Linear description of ultrasound imaging systems

Transcription

1 Linear desription of ultrasound imaging systems Notes for the International Summer Shool on Advaned Ultrasound Imaging Tehnial University of Denmark July 5 to July 9, 1999 Release 1.01, June 29, 2001 Jørgen Arendt Jensen June 10, 1999 Ørsted DTU, Build. 348, Tehnial University of Denmark DK-2800 Lyngby, Denmark jaj@oersted.dtu.dk Web:

2 CONTENTS 1 Introdution 1 2 Desription of ultrasound fields Fields in linear aousti systems Basi theory Calulation of spatial impulse responses Apodization and soft baffle Examples of spatial impulse responses Calulation of the sattered signal Ultrasound imaging Fourier relation Fousing Fields from array transduers Imaging with arrays Simulation of ultrasound imaging Syntheti phantoms Anatomi phantoms Ultrasound systems for flow estimation Introdution Measurement of flow signals Calulation and display of the veloity distribution Making images of veloity Flow types Stationary paraboli flow Simulation of flow imaging systems i

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4 CHAPTER ONE Introdution These notes have been prepared for the international summer shool on advaned ultrasound imaging sponsored by The Danish Researh Aademy. The notes should be read in onjuntion with the notes prepared by Anderson and Trahey 1. The intended audiene is Ph.D. students working in medial ultrasound. A knowledge of general linear aoustis and signal proessing is assumed. The notes give a linear desription of general ultrasound imaging through the use of spatial impulse responses. It is shown in Chapter 2 how both the emitted and sattered fields for the pulsed and ontinuous wave ase an be alulated using this approah. Chapter 3 gives a brief overview of modern ultrasound imaging and how it is simulated using spatial impulse responses. Chapter 4 finally, gives a brief desription of both spetral and olor flow imaging systems and their modeling and simulation. Jørgen Arendt Jensen Tehnial University of Denmark June, M.E. Anderson and G.E. Trahey: A seminar on k-spae applied to medial ultrasound, Dept. of Biomedial Engineering, Duke University, June 21,

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6 CHAPTER TWO Desription of ultrasound fields This hapter gives a linear desription of aousti fields using spatial impulse responses. It is shown how both the pulsed emitted and sattered fields an be aurately derived using spatial impulse responses, and how attenuation and different boundary onditions an be inorporated. The hapter goes into some detail of deriving the different results and explaining their onsequene. Different examples for both simulated and measured fields are given. The hapter is based on the papers [1], [2] and [3] and on the book [4]. 2.1 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 as shown in Fig 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 + h(θ)x(t θ)dθ, (2.1) 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 1, and moving it away from the enter 1 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 Figure 2.1: Measurement of impulse response for a linear eletri system. 3

7 Figure 2.2: A linear aousti system. Figure 2.3: 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. 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 ( p s ( r 1,t) = δ t r ) ( 2 r 1 = δ t r ) (2.2) where r 1 indiates the point in spae, r 2 is the point on the transduer surfae, 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 derease the signal. 4 Chapter 2. Desription of ultrasound fields

8 2.2 Basi theory Figure 2.4: Position of transduer, field point, and oordinate system. In this setion the exat expression for the spatial impulse response will more formally be derived. The basi setup is shown in Fig The triangularly 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 aperture is then found by the Rayleigh integral [5] p( r 1,t) = ρ Z 0 2π S v n ( r 2,t r 1 r 2 ) t r 1 r 2 ds, (2.3) where v n is the veloity normal to the transduer surfae. The integral is a statement of Huyghens 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 p( r 1,t) = ρ 0 2π Z S v n ( r 2,t r 1 r 2 r 1 r 2 t ) It is onvenient to introdue the veloity potential ψ that satisfies the equations [6] v( r, t) = ψ( r, t) ds. (2.4) ψ( r,t) p( r,t) = ρ 0. (2.5) 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 ) 2π r 1 r 2 ds (2.6) 2.2. Basi theory 5

9 The exitation pulse an be separated from the transduer geometry by introduing a time onvolution with a delta funtion as Z Z v n ( r 2,t 2 )δ(t t 2 r 1 r 2 ) ψ( r 1,t) = dt 2 ds, (2.7) 2π r 1 r 2 where δ is the Dira delta funtion. S T Assume now that the surfae veloity is uniform over the aperture making it independent of r 2, then: ψ( r 1,t) = v n (t) where denotes onvolution in time. The integral in this equation h( r 1,t) = Z S Z S δ(t r 1 r 2 ) ds, (2.8) 2π r 1 r 2 δ(t r 1 r 2 ) ds (2.9) 2π r 1 r 2 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) (2.10) 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 (2.10). The reeived response for a olletion of satterers an also be found from the spatial impulse response [7], [1]. This is derived in Setion 2.6. 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 Huyghens priniple of summing ontributions from all areas of the aperture. An alternative interpretation is found by using the aousti reiproity theorem [8]. 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. 2.5, 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. 6 Chapter 2. Desription of ultrasound fields

10 z y x Figure 2.5: Emission of a spherial wave from the field point and its intersetion of the aperture. Field point y r 1 Aperture x r 2 Figure 2.6: Intersetion of spherial waves from the field point by the aperture, when the field point is projeted onto the plane of the aperture Basi theory 7

11 Figure 2.7: Definition of distanes and angles in the aperture plan for evaluating the Rayleigh integral. 2.3 Calulation of spatial impulse responses The spatial impulse response is found from the Rayleigh integral derived earlier h( r 1,t) = Z S δ(t r 1 r 2 ) ds (2.11) 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 (2.12) 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. 2.7). Introduing the substitution 2RdR = 2rdr gives h( r 1,t) = 1 2π Z Θ2 Z R2 Θ 1 δ(t R ) dr dθ (2.13) 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) = Z Θ2 Z t2 δ(t t )dt dθ (2.14) 2π Θ 1 t 1 For a given time instane the ontribution along the ar is onstant and the integral gives h( r 1,t) = Θ 2 Θ 1 (2.15) 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. 8 Chapter 2. Desription of ultrasound fields

12 2.3.1 A simple alulation proedure From the derivation in the last setion it an be seen that the spatial impulse response in general an be expressed as h( r 1,t) = N(t) [ ] 2π Θ (i) 2 (t) Θ(i) 1 (t) (2.16) i=1 where N(t) is the number of ar segments that rosses the boundary of the aperture for a given time and Θ (i) 2 (t),θ(i) 1 (t) are the assoiated angles of the ar. This was also noted by Stepanishen [9]. The alulation an, thus, be formulated as finding the angles of the aperture edge s intersetions with the projeted spherial wave, sorting the angles, and then summing the ar angles that belong to the aperture. Finding the intersetions an be done from the desription of the edges of the aperture. A triangle an be desribed by three lines, a retangle by four, and the intersetions are then found from the intersetions of the irle with the lines. This makes it possible to devise a general proedure for alulating spatial impulse responses for any flat, bounded aperture, sine the task is just to find the intersetions of the boundary with the irle. The spatial impulse response is alulated from the time the aperture first is interseted by a spherial wave to the time for the intersetion furthest away. The intersetions are found for every time instane and the orresponding angles are sorted. The angles lie in the interval from 0 to 2π. It is then found whether the ar between two angles belongs to the aperture, and the angle differene is added to the sum, if the ar segment is inside the aperture. This yields the spatial impulse response aording to Eq. (2.16). The approah an be desribed by the flow hart shown in Fig The only part of the algorithm speifi to the aperture is the determination of the intersetions and whether the point is inside the aperture. Setion shows how this is done for polygons, Setion for irles, and Setion for higher-order parametri boundaries. All the intersetions need not be found for all times. New intersetions are only introdued, when a new edge or orner of the aperture is met. Between times when two suh orners or edges are enountered the number of intersetions remains onstant and only intersetions, whih belong to points inside the aperture need to be found. Note that an aperture edge gives rise to a disontinuity in the spatial impulse response. Also testing whether the point is inside the aperture is often superfluous, sine this only needs to be found one after eah disontinuity in the response. These two observations an signifiantly redue the number of alulations, sine only the intersetions affeting the response are found. The proedure first finds the number of disontinuities. Then only intersetion influening the response are alulated between two disontinuity points. This an potentially make the approah faster than the traditional approah, where the response from a number of different retangles or triangles must be alulated Solution for polygons The boundary of any polygon an be defined by a set of bounding lines as shown in Fig The ative aperture is then defined as lying on one side of the line as indiated by the arrows, and a point on the aperture must be plaed orretly in relation to all lines. The test whether a point is on the aperture is thus to go through all lines and test whether the point lies in the ative half spae for the line, and stop if it is not. The point is inside the aperture, if it passes the test for all the lines. The intersetions are found from the individual intersetions between the projeted irle and the lines. They are determined from the equations for the projeted spherial wave and the line: r 2 = (x x 0 ) 2 + (y y 0 ) 2 y = αx + y 1 (2.17) r 2 = (t) 2 z 2 p 2.3. Calulation of spatial impulse responses 9

13 Figure 2.8: Flow hart for alulating the spatial impulse response. 10 Chapter 2. Desription of ultrasound fields

14 Figure 2.9: Definition of bounding lines for polygon transduer. The arrows indiate the half-planes for the ative aperture. Here (x 0,y 0 ) is the enter of the irle, α the slope of the line, and y 1 its interset with the y-axis. The intersetions are given from the solutions to: 0 = (1 + α 2 )x 2 + (2αy 1 2x 0 2y 0 α)x + (y y x 2 0 2y 0 y 1 r 2 ) = Ax 2 + Bx +C (2.18) D = B 2 4AC The angles are ( ) y y0 Θ = artan (2.19) x x 0 Intersetions between the line and the irle are only found if D > 0. A determinant D < 0 indiates that the irle did not interset the line. If the line has infinite slope, the solution is found from the equation: x = x 1 0 = y 2 2y 0 y + y (x 1 x 0 ) 2 r 2 (2.20) = A y 2 + B y +C in whih A,B,C replae A,B,C, respetively, and the solutions are found for y rather than x. Here x 1 is the line s intersetion with the x-axis. The times for disontinuities in the spatial impulse response are given by the intersetions of the lines that define the aperture s edges and by the minimum distane from the projeted field point to the lines. The minimum distane is found from a line passing through the field point that is orthogonal to the bounding line. The intersetion between the orthogonal line and the bounding line is: x = αy p + x p αy 1 α y = αx + y 1 (2.21) where (x p,y p,z p ) is the position of the field point. For an infinite slope line the solution is x = x 1 and y = y p. The orresponding time is: t i = (x x p ) 2 + (y y p ) 2 + z 2 p (2.22) The intersetions of the lines are also found, and the orresponding times are alulated by (2.22) and sorted in asending order. They indiate the start and end time for the response and the time points for disontinuities in the response Calulation of spatial impulse responses 11

15 Figure 2.10: Geometry for determining intersetions between irles. The top graph shows the geometry when the field point denoted by O 2 is outside the aperture, and the bottom graph when it is inside Solution for irular surfaes The other basi shape for a transduer apart from retangular shapes is the flat, round surfae used for single element piston transduers and annular arrays. For these the intersetions are determined by two irles as depited in Fig Here O 1 is the enter of the aperture with radius r a and the projeted spherial wave is entered at O 2 with radius r b (t) = (t) 2 z 2 p. The length h a (t) is given by [10, page 66] h a (t) = 2 p(t)(p(t) a)(p(t) r a )(p(t) r b (t)) a a = O 1 O 2 p(t) = a + r a + r b (t) 2 (2.23) 12 Chapter 2. Desription of ultrasound fields

16 In a oordinate system entered at O 1 and an x-axis in the O 1 O 2 diretion, the intersetions are at y = h a (t) (2.24) l = ± rb 2(t) h2 a(t) The sign for l depends on the position of the intersetions. A negative sign is used if the intersetions are for negative values of x, and positive sign is used for positive x positions. When the field point is outside the ative aperture the spatial impulse response is h( r 1,t) = Θ 2 Θ 1 = ( ) 2π π artan ha (t) l ( ) ha (t) Θ 2 = artan = Θ 1 l (2.25) It must be noted that a proper four-quadrant ar-tan should be used to give the orret response. An alternative formula is [11, page 19] ( h( r 1,t) = 2π arsin 2 ) p(t)(p(t) a)(p(t) r a )(p(t) r b (t)) rb 2(t) (2.26) = ( ) 2π arsin aha (t) rb 2(t) The start time t s for the response is found from and the response ends at the time t e when r a + r b (t) = O 1 O 2 t s = rb 2(t) + z2 p = ( O 1 O 2 r a ) 2 + z 2 p r b (t) = r a + O 1 O 2 t e = rb 2(t) + z2 p ( O 1 O 2 + r a ) 2 + z 2 p = (2.27) (2.28) When the field point is inside the aperture, the response is h( r 1,t) = for z p t thereafter the ar lying outside the aperture should be subtrated, so that The response ends when (r a O 1 O 2 ) 2 + z 2 p (2.29) h( r 1,t) = 2π Θ 2 Θ 1 (2.30) 2π r b (t) = r a + O 1 O 2 ( O 1 O 2 + r a ) 2 + z 2 p t e = (2.31) The determination of whih part of the ar that subtrats or adds to the response is determined by what the ative aperture is. One ring in an annular array an be defined as onsisting of an ative aperture outside a irle ombined with an ative aperture inside a irle for defining the inner and outer rim of the aperture. A irular aperture an also be ombined with a line for defining the ative area of a split aperture used for ontinuous wave probing Calulation of spatial impulse responses 13

17 2.3.4 Solution for a irular onave surfae For referene the expression for a onave transduer, whih is a type often used in medial ultrasonis, is given. A derivation of the solution an be found in [12] and [13]. The spatial impulse response is [12]: h ( r 1,t) = Region I Region II z < 0 z > 0 0 t < r 0 r 0 < t t < r 1 R r r 0 < t < r 1 r 2 < t < r 0 R 1 r π aros η(t) σ(t) r 1 < t < r 2 r 1 < t < r 2 r 1 < t < r 2 0 r 2 < t t < r 1 r 2 < t (2.32) where: { 1 d/r η(t) = R sinθ + 1 ( R 2 + r 2 2 t 2 )} tanθ 2rR ( R σ(t) = R r 2 2 t 2 ) 2 (2.33) 2rR r = r 1. The variables are defined in Fig Solution for other surfaes Analytial solutions for spatial impulse response have been derived for a number of different geometries by various authors. The response from a flat retangle an be found in [14, 15], and for a flat triangle in [16]) Solution for parametri surfaes For ellipses or other higher order parametri surfaes it is in general not easy to find analyti solutions for the spatial impulse response. The boundary method desribed above an, however, be used for providing a simple solution to the problem, sine the intersetions between the projeted spherial wave and the edge of the aperture uniquely determine the spatial impulse response. It is therefore possible to use root finding for a set of (non-linear) equations for finding these intersetions. The problem is to find when both the spherial wave and the aperture have rossing ontours in the plane of the aperture, i.e., when (t) 2 z 2 p (x x p ) 2 (y y p ) 2 = 0 (2.34) S(x, y) = 0 in whih S(x, y) = 0 defines the boundary of the aperture. The problem of numerially finding these roots is in general not easy, if a good initial guess on the position of the intersetions is not found [17, pages ]. Good initial values are however found here, sine the intersetions must lie on the projeted irle and the intersetions only move slightly from time point to time point. An effiient Newton-Raphson algorithm an therefore be devised for finding the intersetions, and the proedure detailed here an be used to find the spatial impulse response for any flat transduer geometry with an arbitrary apodization and both hard and soft baffle mounting. 14 Chapter 2. Desription of ultrasound fields

18 Region I Field point x r 0 r 1 r 1 Θ d z r 2 R Aperture Region II r 1 Field point x r 0 r 1 Θ d r 2 z R Aperture Figure 2.11: Definition of variables for spatial impulse response of onave transduer Calulation of spatial impulse responses 15

19 2.4 Apodization and soft baffle Figure 2.12: Definition of angle used for a soft baffle. Often ultrasound transduers do not vibrate as a piston over the aperture. This an be due to the lamping of the ative surfae at its edges, or intentionally to redue side-lobes in the field. Applying for example a Gaussian apodization will signifiantly lower side lobes and generate a field with a more uniform point spread funtion as a funtion of depth. Apodization is introdued in (2.12) by writing [18] h( r 1,t) = Z Θ2 Z d2 Θ 1 a p (r,θ) δ(t R ) d 1 2πR r dr dθ (2.35) in whih a p (r,θ) is the apodization over the aperture. Using the same substitutions as before yields Z Θ2 Z t2 h( r 1,t) = a p1 (t,θ)δ(t t ) dt dθ (2.36) 2π Θ 1 t 1 where a p1 (t,θ) = a p ( (t ) 2 z 2 p,θ). The inner integral is a onvolution of the apodization funtion with a δ-funtion and readily yields h( r 1,t) = Z Θ2 a p1 (t,θ) dθ (2.37) 2π Θ 1 as noted by several authors [18, 19, 20]. The response for a given time instane an, thus, be found by integrating the apodization funtion along the fixed ar with a radius of r = (t) 2 z 2 p for the angles for the ative aperture. Any apodization funtion an therefore be inorporated into the alulation by employing numerial integration. Often the assumption of an infinite rigid baffle for the transduer mounting is not appropriate and another form of the Rayleigh integral must be used. For a soft baffle, in whih the pressure on the baffle surfae is zero, the Rayleigh-Sommerfeld integral is used. This is [21, pages 46 50] h s ( r 1,t) = Z S δ(t r 1 r 2 ) osϕ ds (2.38) 2π r 1 r 2 assuming that r 1 r 2 λ. Here osϕ is the angle between the line through the field point orthogonal to the aperture plane and the radius of the spherial wave as shown in Fig Chapter 2. Desription of ultrasound fields

20 Spherial waves y h x t Aperture Figure 2.13: Shemati diagram of field from retangular element. The angle ϕ is onstant for a given radius of the projeted spherial wave and thus for a given time. It is given by osϕ = z p R = z p t Using the substitutions from Setion 2.2, the Rayleigh-Sommerfeld integral an then be rewritten as (2.39) Using the property of the δ-funtion that h s ( r 1,t) = z p 2π (Θ 2 Θ 1 ) Z + Z t2 t 1 δ(t t ) t dt (2.40) g(t )δ(t t ) dt = g(t) (2.41) then gives h s ( r 1,t) = z p Θ 2 Θ 1 = z p t 2π t h( r 1,t). (2.42) The spatial impulse response an, thus, be found from the spatial impulse response for the rigid baffle ase by multiplying with z p /(t). 2.5 Examples of spatial impulse responses The first example shows the spatial impulse responses from a 3 5 mm retangle 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 in 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 Examples of spatial impulse responses 17

21 h [m/s] x Lateral distane [mm] Time [s] Figure 2.14: Spatial impulse response from a retangular aperture of 4 5 mm at for different lateral positions 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. 2.6 Calulation of the sattered signal In medial ultrasound, a pulsed field is emitted into the body and is sattered and refleted by density and propagation veloity perturbations. The sattered field then propagates bak through the tissue and is reeived by the transduer. The field is onverted to a voltage signal and used for the display of the ultrasound image. A full desription of a typial imaging system, using the onept of spatial impulse response, is the purpose of the setion. The reeived signal an be found by solving an appropriate wave equation. This has been done in a number of papers (e.g. [22], [23]). Gore and Leeman [22] onsidered a wave equation where the sattering term was a funtion of the adiabati ompressibility and the density. The transduer was modeled by an axial and lateral pulse that were separable. Fatemi and Kak [23] used a wave equation where sattering originated only from veloity flutuations, and the transduer was restrited to be irularly symmetri and unfoused (flat). The sattering term for the wave equation used here is a funtion of density and propagation veloity perturbations, and the wave equation is equivalent to the one used by Gore and Leeman [22]. No restritions are enfored on the transduer geometry or its exitation, and analyti expressions for a number of geometries an be inorporated into the model. The model inludes attenuation due to propagation and sattering, but not the dispersive attenuation observed for propagation in tissue. This an, however, be inorporated into the model as indiated in Setion The derivation is organized as follows. The following setion derives the wave equation and desribes the different 18 Chapter 2. Desription of ultrasound fields

22 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 2.15: 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 Calulation of the sattered signal 19

23 linearity assumptions made. Setion alulates the sattered field and setion introdues the spatial impulse response model for the inident field. Setion ombines the wave equation solution and the transduer model to give the final equation for the reeived pressure field. To indiate the preision of the model, a single example of a predited pressure field ompared to measured field is given in Setion Derivation of the wave equation This setion derives the wave equation. The setion has been inluded in order to explain in detail the different linearity assumptions and approximations made to obtain a solvable wave equation. The derivation losely follows that developed by Chernov (1960). The first approximation states that the instantaneous aousti pressure and density an be written P ins ( r,t) = P + p 1 ( r,t) (2.43) ρ ins ( r,t) = ρ( r) + ρ 1 ( r,t) (2.44) in whih P is the mean pressure of the medium and ρ is the density of the undisturbed medium. Here p 1 is the pressure variation aused by the ultrasound wave and is onsidered small ompared to P and ρ 1 is the density hange aused by the wave. Both p 1 and ρ 1 are small quantities of first order. Our seond assumption is that no heat ondution or onversion of ultrasound to thermal energy take plae. Thus, the entropy is onstant for the proess, so that the aousti pressure and density satisfy the adiabati equation [24]: dp ins dt = 2 dρ ins dt (2.45) The equation ontains total derivatives, as the relation is satisfied for a given partile of the tissue rather than at a given point in spae. This is the Lagrange desription of the motion [6]. For our purpose the Euler desription is more appropriate. Here the oordinate system is fixed in spae and the equation desribes the properties of whatever partile of fluid there is at a given point at a given time. Converting to an Eulerian desription results in the following onstitutive equation [24], [6]: 1 p 1 2 = ρ 1 + u ρ (2.46) t t using that P and ρ do not depend on time and that ρ 1 is small ompared to ρ. Here u is the partile veloity, is the gradient operator, and symbolizes the salar produt. The pressure, density, and partile veloity must also satisfy the hydrodynami equations [24]: ρ ins d u dt = P ins (2.47) ρ ins = (ρ ins u) (2.48) t whih are the dynami equation and the equation of ontinuity. Using (2.43) and (2.44) and disarding higher order terms we an write ρ u t = p 1 (2.49) ρ 1 t Differentiating (2.50) with respet to t and inserting (2.49) gives 2 ρ 1 2 t = (ρ u) (2.50) = (ρ u t ) = ( p 1) = 2 p 1 (2.51) 20 Chapter 2. Desription of ultrasound fields

24 Differentiating (2.46) with respet to t and inserting (2.51) and (2.49) leads to 1 2 p = 2 ρ 1 t 2 + u ρ (2.52) t t 2 p p 1 2 t = 1 ρ ρ p 1 (2.53) Assuming that the propagation veloity and the density only vary slightly from their mean values yields where ρ 0 ρ and 0. 2 p 1 ρ( r) = ρ 0 + ρ( r) ( r) = 0 + ( r) (2.54) 1 2 p 1 1 ( 0 + ) 2 2 = t (ρ 0 + ρ) (ρ 0 + ρ) p 1 (2.55) Ignoring small quantities of seond order and using the approximation ( 1): gives: (2.56) 2 p 1 ( ( p ) = ( ρ) ρ ) t ρ 0 ρ 2 ( ρ) p 1 (2.57) 0 Negleting the seond order term ( ρ/ρ 0 2 ) ( ρ) p 1 yields the wave equation: 2 p p 1 t 2 = 2 2 p t ( ρ) p 1 (2.58) ρ 0 The two terms on the right side of the equation are the sattering terms whih vanish for a homogeneous medium. The wave equation was derived by Chernov [24]. It has also been onsidered in Gore & Leeman [22] and Morse & Ingard [6] in a slightly different form, where the sattering terms were a funtion of the adiabati ompressibility κ and the density Calulation of the sattered field Having derived a suitable wave equation, we now alulate the sattered field from a small inhomogeneity embedded in a homogeneous surrounding. The sene is depited in Fig The inhomogeneity is identified by r 1 and enlosed in the volume V. The sattered field is alulated at the point indiated by r 2 by integrating all the spherial waves emanating from the sattering region V using the time dependent Green s funtion for unbounded spae. Thus, the sattered field is [6], [22]: Z [ 1 p s ( r 2,t) = ( ρ( r 1 )) p 1 ( r 1,t 1 ) ZV ρ 0 where G is the free spae Green s funtion: T 2 ( r 1) p 1 ( r 1,t 1 ) t 2 ] G( r 1,t 1 r 2,t)dt 1 d 3 r 1 (2.59) G( r 1,t 1 r 2,t) = δ(t t 1 r 2 r 1 0 ) 4π r 2 r 1 (2.60) 2.6. Calulation of the sattered signal 21

25 Figure 2.16: Coordinate system for alulating the sattered field d 3 r 1 means integrating w.r.t. r 1 over the volume V, and T denotes integration over time. We denote by the sattering operator. F op = 1 ρ 0 ( ρ( r 1 )) 2 ( r 1) t 2 (2.61) The pressure field inside the sattering region is: p 1 ( r,t) = p i ( r,t) + p s ( r,t) (2.62) where p i is the inident pressure field. As an be seen, the integral an not be solved diretly. To solve it we apply the Born-Neumann expansion [25]. If G i symbolizes the integral operator representing Green s funtion and the integration and F op the sattering operator, then the first order Born approximation an be written: p s1 ( r 2,t) = G i F op p i ( r 1,t 1 ) (2.63) Here p s has been set to zero in (2.62). Inserting p s1 in (2.62) and then in (2.59) we arrive at p s2 ( r 2,t) = G i F op [p i ( r 1,t 1 ) + G i F op p i ( r 1,t 1 )] = G i F op p i ( r 1,t 1 ) + [G i F op ] 2 p i ( r 1,t 1 ) (2.64) It is emphasized here that G i indiates an integral over r 1 and t 1, and not the pressure at point r 1 and time t 1 but over the volume of V and time T indiated by r 1 and t 1. The general expression for the sattered field then is: p s ( r 2,t) = G i F op p i ( r 1,t 1 ) + (2.65) [G i F op ] 2 p i ( r 1,t 1 ) + [G i F op ] 3 p i ( r 1,t 1 ) + [G i F op ] 4 p i ( r 1,t 1 ) + 22 Chapter 2. Desription of ultrasound fields

26 Figure 2.17: Coordinate system for alulating the inident field. Terms involving [G i F op ] N p i ( r 1,t 1 ), where N > 1, desribe multiple sattering of order N. Usually the sattering from small obstales is onsidered weak so higher order terms an be negleted. Thus, a useful approximation is to employ only the first term in the expansion. This orresponds to the first order Born-approximation. Using this (2.59) an be approximated by (note the replaement of p 1 ( r 1,t 1 ) with p i ( r 1,t 1 )): Z [ 1 p s ( r 2,t) ( ρ( r 1 )) p i ( r 1,t 1 ) ZV T ρ 0 2 ( r 1) 2 ] p i ( r 1,t 1 ) 3 0 t 2 G( r 1,t 1 r 2,t)dt 1 d 3 r 1 (2.66) So in order to alulate the sattered field, the inident field for the homogeneous medium must be alulated Calulation of the inident field The inident field is generated by the ultrasound transduer assuming no other soures exist in the tissue. The field is onveniently alulated by employing the veloity potential ψ( r, t), and enforing appropriate boundary onditions [26], [27]. The veloity potential satisfies the following wave equation for the homogeneous medium: 2 ψ 1 2 ψ = 0 (2.67) t and the pressure is alulated from: p( r,t) = ρ 0 ψ( r,t) t (2.68) The oordinate system shown in Fig is used in the alulation. The partile veloity normal to the transduer surfae is denoted by v( r 3 + r 4,t), where r 3 identifies the position of the transduer and r 4 a point on the transduer surfae relative to r 3. The solution to the homogeneous wave equation is [27]: ψ( r 1 + r 3,t) = Z S Z T v( r 3 + r 4,t 3 )g( r 1,t r 3 + r 4,t 3 )dt 3 d 2 r 4 (2.69) 2.6. Calulation of the sattered signal 23

27 Figure 2.18: Coordinate system for alulating the reeived signal. when the transduer is mounted in a rigid infinite planar baffle. S denotes the transduer surfae. g is the Green s funtion for a bounded medium and is g( r 1,t r 3 + r 4,t 3 ) = δ(t t 3 r 1 r 3 r 4 0 ) 2π r 1 r 3 r 4 (2.70) r 1 r 3 r 4 is the distane from S to the point where the field is alulated and 0 the mean propagation veloity. The field is alulated under the assumption of radiation into an isotropi, homogeneous, non-dissipative medium. If a slightly urved transduer is used, an additional term is introdued as shown in Morse & Feshbah [28]. This term is alled the seond order diffration term in Penttinen & Luukkala [13]. It an be shown to vanish for a planar transduer, and as long as the transduer is only slightly urved and large ompared to the wavelength of the ultrasound, the resulting expression is a good approximation to the pressure field [13]. If the partile veloity is assumed to be uniform over the surfae of the transduer, (2.69) an be redued to [29]: ψ( r 1, r 3,t) = Z T v(t 3 ) Z S g( r 1,t r 3 + r 4,t 3 )d 2 r 4 dt 3 (2.71) This is the spatial impulse response previously derived, and the sound pressure for the inident field then is: p( r 1, r 3,t) = ρ 0 ψ( r 1, r 3,t) t = ρ 0 v(t) t h( r 1, r 3,t) t (2.72) or p( r 1, r 3,t) = ρ 0 v(t) t t h( r 1, r 3,t) (2.73) Calulation of the reeived signal The reeived signal is the sattered pressure field integrated over the transduer surfae, onvolved with the eletromehanial impulse response, E m (t), of the transduer. To alulate this we introdue the oordinate system shown in Fig r 6 + r 5 indiates a reeiving element on the surfae of the transduer that is loated at r 5. The reeived signal is: p r ( r 5,t) = E m (t) t Z S p s ( r 6 + r 5,t)d 2 r 6 (2.74) 24 Chapter 2. Desription of ultrasound fields

28 The sattered field is: p s ( r 6 + r 5,t) = 1 2 ZV Z T F op [p i ( r 1,t 1 )] δ(t t 1 r 6+ r 5 r 1 0 ) dt 1 d 3 r 1 (2.75) 2π r 6 + r 5 r 1 Combining this with (2.74) and omparing with (2.9) we see that p r inludes Green s funtion for bounded spae integrated over the transduer surfae, whih is equal to the spatial impulse response. Inserting the expression for p i and performing the integration over the transduer surfae and over time, results in: p r ( r 5,t) = E m (t) t 1 2 ZV F op [ ρ 0 v(t) t t h( r 1, r 3,t) ] t h( r 5, r 1,t)d 3 r 1 (2.76) If the position of the transmitting and the reeiving transduer is the same ( r 3 = r 5 ), then a simple rearrangement of (2.76) yields: p r ( r 5,t) = ρ 0 2 E v(t) m(t) t t t ZV F op [h pe ( r 1, r 5,t)]d 3 r 1 (2.77) where is the pulse-eho spatial impulse response. h pe ( r 1, r 5,t) = h( r 1, r 5,t) t h( r 5, r 1,t) (2.78) The alulated signal is the response measured for one given position of the transduer. For a B-mode san piture a number of san-lines is measured and ombined to a piture. To analyze this situation, the last fator in (2.77) is expliitly written out Z V [ 1 ρ 0 ( ρ( r 1 )) h pe ( r 1, r 5,t) 2 ( r 1) ] h pe ( r 1, r 5,t) t 2 d 3 r 1 (2.79) From setion it is know that H pe is a funtion of the distane between r 1 and r 5, while ρ and only are funtions of r 1. So when r 5 is varied over the volume of interest, the resulting image is a spatial non-stationary onvolution between ρ, and a modified form of the pulse-eho spatial impulse response. If we assume that the pulse-eho spatial impulse is slowly varying so that the spatial frequeny ontent is onstant over a finite volume, then (2.79) an be rewritten Z [ 1 ρ( r 1 ) 2 h pe ( r 1, r 5,t) 2 ( r 1) 2 ] h pe ( r 1, r 5,t) V ρ t 2 d 3 r 1 (2.80) h pe is a funtion of the distane between the transduer and the satterer or equivalently of the orresponding time given by t = r 1 r 5 0 (2.81) The Laplae operator is the seond derivative w.r.t. the distane, whih an be approximated with the seond derivative w.r.t. time. So 2 h pe ( r 1, r 5,t) = 1 2 h pe ( r 1, r 5,t) 2 0 t 2 (2.82) assuming only small deviations from the mean propagation veloity. Using these approximations, (2.77) an be rewritten: p r ( r 5,t) = ρ E m (t) 0 t v 3 (t) t 3 t ZV [ ρ( r1 ) 2 ( r ] 1) h pe ( r 1, r 5,t)d 3 r 1 (2.83) ρ Calulation of the sattered signal 25

29 Symbolially this is written p r ( r 5,t) = v pe (t) t f m ( r 1 ) r h pe ( r 1, r 5,t) (2.84) r denotes spatial onvolution. v pe is the pulse-eho wavelet 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 propagation veloity perturbations whih give rise to the sattered signal. h pe is the modified 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 E m (t) 0 t v 3 (t) t 3 (2.85) f m ( r 1 ) = ρ( r 1) ρ 0 2 ( r 1) 0 (2.86) h pe ( r 1, r 5,t) = h( r 1, r 5,t) t h( r 5, r 1,t) (2.87) Expression (2.84) onsists of three distint terms. The interesting signal, and the one that should be displayed in medial ultrasound, is f m ( r 1 ). We, however, measure a time and spatially smoothed version of this, whih obsures the finer details in the piture. The smoothing onsists of a onvolution in time with a fixed wavelet v pe (t) and a spatial onvolution with a spatially varying h pe ( r 1, r 5,t) Example of pulse eho response To show that the pulse-eho response an be alulated to good auray, a single example is shown in Fig for a onave transduer (r = 8.1 mm) with a fous at R = 150 mm. The measured and simulated responses were obtained at a distane of 60 mm from the transduer surfae. The measured pressure field was aquired by moving a needle pointing toward the transduer in steps of 0.2 mm in the lateral diretion, making measurements in a plane ontaining the aoustial axis of the transduer. The simulated field was alulated by measuring v pe as the response from a planar refletor, and then using (2.32) and (2.84) to alulate the field. The envelope of the RFsignals is shown as a ontour plot with 6 db between the ontours. The plots span 20 mm in the lateral diretion and 4 µs in the axial diretion Attenuation effets The model inludes attenuation of the pulse due to propagation, but not the dispersive attenuation of the wave observed when propagating in tissue. This hanges the pulse ontinuously as it propagates down through the tissue. Not inluding dispersive attenuation is, however, not a serious drawbak of the theory, as this hange of the pulse an be lumped into the already spatially varying h pe. Or, if in the far field and assuming a homogeneous, dispersive attenuation, then an attenuation transfer funtion an be onvolved onto v pe to yield an attenuated pulse. Submerging the transduer into a homogeneously attenuating medium will modify the propagation of the spherial waves, whih will hange ontinuously as a funtion of distane from the transduer. The spatial impulse is then hanged to Z Z h att (t, r) = T S ) a(t τ, r + r 1 ) δ(τ r+ r 1 r + r 1 dsdτ (2.88) when linear propagation is assumed and the attenuation is the same throughout the medium. a is the attenuation impulse response. The spherial wave is onvolved with the distane dependent attenuation impulse response and spherial waves emanating from different parts of the aperture are onvolved with different attenuation impulse responses. 26 Chapter 2. Desription of ultrasound fields

30 8.2 x 10 5 Measured response 8.1 Time [s] x [mm] 8.2 x Simulated response Time [s] x [mm] Figure 2.19: Measured and simulated pulse-eho response for a onave transduer. The axial distane was 60 mm and the small satterer was moved in the lateral diretion. The envelope of the RF signal is shown as 6 db ontours. A model for the attenuation must be introdued in order to solve the integral. Ultrasound propagating in tissue experienes a nearly linear with frequeny attenuation and a ommonly used attenuation amplitude transfer funtion is A ( f, r ) = exp( β f r ) (2.89) where β is attenuation in nepers per meter. We here prefer to split the attenuation into a frequeny dependent and a frequeny independent part term as A( f, r ) = exp( α r )exp( β( f f 0 ) r ) (2.90) α is the frequeny independent attenuation oeffiient, and f 0 the transduer enter frequeny. The phase of the attenuation need also be onsidered. Kak and Dines [30] introdued a linear with frequeny phase response Θ( f ) = 2π f τ b r (2.91) where τ b is the bulk propagation delay per unit length and is equal to 1/. This, however, results in an attenuation impulse response that is non-ausal. Gurumurthy and Arthur [31] therefore suggested using a minimum phase impulse response, where the amplitude and phase spetrum form a Hilbert transform pair. The attenuation spetrum is then given by A( f, r ) = exp( α r )exp( β( f f 0 ) r ) (2.92) β exp( j2π f (τ b + τ m π 2 ) r ) exp( j 2 f β r ln(2π f )) π where τ m is the minimum phase delay fator. Gurumurthy and Arthur [31] suggest a τ m value of 20 in order to fit the dispersion found in tissue. The inverse Fourier transform of (2.92) must be inserted into (2.88) and the integral has to be solved for the partiular transduer geometry. This is learly a diffiult, if not impossible, task and some approximations must 2.6. Calulation of the sattered signal 27

31 Error [db] Lateral displaement [mm] Figure 2.20: Error of assuming a non-varying frequeny dependent attenuation over the duration of the spatial impulse response. be introdued. All spherial waves arrive at nearly the same time instane, if the distane to the field point is muh larger than the transduer aperture. In this ase the attenuation funtion is, thus, the same for all waves and the result is a onvolution between the attenuation impulse response and the spatial impulse response, whih is a Dira impulse. A spatial impulse response other than a Dira funtion indiates that the spherial waves arrive at different times. Multiplying the arrival time with the propagation veloity gives the distane to the points on the aperture ontributing to the response at that time instane. A first approximation is, therefore, to multiply the non-attenuated spatial impulse response with the proper frequeny independent term. This approximation also assumes that the span of values for r + r 1 is so small that both 1/ r + r 1 and the attenuation an be assumed to be roughly onstant. The frequeny dependent funtion will also hange for the different values of the spatial impulse response. A non-stationary onvolution must, thus, be performed. One possible method to avoid this is to assume that the frequeny dependent attenuation impulse response is onstant for the time and, thus, distanes r + r 1 where h is non-zero. The mean distane is then used in (2.92) and the inverse Fourier transform of A( f, r mid ) is onvolved with h(t, r). The auray of the approah depends on the duration of h and of the attenuation. The error in db for a onave transduer with a radius of 10 mm foused at 100 mm and an attenuation of 0.5 db/[mhz m] is shown in Fig The axial distane to the transduer is 50 mm. An example of the influene of attenuation on the point spread funtion (PSF) is shown in Fig A onave transduer with a radius of 8 mm, enter frequeny of 3 MHz, and foused at 100 mm was used. Fig shows point spread funtions alulated under different onditions. The logarithmi envelope of the reeived pressure is displayed as a funtion of time and lateral displaement. The left most graph shows the normalized PSF for the transduer submerged in a non-attenuating medium. The distane to the field point is 60 mm and the funtion is shown for lateral displaements from -8 to 8 mm. Introduing a 0.5 db/[mhz m] attenuation yields the normalized PSF shown in the middle. The entral ore of the PSF does not hange signifiantly, but the shape at -30 db and below are somewhat different from the non-attenuated response. A slightly broader and longer funtion is seen, but the overall shape is the same. An ommonly used approah to haraterize the field is to inlude the attenuation into the basi one-dimensional pulse, and then use the non-attenuated spatial impulse response in alulating the PSF. This is the approah 28 Chapter 2. Desription of ultrasound fields

32 x a x 10-5 b x Time [s] 7.9 Time [s] Time [s] Lateral distane [mm] Lateral distane [mm] Lateral distane [mm] Figure 2.21: Contour plots of point spread funtions for different media and alulation methods. a: Non attenuating medium. b: 0.5 db/[mhz m] attenuation. : 0.5 db/[mhz m] attenuation on the one-dimensional pulse. There is 6 db between the ontour lines. The distane to the transduer is 60 mm. used in the rightmost graph in Fig All attenuation is inluded in the pulse and the spatial impulse response alulated in the leftmost graph is used for making the PSF. The similarity to the enter graph is striking. Apart from a slightly longer response, nearly all features of the field are the same. It is, thus, appropriate to estimate the attenuated one-dimensional pulse and reonstrut the whole field from this and knowledge of the transduer geometry Calulation of the sattered signal 29

33 30

34 CHAPTER THREE Ultrasound imaging 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. Part of the hapter is based on [4] and [32]. 3.1 Fourier relation 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 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 as shown in Fig. 3.1 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 of [8] d p = j ρ 0k 4πr U 0a p (x)e j(ωt kr ) dx, (3.1) 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. 3.1) r = r xsinθ (3.2) 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 r ) r dx. (3.3) 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 (3.2) in (3.3) gives p(r,θ,t) = j ρ 0U 0 k 4πr Z + a p (x)e j(ωt kr+kxsinθ) dx = ρ 0U 0 k 4πr Z + e j(ωt kr) a p (x)e jkxsinθ dx, (3.4) 31

35 Figure 3.1: Geometry for line aperture. 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: P ax (r) = ρ 0U 0 kl 4πr Z + H(θ) = 1 a p (x)e jkxsinθ dx (3.5) 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 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 j2πx f sinθ dx = 1 L This very losely resembles the standard Fourier integral given by Z + a p (x)e j2πx f dx. (3.6) G( f ) = g(t) = Z + Z + g(t)e j2πt f dt G( f )e j2πt f d f (3.7) 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. 1 The term 1/L is inluded to make H(θ) a unit less number. 32 Chapter 3. Ultrasound imaging

36 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 3.2: 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 patterns The first example is for a simple line soure, where the aperture funtion is onstant suh that a p (x) = { 1 x L/2 0 else (3.8) The angular fator is then H(θ) = sin(πl f sinθ ) πl f sinθ = sin( k 2 Lsinθ) k 2 Lsinθ (3.9) 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. (3.10) 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 (3.11) 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 Fourier relation 33

37 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 3.3: 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. Modern ultrasound transduers onsist of a number of elements eah radiating ultrasound energy. Negleting the phasing of the element (see Setion 3.2) 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), (3.12) where a ps (x) is the aperture funtion for the individual elements, d x is the spaing (pith) between the enters of the individual elements, and N is the number of elements in the array. Using the Fourier relationship the angular beam pattern an be desribed by H p (θ) = H ps (θ)h per (θ), (3.13) where N/2 n= N/2 Summing the geometri series gives δ(x d x n) H per (θ) = N/2 n= N/2 e jnd xksinθ. = H per (θ) = sin( (N + 1) k 2 d x sinθ ) N/2 n= N/2 f sinθ j2π e nd x. (3.14) sin ( k 2 d x sinθ ) (3.15) is the Fourier transform of series of delta funtions. This funtion repeats itself with a period that is a multiple of π = k 2 d x sinθ sinθ = π kd x = λ d x. (3.16) 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 (3.17) 34 Chapter 3. Ultrasound imaging

38 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 for a given diretion θ 0 is τ = d x sinθ 0 Steering in a diretion θ 0 an, therefore, be aomplished by using (3.18) sinθ 0 = τ d x (3.19) 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 sin θ d τ x H per (θ) = ( )). (3.20) sin( k2 d x sin θ d τ x Notie that the delay is independent of frequeny, sine it is essentially only determined by the speed of sound. 3.2 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 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 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 (3.21) 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, is the speed of sound, and t i is the alulated propagation time. 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 (3.22) 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 (3.23) 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. A single fous is only possible in transmit and omposite imaging is therefore often used in modern imaging. Here several pulse emissions with fousing at different depths 3.2. Fousing 35

39 in the same diretion are used and the reeived signals are ombined to form one image foused in both transmit and reeive at different depths (omposit imaging). The fousing an, thus, be defined through time lines as: 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 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 3.1 showed 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 1,1 is the amplitude saling value multiplied onto element 1 after time instane t 1. 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 (3.24) L 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 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, 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 3.1. 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 assuming all N elements to be idential. N 1 h a ( r p,t) = i=0 h e ( r i, r p,t), (3.25) 36 Chapter 3. Ultrasound imaging

40 Array elements D r a r e θ Figure 3.4: Geometry of linear array (from [4], Copyright Cambridge University Press). 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 δ(t r i r p ) (3.26) R p i=0 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, then h a ( r p,t) = k N 1 R p i=0 ( δ t r ) a + id r e r p, (3.27) where r e is a unit vetor pointing in the diretion along the elements. The geometry is shown in Fig The differene in arrival time between elements far from the transduer is The spatial impulse response is, thus, a series of Dira pulses separated by t. h a ( r p,t) k N 1 R p t = DsinΘ. (3.28) i=0 ( δ t R ) p i t. (3.29) 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 = DsinΘ. (3.30) The main lobe is found for Θ = 0 and the next maximum in the response is found for ( ) ( ) λ Θ = arsin = arsin. (3.31) f D D 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 Fields from array transduers 37

41 0-5 Normalized amplitude [db] Angle [deg] Figure 3.5: Far-field ontinuous wave beam profile at 3 MHz for linear array onsisting of 64 point soures with an inter-element spaing of 1 mm (from [4], Copyright Cambridge University Press). 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 ( ( )) Rp exp j2π f R p + idsinθ i=0 = exp( j2π R p ) k N 1 R p = i=0 exp ( j2π f DsinΘ ) i (3.32) DsinΘ sin(π f N) sin(π f DsinΘ exp( jπ f (N 1)DsinΘ) k exp( j2π R p ) R p ). The terms exp( j2π R p DsinΘ ) and exp( jπ f (N 1) ) 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 D λ H a ( f ) = sinθ) R p sin(π D λ sinθ). (3.33) The beam profile at 3 MHz is shown in Fig. 3.5 for a 64-element array with D = 1 mm. 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. Examples of simulated responses are shown in Fig The top graph shows an array of 64 point soures exited with a Gaussian 3 MHz pulse with B r = 0.2. The spae between the elements is 1 mm. The maximum of the response at a radial position of 100 mm from the transduer is taken. The bottom graph shows the response when retangular elements of 1 6 mm are used. This demonstrates the somewhat rude approximation of using the far-field point soure CW response to haraterize arrays. Fig. 3.7 shows the different point spread funtions enountered when a phased array is used to san over a 15 m depth. The array onsists of 128 elements eah mm in size, and the kerf between the elements is 0.05 mm. The transmit fous is at 70 mm, and the foi are at 30, 70, and 110 mm during reeption. Quite ompliated 38 Chapter 3. Ultrasound imaging

42 Norm. max. amp. [db] Norm. max. amp. [db] Array of point soures Angle [deg] Array of retangular soures Angle [deg] Figure 3.6: Beam profiles for an array onsisting of point soures (top) or retangular elements (bottom). The exitation pulse has a frequeny of 3 MHz and the element spaing is 1 mm. The distane to the field point is 100 mm (from [4], Copyright Cambridge University Press). 20 x mm reeive fous 12 Time [s] mm reeive fous Transmit fous (70 mm) mm reeive fous Lateral displaement [mm] Figure 3.7: Point spread funtions for different positions in a B-mode image. Onehundredtwentyeight mm elements are used for generating and reeiving the pulsed field. Three different reeive foi are used. The ontours shown are from 0 to -24 db in steps of 6 db relative to the maximum at the partiular field point (from [4], Copyright Cambridge University Press) Fields from array transduers 39

43 Figure 3.8: Linear array transduer for obtaining a retangular ross-setional image. 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 3.9: Eletroni fousing and steering of an ultrasound beam. point spread funtions are enountered, and they vary substantially with depth in tissue. Notie espeially the edge waves, whih dominate the response lose to the transduer. The edge effet an be redued by weighting responses from different elements. This is also alled apodization. The exitation pulses to elements at the transduer edge are redued, and this diminishes the edge waves. More examples are shown below in Setion 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 3.8, 3.10, and 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 3.2. Only one foal zone is 40 Chapter 3. Ultrasound imaging

44 Figure 3.10: Convex array transduer for obtaining a polar ross-setional image. 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. 3.9b. 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. 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 is used for obtaining the elevation fousing essential for an improved image quality. Eletroni beamforming an also be used in the elevation 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 Simulation of ultrasound imaging One of the first steps in designing an ultrasound system is the seletion of the appropriate number of elements for the array transduers and the number of hannels for the beamformer. The fousing strategy in terms of number of foal zones and apodization must also be determined. These hoies are often not easy, sine it is diffiult to determine the effet in the resulting images of inreasing the number of hannels and seleting more or less advaned fousing shemes. It is therefore benefiial to simulate the whole imaging system in order to quantify the image quality. The program Field II was rewritten to make it possible to simulate the whole imaging proess with time varying fo Simulation of ultrasound imaging 41

45 Figure 3.11: 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 3.12: Elevation foused onvex array transduer for obtaining a retangular ross-setional image, whih is foused in the out-of-plane diretion. The urvature in the elevation diretion is exaggerated in the figure for illustration purposes. 42 Chapter 3. Ultrasound imaging

46 z [mm] y [mm] x [mm] y [mm] 5 z [mm] x [mm] z [mm] y [mm] x [mm] Figure 3.13: 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 using and apodization as desribed in [33] and [16]. This has paved the way for doing realisti simulated imaging with multiple foal zones for transmission and reeption and for using dynami apodization. It is hereby possible to simulate ultrasound imaging for all image types inluding flow images, and the purpose of this setion is to present some standard simulation phantoms that an be used in designing and evaluating ultrasound transduers, beamformers and systems. The phantoms desribed an be divided into ordinary string/yst phantoms, artifiial human phantoms and flow imaging phantoms. The ordinary omputer phantoms inlude both a string phantoms for evaluating the point spread funtion as a funtion of spatial positions as well as a yst/string phantom. Artifiial human phantoms of a fetus in the third month of development and an artiial kedney are also shown. The simulation of flow and the assoiated phantoms will be desribed in Setion 4.7. All the phantoms an be used with any arbitrary transduer onfiguration like single element, linear, onvex, or phased array transduers, with any apodization and fousing sheme Simulation model The first simple treatment of ultrasound is often based on the refletion and transmission of plane waves. It is assumed that the propagating wave impinges on plane boundaries between tissues with different mean aousti properties. Suh boundaries are rarely found in the human body, and seldom show on ultrasound images. This is demonstrated by the image shown in Fig Here the skull of the fetus is not learly marked. It is quite obvious that there is a lear boundary between the fetus and the surrounding amnioti fluid. The skull boundary is not visible in the image, beause the angle between the beam and the boundary has a value suh that the sound bounes off in another diretion, and, therefore, does not reah the transduer. Despite this, the extent of the head an still be seen. This is due to the sattering of the ultrasound wave. Small hanges in density, ompressibility, and absorption give rise to a sattered wave radiating in all diretions. The baksattered field is reeived by the transduer and displayed on the sreen. One might well argue that sattering is what makes ultrasound images useful for diagnosti purposes, and it is, as will be seen later, the physial phenomena that makes detetion of blood veloities possible. Ultrasound sanners are, in fat, optimized to show the baksattered signal, whih is onsiderably weaker than that found from refleting boundaries. Suh refletions will usually be displayed as bright white on the sreen, and an potentially saturate the reeiving iruits in the sanner. An example an be seen at the nek of the fetus, where a struture is perpendiular to the beam. This strong refletion saturates the 3.5. Simulation of ultrasound imaging 43

47 Head Amnioti fluid Amnioti fluid Boundary of plaenta Mouth Foot Spine Figure 3.14: Ultrasound image of a 13th week fetus. The markers at the border of the image indiate one entimeter (from [4], Copyright Cambridge University Press). 44 Chapter 3. Ultrasound imaging

48 Figure 3.15: 4 4 m image of a human liver from a healthy 28-year-old man. The ompletely dark areas are blood vessels (from [4], Copyright Cambridge University Press). input amplifier of this sanner. Typial boundary refletions are enountered from the diaphragm, blood vessel walls, and organ boundaries. An enlarged view of an image of a liver is seen in Fig The image has a grainy appearane, and not a homogeneous gray or blak level as might be expeted from homogeneous liver tissue. This type of pattern is alled spekle. The displayed signals are the baksatter from the liver tissue, and are due to onnetive tissue, ells, and fibrious tissue in the liver. These strutures are muh smaller than one wavelength of the ultrasound, and the 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 [34], 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. Although the baksattered signal is haraterized in statistial terms, one should be areful not to interpret the signal as random in the sense that a new set of values is generated for eah measurement. The same signal will result, when a transduer is probing the same struture, if the struture is stationary. Even a slight shift in position will yield a baksattered signal orrelated with that from the adjaent position. The shift over whih the signals are orrelated is essentially dependent on the extent of the ultrasound field. This an also be seen from the image in Fig. 3.15, as the laterally elongated white spekles in the image indiate transverse orrelation. The extent of these spekle spots is a rough indiation of the point spread funtion of the system. The orrelation between different measurements is what makes it possible to estimate blood veloities with ultrasound. As there is a strong orrelation for small movements, it is possible to detet shifts in position by omparing or, more stritly, orrelating suessive measurements of moving struture, e.g., blood ells. 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 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 power of the sattered sig Simulation of ultrasound imaging 45

49 nal. Sine the small strutures reradiate 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. It is, thus, important that the simulation approah models the sattering mehanisms in the tissue. This is essentially what the model derived in Chapter 2 does. Here the reeived signal from the transduer is: p r ( r,t) = v pe (t) t f m ( r) r h pe ( r,t) (3.34) where r denotes spatial onvolution. v pe is the pulse-eho 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 propagation veloity 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 m(t) t 3 v(t) t 3, f m ( r 1 ) = ρ( r) ρ 2 ( r), h pe ( r,t) = h t ( r,t) h r ( r,t) (3.35) 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. This is the approah taken in these notes. The omputer phantoms typially onsist of 100,000 or more disrete satterers, and simulating 50 to 128 RF lines an take several days depending on the omputer used. It is therefore benefiial to split the simulation into onurrently run sessions. This an easily be done by first generating the satterer s position and amplitude and then storing them in a file. This file an then the be used by a number of workstations to find the RF signal for different imaging diretions, whih are then stored in separate files; one for eah RF line. These files are then used to assemble an image. This is the approah used for the simulations shown here in whih 3 Pentium Pro 200 MHz PCs an generate one phantom image over night using Matlab 5 and the Field II program. 3.6 Syntheti phantoms 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 also was 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 46 Chapter 3. Ultrasound imaging

50 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 3.16: Point target phantom imaged for different set-up of transmit and reeive fousing and apodization. See text for an explanation of the set-up Syntheti phantoms 47

51 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 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. 3.7 Anatomi phantoms The anatomi phantoms are attempts to generate images as they will be seen from real human subjets. This is done by drawing a bitmap image of sattering strength of the region of interest. This map then determines the fator multiplied onto the sattering amplitude generated from the Gaussian distribution, and models the differene in the density and speed of sound perturbations in the tissue. Simulated boundaries were introdued by making lines in the satterer map along whih the strong satterers were plaed. This is marked by ompletely white lines shown in the satterer maps. The model is urrently two-dimensional, but an readily be expanded to three dimensions. Currently, the elevation diretion is merely made by making a 15 mm thikness for the satterer positions, whih are randomly distributed in the interval. Two different phantoms have been made; a fetus in the third month of development and a left kidney in a longitudinal san. For both was used 200,000 satterers randomly distributed within the phantom, and with a Gaussian distributed satterer amplitude with a standard deviation determined by the satterer map. The phantoms were sanned with a 5 MHz 64 element phased array transduer with λ/2 spaing and Hanning apodization. A single transmit fous 70 mm from the transduer was used, and fousing during reeption is at 40 to 140 mm in 10 mm inrements. The images onsists of 128 lines with 0.7 degrees between lines. Fig shows the artifiial kidney satterer map on the left and the resulting image on the right. Note espeially the bright regions where the boundary of the kidney is orthogonal to the ultrasound, and thus a large signal is reeived. Note also the fuzziness of the boundary, where they are parallel with the ultrasound beam, whih is also seen on atual ultrasound sans. Fig shows the fetus. Note how the anatomy an be learly seen at the level of detail of the satterer map. The same boundary features as for the kidney image is also seen. The images have many of the features from real san images, but still lak details. This an be asribed to the low level of details in the bitmap images, and that only a 2D model is used. But the images do show great potential for making powerful fully syntheti phantoms, that an be used for image quality evaluation. 48 Chapter 3. Ultrasound imaging

52 Axial distane [mm] Lateral distane [mm] Figure 3.17: Computer phantom with point targets, yst regions, and strongly refleting regions Anatomi phantoms 49

53 Simulated fetus image 20 Fetus satter map Axial distane [mm] Lateral distane [mm] Figure 3.18: Simulation of artifiial fetus. Simulated kidney image Axial distane [mm] Kidney satter map Lateral distane [mm] Figure 3.19: Simulation of artifiial kidney. 50 Chapter 3. Ultrasound imaging

54 CHAPTER FOUR Ultrasound systems for flow estimation Medial ultrasound sanners an be used for both displaying gray-sale images of the anatomy and for visualizing the blood flow dynamially in the body. The systems an interrogate the flow at a single position in the body and there find the veloity distribution over time. They an also show a dynami olor image of veloity at up to 20 to 60 frames a seond. Both measurements are performed by repeatedly pulsing in the same diretion and then use the orrelation from pulse to pulse to determine the veloity. This hapter gives a simple model for the interation between the ultrasound and the moving blood. The alulation of the veloity distribution is then explained along with the different physial effets influening the estimation. The estimation of mean veloities using auto- and ross-orrelation for olor flow mapping is also desribed. Finally the simulation of these flow systems using spatial impulse responses are desribed. The hapter is a revised version of [35] and [32] with parts originating from [4]. 4.1 Introdution It is possible with modern ultrasound sanners to visualize the blood flow at either one sample volume in the body or over a ross-setional region. The first approah gives a display of the veloity distribution at the point as a funtion of time. The seond method gives a olor flow image superimposed on the gray-sale anatomi image. The olor oding shows the veloity towards or away from the transduer, and this an be shown with up to 20 images a seond. Thus, the dynamis of the blood flow in e.g. the vessel and heart and aross heart valves an be diagnosed. The history of ultrasound veloity measurement dates bak to the mid 1950ties, where experiments by Satomura [36], [37] in Japan demonstrated that ontinuous wave (CW) ultrasound was apable of deteting motion. CW ultrasound annot loate the depth of the motion and several pulsed wave solutions where therefore made by Baker [38], Peronneau and Leger [39], and Wells [40] around These systems ould detet the motion at a speifi loation. The Baker system ould show the veloity distribution over time and aross the vessel lumen. This was later extended by Kasai and o-workers [41], [42] to generate atual ross-setional images of veloity in real time. They used an autoorrelation veloity estimator adapted from radar to find the mean veloity from only 8 to 16 pulse-eho lines. Color flow mapping systems using a ross-orrelation estimator was also suggested by Bonnefous and o-workers [43]. From these early experimental systems the sanners have now evolved into systems for routine use in nearly all hospitals, and ultrasound is one of the most ommon means of diagnosing hemodynami problems today. This hapter will give a brief desription of the main features of all these systems. The hapter starts by deriving a basi model for ultrasound s interation with a point satterer, whih shows that the frequeny of the reeived sampled signal is proportional to veloity. Systems for finding the veloity distribution in a vessel are then desribed and finally the newest olor flow mapping method are explained. A more omprehensive treatment of the systems an be found in [4]. 51

55 x z r 2 r 1 v. T prf θ Blood vessel Ultrasound beam Figure 4.1: Coordinate system for finding the movement of blood partiles (from [4], Copyright Cambridge University Press). 4.2 Measurement of flow signals The data for the veloity measurement is obtained by emitting a short ultrasound pulse with 4 to 8 yles at a frequeny of 2 to 10 MHz. The ultrasound is then sattered in all diretions by the blood partiles and part of the sattered signal is reeived by the transduer and onverted to a voltage signal. The blood veloity is found through the repeated measurement at a partiular loation. The blood partiles will then pass through the measurement gate and give rise to a signal with a frequeny proportional to veloity. A oordinate system for the measurement is show in Fig The vetor r 1 indiates the position for one satterer, when the first ultrasound pulse interats with the satterer. The vetor r 2 indiates the position for interation with the next ultrasound pulse emitted T pr f seonds later. The movement of the satterer in the z-diretion away from the transduer in the time interval between the two pulses is D z = r 2 r 1 os(θ) = v os(θ)t pr f, (4.1) where θ is the angle between the ultrasound beam and the partile s veloity vetor v. The traveled distane gives rise to a delay in the seond signal ompared to the first. Denoting the first reeived signal as r 1 (t) and the seond as r 2 (t) gives r 1 (t) = r 2 (t t s ), t s = 2D z = 2 v os(θ) T pr f = 2v z T pr f (4.2) where is the speed of sound. Emitting a sinusoidal pulse p(t) with a frequeny of f 0 gives a reeived signal of p(t) = g(t)sin(2π f 0 t) for 0 < t < M/ f 0 r 1 (t) = p(t t 0 ) t 0 = 2d (4.3) where d is the depth of interrogation, and g(t) is the envelope of the pulse. Repeating the measurement a number of times gives a reeived, sampled signal for a fixed depth of r i (t) = p(t t 0 it s ) 52 Chapter 4. Ultrasound systems for flow estimation

56 x Amplitude Time [s] Time x Amplitude Figure 4.2: RF sampling of simulated signal from blood vessel. The left graph shows the different reeived RF lines, and the right graph is the sampled signal. The dotted line indiates the time when samples are aquired (from [4], Copyright Cambridge University Press). = g(t t 0 it s )sin(2π f 0 t t 0 it s ) (4.4) Here t is time relative to the pulse emission. Taking the measurement at the same time t x relative to the pulse emission orresponding to a fixed depth in tissue gives r i (t x ) = g(t x t 0 it s )sin(2π f 0 (t x t 0 it s )) = asin( 2π f 0 it s + Θ x ) (4.5) Θ x = 2π f 0 (t x t 0 ) assuming the measurement is taken at times when the envelope g(t) of the pulse is onstant. Suh a measurement will yield one sample for eah pulse-eho RF line, and thus samples the slow movement of the blood satterers past the measurement position or range gate as shown in Fig.4.2. The sampled signal r(i) an be written as: showing that the frequeny of this signal is: r(i) = sin( 2π 2v z f 0iT pr f + Θ x ) (4.6) f p = 2v z f 0, (4.7) whih is proportional to the projeted veloity of the blood in the diretion of the ultrasound beam. The reeived signal will not only onsist of a single frequeny, sine an ultrasound pulse is emitted and reeived. This pulse will be sampled, when it moves past the range gate, and the orresponding digital signal will thus have a spetrum determined by the pulse shape. The frequeny axis of the pulse spetrum will also be multiplied by the 2v z / fator due to the sampling operation. An example is shown in Fig The spetrum of the RF pulse is shown on the top, and the resulting frequeny axis for the reeived digital signal is shown on the bottom. This simple model an also be used for determining the effet of veloity aliasing, limited observation time, attenuation, and non-linear propagation. The signal must be sampled aording to the Nyquist limit, so that frequeny omponents in the spetrum are below half the sampling frequeny. Here it is f pr f = 1/T pr f, and the maximum 4.2. Measurement of flow signals 53

57 P(f) f 0 f Original RF frequeny axis 2v f z p = f 0 f prf 2 f Frequeny axis of sampled signal Figure 4.3: Frequeny saling of the reeived RF signal from the sampling for a number of pulse-eho lines (from [4], Copyright Cambridge University Press). veloity, that an be measured, is determined by: f pr f /2 > 2v max f 0 (4.8) Note here that the sampling frequeny is in the kilohertz range, and that the ultrasound frequeny is in the megahertz range, whih does no seem to make sense. It should, however, be kept in mind, that it is not the RF information that is sampled, but rather the slow movement of the blood satterers past the range gate. Obtaining only a limited number of pulse-eho lines will trunate the digital version of the measurement pulse. This orresponds to multipliation with a retangular window, and the resulting spetrum is onvolved with the Fourier transform of the window leading to a broadening of the spetrum. Ultrasound propagating in tissue is attenuated due to sattering and absorption. The attenuation is proportional to depth and frequeny and is typially in the range from 0.5 to 1 db/[mhz m]. This will for a pulsed system lead to a loss of signal energy and to a derease in enter frequeny of the pulse field. Often a shift in frequeny of khz an be experiened on a 5 MHz pulse. The shift is relative to the pulse enter frequeny, and is, thus, multiplied with the saling fator 2v z / giving rise to only a minor shift in the measured frequeny for the sampled signal. This is also the ase for non-linear propagation and sattering effets, whih then also sale proportionally to the veloity. The multi pulse measurement tehnique is, thus, very robust to the different physial effets enountered in medial ultrasound, and is therefore preferred to finding the Doppler shift of the pulse spetrum during the interation with the moving satterer. The veloity an be both towards and away from the transduer, and this should also be inluded in the estimation of veloity. The sign an be found by using a pulse with a one-sided spetrum orresponding to a omplex signal with a Hilbert transform relation between the imaginary and real part of the signal. The one sided spetrum is then saled by 2v z / and has a unique peak in the spetrum from whih the veloity an be found. The omplex signal an be made by Hilbert transforming the reeived signal and using this for the imaginary part of the signal. A Hilbert transform is diffiult to make with analog eletronis and two other implementations are shown in Fig The top graphs makes the demodulation by a omplex multipliation with exp( j2π f 0 t) and then lowpass filtration for removing the peak in the spetrum at 2 f 0. Mathing the bandwidth of the low-pass filters to the bandwidth of the pulse also gives an improvement in signal-to-noise ratio. The seond solution obtains the signal by quadrature sampling with a quarter wave delay between the two hannels. Both implementations gives omplex signals that an readily be used in the estimators for determining the veloity with a orret sign [4]. 54 Chapter 4. Ultrasound systems for flow estimation

58 Conventional analog demodulation os(2π f 0 t) Transduer S/H ADC I hannel Amplifier S/H ADC Q hannel sin(2π f 0 t) RF quadrature sampling Transduer Sample at 2 d 0 / S/H ADC I hannel Amplifier Mathed filter S/H ADC Q hannel Sample at 2 d 0 / + 1/(4f 0 ) Figure 4.4: Demodulation shemes for obtaining a omplex signal for determining the sign of the veloity (from [4], Copyright Cambridge University Press). 4.3 Calulation and display of the veloity distribution Using a number of pulse-eho lines and sampling at the depth of interest, thus, gives a digital signal with a frequeny proportional to veloity. Having a movement of a olletion of satterers with different veloities then gives a superposition of the ontribution from the individual satterers and gives rise to a spetral density of the signal equal to the density of the veloities. Typial profiles for a stationary laminar flow are given by: ( r p0 ] v(r) = v 0 [1 R) (4.9) where r is radial distane, R is the radius of the vessel, v 0 is the maximum veloity at the enter of the vessel and p 0 is an integer. Under uniform insonation of the vessel, the frequenies reeived are: f d (r) = 2v 0 f [ ( 0 r p0 ] 1 os(θ). (4.10) R) and the normalized power density spetrum is [4]: 2 ) G( f d ) = p 0 f max (1 f 1 2 d p0 fmax 0 for f d > f max f max = 2v 0 f 0 os(θ). (4.11) The profiles given in (4.9) are for a stationary flow in a rigid tube. More ompliated flow patterns are found in the human body sine the flow is pulsatile and the vessels urve and branh (see Setion 4.5). The power 4.3. Calulation and display of the veloity distribution 55

59 Figure 4.5: Duplex san showing both B-mode image and spetrogram of the arotid artery. The range gate is shown as the broken line in the gray-tone image. The square brakets indiate position and size of the range gate (from [4], Copyright Cambridge University Press). density spetrum however still orresponds to the veloity distribution, and ultrasound measurements an be used for revealing the veloity distribution over time. An example for the arotid artery supplying the brain is shown in Fig The left part shows the anatomi gray sale image of the artery with the plaement of the range gate. The right side shows the spetrogram, whih gives the veloity distribution over time. The spetrogram is found from the Fourier transform of the measured sampled signal. Usually a new spetral estimate is alulated every 5 to 10 ms and the estimates are presented in a rolling display with the gray level orresponding to the number of satterers with the partiular veloity. The non-stationary nature of the distribution is learly seen as is the periodiity with the heart yle. 4.4 Making images of veloity Images of veloity an also be made using ultrasound. This is done by aquiring 8 to 16 lines of data in one diretion and then estimate the veloities along that diretion. The beam is then moved to another diretion and the measurement is repeated. Doing this for all diretions in an image gives a mapping of the veloity and that is displayed in olor flow mapping (CFM) systems. A system diagram for a CFM system is shown in Fig The RF signal from the transduer is demodulated to yield a omplex signal, whih is sampled by the ADC s at all the positions along the diretion where the veloity must be found. The demodulation step also serves as a mathed filter to suppress noise. The digital signal is then passed on to the filter for removing stationary signals. The delay line aneler (DLC) subtrats the samples from the previous line from the urrent line to remove the stationary signal. The tissue surrounding the vessels often generate a sattered signal that is 40 db larger than the blood signals, and this will seriously deteriorate the veloity estimates. The tissue signal is, however, stationary and an therefore be removed by subtrating samples at the same depth in tissue for onseutive pulse-eho lines. The sampled data for one diretion is then divided into a number of segment, one for eah pixel in the veloity image. Eah segment holds 8 to 16 omplex samples from whih the veloity must be estimated. The mean 56 Chapter 4. Ultrasound systems for flow estimation

60 os(2π f 0 t) Depth Transduer S/H ADC DLC Estimator v TGC amplifier S/H ADC DLC Pulser sin(2π f 0 t) Figure 4.6: Blok diagram of olor flow mapping system (from [4], Copyright Cambridge University Press). veloity an be found from mean angular frequeny determined by: ω = R + R + ωp(ω)dω. (4.12) P(ω)dω P(ω) is the power density spetrum of the reeived, omplex signal. The veloity is then given by: v z = ω 4π f 0 (4.13) The veloity an also be alulated from the phase shift between the pulse emissions. The omplex digital signal an be written r(i) = x(i) + jy(i) (4.14) where x is the real signal and y the imaginary signal. Using an autoorrelation estimator the veloity estimate is [42] ( ) N 1 i=1 y(i)x(i 1) x(i)y(i 1) v z = artan, (4.15) 4π f 0 T pr f x(i)x(i 1) + y(i)y(i 1) N 1 i=1 where N is the number of samples. This implementation is very effiient, sine only few alulations are performed for eah estimate, and this is the approah preferred in most sanners. The method is also robust in terms of noise, whih is ritial sine the sattering from blood is quite weak ompared to the surrounding tissue. An example of a olor flow image of the artery supplying the brain is shown in Fig Both positive and negative veloities an be seen. The jugular vein on the top has a blue olor oding showing that the movement is towards the transduer, whereas the arotid artery below ontains red olors for movement away from the transduer and to the brain Color flow mapping using ross-orrelation The estimation of veloity an also be done by finding the time-shift between two onseutive RF signals diretly [43], [44], [45]. This is the approah taken in the ross-orrelation system shown in Fig The reeived RF signal is amplified and filtered with a mathed filter to remove noise. An RF sampling at 20 to 30 MHz for a 5 MHz transduer is then performed and the removal of stationary signals is done by subtrating two onseutive RF lines. The data is divided into segments as shown in Fig The veloity is found in eah segment by ross-orrelation with data from the previous pulse-eho line. The reeived signal an be written as: r(t) = p(t) s(t) (4.16) 4.4. Making images of veloity 57

61 Figure 4.7: Color flow mapping image of the arotid artery and jugular vein. Blue olors (top of the olor bar) indiates veloities towards and red away from the transduer (from [4], Copyright Cambridge University Press). Depth Transduer S/H ADC DLC Estimator v TGC amplifier Pulser Mathed filter RF sampling Figure 4.8: System for ross-orrelation flow estimation (from [4], Copyright Cambridge University Press). Pulse emission T prf Segment number 1 t r 0 r 1 r 2 Figure 4.9: Segmentation of RF data prior to ross-orrelation (from [4], Copyright Cambridge University Press). 58 Chapter 4. Ultrasound systems for flow estimation

62 R v(r) r v o Figure 4.10: Laminar flow in a long, rigid tube. where p(t) is the emitted pulse and s(t) is the sattering signal from the blood. Cross-orrelation two reeived signals then gives: R 12 (τ) = E{r 1 (t)r 2 (t + τ)} = E{(p(t) s(t))(p(t) s(t t s + τ))} = R pp (τ) R ss (τ t s ) (4.17) where R pp is the autoorrelation of the pulse and R ss is the autoorrelation of the blood sattering signal. s(t) an be assumed to be a white, Gaussian stohasti proess, so that the ross-orrelation an be written as: R 12 (τ) = σ 2 s R pp (τ t s ) (4.18) where σ 2 s is the power of the blood signal. The autoorrelation of the pulse has a unique maximum at R pp (0), and the time-shift an thus be found from the unique maximum in the ross-orrelation funtion. The veloity is then given by: t s v z = (4.19) 2 T pr f The veloity estimates are then presented by the same method as mentioned for the autoorrelation approah. The ross-orrelation approah is optimized by sending out a narrow pulse, whih also makes is easier to obtain a higher resolution in the olor flow mapping images. The prie is, however, that the peak intensities must be limited due to the safety limits imposed on the sanners. Therefore ross-orrelation sanners an send out less energy and are often less sensitive than the sanners using the autoorrelation approah. An other disadvantage of the ross-orrelation approah is the large amount of alulations that must be performed per seond. For a real time system it an approah one billion alulations per seond making it neessary to use only the sign of the signals in the alulation of the ross-orrelation [46]. 4.5 Flow types Several different flow patterns an be found in the body and this setion will give a brief overview of how to model a simple pulsatile flow. 4.6 Stationary paraboli flow The steady flow of a Newtonian fluid in a long, rigid tube is shown in Fig The flow will be laminar, so that 4.5. Flow types 59

63 the veloity at a radial position is independent of position along the tube. The distribution of veloities is paraboli and given by ) v(r) = (1 r2 R 2 v 0, (4.20) where r denotes radial position, R is the radius of the tube, and v 0 is the maximum veloity attained at the enter of the tube. Note that the veloity is zero at the walls. This is, in general, true for any kind of veloity profile for a visous fluid. The visosity of the fluid will give a resistane to flow. A pressure differene is, thus, needed to overome this for maintaining a steady flow. The relation between veloity and pressure differene is given by P = R f Q, (4.21) where R f is the visous resistane. The relation is named after Poiseuille, who studied flow in apillary tubes and published his findings in 1846 [47]. For a laminar flow with a paraboli veloity profile the visous resistane is R f = 8µl πr 4, (4.22) where l is the distane over whih the pressure drop is found. The resistane is highly dependent on vessel radius. Dereasing the radius by a fator of two inreases the resistane by a fator of 16. The resistane given in the equation is only valid for a paraboli profile. Different values are enountered for other profiles. Poiseuille s law is equivalent to Ohm s law for eletrial iruits, and P is equivalent to voltage and Q to urrent Pulsatile flow in rigid tubes The flow in the human body is generated by the pulsatile ation of the heart, and a pulsatile rather than steady flow is enountered throughout the irulatory system. The veloity at a point in a vessel is, therefore, dependent on time and experienes aeleration and deeleration. This affets the veloity profile, whih annot be onsidered paraboli, even when a steady state of pulsation is reahed. Poiseuille s law an be reformulated for pulsatile flow by assuming linearity, i.e., that the flow pattern an be deomposed into sinusoidal omponents and then added to obtain the veloity variation in time and spae. This assumes that the fluid is Newtonian. Further, entrane effets are disarded, and it is assumed that the flow has attained a steady state of pulsation. The relation between pressure differene and volume flow rate is then for a single, sinusoidal omponent [48]: Q(t) = 8 M 10 R f α 2 Psin(ωt φ + ε 10) ρ α = R µ ω (4.23) R f = 8µL πr 2 when the pressure differene is Pos(ωt φ). Here ε 10 determines the phase of the wave and M 10 its amplitude. They are both omplex funtions of α. The expressions are given in Appendix A of [4]. Womersley s number α indiates how far removed the pulsatile flows pressure-flow rate relation is from that of steady flow. For α < 1 the flow rate and pressure is in phase, whereas for α > 1 the flow rate will lag the pressure differene. The evolution of α 2 /M 10 and ε 10 as a funtion α is shown in Fig α2 /M 10 beomes equal to 8 for α approahing zero and ε 10 approahes 90, so that (4.23) takes the form of the Poiseuille s equation for steady flow. 60 Chapter 4. Ultrasound systems for flow estimation

64 α 2 /M α α 2 /M α ε α Figure 4.11: The Womersley s parameters α 2 /M 10 and ε 10 as a funtion of α. An inrease in frequeny will inrease α 2 /M 10 and demand a larger pressure differene to obtain the same volume flow rate. The phase differene between pressure and flow rate will also inrease and approah 90 for high frequenies. Assuming a heart rate of 62 beats per minute, α is equal to 10.2 for the first harmoni in the aorta, and 1.69 in a small peripheral vessel (R = 0.5 m). The pressure differene and volume flow rate will, thus, be nearly 90 out of phase in the aorta and will be in phase in the peripheral vessel. Higher harmonis of the flow rate in the aorta will be severely attenuated due to visosity, as an be seen in the graph for α 2 /M 10. A disussion of the validity of the Womersley equations is given by Nihols and O Rourke (1990). From a knowledge of volume flow rate, Evans (1982b) has shown that it is possible to alulate the veloity profile for the steady-state pulsatile flow, when entrane effets are negleted. The relation is given by: v m (t,r/r) = 1 πr 2 Q m ψ m (r/r,τ m ) os(ω m t φ m + χ m ) ψ m (r/r,τ m ) = τ αj 0 (τ m ) τ α J 0 ( R r τ m) τ α J 0 (τ m ) 2J 1 (τ m ) χ m = ψ(r/r,τ m ) ρ τ m = j 3/2 R µ ω m, (4.24) where J n (x) is the nth order Bessel funtion. Also, ψ( R r,τ m) denotes the angle of the omplex funtion ψ, and ψ denotes its amplitude. The funtion ψ is dependent on radial position in the vessel, angular frequeny, and the fluid. It desribes how the veloity hanges with time and position over a yle of the sinusoidal flow. Thus, from one measurement of the volume flow rate the whole veloity profile an be alulated. The evolution of ψ and ψ for different radial positions and values of α is shown in Fig Note that the veloity is always zero at the vessel boundary. For α low a nearly paraboli profile is obtained, and the profile beomes progressively more blunt with an inrease in α, so the entral ore of the fluid will move as a whole. The volume flow rate is related to the mean spatial veloity by: Q(t) = A v(t). (4.25) 4.6. Stationary paraboli flow 61

65 2 abs(ψ) α=2 α=5 α=10 α= Normalized radius 40 angle(ψ) 20 0 α=2 α=5 α=10 α= Normalized radius Figure 4.12: Graph of ψ for different values of r/r and α. Assuming that the fluid is Newtonian, so linearity an be assumed, it is possible to superimpose the different sinusoidal omponents to obtain the time evolution of the pulsatile flow. The reverse proess is also possible. Here Q(t) or v(t) is observed and the individual sinusoidal omponents found by Fourier deomposition: V m = 1 T The spatial average veloity is then given by: Z T 0 v(t) = v 0 + φ m = V m. v(t) exp( jmωt)dt. (4.26) m=1 V m os(mωt φ m ) (4.27) Using (4.24) and (4.27) it is now possible to reonstrut the time evolution of the veloity profile for the pulsatile flow by: ( ) r 2 v(t,r/r) = 2v 0 (1 + V m ψ m os(mωt φ m + χ m ). (4.28) R) The first term is the steady flow. m=1 By observing the spatially averaged flow volume, the whole profile an be reonstruted. This an be attained by ultrasound systems insonifying the whole vessel and proessing the returned signal. Examples of the mean veloities from the ommon femoral and arotid arteries are shown in Fig and the different values for V p and φ p are given in Table 4.1. The resulting veloity profiles are shown in Fig Profiles for the whole ardia yle are shown with time inreasing toward the top. The dotted lines indiate zero veloity. 4.7 Simulation of flow imaging systems The derivation given in this hapter generally assumes that the single satterer stays within a region of uniform insonation. This is quite a rude assumption for realisti beams, as shown in Chapter 2. Both ontinuous and 62 Chapter 4. Ultrasound systems for flow estimation

66 1 Veloity [m/s] Phase [deg.] Veloity [m/s] Phase [deg.] Figure 4.13: Spatial mean veloities from the ommon femoral (top) and arotid arteries (bottom). The phase indiated is that within a single ardia yle (from [4], Copyright Cambridge University Press). Common femoral Common arotid Diameter = 8.4 mm Diameter = 6.0 mm Heart rate = 62 bpm Heart rate = 62 bpm Visosity = kg/[m s] Visosity = kg/[m s] m f α V m /v 0 φ m m f α V m /v 0 φ m Table 4.1: Fourier omponents for flow veloity in the ommon femoral and ommon arotid arteries (data from [50]) 4.7. Simulation of flow imaging systems 63

67 Profiles for the femoral artery Profiles for the arotid artery Veloity Veloity Relative radius Relative radius Figure 4.14: Veloity profiles from the ommon femoral and arotid arteries. The profiles at time zero are shown at the bottom of the figure and time is inreased toward the top. One whole ardia yle is overed and the dotted lines indiate zero veloity (from [4], Copyright Cambridge University Press). pulsed fields vary with position and this needs to be taken into aount using the Tupholme Stepanishen field model. The reeived voltage trae was stated in Setion to be p r ( r 1, r 2,t) = v pe (t) t f m ( r 1 ) r h pe ( r 1, r 2,t), (4.29) where f m aounts for the sattering by the medium, h pe desribes the spatial distribution of the ultrasound field, and v pe is the one-dimensional pulse emanating from the pulse exitation and onversion from voltage to pressure and bak again. The model in Setion 4.2 states that the satterer will move during the interation with the ultrasound giving rise to a Doppler shift. The most important feature is, however, the interpulse movement, beause this is used by the pulsed sanners for deteting veloity. The approximation is then to inlude the small Doppler shift into the one-dimensional pulse v pe, shifting its frequeny ontent to f = (1 + 2v z /) f, and assuming that the field interating with the satterer stays onstant. Usually the pulse duration is a few miroseonds, so that the satterers only move a fration of a millimeter, even for high blood veloities, during the interation, and the field an safely be assumed to be onstant over this distane. The reeived voltage trae is then found diretly from (4.29) with r 2 indiating the position of the satterers. Note that a spatial onvolution takes plae and that the reeived response is a summation of ontributions from numerous satterers. The satterers move to the position r 2 (i + 1) = r 2 (i) + T pr f v( r 2 (i),t) (4.30) when the next field from the subsequent pulse emission impinges on the satterers. Here i denotes the pulse or line number and v( r 2 (i),t) is the veloity of the satterer at the position indiated by r 2 (i) at time t. This assumes that the satterers do not aelerate during the interation. The movement between pulses is T pr f v( r 2 (i),t), and the new position of the satterers, r 2 (i + 1), an then be inserted in (4.29) and used for alulating the next voltage trae. The reeived signals for multiple pulse emissions an, thus, be found by these two equations. The atual alulation is rather ompliated in three dimensions, but is easy to handle by a omputer. The different veloities for the satterers neessitate that separate Doppler shifts are inluded in v pe for eah satterer. This an be done in a omputer simulation of eah satterer before the ontributions for all satterers are summed, or a mean Doppler shift an be used. The effet of this Doppler shift is minor in pulsed systems and an, at least to a first approximation, be negleted. A fairly realisti simulation should, thus, be possible with this approah. 64 Chapter 4. Ultrasound systems for flow estimation

68 4.7.1 Examples of flow simulations A phantom was developed for evaluating olor flow imaging. It generates data for flow in vessels with properties like the arotid artery, The veloity profile is lose to paraboli, whih is a fairly good approximation during most of the ardia yle [4] for a arotid artery. The phantom generates 10 files with positions of the satterers at the orresponding time step. From file to file the satterers are then propagated to the next position as a funtion of their veloity and the time between pulses. The ten files are then used for generating the RF lines for the different imaging diretions and for ten different times. A linear san of the phantom was made 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 Setion 3.6. A single transmit fous was plaed at 70 mm, and reeive fousing was done at 20 mm intervals from 30 mm from the transduer surfae. The resulting signals have then been used in a standard autoorrelation estimator [4] for finding the veloity image. The resulting olor flow image is shown in Fig Note how the vessel is larger at the bottom than the top. An other phantom is used for simulating data for a spetral Doppler system. It alulates RF data as measured from arteria femoralis in the upper leg. The phantom generates a number of files for partiular time instanes. The satterers are then propagated between pulses aording to the Womersley model mentioned in Setion 4.5 thereby generating a three-dimensional, pulsed model of the flow in the femoral artery. The satterer position files are all generated before the simulation of the fields take plae. For a pulse repetition frequeny of 5 khz, 5000 files are generated per seond of simulation time, so a large amount of data is involved here. The satterer position files are then used during simulation for generating one RF signal for eah pulse emission. These files an then be used for doing either spetral analysis or for estimating the profile in the artery. The simulation is made so multiple workstations an simutaneously work on the problem, as long as they have aess to the same diretory. This is done by looking at whih RF files that have been generated. The first one not simulated is reserved by the program by writing a dummy file, and then doing the simulation. Multiple workstations an then work simultaneously on the problem and generate a result quikly. The resulting spetrogram is shown in in Fig Simulation of flow imaging systems 65

69 30 40 Axial distane [mm] Lateral distane [mm] Figure 4.15: Color flow image of vessel with a paraboli flow profile. 66 Chapter 4. Ultrasound systems for flow estimation