- Email: [email protected]
Estimating ultrasound propagation velocity in tissues from unwrapped phase spectra
F?ASONIC
IMAGING
2,
223-231
(1980)
ESTIMATING ULTRASOUND PROPAGATION VELOCITY IN TISSUES V. N. Gupta,
FROM UNWRAPPED PHASE SPECTRA P. K. Bhagat,
and A. M. Fried'
Wenner-Gren
Biomedical Research Laboratory University of Kentucky Lexington, Kentucky 40506
A digital computer-based technique for estimating the velocity of sound in tissue is proposed. The technique utilizes the unwrapped phase spectra of fullband reference and attenuated ultrasonic pulses normally used in In addition, it does not require explicit meaattenuation measurements. surement of pulse-transit-times. The accuracy of the technique was tested on several computer simulated models, on specimens of known velocities (aluminum and plexiglass), and on formalin-fixed rabbit kidneys. The results indicate that the technique has an accuracy comparable to the pulse-transit-time method. Key words:
1.
Phase
spectra;
ultrasound;
velocity.
INTRODUCTION
Systematic study of the interaction of biological systems with applied energy provides insight into the organization and structure of the tissue Ultrasound at low intensity levels has been used to visualize medium. the tissue interfaces. The attenuation coefficient and propagation velocity are the two acoustic parameters most commonly used in quantitative tissue characterization. The attenuation coefficient, for a given tissue specimen, is generally estimated by computation of the difference between the log-magnitude spectra of the reference and that of the tissue attenuated signals [l-4]. Velocity, on the other hand, can be computed using one of the several available techniques [5]. Notable among these are the interferometric, pulse-transit-time, and the sing-around techniques. The interferometric technique is limited mostly to liquids and quasi-liquids. It is based on the principle that when the distance between the transducer and the reflector, within the medium under investigation, is an integral multiple of half-wavelengths, the detecting system indicates a corresponding minimum in the driving voltage [6]. In the pulse-transit-time method, two distinct variations are available. In the first, the pulse is transmitted by one transducer and detected by another, the axes of the two transducers being coincident. In the second, only one transducer is used, alternately, in the transmitand receive-mode, with a reflector positioned normal to the transducer axis. The pulsetransit-times together with specimen thickness and velocity in the coupling medium is used to compute the velocity in tissue [7]. A.M. Fried is with Department of Diagnostic Radiology, Kentucky Medical Center, Lexington, KY 40536.
University
of
0161-7346/80/030223-09$02.00/O
223
Copyright :U 1980 b-vAcademic Press, Inc. All righrs of reproduction in any form reserved.
GUPTA, BHAGAT, AND FRIED
The sing-around technique is a variant of the pulse-transit-time Instead of measuring the pulse-transit-time across a known technique. length of the tissue specimen, the number of times the pulse travels back and forth through the tissue specimen is counted over a known period of time [8]. In all the techniques discussed above, one has to conduct separate measurements for velocity estimation. In the present study, we propose a digital computer-based technique for measurement of ultrasonic propagation velocity which does not require explicit measurement of transit times. Instead, propagation velocity is estimated from the phase spectra of reThe underlying hypothesis implicit ference and tissue-attenuated signals. in this technique is the premise that for nondispersive media the phase of a traveling wave is proportional to the frequency for a given propagation velocity. Measurements in our laboratory [9] and elsewhere [lo] indicate that biological media (soft tissue) exhibit negligible dispersion over the frequency range l-10 MHz.
2.
THEORETICAL
DETAILS
When an ultrasonic beam propagates through a weakly scattering, dissipative medium, it suffers a loss of energy due to attenuation. The term attenuation is defined here to include losses due to reflection, scattering, refraction and absorption. Consider the pulse-echo system of figure 1, where a tissue specimen positioned between the ultrasonic transducer and a plane plexiglass reflector is immersed in a dissipationless coupling medium. Let the transducer emit a short, bandlimited burst of ultrasonic energy p.'(t) and let pi(t) and p,(t) be the echoes reflected from the reflector wlthout and with the tissue specfmen in position. The relationships between pi'(t), pi(t) and p,(t) can be expressed as Pi(t)
= ' p Im Pi'(w) -co
Pr(t)
= ap
e-j2wao'vo
ejwt
dw,
and
jm p l(w) e-2a(~)ae-j2[(~o-~)/vot~/v] i -co
,jut
dw,
where
Pi’(w)
=k
joD pi'(t)
e-jut
dt,
-m = angular frequency in radian/s, II = distance between the transducer and the Lo = tissue-specimen thickness, = reflection coefficient for the plexiglass aP = sound velocity in the coupling medium, ;o = sound velocity in tissue, and a(w) = frequency dependent attenuation. w
receiver, reflector,
In deriving Eqs. (1) and (Z), we have implicitly assumed that the coupling medium and tissue specimen are non-dispersive, i.e., the group velocity is the same as the phase velocity. This is based on results published from this laboratory [g] and elsewhere [lo], where maximum dis-
224
UNWRAPPED
PHASE
SPECTRA
TISSUE
ULTRASONIC
Fig.
1.
A schematic
representation
of the
persion was observed to be less than 2 percent over the frequency range 1 to 10 MHZ. Further, the reflection losses at the assumed to be negligible since the acoustic water are approximately equal. Fourier
Equations (1) transforms Pi(w)
and (2) as
= ap Pi'(w) = 1 Pi(w)1
pulse-echo
for
several
tissue-water impedances
may be rewritten
in terms
system. tissue
specimens,
interfaces are of the tissue and of their
respective
e-j2wao'vo (4)
e-jeo(@)
and P,(u)
= aP Pi'(w) = 1 P,(w)1
e -2aa(w)
,-j2u[(k,-a)lv,+alv]
e-jel(,).
(5)
Defining sO
= 2i?o/vo
(61
= 2mo-a)/v,+a/v1,
(7)
and s1 it
can be shown that
a(w)
=
[ln 1 Pi(w)]
-
ln I Prb)l
1 l(2a)
(8)
and V
= [(so-S, f/T,,+11 vos
(9)
where T = 22/v is defined to be the duration of the backscattered signal The use d?iginating from the entire depth of the tissue specimen. of Eq. (8) for attenuation measurements has been reported earlier [l]. Equation (9) provides velocity estimates using the phase components of the Fourier transforms of pi(t) and p,(t), namely, e,(w) and e,(w).
225
GUPTA, BHAGAT, AND FRIED
3.
IMPLEMENTATION
OF VELOCITY ESTIMATIONALGORITHM
As defined in the previous section, the pulse propagation velocity can be estimated from the phase components of the measured reference and attenuated spectra. However, the computations of the propagation-lags s and s are compounded by the fact that:(i) the phase spectra e (w) and e:(w), ai estimated on a digital computer, are discontinuous funceions; (this is due to the fact that arc-tangent routines in a computer provide both the reonly the principal values lying between -IT and 71); and (ii) ference and the subsequently attenuated echo signals are bandlimited, being :0p~5et~~lo:~:o~~eo~~~~~e~~~ w 5 w*.and ;ero elsewhere. The presence of he band idth w2-w,) affects the computations of so and s,. Elimination of (i) is (ii) is known as conversion for which algorithms exist of a band-limited signal to frequency-scaling operation w -w1) i--w2-w,)'
w=lT as proposed
known as phase unwrapping and elimination of of a bandlimited signal to fullband signal in the literature [ll, 123. The conversion a fullband signal can be achieved through a
O
by Tribolet
(10)
[12].
Defining w2
=n-
'2 D2
(11)
and w, = (1 - ;)
w2,
(12)
where I I D and D are integers, Tribolet has proposed that Eq. (10) can be I $pl$ien!.ed usizg digital interpolation/decimation algorithms Pa. With this implementation, the digitized reference and attenuated pulses, p.(n) and p (n) (corresponding to p.(t) and p (t) respectively), can be canverted t)S fullband discrete signals pi(n) bnd p,(n) respectively. Note that s and s in Eq. (9) represent the propagation-lags corresponding to the yinear bhase components of the bandlimited pulses p.(n) and p (n) respectively. Conversion of these pulses to fullband in&oducesra constant K=(D D )/(I I ) in the estimation of s and s If we define n and n as the2inte4eF time-lags (as computed Prom thd'unwrapped phase eseimates 1 for the fullband pulses pi(n) and p,(n) respectively, then so and s, can be expressed as sO
= Kn,T and s,
= Kn,T
,
(13)
where T = sampling interval. The origin of K can be explained as follows. The implementation of Eq. (10) is achieved through four elementary mappings [12] two of which perform decimation by the ratio I /D and I /D and the other two modulation (frequency shifts by IT). The goniequenc & o# the two stages of decimation is that the propagation lags s 'and s are reduced by the reciprocal of which is tile cons i ant K of Eq. T,;Tctor of I, 12/D,D2,
226
UNWRAPPEDPHASE SPECTRA
By substituting as
puted
Eq.
v = [&p
SC 4.
MATERIALS
(13)
in Eq.
(9),
velocity
in tissue
can be com-
+ 11 vO*
(14)
AND METHODS
A detailed description of the experimental apparatus and procedure Briefly, the samples are mounted on a plexihas appeared elsewhere [l]. glass holder and placed in the far field of an ultrasonic transducer, resonant at 5 MHz, with 80 percent bandwidth (3-7 MHz). A plane plexiglass reflector is used to reflect the ultrasonic pulses. The transducer, specimen holder and plane reflector are housed in a plexiglass tank filled with Hank's Balanced Salt Solution which provides dissipationless coupling between transducer, specimen and the reflector. An MP 203 pulser (Metrotek, Inc.) and an UTA-3 (Aerotech Labs.) transducer analyzer were used to excite the transducer and gate and capture the The echoes were digitized at 50 MHz by reference and attenuated echoes. The onset of the digitization means of a Biomation 8100 transient recorder. process was controlled by the 'trigger-sync' signal from the UTA-3. The digitized sweeps of 2048 points were stored on a floppy diskette attached to the Altair 8800 microcomputer and subsequently transferred to a remote PDP 11/34 minicomputer over a serial RS 232 interface. The entire processing was accomplished on a PDP 11/34 minicomputer. chosen
The cut-off to be
frequencies
w1 and w2 of the
transducer
bandwidth
were
which, given a sampling interval of 20 ns, correspond to a passband from 2.85714 MHz to 7.14285 MHz. Thus the cut-off frequencies chosen are approximately equal to those prescribed by the manufacturer (3 and 7 MHz). The discrete bandlimited signals p.(n) and p (n) were converted to fullband signals using the interpolatioa/decimati& algorithms [13]. A 2048-point FFT routine in conjunction with the phase unwrapping algorithm [11] was used to compute the linear, integer phase shifts no and n . The of the backscattered signal was measured from an osci -1 loduration T scope dispfsy of the entire backscattered sweep. Finally, Eq. (14) was used to compute the velocity of sound in the specimen. The system calibration was performed by measuring the velocity of sound in plexiglass and aluminum and subsequently in fixed rabbit kidneys, Velocities were also computed from a knowledge of transit-times [7]. 5.
RESULTS AND DISCUSSION
The accuracy of the proposed technique was first tested on several computer simulated models. An experimentally recorded pulse (Fig. 2A) was used as the reference echo which was time-displaced by known amounts (Figs. 2B-E) to form the hypothetical attenuated echoes. The developed
227
GUPTA, BHAGAT, AND FRIED
E.
, 0
TIME
Fig.
2.
(nT,T=20ns)
Simulated
pulse
600 separations.
technique was tested using computer generated data to measure known separations. Table 1 shows simulated and computed separations. The results indicate that separations as low as 0.12 ~J.S (six samples at a sampling If the signals were fullband to begin interval of 20 ns) can be detected. with, then the technique could, theoretically, detect a separation of one Finite transducer bandwidth implied by the presence of the consample. stant K (discussed earlier) is the limiting factor in the measurement of pulse separations. The minimum separation (number of samples) detectable is the smallest integer equal to or greater than K. Thus, for the 5 MHz transducer (3-7 MHz bandwidth) used in the study and the sampling frequency of 50 MHz, K was determined to be 35/6. Therefore, the minimum detectable separation is six samples. This can be reduced by lowering the sampling frequency and thereby lowering the constant K. However, lowering the sampling frequency results in increased aliasing; therefore, greater precision in velocity can only be attained at the cost of increased aliasing. Measurements were next made using aluminum and plexiglass. For comparative purposes, velocity data were also obtained using the pulse-transit Since the velocities for HBSS, plexiglass and aluminum time technique. are appreciably different, one would expect the unwrapped phase estimates
228
UNWRAPPED
Table
I.
Time lag
PHASE
SPECTRA
measurements--simulated
Simulated Time Lag (PSI 1.
2. 3. 4.
and measured. Measured Time Lag (us)
.120 .500 2.100 6.840
.117 .467 : 2.100 6.767
for the reference and attenuated echoes from aluminum and have different slopes. That this is indeed the case, can figure 3 which shows the unwrapped phase estimates for the In addition, the linearity aluminum and plexiglass echoes. estimates also indicates the nondispersive nature of HBSS, plexiglass.
plexiglass to be seen from reference, of the phasealuminum and
Finally, the technique was used to measure the sound velocity in six Table II summarizes the results obtained formalin-fixed rabbit kidneys. in this study as well as the published velocity data [5, 141. The results indicate that the proposed technique is at least as accurate as the pulseThe minor differences in the reported and published transit-time method. results can be attributed to different experimental conditions that may have been used in making the reported measurements. The significance eliminates the need thereby eliminating duce human bias in technique utilizes estimates. Further,
of the proposed technique is twofold: first, it for separate transit-time or distance measurements, the need for setting trigger thresholds which introthe conventional velocity measurements; second, the the measurements normally conducted for attenuation the term (2~) in Eq. (8) can be replaced by v*T,, to
FREQUENCY
Fig.
3.
Unwrapped
229
(MHZ)
phase
estimates.
GUPTA, BHAGAT, AND FRIED
Table
Specimen
II.
Velocity
Pulse-Transit Method (m/s)
Aluminum[5] Plexiglass[5] Rabbit kidneys[l4] avelocity reported name for the grade is not known.
data.
Phase-Estimate Method (m/s)
Published Values (m/s)
6397 2746 1545
6395 2746 1543 in [5] of the
6400 2680a 1566
is for polymethylmethacrylate; plexiglass specimen used
in the
the chemical present study
enhance the resolution of attenuation estimates. An interesting application of the technique might be phase tomography along the lines of attenuation tomography as proposed by Kak and Dines [15]. We are currently exploring this possibility. 6.
CONCLUSIONS
In summary, a new technique for estimating velocity of sound in tissue has been proposed. The technique utilizes the phase spectra of the reference and attenuated pulses normally used in estimating attenuation. The technique is simple to use2 and is, at least, as accurate as the conventional pulse-transit-time method. ACKNOWLEDGMENTS The authors acknowledge the technical assistance provided by Mr. V. C. Wu and Mr. M. Shafer and also wish to thank Dr. C. F. Knapp for the use of his minicomputer facilities. This study was supported NASA Contract NAS9-15452.
in part
by an NIH-BRSG Grant
RR05374 and
REFERENCES Cl]
Kadaba, M.P., Bhagat, tering of ultrasound Biomed. Eng. BME-27,
P.K., and Wu, V.C., Attenuation in freshly excised animal tissues, 76-83 (1980).
and backscatIEEE Trans.
[21
Kak, A.C. and Dines, K.A., Signal processing of broadband ultrasound: measurement of attenuation of soft biological IEEE Trans. Biomed. Eng. BME-25, 321-344 (1978).
[3]
Mimbs, J.W., Yukas, D.E., Miller, Detection of myocardial infarction ation of ultrasound, Circulation
pulsed tissues,
J.G., Weiss, R.N., and Sobel, B.E., in vitro based on altered attenuResearch 5, 192-198 (1977).
2
The Fortran listings can be obtained from
of the computer the first author.
230
programs
for
a PDP 11/34
minicomputer
UNWRAPPEDPHASE SPECTRA
[4]
M., Estimating the acoustic KUC, R., and Schwartz, ficient slope for liver from reflected ultrasound Sonics Ultrasonics SU-26, 353-362 (1979).
[5]
W;,
[6]
Fry, Sot.
[71
Sollish, B. D., A Device for Measuring Ultrasonic in Tissue, in Ultrasonic Tissue Characterization National Bureau of Standards Soec. Publ. 525, Printing Office, Washington, DC, 1979).
[8]
Biomedical
P.N.T., W.J., Amer.
The double 11, 17-28
Ultrasonics,
crystal (1949).
acoustic
Ficken, G.W., Jr. and Hiedemann, around' method for determination (1956). --Amer. 28, 921-923 Bhagat, P.K., Kadaba, M.P., based system for ultrasonic mentation (In Press 1980).
[lo]
Wells, P.N.T., Review: absorption biological tissue, Ultrasound Med.
[ll]
Tribolet, Speech,
J.M.,
Press,
New York,
Gupta, tissue
NY,
J. Acoust.
interferometer,
Propagation Velocity M. Linzer, ed., 53-56, (U.S. Government
II, PP.
E.A., Simple form of the 'singof sound velocities, J. Acoust.
[9]
Signal
(Academic
attenuation coefsignals, IEEE Trans.
Sot.
V.N., and Wu, W.C., Microprocessorcharacterization, Medical Instruand dispersion of ultrasound Biol. 1, 369-376 (1975).
A new phase unwrapping algorithm, IEEE Trans. Processing, ASSP-25, 170-177 (1977). Seismic Applications of Homomorphic Inc., Englewood Cliffs, NJ, 1979).
Signal
in Acoust.
Processing,
[12]
Tribolet, J.M., (Prentice-Hall,
[13]
Crochiere, R.E., A General Program to Perform Sampling Rate Conversion of Data by Rational Ratios, in Programs for Digital Signal Processing, edited by Digital Signal Processing Committee IEEE Acoustics, Speech, and Signal Processing Society (IEEE Press, pp. 8.2-l-8.2-7, 1979).
[14]
Goss, S.A., Johnston, of empirical ultrasonic Sot. Amer. 64, 423-457
[15]
Dines, K.A., soft tissues,
R.L., and Dunn, F., Comprehensive properties of mammalian tissues, (1978).
and Kak, A.C., Ultrasonic attenuation Ultrasonic Imaginql, 16-33 (1979).
231
compilation
J..Acoust.
tomography
of
IMAGING
2,
223-231
(1980)
ESTIMATING ULTRASOUND PROPAGATION VELOCITY IN TISSUES V. N. Gupta,
FROM UNWRAPPED PHASE SPECTRA P. K. Bhagat,
and A. M. Fried'
Wenner-Gren
Biomedical Research Laboratory University of Kentucky Lexington, Kentucky 40506
A digital computer-based technique for estimating the velocity of sound in tissue is proposed. The technique utilizes the unwrapped phase spectra of fullband reference and attenuated ultrasonic pulses normally used in In addition, it does not require explicit meaattenuation measurements. surement of pulse-transit-times. The accuracy of the technique was tested on several computer simulated models, on specimens of known velocities (aluminum and plexiglass), and on formalin-fixed rabbit kidneys. The results indicate that the technique has an accuracy comparable to the pulse-transit-time method. Key words:
1.
Phase
spectra;
ultrasound;
velocity.
INTRODUCTION
Systematic study of the interaction of biological systems with applied energy provides insight into the organization and structure of the tissue Ultrasound at low intensity levels has been used to visualize medium. the tissue interfaces. The attenuation coefficient and propagation velocity are the two acoustic parameters most commonly used in quantitative tissue characterization. The attenuation coefficient, for a given tissue specimen, is generally estimated by computation of the difference between the log-magnitude spectra of the reference and that of the tissue attenuated signals [l-4]. Velocity, on the other hand, can be computed using one of the several available techniques [5]. Notable among these are the interferometric, pulse-transit-time, and the sing-around techniques. The interferometric technique is limited mostly to liquids and quasi-liquids. It is based on the principle that when the distance between the transducer and the reflector, within the medium under investigation, is an integral multiple of half-wavelengths, the detecting system indicates a corresponding minimum in the driving voltage [6]. In the pulse-transit-time method, two distinct variations are available. In the first, the pulse is transmitted by one transducer and detected by another, the axes of the two transducers being coincident. In the second, only one transducer is used, alternately, in the transmitand receive-mode, with a reflector positioned normal to the transducer axis. The pulsetransit-times together with specimen thickness and velocity in the coupling medium is used to compute the velocity in tissue [7]. A.M. Fried is with Department of Diagnostic Radiology, Kentucky Medical Center, Lexington, KY 40536.
University
of
0161-7346/80/030223-09$02.00/O
223
Copyright :U 1980 b-vAcademic Press, Inc. All righrs of reproduction in any form reserved.
GUPTA, BHAGAT, AND FRIED
The sing-around technique is a variant of the pulse-transit-time Instead of measuring the pulse-transit-time across a known technique. length of the tissue specimen, the number of times the pulse travels back and forth through the tissue specimen is counted over a known period of time [8]. In all the techniques discussed above, one has to conduct separate measurements for velocity estimation. In the present study, we propose a digital computer-based technique for measurement of ultrasonic propagation velocity which does not require explicit measurement of transit times. Instead, propagation velocity is estimated from the phase spectra of reThe underlying hypothesis implicit ference and tissue-attenuated signals. in this technique is the premise that for nondispersive media the phase of a traveling wave is proportional to the frequency for a given propagation velocity. Measurements in our laboratory [9] and elsewhere [lo] indicate that biological media (soft tissue) exhibit negligible dispersion over the frequency range l-10 MHz.
2.
THEORETICAL
DETAILS
When an ultrasonic beam propagates through a weakly scattering, dissipative medium, it suffers a loss of energy due to attenuation. The term attenuation is defined here to include losses due to reflection, scattering, refraction and absorption. Consider the pulse-echo system of figure 1, where a tissue specimen positioned between the ultrasonic transducer and a plane plexiglass reflector is immersed in a dissipationless coupling medium. Let the transducer emit a short, bandlimited burst of ultrasonic energy p.'(t) and let pi(t) and p,(t) be the echoes reflected from the reflector wlthout and with the tissue specfmen in position. The relationships between pi'(t), pi(t) and p,(t) can be expressed as Pi(t)
= ' p Im Pi'(w) -co
Pr(t)
= ap
e-j2wao'vo
ejwt
dw,
and
jm p l(w) e-2a(~)ae-j2[(~o-~)/vot~/v] i -co
,jut
dw,
where
Pi’(w)
=k
joD pi'(t)
e-jut
dt,
-m = angular frequency in radian/s, II = distance between the transducer and the Lo = tissue-specimen thickness, = reflection coefficient for the plexiglass aP = sound velocity in the coupling medium, ;o = sound velocity in tissue, and a(w) = frequency dependent attenuation. w
receiver, reflector,
In deriving Eqs. (1) and (Z), we have implicitly assumed that the coupling medium and tissue specimen are non-dispersive, i.e., the group velocity is the same as the phase velocity. This is based on results published from this laboratory [g] and elsewhere [lo], where maximum dis-
224
UNWRAPPED
PHASE
SPECTRA
TISSUE
ULTRASONIC
Fig.
1.
A schematic
representation
of the
persion was observed to be less than 2 percent over the frequency range 1 to 10 MHZ. Further, the reflection losses at the assumed to be negligible since the acoustic water are approximately equal. Fourier
Equations (1) transforms Pi(w)
and (2) as
= ap Pi'(w) = 1 Pi(w)1
pulse-echo
for
several
tissue-water impedances
may be rewritten
in terms
system. tissue
specimens,
interfaces are of the tissue and of their
respective
e-j2wao'vo (4)
e-jeo(@)
and P,(u)
= aP Pi'(w) = 1 P,(w)1
e -2aa(w)
,-j2u[(k,-a)lv,+alv]
e-jel(,).
(5)
Defining sO
= 2i?o/vo
(61
= 2mo-a)/v,+a/v1,
(7)
and s1 it
can be shown that
a(w)
=
[ln 1 Pi(w)]
-
ln I Prb)l
1 l(2a)
(8)
and V
= [(so-S, f/T,,+11 vos
(9)
where T = 22/v is defined to be the duration of the backscattered signal The use d?iginating from the entire depth of the tissue specimen. of Eq. (8) for attenuation measurements has been reported earlier [l]. Equation (9) provides velocity estimates using the phase components of the Fourier transforms of pi(t) and p,(t), namely, e,(w) and e,(w).
225
GUPTA, BHAGAT, AND FRIED
3.
IMPLEMENTATION
OF VELOCITY ESTIMATIONALGORITHM
As defined in the previous section, the pulse propagation velocity can be estimated from the phase components of the measured reference and attenuated spectra. However, the computations of the propagation-lags s and s are compounded by the fact that:(i) the phase spectra e (w) and e:(w), ai estimated on a digital computer, are discontinuous funceions; (this is due to the fact that arc-tangent routines in a computer provide both the reonly the principal values lying between -IT and 71); and (ii) ference and the subsequently attenuated echo signals are bandlimited, being :0p~5et~~lo:~:o~~eo~~~~~e~~~ w 5 w*.and ;ero elsewhere. The presence of he band idth w2-w,) affects the computations of so and s,. Elimination of (i) is (ii) is known as conversion for which algorithms exist of a band-limited signal to frequency-scaling operation w -w1) i--w2-w,)'
w=lT as proposed
known as phase unwrapping and elimination of of a bandlimited signal to fullband signal in the literature [ll, 123. The conversion a fullband signal can be achieved through a
O
by Tribolet
(10)
[12].
Defining w2
=n-
'2 D2
(11)
and w, = (1 - ;)
w2,
(12)
where I I D and D are integers, Tribolet has proposed that Eq. (10) can be I $pl$ien!.ed usizg digital interpolation/decimation algorithms Pa. With this implementation, the digitized reference and attenuated pulses, p.(n) and p (n) (corresponding to p.(t) and p (t) respectively), can be canverted t)S fullband discrete signals pi(n) bnd p,(n) respectively. Note that s and s in Eq. (9) represent the propagation-lags corresponding to the yinear bhase components of the bandlimited pulses p.(n) and p (n) respectively. Conversion of these pulses to fullband in&oducesra constant K=(D D )/(I I ) in the estimation of s and s If we define n and n as the2inte4eF time-lags (as computed Prom thd'unwrapped phase eseimates 1 for the fullband pulses pi(n) and p,(n) respectively, then so and s, can be expressed as sO
= Kn,T and s,
= Kn,T
,
(13)
where T = sampling interval. The origin of K can be explained as follows. The implementation of Eq. (10) is achieved through four elementary mappings [12] two of which perform decimation by the ratio I /D and I /D and the other two modulation (frequency shifts by IT). The goniequenc & o# the two stages of decimation is that the propagation lags s 'and s are reduced by the reciprocal of which is tile cons i ant K of Eq. T,;Tctor of I, 12/D,D2,
226
UNWRAPPEDPHASE SPECTRA
By substituting as
puted
Eq.
v = [&p
SC 4.
MATERIALS
(13)
in Eq.
(9),
velocity
in tissue
can be com-
+ 11 vO*
(14)
AND METHODS
A detailed description of the experimental apparatus and procedure Briefly, the samples are mounted on a plexihas appeared elsewhere [l]. glass holder and placed in the far field of an ultrasonic transducer, resonant at 5 MHz, with 80 percent bandwidth (3-7 MHz). A plane plexiglass reflector is used to reflect the ultrasonic pulses. The transducer, specimen holder and plane reflector are housed in a plexiglass tank filled with Hank's Balanced Salt Solution which provides dissipationless coupling between transducer, specimen and the reflector. An MP 203 pulser (Metrotek, Inc.) and an UTA-3 (Aerotech Labs.) transducer analyzer were used to excite the transducer and gate and capture the The echoes were digitized at 50 MHz by reference and attenuated echoes. The onset of the digitization means of a Biomation 8100 transient recorder. process was controlled by the 'trigger-sync' signal from the UTA-3. The digitized sweeps of 2048 points were stored on a floppy diskette attached to the Altair 8800 microcomputer and subsequently transferred to a remote PDP 11/34 minicomputer over a serial RS 232 interface. The entire processing was accomplished on a PDP 11/34 minicomputer. chosen
The cut-off to be
frequencies
w1 and w2 of the
transducer
bandwidth
were
which, given a sampling interval of 20 ns, correspond to a passband from 2.85714 MHz to 7.14285 MHz. Thus the cut-off frequencies chosen are approximately equal to those prescribed by the manufacturer (3 and 7 MHz). The discrete bandlimited signals p.(n) and p (n) were converted to fullband signals using the interpolatioa/decimati& algorithms [13]. A 2048-point FFT routine in conjunction with the phase unwrapping algorithm [11] was used to compute the linear, integer phase shifts no and n . The of the backscattered signal was measured from an osci -1 loduration T scope dispfsy of the entire backscattered sweep. Finally, Eq. (14) was used to compute the velocity of sound in the specimen. The system calibration was performed by measuring the velocity of sound in plexiglass and aluminum and subsequently in fixed rabbit kidneys, Velocities were also computed from a knowledge of transit-times [7]. 5.
RESULTS AND DISCUSSION
The accuracy of the proposed technique was first tested on several computer simulated models. An experimentally recorded pulse (Fig. 2A) was used as the reference echo which was time-displaced by known amounts (Figs. 2B-E) to form the hypothetical attenuated echoes. The developed
227
GUPTA, BHAGAT, AND FRIED
E.
, 0
TIME
Fig.
2.
(nT,T=20ns)
Simulated
pulse
600 separations.
technique was tested using computer generated data to measure known separations. Table 1 shows simulated and computed separations. The results indicate that separations as low as 0.12 ~J.S (six samples at a sampling If the signals were fullband to begin interval of 20 ns) can be detected. with, then the technique could, theoretically, detect a separation of one Finite transducer bandwidth implied by the presence of the consample. stant K (discussed earlier) is the limiting factor in the measurement of pulse separations. The minimum separation (number of samples) detectable is the smallest integer equal to or greater than K. Thus, for the 5 MHz transducer (3-7 MHz bandwidth) used in the study and the sampling frequency of 50 MHz, K was determined to be 35/6. Therefore, the minimum detectable separation is six samples. This can be reduced by lowering the sampling frequency and thereby lowering the constant K. However, lowering the sampling frequency results in increased aliasing; therefore, greater precision in velocity can only be attained at the cost of increased aliasing. Measurements were next made using aluminum and plexiglass. For comparative purposes, velocity data were also obtained using the pulse-transit Since the velocities for HBSS, plexiglass and aluminum time technique. are appreciably different, one would expect the unwrapped phase estimates
228
UNWRAPPED
Table
I.
Time lag
PHASE
SPECTRA
measurements--simulated
Simulated Time Lag (PSI 1.
2. 3. 4.
and measured. Measured Time Lag (us)
.120 .500 2.100 6.840
.117 .467 : 2.100 6.767
for the reference and attenuated echoes from aluminum and have different slopes. That this is indeed the case, can figure 3 which shows the unwrapped phase estimates for the In addition, the linearity aluminum and plexiglass echoes. estimates also indicates the nondispersive nature of HBSS, plexiglass.
plexiglass to be seen from reference, of the phasealuminum and
Finally, the technique was used to measure the sound velocity in six Table II summarizes the results obtained formalin-fixed rabbit kidneys. in this study as well as the published velocity data [5, 141. The results indicate that the proposed technique is at least as accurate as the pulseThe minor differences in the reported and published transit-time method. results can be attributed to different experimental conditions that may have been used in making the reported measurements. The significance eliminates the need thereby eliminating duce human bias in technique utilizes estimates. Further,
of the proposed technique is twofold: first, it for separate transit-time or distance measurements, the need for setting trigger thresholds which introthe conventional velocity measurements; second, the the measurements normally conducted for attenuation the term (2~) in Eq. (8) can be replaced by v*T,, to
FREQUENCY
Fig.
3.
Unwrapped
229
(MHZ)
phase
estimates.
GUPTA, BHAGAT, AND FRIED
Table
Specimen
II.
Velocity
Pulse-Transit Method (m/s)
Aluminum[5] Plexiglass[5] Rabbit kidneys[l4] avelocity reported name for the grade is not known.
data.
Phase-Estimate Method (m/s)
Published Values (m/s)
6397 2746 1545
6395 2746 1543 in [5] of the
6400 2680a 1566
is for polymethylmethacrylate; plexiglass specimen used
in the
the chemical present study
enhance the resolution of attenuation estimates. An interesting application of the technique might be phase tomography along the lines of attenuation tomography as proposed by Kak and Dines [15]. We are currently exploring this possibility. 6.
CONCLUSIONS
In summary, a new technique for estimating velocity of sound in tissue has been proposed. The technique utilizes the phase spectra of the reference and attenuated pulses normally used in estimating attenuation. The technique is simple to use2 and is, at least, as accurate as the conventional pulse-transit-time method. ACKNOWLEDGMENTS The authors acknowledge the technical assistance provided by Mr. V. C. Wu and Mr. M. Shafer and also wish to thank Dr. C. F. Knapp for the use of his minicomputer facilities. This study was supported NASA Contract NAS9-15452.
in part
by an NIH-BRSG Grant
RR05374 and
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