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### Chapter 7

Ultrasonics

#### 7.1 Introduction

Ultrasound, as currently practiced in medicine, is a real-time tomographic imaging modality. Not only does it produce real-time tomograms of scattering, but it can also be used to produce real-time images of tissue and blood motion, elasticity, and flow in the tissue (perfusion).

The most common form of imager is the B-scanner (the B merely stands for brightness). An ultrasonic pressure pulse about a microsecond long is launched into the tissue by a transducer consisting of an array of individually pulsed piezoelectric elements. This pulse is reflected from the various scatterers and reflectors within the tissue under investigation. The scattered pressure wave is detected by the transducer array and focused using electronic beam forming. The resulting signals are used to make an image that correlates to the scatterers and reflectors within the region from which the pressure pulse signal was reflected.

Because ultrasound is currently only a two-dimensional imaging modality, many investigators are researching multi-dimensional modalities such as three-dimensional imaging or are combining flow with scattering images in several dimensions. The combination of ultrasound imaging with therapeutics such as hyperthermia or drug injections, or with ultrasound ablation, is developing quickly. Intravascular and intracavitary imaging methods for both imaging and therapeutics are being investigated at a fast pace in both commercial development and basic research laboratories. Even ultrasonic microscopy, not covered here, is developing to surpass a resolution of 1 micron.

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Technical advances in ultrasound supported by mathematics include computed tomography (inverse scattering), scatterer number density calculations (statistics), wave elastic tissue interaction (viscoelasticity), ferroelectric transducer development (ceramic physics), and wave equation modeling of ultrasound in viscoelastic materials such as tissue. The mathematical model behind ultrasound computed tomography is presented in section 14.1.3.

Mathematics and physics have greatly influenced the development of ultrasonic imaging, and many challenging problems from physics and the mathematical sciences remain to be solved. Beam forming in nonhomogeneous and usually nonisotropic materials such as biologic tissue is not at all well developed theoretically. Adaptive beam forming, which corrects for variations in refractive index within the imaged field, is a problem that is not solved. Inverse scattering is not a solved problem for geometries in which the sound is either traversing (forward scattering) or reflecting (backscattering) from the object. Acoustic models for the behavior of transducers with the complicated geometries of today's scanners are not well developed.

Following are some viewpoints on the various aspects of ultrasonic imaging.

#### 7.2 Instrumentation

The instrumentation of ultrasonic imaging consists of the transducer, a beam former, and a signal analysis and display component.

##### 7.2.1 Transducers

There are three important areas of development pertaining to transducers: (1) field distributions, (2) acoustics and vibration, and (3) electromechanical properties of piezoelectric and ferroelectric materials.

###### Field Distributions

The seminal references in this area are, of course, related to the classic diffraction theory of Huygens and various analyses by Rayleigh, Kirchoff, and Sommerfeld. This theory is particularly applicable to monochromatic radiation and enables accurate calculations of the field distribution from ultrasound radiators under certain conditions. Because it is limited to monochromatic ultrasonic energy, this approach is not well suited to wideband ultrasound imaging.

A method called the impulse response model was developed first by Oberhettinger in 1960 and subsequently refined by Stepanishen in the early 1970s

**Suggested Citation:**"7 ULTRASONICS." National Research Council. 1996.

*Mathematics and Physics of Emerging Biomedical Imaging*. Washington, DC: The National Academies Press. doi: 10.17226/5066.

Page 123

into a linear systems model of beam propagation. This approach essentially transformed the problem from one of wave propagation to one of geometry. The great advantage of the impulse response approach was that analytic solutions existed for important classes of radiators such as spherical, rectangular, and annular apertures. It is now possible to make accurate calculations of wideband ultrasound field distributions for most relevant ultrasound transducer geometries.

###### Acoustics and Vibration

The mathematical foundations were laid for acoustics and vibration analysis by Lord Rayleigh in his book *The Theory of Sound,* which was first published in 1896. From the transducer point of view, notable contributions were made by the Curie brothers, who discovered piezoelectricity in 1882, and by some of the great theoreticians such as A.E.H. Love, who made important contributions to the mathematical theory of elasticity. The theory of acoustics is well summarized in the book by Morse and Ingard listed in the suggested reading. The science of crystallography, exemplified in the work of Cady, had an important influence on the development of materials for ultrasound transducers. Similarly, the science of solid-state physics was critical in the development of an appreciation for the underlying physics of piezoelectricity and ferroelectricity. Recently, the classical theory of vibration has been extended by the development of finite-element models.

###### Electromechanical Properties of Ferroelectric Materials

Two classes of models have been particularly useful in modeling electromechanical properties of ultrasound transducers. These are the Mason model, in which the transducer is modeled as a lumped element system, and the KLM (Krimholtz, Leedom, and Matthei) model, in which the transducer is modeled as a transmission line. In recent years, the KLM model has been particularly useful for transducer design and is currently used widely in this regard.

##### 7.2.2 Ultrasonic Beam Forming

Phased array transducers contacting the surface of an object can be electronically scanned and focused to produce real-time images of internal structures. Unlike conventional radar phased arrays operating in the far field, electronic data processing of an ultrasound array emulates the diffraction integral of a lens focused to a particular point within the object. Early work in acoustical holography in the 1960s resulted in the first phased array ultrasonic

**Suggested Citation:**"7 ULTRASONICS." National Research Council. 1996.

*Mathematics and Physics of Emerging Biomedical Imaging*. Washington, DC: The National Academies Press. doi: 10.17226/5066.

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systems in the early 1970s. Continuous progress on real-time processing has led to modern digital beam formers, which are summarized in the article by O'Donnell listed in the suggested reading. To date, all beam-forming systems have assumed a homogeneous distribution of the acoustical index of refraction (i.e., sound velocity). Imaging in highly inhomogeneous media, such as the human body, is severely degraded by this assumption.

##### 7.2.3 Signal Processing

Ultrasound imaging has developed as an analog to radar imaging, but there are a number of areas where the two modes differ, and a more complete theory specific to the interaction of ultrasound and biological materials is required. The signal processing involved in forming images from A-mode data is fundamentally very simple and is based largely on linear (convolutional) models for sound transmission and reflection in tissue. Important early developments to form gray-scale, real-time images included the digital scan converter and the related software for preprocessing and postprocessing of images, along with signal processing schemes to achieve focusing and steering in ultrasonic arrays and to process the results. Currently a number of simple theories are used to model pieces of the imaging system: geometric optics for fixing transmit aperture (f-number), Fourier optics for apodization and control of side lobes, sampling theory for array design and system electronics, and statistical considerations for setting system dynamic range and color flow estimates. However, specific methods used by equipment manufacturers to form images appear to be fairly well guarded trade secrets.

Advances in real-time medical ultrasound imaging can only be achieved with a thorough understanding of the physics and physical mechanisms inherent to the imaging modality of interest, and the interactions of ultrasound with the body. The basis of any algorithm development must be an improved understanding of the underlying physics and physiological constraints of the problem. The literature is filled with examples of unsuccessful, ad hoc signal processing schemes tried on ultrasound images only because it was possible to do so. In years past, manufacturers have also periodically tried, and dropped, signal processing methods developed in an ad hoc manner.

Two fundamental effects that limit the quality of ultrasound imaging are speckle and phase aberration. Speckle is due to the coherent nature of the ultrasound signal, and phase aberration is caused by spatial variations of the propagation speed in tissue. The current state of affairs is to apply the multislice methods of x-ray computed tomography (CT) or magnetic resonance

**Suggested Citation:**"7 ULTRASONICS." National Research Council. 1996.

*Mathematics and Physics of Emerging Biomedical Imaging*. Washington, DC: The National Academies Press. doi: 10.17226/5066.

Page 125

imaging (MRI) to form volumetric or surface rendered versions of tissue. Ultrasound, however, is a real-time imaging method, and it is certainly going to be a challenge to make the multislice methods operate in real time. The development of novel data collection and processing methods for generating volumetric data sets should be given some attention. Methods that use a reduced amount of input data and rely on the brain's ability to sort out large amounts of data (perhaps combined with virtual reality display methods) would hold promise for developing practical three-dimensional real-time systems. Continuous wave techniques of computing motion (e.g., blood flow) are based on computing the Doppler shift in a reflected sinusoid that is performed by any of a number of well-known spectral estimation techniques now considered part of the signal processing literature.

A combined signal processor could have economic and performance advantages wherein it could take into account ''speckles" in a two-dimensional space and track both blood movement and probe movement simultaneously. Tracking the probe movement provides a "compounding effect" to improve gray scale images and reduces noise in color flow imaging. Tracking the blood or tissue movement produces color Doppler and tissue motion imaging with direction, speed, and Doppler power information.

#### 7.3 Scattering

Since the major sources of reflected and scattered waves are changes that occur at organ boundaries, even the crude early systems were able to produce images that correlated with anatomy and offered the first views of soft tissue structures that were invisible to simple transmission x-rays. Further refinements have used empirical techniques combined with medical knowledge to enhance the ability to image details, with image resolution as the main criterion. There is currently little or no ability to characterize tissue or tissue structures on any basis other than anatomical geometry or the flow patterns revealed by the Doppler effect. Since ultrasound instruments are widely used and distributed, there is considerable potential for improving diagnosis and treatment if better characterization can be done. The path to this goal is not clear. The benefits to be gained from a deeper understanding of the scattering process are difficult to predict, given the success of current methods and technology (see, for instance, the paper by Feleppa, Lizzi, and Coleman listed in the suggested reading).

All soft tissues except lung tissue have nearly the same composition,

**Suggested Citation:**"7 ULTRASONICS." National Research Council. 1996.

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being mainly water with small amounts of protein. Therefore, the differences in bulk elastic properties are also small. This has led to the use of pulse-echo techniques for observing the weak scattered signals. The considerable energy losses in tissue also lead to frequency-dependent absorption. This absorption led to the early use of ultrasound as a source of therapeutic heat.

The imaging signal used by present-day clinical instruments is the product of the transmitted wave with scattering and absorption factors, and one factor cannot be measured without knowledge of the other. Laboratory measurement systems are needed to perform measurements free of this restriction. The frequency dependence of attenuation sets the resolution limit. Since the loss is inversely proportional to ultrasonic frequency, the size of the imaged region is proportional to wavelength. The size of the resolution cell is also proportional to wavelength. The images thus have a nearly constant number of resolution elements or pixels in depth and a constant relative resolution. Better resolution cannot be obtained at high frequencies without putting the transducers closer to the tissue to be imaged, and an understanding of absorption is quite important to our use of scattered waves.

Attenuation is thus a factor that influences the image when using scattered waves. Briefly, the peak pressure loss in tissue varies somewhere between the first and second power of frequency, depending on the tissue. The sound speed that is associated with this attenuation characteristic is almost constant with frequency, which makes simple pulse-echo imaging possible. There is evidence that attenuation is a property of the proteins, probably through vibration of the side groups of amino acids. It has not been possible to build up the attenuation characteristic of proteins from those of the constituent amino acids. This frequency dependence of attenuation at low frequencies violates causality, and the attenuation for unit wavelength must decrease at low and high frequencies. The only protein whose behavior has been studied to confirm this is hemoglobin, the attenuation of which must decrease below 50 kHz and above 300-500 MHz.

The propagation and scattering properties of liquids are easier to measure than are those of solids, and so most work has been done on blood. These results must apply to solid tissues to some degree, but more work is needed. The dispersion of sound speed in blood is very small and is in accord with predictions. The scattering is due to the formed elements, and their compressibility and density contributions have been separated. The density difference between red cells and plasma has been found to contribute to a "relative motion" absorption term at high frequencies.

These results for blood used the popular discrete model for scattering, to-

**Suggested Citation:**"7 ULTRASONICS." National Research Council. 1996.

*Mathematics and Physics of Emerging Biomedical Imaging*. Washington, DC: The National Academies Press. doi: 10.17226/5066.

Page 127

gether with the fluid properties of density and compressibility. Investigations correlated tissue properties with their water, lipid, and protein content. Further advances could result from applying the continuous model or the Lame constants used in the theory of viscoelasticity. One problem in investigation is biological in nature: traditional models of molecular structure describe elements too small to affect scattering except in the large. At this level there are cells that are still much smaller than the wavelength, the current resolution-limiting factor. The structures that are of interest at this level (1000 to 100 microns) are not as well described in structure as are entire organs. There is considerable evidence that such structures are important and contribute to the measured anisotropy of propagation and scattering. Future correlation of scattering with structure at these size levels could yield new insights.

The investigation of tissue properties at the levels suggested above could be done by further development and application of diffraction tomography using laboratory techniques. The future clinical application of these techniques could follow but would not be the initial reason for development. Because of the wavelength-based resolution limits of even experimental measuring systems, only the effective scattering cross section, or function averages over the resolution cell, can be measured. Complete computer-based algorithms for making such measurements corrected for system factors have been developed.

#### 7.4 Ultrasonic Tomography

The fundamental groundwork for transmission tomography came from the mathematician Johann Radon in 1920. The inverse scattering problem in ultrasound, however, has not been solved, although accurate iteration methods have been developed. The problem can be broken into transmission tomography and reflection tomography, and this section deals only with the former. An approach using the Rytov approximation was developed by physicists Iwata and Nagata, and mathematicians have developed some methods using novel bases (see, for instance, the paper by Ball, Johnson, and Stenger in the suggested reading). Optical approaches have produced several ideas such as back propagation and diffraction tomography. Perturbation theory has been used as well (for instance, in the paper by Mueller, Kaveh, and Wade listed below).

Ultrasonic tomography has been applied to breast imaging, and a com-

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#### 7.5 Research Opportunities

Many fruitful areas exist in ultrasonic imaging for further mathematical and physics research. The following is a list of areas of research that could produce a large impact on the field.

· Development of effective wideband models that include the physical acoustical properties of the tissues in which the beam is propagating, such as absorption and aberration-inducing effects. This is effectively the forward wave propagation problem.

· Further study of the dielectric and ferroelectric properties of materials, with particular emphasis on the development of new ferroelectrics with high dielectric constants, such as the relaxer ferroelectrics; of ferroelectric functions; and of the role of grain and domain boundaries in the piezo- and ferroelectric properties of materials.

· Development of four-dimensional finite-element models of single transducer elements and multielement arrays.

· Development of superbroadband transducers, array modeling, and effective modeling for multimode vibrations of transducer elements.

· Investigation of theories and development of faster beam-forming algorithms that will be flexible enough to allow adaptive corrections for phase and amplitude distortions attributable to both tissue and system effects. Development of algorithms that ensure that diagnostic information can be obtained from any one patient regardless of intervening tissue components.

· Development of statistical and/or adaptive methods to preserve image detail associated with high-level echoes (such as those that occur when surgical tools interact with tissue) and sharp boundaries around vessels and ducts while improving contrast detail associated with low-contrast or small objects. Development of contrast enhancement routines based on knowledge of underlying tissue types (both healthy and diseased) and their interaction with ultrasound. Investigation of genuine contrast improvement procedures through tissue characterization/parametric imaging (including texture analysis, tissue elasticity, and soon).

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· Development of algorithms based on known differentiation between tissue types and healthy and diseased states.

· Improvement of methods for three-dimensional imaging capabilities. Development of better methods for displaying three-dimensional flow data in a format that is easily comprehensible. Development of image processing routines that overcome the specific problems of three-dimensional ultrasound (e.g., acquisition issues, coherent image formation (speckle), inhomogeneous media, non-transmission modes) and thus allow optimal utilization of existing three-dimensional reconstruction techniques. Development of methods for region segmentation as a preprocessing step to measurements of both area and volume, and application of three-dimensional image data display methods and automatic methods for measuring areas and volumes.

· Development of future imaging systems that use scattered waves more effectively. The present pulse-echo systems use a fractional bandwidth greater than 50%. Because the waves are basically oscillatory and coherent, the images show "speckle," the multiplicative random noise noted in laser images. If a wider bandwidth could be obtained by extending the response to low frequencies, a speckle-free system could result.

· Development of physics for the inverse problem to include the complete viscoelastic wave equation and its solutions for both transmission and reflection tomography.

#### 7.6 Suggested Reading

1. Ball, J., Johnson, S.A., and Stenger, F., Explicit inversion of the Helmholtz equation for ultra-sound insonification and spherical detection, in *Visualization and Characterization,* K.Y. Wang, ed., Plenum Press, New York, 1980, 451-461 (volume 9 in Acoustical Imaging series).

2. Borup, D.T., Johnson, S.A., Kim, W.W., and Berggren, M.J., Nonperturbative diffraction tomography via Gauss-Newton iteration applied to the scattering integral equation, *Ultrasonic Imaging* **14** (1992), 6985.

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3. Cady, W.G., *Piezoelectricity: An Introduction to the Theory and Applications of Electromechanical Phenomena in Crystals,* Dover Publications, New York, 1964.

4. Devaney, A.J., A filtered backpropagation algorithm for diffraction tomography, *Ultrasonic Imaging* **4** (1982), 336-350.

5. Feleppa, E.J., Lizzi, F.L., and Coleman, D.J., Ultrasonic analysis for ocular tumor characterization and therapy assessment, *News Physiol. Sci.* **3** (1988), 193-197.

6. Fredrich, W., Kaarmann, H., and Lerch, R., Finite element modelling of acoustic radiation from piezoelectric phased array antennas, *Proc. Ultrasonic Symp.* **1** (1990), 763.

7. Iwata, K., and Nagata, R., Calculation of refractive index distribution from interferograms using Born and Rytov's approximation, *Jpn. J. Appl. Physiol.* **14** (Suppl.) (1975), 379-383.

8. Kirchoff, G., Zur theorie der lichstrahlen, *Wiedemann Ann.* **18** (1883), 663.

9. Krimholtz, R., Leedom, D.A., and Matthei, G.L., New equivalent circuits for elementary piezoelectric transducers, * Elect. Lett.* **6** (1970), 398-399.

10. Leedom, D.A., Krimholtz, R., and Matthei, G.L., Equivalent circuits for transducers having arbitrary even or odd symmetry piezoelectric excitation, *IEEE Trans. Sonics Ultrason.* ** SU-18** (1971), 128-141.

11. Love, A.E.H., *Mathematical Theory of Elasticity,* Dover, New York, 1944.

12. Mason, W.P., *Piezoelectric Crystals and Their Application to Ultrasonics,* H.M. Stationery Office, London, 1950.

13. Morse, P.M., and Ingard, K.U., *Theoretical Acoustics,* reprinted by Princeton University Press, Princeton, N.J., 1986.

14. Mueller, R.K., Kaveh, M., and Wade, G., Reconstructive tomography and applications to ultrasonics, *Proc. IEEE* ** 67** (1979), 567-587.

15. Oberhettinger, F., On transient solutions of the "baffled piston" problem, *J. Res. Natl. Bur. Stand., Sect. B* ** 65B** (1961), 106.

**Suggested Citation:**"7 ULTRASONICS." National Research Council. 1996.

*Mathematics and Physics of Emerging Biomedical Imaging*. Washington, DC: The National Academies Press. doi: 10.17226/5066.

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16. O'Donnell, M., Applications of VLSI circuits to medical imaging, *Proc. IEEE* **76** (1988), 1106-1114.

17. Rayleigh, J.W.S., *The Theory of Sound,* MacMillan, New York, 1896.

18. Sommerfeld, A., Mathematische theorie der diffraction, * Math. Ann.* **47** (1896), 317.

19. Stepanishen, P.R., Transient radiation from pistons in an infinite planar baffle, *J. Acoust. Soc. Am.* **49** (1971), 1629-1638.