Autocorrelation of integrated power Doppler signals and its application

Copynght

ELSEVIER

PII: SO301-5629(

Ultraxxmd Q 1996

in Med. 8; Biol.. Vol. 77. No. 8. pp 1057- llri7. 1996 World Federation for Ultrasound in Medicine B Biology Printed in the USA. All ri_ehts rewrved 03(1 I -562YiYh $ IS 00 -+ 00

96)00141-X

l Origina1 Contribution AUTOCORRELATION JIAN-FENG

CHEN,?

OF INTEGRATED POWER AND ITS APPLICATION

DOPPLER

SIGNALS

J. BRIAN FOWLKES,: PAUL L. CARSON,: JONATHAN M. RUBIN” and RONALD S. ADLER’~

‘SiemensMedical Systems,Inc., Issaquah,WA; and ‘Departmentof Radiology. University of Michigan Medical Center, Ann Arbor, MI iReceived

9 September

1995: it1 jfirul form

I Ju/.v 1996)

Abstract-The integrated power Doppler signal arising from blood flow is a random process. In this article, a general approach to model this random process is studied. Both the theoretical and experimental results show that the temporal autocorrelation function of the integrated power Doppler signals is directly related to properties of the insonified medium, such as the scattering strengths and velocities of all moving scatterers, and as well as the properties of the Doppler imaging system,such as the point spread function (psf) of the power Doppler images. Some initial experiments are performed to test the proposed model. Its potential application for flow measurement, such as perfusion evaluation, is also discussed. Copyright 0 1996World

Federationfor Ultrasoundin Medicine & Biology Key Words: Autocorrelation

function,

Power

Doppler

signal, Ultrasound.

INTRODUCTION

The integrated power Doppler signal scatteringfrom blood flow is a random process.As shown in a previous paper (Adler et al. 1995) , the inherent stochastic nature of power Doppler signalscan provide additional useful information, such aslocal tissueperfusion, through solving a stochasticdifferential equation derived from a flow model with noise via a fluctuation dissipation theorem (Papoulis 1965) , and it was shown in that article that blood flow and the rate of decorrelation within a region of interest were directly related in a predictable manner. However, becauseof the lack of explicit tissueand Doppler image system dependencies,it is difficult to quantify the blood flow basedon the Auctuation dissipation theorem. In this article, a generalized approach to model this random processis studied. Both the theoretical and experimental resultsshow that the temporal autocorrelation function of the power Doppler signals is directly related to the scatteringstrength. and the velocity of individual moving scatterersas well as to the properties of the Doppler imaging system, such as the resolution cells of the power Doppler flow image. Its application for flow measurementwill also be discussedin what follows. In the first part of this study, a generalized expression for the temporal autocorrelation function of the integrated power Doppler signals for a collection of moving scatterers is presented. This expression contains all factors that significantly contribute to power

Ultrasound Doppler techniquesare currently used for investigating blood flow in the cardiovascular system. The physical basisfor such techniques is the frequency shift causedby the motion of scatterers(red blood cells) in blood vessels. The integrated power Doppler signals. which are defined asthe total frequency-integrated Doppler power spectrum due to blood flow, can be described in the first approximation, however, as resulting from all red blood cells moving as scatterersin blood. The utility of color encoded maps of integrated power on commercially available systems has been demonstrated(Rubin and Adler 1993; Rubin et al. 1994), referring to it as power Doppler flow imaging and, more importantly, a significant improvement in signal-to-noiseratio (SNR) relative to mean-frequency maps in depicting flow has been shown. One example consistsof displaying the soft tissue blush in the renal cortex, which typically requires administration of an intravascular contrast agent. This particular feature is currently being made commercially available by manufacturersas a meansof increasingflow sensitivity and more accurately representingvessel morphology.

Medical sqi.com

Address correspondence to: Dr. Jian-Feng Systems, Inc., Issaquah. WA 98029.

Chen, E-mail:

Siemens chenjf@

1053

i OS.4

1Utrasound

in Medicine

and

Biolog)

Doppler signals, including the medium properties. such ;I< the scattering strength and velocity information, as well as imaging system properties, such as the resolution cell of the power Doppler flow image. Then, some initial experimental results are presented to verify the proposed model. THEORY We will generate a model that will predict the integrated power Doppler signals in a reasonably efficient manner. In order to keep the model as simple as possible, we will make the following assumptions: the moving red blood cells are randomly distributed in the blood vessel and multiple scattering is neglected. We consider a situation where a pulsed transducer is used to insonify a medium which contains randomly distributed moving scatterers. The complex echo signal with a specific time delay t at a specific time 7, U,(T), can be expressed as (Chen et al. 1994) :

U,(T) = 5 u,(t, 7) ,=I

(1)

where M is the total number of scatterers which significantly contribute to the echo signals at a specific delay time t. u,( t,r); is the complex echo signal from the ith scatterer. At first, Doppler signal processing is applied, which includes quadrature detection and mixing of the signal: a wall (high pass) filter and a low pass filter are added to eliminate the signaIs at or near the carrier frequency which are above half the sampling frequency. Then, special wall filters, which are just variations of high pass filters after basebanding, are applied to eliminate specific motion artifacts. Finally, the complex Doppler signal is determined and given by:

U:(r)

=

; ,=I

uj(t,

7-J

(2)

where U’,(t) is the complex echo signal from the moving scatterers (red blood cells) with a velocity v 2 AwccosBlw,,, which significantly contribute to the complex echo signals at a specific time delay t. Here, Aw is the cut-off angular frequency of the wall filter, and ic?,,is the central angular frequency of the ultrasonic pulses and 8 is the angle between the velocity direction and the uhrasonic beam axial direction. M, is the number of these moving scatterers. Then, the time-fluctuating integrated power Doppler signal and its autocorrelation function are defined by:

Volume

37. Number

8. IYYh

P(T)

= u!(t)u!“:(r,

(3)

and R,,(AT)

=
+ Jr,

>.

(-1-l

respectively. Here ( * * * ); denotes a temporal average. Using the moment theorem for the jointly normal. zero mean Gaussian random variables (Wagner et al. 1987). eqn (4) can be simplified to: &CAT)

= :(

1 + llp(A~)jl’,

- : :

= IIP(A~W

(5)

or RJAT)

(5’)

where p(Ar) = < U:(t)U:T*,(t)>./ r is a normalized autocorrelation function for the complex Doppler signals U:(t) , and /I 11denotes a modulus. When acoustic scattering is only from a collection of randomly distributed, pointlike moving scatterers (red blood cells), M, , which are in and extending well beyond the resolution cell of the power Doppler image and significantly contribute to the Doppler signals at the specific delay time, t, the autocorrelation function can be simplified to:

Clbill’ ,=I

(6)

where pi(Ar) = IlaiJ(%(-CiA~) @ h*(i$A~), 8 denotes a convolution operator, (la, 11’and ci are the scattering strength and velocity for the ith scatterer, as shown in the Appendix. h(iti ,Ar ) is a normalized three-dimensional point spread function to represent a resolution cell for the power Doppler image. In that case, we have: %(AT)

- ; ; 2 lla;ll’h( -CiAT)

@ h*(DiAT)

’

= (7) If all moving scatterers

moving through a region

Power

Doppler

autocorrelation

of interest have a single velocity and the point spread function does not change significantly, the autocorrelation function of the complex Doppler signals is approximately simplified to p( AT) = h( -CAt) @ h*(CA7). In that case, the normalized autocorrelation function of the integrated power Doppler signal is given by: /?,(a~)

- :
Based on eqn (7 ’ ) , it is possible quantitatively to determine the velocity,D, if the Doppler imaging system’s point spread function, h(CAr ), is known. EXPERIMENTAL

METHODS

0 J.-F. CHEN

et nl.

1055

original 3-D field. The pixel values were then converted to a linear scale. That is: P,

(T,)

=

1()“‘r,)“06.67

(8)

where Pi (7,) was the integrated power Doppler signal intensity at the specific time Tj and the specific pixel i in the region of interest of the power Doppler how image, f, (7 i) is the raw data at the same image pixel. The factor 106.67 was used to convert the O-255 log scale for intensity to a linear scale. The original data were displayed in a 24-dB dynamic range. The temporal mean value and the normalized auto-correlation function of the integrated power Doppler signal were calculated using following expressions:

AND RESULTS

Some initial experiments were performed to test the proposed model for describing the autocorrelation function of power Doppler signals. A simple flow experiment using a gum rubber tube with internal diameter 1.0 cm that was positioned so that the angle of incidence between the flow stream and the ultrasound field was 0” (Adler et al. 1995 ) . This geometry maximizes the Doppler frequency shift for any flow velocity. The flowing medium was a mixture of corn starch in water (about 2 g in 1 L). A pulsatile pump with the fluid reservoir acting as a capacitance chamber was used to produce a constant flow stream. Knowing the size of the pixel used in sampling the power mode color and the input volume flow rate to the tube, we could estimate how many samples would be needed per pixel in order to observe the decorrelation time at various positions within the tube, if a parabolic flow velocity profile across the tube is assumed. We could also change the effective pixel sizes by merely summing pixels in a specific direction related to the how direction (Adler et al. 1995). Using software made available by Diasonics for our experimental use on a Spectra machine, a buffer was filled with sequential frames of power mode data which had not been temporally averaged. These data were then descrambled out of Diasonics’ proprietary format and loaded into an image processing software package (Advanced Visual Systems, Inc., Waltham, MA), in the form of a 3-D (two dimensions in spatial, and one dimension in temporal) scalar byte field. Each byte represented a decibel scale value of the integrated Doppler signal amplitude at that location. For each experiment, a region of interest was selected within an area of constant scatterer velocity. The pixel values within this region were each written to one line of a 2-D field. Each line of this field is sequentially filled from the same region-of-interest of each frame in the

and N,N, ; ;i, Pi(rj)Pj(r, ;=I CNI N,

+ AT,

j=l

>’

c /=I c Pi(Tj)j 1r=l

- 1.0 (9’)

where the sums were over the N, points in the region of interest and NZ time samples. The data used in a previous article (Adler et al. 1995) were reanalyzed in the current study. Figure 1 shows the results of the autocorrelation functions for the cases of one-pixel, three pixels and five pixels averaged in the flow direction. As the effective size of the resolution cells (the number of pixels) of the power Doppler flow image increases, the full width at halfmaximum (FWHM) of the autocorrelation function also increases. For example, the values of FWHM of the autocorrelation function are about 0.06 s for single pixel, 0.15 s for three pixels and 0.24 s for five pixels. That means the FWHM value of the autocorrelation function is linearly proportional to the number of pixels used for averaging in the region of interest. The results were the same as those predicted in a previous article (Wagner et al. 1987), in which a fluctuation dissipation theorem was applied. Figure 2 shows the results of the autocorrelation function for the different flow velocities in the different regions of interest. One was in the central part of the tube with a relatively higher flow velocity, and the other was near the edge of the wall with a relatively lower flow velocity. The results showed that the autocorrelation function of the integrated power Doppler signals decreased much faster

I rm

Clltrasound

000

0.2

0.4

in Medicine

0.6 0.8 1 Time (set)

1.2

and

Biology

1.4

1.6

Fig. 1.The experimentalresultsof the temporalautocorrelation functions for the casesof single-pixel, three-pixel and five-pixel averagingin the flow direction basedon power estimatesin displayedpixels of a color power imageof a flow tube (see Adler et al. 1995).

for the region with higher flow velocity, as predicted by the theory that increasing the quantity of jll;A~li in the high flow velocity region. Because the autocorrelation function of the power Doppler signals depends on the properties of both the flow imaging system (point spread function) and the scattering medium insonified (the velocity of scatterers), the different results of this autocorrelation function could be obtained by using different scannersfor the sametest object. In general cases,this autocorrelation function could be more complex than a simple autocorrelation function as demonstrated in the previous article (Wagner et al. 1987).

DISCUSSION

Volume

21.

Number

8. IWh

velocity of moving scatterers in the medium il’ the velocities are the same. and the perfusion bu~d OII rhz time-averaged value of the integrated powt~ IIopplcr signals can be determined. Furthermore. if rhc paircorrelation among the moving scatterers ( called paching factor) is negligible. the mean value of 111~‘ integrated power Doppler signals will be linearly pr”portional to the number of moving scatterns in the resolution cell of the image system and could be LIW~ to quantify the perfusion measurement (MO et al. 1992). Obviously, the velocities across a vessel are not the same.This adds an additional, but realistic complication to the analysis. However, if one assumes that the scatterer strength is the same for each velociry component in the sample volume, then eqn (6) bays that the measured amplitude decorrelation rate is just the mean decorrelation rate of all velocities sampled. Hence, eqn (7 ) measuresthe squareof the mean amplitude decorrelation rate, and by the same token, if it is known, then the average velocity can be determined. The assumption of equal scatterer strength seems reasonablesince perfusion measurementswill be made in areas with small vessels where the shear rate ia relatively high. With high shear rates, rouleaux formation will be minimal. and thus the assumption of scattering off of groups of independent. random red blood cells should approximately hold.

AND CONCLUSIONS

We have shown that the autocorrelation function of integrated power Doppler signals is directly related to the properties of both the scattering medium and the image system. The fluctuations of the integrated power Doppler signals are not due to individual moving scatterers (such as red blood cells) but are caused by change in coherent scattering as groups of moving scatterers in the region of interest. For a well-defined clinical scanner, it should be possible quantitatively to measure the flow parameters for a medium using the autocorrelation function. For an example, the FWHM of the autocorrelation function is proportional to the

Center of Tube

0

0.2

0.4

0.6 0.8 1 Time (set)

1.2 1.4

1.6

Fig. 2. The experimentalresultsshowthe effect on temporal autocorrelationfunctionsfor the increasedflow in the center of the vesselcomparedto the flow nearthe walls (seeAdler et al. 1995). The decorrelation plots were derived from Doppler spectrafrom flow through a singleDoppler sample volume.

Power

Doppler

autocorrelation

The method using the autocorrelation function of the power Doppler signals to estimate the velocity of the moving scatterers has several advantages over the conventional A-mode methods. The power Doppler methods can be easily implemented, because the power Doppler signals are easy to obtain through simple quadrature detection and mixing of the signal and both a wall (high pass) filter and a low pass filter are present in current commercial ultrasound scanners. Because the scattering signals from static soft tissue have already been removed from the integrated power Doppler signals, the signal-to-noise ratio (SNR) for our flow image could be improved. The power Doppler imaging technique based on the integrated power Doppler signals also improves the Ilow sensitivity compared with the technique used in conventional color Doppler imaging. Finally, the integrated power Doppler signal is almost angle independent, and is not subject to aliasing artifacts ( Rubin et al. 1994). If the flow is turbulent, there is no longer a simple relationship between the autocorrelation function of the power Doppler signals and the properties of both the imaging system and the medium as shown in eqn (6). because, in turbulent flow, there is a wide distribution of scatterer velocities. In that case, the FWHM value of the autocorrelation function cannot be used for estimating the mean flow velocity. However, if the how is isotropic in a region of interest, it could still be quantitatively possible to estimate how from this autocorrelation function. Ackrlo~lpyrmenrs-This search Grant DAMD

research 17-94-5-4144.

was supported

by U.S. Army

Re-

REFERENCES Adler RS, Rubin JM, Fowlkes JB, Carson PL, Pallister JE. Ultrasonic estimation of tissue perfusion: a stochastic approach. Ultrasound Med Biol 1995;21:493-500. Chen J-F, Zagzebski JA. Madsen EL. Non-Gaussian versus nonRayleigh statistical properties of ultrasound echo signals. IEEE Trans Sonics Ultrason 1994:41:435-440.

0 J.-F. CHEN er (11.

1057

MO LY, Cobbold RS. A unified approach to modeling the backSCattered Doppler ultrasound from blood. IEEE Tram Biomed Eng 1992:39:450-461. Papoulis A. Probability, random variables and stochastic processes. New York: McGraw-Hill, 1965. Rubin JM, Adler RS. Power Doppler expands standard color capability. Diag lmag 1993:66-69. Rubin JM, Adler RS, Fowlkes JB, Spratt S, Pallister JE. Chen J-F. Carson PL. Fractional moving blood volume estimation with power Doppler US. Radiology 1995: 197: 18% 190. Wagner RF, Insana MF, Brown DC. Statistical properties of radiofrequency and envelope-detected signals with application to medical ultrasound. J Opt Sot Am A 1987;4:910-922. Wagner RF, Insana MF. Smith SW. Fundamental correlation lengths of coherent speckle in medical ultrasonic images. IEEE Tram Ultrason Ferroelec Freq Control 1988;35:34-33.

APPENDIX As stated in the text, p( AT ), the normalized autocorrelation function for the complex Doppler signals, I/‘,(/ ). is defined by: I

=
(t)

>,/<

U!(r)U:*(t)i,

(A-l)

where (. . a) denotes an expectation value. If red blood cells can be considered to be individual, pointlike moving scatterers located within and well outside of the sample volume, the expectation value can be written as:


> = c c < U,(?. ,=, j=, *r,

T)fr:(r.

T + AT)>-

where M, is the total number of moving scatterers which significantly contribute to the echo signals at a specific delay time, I. p,( AT ) = lia,ll’h(Ai~(~))~9’(Ai,(~+Ai))(Wagneretal.l987),AT,(7) = $,(~)AT and A?,(T + AI-) = Q,( 7 + Ar)Ar. Here, Ilnil( and is, (Q-) are the backscattered amplitude and velocity of the ith scatterer, h( A?) is a normalized point spread function. That means, p, (0) = JJa,jJ* . and consequently: I,

Finally,

we have eqn (6).