POST BEAMFORMING ADAPTIVE SECOND ORDER VOLTERRA FILTER (ASOVF) FOR PULSE-ECHO ULTRASONIC IMAGING

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1 POS BEAMFORMING ADAPIVE SECOND ORDER VOERRA FIER (ASOVF) FOR PUSE-ECHO URASONIC IMAGING Mamoun Fawzi Al-Mistarii Jordan University of Science and ecnology, Faculty of Engineering, Department of Electrical Engineering, Irbid 110, Jordan Key words: Beamforming, Ultrasound Contrast Agents, Volterra Filter ABSRAC We ave previously introduced post-beamforming second order Volterra filter (SOVF) for decomposing te pulse eco ultrasonic radio-frequency (RF) signal into its linear and quadratic components. Using singular value decomposition (SVD), an optimal minimum-norm least squares algoritm for deriving te coefficients of te linear and quadratic kernels of te SOVF was developed and verified. However, te agent specificity of te standard SVD-based quadratic kernel is sometimes compromised by sensitivity to nonlinear ecoes from tissue. In tis paper, we present an adaptive second-order Volterra filter (ASOVF) designed to obtain te optimum filter coefficients minimizing te cost function to produce images wit ig sensitivity to nonlinear oscillations (0-30 db below te fundamental) from microbubble ultrasound contrast agents (UCA) wile maintaining ig levels of noise rejection. e least-squares approac of a second-order Volterra model and its adaptive filtering algoritm based on recursive least-squares are introduced. I. INRODUCION Microbubble ultrasound contrast agents (UCA) are being investigated for use in clinical imaging applications for tissue function and for targeted terapeutic applications. e objective is to detect minute concentrations of UCA in te microvasculature during ultrasonic exams tus providing a view of te perfusion in te tissue. is functional form of ultrasonic imaging is seen an important component for te continued use of ultrasound as a medical imaging modality. For example, many tumors witout distinguising caracteristics on conventional ultrasound ave caracteristic blood perfusion patterns tat allow for easy detection if a perfusion sensitive imaging is available. Interaction between microbubbles UCAs and acoustic wave result in nonlinear armonic eco generation. is penomenon can be exploited to enance te ecoes from te microbubbles and, terefore, reject fundamental components resulting largely from tissue. Imaging tecniques based on nonlinear oscillations ave been designed for separating and enancing nonlinear UCA ecoes from a specified region of interest witin te imaging field, including second armonic (SH) imaging and pulse inversion (PI) Doppler imaging []. e SH imaging employs a fundamental frequency transmit pulse and produces images from te second armonic component of received ecoes by using a second armonic bandpass filter (BPF) to remove te fundamental frequency. In order to increase UCA detection sensitivity in te limited transducer bandwidt condition, spectral overlap between fundamental and second armonic parts need to be minimized by transmitting narrow-band pulses resulting in an inerent tradeoff between contrast and spatial resolution. In PI imaging a sequence of two inverted acoustic pulses wit appropriate delay is transmitted into tissue. Images are produced by summing te corresponding two backscattered signals. In te absence of tissue motion, te resulting sum can be sown to contain only even armonics of te nonlinear ecoes. e PI imaging overcomes te tradeoff between contrast and spatial resolution because it utilizes te entire bandwidt of te backscattered signals. As a result, superior spatial resolution can be acieved wen compared wit SH imaging. However, te subtraction process results in significant reduction of signal to noise as te armonics are typically 0-30 db or more below te (cancelled) fundamental component. e SOVF-based quadratic kernels provide ig sensitivity to armonic ecoes comparable to PI wit a significant increase in dynamic range due to inerent noise rejection of quadratic filtering [3]. An algoritm for deriving te coefficients of te kernel using singular values decomposition (SVD) of a linear and quadratic prediction data matrix was proposed and experimentally validated in [4]. Imaging results and comparisons wit SH and PI images ave sown tat quadratic imaging is

2 superior to SH and compares favorably wit PI witout te need for multiple transmissions. However, due to reliance on linear and quadratic prediction, te quadratic kernel as sensitivity to te fundamental tat limits its ability to detect UCA in te microvasculature. In tis paper, we present an adaptive second-order Volterra filter (ASOVF) designed to obtain te optimum filter coefficients minimizing te cost function to produce images wit ig sensitivity to nonlinear oscillations (0 30 db below te fundamental) from microbubble ultrasound contrast agents (UCA) wile maintaining ig levels of noise rejection. e least-squares approac of a second-order Volterra model and its adaptive filtering algoritm based on recursive least-squares are introduced. e approac is demonstrated experimentally using images from in vivo kidney after bolus injection wit UCA. Illustrative images of te kidney of a juvenile pig were obtained before and after infusion of contrast agent (SonoVue, Bracco, Geneva, Switzerland) at various concentrations. Imaging results of ASOVF data sow a significant increase in armonic sensitivity and reduction in noise levels. e imaging results given in tis paper indicate tat a signal processing approac to tis clinical callenge is feasible. II. HEORY In tis section, we present te basis of an adaptive secondorder Volterra filter (ASOVF) designed to. However, for te sake of continuity, we summarize te minimum-norm least-squares (MNS) approac ere (for full detail, please see [3-4]). A. MNS Estimation of SOVF Coefficient e algoritm in tis section is based on [3-4], wic ave sown te validity of a SOVF as a model for pulseeco ultrasound data from tissue mimicking media. e response of a quadratically nonlinear system wit memory y ˆ ( n + 1), can be predicted by a second order Volterra model of m past values as follows: yˆ ( n + 1) = y( n i) ( i) () + i= 0 j= 0 k = j y ( n j) y( n k) ( ) j, k were i and ( j, k are te linear and quadratic filter coefficients respectively. It is easy to see tat (1) is a nonlinear equation in terms of te beamformed input data. However, it is a linear equation in terms of te unknown ) (1) filter coefficients (i.e., linear and quadratic Volterra kernels) i j, k. Hence, (1) can be and rewritten in vector form: ( n +1) = Y ( n)h yˆ () were past data vector Y ( n) is defind at time n as: Y ( n) = [ y( n), y( n 1 ), y( n ),..., y( n m + ) y 1, ( n), y( n) y( n 1 ),..., y ( n m + 1)] and te filter coefficient vector H is defined as: H = [ ( 0), ( 1 ),,..., ( m 1 ), ( 0,0), ( 0,1 ),..., ( m 1, m 1)] Note tat m is te system order, N is te total number of filter coefficients [3], wic is equal to ( m + 3m) assuming symmetrical quadratic kernels (i.e., j, k = k j ), and superscript is te transpose, of a vector or a matrix. Similarly, y ˆ ( n + ), yˆ ( n + 3 ),..., yˆ ( n + m) can be represented in te form of () and expressed in a matrix form. A system of linear equations is formed in order to find filter coefficients as follows: F = GH (3) were te vector F and te data matrix G are: [ y( n + 1 ), y( n + ) y( n ) ] [ Y ( n), Y ( n + 1 ),..., Y ( n + )] F =,..., + G = 1 were is te number of linear equations (observations). Using a segment of te RF data, a system of linear equations are formed and solved for elements of te quadratic kernel. Details of te algoritm to determine te quadratic kernel tat provide maximum contrast enancement ave been described in [3-4]. B. Adaptive Second-order Volterra Filering In time domain adaptive filters, te new filter coefficients est n +1 are updated using old filter coefficients ( n) est, and te estimation error, multiplied by te weigting vector e ( n + 1) w ( n +1), as follows: ( n + 1 ) = ( n) + w( n 1) e( n ) est est (4)

3 were e ( n + 1 ) = y( n + 1) ( n) Y ( n + 1) est (5) tat is, te difference between te true observed output (or response) at time n +1 and te output "predicted" by te Volterra model. In general, te weigting vector w ( n) is determined so tat n can converge to te unique est optimum filter coefficients wic minimize te quadratic cost function. A block diagram of a second-order Volterra filter to estimate an unknown quadratic nonlinear system is sown in Figure 1, in wic te linear and quadratic filter coefficients are recursively updated. ere are two well-known algoritms in digital signal processing wic guarantee te convergence of te filter coefficients: te least-mean-squares (MS) algoritm and te recursive least-square (RS) algoritm [6-7]. Figure 1 Recursive estimation of an unknown pysical system using a second-order Volterra filter, in wic a quadratic cost function is minimized and te filter coefficients are recursively estimated using an adaptive filtering algoritm. In te MS algoritm, wic is an implementation of te stocastic gradient metod, te weigting vector is determined by ( n) Y( n) w µ = (6) were µ satisfies te convergence condition 1 0 < µ < (7) λ max and λ is te largest eigenvalue of te matrix GG. If max µ is large wile satisfying te convergence condition, te convergence speed of te algoritm is fast, but wit large variance (steady-state error). On te oter and, a small µ value reduces te variance, but wit slow convergence speed. erefore, in order to acieve fast convergence and small steady-state error, one migt use a "gear sift" tat selects relatively large µ initially, and ten reduces µ to a small value after te error decreases below a certain level. Since te MS algoritm can be easily implemented wit O ( N ) computational complexity, it as been widely used in various signal processing applications. However, te MS algoritm suffers from slow convergence of te filter coefficients, especially wen te input is correlated. In te RS algoritm, wic is an implementation of te stocastic Gauss-Newton metod, te weigting vector is determined by ( GG ) 1 G w n = (8) In fact, te RS algoritm computes w n opt ( n) at every instant of time by computing. Hence, te filter coefficients converge quite rapidly, but at te cost of eavy computational complexity, i.e., O ( N ). us, various fast RS (FRS) algoritms, wic are computationally more efficient, ave been developed. Since a second-order Volterra filter using te RS algoritm is more complicated tan a linear filter using te RS algoritm and requires a large number of filter coefficients, te fast implementation of te RS algoritm for a second-order Volterra filter is desirable in order to reduce te computational complexity, and is often necessary for real-time applications. However, in tis paper, te RS algoritm for a second-order Volterra filter is utilized, instead of te fast RS algoritm, to obtain te optimum filter coefficients minimizing te cost function to produce images wit ig sensitivity to nonlinear oscillations (0-30 db below te fundamental) from microbubble ultrasound contrast agents (UCA) wile maintaining ig levels of noise rejection. e details of te development of te algoritms are beyond te scope of tis paper, but may be found in [6-7]. III. MAERIAS AND MEHODS A. Experimental setup We evaluated te algoritm wit RF data acquired from experiments conducted in vivo on a juvenile pig. Bolus injections of SonoVue M (Bracco Researc SA, Geneva, Switzerland), an UCA consisting of sulpur exafluoride gas bubbles coated by a flexible pospolipidic sell, were administered wit two different concentrations (0.01 m/kg and m/kg). ree- and four-cycle pulses at 1.56 MHz were transmitted using a convex array probe (CA430E) wit mecanical indices (MIs) of and 0.15, respectively, to scan a kidney. ecnos MPX ultrasound system (ESAOE S.p.A, Genova, Italy) was modified so tat a pair of inverted pulses wit te appropriate time delay was subsequently transmitted to produce images

4 wit te PI tecnique. In addition, in order to remove low frequency components due to tissue motion artifacts and retain armonic frequency components from UCAs, RF data from PI imaging were filtered using te linear igpass filter wit cutoff frequency.3 MHz. For eac setup, tree frames of RF data from te PI tecnique were collected wit 10 s and 15 s delays after te injection of 0.01 m/kg and m/kg UCAs, respectively. RF data were acquired wit 16-bit resolution at 0-MHz sampling frequency witout GC compensation and saved for off-line processing. B. Contrast measurements As a comparison of contrast enancement between images from different tecniques, we compute contrast-to-tissue ratio (CR) from data in te RF domain before scan converted. CRs of images can be calculated wit ecoes from two referenced regions: First, te contrast region inside te kidney (bottom-left part). Second, te tissue region outside te kidney (on te left and side of te contrast region wit te same dept). Bot regions are composed of 1 connected A-lines wit 7.5-mm axial extent. IV. RESUS AND DISCUSSION B-mode image), ecogenicity from contrast regions is sligtly lower tan tat from surrounding tissue regions, wic agrees wit te CR value (-.1 db) wereas te PI image provides CR db, ecogenicity of te contrast region from te PI image appears brigter tan tat from surrounding tissue regions. Please note tat te CR value for te PI image witout SH filtering was only 10.3 db, i.e. tere is a 4.13 db gain due to te removal of tissue components introduced by motion. It is also wort noting tat te SH image on te B-mode data suffers from significant loss in spatial resolution. e quadratic image from twice D correlation of 38 t singular mode provides CR 3.13 db, wic sows a contrast enancement over bot standard B-mode, PI and SHPI images. e quadratic image from te ASOVF of te B-mode data is obtained using te algoritm described in section wit te use of a 5.6-mm contrast A-line and a system order 15, provides CR db, wic sows a contrast enancement over te oter four images. We can clearly see not only te kidney s sape and boundary due to UCA ecoes but also large vascular structures as it is seen in te PI, SHPI, quadratic from twice D correlation of 38 t singular mode images too. Also we can see tat te kidney s sape and boundary due to UCA ecoes are te clearest in te ASOVF image compared wit te oter four images due to te efficient removal of noise trougout te spectrum wile maintaining quadratic data, even below te noise floor. Figure Average spectra from te contrast (darker, black) and tissue (ligter, red) regions of te kidney: Standard B-mode, and PI. Figure sows te average spectra of typical ecoes from te contrast and tissue regions of te kidney described from te standard B-mode and PI data in Section 3. e average spectra are calculated by averaging windowed periodogram of every eco line in regions described in Section 3.. e ecoes from te UCA region exibit broader bandwidt tan tose from tissue region Figure 3 sows images obtained using a standard B-mode image of te kidney after te injection of 0.01 m/kg UCAs acquired using 3-cycle transmission, PI, second armonic on PI data (SHPI), quadratic image from twice D correlation of 38 t singular mode of te B-mode data, and quadratic image from te ASOVF of te B-mode data. Due to differences in dynamic ranges, eac image is displayed wit its full dynamic range as can be seen from te db-level scale bars. Due to low microbubble populations in te perfused tissue of te kidney (Standard Figure 3 Images of te: Standard B-mode image of te kidney at 10 s after te injection of 0.01 m/kg, PI, SHPI, quadratic from twice D correlation of 38 t singular mode of te B-mode data, and quadratic from te ASOVF of te B-mode data.

5 Figure 4 sows te gray-level istograms produced from te standard B-mode, PI, SHPI, quadratic from twice D correlation of 38 t singular mode of te B-mode data, and quadratic from te ASOVF of te B-mode data images. In eac case, te istogram from te UCA region is plotted wit ligt solid line (red), wile te istogram from tissue is plotted wit darker solid line (black). One can see te degree of overlap between te istograms is igest for te standard B-mode image, wereas it is te lowest for te ASOVF image. 7.. jung and. Söderström, "eory and Practice of Recursive Identification," Cambridge, MA: M.I.. Press, V. CONCUSION We ave proposed te new algoritm of postbeamforming adaptive second order Volterra filter (ASOVF) for deriving te quadratic kernels wic give te igest CRs, efficiently remove noise trougout te spectrum, maintaining quadratic data, even below te noise floor, and extract quadratic components from UCA nonlinear ecoes wit single transmission. Compared wit te PI image processed from te same RF data, te quadratic images sow comparable performance in terms of bot contrast and spatial resolution. e results sown in tis paper indicate clearly tat ASOVF imaging is far superior to eiter PI or quadratic imaging. In addition, compared to PI imaging, te Volterra filter approac does not require multiple transmissions for acquiring one image line. erefore, tis approac preserves te frame rate of te original B-mode system, an important advantage of ultrasound imaging over oter medical imaging modalities. REFERENCES 1. J. Opir and K. J. Parker, Contrast agent in diagnostic ultrasound, Ultrasound in Med. & Biol., vol. 15, no. 4, pp , Nov D. H. Simpson, C.. Cin, and P. N. Burns, Pulse inversion Doppler: A new metod for detecting nonlinear ecoes from microbubble contrast agent, IEEE rans. Ultrason., Ferroelect., Freq. Contr., vol. 46, no., pp , Mar P. Pukpattaranont and E. S. Ebbini, Postbeamforming second-order Volterra filter for nonlinear pulse-eco ultrasonic imaging, IEEE rans. Ultrason., Ferroelect., Freg. Contr., vol. 50, no. 8, pp , Aug P. Pukpattaranont, M. F. Al-Mistarii and E. S. Ebbini, Post-beamforming Volterra filters for contrast-assisted ultrasonic imaging: In-vivo results, in Proc. IEEE Ultrason. Symp., G. H. Golib and C. F. Van oan, "Matrix Computations," Jons Hopkins University Press, Baltimore, MD, nd edition, B. Widrow and S. D. Steams, "Adaptive Signal Processing," Englewood Cliffs, NJ: Prentice-Hall, Figure 4 Gray-level istograms produced from images sown in Figure 1. Histograms are produced from te contrast region (ligter, red) and te tissue region (darker, black).