimatingthe temperature induced strain from the ultrasoundbackscatter.

3. The shifted RF line acquired at t � tcool (step 2) andthe reference RF line are both segmented into Nnonoverlapping segments and the normalized zero-lagcorrelation coefficient for each segment i is computedusing eqn 19. A fixed value of N is chosen. For a 1 cmscanline segment, a typical value of N is 15. In eqn19, m is total number of RF samples (typically m �40) in the segment.

�i �

�j�1

j�m

RFt�t0(j, i) * RFt�tcool(j, i)

�j�1

j�m

[RFt�t0(j, i)]2 * �j�1

j�m

[RFt�tcool(j, i)]2

(19)

4. The total squared error err � �i�1

i�N �1 � �i�2

is then computed.5. The parameters (smax, R) are updated iteratively

until the value of err is less than 0.002.

timate smax and R in eqn 17.

The values of smax and R obtained after minimiza-

for u


tion of err are called the best-fit parameters. Plots of thezero-lag cross-correlation before and after convergenceare shown in Fig. 2 (bottom panel). This procedure isrepeated for all the RF frames collected during the cooldown phase (after HIFU has been turned off) to obtain aset of values of R(t) corresponding to each frame. Theoptimization algorithm was implemented using theMATLAB (The Mathworks Inc., Natick, MA, USA)function fminsearch, which is based on the Nelder-MeadSimplex search technique (Nelder and Mead 1965).

Phantom experimentsExperiments were performed in a tissue-mimick-

ing phantom made of alginate based hydrogel contain-ing 95% water by weight. The material is simple toprepare, nontoxic, and provides uniform scatterer dis-tributions ideal for speckle tracking studies. The phan-tom is prepared by dissolving 16 gm of Jeltrate™dental impression material (Dentsply Caulk Inc., Mil-ford, DE, USA) in 300 ml water while stirring con-tinuously to ensure a homogenous mix. The solution isdegassed under vacuum and allowed to settle for 20min in specially designed sample holders of size 5 �5 � 6.5 cm3 in which the solution solidifies. Thesound speed and the acoustic attenuation of the phan-tom at 25°C are measured using the sample replace-ment technique (Bloch et al. 1998; Madsen et al. 1999)with a pair of 7 mm diameter PVDF transducers(Sonic Concepts, Woodinville, WA, USA); measuredbulk values for the sample used in this example werec � 1483 m/s and � � 0.35 dB/cm/MHz, respectively.The schematic diagram of the experimental set-up

Fig. 3. Schematic of experimental set-up

(calipers not shown) is presented in Fig. 3. A 5 MHz

HIFU therapy transducer (SU-104, Sonic Concepts,Woodinville, WA, USA) with an aperture diameter of33 mm and a focal depth of 35 mm was used to deliverthe HIFU heating pulse. The HIFU transducer wasrigidly attached to a three-dimensional translationstage that allows controlled motion in increments of 1mm. The driving electronics for the HIFU transducerconsisted of a signal generator (HP 33120, HewlettPackard, Palo Alto, CA, USA) driving a power am-plifier (A300, ENI, Rochester, NY, USA). The imag-ing probe used was the ATL CL10-5 (Philips MedicalSystems, Bothell, WA, USA) with a bandwidth of 5 to10 MHz. Vertical alignment pins affixed to the base ofthe sample holder were used to co-register the HIFUbeam propagation plane with the imaging plane. Spe-cifically, as part of the alignment procedure, the HIFUtransducer is operated in pulse-echo mode and trans-lated using the motion stage until the received echoamplitude from the proximal alignment pin is maxi-mized thereby ensuring that the pins are in the HIFUfocal plane; the HIFU transducer is then raised toplace the focal region above the pins. The imagingtransducer is mounted on the sample holder such thatalignment pins are visible on B-mode images asstraight lines, thus, assuring that the HIFU focal zonelies within the imaging plane.

RF data frames time-synchronized with the HIFUdelivery were acquired using an ATL HDI-1000 ultra-sound scanner (Philips Medical Systems, Bothell,WA, USA) during the HIFU heating phase lasting 5 sand the cool down phase lasting 15 s. The RF dataframes were acquired at a frame rate of 1 frame per

ltrasound thermal diffusivity estimation.

second. A personal computer running a software pro-


gram developed in LabVIEW™ (National Instru-ments, Austin, TX, USA) controlled the operation ofthe entire system (Anand et al. 2003; Anand andKaczkowski 2004). The RF data were transferred tothe PC at the end of the experiment for offline dataanalysis. HIFU acoustic intensities were chosen suchthat the maximum temperature rise induced was ap-proximately 10°C. The RF data sets were analyzedusing the algorithm described in the previous sectionto compute R(t) and a least square fit was performedusing eqn 7 to estimate the thermal diffusivity K. Theexperiment and estimation procedure was repeated onfive phantom samples prepared using the same recipe.

The thermal diffusivity (K) was also independentlymeasured for the phantom samples using the transienthot-wire technique (Carslaw and Jaeger 1959; Prokop etal. 2003), a standard method reported in the heat transferliterature for measuring the thermal properties of solids.The estimates obtained by this method were comparedwith those obtained using the noninvasive ultrasound-based estimation procedure. A schematic representationof the imaging plane view of the transient hot-wireexperiment set-up is shown in Fig. 4. A nickel-chromium(Ni-Cr) heating wire (Omega Engineering Inc., Stam-ford, CT, USA) was pulled taut through the center of thephantom sample placed in a custom designed aluminumholder. Three 0.2-mm diameter T-type thermocouples

Fig. 4. Schematic of experimental set-up for transient hot-wiretechnique

with copper-constantan junctions (HYP0-33-1-T-G-60-

SMPW-M, Omega Engineering Inc., Stamford, CT,USA) were inserted into the phantom under ultrasoundB-mode guidance to ensure that the thermocouple stayedparallel to the heating wire along its length. The thermo-couples were placed at different radial distances from thewire, ranging from 2 to 5 mm. The distances from the tipof the thermocouples to the heating wire were measuredusing the distance measurement feature with L11-5transducer (Philips Medical Systems, Bothell, WA,USA) on the HDI-1000 clinical ultrasound scanner witha precision of �0.1 mm. A programmable DC powersupply (PSP-2010, Goodwill Instek Ltd, Chino, CA,USA) was connected to the heating element through aseries relay switch used to turn the heating circuit on andoff remotely. The thermocouple terminals were con-nected to a data acquisition module (HP 34970, HewlettPackard Inc., Palo Alto, CA, USA) to store the temper-ature readings in digitized form. A software-based con-trol program developed in LabVIEW™ (National Instru-ments, Austin, TX, USA) controlled the delivery ofelectrical power to the heating wire, and the thermocou-ple data acquisition. The heating power was obtainedusing the voltage drop across and current through thewire, monitored with a pair of multimeters throughoutthe experiment; the measured values were 2 � 0.01 Vand 7 � 0.01 A, respectively. The total heating time was90 s and thermocouple data was acquired at 1 Hzthroughout this interval. The maximum temperature risewas no greater than 10 to 15°C to prevent irreversiblechanges to the sample under test and to prevent forma-tion of gas bubbles at the wire-gel interface. The exper-iments were performed in four samples prepared usingthe same recipe with three data sets acquired in eachsample.

The spatiotemporal temperature distribution radi-ally outward from the heating wire during heating with aconstant heat source is approximately described by thefollowing equation in axisymmetric cylindrical coordi-nates (Carslaw and Jaeger 1959),

T(r, t) � � � q

4�� r2

4Kt

�

e�u

udu (20)

where the first term, A�q

4�, is a constant and does not

vary with time, and the second term is the exponentialintegral Ei� � r2⁄4Kt�.q represents the power (W) deliv-ered per unit length of the wire, � represents the thermalconductivity (W/m/°C), K is the thermal diffusivity (m2/s), r represents the radial distance away from the wire(m), and t represents the time (s) after heating com-menced. It is important to note that in the transient
hot-wire method the thermal diffusivity is measured dur-


ing the application of a constant heat source (using anembedded hot wire) and using eqn 20, while in theultrasound based estimation approach the diffusivity ismeasured during cooling after the application of a shortfocused ultrasound heating pulse using eqn 17. The ther-mal diffusivity estimates obtained using the two inde-pendent methods are then compared in this article. Whileeqns 20 and 17 may appear similar in form, they arederived for very different physical conditions. Equation(20) assumes that the heat source is infinitely long withnegligible diameter surrounded by an infinite solid, andthese assumptions are reasonable for this experimentalarrangement. The exponential integral in eqn 20 is eval-uated using the function “expint” included in MAT-LAB™ (The Mathworks Inc, Natick, MA, USA). Un-known parameters A and thermal diffusivity K are ob-tained by finding the best fit to eqn 20 for thethermocouple data. The hot-wire method provides a ref-erence estimate of the thermal diffusivity for the phan-tom material that is independent of the values obtainedusing the new backscattered ultrasound technique de-scribed in this work, though experimental uncertaintyprimarily due to thermocouple placement error is actu-ally larger than that for the ultrasonic approach.

In vitro turkey breast muscle experimentsIn vitro experiments on turkey breast muscle were

performed using the same physical set-up describedabove for the phantom study. Prior to the experiments,store bought tissue was cut into convenient sizes so thatpieces could be placed in the sample holders of dimen-sions 5 � 5 � 6.5 cm. The cut samples were thenimmersed in de-ionized water and degassed under vac-uum. The degassing process was continued until novisible outgassing from the sample was evident. Thisprocess typically lasted 30-40 min. After degassing, thetissue sample was placed near the center of the holderand suspended vertically with the aid of a needle andsewing thread as shown in Fig. 5. Chunks of tissue wherethe muscle fiber orientation could be visually recognizedto be straight were carefully selected for use in theexperiment. This orientation is important because of theanisotropy of muscle fibers. In muscle, the attenuationalong the fibers exceeds that across the fibers by a factorof 2 or 3 (Duck 1990). The sample was then carefullysuspended in the holder ensuring that the direction ofpropagation of the ultrasound therapy and imaging beamwere both perpendicular to the fiber orientation. With thesample held in this position, alginate gel was poured intothe holder to encase the tissue sample. This arrangementensured that the tissue sample was positioned in thecenter of the holder allowing easy insertion of thermo-couples into the tissue. It also ensured that the ultrasound
therapy beam enters through a relatively flat front sur-
face. Not doing so may lead to significant refraction andmisalignment of the imaging and therapy beams. Thepropagation distance through the gel was approximately1 cm. The attenuation and sound speed was measuredusing the sample replacement technique for each of thesamples in the experiment. The average sound speed andattenuation measured over four samples was 1567.8 �12.4 m/s and 1.24 � 0.18 dB/cm/MHz, respectively.These attenuation measurements were performed withthe muscle fibers oriented perpendicular to the beampropagation direction. The attenuation values measuredwith the fibers oriented parallel to the beam propagationdirection were 2.1 � 0.16 dB/cm/MHz. The reportedvalues for attenuation in muscle perpendicular and par-allel to the fibers are in the range of 0.9-1.4 dB/cm/MHzand 1.6-2.8 dB/cm/MHz, respectively (Duck 1990). Theattenuation of the alginate gel phantom was also inde-pendently measured to be 0.34 � 0.032 dB/cm/MHz. Itmay be noted that the attenuation and sound speed valuesare not used in the thermal diffusivity computation, butdo affect the heating rate and imaging signal-to-noiseratio.

A brief HIFU exposure at an in situ intensity ofabout 30 W/cm2 was applied to estimate the thermaldiffusivity (K). The exposures were typically 5 s induration and the diffusivity was estimated during thecool down period. The actual temperature in the tissue isnot needed to measure the spatial diffusion rate of theheat pulse.

RESULTS

Phantom experimentFig. 6 presents model-derived curves for cumulative

acoustic strain s(r,t) (Fig. 6a) and local acoustic strain�(r,t) (Fig. 6b) at various times after the HIFU heatingpulse has been turned off for a RF data set acquired in the

Fig. 5. (a) Tissue sample suspended by a sewing thread from aclamp and held in place near the center of the metallic sample

holder. (b) Alginate gel is then poured to encase the tissue.

alginate phantom. The local strain �(r,t) is Gaussian as in


eqn 16 and s(r,t) has the form of a error function erf(x)as described in eqn 17. It can be clearly seen that thewidth scale parameter R(t) increases as coolingprogresses and the maximum amplitude decreases due tothermal diffusion and conservation of energy. Fig. 6cincludes plots of the zero-lag cross-correlation coeffi-cient �i at initial iteration 0 and after final convergence ofthe iterative K-estimation algorithm. At iteration 0 of thealgorithm, proximal to the HIFU focal zone, �i is close tounity because no shift in the RF signal due to thermalchange appears before the heated zone. Correlation re-duces sharply approaching the HIFU focal region andstays low beyond the heated zone. This is consistent withthe temperature induced echo-shift model represented byeqn 17. After final convergence of the iterative strainestimation procedure, �i is close to unity over all depths,illustrating that the model properly accounted for theshifts in the post-heated RF scanline caused by temper-ature rise. As noted before, s(r,t) does not include theinfluence of the thermal lens effect, yet the methodproduces highly correlated results even for regions distalto the HIFU focus. This case illustrates how to evaluatethe degree to which the thermal lens effect may interferewith the method.

The estimated R(t) values are plotted as a functionof time t in Fig. 7a and a least squares linear fit to the datapoints based on eqn 18 is shown. Note the good agree-ment (r � 0.96) of the data with the straight line model,indicating a constant value of K for the experiment. Thelinearity of the model also indicates that the strain coef-

Fig. 6. (a) s(r,t) and (b) �(r,t) from t � 1 s (after start of cooldown) to t � 15 s at intervals of 2 s for RF scanline passingthrough center of HIFU focal region (c) Zero-lag cross-corre-lation �i versus depth at iteration 0 (o) and at final convergence(�) of the model-based strain estimation algorithm at time t�

1 s after start of cooling.

ficient k is constant over this range of temperatures,

within experimental accuracy. Thermal diffusivity calcu-lated using eqn 18 is 1.72 � 10�7 m2/s for this phantomsample. The mean estimate of K obtained noninvasivelyfor the five phantom samples analyzed is 1.7 � 0.07 �10�7 m2/s. Fig. 7b is a plot of the mean value of R(t) forthe five different samples as a function of time. The errorbars represent the standard deviation in the R(t) estimateat each time instant. The good agreement between thedata points and the linear fit demonstrates the robustnessof the technique for the experimental conditions in thiswork.

Fig. 8a illustrates the typical temperature profilesobtained during the transient hotwire experiments andthe superimposed analytical fit to the temperature datausing eqn 20. Fig. 8b shows the residual error in the fit.The mean value of K estimated using this technique infour samples is (1.44 � 0.2) � 10�7 m2/s.

Fig. 7. (a) Plot of R(t) versus time t estimated from a RF dataset acquired in the alginate phantom. The straight line repre-sents a least squares linear fit to the data points. (b) Plot of themean value of R(t) versus t from data collected in five inde-pendent phantom samples. The error bars represent the standard

deviation in R(t).


SimulationsFig. 9 illustrates the histogram of the ratio of esti-

mated thermal diffusivity (Kest) and the true diffusivity(Ktrue) for 50 realizations in the simulation. The esti-mated mean value was 0.994 and the standard deviationwas 0.089. This close match between the noninvasiveestimate and true value illustrates the precision of theestimator used in the ultrasound based noninvasive tech-nique for estimating K and suggests very little bias in themethod.

Fig. 8. (a) Temperature profiles measured by thermocouplesplaced at radial distance of 2, 3 and 5 mm from the nichromeheating wire in the transient hotwire experiment (o) and ana-lytical fit to the data. (b) Error residuals (in degrees Celsius)between the experimental and fitted values. Note the limits

along the vertical axis.

Fig. 9. Histogram illustrating distribution of estimated thermalconductivity (Kest) relative to true value (Ktrue) under simulatedconditions at an ultrasonic SNR of 20 dB over 50 independent

trials.

In vitro turkey breast muscle experimentsA plot of R(t) as a function of time during cool

down (beginning immediately after heating was turnedoff) is shown in Fig. 10a for one of the samples tested.The slope of the linear fit to the data points leads to thethermal diffusivity: 1.42 � 10�7 m2/s for this sample.The uncertainty in this estimate at the 95% confidencelevel is 0.3 � 10�7 m2/s. The thermal diffusivity esti-mates obtained for the remaining samples tested are 1.18� 10�7, 1.3 � 10�7 and 1.55 � 10�7 m2/s. The variationin the diffusivity values is possibly due to the localheterogeneities and inter-sample variability that wouldbe expected between independent tissue samples. Fig.10b is a plot of the mean value of R(t) for the fourdifferent tissue samples as a function of time. The errorbars represent the standard deviation in the R(t) estimateat each time instant. For comparison, the reported range

Fig. 10. (a) Plot of R(t) versus time t estimated from a RF dataset acquired in turkey breast muscle. The straight line repre-sents a least squares linear fit to the data points. (b) Plot of themean value of R(t) vs. t from data collected in four turkeybreast tissue samples. The error bars represent the standard

deviation in R(t).

of thermal diffusivity values in the literature for muscle


is 1.25-1.54 � 10�7 m2/s (Duck 1990). The valuesestimated from our experimental noninvasive measure-ments for turkey breast tissue fall in this range.

DISCUSSION

The noninvasive estimation of tissue thermal andacoustic properties using ultrasound backscatter has beenpreviously proposed. Yao and Ebbini (Hui et al. 2004)demonstrated that the ultrasound backscatter providesinformation to estimate the acoustic absorption and thetissue perfusion following a short ultrasound heatingpulse. However, no quantitative estimates of these pa-rameters were presented. Recently, Sumi andYanagimura (2005 and 2007) proposed a technique forreconstructing the thermal properties such as thermalconductivity and diffusivity. The temperature rise in thesample was induced using a heated waterbath. Practicalapplication of the methods in noninvasive therapy mo-dalities such as HIFU was not presented in that study.Unique to this article is the development of a noninva-sive ultrasound based approach to obtain quantitativeabsolute estimates of the thermal diffusivity by focusedultrasound heating. Furthermore, ultrasound strain esti-mation is performed using a new iterative heat transfermodel-based signal processing algorithm. The method isintended to facilitate HIFU therapy planning but thesmall transverse beam width of the HIFU focus and theease of delivering a short duration heating pulse presentdistinct advantages for general estimation of thermaldiffusivity, and perhaps for capillary bed perfusion. Inour experimental set-up, the ultrasound imaging scan-lines are oriented transverse to the HIFU beam (Fig. 1).It is advantageous to image the temperature evolution inthis direction since the thermal gradients are greater(compared with imaging in parallel with the HIFUbeam), thus, providing greater sensitivity to spatial ratesof thermal dissipation. The estimation approach wasdemonstrated in an in vitro experimental set-up in a gelphantom and in turkey breast tissue. Using a short ultra-sound heating pulse permits simplification of the solutionto the BHTE such that the diffusivity and the tissueperfusion can be separately estimated, provided the un-derlying BHTE model in eqn 1 is valid. The presence oflarge blood vessels in the region of interest would inval-idate assumptions of homogeneity of acoustic and ther-mal parameters near the HIFU focus.

The results from the numerical simulations of HIFUinduced acoustic strain demonstrated that under realisticultrasound signal-to-noise conditions, K can be estimatedcompared to the ground truth value input in the simula-tions with a variation of less than 10%. The simulationsprovide a means to select the signal processing parame-
ters and predict the lower bound of the estimation accu-
racy that can be expected with the algorithm. Experimen-tal variation in the estimate of diffusivity would beexpected to be greater than that obtained through thesesimulations which used Gaussian particle distributionstatistics. In phantoms, the estimates of K independentlyobtained using the invasive transient hot-wire methodand the noninvasive method showed a difference ofapproximately 15%. The hot-wire method was used forcomparison because it is a standard method of estimatingthe thermal properties of solid materials in the heattransfer literature. However, there exist significant dis-crepancies between the governing equation describingthe heat flow (eqn 20) and the experimental arrangementfor the hot-wire method. For example, the estimationprocedure includes a number of assumptions regardingthe geometry of the heat source and sample, thermalproperties of the heating wire, and heat transfer betweenthe heating wire and the sample being tested (indeed,boiling at the wire surface is difficult to avoid). Based ona detailed quantitative error analysis incorporating modelerrors and experimental measurement errors, Hammer-schmidt and Sabuga (2000) showed that the expecteduncertainty in the thermal diffusivity from a hot-wiretechnique can be on the order of 30%. Another majorsource of uncertainty in the hotwire method specific toour experimental set-up is the distance between the ther-mocouples and the heating wire and orientation of thethermocouples. Since r2/4K appears as a lumped term ineqn 20, any errors in the measured distance stronglyinfluences the diffusivity estimate. For example, an errorof 0.2 mm in a thermocouple placed 2 mm from the wireresults in an error of 20% in the estimated diffusivity.Given these challenges with the hot-wire method, we canonly conclude that the two independent methods are inexperimental agreement. In fact, the ultrasonic estima-tion technique using a HIFU source has significant ad-vantages: i) the diffusivity estimation is based on datacollected at multiple points along a complete scanlinecompared with a set of discrete thermocouple measure-ments, ii) the presence of the heating source (HIFUbeam) does not affect the integrity of the sample as longas the temperature rise is small to prevent irreversibledamage, and iii) the heat transfer from the HIFU focus tothe surrounding medium is continuous. Finally, the ul-trasonic approach is noninvasive. Experiments were alsoperformed in excised turkey breast. The hot-wire methodwas not used in these turkey breast experiments becauseof experimental difficulty in visualizing the thermocou-ples amid the field of bright tissue scatterers present inthe background. The estimates of K obtained in excisedturkey breast samples showed good agreement with thepublished literature values.

Accurate estimation of echo location shifts s(r,t) is
important to minimize errors in the ultrasonic estimates


of K. In this study, no attempt was made to optimize thesignal processing parameters such as RF data samplingrate, imaging and HIFU transducer frequency, cross-correlation window length (N) and thresholds in theoptimization routines used in the estimation technique. Adetailed error analysis would be necessary in a futurestudy to understand how choice of these parametersaffect s(r,t) estimates and, consequently, the uncertain-ties in K. Nevertheless, the results of this study clearlydemonstrate that the local thermal diffusivity can benoninvasively estimated from the RF data acquired dur-ing brief HIFU exposures at subablative intensities.

A scanline based (one-dimensional [1D]) RF pro-cessing scheme was adopted in this article to estimate Kwith the lateral scan direction of the imaging transduceroriented parallel to the HIFU beam propagation direc-tion. RF lines near the center of the HIFU focal regionwere used in the analysis. This is consistent with theorientation recommended by Simon and Ebbini (Simonet al. 1998) to minimize the acoustic thermal lens effect(decorrelation in the RF signal due to refraction andrefocusing of the US beam during heating) because thethermal gradients across the imaging transducer apertureare smaller near the center as compared to the edges ofthe HIFU focal region. Extension of the 1D based ap-proach to a 2D estimation technique performed overmultiple adjacent scan lines in the heated region mightbenefit from incorporating the thermal lens effect in anappropriate numerical model (Alaniz et al. 2002) tominimize model inconsistencies. Additional signal pro-cessing and filtering could be performed on the rawultrasound backscatter data to reduce the spatial rippleintroduced by the thermal lens effect (Simon et al. 1998).In another recent approach proposed to mitigate theeffect, the conventional beamforming on transmit wasreplaced by multiple steered plane wave insonificationsusing several subapertures. The results demonstrated thatthe artifact is reduced by averaging the 2D maps from themultiple steered plane wave insonifications. The aver-aged temperature induced strain map could be combinedwith the proposed approach presented in this article toestimate R(t) and the thermal diffusivity (Pernot et al.2004). Nevertheless, in the experiments presented here,thermal lens aberration was not severe enough to pre-clude strain estimation using the new BHTE-constrainedalgorithm, both proximal and distal to the HIFU focalzone.

In this analysis, the medium was considered to bespatially homogenous and isotropic and the anisotropy ofthermal properties in muscle is unknown and was ne-glected. In heterogeneous media, spatial variations in thethermal properties can be expected. To account for thesespatial variations, measurements of the local thermal
diffusivity at a number of spatial locations within the
region of interest can be repeated to construct a spatialmap of thermal properties. The maximum temperaturerise induced in the sample to perform the thermal diffu-sivity estimation was less than 10°C and it was assumedthat the thermal diffusivity remains constant with respectto temperature in this range. For the in vitro experimentalresults presented in this study, the excellent fit to thelinear model (Fig. 7 and Fig. 10) provide confidence inthe validity of this key assumption. The threshold forthermal ablation is typically close to 60-70°C (i.e., 20-30°C temperature rise from 37°C). Further experimentalstudies are required to understand whether the tempera-ture dependence of thermal diffusivity is significant inthe regimes involved in ablative thermal therapy in vivo.The proposed methods in this article could be used toperform repeat measurements of the diffusivity to under-stand its temperature sensitivity at elevated temperatures.The diffusivity can be estimated for small local temper-ature rises starting from an elevated baseline temperatureand these measurements can be carried out successivelyover small temperature ranges for which the diffusivity isconstant until the coagulation threshold is reached. Thetemperature dependence beyond the ablation threshold isless relevant in HIFU therapy since the clinical goal is tobe able to monitor the treatment accurately until ablationhas been reached. The temperature dependence of diffu-sivity was also not considered by a number of previousauthors who developed simulation programs to predictthe temperature rise during HIFU (Meaney et al. 1998;Curra 2001). Their results showed good agreement be-tween simulated temperature profiles and measured tem-perature estimates from thermocouples in vitro.

The methods developed in this article can be used intherapy planning applications to compute the thermaldose at the treatment site for a given exposure protocol.Currently, these therapy planning procedures typicallyuse standard values for the thermal parameters derivedfrom the literature. In situ estimates can reduce theuncertainty in the knowledge of these parameters andthus improve the accuracy of the predicted results, andthe new method is performed noninvasively without ab-lating the tissue. The experimental results in this articlewere limited to the estimation of thermal diffusivity. Inan in vivo setting, tissue perfusion and its variabilitywould play a significant role in the local temperatureevolution. Future work would address the estimation oftissue perfusion in vivo by extending the methods devel-oped in this article. In the absence of large blood vesselsresulting in a directional heat sink, an exponential decayterm would adequately model the perfusion in capillarybeds and predict the temperature evolution. The methodpresented here improves upon conventional elasto-graphic processing techniques for estimating strain be-
cause it constrains the solution using the heat transfer


model. This approach has application in noninvasiveestimation of temperature rise as well and we use thisthermal diffusivity estimation algorithm as part of acalibration procedure for HIFU therapy monitoring(Kaczkowski and Anand 2004).

This study examined the feasibility of a noninvasiveultrasonic estimation of thermal diffusivity under in vitroconditions. Convective heat loss due to proximity to ablood vessel requires detailed study and the next step inthe research is to extend the method to in vivo conditions.In addition to heat loss from vascularity and perfusion,potential complications include motion artifacts fromrespiration, cardiac pulsation and bulk motion of trans-ducer assembly (especially for freehand scanning) thatcan preclude cross-correlation analysis. Although theexperimental methods developed in this article employedHIFU devices, the same techniques can be applied toother forms of thermal ablation including radio-fre-quency ablation and microwave therapy. The heat trans-fer equations would need to be modified accordingly toinclude the geometry of the heat source and appropriateboundary conditions.

CONCLUSIONS

A noninvasive technique for estimating local ther-mal diffusivity using ultrasound backscatter data is de-veloped and presented in this article. The method uses ahighly focused ultrasonic source to develop a localizedheat impulse in a medium and deduces the diffusivityfrom ultrasonic backscatter measurements of spatial andtemporal rates of heat dissipation. The maximum tem-perature rise is small (less than 10°C) and the soundspeed is assumed to vary linearly with temperature; thus,temperature variations induce proportional apparent dis-placements of scatterers in the medium that shift thebackscattered waveforms over the dissipation time. Aheat transfer model that takes advantage of the impulsiveheat source is used to develop a spatio-temporal templatefor the dissipation of the heat pulse. A significant ad-vance in acoustic strain estimation provided by themethod presented here is the use of the heat transfermodel to constrain the estimation of the heat diffusivityby fitting the ultrasonic RF waveform curves using themodel templates and, thus, optimally compensate for thetemperature induced strain. The applicability of this tech-nique to estimate the diffusivity parameter can be ex-tended to assess the heterogeneity of biological tissueand to construct spatial maps of the variation of this andother tissue specific thermal parameters. The techniquesdeveloped in this article have potential applications inthermal therapy planning and monitoring, by noninva-sively and nondestructively determining tissue thermal
properties specific to the patient and organ at the site of
interest. Extension of the method to in vivo conditions isnontrivial, as motion artifacts and variable vasculariza-tion and perfusion rates will require modification of themodel. Furthermore, validation of the method in vivowill require significantly more complicated experimentalarrangements.

Acknowledgments—The authors thank Andrew Proctor for help withthe experiment set-up. This work was supported in part by U.S. ArmyMRMC/TATRC (DAMD 17-002-0063, DAMD 17-02-2-0014), andOffice of Naval Research (N00014-01-G-0460), and NIH-NCI (R01CA109557-01).

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Noninvasive Measurement of Local Thermal Diffusivity Using Backscattered Ultrasound and Focused Ultrasound Heating