Ultrasound imaging and its modeling

Transcription

1 Preprint of: Ultrasound imaging and its modeling Chapter in. Fink et al. (Eds.): Imaging of Complex Media with Aousti and Seismi Waves, Topis in Applied Physis, vol. 84, pp , Springer Verlag, 2002 Jørgen Arendt Jensen Department of Information Tehnology, Build. 344 Tehnial University of Denmark, DK-2800 Lyngby, Denmark Published by Springer Verlag,

2 Contents 1 Ultrasound imaging and its modeling Fundamental ultrasound imaging Imaging with arrays Fousing Ultrasound fields Derivation of Fourier relation Beam patterns Spatial impulse responses Fields in linear aousti systems Basi theory Geometri onsiderations Calulation of spatial impulse responses Examples of spatial impulse responses Pulse-eho fields Fields from array transduers Examples of ultrasound fields Summary

3 Chapter 1 Ultrasound imaging and its modeling Modern medial ultrasound sanners are used for imaging nearly all soft tissue strutures in the body. The anatomy an be studied from gray-sale B-mode images, where the refletivity and sattering strength of the tissues are displayed. The imaging is performed in real time with 20 to 100 images per seond. The tehnique is widely used, sine it does not use ionizing radiation and is safe and painless for the patient. This hapter gives a short introdution to modern ultrasound imaging using array transduers. It inludes a desription of the different imaging methods, the beamforming strategies used, and the resulting fields and their modeling. 1.1 Fundamental ultrasound imaging The main units of a modern B-mode imaging system is shown in Fig A multi-element transduer is used for both transmitting and reeiving the pulsed ultrasound field. The entral frequeny of the transduer an be from 2 to 15 MHz depending on the use. Often advaned omposite materials are used in the transduer, and they an attain a relative bandwidth in exess of 100 %. The resolution is, thus, on the order of one to three wavelengths. The mean speed of sound in the tissue investigated varies from 1446 m/s (fat) to 1566 m/s (spleen) [1, 2], and an averaged value of 1540 m/s is used in the sanners. This gives an wavelength of mm and a resolution in the axial diretion of 0.3 to 1 mm. The emission of the beam is ontrolled eletronially as desribed in Setion 1.3, and is for a phased array system swept over the region of interest in a polar san. A single fous an be used in transmit, and the user an selet the depth of the fous. The refleted and sattered field is then reeived by the transduer again and amplified by the time gain ompensation amplifier. This ompensates for the loss in amplitude due to the attenuation experiened during propagation of the sound field in the tissue. Typial attenuation values are shown in Table 1.1. A typial 3

4 value used, when designing the sanner, is 0.7 db/[mhz m], indiating that the attenuation inreases exponentially with both depth and frequeny. This is the one-way attenuation and a 5 MHz wave measured at a depth of 10 m would, thus, be attenuated 70 db. After amplifiation the signals from all transduer elements are passed to the eletroni beamformer, that fouses the reeived beam. For low-end sanners this is done through analog delay lines, whereas more modern high-end sanners employ digital signal proessing on a sampled version of the signal from all elements. Hereby a ontinuous fous an be attained giving a very high resolution image. Often the beamformer an handle 64 to 192 transduer elements, and this is the typial element ount in modern sanners. There is a ontinuous effort to expand the number of hannels to improve image resolution and ontrast. The beamformed signal is envelope deteted and stored in a memory bank. A san onversion is then performed to finally show the ultrasound image on a gray-level display in real time. The images an over an area of 15 by 15 m, and a single pulse-eho line then takes /1540 = 195µs to aquire. Sine an image onsists of roughly 100 polar lines, this gives a frame rate of 51 images a seond. Often smaller images are seleted to inrease the frame rate, espeially for blood veloity imaging [3]. A typial ultrasound image is shown in Fig. 1.2 for a fetus in the 13th week. The head, mouth, legs and the spine is learly identified in the image. It is also seen that the image has a grainy appearane, and that there are no lear demarations or refletion between the plaenta and the amnioti fluid surrounding the fetus. There are, thus, no distint refletions from plane boundaries as they seldom exist in the body. It is the ultrasound field sattered by the onstituents of the tissue that is displayed in medial ultrasound, and the medial sanners are optimized to display the sattered signal. The sattered field emanates from small hanges in density, ompressibility, and absorption from the onnetive tissue, ells, and fibrous tissue. These strutures are muh smaller than one wavelength of the ultrasound, and the resulting spekle pattern displayed does not diretly reveal physial struture. It is rather the onstrutive and destrutive interferene of sattered signals from all the small strutures. So it is not possible to visualize and diagnose mirostruture, but the strength of the signal is an indiation of pathology. A strong signal from liver tissue, making a bright image, is, e.g., an indiation of a fatty or irrhoti liver. As the sattered wave emanates from numerous ontributors, it is appropriate to haraterize it in statistial terms. The amplitude distribution follows a Gaussian distribution [4], and is, thus, fully haraterized by its mean and variane. The mean value is zero sine the sattered signal is generated by differenes in the tissue from the mean aousti properties. Sine the baksattered signal depends on the onstrutive and destrutive interferene of waves from numerous small tissue strutures, it is not meaningful to talk about the refletion strength of the individual strutures. Rather, it is the deviations from the mean density and speed of sound within the tissue and the omposition of the tissue that determine the strength of the returned signal. The magnitude of the returned signal is, therefore, desribed in terms of the 4

5 power of the sattered signal. Sine the small strutures re-radiate waves in all diretions and the sattering strutures might be ordered in some diretion, the returned power will, in general, be dependent on the relative position between the ultrasound emitter and reeiver. Suh a medium is alled anisotropi, examples of whih are musle and kidney tissue. By omparison, liver tissue is a fairly isotropi sattering medium, when its major vessels are exluded, and so is blood. 1.2 Imaging with arrays Basially there are three different kinds of images aquired by multi-element array transduers, i.e. linear, onvex, and phased as shown in Figures 1.3, 1.5, and 1.6. The linear array transduer is shown in Fig It selets the region of investigation by firing a set of elements situated over the region. The beam is moved over the imaging region by firing sets of ontiguous elements. Fousing in transmit is ahieved by delaying the exitation of the individual elements, so an initially onave beam shape is emitted, as shown in Fig The beam an also be foused during reeption by delaying and adding responses from the different elements. A ontinuous fous or several foal zones an be maintained as explained in Setion 1.3. Only one foal zone is possible in transmit, but a omposite image using a set of foi from several transmissions an be made. Often 4 to 8 zones an be individually plaed at seleted depths in modern sanners. The frame rate is then lowered by the number of transmit foi. The linear arrays aquire a retangular image, and the arrays an be quite large to over a suffiient region of interest (ROI). A larger area an be sanned with a smaller array, if the elements are plaed on a onvex surfae as shown in Fig A setor san is then obtained. The method of fousing and beam sweeping during transmit and reeive is the same as for the linear array, and a substantial number of elements (often 128 or 256) is employed. The onvex and linear arrays are often too large to image the heart when probing between the ribs. A small array size an be used and a large field of view attained by using a phased array as shown in Fig All array elements are used here both during transmit and reeive. The diretion of the beam is steered by eletrially delaying the signals to or from the elements, as shown in Fig. 1.4b. Images an be aquired through a small window and the beam rapidly sweeped over the ROI. The rapid steering of the beam ompared to mehanial transduers is of espeial importane in flow imaging [3]. This has made the phased array the hoie for ardiologial investigations through the ribs. More advaned arrays are even being introdued these years with the inrease in number of elements and digital beamforming. Espeially elevation fousing (out of the imaging plane) is important. A urved surfae as shown in Fig. 1.7 is used for obtaining the elevation fousing essential for an improved image quality. Eletroni beamforming an also be used in the ele- 5

6 vation diretion by dividing the elements in the elevation diretion. The elevation fousing in reeive an then be dynamially ontrolled for e.g. the array shown in Fig Fousing The essene of fousing an ultrasound beam is to align the pressure fields from all parts of the aperture to arrive at the field point at the same time. This an be done through either a physially urved aperture, through a lens in front of the aperture, or by the use of eletroni delays for multi-element arrays. All seek to align the arrival of the waves at a given point through delaying or advaning the fields from the individual elements. The delay (positive or negative) is determined using ray aoustis. The path length from the aperture to the point gives the propagation time and this is adjusted relative to some referene point. The propagation time t i from the enter of the aperture element to the field point is t i = 1 (x i x f ) 2 + (y i y f ) 2 + (z i z f ) 2 (1.1) where (x f,y f,z f ) is the position of the foal point, (x i,y i,z i ) is the enter for the physial element number i, and is the speed of sound. A point is seleted on the whole aperture as a referene for the imaging proess. The propagation time for this is t = 1 (x x f ) 2 + (y y f ) 2 + (z z f ) 2 (1.2) where (x,y,z ) is the referene enter point on the aperture. The delay to use on eah element of the array is then t i = 1 ( ) (x x f ) 2 + (y y f ) 2 + (z z f ) 2 (x i x f ) 2 + (y i y f ) 2 + (z i z f ) 2 (1.3) Notie that there is no limit on the seletion of the different points, and the beam an, thus, be steered in a preferred diretion. The arguments here have been given for emission from an array, but they are equally valid during reeption of the ultrasound waves due to aousti reiproity. At reeption it is also possible to hange the fous as a funtion of time and thereby obtain a dynami traking fous. This is used by all modern ultrasound sanners, Beamformers based on analog tehnology makes it possible to reate several reeive foi and the newer digital sanners hange the fousing ontinuously for every depth in reeive. The fousing an, thus, be defined through time lines as: 6

7 From time Fous at 0 x 1,y 1,z 1 t 1 x 1,y 1,z 1 t 2 x 2,y 2,z 2.. For eah foal zone there is an assoiated foal point and the time from whih this fous is used. The arrival time from the field point to the physial transduer element is used for deiding whih fous is used. Another possibility is to set the fousing to be dynami, so that the fous is hanged as a funtion of time and thereby depth. The fousing is then set as a diretion defined by two angles and a starting point on the aperture. Setion 1.4 shows that the side and grating lobes of the array an be redued by employing apodization of the elements. Again a fixed funtion an be used in transmit and a dynami funtion in reeive defined by: From time Apodize with 0 a 1,1,a 1,2, a 1,Ne t 1 a 1,1,a 1,2, a 1,Ne t 2 a 2,1,a 2,2, a 2,Ne t 3. a 3,1,a 3,2, a 3,Ne. Here a i, j is the amplitude saling value multiplied onto element j after time instane t i. Typially a Hamming or Gaussian shaped funtion is used for the apodization. In reeive the width of the funtion is often inreased to ompensate for attenuation effets and for keeping the point spread funtion roughly onstant. The F-number defined by F = D L (1.4) where L is the total width of the ative aperture and D is the distane to the fous, is often kept onstant. More of the aperture is often used for larger depths and a ompensation for the attenuation is thereby partly made. An example of the use of dynami apodization is given in Setion Ultrasound fields This setion derives a simple relation between the osillation of the transduer surfae and the ultrasound field. It is shown that field in the far-field an be found by a simple one-dimensional 7

8 Fourier transform of the one-dimensional aperture pattern. This might seem far from the atual imaging situation in the near field using pulsed exitation, but the approah is very onvenient in introduing all the major onepts like main and side lobes, grating lobes, et. It also very learly reveals information about the relation between aperture properties and field properties Derivation of Fourier relation Consider a simple line soure of length L as shown in Fig. 1.9 with a harmoni partile speed of U 0 exp( jωt). Here U 0 is the vibration amplitude and ω is its angular frequeny. The line element of length dx generates an inrement in pressure at r of [5] d p = j ρ 0k 4πr U 0a p (x)e j(ωt kr ) dx, (1.5) where ρ 0 is density, is speed of sound, k = ω/ is the wavenumber, and a p (x) is an amplitude saling of the individual parts of the aperture. In the far-field (r L) the distane from the radiator to the field points is (see Fig. 1.9): r = r xsinθ (1.6) The emitted pressure is found by integrating over all the small elements of the aperture p(r,θ,t) = j ρ 0U 0 k 4π Z + a p (x) e j(ωt kr ) r dx. (1.7) Notie that a p (x) = 0 if x > L/2. Here r an be replaed with r, if the extent of the array is small ompared to the distane to the field point (r L). Using this approximation and inserting (1.6) in (1.7) gives p(r,θ,t) = j ρ 0U 0 k 4πr Z + Z + e j(ωt kr) a p (x)e jkxsinθ dx, (1.8) sine ωt and kr are independent of x. Hereby the pressure amplitude of the field for a given frequeny an be split into two fators: a p (x)e j(ωt kr+kxsinθ) dx = j ρ 0U 0 k 4πr P ax (r) = ρ 0U 0 kl 4πr Z + H(θ) = 1 a p (x)e jkxsinθ dx (1.9) L P(r,θ) = P ax (r)h(θ) The first fator P ax (r) haraterizes how the field drops off in the axial diretion as a fator of distane, and H(θ) gives the variation of the field as a funtion of angle. The first term drops 8

9 off with 1/r as for a simple point soure and H(θ) is found from the aperture funtion a p (x). A slight rearrangement gives 1 H(θ) = 1 L Z + a p (x)e This very losely resembles the standard Fourier integral given by sinθ j2πx f dx = 1 Z + a p (x)e j2πx f dx. (1.10) L G( f ) = g(t) = Z + Z + g(t)e j2πt f dt G( f )e j2πt f d f (1.11) There is, thus, a Fourier relation between the radial beam pattern and the aperture funtion, and the normal Fourier relations an be used for understanding the beam patterns for typial apertures Beam patterns The first example is for a simple line soure, where the aperture funtion is onstant suh that { 1 x L/2 a p (x) = 0 else (1.12) The angular fator is then H(θ) = sin(πl f sinθ ) πl f sinθ = sin( k 2 Lsinθ) k 2 Lsinθ (1.13) A plot of the sin funtion is shown in Fig A single main lobe an be seen with a number of side lobe peaks. The peaks fall off proportionally to k or f. The angle of the first zero in the funtion is found at sinθ = L f = λ L. (1.14) The angle is, thus, dependent on the frequeny and the size of the array. A large array or a high emitted frequeny, therefore, gives a narrow main lobe. The magnitude of the first sidelobe relative to the mainlobe is given by H(arsin( 3 2L f )) H(0) = L sin(3π/2) /L = 2 3π/2 3π (1.15) 1 The term 1/L is inluded to make H(θ) a unit less number. 9

10 The relative sidelobe level is, thus, independent of the size of the array and of the frequeny, and is solely determined by the aperture funtion a p (x) through the Fourier relation. The large disontinuities of a p (x), thus, give rise to the high side lobe level, and they an be redued by seleting an aperture funtion that is smoother like a Hanning window or a Gaussian shape. Modern ultrasound transduers onsist of a number of elements eah radiating ultrasound energy. Negleting the phasing of the element (see Setion 1.3) due to the far-field assumption, the aperture funtion an be desribed by a p (x) = a ps (x) N/2 n= N/2 δ(x d x n), (1.16) where a ps (x) is the aperture funtion or apodization for the individual elements, d x is the spaing (pith) between the enters of the individual elements, and N + 1 is the number of elements in the array. Using the Fourier relationship the angular beam pattern an be desribed by where N/2 δ(x d x n) H per (θ) = n= N/2 Summing the geometri series gives H p (θ) = H ps (θ)h per (θ), (1.17) N/2 e jnd xk sinθ = n= N/2 H per (θ) = sin( (N + 1) k 2 d x sinθ ) N/2 f sinθ j2π e nd x. (1.18) n= N/2 sin ( k 2 d x sinθ ), (1.19) whih is the Fourier transform of a series of delta funtions. This funtion repeats itself with a period that is a multiple of π = k 2 d x sinθ sinθ = 2π kd x = λ d x. (1.20) This repetitive funtion gives rise to the grating lobes in the field. An example is shown in Fig The grating lobes are due to the periodi nature of the array, and orresponds to sampling of a ontinuous time signal. The grating lobes will be outside a ±90 deg. imaging area if λ d x = 1 d x = λ (1.21) 10

11 Often the beam is steered in a diretion and in order to ensure that grating lobes do not appear in the image, the spaing or pith of the elements is seleted to be d x = λ/2. This also inludes ample margin for the modern transduers that often have a very broad bandwidth. An array beam an be steered in a diretion by applying a time delay on the individual elements. The differene in arrival time between elements for a given diretion θ 0 is τ = d x sinθ 0 Steering in a diretion θ 0 an, therefore, be aomplished by using (1.22) sinθ 0 = τ d x (1.23) where τ is the delay to apply to the signal on the element losest to the enter of the array. A delay of 2τ is then applied on the seond element and so forth. The beam pattern for the grating lobe is then replaed by ( ( )) sin (N + 1) 2 k d x sinθ d τ x H per (θ) = ( ( )). (1.24) sin k2 d x sinθ d τ x Notie that the delay is independent of frequeny, sine it is essentially only determined by the speed of sound. 1.5 Spatial impulse responses The desription in the last setion is stritly only valid for the far-field, ontinuous wave ase, whereas the fields employed in medial ultrasound are pulsed and in the near field. A more aurate and general solution is, thus, needed, and this is developed in this setion. The approah is based on the onept of spatial impulse responses developed by Tupholme [6] and Stepanishen [7, 8] Fields in linear aousti systems It is a well known fat in eletrial engineering that a linear eletrial system is fully haraterized by its impulse response. Applying a delta funtion to the input of the iruit and measuring its output haraterizes the system. The output y(t) to any kind of input signal x(t) is then given by y(t) = h(t) x(t) = Z + 11 h(θ)x(t θ)dθ, (1.25)

12 where h(t) is the impulse response of the linear system and denotes time onvolution. The transfer funtion of the system is given by the Fourier transform of the impulse response and haraterizes the systems amplifiation of a time-harmoni input signal. The same approah an be taken to haraterize a linear aousti system. The basi set-up is shown in Fig The aousti radiator (transduer) on the left is mounted in a infinite, rigid baffle and its position is denoted by r 2. It radiates into a homogeneous medium with a onstant speed of sound and density ρ 0 throughout the medium. The point denoted by r 1 is where the aousti pressure from the transduer is measured by a small point hydrophone. A voltage exitation of the transduer with a delta funtion will give rise to a pressure field that is measured by the hydrophone. The measured response is the aousti impulse response for this partiular system with the given set-up. Moving the transduer or the hydrophone to a new position will give a different response. Moving the hydrophone loser to the transduer surfae will often inrease the signal 2, and moving it away from the enter axis of the transduer will often diminish it. Thus, the impulse response depends on the relative position of both the transmitter and reeiver ( r 2 r 1 ) and hene it is alled a spatial impulse response. A pereption of the sound field for a fixed time instane an be obtained by employing Huygens priniple in whih every point on the radiating surfae is the origin of an outgoing spherial wave. This is illustrated in Fig Eah of the outgoing spherial waves are given by ( ) ( ) δ t r 2 r 1 δ t r p s ( r 1,t) = k p = k p (1.26) r 2 r 1 r where r 1 indiates the point in spae, r 2 is the point on the transduer surfae, k p is a onstant, and t is the time for the snapshot of the spatial distribution of the pressure. The spatial impulse response is then found by observing the pressure waves at a fixed position in spae over time by having all the spherial waves pass the point of observation and summing them. Being on the aoustial axis of the transduer gives a short response whereas an off-axis point yields a longer impulse response as shown in Fig Basi theory In this setion the exat expression for the spatial impulse response will more formally be derived. The basi setup is shown in Fig The triangular shaped aperture is plaed in an infinite, rigid baffle on whih the veloity normal to the plane is zero, exept at the aperture. The field point is denoted by r 1 and the aperture by r 2. The pressure field generated by the 2 This is not always the ase. It depends on the fousing of the transduer. Moving loser to the transduer but away from its fous will derease the signal. 12

13 aperture is then found by the Rayleigh integral [9] p( r 1,t) = ρ Z 0 2π S v n ( r 2,t r 1 r 2 ) t r 1 r 2 ds, (1.27) where v n is the veloity normal to the transduer surfae. The integral is a statement of Huygens priniple that the field is found by integrating the ontributions from all the infinitesimally small area elements that make up the aperture. This integral formulation assumes linearity and propagation in a homogeneous medium without attenuation. Further, the radiating aperture is assumed flat, so no re-radiation from sattering and refletion takes plae. Exhanging the integration and the partial derivative, the integral an be written as Z p( r 1,t) = ρ 0 S 2π v n ( r 2,t r 1 r 2 r 1 r 2 t ) ds. (1.28) It is onvenient to introdue the veloity potential ψ that satisfies the equations [10] v( r,t) = ψ( r,t) ψ( r,t) p( r,t) = ρ 0. (1.29) t Then only a salar quantity need to be alulated and all field quantities an be derived from it. The surfae integral is then equal to the veloity potential: Z ψ( r 1,t) = S v n ( r 2,t r 1 r 2 ) ds (1.30) 2π r 1 r 2 The exitation pulse an be separated from the transduer geometry by introduing a time onvolution with a delta funtion as where δ is the Dira delta funtion. Z Z v n ( r 2,t 2 )δ(t t 2 r 1 r 2 ) ψ( r 1,t) = dt 2 ds, (1.31) S T 2π r 1 r 2 Assume now that the surfae veloity is uniform over the aperture making it independent of r 2, then: Z δ(t r 1 r 2 ) ψ( r 1,t) = v n (t) ds, (1.32) S 2π r 1 r 2 where denotes onvolution in time. The integral in this equation Z h( r 1,t) = S δ(t r 1 r 2 ) ds (1.33) 2π r 1 r 2 13

14 is alled the spatial impulse response and haraterizes the three-dimensional extent of the field for a partiular transduer geometry. Note that this is a funtion of the relative position between the aperture and the field. Using the spatial impulse response the pressure is written as p( r 1,t) = ρ 0 v n (t) t h( r 1,t) (1.34) whih equals the emitted pulsed pressure for any kind of surfae vibration v n (t). The ontinuous wave field an be found from the Fourier transform of (1.34). The reeived response for a olletion of satterers an also be found from the spatial impulse response [11], [12]. Thus, the alulation of the spatial impulse response makes it possible to find all ultrasound fields of interest Geometri onsiderations The alulation of the spatial impulse response assumes linearity and any omplex-shaped transduer an therefore be divided into smaller apertures and the response an be found by adding the responses from the sub-apertures. The integral is, as mentioned before, a statement of Huygens priniple of summing ontributions from all areas of the aperture. An alternative interpretation is found by using the aousti reiproity theorem [5]. This states that: If in an unhanging environment the loations of a small soure and a small reeiver are interhanged, the reeived signal will remain the same. Thus, the soure and reeiver an be interhanged. Emitting a spherial wave from the field point and finding the wave s intersetion with the aperture also yields the spatial impulse response. The situation is depited in Fig. 1.15, where an outgoing spherial wave is emitted from the origin of the oordinate system. The dashed urves indiate the irles from the projeted spherial wave. The alulation of the impulse response is then failitated by projeting the field point onto the plane of the aperture. The task is thereby redued to a two-dimensional problem and the field point is given as a (x,y) oordinate set and a height z above the plane. The three-dimensional spherial waves are then redued to irles in the x y plane with the origin at the position of the projeted field point as shown in Fig The spatial impulse response is, thus, determined by the relative length of the part of the ar that intersets the aperture. Thereby it is the rossing of the projeted spherial waves with the edges of the aperture that determines the spatial impulse responses. This fat is used for deriving equations for the spatial impulse responses in the next setion. 14

15 1.5.4 Calulation of spatial impulse responses The spatial impulse response is found from the Rayleigh integral derived earlier Z h( r 1,t) = S δ(t r 1 r 2 ) ds (1.35) 2π r 1 r 2 The task is to projet the field point onto the plane oiniding with the aperture, and then find the intersetion of the projeted spherial wave (the irle) with the ative aperture as shown in Fig Rewriting the integral into polar oordinates gives: h( r 1,t) = Z Θ2 Z d2 Θ 1 δ(t R ) d 1 2πR r dr dθ (1.36) where r is the radius of the projeted irle and R is the distane from the field point to the aperture given by R 2 = r 2 + z 2 p. Here z p is the field point height above the x y plane of the aperture. The projeted distanes d 1,d 2 are determined by the aperture and are the distane losest to and furthest away from the aperture, and Θ 1,Θ 2 are the orresponding angles for a given time (see Fig. 1.17). Introduing the substitution 2RdR = 2rdr gives h( r 1,t) = 1 2π Z Θ2 Z R2 Θ 1 δ(t R ) dr dθ (1.37) R 1 The variables R 1 and R 2 denote the edges losest to and furthest away from the field point. Finally using the substitution t = R/ gives h( r 1,t) = 2π Z Θ2 Z t2 Θ 1 t 1 δ(t t )dt dθ (1.38) For a given time instane the ontribution along the ar is onstant and the integral gives h( r 1,t) = Θ 2 Θ 1 (1.39) 2π when assuming the irle ar is only interseted one by the aperture. The angles Θ 1 and Θ 2 are determined by the intersetion of the aperture and the projeted spherial wave, and the spatial impulse response is, thus, solely determined by these intersetions, when no apodization of the aperture is used. The response an therefore be evaluated by keeping trak of the intersetions as a funtion of time. 15

16 1.5.5 Examples of spatial impulse responses The first example shows the spatial impulse responses from a 3 5 mm retangular element for different spatial positions 5 mm from the front fae of the transduer. The responses are found from the enter of the retangle (y = 0) and out in steps of 2 mm in the x diretion to 6 mm away from the enter of the retangle. A shemati diagram of the situation is shown in Fig for the on-axis response. The impulse response is zero before the first spherial wave reahes the aperture. Then the response stays onstant at a value of. The first edge of the aperture is met, and the response drops of. The derease with time is inreased, when the next edge of the aperture is reahed and the response beomes zero when the projeted spherial waves all are outside the area of the aperture. A plot of the results for the different lateral field positions is shown in Fig It an be seen how the spatial impulse response hanges as a funtion of relative position to the aperture. The seond example shows the response from a irular, flat transduer. Two different ases are shown in Fig The top graph shows the traditional spatial impulse response when no apodization is used, so that the aperture vibrates as a piston. The field is alulated 10 mm from the front fae of the transduer starting at the enter axis of the aperture. Twenty-one responses for lateral distane of 0 to 20 mm off axis are then shown. The same alulation is repeated in the bottom graph, when a Gaussian apodization has been imposed on the aperture. The vibration amplitude is a fator of 1/exp(4) less at the edges of the aperture than at the enter. It is seen how the apodization redues some of the sharp disontinuities in the spatial impulse response, whih an redue the sidelobes of the field Pulse-eho fields The sattered field and reeived signal by the transduer an also be desribed using the spatial impulse response. The reeived signal from the transduer is [12]: p r ( r,t) = v pe (t) t f m ( r) r h pe ( r,t) (1.40) where r denotes spatial onvolution and t denotes temporal onvolution. v pe is the pulseeho impulse, whih inludes the transduer exitation and the eletro-mehanial impulse response during emission and reeption of the pulse. f m aounts for the inhomogeneities in the tissue due to density and speed of sound perturbations, whih give rise to the sattered signal. h pe is the pulse-eho spatial impulse response that relates the transduer geometry to the spatial extent of the sattered field. Expliitly written out these terms are: v pe (t) = ρ 2 2 E 3 v(t) m(t) t t 3, f m ( r 1 ) = ρ( r) ρ 16 2 ( r), h pe ( r,t) = h t ( r,t) h r ( r,t) (1.41)

17 Here ρ are the perturbations in density and in speed of sound, and h t ( r,t) and h r ( r,t) are the spatial impulse responses for the transmitting and reeiving apertures, respetively. E m (t) is the eletro-mehanial impulse response of the transduer during reeption. So the reeived response an be alulated by finding the spatial impulse response for the transmitting and reeiving transduer and then onvolving with the impulse response of the transduer. A single RF line in an image an be alulated by summing the response from a olletion of satterers in whih the sattering strength is determined by the density and speed of sound perturbations in the tissue. Homogeneous tissue an thus be made from a olletion of randomly plaed satterers with a sattering strength with a Gaussian distribution, where the variane of the distribution is determined by the baksattering ross-setion of the partiular tissue. 1.6 Fields from array transduers Most modern sanners use arrays for generating and reeiving the ultrasound fields. These fields are quite simple to alulate, when the spatial impulse response for a single element is known. This is the approah used in the Field II program [13], and this setion will extend the spatial impulse response to multi-element transduers and will elaborate on some of the features derived for the fields in Setion 1.4. Sine the ultrasound propagation is assumed to be linear, the individual spatial impulse responses an simply be added. If h e ( r p,t) denotes the spatial impulse response for the element at position r i and the field point r p, then the spatial impulse response for the array is h a ( r p,t) = assuming all N elements to be idential. N 1 h e ( r i, r p,t), (1.42) i=0 Let us assume that the elements are very small and the field point is far away from the array, so h e is a Dira funtion. Then h a ( r p,t) = k N 1 R p i=0 δ(t r i r p ) (1.43) when R p = r a r p, k is a onstant of proportionality, and r a is the position of the array. Thus, h a is a train of Dira pulses. If the spaing between the elements is d x, then h a ( r p,t) = k N 1 R p i=0 ( δ t r ) a + id x r e r p, (1.44) where r e is a unit vetor pointing in the diretion along the elements. The geometry is shown in Fig

18 The differene in arrival time between elements far from the transduer is t = d x sinθ. (1.45) The spatial impulse response is, thus, a series of Dira pulses separated by t. h a ( r p,t) k N 1 R p i=0 ( δ t R ) p i t. (1.46) The time between the Dira pulses and the shape of the exitation determines whether signals from individual elements add or anel out. If the separation in arrival times orresponds to exatly one or more periods of a sine wave, then they are in phase and add onstrutively. Thus, peaks in the response are found for n 1 f = d x sinθ. (1.47) The main lobe is found for Θ = 0 and the next maximum in the response is found for ( ) ( ) λ Θ = arsin = arsin. (1.48) f d x d x For a 3 MHz array with an element spaing of 1 mm, this amounts to Θ = 31, whih will be within the image plane. The reeived response is, thus, affeted by satterers positioned 31 off the image axis, and they will appear in the lines aquired as grating lobes. The first grating lobe an be moved outside the image plane, if the elements are separated by less than a wavelength. Usually, half a wavelength separation is desirable, as this gives some margin for a broad-band pulse and beam steering. The beam pattern as a funtion of angle for a partiular frequeny an be found by Fourier transforming h a )) H a ( f ) = k N 1 R p i=0 ( ( Rp exp j2π f + id x sinθ = exp( j2π f R p ) k R p = sin(π f d x sinθ N) sin(π f d x sinθ N 1 exp i=0 ( j2π f d x sinθ ) exp( jπ f (N 1)d x sinθ ) i (1.49) ) k R p exp( j2π f R p ). The terms exp( j2π f R p ) and exp( jπ f (N 1) d x sinθ ) are onstant phase shifts and play no role for the amplitude of the beam profile. Thus, the amplitude of the beam profile is k sin(nπ dx H a ( f ) = λ sinθ) R p sin(π d x λ sinθ), (1.50) 18

19 whih is onsistent with the previously derived result. Several fators hange the beam profile for real, pulsed arrays ompared with the analysis given here. First, the elements are not points, but rather are retangular elements with an off-axis spatial impulse response markedly different from a Dira pulse. Therefore, the spatial impulse responses of the individual elements will overlap and exat anellation or addition will not take plae. Seond, the exitation pulse is broad band, whih again influenes the sidelobes. The influene of these fators is shown in a set of simulations in the next setion. 1.7 Examples of ultrasound fields The field examples are generated using omputer phantoms and the Field II simulation program, that is based on the spatial impulse approah [14, 13]. The first syntheti phantom onsists of a number of point targets plaed with a distane of 5 mm starting at 15 mm from the transduer surfae. A linear sweep image of the points is then made and the resulting image is ompressed to show a 40 db dynami range. This phantom is suited for showing the spatial variation of the point spread funtion for a partiular transduer, fousing, and apodization sheme. Twelve examples using this phantom are shown in Fig The top graphs show imaging without apodization and the bottom graphs show images when a Hanning window is used for apodization in both transmit and reeive. A 128 elements transduer with a nominal frequeny of 3 MHz was used. The element height was 5 mm, the width was a wavelength and the kerf 0.1 mm. The exitation of the transduer onsisted of 2 periods of a 3 MHz sinusoid with a Hanning weighting, and the impulse response of both the emit and reeive aperture was also a two yle, Hanning weighted pulse. In the graphs A C, 64 of the transduer elements were used for imaging, and the sanning was done by translating the 64 ative elements over the aperture and fousing in the proper points. In graph D and E 128 elements were used and the imaging was done solely by moving the foal points. Graph A uses only a single foal point at 60 mm for both emission and reeption. B also uses reeption fousing at every 20 mm starting from 30 mm. Graph C further adds emission fousing at 10, 20, 40, and 80 mm. D applies the same foal zones as C, but uses 128 elements in the ative aperture. The fousing sheme used for E and F applies a new reeive profile for eah 2 mm. For analog beamformers this is a small zone size. For digital beamformers it is a large zone size. Digital beamformer an be programmed for eah sample and thus a ontinuous beamtraking an be obtained. In imaging systems fousing is used to obtain high detail resolution and high ontrast resolution preferably onstant for all depths. This is not possible, so ompromises must be made. As an example figure F shows the result for multiple transmit zones and reeive 19

20 zones, like E, but now a restrition is put on the ative aperture. The size of the aperture is ontrolled to have a onstant F-number (depth of fous in tissue divided by width of aperture), 4 for transmit and 2 for reeive, by dynami apodization. This gives a more homogeneous point spread funtion throughout the full depth. Espeially for the apodized version. Still it an be seen that the omposite transmit an be improved in order to avoid the inreased width of the point spread funtion at e.g. 40 and 60 mm. The next phantom onsists of a olletion of point targets, five yst regions, and five highly sattering regions. This an be used for haraterizing the ontrast-lesion detetion apabilities of an imaging system. The satterers in the phantom are generated by finding their random position within a mm ube, and then asribe a Gaussian distributed amplitude to the satterers. If the satterer resides within a yst region, the amplitude is set to zero. Within the highly sattering region the amplitude is multiplied by 10. The point targets has a fixed amplitude of 100, ompared to the standard deviation of the Gaussian distributions of 1. A linear san of the phantom was done with a 192 element transduer, using 64 ative elements with a Hanning apodization in transmit and reeive. The element height was 5 mm, the width was a wavelength and the kerf 0.05 mm. The pulses where the same as used for the point phantom mentioned above. A single transmit fous was plaed at 60 mm, and reeive fousing was done at 20 mm intervals from 30 mm from the transduer surfae. The resulting image for 100,000 satterers is shown in Fig A homogeneous spekle pattern is seen along with all the features of the phantom. 1.8 Summary Modern ultrasound sanners has attained a very high image quality through the use of digital beamforming. The delays on the individual transduer elements and their relative weight or apodization is hanged ontinuously as a funtion of depth. This yields near perfet foused images for all depths and has inreased the ontrast in the displayed image, thus, benefitting the diagnosti value of ultrasoni imaging. The development of the fousing strategies is nearly exlusively based on linear aoustis, and the high suess of the approah attest to the validity of using linear aoustis. It is, thus, appropriate to haraterize the medial ultrasound systems using linear aoustis. This hapter has developed a omplete linear desription of all the fields enountered in medial ultrasound. The various imaging methods were desribed, and then the onept of spatial impulse responses was developed. This ould be used for desribing both emitted and pulse-eho fields for both pulse emission and ontinuous wave systems using linear systems theory. Examples of the influene of digital beamforming and apodization were also shown. 20

21 Bibliography [1] S. A. Goss, R. L. Johnston, and F. Dunn. Comprehensive ompilation of empirial ultrasoni properties of mammalian tissues. J. Aoust. So. Am., 64: , [2] S. A. Goss, R. L. Johnston, and F. Dunn. Compilation of empirial ultrasoni properties of mammalian tissues II. J. Aoust. So. Am., 68:93 108, [3] J. A. Jensen. Estimation of Blood Veloities Using Ultrasound: A Signal Proessing Approah. Cambridge University Press, New York, [4] R. F. Wagner, S. W. Smith, J. M. Sandrik, and H. Lopez. Statistis of spekle in ultrasound B-sans. IEEE Trans. Son. Ultrason., 30: , [5] L. E. Kinsler, A. R. Frey, A. B. Coppens, and J. V. Sanders. Fundamentals of Aoustis. John Wiley & Sons, New York, third edition, [6] G. E. Tupholme. Generation of aousti pulses by baffled plane pistons. Mathematika, 16: , [7] P. R. Stepanishen. The time-dependent fore and radiation impedane on a piston in a rigid infinite planar baffle. J. Aoust. So. Am., 49: , [8] P. R. Stepanishen. Transient radiation from pistons in an infinte planar baffle. J. Aoust. So. Am., 49: , [9] A. D. Piere. Aoustis, An Introdution to Physial Priniples and Appliations. Aoustial Soiety of Ameria, New York, [10] P. M. Morse and K. U. Ingard. Theoretial Aoustis. MGraw-Hill, New York, [11] P. R. Stepanishen. Pulsed transmit/reeive response of ultrasoni piezoeletri transduers. J. Aoust. So. Am., 69: , [12] J. A. Jensen. A model for the propagation and sattering of ultrasound in tissue. J. Aoust. So. Am., 89: , 1991a. 21

22 [13] J. A. Jensen. Field: A program for simulating ultrasound systems. Med. Biol. Eng. Comp., 10th Nordi-Balti Conferene on Biomedial Imaging, Vol. 4, Supplement 1, Part 1: , 1996b. [14] J. A. Jensen and N. B. Svendsen. Calulation of pressure fields from arbitrarily shaped, apodized, and exited ultrasound transduers. IEEE Trans. Ultrason., Ferroele., Freq. Contr., 39: , [15] M. J. Haney and W. D. O Brien. Temperature dependeny of ultrasoni propagation properties in biologial materials. In J. F. Greenleaf, editor, Tissue Charaterization with Ultrasound. CRC Press, Boa Raton, Fla.,

23 Tables Attenuation Tissue db/[mhz m] Liver Kidney Spleen Fat Blood Plasma 0.01 Bone Table 1.1: Typial attenuation values for human tissue (assembled from the ompilation in [15]). 23

24 Figures Figure 1.1: Real-time B-mode ultrasound imaging system. 24

25 Figure 1.2: Ultrasound image of a 13th week fetus. The markers at the border of the image indiate one entimeter. Figure 1.3: Linear array transduer for obtaining a retangular ross-setional image. 25

26 Eletroni fousing Beam steering and fousing t Exitation pulses t Exitation pulses 0 Transduer elements 0 Transduer elements Beam shape Beam shape (a) (b) Figure 1.4: Eletroni fousing and steering of an ultrasound beam. Figure 1.5: Convex array transduer for obtaining a polar ross-setional image. 26

27 Figure 1.6: Phased array transduer for obtaining a polar ross-setional image using a transduer with a small foot-print. 5 z [mm] y [mm] y [mm] x [mm] 5 z [mm] x [mm] z [mm] y [mm] x [mm] Figure 1.7: Elevation foused onvex array transduer for obtaining a retangular rosssetional image, whih is foused in the out-of-plane diretion. The urvature in the elevation diretion is exaggerated in the figure for illustration purposes. 27

28 z [mm] y [mm] x [mm] y [mm] 5 z [mm] x [mm] z [mm] y [mm] x [mm] Figure 1.8: Elevation foused onvex array transduer with element division in the elevation diretion. The urvature in the elevation diretion is exaggerated in the figure for illustration purposes. Figure 1.9: Geometry for line aperture. 28

29 1 Beam pattern as a funtion of angle for L = 10 λ H(θ) [m] θ [deg] 1 Beam pattern as a funtion k sin(θ) for L = 10 λ H(θ) [m] k sin(θ) [rad/m] x 10 4 Figure 1.10: Angular beam pattern for a line aperture with a uniform aperture funtion as a funtion of angle (top) and as a funtion of k sin(θ) (bottom). Beam pattern for 8 element array of point soures 8 H(θ) [m] k sin(θ) [rad/m] Beam pattern for 8 element array. 1.5 λ element width and 2λ spaing Main lobe x 10 4 H(θ) [m] 6 4 Grating lobe Angular beam pattern for one element k sin(θ) [rad/m] x 10 4 Figure 1.11: Grating lobes for array transduer onsisting of 8 point elements (top) and of 8 elements with a size of 1.5λ (bottom). The pith (or distane between the elements) is 2λ. 29

30 Figure 1.12: A linear aousti system. Figure 1.13: Illustration of Huygens priniple for a fixed time instane. A spherial wave with a radius of r = t is radiated from eah point on the aperture. 30

31 Figure 1.14: Position of transduer, field point, and oordinate system. z y x Figure 1.15: Emission of a spherial wave from the field point and its intersetion of the aperture. 31

32 Field point y r 1 Aperture x r 2 Figure 1.16: Intersetion of spherial waves from the field point by the aperture, when the field point is projeted onto the plane of the aperture. 32

33 Figure 1.17: Definition of distanes and angles in the aperture plan for evaluating the Rayleigh integral. Spherial waves y h x t Aperture Figure 1.18: Shemati diagram of field from retangular element. 33

34 h [m/s] x Lateral distane [mm] Time [s] Figure 1.19: Spatial impulse response from a retangular aperture of 4 5 mm at for different lateral positions. 34

35 Response from irular, non apodized transduer h [m/s] x Lateral distane [mm] Time [s] Response from irular, Gaussian apodized transduer h [m/s] x Lateral distane [mm] Time [s] Figure 1.20: Spatial impulse response from a irular aperture. Graphs are shown without apodization of the aperture (top) and with a Gaussian apodization funtion (bottom). The radius of the aperture is 5 mm and the field is alulated 10 mm from the transduer surfae. 35

36 Figure 1.21: Geometry of linear array. 36

37 A B C D E F Axial distane [mm] Lateral distane [mm] A B C D E F Axial distane [mm] Lateral distane [mm] Figure 1.22: Point target phantom imaged for different set-up of transmit and reeive fousing and apodization. See text for an explanation of the set-up. 37

38 Axial distane [mm] Lateral distane [mm] Figure 1.23: Computer phantom with point targets, yst regions, and strongly refleting regions. 38