Assessment of ultrasound tomography as a technique for
quantitative tissue characterization
Citation for published version (APA):
Rietsema, J. (1993). Assessment of ultrasound tomography as a technique for quantitative tissue characterization. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR398895
DOI:
10.6100/IR398895
Document status and date: Published: 01/01/1993
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ASSESSMENT OF ULTRASOUND TOMOGRAPHY
AS A TECHNIQUE FOR
CIP-DATA KONINKLIJKE BffiLIOTHEEK, DEN HAAG
Rietsema, Jan
Assessment of ultrasound tomography as a technique for
quantitative tissue characterization
I
Jan Rietsema.
-[S.l. : s.n.]. - Fig., photos, tab.
Thesis Eindhoven. - With ref. - With summary in Dutch.
ISBN 90-9006207-6
NUGI 743
Subject headings
:
ultrasound computerised tomography
I
ultrasonic propagation
I
tissue characterization.
ASSESSMENT OF ULTRASOUND TOMOGRAPHY
AS A TECHNIQUE FOR
QUANTITATIVE TISSUE CHARACTERIZATION
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de
Technische Universiteit Eindhoven, op gezag van
de Rector Magnificus, prof
.
dr.
J
.
H. van Lint, voor
een conunissie aangewezen door het College
van Dekanen in het openbaar te verdedigen op
vrijdag 9 juli 1993 om 16
.
00 uur
door
JAN RIETSEMA
Dit proefschrift is goedgekeurd door de promotoren
prof. dr
.
ir
.
J.E
.
W
.
Beneken
en
prof. dr
.
ir. N. Bom
Things should be made
as simple as possible,
but no simpler.
Table of contents
List of symbols
List of abbreviations
I
Ll 1.1.1 1.1.2 1.1.3 1.1.4 1.2 1.311
11.1 11.2 11.3 11.3.1 11.3.2 11.3.3 11.3.4 11.4 11.4.1 11.4.2 11.4.3Ill
ill.l m.2 111.3 III.4IV
IV.1 IV.l.l IV.1.2 JV.2 IV.2.1 JV.2.2General introduetion
Ultrasound tissue characterizationTransmission tomography Reflection tomography Quantitative B-mode imaging
Other noninvasive tissue characterization techniques Fundamentals of medica! ultrasound
Scope of this thesis
Measurement techniques and measurement device
Principles of computerized tomographyMeasurement device
Measurement of the acoustic parameters Speed of sound
Attenuation coefficient Attenuation slope Reflectivity Reconstruction principles
Filtered backprojection algorithm Direct Fourier inversion algorithm Hartley transform in image reconstruction
Assessment of the backscatter measurement
Theoretica! foundationImplementation Experiments Discussion
Assessment of the performance of the tomograph
Measurements on phantomsTissue-mimicking phantoms
Standard measurement protocol for phantom studies
Phantom study on spatial resolution Introduetion Results
viii
x
1
3 4 6 7 9 10 1416
16 19 20 20 22 2426
28 30 31 32 3535
40 41 4547
47 47 49 50 50 52IV.2.3 IV.3 IV.3.1 IV.3.2 IV.3.3 IV.4 IV.4.1 IV.4.2 IV.5 IV.5.1 IV.5.2 IV.6
V
V.1 V.l.l V.1.2 V.1.3 V.2 V.2.1 V.2.2 V.2.3 V.3 V.4 V.5VI
VI.l VI.2 Vl.3 Vl.4References
Summary
Samenvatting
Nawoord
DiscussionPhantom study on contrast resolution Introduetion
Results Discussion
Phantom study on quantitative accuracy
Quantitative characterization using an acoustic macroscope
Relationship between size and quantitative accuracy Dominant sourees of artifacts influencing the image quality
Refraction
Interference and phase cancellation General Discussion
Imaging bone-containing tissue structures
Measurements on a 'bone phantom'Method Results Discussion
Current measurement techniques for osteoporosis Osteoporosis
Diagnostic techniques
Ultrasonic assessment of bone
Ultrasound tomography measurements on bone in vitro In vivo measurements of human extremities
Conclusion
Clinical perspective of ultrasound tomography
Tissue classification using acoustic parametersClinical applicability of ultrasound tomögràphy Recommendations General conclusion
Curriculum vitae
53 54 54 55 56 56 5764
6666
70 7980
81 8183
85 87 87 88 90 91 94 9799
99 101 104 105106
123
128
133
134
List of symbols
a b c d f f(x,y) g(v) k Pe(v) r x x,y v,w A BIA E F(X,Y) F(R,<I>) y H(X,Y) I M N P(r,t) Q(r,t) RuP
Vw
z
a
ö 1( À !l llbs p (J2 aa radius (e.g. of a scatterer) a constant speed of sound thickness frequency a two-dimensional function a filter function wave number (=27t/À) projection at angle
e
a ra<lius time distance, depth cartesian coordinates rotated cartesian coordinates signa! amplitudenonlinearity parameter elastic modulus
Fourier transfarm of f(x,y)
Fourier transfarm in polar coor<linates Fourier transfarm operator
Hartley transfarm of f(x,y) signa! intensity
an integer an integer
pressure in the acoustic wave scattering strength of a medium reflection coefficient
block pulse with the duration equal to half the period time of the received ultrasound pulse
voltage, volume signa! power acoustic impedance
amplitude attenuation coefficient Dirac delta function
compressibility wavelength
intensity (or power) attenuation coefficient backscattering coefficient
density of a medium
effective scattering cross-section width of ultrasound pulse angular frequency (=2nf) small tluctuation an angle an angle convolution differential operator Laplacian operator
List of abbreviations
2-D AID A-mode B-mode BUA C-D diagram DC DHT DPA DXAEIT
EPC
FFT
FHT FT HTm
MRI MRS MTFNDT
PET
PMMAQCT
RF ROl SAM SLAM SPASPE CT
TGC
TOF TPCTIS
XCT two dimensional analog to digital amplitude mode brightness modebroadband ultrasound attenuation contrast detail visibility diagram direct current
discrete Hartley transfarm dual photon absorptiometry dual-energy X-ray absorptiometry electrical impedance tomography
echo pulse counts, number of transmitted pulses for one echo sample
fast Fourier transfarm fast Hartley transfarm Fourier transfarm Hartley transfarm integrated backscatter magnetic resonance imaging magnetic resonance spectroscopy modulation transfer function non-destructive testing positron emission tomography polymethylmethacrylate
quantitative computerized tomography radio-frequency
region of interest
scanning acoustic microscopy scanned laser acoustic microscopy single photon absorptiometry
single photon emission computerized tomography time gain compensation
time of flight
TOF pulse counts, number of reverberation cycles TOF time scaler
I General introduetion
Ultrasound techniques are widely used as diagnostic techniques in any modern hospita!.
Ultrasound is the first diagnostic technique to be used for a great majority of the patients and often it wiJl even be the only one used. Ultrasound is used in almost all medica! specialisms. Generally known are the pictures of unborn children made by B-mode echography. On a worldwide basis ultrasound covers nearly 25% of all imaging studies perfonned (Leopold, 1991). Of great importance for the popularity of ultrasound is the fact that it is considered to be harmless for biologica! tissue, at least in the intensity range used in medica! applications.
Besides the above-mentioned imaging techniques, ultrasound is also used for flow measurements. In the last few years these so-called Doppier techniques have gained in
importance by the development of the colour Doppier technique (Evans et al., 1989).
The fust diagnostic application of ultrasound was reported by Dussik (1942; Dussik et al.,
1947). He described a transmission technique to produce images of the head. A similar
scanner was constructed by Ballantine and co-workers (1950; Hueter and Bolt, 1951). In
the first instanee the results seemed to be very promising. It was assumed that the images depicted intracerebral structures, particulary the geometry of the ventri cl es of the brain and the technique was meant to be used in the detection of brain tumours. In later publications
(Güttner et al., 1952; Ballantine et al., 1954) it was proved that these images of the head did not contain any clinical infonnation about the brain, but that only artifacts were imaged.
In the meantime, stimulated by new transducer technologies, a pulse-echo technique was developed for non-destructive material testing (NDT). The first clinical pulse-echo application was detecting gallstones by means of such a NDT apparatus (Ludwig and Struthers, 1950). In the following decades the pulse-echo techniques developed rapidly and
nowadays applications can be found in various fields of medica) diagnostics. Results are
publisbed in leading journals, for instanee the 'Journal of Clinical Ultrasound'.
In spite of the discouraging results, developments continued on transmission ultrasound
systems. To record the transmitted beam, research was done on ultrasound image converters. The sound pattem converted into a corresponding voltage by a piezoelectric
plate produces a visible image, by use of relevision scanning techniques (Smyth et al.,
1963; Jacobs, 1965). This idea of the ultrasound camera was improved by Green et al.
(1974). By use of a system of acoustic lenses the sound field was scanned past a linear
array of receiving elements. The quality of the in vivo images was degraded by artifacts
due to refraction, reflection and diffraction.
The Doppier technology also started with a transmission mode (Baldes et al., 1957) but
soon the development switched to the reileetion mode (Satomura, 1957). After a short
period the Doppier technique was used for clinical applications. The Doppier method shall
not be mentioned further in this thesis, because it is outside its scope.
This technique uses frequencies in the gigahertz range to produce extremely high resolution images of the acoustic properties of tissue samples. In 1949 the first artiele on acoustic microscopy appeared (Sokolov, 1949). The further development of this technique started only in 1959 with a publication about ultrasonic absorption microscopy (Dunn and Frey, 1959). Korpel and co-workers (1971) combined acoustic mîcroscopy with laser technology into a Scanned Laser Acoustic Microscope (SLAM). In 1974 Lemons and Quate (1974, 1975) described a technique in which a tissue sample mounted on a plastic film is scanned with a highly focused (by sapphire lenses) ultrasonic beam. This technique is called Scanning Acoustic Microscopy (SAM). Acoustic microscopy is not practicabie as a noninvasive technique but may be very useful in the field of histopathology.
A new ultrasonic imaging method called time delay speetrometry was proposed in 1974 (Heyser and Le Croisette, 1974). This technique involves the transmission of an ultrasound beam with a repetitive linear frequency sweep through the object under investigation. The transmitted frequency is a linear function of time and therefore the difference between the transmitted and received frequencies is constantfora given transit time. The signals that do not arrive by a straight path could be separated from those that are of interest. A time-of-flight image and an absorption image could be produced with this technique. The method was never developed for clinical application, however.
In the early seventies Greenleaf and co-workers (1974, 1975) developed the transmission ultrasound computerized tomography. This technique appears to be very useful for quantitative tissue characterization. The bistorical perspective of transmission tomography wil! be discussed in detail in the next section (1.1.1 ).
After this short survey of the history of transmission techniques in diagnostic ultrasound, the next section will place emphasis on ultrasound tissue characterization. Transmission tomography, reflection tomography and quantitative B-mode imaging wil! be discussed. Developed methods and publisbed results will be summarized. Because of a growing interest in tissue characterization more techniques are developed outside the range of ultrasound. At the end of that section these methods for tissue characterization will be mentioned.
To understand the complex interactions between ultrasound and biologica! tissue it is essential to know more about the physics of ultrasound. The second section deals with the fundamentals of medica] ultrasound. This chapter will conclude with a description of the aim and an outlîne of the thesis.
1.1
Ultrasound tissue characterization
Current clinical ultrasound imaging systems produce qualitative images of tissue interfaces and scattering and although distances between interfaces may be relatively accurate, quantitative evaluation of fundamentál acoustic properties is not possible. The latter is the field of quantitative tissue characterization. Tissue characterization was defined by Chivers ( 1981) as the identification of one or more physical parameters of a small volume of tissue that are sufficiently well correlated with the type or condition of the tissue that the measurement of these physical parameters may alone be used as an effective index of the type or condition of that volume of tissue. The present section describes ultrasound tissue characterization techniques in detail and at the end other tissue characterization techniques will be considered briefly.
As a result of a large number of studies the acoustic properties in various types of mammalian tissue are known (Chivers and Parry, 1978; Gosset al., 1978, 1980). Table I.1 presents typical values of acoustic properties in various types of human tissue. The objective is to gain an impression of the range of the values. The experimental data are dependent among other things on the condition of the tissue, the temperature, the measurement method and the frequency used. The diversity in purpose of measurements makes it difficult to compare the reported data.
tissue type skull bone speed of sound (mis) 2770
.
.
attenuation coefficient * (dB/cm) 13...
...
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..o
~o
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oooo•••••••••••••••••••
:::~~:.":l~::(::'~~~::=::J,:):::::::·:::::J:::::::::
::
:::l:::l;~i.:.~:::::::::::::::t::::::::::::::::::' :~:~:::::~:::::::::::::::
kidney
l
1570i
0.8 (1.5)...
;·~~:;···,···~-~~~···
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:
~·-~·;
·
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···
···~~~~···~~·~;~~:;~·~:;···r···~·~;~···r···~·:~··· Table 1.1: Typical values of acoustic properties in various types of human tissue as mentioned in the literature. Souree of data: Goss et al., 1978.• Values measured at I MHz unless other frequency mentioned between brackets (in MHz).
The range of values of the speed of sound in normal and pathological tissues within one specific organ are presented in table 1.2. The breast was chosen because the data was readily available and the nature of the internal tissues vary. This compilation show that
tissue type
I
speed of sound (mis) subcutaneous fat 1350-1487 . . . ... . . uooooooo . . . .. . . .. . . . parenchyma 1445-1609...
.
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fibroadenoma 1540-1582...
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benign solid 1514-1576...
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cyst 1500-1578 ···~~·~~;~~:~... r ...
~·~·~~·~·~·~~·~···Table 1.2: The range of values of speed of sound in various breast tissues as mentioned in the literature. Sourees of data: Glover, 1977; Greenleaf and Bahn, 1981; Scherzinger et al., 1989.
1.1.1 Transmission tomography
As a result of a series of publications on the algorithms to obtain a cross-sectional image of the distribution of a physical quantity in an object from a set of one-dimensional data (projections), Greenleaf and co-workers (1974, 1975) introduced the principles of transmission ultrasound computerized tomography.
Radon (1917) was the first to treat the relationship between a two-dimensional function and the set of projections. At that time, there was no reason for stating and solving this problem other than that it was a very interesting mathematica! problem and a challenge to solve it. In 1954 Braceweil (Bracewell and Roberts, 1954; Bracewell, 1956) was the first to present a practical application of the inverse Radon transfonn. He applied the mathematica! theory in radio-astronomy mapping solar microwave activity from a series of strip detector signals. The first known application to biology was made by De Rosier and Klug (1968) who obtained the cross-sectional structure of the tail of a bacteriophage from projections obiained with an electron microscope.
In 1971 the principle of image reconstruction from projections was commercially introduced into medica! radiology by Hounsfield of EMI Ltd. (1972, 1973 ), who produced a clinically useful machine specifically for imaging the inside of the head. The technique was immediately seen to be useful and the development of X-ray computerized
tomography is progressing by storm. Nowadays the X-ray CT scanner is almost a basic device in every fair-sized hospita! in the western world.
As already stated, shortly after the development of X-ray computerized tomography, Greenleaf and co-workers presented the transmission ultrasound computerized tomography. They showed the reconstructibility of the two-dimensional distribution of the attenuation and the speed of sound (Greenleaf et al., 1974, 1975). The objective of ultrasound tomography is to obtain parametrie images of quantitative values of material properties for evaluating the state of tissue.
The acoustic tissue properties under study are the speed of sound and the attenuation. Speed of sound or refractive index imaging is done by measuring the time of flight from
a sound pulse (Greenleaf et al., 1975; Glover and Sharp, 1977). Two main techniques have
been described to measure the attenuation. The first metbod compares the total energies of the received and transmitted ultrasonic waveforms (Greenleaf et al., 1974). The second metbod is based on the idea of using the frequency shifts for estimating the attenuation (Dines and Kak, 1979).
In vitro studies were publisbed as well as results of in vivo clinical studies. The main clinical application is the detection and classification of tumours in the female breast (Glover and Sharp, 1977; Glover, 1977; Carson et al., 1978; Greenleaf et al., 1978; Carson
et al., 1981; Greenleaf and Bahn, 1981; Schreiman et al., 1984; Scherzinger, et al., 1989).
The speed of sound as well as the attenuation for various breast tissues were considered.
Simultaneous measurements of these properties were made across sequentia! corona! plan es
of the breast. The studies were performed with a small number of patients. The findings
of all studies are an increased speed of sound in breast cancer tissue. The attenuation information is used as a supplement in èharacterizing various types of breast tissue. Although all studies mentioned problems with the resolution and accuracy of the
reconstructions, the method is considered a potentially useful diagnostic technique.
The in vitro studies contain an evaluation of structures containing bone (Carson et al., 1977). Carson showed the possibility of imaging the attenuation properties of a leg of lamb containing bone. Dines and co-workers (1981) reported successful imaging ofthe brain in an intact human head. Applications of testiele imaging were also described (Hiller and Ermert, 1981 ). For the investigation of the relation between cardiac function and corresponding changes in cardiac geometry, ultrasound computerized tomography using the speed of sound was used by Mol (1981 ). Transmission ultrasound tomography also can be used to study in vitro the distribution of myocardial ischaemia (Chandrasekaran et al., 1986). This study indicated that ischaemie regions are associated with increased backscatter, decreased attenuation and decreased speed of sound.
The above-mentioned methods all use the assumption that sound behaves analogously to geometrical opties where the energy is assumed to propagate along rays. Another approach is called diffraction tomography (Iwata and Nagata, 1974; Mueller et al., 1979;
Mueller, 1980; Kaveh et al., 1981, 1982; Devaney, 1982; Sponheim et al., 1991). This is
an actvaneed technique in which a wave equation is used as a descriptor of wave propagation rather than a simple straight ray model. The name diffraction tomography was chosen because of the relatively large wavelengtbs associated with typical ultrasonic sources, which result in diffraction of the sound wave when it propagates through the object to be scanned. Using special reconstruction algorithms within the so-called Rytov or Born approximations this metbod is also capable of giving quantitative images of the
speed of sound and the absorption. This thesis only deals with the straight ray
reconstruction tomography and will not go further into diffraction techniques. The principle
of the straight ray reconstruction tomography wil! be described in chapter 11.
In spite of the above results ultrasound computerized tomography bas not yet gained
wide acceptance in clinical practice. Two main reasons account for this. First, the
have been perfonned on artifact reduction in computerized tomography (Klepper et al., 1977; Schmitt et al., 1984b; Fitting et al., 1984). The iniluence of the artifacts on the image quality will be extensively discussed in this thesis.
In the second place it is not yet clear whether ultrasound computerized tomography can compete with or complement existing medica) imaging systems. In the case of breast imaging it is not yet clear whether ultrasound computerized tomography is able to replace or complete the X-ray mammography as a screening or as a diagnosing technique.
1.1.2 Reneetion tomography
Ultrasound tomography is not restricted to systems using transmission of the ultrasound beam. Ultrasound tomography in the reileetion mode has also been described. Reileetion ultrasound computerized tomography images the ultrasonic reilectivity in a medium. Several groups (Wade et al., 1978; Johnson et al., 1978) described methods to make a reconstruction of the reflectivity based on the principle that the received echoes from scatterers at the same range are integrated over the transducer and that the echoes from
greater ranges are received at later times. So, a point on an A-mode like trace can be
interpreled as the line integral of the echoes over a circular are and the received echo train can be considered a projection. A set of these projections can be reconstructed using the conventional reconstruction algorithms. These methods make use of fan beam scanning to obtain the reflected data.
In 1979 a comprehensive theoretica] analysis waspresentedof the problem of using echo data to reconstruct an image of the acoustic reflectivity for the case of a circular transducer array and a circular integration path (Norton and Linzer, 1979a). The assumptions to make a traetabie analysis are that the medium is weakly reflecting, the speed of sound is constant and the absorption is uniform. Additionally the medium is
assumed to contain isolated point scatterers without mutual interactions. Computer
simulations are perfonned to support the theory. Dines and Goss (1987) refined this model in the sense that they account for the practical effects of beam pattem and backscatter di vergence.
Norton and Linzer (1979b) also handled the problem of three-dimensional image
reconstruction from echo data from the point of view of conventional tomography. They
started with the assumption that the backscattered signals provided measurements of surface integrals of a postulated reflectivity function and then applied standard reconstruction concepts, such as backprojection. This direct approach is potentially superior to staclcing multiple transducer sections. Later they solved this problem as an inverse scattering problem (1981).
Ex perimental results of ultrasound reileetion tomography that are very promising were publisbed by Hiller and Ennert (1980, 1981). They make use of animaging system that uses a linear transducer array and special data processing to reconstruct echo data in a tomographic manner. By analogy with the above-described methods, each B-mode image
is integrated over the aperture axis for each point of the time axis. These
'pseudo-projections' are reconstructed. This approach is based upon a plane wave approximation
Another processing method, applied by the same authors, convolves each B-mode image with a filter function and then all filtered images are superimposed to give the final image. This alternative approach is called computer-aided compound scanning. In the
reflection mode one transducer picks up a number of conventional B-scans. Computerized
superposition of all single B-mode images eliminates the poor lateral resolution and makes use of the good axial resolution for all directions. However, a simple summation leads to a blurred image. An algorithm analogous to the conventional filtered backprojection
algorithm is therefore necessary to obtain a high quality image from reflectior6 ultrasound
tomography. A system analysis of these methods was presented in 1984 (Hiller and Ermert, 1984).
Experimental results were also published by Maderlechner et al. (1980). They reported
results of experiments on several objects and excised organs and the influence of several physical effects on image quality in the procedure of reflection ultrasound tomography in
the same two modes as Hiller and Ermert (1980).
After some more in vivo experiments the clinical applications were considered. Hiller and Ermert (1982) described the medica! applications of transmission ultrasound tomography and reflection ultrasound tomography. They considered only organs that can
be viewed from 360 ·, so both modes can be applied. lt appears that ultrasound
tomography is not useful for imaging the whole human body, but it is suitable for imaging
smal! organs such as the breast and testicles. Röhrlein and Ermert (1985) investigated the
possibilities of using the computerized compound scan technique with a limited angle area.
This gives the possibility of imaging organs which do not allow a 360 • imaging range.
Reflection ultrasound tomography works exactly if the speed of sound is constant in
the whole imaged cross-section. The speed of sound image and the absorption image,
obtained with transmission ultrasound tomography, could be used to correct the reflection mode image. Experimental results show that it is possible to avoid distortions in the reconstructions due tospeedof sound differences (Greenleaf et al., 1977; J(jm et al., 1984; Bartelt 1988; Jago and Whittingham, 1991).
A new developed diffraction tomography algorithm in the retlection mode was
described by Roberts and Kak (1985). This algorithm makes use of a very simple scanning
geometry and combined with speetral extrapolation techniques makes possible a reconstruction of objects for which the Born approximation holds.
In the next subsection another tissue characterization technique, quantitative B-mode imaging, will be discussed.
1.1.3 Quantitative B-mode imaging
In conventional B-mode images the amplitude of the received echo signals is displayed. In these B-mode images trained observers can detect large-scale structures.
A few years ago it became clear that the full diagnostic information contained in B-mode images is not accessible by simply viewing the image. This prompted research on methods
to obtain more objective, more quantitative images. Quantitative methods are used to make
To obtain objective, quantitative images it is necessary to be independent of the device and the operator. To be able to make reproducible images, pre-processing has to be done before analysis of the signals. This pre-processing corrects for the acoustic beam
characteristics that induce a depth dependence. In case of a time-gain compensation (TGC),
a recompensation has to be done.
The acoustic properties under study are the attenuation, the backscattering, the speed of sound and the nonlinearity parameter B/A. The latter parameter is not obtained from the backscattered signa!, but it appears to be a useful parameter in quantitative ultrasound tissue characterization. For this reason it will be mentioned in this section. By applying a multi-parameter method, the possibilities of tissue characterization are improved. In spite of the large amount of research, no method is routinely employed in clinical practice.
Different methods are developed to obtain quantitative information about the attenuation coefficient of tissue. One method converts the RF-signals from various depths to power spectra. The log power spectra from two different depths are subtracted from each other and a power law line is fitted to the result. This method is called the speetral difference method (Kuc, 1980). Because attenuation in soft tissue is described by a power law function of the frequency, this method gives the frequency dependent attenuation
coefficient and the power law dependency. Another method is based on the frequency
dependenee of the attenuation coefficient. The higher frequencies are more attenuated than the lower ones. The centre frequency of the spectrum plotted versus depth is used to estimate the attenuation coefficient (Dines and Kak, 1979). For reasons of simplicity several methods are developed that make use of the measured amplitudes rather than
speetral parameters (Taylor et al., 1986; Garra et al., 1987).
In vivo measurements are obtained for several organs. However, most of the clinical
studies are concerned with measurements in the liver. Although the results show a large
variability, it appears that the attenuation coefficient has some ability to separate normal from diseased tissues (Taylor et al., 1986).
The backscatter coefficient appears to correlate with pathologies. Quantitative
information about the scattering in the form of the backscatter coefficient of the medium,
determined by size, concentration, strength and shape of the scatterers, is obtained by analysing the corrected backscattered power spectrum (Lizzi et al., 1983; Madsen et al.,
1984; Lizzi et al., 1986; Romijn et al., 1989a, 1989b ).
In vivo measurements of the speed of sound create great difficulties. Nevertheless,
many methods have been proposed to estimate the velocity by means of backscattered signals. Robinson (1982) described a method in which two images are built up from different directions. From the displacement in these images of a specific scattering structure a sound velocity was calculated. Ophir proposed the so called 'beam tracking method' (1986). The complexity of these methods and the disappointing accuracy decrease researchers' interest.
The nonlinearity parameter B/ A is a measure of the nonlinearity of the pressure-density relation for a medium. This relation describes the distartion within the waveform, due to variations of the speed. These variations occur because the density of the medium changes with pressure. Because the parameter B/A is a property of the medium, it may provide
diagnostic information (Muir and Carstensen, 1980; Law et al., 1985; Errabolu et al.,
This method is based on measuring the phase shift by interaction of two ultrasonic waves with different frequencies and power levels. Later on, improved methods to determine the non-linear parameter B/A of biologica] media were proposed basedon estimating the shift in phase and frequency (Sehgal et al., 1984; Kim et al., 1990).
The image texture in B-mode images can be characterized as coherent speckle. It is due to the interaction of ultrasound and the tissue and therefore it can be used as a parameter for tissue characterization. The quantitative characteristics of texture depend not only on the tissue but also on the equipment, including the transducer. A correction has to be applied prior to the analysis, to increase the reproducibility.
The texture of B-mode images can be analysed by first orderand second order statistics using stochastic signa! analysis (Wagner et al., 1983). First order statistics characterize the histogram of the echo amplitudes. This histogram is quantified by its mean and the standard deviation. Many other features can be extracted from the first order grey level histogram. The second order statistics are characterized by the autocovariance function. The full width at half maximum of this function is an estimate of the average size of the speckle.
Speckle analysis can be used for diagnostic purposes. In a clinical study performed by
Oosterveld et al. (1991) it was shown, for instanee that the first order statistica! parameters could differentiate between several kinds of liver pathology.
1.1.4 Other noninvasive tissue characterization techniques
Magnetic resonance imaging (MRI) is a relatively new imaging technique (Lauterbur, 1973). The contrast on the MR images is basically the result of the relaxation times Tl and T2 and the proton density. Tl is the characteristic time of the process of returning to a state of equilibrium from an excited state following the impulse given by a radio-frequency burst. T2 characterizes the dephasing of spins. The proton density reflects the water content. Because of its high soft tissue contrast it was expected that MRI might be used fortissue characterization. lt was claimed that quantitative measurements of relaxation time could be used as a diagnostic technique for noninvasive tissue characterization (Damadian, 1971). Later this claim was refuted because in vivo relaxation time measurements show great variability (Rinck, 1985; Bottomley et al., 1987). Increase of measurement accuracies change the perspective. There are some successful applications of relaxation time measurements for tissue characterization. For instanee the differential diagnosis of leukaemie bone marrow could be made by relaxation time measurements (lensen et al., 1990; Thomsen et al., 1987). In cirrhosis, also, a prolonged Tl relaxation time is found (Thomsen et al., 1990).
A spectroscopie approach to magnetic resonance (MRS) is also possible and this is
widely used in chemica) and biochemica! research. The spectra are formed by the effect
that not all protons exhibit the same frequency when submitted to the same external
magnetic field. The higher the field strength, the better the spectra will be. Based on such
tissue characterization by magnetic resonance techniques.
As far as known at this moment MRI and MRS are harmless methods for the patients. A limitation of MR measurements is that the equipment is immobile and generally more expensive than other imaging techniques and the examination time is relatively long.
The examination time for X-ray computerized tomography (XCT) is much shorter. XCT measures very accurately the attenuation of X-rays passing through an object. The calculated CT number is related to the linear attenuation coefficient of the material for the effective energy of the X-ray beam. Since linear attenuation coefficients for soft tissue are a direct reflection of volume electron densities and of mass densities, the potential exists to characterize soft tissues by their CT numbers.
Quantitative computerized tomography also offers the possibility of characterizing bone tissue (Ruegsegger et al., 1976; Genant and Boyd, 1977). By simultaneous measurement of a calibration phantom, the bone mineral density can be predicted. A disadvantage of X-ray CT is the use of ionising radiation.
The last technique mentioned in this section is positron emission tomography (PET). Since the position of a positron emitting radioisotope can be determined by the two annihilation photons, which are emitted 180 • from one another, detectors positioned around a patient can determine the line along which the disintegration occurs. PET is an indirect techni<jue. With a labelled tracer, the metabolism in a variety of organs can be studied. From the metabolic information characteristics of the tissue can be identified. Several studies have demonstrated the ability of PET to quantitatively evaluate metabolic function (Phelps et al., 1986).
1.2
Fundamentals of medica) ultrasound
Ultrasound is a mechanica! wave with a frequency that is too high for perception by the human auditory system. For this reason, it took until the end of the nineteenth century before ultrasound was investigated. The lower limit for ultrasound is often taken rather arbitrarily as 20 kHz. This section briefly describes the physics of ultrasound as far as applied in this thesis. A thoroughly study of the physics of medical ultrasonics was published e.g. by Hili (1986).
An ultrasound wave is attenuated, primarily due to absorption and scattering, when it passes through a medium, such as tissue. Absorption is mainly due to relaxation phenomena of biologica! macromolecules (Wells, 1975). Tissue exhibits scattering from histological features such as clusters of cells, small vascular channels and microscopie calcific aggregations.
Scattering of ultrasound by a partiele greatly depends on the ratio of the size of the partiele to the wavelengthof the ultrasound wave. A measure for this ratio is the scattering parameter ka, where k is the wave number and a is the dimension of the particle. Two scattering regimes are of interest. If ka > 1 the scattering is called geometrie scattering. This is determined by the laws of reflection and refraction and will be explained later in this section. If ka< 1 the scattering is called Rayleigh scattering after Lord Rayleigh, who first studied this theory in 1871. Rayleigh scattering is proportional to the fourth power of
frequency and is omnidirectional.
Quantitative measurements of scattering strength would be significant for tissue characterization. The scattering strength can be measured quantitatively in terms of a backscattering coefficient.
The propagation of ultrasound, if absorption effects are negiected, is described by a wave equation, formulated by Morse and lnghard (1968):
where
2.
-
VPoKo·
cP(r,t) is the sound pressure, ~p(r) and ~IC(r) show the small regional variations in density and compressibility around mean values
Po
andKo·
respectively, assuming slight inhomogeneity.The most important simplification to solve this general equation is the Born approximation: the pressure of the sound field can be taken equal to the incident pressure (P;) if it would travel through a homogeneous medium and only tirst-order scatter will be considered. The latter assumption means that. the contri bution of each scatterer to the scattered signa) can then be treated independently, and the resulting scattered signa! represented as a sum of the contributions from all individual scatterers. The solution is (Ueda and Ichikawa, 1981; Hili, 1986)
P(r,t) - P;(r,t) + Ps<r,t), (1.2)
where Plr,t) is the scattered wave from the volume V. P ( ) _ JQ(r,t)Ö(t-lrs r,t 1-rl!c) d v,
v
(
41tIr -
r
11)(1.3)
where r' is a position vector. Q(r,t) is a scattering strength of a medium:
(1.4)
The first term expresses the con tribution of the compressibility fluctuations and the second term expresses the contribution of the density variations.
A measure of the contribution of scattering to the attenuation is the effective scattering cross-section, which is defined as the ratio of the total scattered power to the intensity of the incident wave:
scattered back into unit solid angle, located at 180 ' to the direction of the incident ultrasound.
If the scatterers are randomly positioned and the volume contains sufficient scatterers, the power scattered will be proportional to the volume of the media and the scattering is tenned incoherent. In case of incoherent scattering, the scattering strength can be measured quantitatively in tenns of a backscattering coefficient (!lbs), which is defined as the backscattering cross-section per unit volume.
In the special case of the inhomogeneities being large with respect to the wavelength, the scattering response is quite different from the response from a point-like scatterer. At the interface between two media with different acoustic impedances two phenomena occur: reflection and refraction. This is illustrated in tigure l.I.
Figure 1.2: Behaviour of ultrasound at the boundary between two media with different acoustic impedances.
Reflection follows the law that stipulates that the angle of incidence (e;) equals the angle of reflection (er). The reflected ray also lies in the plane of incidence and on the opposite site of the nonna!. The plane of incidence is the plane that contains the ray of incidence and the normaL
Refraction follows Snell's Law that states that the ratio of the sine of the angle of incidence and the sine of the angle of refraction (e1) equals the ratio of the speed of sound in the two media:
sine; sine1
(1.6)
The energy of the incident wave is shared between the waves reflected and transmitted at
the boundery. The intensity of reflectivity (1/I;) is:
Ir -
[z2
cos ei - ZJ cos eIJ
.
I; ~cose; + Z1cose1
where Z; is the acoustic impedance of a medium (Z; = P;c;). The intensity of transmissivity is:
4Z1 Z2cos8;cos81
(~cos ei+ zl cos
ei
(1.8)
Absorption together with scattering wil! cause attenuation. The attenuation causes an exponential decrease of the power of the wave as a function of the depth (x):
(1.9)
where J.l is the attenuation coefficient, expressed in termsof nepers (Np) per unit distance. For many purposes it is more convenient to express the amplitude attenuation coefficient
(<X) as the ratio of signa! amplitudes in decibels per centimetre. The attenuation m
biologica! tissue is frequency dependent (Pohlman, 1939; Hueter 1948; Wells, 1975):
(1.10)
In biologica! tissue the value of the frequency power n is in general between 1 and 2 (Hili, 1978). Frequency dependent attenuation implies the presence of dispersion (Gurumurthy and Arthur, 1982). This is established by the Kramers-Kronig relations (Kak and Dines, 1978; O'Donnell et al., 1978, 1981).
If tissue exhibits dispersion, the spectrum changes when the pulse passes through the tissue. It has been shown that a Gaussian modulated pulse remains Gaussian when it passes through a dispersive medium, such as biologica! tissue (Ferrari and Jones, 1985).
The ultrasound pulse generated by a common piezoelectric transducer is mostly assumed to have a Gaussian envelope (Dines and Kak, 1979). A Gaussian modulated pulse has a Gaussian shaped spectrum in the frequency domain. This is also the case for the transducers used in the experiments describ~ in this thesis. A typical sound pulse from the transducer used is shown together with its spectrum and the best fitting Gaussian envelopes in figure 1.2.
Diffraction is the term used to describe the spreading out of a wave in space. The shape of the sound field depends highly on the source. The sound field generated by an unfocused ultrasound transducer can basically be divided into the cylindrical near field or Presnel region and the cone shaped far field or Frauenhofer region. Focusing is achieved by using an acoustic lens incorporated in the transducer or by using a concave piezoelectric element. For both options the focus length is fixed.
1.00 I 1.00 I w 0 0.75 a: 0.75 :::1 w 1- ~ ::i 11. 0 :::E 11. < 0 0 0.50 w N 0.50 w ::i N ::i < < :::E :::E a: a: 0 0 0.25 z 0.25 z 0.00 0.00 0 2 0 2 4 8 8 10 TIME (111) FREOUENCY (MHz)
Figure 1.2: Waveform (left) and spectrum (right) of a characteristic ultrasound pulse and the best fitting Gaussian envelopes.
1.3
Scope of this thesis
This study is a continuation of the research done by Stapper and Sollie (Stapper and Soli ie, 1985; Sollie, 1988). Sollie concluded his doctoral dissertation (1988) with the thesis that it is possible to build a transmission ultrasound tomograph using cheap and unsophisticated electronic components and computer equipment. This tomograph rnight be used clinically as an imaging system. It was his expectation that the system would offer the possibility of performing a limited degree of tissue characterization.
The aim of the present study is to extend the prototype ultrasound tomograph with a
measurement of the backscattered signal. This measurement has to be performed simultaneously with the three already existing measurements. Because of the simplicity of the set-up it is being tried to find unsophisticated electronic components to implement the measurement and to use only the personal computer available in the set-up. An experimental evaluation of the imaging characteristics of the system will be performed to investigate whether the system can be used for quantitative tissue characterization. The spatial resolution, contrast resolution and quantitative accuracy of the tomograms measured with this prototype ultrasound tomograph will be determined. The ultimate goal of the project is to develop a clinically useful ultrasound tomograph. The clinical perspeelive of ultrasound tomography will be discussed, with the knowledge of the imaging characteristics determined.
A short overview of the thesis concludes this chapter. In the next chapter computerized tomography using ultrasound will be described. The measurement of the acoustic
parameters with the developed prototype will be discussed and the theory of the reconstruction will be explained. The questions which algorithm is used and why the Hartley transform is used insteadof the well-known Fourier transform will be discussed. In chapter lil the measurement of the reflectivity will be discussed in detail, because this is a completely new extension of the prototype. The similarities and differences with the methods mentioned in the literature will be discussed. Some experiments on tissue-mimicking phantoms to test the validity of the proposed concept will be shown.
The quality of the tomograms will be discussed in chapter IV. A description of the spatial resolution, the contrast resolution and the quantitative accuracy will be given. Phantom studies of these parameters have been performed and the results will be presented. The effects of image quality degradation by artifacts due to refraction, interference and phase cancellation will be discussed in the last part of chapter IV. These image distortions prove to be an important problem in ultrasound tomography.
In chapter V the results of measurements on bone-containing structures will be presented. The first aspect to be discussed is the question whether it will be possible to image soft tissues accurately if there are bones in the scan plane. The second aspect discussed is osteoporosis. Although ultrasound has been used in the diagnosis of osteoporosis, no studies are known on ultrasound tomography in relation to osteoporosis. Preliminary results of tomographic measurements on a rabbit femur in vitro will be discussed. lt was also attempted to measure some parts of the hu man body in vivo. Results of these measurements will be presented.
Finally, in chapter VI a discussion ofthe clinical perspective of ultrasound tomography will be given tagether with the final condusion of the project and a number of recommendations conceming ultrasound tomography.
11 Measurement techniques and measurement device
In this chapter computerized tomography will be described, concentrating on ultrasound computerized tomography. The latter technique is subdivided into transmission ultrasound
tomography and reflection ultrasound tomography. In transmission ultrasound tomography
the sound pulse propagates from a transmitting transducer to an opposite receiving transducer. Reflection ultrasound tomography uses only one transducer for transmitting a pulse and receiving the backscattered signal.
After the description of the principles of computerized tomography, a description of the
set-up used will be given. In this set-up, transmission ultrasound tomography and reflection
ultrasound tomography are combined. A photograph of this tomograph is shown in figure 11.1.
Subsequently, the measurements implemented in the experimental ultrasound tomograph
will be described. The measurement principles, the reconstruction conditions, the
implementation and the theoretica) precision of the implemented measurements will be
considered. The set-up providesus with four acoustic parameters: the speed of sound, the
attenuation coefficient, the attenuation slope and the reflectivity. The measurement of the latter parameter will only be discussed in general terms. Because this is an entirely new extension of the prototype, the physical meaning of this measurement will be discussed
separately in detail in chapter lil.
In section Il.4, the image reconstruction will be explained. Two reconstruction
algorithms used in this study will be described: the filtered backprojection algorithm and
the direct Fourier inversion algorithm. The filtered backprojection algorithm perfarms the
reconstruction in the spatial domain. The direct Fourier inversion algorithm perfarms the
reconstruction via the frequency or Fourier domain. The fundamental principle of both
algorithms is the same: the 'Centra! Section Theorem' or 'Projection-slice Theorem'. This
theorem will be explained. Since the projections are real signals, the Hartley transform
instead of the Fourier transfarm can be used. The consequences for the computing time
and memory space will also be discussed.
11.1
Principles of computerized tomography
The goal of computerized tomography is to make a cross-sectional image depicting intemal
detail of an object, such as a part of the hu man body. To be able to reconstruct such a
cross-sectional image, a set of one-dimensional data (projections) is necessary. These projectionscan be obtained by means of X-rays (XCT), nuclear magnetic resonance (MRI), pboton emission (SPECT), positron emission (PET), electrical impedance (EIT) or ultrasound.
Figure D.l: The experimental ultrasound tomograph. Overview of the set-up (top) and top view of
In the case of ultrasound computerized tomography two transducers are mounted opposite to each other and submerged in a water tank. The object to be imaged is placed between the two transducers. The transmitting transducer sends an ultrasound pulse, which travels through the object to the receiving transducer. Part of the energy of this pulse is scattered back to the transmitting transducer. Infonnation about the acoustic properties of the object, along the sound beam, is extracted from the transmitted pulse and from the backscattered signal.
Along a linear scan, perpendicular to the sound beam, the acoustic properties are detennined with a constant sampling interval. The data obtained along such a scan make up a so-called projection. These projections are measured at a variabie number of angles, equally divided over 180 '. This is elucidated in figure 11.2. Several features of the pulses received are measured simultaneously and each feature fonns its own set of projections. For the tomograph used in this study this will be explained in section 11.3. Tomograms of the acoustic parameters are reconstructed from these sets of projections. This reconstruction is done using one of the available reconstruction algorithms.
SCANNING MOVEMENTS
Figure 11.2: Schematic diagram of the measurement principle of ultrasound computerized tomography.
The idea behind the reconstruction algorithms is that every projection of a rotational scan has the infonnation in one direction about an acoustic property of the same slice. The reconstruction algorithms provide us with the local value of this acoustic property. A set of projections must satisfy some conditions to be reconstructible. These conditions follow from the denvation of the reconstruction algorithms, which is described in section 11.4. The conditions will be given here in advance. In section 11.3 it will be shown that these conditions are satisfied in the measurement methods implemented. The first condition is that the measured data must have an one-to-one relationship with a line integral of the two-dimensional distribution of the acoustic property in the scan plane. The second condition is that the measured object must be fully covered by a projection and the projection has to be zero outside the object. The third condition is that the distribution of the acoustic
property has to be isotropie and the fourth condition says that the set of projections must be consistent. This means that all projections have to be a projection of the same two-dimensional distribution of an acoustic property.
11.2
Measurement device
The measurement device consists of a personal computer (PC-AT, 80286) which is interfaced using an IEEE-488 bus both to the tomograph interface, based on a single chip microcontroller (lntel 8751), and to a stepper motor interface. The microcontroller is responsible for the measurement and for the control of the measuring electronics. All the measuring electronics was designed and built in our own group. It consists of a pulse generator, a receiver and the electronics to determine and process the features of the ultrasound pulse received, as will be described in the next section. The microcontroller reads the counters and AID converters and sends the data to the personal computer. The microcontroller has a 4 kByte EPROM programming memory, into which the program is loaded that carries out the measurements.
CAN PLATFORM WITH STEPPER MOTORS AND END SWITCHES
Figure 11.3: Schematic diagram of the measurement device.
The transducers are mounted on a platform. The central part of the platform can be rotated (figure 111). On this central part the two sets of parallel, straight rails which guide the transducer carriers are mounted. The lateral and rotational scanning movements are driven by stepper motors. During a measurement these stepper motors are controlled by the microcontroller. End switches are mounted to give position information. During the measurement of one sample, the transducers keep moving to avoid mechanica! vibrations. Small deviations in the parallel, straight rails cause variations in the distance between the transducers and in the direction of the sound beam duringa linear scan. Because of this inaccuracy a 'track-error' occurs in the projections. To correct this 'track-error', one
projection is measured at the end of each complete rotational scan, without an object between the transducers. This projection is subtracted from the projections, that have been measured with an object between the transducers. The requirement that the projections are zero outside the object is also satisfied by this 'track-error' correction.
The personal computer acts as the user-interface and data storage. In fact the personal computer is the system controller of the IEEE-488 bus, so the user can always interrupt a measurement and take over the control function of the microcontroller. The personal computer is also used for image reconstruction after a complete measurement.
The platform with the transducers is placed over a water tank during the measurement. The inner wall of this tank is covered with artificial grass to reduce unwanted reflections.
11.3
Measurement of the acoustic parameters
In this section the measurement of the four acoustic parameters, as implemented in the ex perimental tomograph will bedescri bed. Information about three parameters is extracted from the pulse propagating through the object. A fourth parameter is derived from the backscattered signa! received. Transmission ultrasound tomography and reflection ultrasound tomography are combined in one set-up.
11.3.1 Speed of sound
Speed of sound is one parameter that can be useful for tissue characterization. For this reason velocity data of biologica! tissues were estimated by several authors. Reviews of these data are compiled by Wells (1975), Chivers and Parry (1978) and Goss and co-workers (1978, 1980). Greenleaf et al. (1975) firstly demonstrated the computerized tomography concept with time-of-flight (TOF) measurements. Speed of sound can easily be determined from the time of flight of a sound pulse. The time of flight is the time between transmitting a sound pulse and receiving the sound pulse on the opposite transducer.
The time of flight is determined by the speed of sound in water and the speed of sound in the object between the two transducers. In fact it is the line integral of the inverse of the loc al speed of sound c(v, w) on the path Lve of the sound pul se between the transducers. The definitions of the co-ordinates are elucidated in figure Il.9.
TOF(v) -
J _
1_ dw. L c(v,w)ve
(ll.l)
By subtracting the projection measured with only water between the transdoeers from the projections measured with object, the condition that the projection has to be zero outside the object is fulfilled.
In the set-up used, the TOF measurement is done using a reverberation technique (Stapper and Sollie, 1985). Each time the receiving transducer receives an ultrasound pulse, a putse generator is triggered and another ultrasound pulse is generaled by the transmitting
transducer. As mentioned before, the transducers keep moving during one sample. The duration of an adjustable number (TPC) of reverberation cycles can easily be measured. The reverberation method is simpler to implement than the methods described in the literature, which use the digitized RF-signa! and extensive signa] processing (Greenleaf et al, 1975; Carson et al., 1977; McK.innon et al., 1984) or very fast, dedicated electronics (Giover and Sharp, I 977). Using the reverberation technique the precision can be improved by increasing the number of reverberation cycles. Then the precision is of the same magnitude as that of the other methods.
The detection of the sound pulse is performed by a level detector and a zero-crossing detector. The amplifier connected with the receiving transducer has a very simple automatic gain control and the output signa) will not be constant over the full range of the gain controL To prevent the level detector missing one period of the sound pulse received, the threshold level is set relative to the amplitude of the sound pulse received. The time of arrival is defined as the time of the first zero-crossing after exceeding the threshold. This moment has been chosen because its position in time remains unaffected by changes of amplitude. It is only relatively little affected by the frequency-dependent attenuation in biologica) tissue.
The duration of TPC reverberation cycles is counted by a software adjustable counting frequency. This frequency is derived from a 100 MHzoscillator by first dividing it by 16. This 6.25 MHz signa) is divided again by a programmabie divider which decrease the counting frequency by a factor called TOF Time Scaler (TTS). A 16 bits counter is counting during the TPC pulses. TPC itself is counted with another divider. After TPC reverberation cycles the 16 bit counter is read by the microcontroller and the value is sent to the personal computer. This is illustrated in a simplified schematic diagram in tigure Il.4. 6 25
- -
~Hz 6.25 MHz 1ns
-TTS COUNTER
___..
LATCH ~ CLK4
CONTROLLER MICRO-DETECTED 1-PULSES TPC
To avoid stopping of the reverberation in case a pulse is not detected, the system has a start-up oscillator. This oscillator triggers the pulse generator to send a pulse when after detection of a pulse the next pulse is not detected within a certain time, which is Jonger than the longest expected cycle time.
A spurious pulse with an amplitude exceeding the threshold also can trigger the pulse generator. This will cause reverberation at a higher frequency than the original reverberation frequency. To suppress the possibility of receiving spurious pulses, a time window is adqed to the circuit. During this window the receiver is disabled. If the duration of the blocicing window is langer than half the expected cycle time, any spurious pul se wil! die out. If the window is set to three-quarters of the reverberation cycle time, the error caused by one spurious pulse is at worst (0.25 I TPC)
*
I 00% and the probability that a pulse actually will introduce an error is maximally 25%.The counting frequency is (6.25 I TTS) MHz. The precision of the time-of-flight values will increase by increasing the number of reverberation cycles (TPC). The precision in relation to the values of TTS and TPC is at worst:
precision - ---~---
*
100%.6·
25*
106 TPC ll l .--===-- *
*
sma est cyc e ttmeITS
(ll.2)
If TPC = 600, TTS = 12 and the smallest cycle time is 160 J.ls (common values), then the quantization error of the counter is 0.002%, which means 3.2 ns.
11.3.2 Attenuation coefficient
The measurement of the attenuation coefficient by a comparison of the received and transmitted energy of the sound pulse is the first application of transmission ultrasound tomography described (Greenleaf et al., 1974). An error is introduced, because only that part of the ultrasonic beam which is captured by the receiver is measured. The signa! loss is therefore not only due to absorption but also due to refraction, reflection and diffraction. In addition, it is stated that this kind of attenuation measurement is fundamentally inaccurate for frequency-dependent media, such as tissues (Kak and Dines, 1978; Dines and Kak, 1979). In spite of it, Sollie expected additional information from measuring peak values and it was decided to include an amplitude measurement in the experimental ultrasound tomograph (Sollie, 1988). From the preliminary results it appears that the attenuation coefficient tomograms give at least qualitative, geometrical information. For this reason the measurement is maintained in the experimental tomograph.
The reconstructibility of the amplitude measurement is based on the following equation,
which can easily be derived from the attenuation equation for the signa! amplitudes (analog to 1.9):
Ar
f
-Jn_ - !l(w) dw.
Ar L._e
