Echo decorrelation from displacement gradients

ELSEVIER

Ultrasonics 36 ( 1998) 661-666

Echo decorrelation from displacement gradients E. Ignacio CCspedes a,b**, Chris L. de Korte a, Anton

F.W. van der Steen a,c

a Erusmus Universit?, Rotterdam, ’ Interuniversity

Thoruscenter, Ee 2302. Postbus 1738, 3000 DR Rotterdam, The Nctherlund,s b EndoSonics Corp., Rancho Cordova. CA, USA Cardiology Institute of‘ the Netherlund.s, Utrecht, The Netherland,s

Abstract Several ultrasonic techniques for the estimation of blood velocity and tissue motion and elasticity are based on the estimation of displacement through echo time-delay analysis. A common assumption is that tissue displacement is constant within the observation time. The precision of time delay estimation (TDE) is mainly limited by noise sources corrupting the echo signals. In addition to electronic and quantization noise, a substantial source of TDE error is the decorrelation of echo signals as a result of displacement gradients within the observation time. We present a theoretical model that describes the mean changes of the crosscorrelation function as a function of observation time and displacement gradient. The gradient is assumed to be small and uniform; the decorrelation introduced by the lateral and elevational displacement components are assumed to be small compared to the decorrelation due to the axial component. The decorrelation model predicts that the expected value of the crosscorrelation function is a low-pass filtered version of the autocorrelation (i.e.. the crosscorrelation obtained without gradients). The filter is a function of the axial gradient and the observation time. This theoretical finding is corroborated experimentally. Limitations imposed by and potential uses of decorrelation in medical ultrasound are discussed. 0 1998 Elsevier Science B.V. Ke_v~vords: Ultrasound:

Strain:

Decorrelation;

Crosscorrelation;

1. Introduction Additions to the usefulness of ultrasonic imaging have been proposed and implemented in relation to the capability of this modality to assess tissue dynamics. A major contribution to modern ultrasonic instrumentation has resulted from the incorporation of blood flow measurement and imaging. Also, during the past 15 years, tissue motion and elasticity imaging of soft tissue for medical diagnosis has become a field of increasing research [l-6]. In brief, these techniques measure local displacement or velocity of tissue based on the evolution or dynamics of received echo signals. For example, tissue strain may be computed by finite difference of locally assessed tissue displacements measured at subsequent compression levels; displacements may be estimated from the relative time delays (or shifts) between preand post-compression echo signals. Common to both elasticity and blood velocity estimation, is the presence of displacement gradients. In elastic* Correspondingauthor. Fax: (31 ) 10-436-5191; e-mail:cespedes(k$ch.fgg.eur.nl 004l-624>(/98,‘$19.00 0 1998 Elsevier Science B.V. All rights reserved. P/I SOO4l-624X(97)00128-5

Displacement

gradient

ity assessment, the local displacement of tissue varies in space: the displacement gradient along a given direction is by definition the strain component along that direction. In blood velocity assessment, common flow conditions give rise to velocity gradients usually measured by the shear rate, or rate of velocity change in the vessel lumen [7]. Velocity gradient result in corresponding displacement gradients [8]. Recent work has demonstrated that displacement gradients due to blood velocity gradients impose constraints on the choice of observation window in intravascular ultrasound flow assessment [9]. Although in general strain and velocity gradients is vary in space, a piece-wise linear approximation reasonable within a small ultrasonic sample volume of interest. Thus, in the remainder of this paper we will assume constant strains and velocity gradients. In practice the precision of time delay estimation (TDE) is limited by noise corrupting the echo signals. In addition to electronic and quantization noise, a main source of error in ultrasonic TDE is the decorrelation of the echo signal as a result of the aforementioned displacement gradients (termed decorrelation noise). As a result of displacement gradients, scattering particles

E. I. Cisprdrs et al. / Ulirasonics 36 (1998) 661-666

662

change their relative positions along the axial direction, with concomitant deformation of the corresponding echo signals. The effect of displacement gradients on the echo signals is illustrated in Fig. 1, which shows echo signal pairs obtained from an artery wall undergoing compression. These signals demonstrate not only a overall temporal compression but also localized high decorrelation. Wear and Popp [lo] investigated the correlation properties of ultrasonic echoes from moving tissue in the development of a method for the measurement of contractile velocity of the myocardium. Assuming a uniform displacement pdf, the theory and a set of experiments (the elastic band experiments) were compared, showing an overall agreement of theoretical and experimental results; however, the theoretical estimates of the correlation function fell within three to four standard deviations from the experimental estimates. A disadvantage of the approach taken by Wear and Popp is that it does not provide a direct expression for the crosscorrelation function of the compressed scatterers (unknown) in terms of the autocorrelation function of the system (which can be estimated). Meunier and Bertrand [ 1 l] have investigated the changes of ultrasound images when tissue undergoes rotation, translation and biaxial deformation. In [ 111, expressions were derived for the decorrelation of radio frequency and demodulated echo signals when these signals are previously compensated for the motion and the point spread function is Gaussian; simulation results obtained using large strains (l-10%) showed good agreement with the theory. Walker and Trahey [ 121 derived a theoretical expression for the correlation coefficient based on the strain and the data length. In the limit of small strains, the expression derived by Walker and Trahey is similar to a preliminary version [ 131 of the development presented

here. Foster et al. investigated the error introduced by displacement gradients on blood velocity estimates and report on gradient related errors that can exceed all other sources in some situations [8]. However, to date, little has been done to characterize the effect of displacement gradients on the crosscorrelation function. Such information would be useful in order to better understand limiting factors and potential applications of decorrelation in elasticity and blood flow imaging. In the present report, we describe a theoretical model for the mean changes of the crosscorrelation function as a result of an axial displacement gradient. Hereafter, we proceed with the analysis in the context of tissue strain since this application is better suited for experimentation with controlled displacement gradients. However, the results are also applicable to decorrelation effects due to velocity gradients. The development is valid for small and uniform gradients. Furthermore, we assume that the decorrelation introduced by the lateral and elevational components of displacement are small compared to the decorrelation due to the axial component. This assumption is reasonable when the target is compressed with a compressor that is large compared to the transducer [ 141 and the imparted strain results in lateral and elevational displacements that are small compared to the spatial resolution of the ultrasound system in the corresponding directions. When these assumptions are not valid, the lateral and elevational decorrelation must also be considered and it may be possible to extend the approach developed here for axial decorrelation to the other directions. The decorrelation model for axial displacement gradient predicts that the expected value of the crosscorrelation function is a low-pass filtered version of the autocorrelation (i.e., the crosscorrelation obtained without a displacement gradient). The filter is a function of the axial gradient and the observation window length. This theoretical finding is corroborated experimentally. Without loss of generality, experimental data was obtained utilizing intravascular ultrasound equipment and frequencies, and the results of this study can be generalized to general or cardiac ultrasound environments by virtue of frequency scale transformations.

2. Theory

I , 0

01

I

I

I

I

1

1

02

03

04

0.6

06

07

I

06

09

1

EshcdspUl@S,

Fig. 1. Echo signal pair from an iliac artery specimen obtained before and after compression illustrating the nonstationarity of the time shift. Increased local decorrelation is apparent between 0.55-0.65 ps.

An estimate of the time delay between a broad-band signal and a delayed replica of the signal, in the presence of additive noise, is commonly obtained by crosscorrelation techniques. The use of these techniques was introduced for sonar applications where the crosscorrelation is usually performed between a signal and a time-scaled version of the signal (due to Doppler shifts) in the presence of additive noise [ 151; by definition, such signals are jointly nonstationary. In this case, estimates

E.I. Gspedes et al. ! Ulrrusonics 36 (199X) 661-666

of differential Doppler and relative time-delay are found with time-companding crosscorrelators and appropriate signal filtering [ 16,171. The effect of uncompensated time-scaling on the correlation in sonar applications has been investigated [ 15,16,18,19]. In medical Doppler velocimetry, the decorrelation due to gradients has been recognized as a contributor to Doppler spectral broadening [20,21]. Displacement gradients also lead to echo signals that are jointly nonstationary as a result of an essentially different situation from Doppler, however. Due to the displacement gradient, the location of the scatterers, rather than the frequency of the signal, is scaled. Consequently, the crosscorrelation between echo signals is generally a distorted version of the autocorrelation function. The validity of the analysis presented here is restricted to small, one-dimensional displacement gradients such that the fundamental shape of the correlation is maintained; for larger gradients, the analysis of time delay in the presence of correlation peak ambiguities is fundamentally different and is beyond the scope of this paper. We develop an analytical expression of the crosscorrelation function in terms of the autocorrelation function when displacement gradients are present. In practice, the autocorrelation function can be estimated or obtained empirically. The analysis is similar to the work of Betz [ 161 for sonar signals that are temporally scaled due to the Doppler effect. However, in Dopplerbased velocity estimation, time compression or expansion (termed compansion) occurs in addition to the decorrelation effect from gradients, but its contribution fairly small for short data lengths and thus not considered in this work [22]. The pre- and post-displacement echo-signals are assumed to be zero mean, bandpass signals with additive noise modeled by r,(t)=.s,(t)+n,(t)=s(t)*p(r)+n,(t)

‘I

R,,(t)=

lim L r,(f)r,(t+s) 7.-a,T i o

d/

(4)

[23]. Since signal and noise in the model are zero mean, the crosscorrelation and crosscovariance functions are identical, R,,(t) = C,,(T). The correlation coefficient function is defined by (5) where (TV and (TV are the standard deviations of the correspondingly indexed signal amplitudes [ 231. The correlation coefficient, pi2, is the peak value of the correlation coefficient function, plZ = ail. When using finite length echo signals, the crosscorrelation function can be estimated with k,,(T)=

’

T

s T:2

YI(t)r,(t+T)

(6)

dr

-~,z

where T is the observation time and ri,,(z) denotes the estimate of R,,(T) [23]. Because of the finite integration time, the crosscorrelation function estimate in Eq. (4) varies for each pair of signals: it is in effect a random variable, and must be described statistically. The average crosscorrelation function is defined as the expected value of k12(5), R,,(T)=E[~,,(T)] where E[] denotes expectation. It is interesting to note that when the crosscorrelated signals are jointly stationary, fi,,(t)-+R,,(~)

(2)

where s(t) is the backscatter function representing the tissue, p(t) is the impulse response of the bandpass ultrasonic system, ni(t) and n,(t) are zero-mean uncorrelated random noises, t, is the time delay between the center portion of the echo-signal segments, and * denotes convolution. We define the constant I U=---Zl+E l--E

scattering function in Eq. (2) can be expressed as s~(~)=s(u~-ftO)=~{t-[(l -u)t-rfO]), which evidences the displacement gradient by depicting the original scattering function s(t) shifted (i.e., displaced) by constant and time dependent factors [( 1 - cl)t - to]. The crosscorrelation function between two signals, r,(t) and r2(t) is defined by

as T---x..

(7)

(1)

and rz(t)=.s,(t)+n,(t)=S(ut-~“)~p(t)+n,(t)

663

Conversely, when there are displacement gradients, subsequently acquired echo signals are not jointly stationary and therefore the above limit for infinite integration time does not hold. In fact, it will be shown that for nonzero strain, E[~',,(T)] vanishes as T-t a. Alternatively, in the nonstationary case the expected value of the crosscorrelation can be calculated by considering an ensemble of echo-signal pairs, r,(t) and rz(t), viz. E[r,(f)r,(t+T)l

dt

(3)

that is close to unity, where E is a constant proportional to the displacement gradient and E co.02 (< 2%). Positive values of LIindicate positive strain (i.e. compression) or a decrease in velocity, and vice versa. The

1

T!Z E[sl(t)s2(t

=-sT

+

r)]

dt

(8)

-T/2

where the last expression above was obtained by taking into consideration that signals and noise are mutually

E.I. Cispedes et al. / Uhusonics

664

uncorrelated. Also, Eq. (8) assumes stationarity during the observation time interval T, a fact that is clearly an approximation. Following the algebraic steps detailed in [24], we obtain (9)

h(t)

1, 14I k/2,

=

0,

(10)

otherwise.

Inspection of Eq. (9) elucidates that the effect of the displacement gradient is equivalent to low-pass filtering the companded (i.e., compressed or expanded) autocorrelation function; the impulse response of the filter, h(z), is defined by the gradient and the observation time (Eq. ( 10)). It should be noted that this low-pass filter is not the integrating filter in the correlation operation of Eq. (4); it is effectively an additional low-pass filter that is introduced by the scatterer displacement gradient observed over the observation time interval T. Alternatively, we can express Eq. (9) in the form of a convolution integral as follows, m

m,(91=

R,,(at)h[r-(t+t,)]

dt

s -cc +rrtT/Z

- f,,

RI, (at) dt.

=

(11)

s r-urT/Z-t,

Using Eq. (5) we obtain coefficient,

the average

correlation

EU42(~0)1 Pl2

=

-wlZ)

1 =-srs2

=

E2br(0)l

T/2 -to

For the relatively low displacement gradients of interest in this paper, u z 1. Therefore, by setting u = 1, the decorrelation function can be approximated by 1 1 p12 = -

rc7‘12

J

CT2 -ET/Z

R,,(t) dr.

(14)

3. Experimental method In order to test the theory presented, we conducted experiments using a tissue mimicking test object. The gel test object consists of a cylinder 38 mm in diameter and 20 mm high with uniform acoustic and mechanical properties. The test object was made with 8% by weight porcine skin gelatin (G-2500, Sigma Chemical, St Louis, MO), 2% agar-agarose, and 2% by weight Carborundum (SIC) particles for scattering according to the method described by de Korte et al. [25]. A water tank with a micromanipulator (resolution 1 pm per division) fixed on a support bar was used for controlled compression of the test object (see Fig. 2). The gel was subjected to uniaxial deformation along the direction of the ultrasound beam. A 50 mm compression plate was securely attached to the transducer assembly to form a large compressing surface. The transducer assembly is attached to the compressor through a central opening which is covered at the level of the compressor surface by a thin, practically sonolucent layer. The water filled spacing between the transducer and the compressor surface was used to offset the near field. The compression plate was designed to be larger than the test object to obtain a uniform distribution of deformation [14]. By

mT/Z

Rll(at-t,) -&T/2-t,,

36 (lVYX) 661-666

dt= f

Rll(at)

dt.

s -&T/Z (12)

Thus, the mean correlation coefficient is given by the average of the autocorrelation function over an interval of duration UTE centered around the peak and normalized by the square of the standard deviation of the signal. Eq. (12) provides the decorrelation value as a function of T and e and is called the decorrelation function. It is interesting to consider some special cases; for zero gradient, e=O, h(z) becomes a delta function and a+ 1, and therefore from Eq. (9) the crosscorrelation is

E[&z(d=R&-kJ.

(13)

For any nonzero gradient and T+ cc, the time integration in Eq. (11) amounts to zero and therefore, for such theoretical situation, there is no correlation between the pre- and post-compression echo-signals.

Fig. 2. Illustration

of the experimental

set up.

E.I. Cbprdrs

665

rt (11.; Ultrasonics 36 (199X) 661-666

insonifying the central part of the test object we minimized decorrelation components due to lateral and elevational motions. Strains ranging from 0% to 2.5% in steps of approximately 0.25% were applied. Independent echo signals (N= 9) were obtained by shifting the test object laterally to positions around the center spaced approximately 1 mm. The actual amount of compression and the sample were measured ultrasonically at each compression step from the travel time of the echo from the bottom of the tank and the acoustic window. The standard deviation of the applied strain was below 0.1%. The test object was examined using a 28 MHz (30%, 10 dB bandwidth) unfocussed piston transducer with an aperture of 1 mm. At the center frequency of the transducer, the wavelength is approximately i,, = 50 urn. The pulse length is approximately 3.5 i,,. Echo signals were recorded from the far field of the transducer at a depth of 15 mm. The transducer was pulsed using a modified intravascular ultrasound scanner (EndoSonics/Du-Med, Rijswijk, The Netherlands). Radio frequency (rf ) signals were digitized at 100 MHz and 8 bits with a LeCroy 9400 ( LeCroy, Spring Valley, NY ) digital oscilloscope; ten rf signals obtained at a pulse repetition frequency of 200 Hz were averaged in each case to improve the signal-to-noise ratio. Note that Eq. (12) predicts the value of the mean crosscorrelation function. Correlation coefficient estimates are not normally distributed. Therefore, mean and standard deviations of the correlation estimates were calculated utilizing the Fisher-z transform as described in [24], which transforms the skewed distribution to approximate normality [26]. This procedure allowed to include all the available data without biasing the statistics.

-1 -80

-60

40

-20

0

20

TimZ”[nsl

@I

Fig. 3. Mean crosscorrelation functions at zero (solid line), I % (dashed line) and 2% (dotted line) strain (observation time 1 gs, center frcquency 28 MHz).

0.3 “2

-----I Thmrsticalfunction

I

Measured correlationvalue

Fig. 4. Theoretical and experimental mean correlation coetticient values as a function of strain for observation times 0.75 ps. The error bars indicate 65% confidence intervals calculated using the Fisher-z transform.

5. Discussion and conclusions

4. Results The mean crosscorrelation function at zero strain is shown in Fig. 3. This correlation function is used as an estimate of the autocorrelation, and was used to compute the theoretical decorrelation coefficient as a function of strain and observation time using Eq. ( 12). Fig. 3 also shows the mean correlation functions for 1% and 2% strain with 1 us observation time. Experimental and theoretical decorrelation functions were appropriately normalized and are plotted in Fig. 4 for an observation time of 0.75 us. A generally increasing variance of the estimate can be observed for increasing values of strain. By analogy to the corresponding situation in stationary decorrelation [23], this is an expected result in the nonstationary situation as well.

We have developed a theoretical model for the crosscorrelation function of echo signals from tissue obtained under the influence of displacement gradients. The model can be applied to echo signals of arbitrary bandlimited spectra. However, an approximate decorrelation model appears to be adequate for a range of typical spectra in ultrasound imaging including the case of the experimental conditions of this work. The effect of displacement gradients can be modeled as a low-pass filtering of the autocorrelation function, where the impulse response of the filter is defined by the gradient and the observation time. Higher gradients and longer observation times result in heavier filtering of the autocorrelation function and a reduction of the correlation coefficient (increased decorrelation). A compression experiment using a gel test object was conducted to test the validity of this decorrelation model. Fig. 4 shows good general agreement between the experiment and the

E.I. Chprdes et (11./ Ultrasonics 36 (1998) 661-666

666

theory with more significant deviations at higher timestrain products. Although echo decorrelation has been presented so far as a deleterious effect, it carries information that can be rendered useful if appropriately utilized. Based on the relationship between the correlation coefficient and strain (for a fixed observation time), decorrelation could be used for strain estimation within the constraints of the variance of the estimate. Averaging appears to yield the expected improvement and therefore the mean decorrelation over a number of steps may prove to be an adequate estimator. Additionally, decorrelation can be used as a subsidiary indicator of quality in time delay estimation. This approach is currently being investigated for selection of acceptable strain estimates in intravascular elastography [27] and intravascular flow assessment

191. It is important

to note that the theoretical axial decorrelation described here is the minimum level of decorrelation that can be expected as a result of an axial displacement gradient. In general, other decorrelation components from displacement and gradients across the ultrasound beam will contribute with additional decorrelation.

Acknowledgement We wish to acknowledge the contribution of J. Ophir to a preliminary version of this work. We acknowledge useful comments from Dr. K. Alam, the assistance of G. van Dijk from the Laboratory Seismics and Acoustics at Delft University Technology for the construction of the transducer lized in the experiment.

Dr. also and of of uti-

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