Modeling the modulated acoustic radiation force distribution in a viscoelastic medium driven by a spherically focused ultrasound transducer

Transcription

1 Modeling the odulated aousti radiation fore distribution in a visoelasti ediu driven by a spherially foused ultrasound transduer F.C. Meral a, T. J Royston b and R. L Magin a a University of Illinois, 84 West Taylor St. MC 51, Chiago, IL 60607, USA b University of Illinois at Chiago, 84 W. Taylor St. RF 107, Chiago, IL 60607, USA feral@ui.edu 6185

2 Aousti radiation fore reated by foused ultrasound transduers is gaining great interest in edial diagnosis. This study ais to larify the aousti power delivery by eans of a odulated foused transduer and to predit the perforane of suh systes. A spherial-ap shaped transduer, ade of piezoeletri aterial, is used to reate ultrasoni waves at a foal point. Different odulation ethods, given in the literature and reviewed here, are available for reating a onentrated alternating fore due to the aousti radiation pressure. The relationship between the voltage input to the piezoeletri transduer and its resulting ehanial deforation is exained using a finite eleent odel (FM) for high frequeny haroni exitation (3MHz 10MHz). The osillating surfae of the transduer drives the ontating edia, whih exerts an aousti load on the transduer that is also onsidered in the F analysis. Also, the otion of resulting aousti waves in a lossy ediu is studied for a ore aurate estiation of the indued fore distribution and energy dissipation within the ediu. Ultiately, the intention is to relate the eletrial power input to the transduer to the resulting dynai fore generated in the oupling ediu. 1 Introdution The shear elasti oduli of soft tissue, whih an be used to distinguish between healthy and pathologi tissue, varies about four orders of agnitude. Tehniques suh as vibroaoustography [1] or agneti resonane elastography [] an be used to differentiate tissue ehanial properties and, thus an be used as edial diagnosti tools. These tehniques have eployed the aousti radiation fore of foused ultrasound to provide dynai exitation inside the tissue. Vibro-aoustography ais to iage an objet in ters of variation in its ehanial properties. A highly loalized fore is used to osillate the objet; this fore is reated by a foused ultrasoni transduer. The aousti radiation pressure of ultrasound is odulated to obtain the yli radiation fore inside the objet. The sound eitted by the objet whih is a funtion of objet s ehanial properties and loation, is aptured using a hydrophone. Magneti Resonane lastography (MR) iages the shear waves inside the tissue reated by an external haroni fore. Foused ultrasound an be used as an external ehanial atuator to indue shear waves [3]. The propagation and attenuation of shear waves, whih an be extrated fro shear wave iages, are funtions of tissue visoelasti properties, so they an be used to obtain the aterial properties. The ultrasound transduer is a spherially-foused ap resonating in its thikness ode (Fig. 1). The aousti radiation due to its otion is onentrated at the geoetri enter of the sphere whih is also alled the foal point of the transduer. Different odulation ethods are disussed in the literature [4], suh that the radiation fore field reates a yli fore. These inlude Aplitude Modulation (AM), onfoal arrangeent and X-Foal arrangeent. These odulation ethods with their orresponding radiation field siulations for the lossless edia are also presented. The radiation field of foused ultrasound in dissipative edia is different than the radiation field in lossless edia. The forulations for aousti radiation fore for linear and nonlinear ultrasound in a dissipative ediu are also available in literature [5]. This study ais to larify the aousti power delivery by eans of a odulated foused transduer and to predit the perforane of suh systes. The ultrasound transduer is a piezoeletri disk and the relationship between the voltage input to the piezoeletri transduer and its resulting ehanial deforation is exained in the transduer dynais setion for high frequeny haroni exitation (3MHz 10MHz). The osillating surfae of the transduer drives the ontating edia, whih exerts an aousti load on the transduer; this is also onsidered in the transduer dynai analysis. Motion of the ontat surfae between the ediu and transduer reates an aousti field foused at the foal point. The aousti radiation fore field inside the dissipative ediu is reviewed under the radiation fore setion. A relationship between the eletrial power input to the transduer and the resulting dynai fore generated in the oupling ediu is presented. Theory.1 Transduer Dynais A ross setional view of a spherially foused ultrasoni transduer showing the geoetri paraeters are given in Figure 1. a a 1 a z Figure 1. Spherially foused transduer. R is the foal length, a is the disk radius, a 1 and a are radii defining onentri disks (onfoal arrangeent). The foused ap transduer an be assued to be a planar disk for saller radius versus foal length ratios, a/r. Also, the effet of the transduer edges is negleted. These R r 6186

3 assuptions siplify the ap shaped transduer into an infinite plane, as shown in Figure. In this figure, l denotes the thikness of the transduer where z is the position vetor fro the enter of the transduer. l z z=0. Daping Material daping for piezo aterials are available in the literature as ehanial quality fators (Q eh ). Using the onstants fro q (8) agnitude response an be plotted for the undaped ase, and the daping ter an be added to these onstants to obtain the desired daping. Figure. For saller disk radius versus foal length ratios the transduer is assued to be a planar disk; also, effets at the edges are negleted. Transduer partile veloity, v, is related to stress, T, and strain, S, as follows. dt dz dv = jωv, jωs dz = (1) Here, is the ean aterial density for the transduer, ω is the irular frequeny of osillations and j is the iaginary nuber. Piezoeletri onstitutive relations are S D = ε es () T = S e (3) where D denotes the eletri displaeent, is the eletri field, ε S is the dieletri perittivity with zero or onstant strain, is the elasti onstant in the presene of a onstant or zero field, and e is the piezoeletri stress onstant. Veloity v satisfies the wave equation and has the solution d v ω + v = 0 dz ( z) Asin( kz) B os( kz) v = + (5) where wave nuber k and wave speed are defined as This leads to T = 1 j = ω k = ~, = (4) ~ (6) [ kaos( kz) kb sin( kz) ] e ω (7) [ A os( kz) B sin( kz) ] e Applying the free-free boundary onditions to the both ends of the transduer, T(l/)=T(-l/)=0, the onstants A and B of q (5) are found l l Aos( k ) B sin( k ) 0 e = l l Aos( k ) + B sin( k ) = 0 e e, B = 0 (8) A = l os( k ) Veloity Magnitude MHz Frequeny Figure 3. Veloity frequeny response funtion of the transduer surfae for unity eletrial field (V/) aross the transduer, for free boundary onditions. - - undaped, daped. The piezo-eletri aterial properties and transduer geoetry is given in Table 1. Transduer (PZT-5A) Radius (a) Foal Length (R) Thikness (l) Stress onstant (e) 15.8 C/ lasti onstant ( ) 11.1 GPa Density () 7750 kg/ 3.3 Aousti Loading on the transduer However the free-free boundary ondition is not realisti; so, the loading on the surfae due to aousti pressure is also onsidered. Aousti ipedane of the ediu is written as Z =. ~ (9) 0 where is the ediu density and is the opression wave speed in the ediu. Assuing a Voigt odel for the ediu, an be alulated as ~ λ + μ = (10), so Z 0 = ( λ + μ) (11) The pressure on the transduer surfae due to the aousti radiation is, 6187

4 p = z. v (1) l l p = ( λ + μ) Asin( k ) + B os( k ) (13) This an be ipleented as a new boundary ondition, T(l/)=p and T(-l/)=0 = l l A os( k ) B sin( k ) e = p l l Asin( k ) + B os( k ) l l A os( k ) + B sin( k ) = 0 e.( λ + μ) The solution for A and B (using Matlab): A = e l j sin( k ) l l sin( k ) os( k l + j os( k ).( λ + μ).( λ + μ) ).( λ + μ) l ) os( k (14a,b) l ).( λ + μ) sin( k B = e l l sin( k ) os( k ). ( λ + μ) j l + j os( k ).( λ + μ) (15a,b) where A and B are onstants fro (5). The frequeny response for the transduer an be obtained using the onstants A and B. The wave nuber k is related to the angular veloity ω and thus the transfer funtion (veloity/voltage) an be obtained for different frequenies. Veloity Magnitude MHz MHz Frequeny Figure 4. Veloity frequeny response funtion of the transduer surfae for aoustially loaded boundary ondition, - - undaped, daped. The visoelasti ediu properties are given in Table. j Mediu (0.5% Agar) Density 1000 kg/ 3 Volue lastiity (λ 1 ).6 GPa Volue Visosity (λ ) 0 Shear lastiity (μ 1 ) Shear Visosity (μ ) 4.89 kpa 0.3 Pa.s Table. Transduer and oupled ediu properties..4 Radiation Fore aplitude odulation ase. The ase of aplitude odulated (AM) of a single transduer is onsidered. The developed theory of Rudenko et al. (96) is utilized and the odulation ter is added. The transduer, with resonating frequeny ω is odulated at frequeny Δω/. The aousti pressure field that is reated is given by: where p ( x r, t) p 0 r = exp αx f x sin ω t ~ + φ, ( x) a f ( x) Δωt ( x, r) os x dif (16) x x f ( x) = 1 + (17) R x / Rdif r xdif d φ ( x, r) = artan + H ( x R) { ln[ f ( x) ]} 1 x R π (18) a dx bω = (19), b = λ μ χ (0) 3 C v C p α 3 ~ ωa = (1) x dif ~ Here, p 0 denotes the aousti pressure at the transduer surfae, and respetively denote the opression wave speed and density of the ediu, H denotes the Heaviside step funtion, λ and μ respetively denote the ediu volue and shear visosities, χ is the theral ondutivity of the ediu, and C p and C v are the heat apaities of the ediu at onstant pressure and volue, respetively. And, α is an absorption oeffiient. 'b is effetive dissipation. x dif is Rayleigh Distane. Modifying Rudenko et al. (96) for the ase of a slowly odulated aousti pressure and applying short ter tie averaging (given that Δω << ω, the odulated radiation fore under a linear approxiation takes the following for: F ( x r). ~ p α ~ 0 1 Δωt os 1 r, = exp αx f ( x) a f ( x) () 6188

5 3 Siulations 3.1 Radiation Fore alulations The radiation fore field for a 3.5 radius and 7 foal length transduer is alulated using q () in a dissipative ediu assuing linear ultrasound. The fore intensity signifiantly inreases at the foal point, ating like a point fore inside the ediu. Figure 5. Noralized Radiation fore alulations for 0.5% agar gel. Fore values are noralized with respet to radiation fore on the transduer surfae 3. Displaeent alulations using FA Aplitude odulation of the radiation fore results an osillating fore between zero and its axiu value (not between positive axiu and negative axiu). The alulated fore field is used as a haroni input for finite eleent analysis. Figure 6. Displaeent FA results due to radiation fore in 0.5 % agar gel in db sale. Displaeent values are noralized with respet to the axiu displaeent (at the foal point). Axisyetri planar linear eleents are used in the FA. Utilization of linear eleents is allowable sine very sall displaeents are onsidered. Haroni analysis is onduted in different frequenies. Figure 5 shows displaeent results at two different frequenies. 4 Results and Disussion An exaple of a transduer resonating at 10 MHz on a visoelasti ediu (properties given in Table 1) is onsidered, to alulate values of displaeent inside the ediu. For an input of unity eletri field (1 V/) aross the piezo transduer s two poles, the agnitude of resulting veloity at resonane at the transduer surfae is alulated as 4μ/s. The aousti pressure at the transduer surfae for the oupled ediu is alulated using q (11) and found to be 6.5 Pa. The noralized radiation field is also realulated to find the radiation fore distribution inside 0.5% agar. The axiu radiation fore obtained inside the gel is 0.3 μn observed at the foal point. The axiu values drops to saller values as one oves away fro the foal point. The area ontaining the fore values greater than half axiu is alulated as 4.6x10 -, and the average pressure ating on this saller area is alulated as 1.08 kpa. When the ultrasound is odulated with a frequeny equal to 1000 Hz, the axiu resulting displaeent at the foal point is also alulated using the sae F odel, and it is found to be around.6 n (for 1 V/ input to the piezo). Note, the depolarization voltage for this transduer is higher than 100 V. If one assues that this transduer ould be driven at 50 V peak aplitude, the resulting axiu displaeent in the radiation field is alulated to be Conlusion The radiation fore of foused ultrasound, whih is used to reotely reate exitation inside visoelasti edia is onsidered. The ai of this study is to oe up with an analysis ethod obining the different physial doains assoiated with this transdution proble. Governing piezo-eletri equations are used to derive the response funtion for aoustially loaded and unloaded onditions. Daping, whih plays an iportant role in ehanial response is also onsidered as ehanial quality fator. Transduer otion on the visoelasti ediu results in an aoustial pressure field inside the ediu, whih in turn loads the transduer. The fore field due to aousti radiation is alulated and resulting displaeents inside the visoelasti edia are alulated using FA. It is shown that for a given transduer, the resonane frequeny and the response at this frequeny an be approxiately alulated, and the resulting pressure field and displaeent field an be estiated. This researh was initiated by the otivation of iproving the use of foused ultrasound in MR; results disussed here see supportive for further MR utilization of spherial ultrasound transduers. Aknowledgents This researh is supported by U.S. NIH Grant # B

6 Referenes [1] M. Fatei, J.F. Greenleaf, Vibro-aoustography: An iaging odality based on ultrasound-stiulated aousti eission, Pro. Natl. Aad. Si. USA 96, (1999) [] R. Muthupillia, P. J. Rossan, D. J. Loas, J.F. Greenleaf, S. J. Riederer, R. L. han, Magneti resonane iaging of transverse aousti strain waves, Magneti Resonane in Mediine 36:66-74 (1996) [3] T. Wu, J.P. Fellee, J,F, Greenleaf, S.J. Riederer, R.L. han, MR Iaging of Shear Waves Generated by foused Ultrasound, Magneti Resonane in Mediine 43: (000) [4] S. Chen, M. Fatei, R. Kinnik, J.F. Greenleaf, Coparison of Stress Field Foring Methods for Vibro-aoustography, I Transations on Ultrasonis, Ferroeletris, and Frequeny Control 51: (004) [5] O.V. Rudenko, A.P. Sarvazyan, S.Y. elianov, Aousti radiation fore and streaing indued by foused nonlinear ultrasound in a dissipative ediu. J Aoust So. A. 99, (1996) 6190