Theory of ultrasound Doppler-spectra velocimetry for arbitrary beam and flow configurations

740 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 35, NO. 9, SEPTEMBER 1988

Theory of Ultrasound Doppler-Spectra Velocimetry for Arbitrary Beam and Flow Configurations

DAN CENSOR, VERNON L. NEWHOUSE, FELLOW, IEEE, THOMAS VONTZ, AND HECTOR

Abstract-In conventional ultrasound Doppler systems, the velocity component along the beam axis is derived from the observed frequency shift. Recently, it was verified that by using a pulsed-Doppler system with the beam transversely oriented with respect to the flow, the velocity component transverse to the beam can be derived from the edges of the spectrum. Presently, these results are generalized to take into account arbitrary angles of incidence, effects of velocity gradients, arbitrary apertures, and arbitrary source pulses. For uniform apertures and transverse flow, it has been previously shown that the Doppler output spectrum is symmetrical about zero frequency, with its width depending on the Doppler effect due to the transverse velocity and the geometry of the problem. For a beam direction oblique to the velocity, it is shown that the spectrum is now shifted, and is centered about the classical Doppler frequency. I t is shown here that for velocity gradients and transverse flows the spectrum remains symmetrical, with the edges corresponding to the maximal velocity; however, the profile becomes peaked at the center. For oblique incidence, an asymmetrical spectrum is obtained and its edges are related to the maximal and minimal velocities within the sampling volume.

For the simple case of a long strip transducer discussed previously, it was shown that the Doppler system output spectrum is essentially obtained by convolving the transmitter and receiver aperture functions. The general discussion given here, even for the single particle, is more complicated. Nevertheless, using the reciprocity theorem it is shown that the output spectrum is obtained by convolving the particle excitation spectrum due to monochromatic excitation, with the receiver input spectrum due to a moving monochromatic source, all this shifted to the classical Doppler frequency mentioned above. I t is shown that when the excitation possesses an arbitrary (narrow band) spectrum, this spectrum, replicated in a prescribed manner, has to be convolved with the spectra derived above. Following the single scatterer analysis, the effects produced by an ensemble of particles are considered. Numerical computations of theoretical results are given, sup- porting the semi-quantitative graphical interpretations for various configurations. To further substantiate the results analytical treat- ments of simplified models are given.

The results contribute to our understanding of Doppler spectra and their use, mainly in medical ultrasound diagnostics.

INTRODUCTION LTRASOUND DOPPLER techniques are currently U used for analyzing motional effects, e.g., in clinical

Manuscript received November 23, 1987; revised April 18, 1988. This work was supported by NSF Grant ECS-8219736 and by NIH Grant R01- HL 34986.

D. Censor is with the Department of Electrical and Computer Engi- neering, Ben Gurion University of Negev, Beer Sheva, Israel.

V. L. Newhouse and H. V. Ortega are with the Department of Electrical and Computer Engineering and the Biomedical Engineering and Science Institute, Drexel University, Philadelphia, PA 19104.

T. Vontz is a Visiting Scientist at the Biomedical Engineering and Sci- ence Institute, Drexel University, Philadelphia, PA 19104, on leave from the Fraunhofer Institut fuer zerstoerungsfreie Pruefverfahren, Saar- bruecken, Federal Republic of Germany.

IEEE Log Number 8822270.

V. ORTEGA

diagnostics of the cardiovascular system. For discussion of principles and references, see, for example, Fish [l]. The common base for such techniques is the phenomenon of frequency shift caused by the motion of scatterers relative to the ultrasound beam axis. Thus, for an object moving with velocity U = v6 at an angle 8 with respect to the propagation vector k = kk of a plane wave where 6, k are unit vectors, and k = w o / c , wo being the (angular) frequency, and c is the speed of sound, we get, to the first order in v / c a scattered wave with a frequency shift 111

wd = (2wov /c ) cos 8 , cos 8 = k 6. (1) In (1) the factor 2 is a result of the fact that a monostatic configuration is considered, whereby the back-scattered wave propagates in the direction -6. Only the first order effects in v / c are significant. Higher-order effects are negligible for low velocities and in any case it must be remembered that the whole subject of linear acoustics is based on approximations of the first order in the Mach number. The formula (1) has been derived on the basis of a time harmonic (monochromatic) signal, but is nevertheless used for pulsed Doppler systems, provided the length of individual pulses is much larger than the wavelength of the carrier wave. The effect vanishes for motion perpendicular to the beam axis. However, since all real beams have a finite cross section, i.e., are not a planewave of infinite extent, or equivalently, since transducer apertures are finite, Doppler spectra are produced even in the transverse mode. The relation of Doppler spectra to beam geometry has been discussed recently [2], [3] on the basis of ray geometrical considerations. Recently, it has been shown experimentally [4] and verified theoretically [6] that for beam axis and flow at right angles the amplitude Doppler spectrum produced by a uniformly insonified aperture of a long strip transducer possesses a triangular profile, and that the edge frequencies of the spectrum profile, which are relatively easy to measure, are related to the velocity by means of a simple formula.

In order to better understand the Doppler spectra discussed below, a short review of the principles and properties of typical Doppler ultrasound systems is appropriate [7]. Continuous Doppler systems emit a frequency wo and receive Doppler shifted spectral components of various frequencies w. In the simplest case wd = wo - w is given by (1). The received signal is multiplied by the carrier signal with frequency wo and low-pass filtered, thus

0018-9294/88/0900-0740$01 .OO O 1988 IEEE

CENSOR: et al.: DOPPLER-SPECTRA VELOCIMETRY FOR ARBITRARY BEAM AND FLOW CONFIGURATIONS 74 1

yielding wd. In pulsed Doppler systems, periodic pulses of center frequency oo and a repetition (angular) frequency are transmitted. At the receiver, after down- shifting the signal by wo, it is sampled at the pulse repetition frequency 62 with a gate pulse whose time delay relative to the transmit signal determines the distance of the so called range cell or measurement volume. It has been shown [8] that, if the beam is almost parallel to the flow and the range cell is short enough, the Doppler system output amplitude spectrum Ao( p ) is a frequency scaled replica of the echo amplitude spectrum A ( p ) , given by

with the scaled longitudinal Doppler spectrum given in

Clearly there are severe restrictions that must be applied, in order to be able to define tractable problems. In the present study, a laminar and steady flow is assumed. Spectral effects resulting from turbulence and flow accel- eration are therefore beyond the scope of this study. Also, assumptions are made on the propagation medium, which are not always realistic. It is assumed that the medium is linear, homogeneous and lossless. All these are interest- ing effects in the context of measurements and diagnostics of the cardiovascular system. There is, therefore, a need to further analyze the problem, tackling the complicated aspects one at a time in hope that in time a better understanding of the question of Doppler spectra will emerge.

(2).

This then is the effect of the pulsed Doppler system in- strumentation when the effect of the transmitter and receiver's finite aperture can be neglected.

Previously [6], transverse motion and constant velocity have been assumed. Furthermore, in the past only monochromatic (CW) signals were considered, which is adequate for very long pulses (i.e., pulses containing many cycles of the carrier wave). The model used for the present analysis is more realistic and general, addressing the questions of arbitrary flow angles, various transducer aperture geometries and arbitrary transmitter pulse shapes. The cases treated earlier [6] are special cases. It is assumed in the present study, as was done previously, that the particles producing the Doppler effects are small, that their positions are uncorrelated, and that their motion co- incides with the ambient fluid's velocity field.

While studying the transverse flow case [6], it became clear that there are two main considerations underlying the theory for Doppler spectra. The first is the well known reciprocity principle [9], which states that the spatial properties (i.e., the radiation pattern) of a transducer are the same whether it acts as a transmitter or a receiver. Secondly, for the special case of constant velocity and long strip transducer, it was shown that the Doppler amplitude spectrum has a shape which is obtained by convolving the transmitter and receiver aperture functions. A more general formulation is given below, again involving convolution of two spectra which are determined by the transverse velocity component and the transmitter and receiver's radiation patterns.

The remainder of the paper is arranged as follows. Ex- ploiting the results obtained from the reciprocity and spectra convolution principles, and assuming monochromatic signals, Doppler spectra produced by transverse and oblique flows are analysed for various transducer apertures and flow configurations. Numerical calculations provide us with graphical results which are illustrated and substantiated by analytical discussions in Appendixes B- D. In order to facilitate analysis, the examples chosen are highly idealized. For arbitrary narrow band pulses, as found in pulsed Doppler systems, it is shown that the single frequency excitation spectrum has to be convolved

EXTENDED THEORY AND THE DOPPLER SPECTRUM CONVOLUTION THEOREM

The Particle Excitation Field

It is well known that at large distances from the source, the solution of the wave equation can be approximated by a spherically outgoing wave, multiplied by the so called scattering amplitude [lo], or directivity function. Thus, we have

(3 ) 1 r i = po - ei"-'"tg(P), k = @ I C

where r is the distance from the origin, w is the (angular) frequency, c is the sound speed and 3 = 3 ( 8 , 4) is a unit vector depending on directions 8, the polar angle, and 4 the azimuthal angle, as shown in Fig. 1. The quantity 9' can be any solution of the acoustical wave equation, e.g., the velocity potential. The constant io, which also takes care of units involved, is arbitrary and can be set equal to unity. We can also represent \k in terms of the source amplitude or aperture function \ka ( p ), dependent on the radius vector p,

\k = e-''"' 1 e i k R i , ( p ) dv( p) , R = I r - P I (4)

and for cases where the source is distributed on a surface, (4) becomes a surface integral with ds ( p ) replacing dv( p). Furthermore, it is now assumed that this surface is planar and perpendicular to the ray 8 = 0, as in Fig. 1 . In the far field

R = r - p . f = r - p - r / r = r - ( x / r - q y / r ( 5 )

leading to the Fraunhofer diffraction approximation [l 11 of (4):


exciting the particle is therefore

( 1 1 ) q e = - erkro-iwr ' 1 e- i@G,(p)dp. r0

In response to a unit amplitude excitation at frequency p' = w + p , the particle radiates a field

, k' = p ' / c (12) i k ' r ' - i p ' t 4-4 r, e

Fig. 1 . Source-observer geometry.

Comparison of (3), (6) shows [12] that qa, g constitutes a two dimensional Fourier transform pair.

Consider now a single particle moving according to

r = r, + ut

1 r 1 = r = [ ( r , + Ut) - (ro + = ro + v i o l

3 = r / r = to + vt/r , - to(t0 * U ) t / r o

= to + u + t / r o (7)

where r' is the distance from the particle. It is assumed that the particles are small and sparse, and therefore, the scattering is isotropic, i.e., the scattering amplitude a( p' ) is constant with respect to directions, but depends on the excitation frequency (e.g., according to the Ray- leigh scattering theory [ l o ] , [ l l ] ) . According to the reciprocity principle [ 9 ] , if the particle acts as a source at frequency p' , it will produce in the transducer, now act- ing as receiver, a field depending on the location of the source within the transducer's radiation pattern gpr ( 3 ). Since obviously r' = r = r,, at the receiver we obtain the signal

1 -ip't ' where r,, U are constants and the time origin is immate- rial, as explained below, and U, is the vector component (13)

for short-time intervals only, such that the leading terms where t' is the retarded time referred to the particle's po- retained in (7) provide an adequate approximation. This sition. Thus, in (13) we have is applicable to focused transducers, where ro = Fo be-

rectivity function is very sharp; hence, the distance v + t the particle traverses in the insonified focal region is small compared to Fo. Substituting (7) in (3) yields g ( ? [ t ] ) , and substitution of (7) in (6)' noting that e- lkpo'p = 1, and

\k, = a( p ' ) - e g p g ( P ) , t' = t - ro/c of the velocity v perpendicular to r,. This analysis is valid r0

comes the focal length and U + t / r o << 1 because the di- s , t ( r [ t ' l ) = 1 e- ipttGpv( p ) dp ( 1 4 )

where for p' = U, gpt (3 ) and g, (3 ) become identical. Combining (11)-(14), the received field is given by

n s 3, p = 0 results in $R = e ikro [ d p ~ , ( p ) a (p ' )e - ' " ' ' '

g ( f [ t ] ) = \ e - i k v + . p t / r o * a ( P ) d 4 P ) . ( 8 )

Thus (8) yields g (3[ t ] ) as a function of t and provides a - _ eikm - i o t ' relation between the field exciting the particle and the ap-

erture function qa( p). Corresponding to (8), we define a -

4 spectrum

G , ( d = F g ( W 1 ) ( 9 ) 1 dpe-ipt'G,t( p ) ] . ( 1 5 )

where F denotes the Fourier transform and w indicates the excitation frequency in (8). For the special case of the long strip transducer it is shown below that G,( p ) rep- licates the aperture function.

The Doppler Spectrum Convolution Theorem Essentially, this theorem follows from the well known

reciprocity principle [9] for the interchange of source and observer mentioned above. For simplicity let us first assume U - r = 0, i.e., transverse motion. For this case r = r, as in (3). The spectrum exciting a particle is given by

As (15) stands, the two integrations are not separable since p of the first integral appears in Gpl of the second one. However, since the creation of the spectrum is already a first order velocity effect, we are justified in neglecting higher order effects appearing in (15). In other words, the approximation is justified because a( p' ), Gpt are narrow- band functions with respect to p' . Thus, we write

aa a( p ' ) = . (U) + - p + * - = . (U)

aw

& ( C L - 4 * G U b ) ( l o ) where 6 ( p - w ) is due to the factor e-""'in (3). The field

Since w is a constant for monochromatic excitation a( w ) is taken to be a constant, and GPO = G,. Consequently

CENSOR: et al.: DOPPLER-SPECTRA VELOCIMETRY FOR ARBITRARY BEAM AND FLOW CONFIGURATIONS 143

( 15) becomes 2

* R - - Ke-'ot' [ s d P e - i % d ] (17)

where K absorbs the extra terms in (15). Inasmuch as (17) describes a product of time domain functions, the amplitude spectrum of the received signal is given by

A h ) = S ( P - 4 * Gotcl) * GJCL). (18) The Doppler spectrum convolution theorem (1 8) states

that the spectrum A ( p ) measured at the receiver is obtained by convolving the excitation spectrum of the moving particle by itself, then shifting the zero reference to the transmitter frequency w . As explained above, in an ultrasound Doppler system, the frequency is downshifted by w , yielding Ao( p ) , i.e.,

Ao(I.4 = G o ( d * G o ( P ) . (19)

Thus far, in calculating the effect of the particle velocity on its excitation, only the velocity effect on g(P[t]) was considered. But when v r # 0, then the spherical wave part of (3), (6) is affected, too. According to (7) we obtain

/ ( r o + v * ~ ~ t ) = erkm-iolt /r0 eik(m + 1 ) . Po. t ) - iot

U ] = w(1 - v - i o ) / .

(20) where in the denominator the velocity effect is neglected, due to the fact that the observation time t is restricted, i.e., when the flow moves the particle outside the focal region, it is eventually lost, as far as the performance of the system is concerned. However, in the exponent a Doppler frequency shift is obtained. This means that instead of (10) we now have

- U11 * Go,(CL) = S ( P - w1) * G o ( d (21)

where w1 is defined in (20), and the same argument leading to the approximation (16) is applied in (21). Similarly (18), the echo amplitude spectrum, is now modified

A ( P ) = S ( P - w2) * G W ( L . 4 * Go(l.4

w2 = w(1 - 2v * Po/c) (22)

and in the geometry of Fig. 1 v - ?o = v 2. Instead of monochromatic excitation, consider now a

transmitter pulse whose spectrum H ( w ) is given. In a more general context, we would also have to include the transmitting and receiving transducers spectral transfer functions. In order to keep the argument simple, we do not attempt here to go into all the details. The attribution of H ( w ) to the pulse shape will be enough to bring out the salient effects on the Doppler spectrum. The contributions of all spectral components must be properly added, i.e., the received field is given by multiplying (15) by H ( w ) and integrating over all transmitter pulse frequencies. Consider first the case of transverse motion. In-

stead of qR (15), we have the integral

1 dwH( U ) \ k R . (23)

Inasmuch as qR (15) or its approximate form (17) involve integration, (23) becomes a nonseparable integral. Only for narrow-band pulses, such that most energy is concentrated in the vicinity of the carrier frequency wo, will the general expression (23) become separable. Again we have a narrow-band approximation as in (16)

. ( U - W O ) + = G,(p) (24)

and similarly for the functions Gp,, a( p' ) in (15), and instead of (17) we now have

(25)

constituting independent integrals. Corresponding to (1 8), the spectrum is given by

N P ) = H ( P ) * G,(cL) * G,(cL)

= H ( P ) * & ( C L ) . (26)

Ao( p ) given by (19). The same considerations apply to oblique motion, with the appropriate modifications discussed below.

For oblique incidence and monochromatic excitation (22) applies. When the transmitter emits a signal having a spectrum H ( p ) , corresponding to a time domain signal h ( t ) = j H ( p ) e-'" dp, then the particle is excited by frequencies pl = p a = p ( 1 - v - Po/c), i.e., by a signal h l ( t ) = j H ( p ) H I ( p l ) e-ip" d p I , where by change of variable

dp =

(27)

Consequently, except for the amplitude factor 1 /CY, the spectrum H ( p / a ) appears as a compressed or expanded replica of the original H ( p ) . The echo signal involves p2 = pp = p ( 1 - 2u - Po/c) and the spectrum at the receiver will be given by

where ( 1 / p ) H ( p / p ) is the spectrum of the time domain function h2 ( t ) picked up by the receiving transducer. For p = 1, i.e., transverse flow, (26) is obtained as a limiting case. The signal processing performed on the pulse Dopp- ler system has been analyzed before [8], and it has been shown that the detected signals may appear time inverted, depending whether > 1 or p < 1. The corresponding spectrum is given in (2). Consequently, the output spec-


trum corresponding to (19) is now modified, yielding

1 A o ( r ) = Y H ( p / y ) * G , ( L L ) * G , ( p ) , r = 11 - P I .

(29) Note that in (29) H( p / y ) depends on the longitudinal velocity component v - Po while G,( p ) depends on v,, the transverse component. A simple proof of (29) is given in Appendix A. By deconvolving ( 1 /y ) H ( p l y ) out of (29), we can restore (19) and we are able to extract information about the velocity. In most practical cases this ad- ditional broadening factor is very small; hence, the de- convolution is not necessary.

Up to now the spectrum produced by a single particle has been discussed. We assume that the particles have random uncorrelated positions, and therefore produce uncorrelated signals, but their velocity is deterministic, co- inciding with the ambient flow velocity field. Hence, all particles possessing the same velocity v produce, according to (19) and (29), the same power spectrum. The total effect of this incoherent process is obtained by adding the intensities (rather than amplitudes, as in the case of co- herent fields), i.e., the power spectra produced by all the individual uncorrelated events. Let us denote the amplitude (i.e., square root of the intensity) spectrum due to one particle having the velocity U by A ( p , U ) . The total power spectrum at a certain velocity depends on the number of events per unit time at this velocity, and is therefore proportional to the density of the particles possessing this velocity, and also on the velocity. The number of particles within the sampling volume could be the same for different velocities, and since (on the average) the same number of particles are entering and leaving the sampling volume, the particle density at a certain velocity is independent of time. However, the larger the velocity, the higher the number of uncorrelated events created in the sampling volume. Taking into account all the velocity components appearing in the measurement volume, the amplitude spectrum is therefore given by

' /2

4 4 = [ s dv[A(C1, .,12f(v)j (30)

where f ( U ) = p ( v) v already absorbs the velocity proportionality factor and the velocity dependent density function with p ( U ) defined as the number of particles having a velocity between v and v + dv crossing unit area per unit time. For a parabolic flow profile, as assumed in the numerical calculation for flow gradients, p ( v ) is constant. Note that the spatial distribution of the velocity field does not appear in (30), i.e., the system cannot resolve positions within the range cell.

Thus far the theory for analyzing Doppler spectra has been presented. Special cases of interest will be discussed below. In the examples analyzed in the appendixes below, arbitrary f ( U ) have been chosen, in order to facilitate the relevant integrations. It is assumed that the ingredients mentioned above are already absorbed into f ( U ) .

TRANSVERSE MOTION, CONSTANT VELOCITY The Long Strip Transducer: This case has been ana-

lyzed previously [6]. It is included here to serve as a simple example for the use of the theory and to provide con- tinuity. Let ro = Fo be the focal length, and v in the x direction perpendicular to the beam axis and the direction y along the strip. According to (8), (9) we obtain for (10)

S ( P - U ) * G ~ ( P ) = S ( P - U ) * * a ( k v E / F )

For a uniformly insonified aperture \k is constant in the range - W/2 I 4 I W/2 and zero for 4 I > W/2; hence, \k, and A ( p ) from (18) are as shown in Fig. 2(a) and (b), respectively. Clearly the edges of the spectrum correspond to the Doppler effect produced by the edges of the transducer, and these extreme frequencies can be measured. If W, Fo are known, the velocity can be derived. For this case of constant velocity the computation of A ( p ) as in (30) is automatically taken into account; hence, (31) applies to a single particle, as well as to a stream of uncorrelated particles distributed at arbitrary locations y along the strip.

The Rectangular Aperture Transducer: In certain cases qa( p ) (8) is separable. For example, in rectangular co- ordinates in (31) we will have a product \k , (kv ,E/F) * \ k Y ( k v y q / F ) instead of \k , (kut ; /F) . In such a case, we obtain the Doppler spectrum as a convolution of two spectra; hence, roughly speaking the bandwidth of the combined spectrum will be the sum of the individual band- widths. This can be exploited for measuring the components U,, uy of the velocity: by rotating the transducer about the beam axis through a right angle the above expression now becomes qx ( kvy q / F ) * qy ( kv, 4 / F ) , providing two equations for the unknowns U,, vy.

The Circular Aperture Transducer: Another case of interest, as analyzed previously [6], is derived from (6) when the aperture has angular symmetry, i.e., = 9,( p ) . The scatterer trajectory is now written as

r = j a + Rut (32)

where a is the distance of the particle from the beam axis at time t = 0. Consequently (3), (6) now prescribe

g ( j a + Rut) = jr dp 1; pdpe-ikPCOS(~-d)sineq a ( p )

bo

= 2~ Io dpp*a( P ) J o ( b sin 0) (33)

(which is usually referred to as the Hankel transform). Here c$ can be chosen t$ = 0, i.e., the origin of c$ is im- material because of the angular symmetry, and

2 1/2 a 2 + (u t ) sin 0 = [ r2 ] . (34)

For the uniformly insonified circular aperture (6) yields


4 AMPLrmM

col 1 - (v/c) (W/ 2 F 0 ) ] w[ I + (v/c) (W/ 2Fo)]

FREOUENCY

(a)

(I) [ l -(v/c) (W/Fo)] w w [I+(v/c) (W/Fo )]

(b) Fig. 2. Long strip transducer with uniformly insonified aperture. (a) Am-

plitude spectrum of the field exciting the particle; (b) amplitude spectrum for the echo signal.

FREQUENCY

( 3 5 ) J1 ( bok sin 0) - iwt

= 2 r b ; - I-, bok sin 0

where q5 = 0 is assumed and bo, J 1 denote the aperture radius and the nonsingular Bessel function of order one, respectively. The spectrum exciting a single particle is therefore given by

6 ( P - 4 * G(P7 Y )

= 6 ( p - U) * F

J,(bok[v2t2 + y 2 ] 1 / 2 / [ v 2 t 2 + y 2 + Fg]1'2)

bok[u2t2 + y 2 ] 1 / 2 / [ v 2 t 2 + y 2 + Fg]' /2

( 3 6 ) From (18) it follows that the received echo due to particles with a given offset y is proportional to

A b , Y ) = S ( P - 4 * G J P , Y ) * G J P , Y ) (37 ) and if the density with respect to y is constant, then according to (29) we obtain

The transformation in (36) and the integration (38) have been numerically evaluated. In Appendix B the results are supported by analysis of a special idealized model.

It is clear from (36) that the spectrum exciting a single particle depends on y . For y >> vt the spectrum becomes proportional to 6 ( p - U), i.e., all Doppler spectra effects disappear. For y = 0, i.e., for a particle going through the beam's axis, the time modulation is more pronounced; hence, the spectrum is broader. For different y such that y1 < y2 < y3 we expect from (37) increasingly broader spectra, as shown in Fig. 3. Note that the experiments

- A\

i ' (Y1 AMPLITUDE

1-1 y4

FREQUENCY 0

(a)

/,f\, AMPLITUDE 1 -

0 FREOUENCY

(C) Fig. 3. (a) Calculated amplitude spectra for y , < y , < ys < y4 for circular

aperture; (b) peaked amplitude spectrum due to the combined effect of particles traversing the focal plane at various off axis distances; (c) smooth curve: calculated amplitude spectrum. Computational parameters for the evaluation of (38): 0 = 90", v = 0.8 m/s, W = I 1 mm, F = 40 mm, c = 1540 m/s; other curve: experimental [6] obtained averaged spectrum. The arrows denote the calculated extrema1 frequencies (40). That the experimental spectra [also Fig. 5(b)] are apparently broader than the calculated ones is expected. This is due to the fact that in reality we used a pulsed wave, whereas the calculations assumed monochromatic excitation as an approximation. The dip at the origin is due to the wall filter.

have been performed with a pulsed Doppler system, which therefore has a finite pulse spectrum width, but the computations are carried out with a single frequency excitation. This is only justified for pulses whose spectrum is sufficiently narrow, i.e., contain many RF cycles within a single pulse. Future more accurate computations will need also pulse width considerations, according to (23)- (26). The results are also supported by simulations given by Bascom et al. [3], although they consider a CW Dopp- ler system. According to the above explanation, the ends


of the spectrum are due to particles going through the beam axis y = 0, and therefore undergoing the strongest modulation. In (38) this corresponds to

It is easy to show that these extreme frequencies correspond to Doppler effects produced by the transducer's rim. Consider (8) for maximum v - p = up, i.e., (8) describes Doppler shifted waves and the maximum shift Af,,,, at the rim p = bo is given by

U bo c Fo

Af,,, = w - - (39)

exactly as in the case of the long strip transducer. After convolving the signals as in (37), the edges of the spectrum in Fig. 3 will be at frequencies

p, ,=o( l + ; 2 2 ) , p ~ , , = w ( l - ;2$ ) . (40)

Fig. 3(c) shows that the calculation agrees well with experimental [6] data. Some of the features derived above can also be obtained from the analysis of a circular aperture with Gaussian apodization, see Appendix B.

OBLIQUE MOTION, CONSTANT VELOCITY The Long Strip Transducer: Once again we consider

the long transducer, oriented in the y direction. The particles possess now a velocity component in the radial direction, i.e., in the z direction for focused beams (Fig. 1) . From (7) it is clear that this velocity component cannot affect G. The only effect is on the distance r , according to

r = ro + v ?t = Fo + v,t. (41) The effect of (41) on the amplitude of the cylindrical wave is neglected; however, in the exponent we get the Doppler frequency shift described by (20). The transverse velocity component U, has the same effect as in the case v, = 0 discussed above, and the two effects are juxtaposed. We therefore obtain a spectrum as in Fig. 2(b), appropriately shifted according to U , and with endpoints affected by U,, as shown in Fig. 4. Corresponding to (31) we have for the present case

6 ( ~ - ~ 1 ) * * a ( k v x E / F o ) , = w ( 1 - ~ Z / C ) (42) and therefore, according to (22)

A ( P ) = CL - W ) * * a ( k v x E / F O ) * * a ( & x E / F o )

(43 1 and if the pulse spectrum has a significant effect then (43) should be modified according to (28). The spectrum de- picted in Fig. 4 describes a uniformly insonified aperture and provides a representative example.

The Rectangular Aperture Transducer: The situation is essentially identical to the case of the transverse motion

Fig. 4 . Amplitude spectrum for long strip transducer for constant velocity and oblique incidence.

AMPLITUDE 7

0 FREQUENCY

(b) Fig. 5. (a) Amplitude spectrum for circular aperture transducer for con-

stant velocity and oblique incidence. (b) Smooth curve: calculated amplitude spectrum. Computational parameters for the evaluation of (38): 0 = 120", o = 0.3 m/s, W = 11 mm, F = 40 mm, o = 4.55 MHz. Other curve: experimental [6] obtained averaged spectrum. The amws denote the calculated extrema1 frequencies.

above except that in the present case we get also the component U , along the beam axis.

The Circular Aperture Transducer: The same argument used above for the long strip transducer applies here. The two effects of the U, and v, velocity components are compounded. Thus, the spectrum is shifted due to v, and the endpoints are determined by the maximum Doppler effect produced by the transducer's rim.

For the present case, (36) is modified by replacing 6 ( p - w ) by 6 ( p - wl), w1 as in (42) and changing U to v, in the expression in braces (36). Similar modifications are performed in (37) which now contains 6 ( p - w 2 ) as in (22), leading to the appropriate form of (38) which yields the spectrum of Fig. 5 . Fig. 5(b) shows the good agree- ment of the calculations with experimental [6] data.

TRANSVERSE MOTION, VELOCITY GRADIENTS The Long Strip Transducer: The geometry is the same

as for constant velocity; however, here the velocity v (2) is in the x direction and depends on z . For each particle the spectrum is as in Fig. 2. Due to the incoherent field

CENSOR: et al. : DOPPLER-SPECTRA VELOCIMETRY FOR ARBITRARY BEAM AND FLOW CONFIGURATIONS 747

w [ 1 -(2v* /C)] 4 1 -'2vz /c)l

1 -(2Vz 2/C)1 FREQUENCY

Fig. 6. Amplitude spectrum for long strip transducer for transverse motion and velocity gradients. Computational parameters, assuming parabolic flow profile': 8 = 90°, vmax = 0.5 m/s, U,,, = 0.07 m/s, W = 1 1 mm, F = 20 mm, U = 4.55 MHz.

produced by the ensemble of particles, the combined spectrum is computed by adding intensities as in (30). The effect is the same as the peaking of the spectrum in Fig. 3, but the reasons are different. Obviously, the largest velocity component produces the broadest spectrum; hence, the endpoints of the spectrum profile correspond to the maximum velocity. The exact shape depends on the number of particles possessing various velocities. The result of a numerical calculation is shown in Fig. 6. In order to gain more insight, an idealized model is analyzed in Ap- pendix C.

The Circular Aperture Transducer: Numerical calculation results in similar shaped spectra compared to Fig. 6. Peaking of the spectrum is due to the special geometry, as explained above (see Fig. 3), even if the velocity is constant. On the other hand, from the discussion of the long strip transducer for velocity gradients (see Fig. 6), we expect similar peaking effects for the circular aperture transducer as well. We therefore have two mechanisms which affect the spectrum shape in the same way. The spectrum is symmetrical with respect to U = w and the ,endpoints depend on the maximum velocity. The above ,conclusions are supported by the numerical simulations of Bascom et al. [3], although they treat the case of a CW Doppler system.

OBLIQUE MOTION, VELOCITY GRADIENTS The Long Strip Transducer: Consider now oblique mo-

tion and velocity gradients, such that u,/v, = a! remains constant, i.e., the velocity varies but its direction in space is constant. For a single particle the amplitude spectrum is given in Fig. 4 and the endpoint frequencies are

p = .[ 1 - L- %)I.

'For this case, (30) was evaluated. For programming purposes, a variable change from dv to dy was made in order to get a similar expression to (38). This changed the factorf( v ) = p ( U ) v = const . v tof( y ) = const . y ' U.

AMPLITUDE

I * p

w( 1- (vm IC) (W/Fo )] o m[ I + ( vmaX/c) (WEo ) I

FREWENCY

Fig. 7 . Addition of power spectra for long strip transducer, oblique motion, and velocity gradients: spectra for v = 0.5; 0.4; 0.3; 0.2; 0. l m/s. (approaching flow, v, < 0).

The spectrum is symmetrical with respect to p = w ( 1 - 2 v , / c ) . Note that as U, increases, the spectra corresponding to various particles having different velocities are shifted on the p axis, as well as becoming broader. Thus, we have to add the corresponding power spectra as de- picted in Fig. 7. The total amplitude spectrum is expected to have the shape shown in Fig. 8 where particles with the lowest, respectively, highest velocity determine the lower and upper limits of the spectrum for receding flow according to

p = w 1 - ZIZmax,min +/- - . (44) I (: ;)I If U, is negative, i.e., if the particles have a velocity component towards the transducer, then the expected spectrum will be as in Fig. 8(b) and

c1 = U [ 1 - U , ~ " , - ( : -/+ E)]. (45)

In Fig. 8(c) a spectrum is calculated for the case where the sample volume includes the whole vessel, i.e., where all velocities of a parabolic flow profile from zero to vmax make contributions. The flat spectrum shape is in agree- ment to previous results [2], [3]. In Appendix D a semianalytical discussion is given which supports the present conclusions on the asymmetrical shape and the endpoints of the spectra in Fig. 8.

The Circular Aperture Transducer: We have to rely on a verbal argument and on numerical calculations once more. For a single velocity component the spectrum of Fig. 5 is expected, which qualitatively speaking is not different from Fig. 4 for the long strip transducer. Hence, for the present case we expect spectra as in Fig. 8(a)-(c), with W replaced by the diameter 2bo of the circular transducer.

CONCLUDING REMARKS As the reader is well aware, all existing ultrasonic

Doppler systems measure only the velocity component ~ 1 1


I where 4 AMPLITUDE

I I

0

FREQUENCY

(b)

m FREQUENCY

(C)

Fig. 8. (a) Amplitude spectrum for long strip transducer, oblique motion and velocity gradients for the case U, > 0. Computational parameters: 8 = 60", U,,, = 1 m/s, vmin = 0.55 m/s, assuming parabolic profile', W = 1 1 mm, F = 20 mm, w = 4.55 MHz. (b) As in (a), but U, < 0, i.e., 0 = 120". (c) Case U, < 0 but the entire diameter of the vessel lies within the sample volume. Parameters: 8 = 120", U,,, = 0.5 m/s, vmin I: 0.

along the beam axis, using the classical Doppler equation

2Vll wd =

where

vll = COS e and where in practical situations the beam to flow angle 8 has to be smaller than about 60". In the present paper, we have demonstrated that one can also measure U , the velocity component at right angles to the beam, by using what we will refer to as the second Doppler equation (for the spectrum bandwidth at constant flow velocity), namely

2v+ w Bd = WO - -

c F

U + = v sin 8

and W is the width of the transducer. (The same equation in a slightly different notation was derived in [6] for flow transverse to the beam axis.)

In the current paper, we have extended the theory of [6], which was limited to transverse motion, flat velocity profile and monochromatic excitation, to arbitrary beam to flow angles and source pulses, as well as velocity gradients. Expressions have been derived for long strip, circular and rectangular aperture transducers. An important result of the numerical evaluation of the expression for the commonly used circular aperture transducer was that the spectrum edges show the same dependency on velocity (given by the above quoted second Doppler equation) as the idealized model of the long strip transducer. This theoretical result is confirmed by experiments presented above and also by experimental tests of an earlier-and slightly different-version of the second Doppler equation, given in [2].

We have also shown theoretically but without experimental confirmation, that for moderately broadband transmitted pulses the resulting Doppler spectrum can be approximated as a convolution of the spectrum for monochromatic illumination with the spectrum of the pulse.

Use of the second Doppler equation, which has been derived and verified above, improves the accuracy of Doppler measurements. This follows from the fact that the classical Doppler velocity estimate is not accurate for angles approaching the transverse direction because, when using the first Doppler equation, v is proportional to 1 /cos 8 which approaches infinity as 8 approaches 90". On the other hand, the accuracy of determining v using the second Doppler equation improves for angles approaching 90" because v is now proportional to l /sin 8. Joint ex- ploitation of the two Doppler equations extends our abil- ity to measure flow velocity from a very narrow range of angles to all angles if the flow angle is known, and en- ables us to estimate the velocity component normal to the beam as well as that parallel to the beam, even when the beam to flow angle is unknown.

Use of the above techniques should allow blood velocity measurements in arteries and veins inaccessible to conventional Doppler and also help avoid aliasing by making it possible to insonate high speed flows at angles which are sufficiently large to produce only low Doppler frequencies. It should be pointed out, however, that ex- ploitation of the above methods at the present stage of development requires a good signal-to-noise ratio, non- turbulent flow, and the placement of the range cell so as not to intersect moving vessel walls.

Finally, it must be emphasized that the theory is limited by the restrictions enumerated in the introduction; essentially, it is based on the assumption of laminar and steady flows, and propagation in "nice" (homogeneous, linear, lossless, etc.) media. These simplifications facilitated the


analysis, which combined mathematical wave theory, numerical computations, and some verbal arguments which, based on physical plausibility, extended the results to more complicated cases. The removal of these restrictions poses open questions which must be dealt with as better

justified in approximating

* (B2) Jo(kp sin 8) = 1 - ~ = e-(kpsine/2)2 (kp ::

1 dpe-Po2 = - 47r

Hence, (33) becomes

1 theoretical tools and experimental data become available. m

APPENDIX A: A SIMPLIFIED PROOF OF (2)

A detailed treatment is given elsewhere [8], the present discussion is simplified, merely to show how (29) is obtained. The system transmits a train of pulses at a repe-

signal is

6 0 6 l / y + k 2 sin2 8

som d ( pP2)

d a2 + t2’

- -- 1 tition frequency a. A simple model representing such a 47r 6 l / y + k2 sin2 8

h ( t ) = e-im‘ cos nt (AI) p’2 = p 2 ( l / y + k2 sin2 8)/4

i.e., a beat signal with carrier frequency coo and repetition frequency a. Due to the Doppler effect, the echo at the receiver is

h2(t) = e-im@t cos npt (W p being the Doppler factor appearing in (28). The mixing with w0 and subsequent filtering, which takes place in the receiver yields

eimrt cos Qpt (A31 where y = 1 - p as in (29). The signal is then sampled by narrow pulses with the repetition frequency 62. This provides a range gate such that the received echoes orig- inate at a certain distance from the transducer. Because of the low pass filter following the sampler (or in some systems the sample and hold arrangement), only the lowest frequencies of the sampling signal are relevant to our discussion. Thus if (A3) is multiplied by 2 cos Qt and only the lowest frequencies are retained, we obtain

cos nyt’, t’ = -t e-mYt‘ (A41 which is a compressed time inverted replica of (Al). The same relation as between (Al) and (A4) also exists between (27) and (29).

APPENDIX B: CIRCULAR APERTURE WITH GAUSSIAN APODIZATION; TRANSVERSE MOTION AND

CONSTANT VELOCITY

The following analysis deals with a highly idealized model. The results support the conclusions derived above for the uniformly insonified aperture under the same cir- cumstances.

Consider (33) for an aperture function:

47r F$ d = - - k2v2’

and sin 8 is given in (34), and we put ro = Fo, the focal length.

From a table of Fourier transform one finds that the spectrum of d / ( a 2 + t 2 ) is given by ( & / 2 ) ( d / a ) e - a 1 p 1 . It follows that for large y, i.e., larger a, the amplitude at p = 0 is smaller and e - a 1 falls off faster, i.e., the spectrum is narrower. This result agrees with Fig. 3(a) and therefore supports the ensuing spee- trum in Fig. 3(b).

APPENDIX C: LONG STRIP TRANSDUCER WITH GAUSSIAN APODIZATION: TRANSVERSE MOTION AND VELOCITY

GRADIENTS Let the aperture function have the Gaussian form

where scribe

is a constant. For a single particle (8), (9) pre-

J eViprG( p ) dp

where K is a proportionality factor. Therefore,

where y is a constant. The Gaussian apodization facili- tates analysis for certain limiting cases. For the long strip transducer (31), a Gaussian aperture would lead to a Gaussian suectrum. For the circular aoerture we have

We find A ( p, U ) from (18), but instead of convolving (C3), Fe use the transform G ( p’ ) = F { G( p ) } = Ke-@’” . Squaring yields

(33), and f i r small y (Bl) falls off rapidiy; hence, we are [ G( p)I2 = K 2 e -*”p’’ = K2e-@*p”, p* = 20’. (C4)


Hence, A@) ( AMPLITUDE)

is Gaussian once more. In order to compute (30) f ( U ) must be chosen in A ( p )

FREQUENCY

Fig. 9. Relevant to the analysis for oblique incidence and velocity gradients, Appendix D. The result is consistent with the shape obtained in

( c 6 ) Fig. 8(a).

Noting that (dldv) f ( v ) = l / v . Hence, (C6) becomes

= ( 2 D / v 3 ) e-D/u2 we choose

1”’ -DIU2 dv - e K ~ F ~ C

A ( p ) [G dv

In order to check the limit D -+ 0, to ensure that A ( p ) does not become singular, we expand the exponentials near the origin. This yields

(C8) D-0

The results of this analysis support the argument leading to Fig. 6. Near p = w , i.e., near D = 0, A ( p ) (C7) is dominated by the factor 1 /D whic! produces peaking in the spectrum. qince V2 > VI, falls off slowly compared to e-D/v l ; therefore, for large D, i.e., p >> w the shape is dominated by the maximum velocity, as antici- pated in Fig. 6. Also note that A ( p ) , (C7) is symmetrical with respect to p = U , as expected.

APPENDIX D: LONG STRIP TRANSDUCER WITH GAUSSIAN APODIZATION; OBLIQUE MOTION AND VELOCITY

GRADIENTS

By changing the variable we obtain

where K’ is a constant and the sign of du is chosen such that A ( p ) is real (i.e., the expression in braces in (Dl) is positive). By choosingf(av,) = 1 and writing 1:; = 57 - c l , (D2) is representable in terms of the error function. Thus, we have

\ 1 I 2

t - w J

and a similar expression for with U, for u2. It follows that (D3) describes the square root of the difference of two displaced error functions, further multiplied by ( 1 - p/w)-’, as sketched in Fig. 9. By inspection of Fig. 9 it becomes clear that (D3) is consistent with the amplitude spectrum shape derived in Fig. 8.

For the present case (C3), (C5) are modified by changing U to U, = a * U, and in (C6) p - w is changed to p - U( 1 - 2vz/c). Thus, we now have

REFERENCES [l] P. J. Fish, “Doppler methods,” in Physical Principles of Medical

New York: Wiley, 1986, ch. 1 1 , pp. Ultrasonics, C. R. Hill, Ed. 338-376. _ .

[Z] V. L. Newhouse, E. S. Furgasofi, G. F. Johnson, and D. A. Wolf, “The dependence of ultrasound Doppler bandwidth on beam geometry,” IEEE Trans. Sonics Ultrason., vol. SU-27, pp. 50-59, 1980.

[3] P. A. J. Bascom, R. S. C. Cobbold, and B. H. M. Roelofs, “Influ- ence of spectral broadening on continuous wave Doppler ultrasound spectra: A geometric approach,” UltrasoundMed. Biol., vol. 12, pp.

[4] J. A. Cisneros, “Medical applications of Doppler spectral analysis

1 12

387-395, 1986. ( D O D = ’ [ p - w(l - 2 v , / c )

P 20


using a normal orientation of the beam with respect to the direction of flow,” Ph.D. dissertation, Drexel University, Philadelphia, PA 1985.

[5] J. A. Cisneros, V. L. Newhouse, and B. Goldberg, “Doppler spectral characterization of flow disturbances in the carotid with the Doppler probe at right angles to the vessel axis,” Ultrasound Med. Biol., vol.

[6] V. L. Newhouse, D. Censor, T. Vontz, J. A. Cisneros, and B. Gold- berg, “Ultrasound Doppler probing of flows transverse with respect to beam axis,” IEEE Trans. Biomed. Eng., vol. BME-34, pp. 779- 789, 1987.

[7] D. W. Baker, F. K. Foster, and R. E. Daigle, “Doppler principles and techniques,” in Ultrasound: Its Application in Medicine and Bi- ology, vol. I , F. J. Fry, Ed. New York: Elsevier, 1978, pp. 161- 287.

[8] V. L. Newhouse and I. Amir, “Time dilation and inversion properties and the output spectrum of pulsed Doppler flowmeters,” IEEE Trans. Sonics Ultrason., vol. SU-30, pp. 174-179, 1983.

[9] P. M. Morse and K. U. Ingard, Theoretical Acoustics. New York: McGraw-Hill, 1968.

[IO] J. D. Jackson, Classical Electrodynamics. New York: Wiley, 1975. [ 111 M. Born and E. Wolf, Principles of Optics. New York: Pergamon,

[I21 J. W. Goodman, Introduction to Fourier Optics. New York:

11, 319-328, 1985.

1980.

McGraw-Hill, 1968.

Vernon L. Newhouse (M’55-SM’66-F’68) was born in Mannheim, West Germany. He received the B.Sc. and Ph.D. degrees from the University of Leeds, England, in 1949 and 1952, respectively.

From 1952 to 1957, he worked as a Project En- gineer on magnetic memory development and computer circuit design, first at Ferpnti, Ltd. in Manchester, England, and then later at RCA in Princeton and Camden, NJ. From 1957 to 1967, he worked on superconductive film switching and

storage devices at General Electric Research Laboratory. From 1967 to 1981 he was Professor of Electrical Engineering at Purdue University, La- fayette, IN, where he originated a Clinical Engineering program which became the country’s largest. In January 1982, he and his group moved to Drexel University, Philadelphia, PA, where he holds the Disque Chair of Electrical and Computer Engineering. He has since been appointed adjunct Professor of Radiology at Jefferson University Medical School.

During 1950’s and 1960’s, he achieved international recognition for research on computer memories and applied superconductivity. In 1968, he entered the field of acoustics where several of his achievements have been featured in the annual NSF reports to Congress. In the last 15 years, he has concentrated on the application of communication science to acoustic imaging starting with the development of the first random signal blood flow measurement and flaw detection systems. His work in random signal imaging has received recognition in the form of an IEEE award and his work on geometrical broadening and Doppler-field design was the subject of a special conference session. He also has a magnetic film switching phenomenon named after him and has been invited to organize the forthcoming New England Doppler conference.

Dr. Newhouse is listed in American Men of Science, American Men and Women of Science, International Who’s Who in Engineering, Who’s Who in the East, Who’s Who in Technology Today, and Who’s Who in America. He is a U.S. citizen, a former member of the American Physical Society, the American Institute for Ultrasound in Medicine, and a Consultant to NSF and NIH.

Thomas Vontz received the Dipl.Ing degree in material science in 1986 from the University of Saarbriicken, FRG.

Since 1984, he has been with the Fraunhofer Institut fur zerstorungsfreie Priifverfahren, Saar- briicken. He has worked on computer tomography and is now a Visiting Scientist at the Biomedical Engineering and Science Institute, Drexel Uni- versity, Philadelphia, PA.

Dan Censor received the B.Sc. E.E., the M.Sc. E.E., and the D.Sc. Tech. degrees from the Tech- nion-Israel Institute of Technology, Haifa, Israel in 1962, 1963, and 1967, respectively.

He is currently with the Department of Electri- cal and Computer Engineering, Ben Gurion Uni- versity of the Negev, Beer Sheva, Israel, and is presently a Visiting Professor and Stein Family Foundation Fellow at Drexel University, Phila- delphia, PA. From 1967 to 1969, he was with the Department of Information Engineenng, Univer-

sity of Illinois, Chicago, 1969-1975, he was with the Department of Geo- physical and Planetary Sciences, Tel Aviv University, Israel, 1975-1976, he was Guest Lecturer at the Institute of Theoretical Physics, University of Dusseldorf, West Germany, 1979-1980, he was a Visiting Professor with the Department of Electrical Engineering and Computer Science, Uni- versity of California, San Diego, and from 1980 to 1981, he was Senior NSF Associate at the NASA Goddard Space Flight Center, Greenbelt, MD. His current interests are in the areas of wave and ray propagation and scattering in various media, linear and nonlinear, lossless and absorptive, and Doppler effects in moving media and in the presence of moving objects.

Dr. Censor is a member of the URSI Israel National Committee.

Hector V. Ortega received the M.D. degree from the Autonomous National University of Mexico, Mexico City, in 1977.

In 1980 he received a fellowship at Thomas Jef- ferson University Hospital, Philadelphia, PA. He was Chief of the Ultrasound Division and Asso- ciate Professor of Radiology at Juarez Hospital, Mexico City, from 1981 to 1984, and a Visiting Professor at the Autonomous National University of Mexico of postgraduate courses in continuing medical education. Currently, he is a graduate

student at Drexel University, Philadelphia, PA, pursuing the Ph.D. degree in biomedical engineering.

Theory of ultrasound Doppler-spectra velocimetry for arbitrary beam and flow configurations

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Transcript of Theory of ultrasound Doppler-spectra velocimetry for arbitrary beam and flow configurations