Ultrasound in Med. & Biol., Vol. 2, pp. 195-198. Pergamon Press, 1976. Printed in Great Britain
STUDIES OF INHOMOGENEOUS SUBSTANCES BY ULTRASONIC BACK-SCATTERING B. FAY, K. BRENDEL and G. LUDWIG Information from Physikalisch-Technische Bundesanstalt, Braunschweig, GFR
(Firstreceived21 October1975; and in]inal[orm 28 January 1976) Abstract--A method is outlined by which ultrasonic back-scattering measurements may reveal information concerning
both the scattering and absorption properties of inhomogeneous substances. After a description of the principle of the measuring method, experimental studies of a sample consisting of four layers with different scattering properties are discussed. To carry out measurements on substances, the acoustic properties of which are similar to those of biological tissues, inhomogeneous gelatine gels are investigated. The gels are produced using appropriate ethanol-glycerine mixtures. The inhomogeneities were introduced by adding tiny plastic spheres to the gel. It is shown, that the ultrasonic back-scattering method allows separation of the attenuation into scattering and sbroption as functions of the location. In this way recognition of the inhomogeneities is possible. This fact should be helpful in the fieldof medical diagnostics.
Key words: Ultrasound, Scattering, Absorption, Inhomogeneous substances, Gels.
~
INTRODUCTION
When an ultrasonic wave is radiated into an inhomogeneous solid, part of the transmitting energy is scattered. This fact is exploited in the ultrasonic back-scattering method for the determination of the absorption and scattering coefficients (Koppelmann, 1967; Fay, 1973). Since these quantities, as functions of the coordinates, reveal information about the structure of the substance investigated, this measuring method should also be of interest in the field of medical diagnostics (Chivers et al., 1974; Hill, 1974; Lele et al., 1975; Lorenz et al., 1975; White et al., 1975).
•
•
cirnen
¶"11
MEASURING METHOD
1. General considerations
Inhomogeneity
An ultrasonic transducer radiates short acoustic tone bursts through a water contact into a specimen. The angle of incidence is chosen non-zero in order to decrease the high amplitude of the upper and lower direct-surface echoes, which would otherwise be the case for normal incidence. The acoustic pulse on passing through the specimen is attenuated by absorption and by scattering at the inhomogeneities of the medium (Mason et al., 1947, 1948; Papadakis, 1968). The absorbed ultrasonic energy is transformed into heat, whereas the scattered energy travels through the specimen as secondary ultrasonic waves. A small portion of this scattered energy returns to the transducer (Fig. 1). All the received waves which have been scattered by the inhomogeneities within the specimen, generate the back-scattering curve (Fig. 2). The existence of a distinct bottom-echo is surprising because of the inclined incidence of the sound pulse. Theoretical studies have shown that it is produced by scattered waves reflected at the bottom (Fay, 1975). The back-scattering curve can be described by the following expression
L(x) klO(x)~,(x) xexp (- foXa(x)dx + f; a(x)(-dx)),
Ironsducer
Fig. 1. Principle of the ultrasonic back-scattering method. is the intensity scattering coefficient, is the intensity attenuation coefficient (aa is the intensity absorption coefficient), x is the position within the specimen where the scattered waves are produced, G(x) is a sound field correction factor, and kl is a constant. O/s
O~ = /Xs + O~a
To simplify the description of the measuring procedure the factor G (x) is assumed to be constant in the following text. The measurements mentioned below show that this assumption is approximately Valid in this case. The exponential function
exp (-2 foXet(x)dx + fx°a(x)(-dx))
=
where L
(1)
describes the attenuation loss of the incident pulse on its way through the specimen up to the coordinate x, and the corresponding loss of those scattered waves travelling back to the transducer from their origin at x on the same
is the intensity of the received waves scattered at the coordinate x, 195
B. FAYet al.
196
Exciting Pulse
!
~h0 X
.,
Fig. 2. Schematic illustration of a back-scattering curve, h is the averaged intensity of the back-scattered signals, and x the position where the scattered waves are produced. path, but in the opposite direction. The quantity O/, determines that part of the energy of the incident pulse which is scattered within the volume-element at x. For x = O, the intensity L is proportional only to the scattering coefficient in the surface layer. On the sound path between the transducer and the specimen the intensity L is zero because water shows no considerable scattering at these frequencies.
2. Scattering and absorption independent of the xcoordinate If the scattering and absorption coefficients are independent of the x-value and the beam spreading loss is neglected, then the back-scattered intensity shows an exponential decay. When the logarithm of L is plotted as a function of x, we get a straight line in the diagram (Fig. 3) corresponding to the expression log ~
= log O/' - k2ax, s, ref
(2)
O/s, ref
where L. ,of and O/,.ref are interdependent constants, e.g. the scattered intensity L (0) and the scattering coefficient as (0) at the coordinate x = 0. The quantity k2 is also a constant. In the case of strong scattering the ordinate value at x = 0 is larger than in the case of weak scattering. The slope of a curve depends on the respective attenuation coefficient. From both these values, the ordinate, value at x = 0 and the slope, the scattering and absorption coefficients can be determined.
3. Scattering and absorption dependent on the xcoordinate More interesting is the case when absorption and scattering coefficients vary discretely or continuously along the path of the incident sound pulse through the specimen. The logarithm of the back-scattered intensity L can be described by X
x
log ~I,(x) = log a'( a~, ~f) - k E fJo a(x)dx.
(3)
In this expression the term log[o/~(x)/o/,,m] is also a function of x, i.e. both terms on the right-hand side of equation (3) depend on x. It is not generally known in which way the two parts influence the back-scattering curve. Separation of these two parts is possible if we can for instance perform the following two measurements on the same specimen. For measurement 1, the pulse enters the specimen through the surface, for measurement 2, through its underside. Figure 4 shows these measurements applied to a simple two-layer specimen. One layer has a large scattering coefficient, the other layer a small one. FOr measurement 1, the scattered intensity decreases if the incident pulse travels through the boundary of these two different layers. The back-scattering curve therefore sh6ws a steep decrease at the corresponding value of x. For measurement 2, the scattered energy increases if the pulse travels through the boundary in the opposite direction. The back-scattering curve then shows a steep increase at the corresponding x-value. To determine the scattering and absorption coefficients, the sum and difference of the corresponding functional values of log [L (x)/I,. ref]must be formed (Fay, 1973). This operation can easily be performed graphically by inverting curve 2 and re-plotting both curves on the same x-coordinates relative to position in the target specimen. The difference between curves 1 and 2 is a function only of the scattering coefficient at the corresponding x-value. From the slope of the averaged curve (cf. Fig. 4) the attenuation coefficient is obtained as a function of x. The absorption coefficient can be determined from both these coefficients also as a function of x. MEASUREMENTS
The original ultrasonic back-scattering method was developed for investigations of polycrystalline materials. In order to test its applicability to the field of medical diagnostics, we began measurements with gelatine gels of varying inhomogeneity. These gels, made by using high quality gelatine of the "Nienburger Gelatine GmbH" and
0 ii CI
strong scattering
X
-
b
t
log I
/ tog a) weak scattering Fig. 3. Normalized back-scattering curves for the case of homogeneous substances.
2 b
~
x
a
tog-L I Is(a)I a
f--- ,rotated, ...... Jz,og fx! I /.... ~,s/oj t.~ ~'~,1 x b
Fig:4. Illustration of the graphical determination of the scatteringand the attenuation coefficient for the case of an inhomogeneous substance.
197
Studiesof inhomogeneoussubstancesby ultrasonicback-scattering mixtures of ethanol-glycerine, have acoustic properties similar to those of biological tissues. The inhomogeneities necessary for these studies have been produced by adding appropriate amounts of gelatine particles partially soaked, or tiny plastic spheres (dia. about 0.5 mm) to the gel. To achieve sufficient mechanical stability of the samples, the gel was cast into 10 cm long pieces of lucite tube of 15 cmdia. The front surfaces were covered with thin plastic foil to avoid erosion of the gel by the water in contact. Figure 5 shows the sectional view of the specimen with four distinct layers. The number of the scattering centres within the different layers A-D are approximately in the ratio of 0: 4: 2: 1. Within the layer C the concentration of the scattering centres varies more than within the layer B and D due to partial segregation during the casting process of the specimen. In Fig. 6, the original back-scattering curves of measurements 1 and 2 mentioned above have been plotted. The main frequency of the excitant sound pulse was about 1 MHz. For measurement 1, the pulse travelled first through the layer A containing no inhomogeneities, and no back-scattered signal was registered. On entering the layer B, the sound wave is strongly scattered and a high signal is measured. Further along the sound path, the received signal decreases by attenuation and in accordance with the decay in the concentration of the scattering centres within the layers C and D. The peak at the end of the curve is caused by the bottom-echo (cf. Fig. 2). For measurement 2, the pulse took the same path through the specimen, but in the opposite direction. Owing to the low scattering centre concentration in the layer D, the back-scattered signal initially has a small value. It increases with the sound path covered, corresponding to the higher concentration of scattering centres within the layers C and B. From layer A again no signal is received as a result of its homogeneity.
I
I
B
1
I
dB l IO--
/
P
10log s,ref
dB
5 cm 10
0 X
lOl°gIsdB ~
/
2
ls'r el 5
o
B
c
,' ~'/I'/I~ii
X
Fig.6. Originalplotsof the measuredback-scatteringcurves I and 2 of thefour layerspecimen. RESULTS In order to analyse the measured back-scattered signals, the curves of Fig. 6 are plotted in Fig. 7 in the manner explained above. The points drawn additionally in this are half the sum of the ordinate values of the two curves. The dashed line represents the averaged values of these points. The average attenuation coefficient is given by its slope. In our specimen the intensity attenuation coefficient ~ is about 4. I0-~ cm -~, corresponding to an
C
1
D
I
Fig. 5. Cross-sectionof the measuredgelatinegel specimenconsistingof 4 layerswith differentconcentrationsof scatteringcentres (solid plasticspheres).
B. FAYet al.
198
2.10log [Ots/a,,ref] of the layers B and D is about 9dB. Assuming that the scattering coefficient is proportional to the density of the scattering centres, the difference should be 12 dB.
,o_ dB
DISCUSSION Much valuable information about the acoustic properties of tissue-like substances may be obtained by applying the ultrasonic back-scattering method described above. It goes without saying that the ideal condition of samples with two parallel surfaces is not met in the field of medical diagnostics. The reported measurements however, demonstrate that considerable information may also be received by performing measurements from one side only. The back-scattering curve 1 in Fig. 7 and the curve for the scattering coefficient in Fig. 8 appear very similar. For the condition
5-
i
c
I
u
"": -"-tl . . . . " I .--.~
/'
lO,oo .d l
i
" .
o.
'
log~
-10Fig. 7. Superpositionof the measured back-scatteringcurves 1 and 2 for determiningthe scattering coefficienta, and the attenuation coefficienta as functions of the x-coordinate. attenuation of 0.2 dB .cm-', i.e. it is relatively small compared with the variation of the scattering coefficient and so the disturbances caused by the limited accuracy of measurement are obvious. Attenuation measurements using the pulse-echo-method confirmed this value. In Fig. 8, the relative scattering coefficient given by the difference between curve 1 and 2 of Fig. 7 is plotted as a function of the coordinate x. The reference value as.,of was arbitrarily fixed. The peak at x = 0 is due to the bottom-echo and scattering from the surface of the specimen. In layer A, without inhomogeneities, no scattered signal was created. Layer B contains a large number of inhomogeneities causing a high scattering coefficient. It decreases as a function of x, corresponding to the decay in the concentration of the scattering centres within the layers C and D. The minimum of the scattering coefficient, for example, within the boundary region between the layers C and D, is due to the small amount of scattering particles (cf. Fig.. 5). This lack of particles was caused by partial segregation during the step-by-step casting of the specimen. The difference between the average values of
-20
T
- - -- - -
d8
6d
- -[-8
-10
J
cm
10
X
Fig. 8. Logarithmic relative scattering coefficient 2.10 log la,/a,.~f] given by the differencebetween the curves of Fig. 7 as a function of the x-coordinate.
>> k foXC~(x)dx ,
(4)
these two curves mentioned become nearly identical (cf. equation 3). Furthermore the solution of equation (3) is possible, if the absorption coefficient is known. In difficult cases, information about the scattering and absorption coefficients can be gained by performing measurements at different angles of incidence and at different frequencies. Also the use of a reference curve (normal tissue values) is advantageous. Acknowledgements--We are very grateful to Mr. H.-G. Unger for performing the measurements reported in this paper. REFERENCES
Chivers, R. C., Hill, C. R. and Nicholas, D. (1974) Frequency dependence of ultrasonic back-scattering cross-sections: an indicator of tissue structure characteristics. Proceedings 2nd World Congress on Ultrasonics in Medicine, pp. 300-303. Excerpta Medica, Amsterdam. Fay, B: (1973) Theoretische Betrachtungen zur Ultraschallriickstreuung.Acustica 28, 6, 354-357. Fay, B. (1975) Schallausbreitungsverluste bei der Ultraschallriiekstreuung. Fortschritte der Akustik ; Plenarvortriige und Kurzre[erate d. 4. Tagung d. Dt. Arbeitsgemeinschaft f. Akustik, DAGA'75, Braunschweig. pp. 597600. Hill, C. R. (1974) Interactions of ultrasound with tissues. Proceedings 2nd World Congress on Ultrasonics in Medicine, pp. 14-20. Excerpta Medica, Amsterdam. Koppelmann, J. (1967) H~irtetiefenmessungan Stahlwalzen mit Ultraschall. Materialprii[ 9, 401-405. Lele, P. P. and Senapati, N. (1975) Ultrasonic frequency-domain analysis and studies on acoustical scattering for diagnosis of tissue pathology. Abstracts, 2nd European Congress Ultrasonics in Medicine, p. 16. Munchen. Lorenz, W. J., van Kalck, G., Lorenz, A., Doll, J. and Geissler, M. (1975) Computer analysis of the A-scan for the detection of generalized diseases of the liver. Abstracts, 2nd European Congress Ultrasonics in Medicine, p. 40. Mfinchen. Mason, W. P. and Mc Skimin, H. J. (1947) Attenuation and scattering of high frequency sound waves in metals and glasses. J. acoust. Soc. Am. 19, 464--473. White, D. N. and Curry, G. R. (1975) Absorption of ultrasonic energy by the skull. Abstracts, 2nd European Congress Ultrasonics in Medicine, p. 7. MOnchen.
STUDIES OF INHOMOGENEOUS SUBSTANCES BY ULTRASONIC BACK-SCATTERING B. FAY, K. BRENDEL and G. LUDWIG Information from Physikalisch-Technische Bundesanstalt, Braunschweig, GFR
(Firstreceived21 October1975; and in]inal[orm 28 January 1976) Abstract--A method is outlined by which ultrasonic back-scattering measurements may reveal information concerning
both the scattering and absorption properties of inhomogeneous substances. After a description of the principle of the measuring method, experimental studies of a sample consisting of four layers with different scattering properties are discussed. To carry out measurements on substances, the acoustic properties of which are similar to those of biological tissues, inhomogeneous gelatine gels are investigated. The gels are produced using appropriate ethanol-glycerine mixtures. The inhomogeneities were introduced by adding tiny plastic spheres to the gel. It is shown, that the ultrasonic back-scattering method allows separation of the attenuation into scattering and sbroption as functions of the location. In this way recognition of the inhomogeneities is possible. This fact should be helpful in the fieldof medical diagnostics.
Key words: Ultrasound, Scattering, Absorption, Inhomogeneous substances, Gels.
~
INTRODUCTION
When an ultrasonic wave is radiated into an inhomogeneous solid, part of the transmitting energy is scattered. This fact is exploited in the ultrasonic back-scattering method for the determination of the absorption and scattering coefficients (Koppelmann, 1967; Fay, 1973). Since these quantities, as functions of the coordinates, reveal information about the structure of the substance investigated, this measuring method should also be of interest in the field of medical diagnostics (Chivers et al., 1974; Hill, 1974; Lele et al., 1975; Lorenz et al., 1975; White et al., 1975).
•
•
cirnen
¶"11
MEASURING METHOD
1. General considerations
Inhomogeneity
An ultrasonic transducer radiates short acoustic tone bursts through a water contact into a specimen. The angle of incidence is chosen non-zero in order to decrease the high amplitude of the upper and lower direct-surface echoes, which would otherwise be the case for normal incidence. The acoustic pulse on passing through the specimen is attenuated by absorption and by scattering at the inhomogeneities of the medium (Mason et al., 1947, 1948; Papadakis, 1968). The absorbed ultrasonic energy is transformed into heat, whereas the scattered energy travels through the specimen as secondary ultrasonic waves. A small portion of this scattered energy returns to the transducer (Fig. 1). All the received waves which have been scattered by the inhomogeneities within the specimen, generate the back-scattering curve (Fig. 2). The existence of a distinct bottom-echo is surprising because of the inclined incidence of the sound pulse. Theoretical studies have shown that it is produced by scattered waves reflected at the bottom (Fay, 1975). The back-scattering curve can be described by the following expression
L(x) klO(x)~,(x) xexp (- foXa(x)dx + f; a(x)(-dx)),
Ironsducer
Fig. 1. Principle of the ultrasonic back-scattering method. is the intensity scattering coefficient, is the intensity attenuation coefficient (aa is the intensity absorption coefficient), x is the position within the specimen where the scattered waves are produced, G(x) is a sound field correction factor, and kl is a constant. O/s
O~ = /Xs + O~a
To simplify the description of the measuring procedure the factor G (x) is assumed to be constant in the following text. The measurements mentioned below show that this assumption is approximately Valid in this case. The exponential function
exp (-2 foXet(x)dx + fx°a(x)(-dx))
=
where L
(1)
describes the attenuation loss of the incident pulse on its way through the specimen up to the coordinate x, and the corresponding loss of those scattered waves travelling back to the transducer from their origin at x on the same
is the intensity of the received waves scattered at the coordinate x, 195
B. FAYet al.
196
Exciting Pulse
!
~h0 X
.,
Fig. 2. Schematic illustration of a back-scattering curve, h is the averaged intensity of the back-scattered signals, and x the position where the scattered waves are produced. path, but in the opposite direction. The quantity O/, determines that part of the energy of the incident pulse which is scattered within the volume-element at x. For x = O, the intensity L is proportional only to the scattering coefficient in the surface layer. On the sound path between the transducer and the specimen the intensity L is zero because water shows no considerable scattering at these frequencies.
2. Scattering and absorption independent of the xcoordinate If the scattering and absorption coefficients are independent of the x-value and the beam spreading loss is neglected, then the back-scattered intensity shows an exponential decay. When the logarithm of L is plotted as a function of x, we get a straight line in the diagram (Fig. 3) corresponding to the expression log ~
= log O/' - k2ax, s, ref
(2)
O/s, ref
where L. ,of and O/,.ref are interdependent constants, e.g. the scattered intensity L (0) and the scattering coefficient as (0) at the coordinate x = 0. The quantity k2 is also a constant. In the case of strong scattering the ordinate value at x = 0 is larger than in the case of weak scattering. The slope of a curve depends on the respective attenuation coefficient. From both these values, the ordinate, value at x = 0 and the slope, the scattering and absorption coefficients can be determined.
3. Scattering and absorption dependent on the xcoordinate More interesting is the case when absorption and scattering coefficients vary discretely or continuously along the path of the incident sound pulse through the specimen. The logarithm of the back-scattered intensity L can be described by X
x
log ~I,(x) = log a'( a~, ~f) - k E fJo a(x)dx.
(3)
In this expression the term log[o/~(x)/o/,,m] is also a function of x, i.e. both terms on the right-hand side of equation (3) depend on x. It is not generally known in which way the two parts influence the back-scattering curve. Separation of these two parts is possible if we can for instance perform the following two measurements on the same specimen. For measurement 1, the pulse enters the specimen through the surface, for measurement 2, through its underside. Figure 4 shows these measurements applied to a simple two-layer specimen. One layer has a large scattering coefficient, the other layer a small one. FOr measurement 1, the scattered intensity decreases if the incident pulse travels through the boundary of these two different layers. The back-scattering curve therefore sh6ws a steep decrease at the corresponding value of x. For measurement 2, the scattered energy increases if the pulse travels through the boundary in the opposite direction. The back-scattering curve then shows a steep increase at the corresponding x-value. To determine the scattering and absorption coefficients, the sum and difference of the corresponding functional values of log [L (x)/I,. ref]must be formed (Fay, 1973). This operation can easily be performed graphically by inverting curve 2 and re-plotting both curves on the same x-coordinates relative to position in the target specimen. The difference between curves 1 and 2 is a function only of the scattering coefficient at the corresponding x-value. From the slope of the averaged curve (cf. Fig. 4) the attenuation coefficient is obtained as a function of x. The absorption coefficient can be determined from both these coefficients also as a function of x. MEASUREMENTS
The original ultrasonic back-scattering method was developed for investigations of polycrystalline materials. In order to test its applicability to the field of medical diagnostics, we began measurements with gelatine gels of varying inhomogeneity. These gels, made by using high quality gelatine of the "Nienburger Gelatine GmbH" and
0 ii CI
strong scattering
X
-
b
t
log I
/ tog a) weak scattering Fig. 3. Normalized back-scattering curves for the case of homogeneous substances.
2 b
~
x
a
tog-L I Is(a)I a
f--- ,rotated, ...... Jz,og fx! I /.... ~,s/oj t.~ ~'~,1 x b
Fig:4. Illustration of the graphical determination of the scatteringand the attenuation coefficient for the case of an inhomogeneous substance.
197
Studiesof inhomogeneoussubstancesby ultrasonicback-scattering mixtures of ethanol-glycerine, have acoustic properties similar to those of biological tissues. The inhomogeneities necessary for these studies have been produced by adding appropriate amounts of gelatine particles partially soaked, or tiny plastic spheres (dia. about 0.5 mm) to the gel. To achieve sufficient mechanical stability of the samples, the gel was cast into 10 cm long pieces of lucite tube of 15 cmdia. The front surfaces were covered with thin plastic foil to avoid erosion of the gel by the water in contact. Figure 5 shows the sectional view of the specimen with four distinct layers. The number of the scattering centres within the different layers A-D are approximately in the ratio of 0: 4: 2: 1. Within the layer C the concentration of the scattering centres varies more than within the layer B and D due to partial segregation during the casting process of the specimen. In Fig. 6, the original back-scattering curves of measurements 1 and 2 mentioned above have been plotted. The main frequency of the excitant sound pulse was about 1 MHz. For measurement 1, the pulse travelled first through the layer A containing no inhomogeneities, and no back-scattered signal was registered. On entering the layer B, the sound wave is strongly scattered and a high signal is measured. Further along the sound path, the received signal decreases by attenuation and in accordance with the decay in the concentration of the scattering centres within the layers C and D. The peak at the end of the curve is caused by the bottom-echo (cf. Fig. 2). For measurement 2, the pulse took the same path through the specimen, but in the opposite direction. Owing to the low scattering centre concentration in the layer D, the back-scattered signal initially has a small value. It increases with the sound path covered, corresponding to the higher concentration of scattering centres within the layers C and B. From layer A again no signal is received as a result of its homogeneity.
I
I
B
1
I
dB l IO--
/
P
10log s,ref
dB
5 cm 10
0 X
lOl°gIsdB ~
/
2
ls'r el 5
o
B
c
,' ~'/I'/I~ii
X
Fig.6. Originalplotsof the measuredback-scatteringcurves I and 2 of thefour layerspecimen. RESULTS In order to analyse the measured back-scattered signals, the curves of Fig. 6 are plotted in Fig. 7 in the manner explained above. The points drawn additionally in this are half the sum of the ordinate values of the two curves. The dashed line represents the averaged values of these points. The average attenuation coefficient is given by its slope. In our specimen the intensity attenuation coefficient ~ is about 4. I0-~ cm -~, corresponding to an
C
1
D
I
Fig. 5. Cross-sectionof the measuredgelatinegel specimenconsistingof 4 layerswith differentconcentrationsof scatteringcentres (solid plasticspheres).
B. FAYet al.
198
2.10log [Ots/a,,ref] of the layers B and D is about 9dB. Assuming that the scattering coefficient is proportional to the density of the scattering centres, the difference should be 12 dB.
,o_ dB
DISCUSSION Much valuable information about the acoustic properties of tissue-like substances may be obtained by applying the ultrasonic back-scattering method described above. It goes without saying that the ideal condition of samples with two parallel surfaces is not met in the field of medical diagnostics. The reported measurements however, demonstrate that considerable information may also be received by performing measurements from one side only. The back-scattering curve 1 in Fig. 7 and the curve for the scattering coefficient in Fig. 8 appear very similar. For the condition
5-
i
c
I
u
"": -"-tl . . . . " I .--.~
/'
lO,oo .d l
i
" .
o.
'
log~
-10Fig. 7. Superpositionof the measured back-scatteringcurves 1 and 2 for determiningthe scattering coefficienta, and the attenuation coefficienta as functions of the x-coordinate. attenuation of 0.2 dB .cm-', i.e. it is relatively small compared with the variation of the scattering coefficient and so the disturbances caused by the limited accuracy of measurement are obvious. Attenuation measurements using the pulse-echo-method confirmed this value. In Fig. 8, the relative scattering coefficient given by the difference between curve 1 and 2 of Fig. 7 is plotted as a function of the coordinate x. The reference value as.,of was arbitrarily fixed. The peak at x = 0 is due to the bottom-echo and scattering from the surface of the specimen. In layer A, without inhomogeneities, no scattered signal was created. Layer B contains a large number of inhomogeneities causing a high scattering coefficient. It decreases as a function of x, corresponding to the decay in the concentration of the scattering centres within the layers C and D. The minimum of the scattering coefficient, for example, within the boundary region between the layers C and D, is due to the small amount of scattering particles (cf. Fig.. 5). This lack of particles was caused by partial segregation during the step-by-step casting of the specimen. The difference between the average values of
-20
T
- - -- - -
d8
6d
- -[-8
-10
J
cm
10
X
Fig. 8. Logarithmic relative scattering coefficient 2.10 log la,/a,.~f] given by the differencebetween the curves of Fig. 7 as a function of the x-coordinate.
>> k foXC~(x)dx ,
(4)
these two curves mentioned become nearly identical (cf. equation 3). Furthermore the solution of equation (3) is possible, if the absorption coefficient is known. In difficult cases, information about the scattering and absorption coefficients can be gained by performing measurements at different angles of incidence and at different frequencies. Also the use of a reference curve (normal tissue values) is advantageous. Acknowledgements--We are very grateful to Mr. H.-G. Unger for performing the measurements reported in this paper. REFERENCES
Chivers, R. C., Hill, C. R. and Nicholas, D. (1974) Frequency dependence of ultrasonic back-scattering cross-sections: an indicator of tissue structure characteristics. Proceedings 2nd World Congress on Ultrasonics in Medicine, pp. 300-303. Excerpta Medica, Amsterdam. Fay, B: (1973) Theoretische Betrachtungen zur Ultraschallriickstreuung.Acustica 28, 6, 354-357. Fay, B. (1975) Schallausbreitungsverluste bei der Ultraschallriiekstreuung. Fortschritte der Akustik ; Plenarvortriige und Kurzre[erate d. 4. Tagung d. Dt. Arbeitsgemeinschaft f. Akustik, DAGA'75, Braunschweig. pp. 597600. Hill, C. R. (1974) Interactions of ultrasound with tissues. Proceedings 2nd World Congress on Ultrasonics in Medicine, pp. 14-20. Excerpta Medica, Amsterdam. Koppelmann, J. (1967) H~irtetiefenmessungan Stahlwalzen mit Ultraschall. Materialprii[ 9, 401-405. Lele, P. P. and Senapati, N. (1975) Ultrasonic frequency-domain analysis and studies on acoustical scattering for diagnosis of tissue pathology. Abstracts, 2nd European Congress Ultrasonics in Medicine, p. 16. Munchen. Lorenz, W. J., van Kalck, G., Lorenz, A., Doll, J. and Geissler, M. (1975) Computer analysis of the A-scan for the detection of generalized diseases of the liver. Abstracts, 2nd European Congress Ultrasonics in Medicine, p. 40. Mfinchen. Mason, W. P. and Mc Skimin, H. J. (1947) Attenuation and scattering of high frequency sound waves in metals and glasses. J. acoust. Soc. Am. 19, 464--473. White, D. N. and Curry, G. R. (1975) Absorption of ultrasonic energy by the skull. Abstracts, 2nd European Congress Ultrasonics in Medicine, p. 7. MOnchen.
