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On Convolution Model for Ultrasound Echo Signal Processing
 

On Convolution Model for Ultrasound Echo Signal Processing

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    On Convolution Model for Ultrasound Echo Signal Processing On Convolution Model for Ultrasound Echo Signal Processing Document Transcript

    • Proceedings of the IEEE-EMBS International Conference on Biomedical and Health Informatics (BHI 2012) Hong Kong and Shenzhen, China, 2-7 Jan 2012 On Convolution Model for Ultrasound Echo Signal Processing Ming Chen and Cishen Zhang Abstract- Systematic ultrasound signal processing and imag­ ing require a model for the ultrasound echo signal so to extract the tissue information for image construction. Based on careful studies of the classical ultrasound acoustics and traditional con­ volution model and taking into account the echo effects arising from anatomical tissue properties, this paper proposes a new convolution model for the ultrasound pulse-echo process, which presents dominant terms under common practically feasible conditions of medical ultrasound. The proposed convolution model can provide an effective means for processing ultrasound echo signals and constructing images. Index Terms-Convolution, Modeling, Signal processing, Ul­ trasound Imaging. I. INTRODUCTION Ultrasound has been an increasingly important imaging modality in medical diagnosis since late 1930s. It facilitates the non-invasive observation of human soft tissues as the 2nd widely-used tool only next to x-ray. While it is applied as a means of medical imaging without considerably adverse effect being reported, its real-time capability makes it suit­ able in diagnosis, monitoring the anatomical changes during surgery, chemotherapy, rehabilitation therapy, and many oth­ er clinical applications. Unlike the bulky and stationary Com­ puter Tomography (CT) and Magnetic Resonance Imaging (MRI), ultrasound is mobile for medical operators to move easily and conveniently. Moreover, the cost of purchase and operating expenses of ultrasound is incomparably lower than CT and MRI. One of fundamentals to ameliorate the imaging quality of ultrasound lies in the full understanding of the ultrasound echo process. In general, the ultrasound image degradation is caused by the constructive and destructive interferences of backscattered signals due to unresolved tissue inhomogeneity [2]. An accurate mathematical model in view of signal and system can provide fundamental comprehension of this complicated echo process. Since 1980s, convolution models of ultrasound echo pro­ cesses have been studied and applied. Attempts were made to ameliorate the resolution of scanners a posteriori with digital signal processing [3]. The convolution model based on linear system theory and ultrasound acoustics was first systematically formulated in [5], [6], [7]. Using this model, a number of methods and algorithms have be developed to deal with ultrasound signal processing and imaging problems such as despeckling and denoising [8]-[10]. Ming Chen is with School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, anniechenming@gmail.com Cishen Zhang is with Faculty of Engineering and Industrial Sciences, Swinburne University of Technology, Hawthorn VIC 3122, Australia, cishenzhang@swin.edu.au 978-1-4577-2177-9/12/$25 (C) 2012 IEEE 357 Ultrasound Transducer Aperture: 5 Tissue of Interest: V Fig. l. Coordinate system for calculating the received signal. This paper proposes a new model of the pulse-echo pro­ cess in medical ultrasound scanning. Based on the study of classical acoustical equations, the proposed model presents dominant terms under practically feasible conditions of med­ ical ultrasound. With the pulse spread function (PSF) of the ultrasound transducer and the radio frequency (RF) echo signal as the input and output, respectively, the proposed model describes the interaction between the PSF and the scanned tissue, in terms of its acoustical velocity of propa­ gation or mass density, as a convolution in the time-spatial domain. With this model, the objective of the ultrasound signal processing and imaging becomes to identify the spatial distribution of the tissue mass density using the measurement of the RF echo signal. Built upon the acoustical properties of the ultrasound pulse-tissue interactions, the proposed model is computationally feasible and can provide an effective means for medical ultrasound echo signal processing and imaging. Section II introduces fundamentals and expressions of ultrasound acoustics. Section ill provides important assump­ tions and approximations for the rest derivations. The new convolution model of ultrasound echo signal is proposed in Section IV with a brief guideline of its derivation. And Section V concludes the paper. II. ULTRASOUND ACOUSTICS Under the condition that there is no heat conduction or conversion of ultrasound to thermal energy and the first-order Born approximation [5], the two dimensional coordinate system for representing the ultrasound echo process is shown in Figure 1, where the left apparatus is the ultrasound transducer with surface area S and the object on the right is the tissue of interest. Following from the adiabatic and hydrodynamic equations as in [l], the ultrasound echo pro-
    • cess is modelled as Pr(rO,t) (1) where Em is the electrical-magnetic characteristics of med­ ical ultrasound machine and ps(ro+ra,t) is the ultrasound scattered field at the spatial location (ro+ra); ro refers to the location of transducer while ra the location of transducer aperture with respect to roo The ultrasound scattered field can be formulated at the point denoted by r, by integrating all the spherical waves from the scattering region: ps(ro,l) where 1 _ -V'(p(r,). V'(Pi(r"tI)) Po 2c(r,) (}2Pi(r,,t,) - � (}t2o (2) (3) V represents the spatial volume of tissue, T denotes the inte­ gration interval over time, Po means the average mass density of tissue; r, denotes the field point in tissue of interest, v refers to the gradient operation, p(r,) is the perturbation of mass density r, and calculated as the difference between the instantaneous mass density and the mean density of the tissue, c(r,) is the perturbation of propagation velocity of the tissue at r, and calculated as the difference between the instantaneous velocity and the mean propagation velocity of the tissue, Pi(r,,td is the incident pressure at (r,,t,) written as (4) with (5) as the the velocity potential and v(t) as the particle velocity normal to the transducer surface. The integrated functions Fps(r"ro,t,t,) contains tissue information of interest such as ultrasound propagation perturbations and gradient of tissue density perturbations. The analysis and formulation in this paper are based on the ultrasound RF echo signals after beamforming. Usually, commercial ultrasound scanners collect ultrasound radio fre­ quency (RF) echo signals line by line in the axial direction, and then combine the received lines of echo signals into 2- dimensional echo signals. It is noted that, although the final ultrasound images are 2-dimensional, still the ultrasound echo signals are not collected simultaneously in both axial and lateral direction. 358 III. IMPORTANT AND FEASIBLE CONDITIONS A. Typical acoustical data and properties of tissues In addition to the heat exchange and Born approxima­ tion conditions stated earlier, we will incorporate into the proposed convolution model typical data of tissue acoustic properties as given in Table 1. With these data, we exploit the TABLE I DENSITIES AND ULTRASOUND VELOCITY FOR SOME COMMON TISSUES IN NATIONAL PHYSICAL LABORATORY, UK Tissue Density Velocity of sound (kg/m3) (m/s) Liver 1050-1070 1578-1640 Kidney 1050 1560-1580 Muscle 1038-1056 1529-1629 Pancreas 1040-1050 1591 Skin l 1LO-1190 1729+ Spleen 1054 1567-1635 Breast 1020 1430-1570 following anatomical and mechanical properties to facilitate derivation of the new convolution model. B. Acoustical parameters In theoretical acoustics [1], the dependence of the three acoustical parameters: c(rl)' the propagation velocity of ultrasound, p(r,), the mass density and K(r,), the compress­ ibility, at the spatial location r, within the tissue is given as 1 c(r,) = ---;==:===:====:= vp(r,)K(r,) (6) It follows from [2] that, if the mass density of human soft tissue varies in a small range and the typical average propagation velocity is assumed as 1545 mis, the tissue compressibility can be linearized and represented by its mass density: dK K = KO+- � 1/1012(1545-l.1p) m2/Nt. dp Substituting (7) into (6) yields 106 c(rt) = ---;==:===:========:==0:=:= vp(rI)(1545-l.1p(rI)) (7) (8) Let c(rl) = Co+c(rd and p(r) = Po+p(rd where Co and Po are average propagation velocity of ultrasound waves and average mass density of the tissue of interest. It is straight­ forward to use the first order Tailor series approximation at Po to simplify the relationship between c(r,) and p(r,) into where 106 (772.5-l.1po) [Po(1545-l.1po)J3/2' 106 [Po(1545-l.1PO)] , /2· (9) It is noted that, both theoretically and practically, the ultra­ sound propagation velocity depends on the variation of tissue
    • mass density. In another word, if there is no change of tissue mass density, ultrasound propagation velocity should remain unchanged, i.e. c(rl)=O, at p(rl)=O. (10) This together with (S) results in 106 Co=- yl r p=o(;=' IS= 4== S =-=I=.I=po =;= ) and (11) In fact, when the tissue mass density ranges from 900 kg/m3 to 1200 kg/m3, the difference between a1 and Co is quite small. Hence, the omitted term (al- co) can be regarded as the error of linearization. C. Particle Velocity The ultrasound scanning for B-mode imaging is a se­ quence of repetitive operations and each operation is com­ posed of a short duration pulsing time and a long duration listening or measurement time. During the pulsing time, the ultrasound probe produces and transmits an ultrasound pulse to the body tissue controlled by the electronic circuit of the scanner. In commercial ultrasound scanners, the pulses produced are characterized by its centre frequency, typically between 3- 10 MHz. Thus it is practically feasible to model the particle velocity v(t) normal to the surface of the trans­ ducer as a short duration envelop function g(t) modulated by a trigonometry function of the centre frequency. It is noted that the particle velocity v(t) is a different physical quantity from the propagation velocity c(r). Without loss of generality, the particle velocity function can be written as v(t)=g(t)sin(2n fot), (12) where g(t) is the envelop function of the particle velocity and fo is the centre frequency of the ultrasound probe. The envelop function g(t) is of short time duration and its components in the frequency domain are at significantly lower frequency than the centre frequency. It is assumed that the envelop function g(t) satisfies This assumption has been verified to be feasible by the ve­ locities with different shapes of envelope such as sinusoidal function and Gaussian-shaped sinusoidal function. The en­ velop function based formulation of ultrasound echo process has been recently applied to process clinical ultrasound data produced excellent imaging result [11]. 359 IV. NEW CONVOLUTION MODEL OF ULTRASOUND ECHO SIGNALS This section provides a brief outline of the derivation of the proposed convolution model. Incorporating the conditions and results provided in Section III, the integral solution for Fps(r],ro,t,t]) in (3) is S c(rt)po n2 fJ ( cot- rl )--- --g --- x c6 r] Co cot- r] cos(2n fo(-- )) ·Co (14) This is used to solve the function ps(ro+ra,t) in (1) resulting in r r (_ 2c(rJ) d2Pi(rl,t]) ) X iviT c6 d� 8(t- t _ Iro+ra-r11)I co 3 � ---::---,----'-"-.,---dtldr]. (1S) 2nlro+ra- r]1 With the original acoustical model (1)-(S) and all the derivations before, ultrasound echo signals are mainly de­ termined by ultrasound perturbation of propagation velocity and the effects from the gradient of tissue perturbation of mass density are negligible. Hence, ultrasound echo signals can be approximated as Pr(ro,t)=Em(t) * rp,(ro+ra,t)d2ra, (16) tis where * denotes the convolution over time and I r r [_ 2c(rt) d2Pi(rl,tI) ] iviT c6 dt2 8(t- tl _ Ira+ro-r11) ---'-::--;-----' c"'- o -..,--'- dtld3rl. (17) 2nlra+ro- r]1 As a result, the echo signal can be modeled as the following convolution model where * denotes the spatial convolution, r PO d3V(t) vcc(t)=2" ----=;--: Co ot is the pulse-echo wavelet of the excitation, c(rl) fcc(rl)=--- Co (1S) accounts for the inhomogeneities in the tissue due to propa­ gation velocity perturbations which give rise to the scattered signal, and hpe is the modified pulse-echo spatial impulse response that relates the transducer geometry to the spatial extent of the scattered field. In the proposed convolution model (1S), the tissue pa­ rameter is represented in terms of the perturbation of the
    • propagation velocity c(rl) which can be a distribution of the tissue characteristic to be estimated for image construction. In practice, it may be comparatively more convenient to represent c(rl) in terms of the perturbation of tissue mass density p(rl) using relation (11) between the two physical parameters. We have conducted extensive simulation to verify our developed convolution model and the results obtained can verify its feasibility and effectiveness. Remark: The convolution model (18) is derived based on the conditions and approximations described in Section III. This model appears in a similar form as the conventional model in (ref[5]) which has been popularly known for a long time since presented. We would point out that our recent and careful studies have found that there exist sub­ stantial mathematical problems in the original derivations of the convolution model in [5], so this model is not fully theoretically sound and valid. Consequently, the techniques and algorithms for ultrasound echo signal processing and imaging based on such convolution model no longer have a theoretical foundation. Details of our finding and analysis will be further reported. V. CONCLUSION This paper has presented a new convolution model for medical ultrasound echo signals. It is based on careful studies of the classical ultrasound acoustics and traditional convolution model. The proposed model takes the acoustical and practical conditions of medical ultrasound into account and evaluates the echo effects arising from different anatom­ ical tissue properties. In reasonable clinical circumstances, ultrasound echo signals can be characterized in terms of the perturbation of the ultrasound propagation velocity or mass density of the tissue. With the new model, the convolution framework of ultrasound echo signal processes remains valid despite the mathematical flaw in the derivation of the con­ ventional convolution model. REFERENCES [1] Morse, Philip McCord. Theorelical AcouSlics. McGraw-Hill, 1968. [2] Angelsen, Bjorn. Ullrasound imaging waves, signals, and signal processing, Emantec, 2000. [3] Hundt E E and Trautenberg E A, "Digital processing of ultrasonic data by deconvolution", IEEE Trans. Sonics Ultrasonics, Vo1.27, pp.249- 252, 1980. [4] Gore, J.e. and Leeman, S. "Ultrasonic backscattering from human tissue: a realistic model" , Physics in Medicine and Biology, Vo1.22, No.2, pp.317-326, 1977. [5] Jensen J. A. "A model for the propagation and scattering of ultrasound in tissue", J. Acoust. Soc. Am., pp:182-192, 1991. [6] Jensen J. A. , D. Gandhi, and W. D. O'Brien. "Ultrasound fields in an attenuating medium", In Proc. IEEE Ultrason. Symp., pp.943-946, 1993. [7] Jensen J.A., "A Program for Simulating Ultrasound Systems", Medical and Biological Engineering and Computing, Vol.34, pp.351C352, 1996. [8] Taxt T., "Two-dimensional noise-robust blind deconvolution of ul­ trasound images", IEEE Trans. on Ultrasonics, Ferroelectrics and Frequency Control, Vo1.48, Issue:4, pp.861-866, 2001. [9] Michailovich, O.v. and Adam, D., "A novel approach to the 2-D blind deconvolution problem in medical ultrasound", IEEE Trans. on Medical Imaging, Vo1.24, pp.86-104, 2005 360 [10] Yeoh, Wee Sonn and Cishen Zhang. "Constrained Least Squares Fil­ tering Algorithm for Ultrasound Image Deconvolution", IEEE Trans. on Biomedical Engineering, Vo1.53, Issue:10, pp.2001-2007, 2006. [11] Yu, Chengpu, Cishen Zhang and Lihua Xie, An Envelope Signal Based Deconvolution Algorithm for Ultrasound Imaging, to appear Singal Processing, March 2012.