Spectral characterization and attenuation measurements in ultrasound

RASONIC

IMAGING

5,

95-116

(1983)

SPECTRAL CHARACTERIZATION AND ATTENUATION MEASUREMENTS IN ULTRASOUND Stephen

W. Flaxl, Norb?rt .I. Pelcl, Frank D. Gutmann , and Maurice

Gary H. Glgver', McLachlan

1 Applied Science Laboratory Medical Systems Operations General Electric Company P.O. Box 414, NB-924 Milwaukee, WI 53201 2Mt. Sinai Milwaukee,

Medical Center WI 53201

Various means of characterizing ultrasonic attenuation in tissue are A simple method for estimating frequency-dependent attenuation reviewed. via measurement of the zero crossing density of the signal is presented and validated. Both the effects of the frequency dependence of scatter and stochastic variability of the measurement are considered and discussed. animals and humans are presented Results of measurements made in phantoms, The technique is shown to be and compared to the theoretical model. technically feasible. Key words: I.

Attenuation;

computer

model;

scattering;

ultrasound.

INTRODUCTION

survey Clinical ultrasound scanners are most commonly used to the structures. Typically, the images are based upon anatomical backscattered signal produced by acoustic impedance interfaces and by The microand macro-scatter sites inherent throughout the tissue matrix. specular reflections produced by large regular boundaries provide the bright outlines of organs and structures while the lower level scatterers throughout the different tissues. provide a gray level background texture of diagnostic The resolution, ease of operation and image quality Simultaneously, the level of ultrasound equipment has steadily improved. sophistication of clinicians has increased to the point where impressive diagnostic accuracy can be achieved using modern ultrasound equipment. Nevertheless, clinicians using today's equipment are still only able to utilize a fraction of the total information contained in echoes returned from tissues. In fact, so much structure is contained in the returned echoes that the real problem is trying to interpret or glean clinically useful information from the several physical processes involved in the echo production. It is to this end, the quantitative measurement of ultrasonic parameters, that much current research in medical ultrasonics is devoted. Four parameters that are good candidates for quantitative measurement are the speed of sound [l-6], the attenuation of acoustic energy [7-141, the frequency-dependent scatter properties [15-211, and the regularity of structure (causing diffraction patterns) within tissue [22-251. While these parameters may perhaps be studied separately in the laboratory, at some level all the processes are operable simultaneously in clinical situations. As a result, when quantifying a given parameter it is necessary to consider possible interactions with the other processes.

95

FLAX ET AL

This paper is primarily concerned attenuation in tissue. A simple method attenuation is demonstrated. The effects the measurement of attenuation are also the theoretical and practical ramifications Finally a means of placing definitive reliability of the measurement is discussed.

with the characterization of for measuring frequency-dependent of frequency-dependent scatter on considered in order to determine of this interfering effect. error bounds on the statistical

Attenuation of acoustic energy during propagation is a complex phenomenon. Two mechanisms are primarily responsible for the attenuation [15, 18, 19]. These are (1) the scatter of energy out of the acoustic pathway, and (2) absorption, in which acoustic energy is transformed into thermal energy. The two mechanisms interact in biological tissue to produce a net acoustic attenuation which has an approximately linear frequency dependence over the frequency range of typical ultrasound equipment. While mechanisms have been proposed for this behavior, no definitive explanation yet exists. Attenuation is generally specified in decibels of signal power dissipated per centimeter of propagation per megahertz. Two measurement schemes have been used to measure attenuation The loss of signal amplitude ]12-141 and the shift in the --in vivo: spectrum of the rf signals received at various depths [8-111. A noteworthy factor in biological tissue is that the energy loss associated with the scatter mechanism is small compared to absorption [19]. Yet, when observing a backscattered signal in the reflectance mode, the observed signal energy is dependent on the scatter mechanism. It is instructive to consider two cases. First, consider an example in which the scatter cross-section of the tissue varies with depth. Variation in the backscattered signal amplitude as a function of depth will result due to the changing cross-section. Since the scatter mechanism only accounts for a small fraction of the total attenuation, even a sizeable change in the scatter level will produce only a slight change in total attenuation. Thus, the variation in signal amplitude may be completely unrelated to what the case in which the the actual attenuation IS. Second, consider of the scatter cross-section changes. Again, this frequency dependence might not appreciably affect the total signal attenuation, yet it would This shift would produce a spectral shift in the backscattered signal. cause ambiguity when trying to determine tissue attenuation using the spectral shift method. We see then that both methods can be misleading under certain circumstances so one should be cautious when interpreting results. Usually, for these reasons attenuation measurements are made in large, relatively homogeneous regions. Characterization of the attenuation of ultrasound in materials by the was first reported by Serabian [26] and measurement of spectral shifts Merkulova [27] in 1967. Use of the spectral shift method to characterize attenuation in biological tissues was later pioneered by Kuc [8, 91. In this technique, the spectral shape of segments of the rf echo are measured The ratio of the distal and at various depths along an A-mode vector. The proximal spectra reflects the frequency dependence of the attenuation. drawback to this technique is that the individual spectra and the resulting and to compensate for this many measurements must be ratio are noisy, averaged. Kuc [8] and Dines and Kak [7] have both shown that if the input and the attenuation is a linear function of spectrum is Gaussian shaped, the attenuated spectrum is also Gaussian with the peak frequency frequency, If these assumptions shifted linearly downwards as a function of depth. are valid, a significant simplification results since by merely determining the total process is frequency shift of the spectrum the mean One methodological difference between the two approaches is characterized. that while Kuc [8-lo] measured the spectra of the echoes, Dines and Kak 171

96

SPECTRAL CHARACTERIZATION AND ATTENUATION MEASUREMENTS

In the transmission mode the pulse worked in the transmission mode. integrity is maintained which makes spectral estimation much simpler. Unfortunately, clinical ultrasound measurements are usually obtained in the reflectance mode since transmission through most sections of the body is not possible. There is however, no reason why the method proposed by Dines and Kak cannot be applied to a reflectance mode signal although the effect of noise on the estimation of the mean of the spectrum must be incorporated. The work reported here is a further simplification of the spectral shift techniques. Prior methods sampled and Fourier transformed the rf Here the signals, thus requiring complicated and expensive equipment. density of zero crossings of the rf signal., a technique borrowed from cw Doppler flow meters, is used as a spectral estimator [28, 291. The relationship between the zero crossing density and the power spectrum of The use of this the s
II. A.

THEORY Basic

Model

We first the echo received when an ultrasound pulse calculate propagates through a scattering medium. We assume that the tissue is composed of a random ensemble of weak impulse scatterers producing a intensity which is negligible compared to the incident backscattered intensity (Born approximation). We also neglect velocity dispersion in the tissue [l, 61, and further assume that all measurements are made in the focal region of the transducer or beyond so that the diffractive effects of pulses traversing the face of the transducer may be neglected. For now, we also neglect both the frequency dependence of the scatterers and the frequency-dependent attenuation of the medium and so are left with the rather simple model of an unperturbed pulse interacting with the random ensemble of impulse scatterers. The Fourier transform of the resulting signal is:

j$n(f) Y(f)

=

jJ

L,

e

X(f),

(1)

n=O where X(f)

= Fourier

On(f) ,. a

transform

of the n-th =

n

N

of pressure

= frequency-dependent

=

The power S(f) where A(f) scatterers.

amplitude number

density = lY(f)12

phase

of scatterers

with

the

location

scatterer, encountered.

of this

signal

= IA(f)12*IX(f)12

results from If {a 1 and

associated

scatterer,

of the n-th

spectrum

pulse, change

combining the scatter

is: ,

all of the randomly placed sites are randomly distributed,

97

(2) impulse and

FLAXETAL

if any frequency c&pendence is ignored, A(f) is a random variable whose expected value is independent of f. Effectively, then, the input pulse power density spectrum is being modulated by white noise. As in Dines and Kak [7] we assume that the input pulse power density spectrum is Gaussian: -(f-fo)2/2u2 /X(f)]*

= Ce

(3)

3

where f. u C

= = =

transducer center frequency, characteristic width of the amplitude coefficient.

The power

density

spectrum

transducer

of the backscattered

power signal

spectrum, thus

and

becomes

-(f-fo)2/21? S(f) Now, if spectrum stated, spectrum

= l;l(f)j'Ce

(4)

frequency-dependent is simply modulated we are neglecting becomes a(f,il)

S(f)

= e

^ [A(f)/'Ce

attenuation by the velocity

is introduced, the power density attenuation term (since, as already dispersion). Thus, the attenuated

-(f-fo)*/202 (5)

7

where e(f,a) is an attenuation term dependent path length II. If we assume that attenuation linear function of frequency and path length, n(f,L) where

the

and propagation tissues is

a

(6)

= -aofR,

CI is

on frequency in biological we have

attenuation

coefficient.

0

The backscattered S(f) This

spectrum

where

= /A(f)12C.e

f center

1 .

(7)

to -(f-f

S(f)

becomes

-(f-fo)2/2uP -cY fL O )(Ce

= jA(f)j'(e

can be reduced

thus

f 0 - soLa

center)""'

,

, and

(8) (9)

Clearly the C' is the amplitude coefficient for the attenuated spectrum. Gaussian spectral shape has remained unchanged as a result of the attenuation factor. However, the center frequency has now shifted down by While an equivalent result was derived by Dines and Kak [7], the bx,PAs*). complicating factor that makes this backscattered case more difficult t‘S modulated by the noise term IA(f)] . that the shifted spectrum is still Thus the problem reduces to one of estimating the center frequency of a noisy spectrum. While there are a variety of ways to approach the problem, perhaps the most strajghtforward is to simply use the first moment of the measured spectrum [32] T=

I" 0 I" 0

f S(f)df (10) S(f)df

*

98

SPECTRAL CHARACTERIZATION AND ATTENUATION MEASUREMENTS

The expected

value

s.

Estimation

Spectral

f

of 7 is

center'

Using

Zero

Crossing

Density

The relationship between the density of zero crossings of a waveform and the power spectrum of that waveform was first elucidated by Rice [30] and further elaborated on by Papoulis [33]. Following the development by Papoulis 1331, let g(t) be the waveform of interest and let R(t) be its If r is autocorrelation. Consider samples taken at t and t + T. short compared to the width of R we expect g(t") and g& + T ) to be of is negative the same sign. For any T, if the product get )g(t +OT) then an odd number of zero crossings occurred durfng tl?e time interval and the probability of this occurring can be written p(T)

If g(to)

=

p

{g(to)g(to

and g(t

0

p(r)

+

T)

+ T) are = f


(11)

.

jointly

normal,

the

cosine

law results

in

[29]: (12)

'

where R(T) COS(B) = R!O)

.

(13)

Equivalently: co.5 (rp(-r))

RcTc)

= R(O)

(14)

.

For-r small compared to the width of R, P(T) be the probability of a single zero crossing both cos (~rp(.r )) and R(r) about T = 0 using terms to second order we get

COS

(rip,(T))

is small and will essentially occurring, p (T). Expanding a Taylor series, and keeping

(15)

L- 1 -

and R(r) Substituting of a zero

= R(O) + +

Eqs. crossing p,(r)

(15) and occurring

. (16) into Eq. (14) we find during a short time 7 is

that

the

probability

= XT ,

(17)

where

density

Because of the linear of zero crossings.

Using well be shown that

dependence The desired

of p on result'follows.

known properties of Fourier the expected zero crossing

99

transforms density is

T,

X is

the

on Eq. (18), proportional

expected it to

can the

FLAX ET AL

square

root

of the

second

moment of the power

spectrum

(19)

Given a Gaussian simplified to

shaped

x = 2[fZenter

+ $14

If the characteristic center frequency (this have dh z= It

follows

-2aou2

spectrum

as in

Eq.

= 2[(fo-aokJ*)*

bandwidth condition

(l-aoO%/fo)

+

(8),

Eq.

cd

.

(19)

can be further (20)

u is small compared to is usually met in clinical

the transducer transducers) we (21)

= -2aou*.

that c( FI 0

(> --1 252

dh da

(22)

.

The rate of change of the expected the attenuation coefficient c1

zero

crossing

density

is

proportional

to

0’

C.

Statistical

Considerations

Unfortunately, even if all of the approximations and assumptions are valid, there remains th,e fat that the input spectrum is modulated by the 5. random scatter term IA(f)/ (c.f. Eq. (2) This factor or Eq. (8)). affects the precision with which the expected zero crossing density (and thus the spectral mean) can be measured. this factor has the Of course, when the spectral shift is calculated numerically in the same effect frequency domain. There are two ways in which the problem can be controlled. First, measurements on ^ A-mode traces through slightly different paths, thus with different IA(f)1 , can be averaged. Secondly, the time length of the sample can be increased, but this causes spectral smearing due to the fact that the spectrum is shifting downward as a result of attenuation. When possible, the first method is preferred. Optimizing the sample length and determining how many vectors must be to produce an acceptable spectral estimation is a nontrivial averaged turning to the temporal domain and reexamining the problem. However, estimation from the standpoint of changing zero crossing density, we can gain some useful insight into the overall problem. The problem of characterizing the estimation error associated with the zero crossing process has previously been evaluated. While the results are dependent on the specific characteristics of the spectrum producing the temporal waveform, two intuitively satisfying bounds can be placed on the problem. These bounds can be demonstrated conditions. as given in Eq. (17), First, occurrence of a zero crossing in same short to the expected zero crossing density X. second zero crossing event were statistically

100

by considering the fol.lowing the probability for finding the time interval T is proportional Second, if the occurrence of a independent of when the

SPECTRAL CHARACTERIZATION

AND ATTENUATION

MEASUREMENTS

previous event occurred, then these two conditions would define a Poisson of independence is not met, process [33]. However, if the second condition then there will be some conditional information concerning the probability of when the second event will occur given the occurrence of the initial As an extreme example, consider a sine wave of known frequency but event. unknown phase. Since the phase is not known, the probability of finding a zero crossing in a short interval r will still be XT. However, once that event has occurred, the process will be perfectly defined and we will know where all other zero crossings will occur. A band limited signal will produce conditions between these two cases. That is, given that a zero crossing event has occurred, we would expect the occurrence of the next crossing to be consistent with the frequency constraints of the zero signal. There is a very low probability of having the second event occur either very soon or very long after the occurrence of the initial event. Variations of this problem have been considered in several papers [28, 341. Experimental results also confirm these expectations [28]. Thus, using a Poisson process and a sine wave as a worst case and best case to bound the bandlimited zero crossing process, we have the following conditions. First, let us define the expected number of zero crossing events in a long interval t as Fi, and the variance in the process to be S. For a sine wave the variance of counts in a long interval will essentially be zero; thus the noise-to-signal ratio (S/n) will be zero. For a Poisson process, we know that the variance is related to the mean as S Poisson Thus,

the ratio

4 = n

(23)

'

of S to u for

a Poisson

process

is

given

as

-5 S Poisson/lJ

=

(At)

(24)

,

which shows that the estimation improves with the square root of the number of zero crossings counted. However, this is a worst case, and for band limited processes the estimates will be better. Nevertheless, the technique provides a ready means for predicting the system resolution and the statistical accuracy of the estimation. For example, if we wish to measure the attenuation in a homogeneous tissue using zero counting sample periods corresponding to 1 cm lengths, then by knowing the approximate velocity of sound in the tissue and by knowing the approximate center frequency of the spectrum (based on the transducer center frequency) we can determine the approximate number of zero crossings which can occur in the corresponding period. Then, one can calculate how many vectors have to be summed together to achieve the desired total count for a given accuracy. D.

Frequency-Dependent

Scatter

Variations in the frequency dependence of scatter are a fundamental problem to the estimation of tissue attenuation. While Kuc [8] and Dines and Kak [7] showed that the bandwidth and Gaussian characteristics of an ultrasound pulse were not altered during the process of attenuation, the effect of scatter, which has a nonlinear frequency dependence, has a more serious influence. We can introduce the effect of scatter by including a frequency-dependent term, fZ, into Eq. (8): -(f-f S(f)

= jA(f)12C'fZe

Here, z is the characteristic for a specular reflection

center)212a2 scatter to 4

exponent for pure

101

(25) which usually varies Rayleigh scatterers.

from

0 For

FLAK ET AL

biological [17, 181.

tissue,

z

is

experimentally

found

to

be

between

In order to get a sense of how the fz term affects the spectrum we can write the term as an exponent raised power and then approximate the natural log with a Taylor terms to second order we get z JfiZ

r e

Substituting

f-f0 + f

ln(fo) i

(f-f - o 2fo?

0

Eq.

(26)

into

Eq.

(25)

-(f-f' S(f)

center

= l;l(f)/We

)*

.

/(2o'F)

2

the behavior of to a natural log series. Keeping

1

(26)

and combining j2

1 and

terms

we get

,

(27)

where F = f;/(f; frenter

and

+ zc2) =

cfo

C" = amplitude spectrum.

-

(28)

, aou2& + (2za"/fo)]

coefficient

for

F ,

(29)

the attenuated

and backscattered

Observe that the basic spectrum remains Gaussian in nature as long as the three therm Taylor series approximation characterizes the term zln(f). However, the center frequency is now not only dependent on the attenuation term (a ) and depth (a), but also is offset by an amount 2z02 /f . In addition' the bandwidth is also modified by the factor f2/(f2 + 20~) . Assuming U* is small compared to f2 and since z ranges begwee: 1 and 2, then the scaling term can approximately be ignored reducing Eq. (29) to f where center

= aoo*P, + 2za2/fo , f. center the effect of changing scatter properties frequency up or down as a function of tissue

(30) is simply to shift the scatter property, z.

It is evident now that the frequency dependence of scatter produces an additive term to the center frequency of the detected spectrum. In a region where the frequency dependence z is constant, the depth rate of change of the center frequency (e.g. as measured by the zero crossing density) can still be used to measure the attenuation coefficient. In cases where z changes abruptly from one region to another, an abrupt change in the center frequency will be observed. If this abrupt change is recognized, both attenuation and the magnitude of the change in the However, in those frequency dependence of scatter can be characterized. cases in which the frequency dependence of scatter within the sample varies in an unknown manner, the detected echo cannot be used to measure attenuation. III. A.

METHODS Computer

Model

A convolutional temporal waveforms.

model based The generated

on Eq. temporal

102

(1) was developed signal was Fourier

to simulate transformed

SPECTRAL

CHARACTERIZATION

AND

ATTENUATION

MEASUREMENTS

to yield the noise modulated Gaussian spectrum. The number of zero crossings contained within a time window were counted as a function of the temporal position of the window and plotted. Center frequencies of 3.5 MHz and 7 MHz, count windows of 5 ~i.s and 10 us, and a sliding increment of 0.5 us were used. The goal was to gain intuitive insight, to examine the precision that could be expected, and to provide a baseline for comparison to the experimentally collected data. B.

Phantom

Measurements

A custom-made 0.8 db/cm/MHz graphite phantom (Radiation gel Measurements, Inc., Middleton, WI) was used to assess the suitability of using the change of zero crossing density to measure attenuation. The rf output from a static arm B-scanner (Datason, General Electric Co., Milwaukee, WI) was sampled and digitized to 8 bit accuracy at a rate of 20 MHz using a transient waveform analyzer (Biomation 88100, Gould Inc., Santa Clara, CA) and transferred to a hard disk for analysis. A variety of 3.5 The sample period was delayed and 5 MHz clinical transducers were tested. by 40 !!s to reduce the spectral effects of the transducers' surface diffraction. Six to twenty rf vectors at different physical locations were collected for each transducer. The stored rf data were used to compare frequency domain estimation processes with the proposed zero crossing density method. C.

Animal

Studies

The change in zero crossing density as a function of depth was used to study ultrasonic attenuation in the livers, spleens and kidneys of six mongrel dogs. The dogs, ranging in weight from 15 kg to 20 kg, were mildly dehydrated by withholding water for 12 hours prior to the procedure and were anesthetized with methoxyflurane. The abdominal organs were exposed via a long midline incision. The left kidney, liver and spleen were sequentially measured by gently retracting the surrounding organs. Parameters indicating the dog's physiological state were monitored throughout the procedure. Data were collected using a focused (4 to 9 cm), 5 MHz, 13 mm diameter transducer mounted on a 6 cm water stand-off. The stand-off was used so that diffractive effects at the transducer face could be ignored and so the tissue of interest would be sampled with as uniform a beam profile as possible. The transducer was connected to a pulser/receiver (Model 5050, Panametrics, Morristown, NJ) which in turn was connected to the waveform analyzer. Control of data acquisition and storage on magnetic cassette tape was performed by a microcomputer (MicroNova, Data General Corp., Southboro, MA). The recorded data were transmitted from the animal laboratory at Mt. Sinai Medical Center in Milwaukee to the computer laboratory via telephone lines. After transducer placement, the A-mode signal was evaluated to insure that only relatively homogeneous tissue and no complex structure or interfaces were present in the pathway. (This, of course, was not possible to achieve in the kidney.) At that point, 2000 samples were collected with the waveform analyzer as in the phantom studies. Six measurements were made at different physical locations in each organ. Due to slow data transfer rates (300 baud) measurements were separated by at least 2 minutes. D. was

Human Liver

Measurements

Two human volunteers were scanned capable of measuring attenuation

103

to demonstrate that under clinical

the

technique conditions.

FLAX ET AL

Intervening tissue and anatomic identification make this situation very different than the previous experimental one. The previously mentioned static arm B-scanner and waveform analyzer were used. For each subject, the liver was scanned in a survey mode until a slice that clearly showed the vena cava was found. the sonographer slowly retraced At that point, the scan until the ultrasound beam propagated through a relatively uniform portion of the liver. The time delay on the waveform analyzer was adjusted until the last portion of the sampled waveform just reached the vena cava (easily recognizable in the A-mode display). At that point, 2000 samples corresponding to 7.7 cm of tissue were recorded and transferred to the disk. At the same time, that corresponding line in the B-mode image was brightened and the image was recorded. This provided a permanent record of the tissue region actually measured. Ten measurements were made on each subject. IV. A.

RESULTS AND DISCUSSION Computer

Model

and Phantom

Experiments

The spectral and temporal results produced by the convolutional model are qualitatively similar to the data collected with the graphite gel phantom. This can be seen by comparing examples of the simulated data (Fig. 1) with the measured results (Fig. 2). Results of the calculation of the zero crossing density of the simulated waveform as a function of time using two averaging windows are shown in figure 3. Since frequency-dependent attenuation is not included in the model, no change in frequency with time is expected. The fluctuations about a constant value in figure 3 are due to statistical variations. The improvement in precision that results when the length of the averaging window is doubled is clearly demonstrated. Doubling the center frequency of the waveform results in 2 concomitant increase in the density of zero crossings, as shown in figure 4. This verifies that the zero crossing density is an estimation of the center frequency. Experiments on the phantom bridge the gap between the rather idealized mathematical model and the more complex and less controlled physiologic Data from the phantom, stored in the computer, were used to situations. compare different estimation techniques. Figure 5 shows two 15 vs segments separated by 50 IJS from one waveform. A decrease in frequency with depth can be observed visually. The result of calculating zero crossing

Fig. 288

0

SFIMPLE

2

POINTS

LOG

FREQZENCY 6_ MHZ (256 POINT FFT)

OUTPUT

8

10

104

1

Example of the temporal rf waveform produced by the model of computer random impulse scatterers (top) and the corresponding power density spectrum (below). The model is generated using a given 3.5 MHz Gausian shaped pulse and a sampling rate of 20 MHZ. The temporal waveform shows 288 sample points of the log compressed scatter signal.

SPECTRAL CHARACTERIZATION

AND ATTENUATION

I I I’f ‘ITt’l I I I I I I I

288

SAMPLE

0

Fig.

3

2

POINTS

LOG

OUTPUT

8

Fig.

2

10

FREQtiENCY 6_ MHZ (256 POINT FFT)

MEASUREMENTS

Example of the temporal rf waveform as measured from a graphite-gel phantom (MUMiddleton, WI) (top) and the corresponding power density spectrum (below). The measured data was obtained using a 3.5 MHz transducer and a 20 MHz sampling rate. The temporal waveform shows 288 sample points of the log compressed scatter signal.

Plot of the number of zero crossings per interval (zero crossing density) as a function of depth for simulated data. (a) represents a count interval of 200 samples while (b) represents a count interval of 100 samples. In both cases the interval slides 10 samples for each count. Note the relative in count increase variance for the shorter count interval when both curves are normalized.

16

Fig.

.

‘REiAiIliE

bE6TH

1-7

tCNc,

-6

105

4

Comparison of crossing density for two different spectral center (f o and 2fo).

the zero per depth simulated frequencies

FLAX ET AL

Fig. I

I

I 38

I

I

I

_I I

S MPLES I

I

ST I

.

I RT

I

NG I

A

, 2

I

B

5

, 1 I

densities using a 10 ps window is the density of zero crossings frequency-dependent attenuation.

I

An example of the sampled rf waveform from a phantom taken at two different depths. The slight shift in frequency caused by frequency selective attenuation can be visually observed when comparing the two waveforms. The measured data was obtained using a 3.5 MHZ transducer and a 20 MHZ sampling rate. The top waveform shows 300 samples beginning at sample 200. The bottom waveform shows 300 samples beginning at sample 1200.

shown in figure 6. a5 a function

Note of

the decrease time due

in to

Analogous information can be derived in the frequency domain. Power density spectra were calculated using 256 point Fourier transforms. A Hanning window was used and spectra were calculated as the window was moved through the data in 2 us (40 point) steps. The resulting spectra as a should be noted, (1) function of time as shown in figure 7. Two features the shift toward lower frequencies as the samples are taken at greater and (2) the predicted random modulation of the power density spectra depth, The same data displayed by allowing image brightness to (see Eq. (8)). In this represent the amplitude of the spectrum is shown in figure 8. form, the shift to lower frequencies is even more apparent. Fig.

0

DEPTH

INTO

PHANTOR

t CH)

6

106

6

An example of a single plot of the zero crossing density as a function of depth in a graphite gel phantom. The density represents the number of zero crossings occurring in a 200 sample window rate) (at a 20 MHz sample allowed to slide at 10 samples per step through 2000 samples. The data is obtained from a graphite-gel phantom with a measured attenuation of 0.8 dB/cm/MBz and using a 3.5 MHZ transducer. Note the downward trend in zero crossing density with depth, correfrequency sponding to the selective attenuation.

SPECTRAL

CHARACTERIZATION

AND ATTENUATION

A series of point, 256 Hanning windowed, Fourier transforms for the waveform corresponding to that used for figures 5 and 6. In this case, each successive spectrum represents a shift of 40 sample points. The random spectral variations can best be seen in this pseudo three-dimensional format.

Fig.

Fig.

8

MEASUREMENTS

The same spectral information shown in figure 7 except not presented in a pseudo three-dimensional format. In this, amplitude is denoted as brightness. The spectral shift toward lower frequencies can best be seen in this format.

Fig.

107

9

A comparison between the zero crossing density, the spectral first moment, and the spectral second moment, all plotted as a function of depth within the phantom. curves are The three all generated from the same sampled rf wave form. The zero crossing density was measured directly from the sampled data, while the first and second spectral moments were calculated after first performing the 256 point FFT's. Note the qualitatively the three curves track the data equally well.

FLAX

:,

ET

AL

36

z w N

18 0

DEPTH

INTO

PHFlNTOM

(CM)

7

Fig.

10

Results of obtaining six zero crossing density sample vectors from a graphite-gel phantom Phantom (RMI 0.8 dB/cm/MHz). (a) Plot of the six vectors superimposed. (b) Plot of the average value of the six vectors.

In order to quantify the spectral shift observed in figures 7 and 8, In addition, the the first moment of each power spectrum was calculated. square root of the second moments were also calculated since zero crossing densities are mathematically related to the second moments (see Eq. (19)). shown in figure 9. The comparison of the estimation techniques is Qualitatively, the three techniques track each other relatively well. It that none is a very precise predictor of should be noted, however, attenuation, as evidenced by the fact that the variation within each curve is comparable to the change resulting from frequency-dependent attenuation. considering the temporal sample lengths This is not at all surprising involved.

0

DEPTH

INTO

LIUER

5

Fig.

108

11

six obtaining Results of zero crossing density sample vectors from an --in vivo dog liver. (a) Plot of the six vectors superimposed. (b) Plot of the average value of the six vectors.

SPECTRAL CHARACTERIZATION

1

I,

t

DEPTH

0

1

i

INTO

t

I

SPLEEN

I

!


II

AND ATTENUATION

I

MEASUREMENTS

5

Fig.

12

obtaining six Results of zero crossing density sample vectors from an --in vivo dog spleen. (a) Plot of the six superimposed. vectors (b) Plot of the average value of the six vectors.

means of improving the precision of As discussed above, one effective The tissue should be homogeneous the estimated frequency is via averaging. along slightly different paths (so as to be so that measurements Measurements of independent) are representative of the same tissue. average zero crossing densities are particularly well suited to an averaging technique. All that is required is to allow the zero crossing events to accumulate in each sample interval as additional vectors are collected, and to normalize the observed events by the number of vectors n. the estimation error will be reduced If the n measurements are independent, by a factor of G Figure 10 (top) shows plots of six measurements similar The result of averaging these data to figure 6 superimposed on each other. and the obvious impact on precision is also shown in figure 10. B.

Animal

Studies

The data measured on the livers and spleens of the dogs was in many similar to the phantom and simulated data. Composite and ways very averaged plots for the liver and spleen of one animal are shown in figures 11 and 1'2 respectively. The measurement variation is comparable to that in the previous experiments. To within experimental errors, there is a monotonically decreasing center frequency and the measurements are reasonably linear as a function of depth. In figures 13 and 14, the

Fig.

-

DEPTH

INTO

LIVER

109

13

Plot of the averaged zero crossing density variations for the six livers dog scanned.

FLAX ET AL

Fig.

14

Plot

of

crossing

for the scanned.

the averaged zero density variations six dog spleens

averaged data for the livers and spleens, respectively, of all the animals are superimposed. These data, meant to be only a demonstration, suggest that the attenuation in liver is higher than that in spleen, but no attempt has yet been made to demonstrate statistical significance. The data from the kidney scans is less similar to the phantom and simulation studies and that makes it more interesting. The kidney is a nonhomogeneous tissue as evidenced by the anatomy shown in figure 15, and by an example of an A-mode signal through the kidney shown in figure 16 The plot of zero crossing density in these experiments demonstrate (top). an unexpected rise in the zero crossing density in the medullary and papillary regions of the kidney (Fig. 16). When five scans from the same kidney are averaged as in figure 17, it appears that the trend is not a statistical aberration. Further, combining the data from the six animals in the study shows the basic trend holds for the series as can be seen in figure 18. While any single vector has stochastic errors, when all the measurements are included the difference between the proximal to mid and mid to distal regions are statistically significant (P < .Ol). It would be difficult to explain this increase in frequency using attenuation mechanism. It is more realistic to consider changes in frequency dependence of the scatterers, as discussed above and described

Fig.

110

15

an the in

Photograph showing the cross sectional anatomy of a dog kidney. Silicone rubber was injected into the arterial supply to help delineate the anatomy,

SPECTRAL CHARACTERIZATION

PROXIMAL

C&P.

DISTAL

1I I i I 1

AND ATTENUATION MEASUREMENTS

a

CAP

( /

! lb\

I

Fig.

I

16

(a) Typical A-mode scan of an -__ in vivo dog kidney using a water standoff transducer placed directly on the cspsule. Corresponding (b) zero crossing density.

N /

0

DEPTH

(CM)

5

. ^^

Fig.

17

0

DEPTH

INTO

KIDNEY

(CM)

5

Results of five zero crossing density sample vectors from a dog kidney. (a) Plot of the five vectors superimposed. (b) Plot of the value of the five average vectors.

120 :: :: z E a \ ::

90

III

E

N 0

I DEPTH

I

I

INTO

I

I

KIDNEY

II

Fig.

III (CM)

5

111

1S

Plot of crossing for the scanned.

the averaged zero density variations six dog kidneys

FLAX ET AL

Fig.

19

B-mode image of a human liver from a volunteer subject. Note the brightened line indicating the sample vector obtained for one data set. Also, note the vena cava (VC) is used as a landmark.

Eqs. (25) through (30) as being responsible for this effect. It could be hypothesized that scattering in the medullary and papillary regions of the kidney has a higher frequency dependence (larger z) than that in the cortical regions. It must be emphasized that these results are presented as an interesting observation only. Additional work needs to be carried out to understand and distinguish the frequency dependence of scatter and attenuation in tissues. C.

Human Liver

Measurements

Since only two human volunteers are included in this report, the data can only suggest that the technique can be applied to clinical situations. The data do demonstrate that performance comparable to that in the previous experiments can be achieved. Figure 19, an example of one of the B-mode images obtained from the first subject, shows a cross-section of the liver with the vena cava at the bottom. The brightened line through the liver represents one of the sample vectors obtained. Figure 2@ shows the zero crossing density data for the brightened vector in figure 19. Composite data for this subject is shown in figure 21. While the averaged data is well behaved, there is more variation among the 10 scans that there was in the phantom or animal experiments. This can be explained by the fact that the path length to the vena cava varied depending on where the scans were made. Thus, the amount and type of intervening attenuating tissue in each pathway varied and so the effective starting frequency is slightly different for each scan. This effect will not cause variation in the estimated slope if the frequency dependence is linear. Qualitatively this appears to be the case.

70

\ :: E40 N

- iI 0

I

I

I

I

RELATIVE

I

t

I

I

I

t

DEPTH

I

I

Fig.

I 1 (CM: )

7

112

20

Zero crossing for the scan figure 19.

density vector

plot from

SPECTRAL CNANACTERIZATION

0

DEPTH

INTO

LIVER

AND ATTENUATION MEASUREMENTS

(CM)

7

Fig,

r z5 \ cs $j 40’

II

II

II

I I

II

II

II

N 0

V.

JEPtH

‘INtO

LIUER

(:M)

I

I

II

,

,,

,

21

obtaining ten Results of zero crossing density sample vectors from a human volunteer subject's liver. (a) Plot of the ten vectors superimposed. (b) Plot of the ten average value of the vectors.

SUMMARY AND CONCLUSIONS

The theoretical background of ultrasonic parameter estimation by measurement of the frequency of the rf signal, much of it previously described by Kuc [&lo] and Dines and Kak [7], was reviewed. Of particular interest is the fact that the change in the center frequency of the spectrum of the ultrasonic wave can be used to measure frequency-dependent attenuation. However, when measurements are made in the reflectance mode, as is usual in clinical settings, the effects of backscatter make the interpretation considerably more complicated. Both the random spatial distribution of scatterers and their frequency-dependent properties must be considered before inferences can be made. Rice [30] has shown that the density of zero crossings of the rf signal is proportional to the square root of the second moment of the rf power spectrum. For applications in ultrasound where the bandwidth of the spectrum is smaller than the center frequency, the .density of zero crossings can be used to estimate the spectral center frequency. The major advantage of this estimating technique is that it is much simpler to implement and much faster than frequency domain methods. Statistical errors in measurements made in the reflection mode were These distribution of the discussed. errors are due to the random scatterers producing the detected echo which cause the spectrum to be modulated by noise. This effect limits the precision in the measurements and can be reduced by the use of averaging techniques. It was shown that the statistical error in the measured number of zero crossings in a given time period is lower than it would be if this was a Poisson process. As a result, with this technique, Poisson statistics can be used as a bound on the precision of the measurement and can thus guide the choice of the number of averages and time interval length required to achieve the desired precision. The application of the zero crossing technique was validated in a series of computer simulation and experimental studies. Measurements on the livers and spleens of six dogs were shown to be qualitatively similar

113

FLAX ET AL

to measurements on a phantom and to the simulated data. Measurements on the kidneys of the dogs showed that the echoes from the medullary and papillary regions had a higher zero crossing density than echoes from the cortical region. This interesting effect is probably due to a higher frequency dependence of the ultrasonic scatter in these two regions. Finally, scans of two human volunteers showed that, from a technical standpoint, the technique can be applied in clinical situations. In summary, the zero crossing technique has been shown to be a simple and practical method of estimating the frequency of an ultrasonic signal. This technique may make it possible to quantify several tissue parameters It remains to be conclusively shown, however, whether these --in vivo. parameters are relevant to clinical prohlems. Effort is now underway to try to answer this critical question. ACKNOWLEDGEMENTS The authors would like to thank Sandy Maleski and Ed Steinike for their excellent technical assistance, Rita Edwards and Robert Wurm for their invaluable assistance with the experimental animal preparations, and Mary Koch for typing the manuscript. REFERENCES

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