NUMEV Seminar: “Acoustic Tomography: Background, Current Status, Future Prospects, and Experimental Research Tools”

The NUMEV Seminars are open to a wide audience of students and researchers from all disciplines who wish to learn more about the current research areas of the NUMEV-MIPS community (Mathematics, Computer Science, Physics, and Systems) or about opportunities to develop their skills and expertise.

Acoustic Tomography
Background, Current Status, Future Prospects, and Experimental Research Tools

Philippe Lasaygues
, Aix-Marseille University, CNRS, Centrale Marseille,
LMA UMR 7031, Marseille, France

The goal of tomography is to produce cross-sectional images of objects. X-ray tomography (or computed tomography), the gold standard in this field, achieves this by using X-rays. Acoustic tomography attempts to do the same using acoustic waves: either ultrasonic waves in the biomedical and nondestructive testing fields, or sound waves—or even infrasonic waves (depending on the scales of interest)—in the field of geophysics (Lefebvre et al. 2009). To do this, acoustic probing signals (generally broadband pulses) are transmitted toward the object to be imaged from transmitters positioned at a number of points on the object’s exterior; the field transmitted or diffracted by the object is detected using transducers also placed outside the object, and the data is inverted using ad hoc algorithms.

But what is the difference between ultrasound (long referred to as echotomography) in the medical and geophysical fields, and acoustic tomography? The key difference lies in how the problem is formulated: in tomography, it is framed as an inverse problem, whereas in ultrasound it is framed in terms of physical wave focusing, and in seismic reflection as synthetic focusing. As for the methods used, they are mathematical and numerical in the inverse problem and synthetic focusing (a technique that has, incidentally, been successfully adapted from geophysics to ultrasonic nondestructive testing) (Lefebvre 1994), and physical (lenses) and/or electronic (antenna processing) in ultrasound. Finally, we show that the difference between synthetic focusing and tomography is quite subtle: the former is in fact merely an approximate, degraded—but often sufficient—version of the latter.

In tomography, a distinction is made between transmission tomography (where the transmitter and receiver are on opposite sides of the object) and reflection and diffraction tomography (where the transmitter and receiver are on the same side of the object) (Lefebvre et al. 2009). Both are based, at least in terms of their basic formalism, on the assumption of low heterogeneity in the object and low acoustic impedance contrast relative to the host medium, so that a straight-line approximation can be used to model transmission and a Born approximation to model diffraction. Such an approximation is particularly relevant in the biomedical field for soft tissues, which are known to consist primarily of water; hence the large number of studies—dating back many years—in the field of biomedical ultrasound tomography, particularly for breast imaging in women (Lasaygues et al. 2002; Mensah et al. 2011).

For several years now, however, the scope of ultrasound imaging has been expanding to include the musculoskeletal tissues of the lower limbs (arms) and upper limbs (legs), which contain one or two long bones (Lasaygues et al. 2022). The linear approximations previously used for soft-tissue imaging are no longer valid and severely limit the use of tomography (as well as ultrasound). Nevertheless, in children, whose bones are still immature and cartilaginous, certain limitations can be overcome through nonlinear inversion approaches, and ultrasound tomography imaging is feasible (Doveri et al. 2021), and may even extend to the parameterization of maps by associating a grayscale value with one or two acoustic parameters, such as ultrasound velocity or attenuation (Doveri et al. 2022).

In this presentation, the introductory lecture will cover the fundamentals of acoustic propagation and diffraction, related electroacoustics (transducers, acoustic fields, focusing), and ultrasound and tomographic imaging. The basic mathematical formulations will be reviewed and expanded upon to address imaging of low-contrast media, such as fibrous breast tissue, as well as higher-contrast media, such as musculoskeletal tissues in children.

“Ultrasound Computed Tomography Using Linear and Nonlinear Inversion Schemes
From soft tissue (breast) to hard tissue (pediatric bones) imaging”

Philippe Lasaygues
Laboratory of Mechanics and Acoustics -Laboratory of Mechanics and Acoustics
LMA – CNRS / Aix-Marseille University / Centrale Méditerranée
Aix-Marseille University, CNRS, Centrale Marseille, LMA UMR 7031, Marseille, France

This conference presents the theoretical, numerical, and experimental frameworks for Ultrasound Computed Tomography (USCT) for imaging soft and hard tissues. The challenges posed by these two applications are somewhat different. In soft tissues, the very small fluctuations to be quantified are affected by their extremely low impedance contrast. This low echogenicity generally results in a low probability of detection, for example in the case of large, diffuse masses. In hard tissue imaging, the challenges stem from the very high impedance contrast, which alters the propagation of ultrasonic waves. Solutions involve optimally accounting for these nonlinear effects through an iterative approach aimed at local linearization. The linear USCT method, based on the first-order Born approximation and applied to a homogeneous and constant background, is described. The unknown object function, which is assumed to be weakly heterogeneous, is linearly related to the measured field via a tool known as the elliptical Fourier transform. The inverse problem is addressed using an extension of the Filtered Back-Projection (FBP) algorithm. This technique is suitable for inspecting soft tissues, such as breast tissue, where the probe is either in contact with the skin or located within a near-field distance when using a coupling device (water bag or water tank). The first-order Born USCT has some limitations when dealing with highly contrasting scatterers, such as bones.

When the problem can be reduced to the study of a fluid-like cavity embedded in an elastic cylinder surrounded by water or soft tissues, the Compound USCT is proposed as an extension of the classical USCT, by taking into account physical phenomena such as wave refraction.

The main limitation of the method is the high cost of the experiments involved (multiple iterative experiments) [1]. We have therefore proposed a purely numerical nonlinear inversion algorithm, and the minimization problem between the full recorded and simulated data is solved using a conjugate-gradient method—primarily developed in the field of nondestructive testing—or an efficient quasi-Newton technique—primarily developed in seismology (full waveform imaging method).

An overview of the performance and limitations of these tomography methods as applied to breast and pediatric musculoskeletal imaging is presented and discussed.

References

Doveri E, Baldisser J, Sabatier L, Long V, Espinosa L, Guillermin R, et al. Quantitative anatomical imaging by ultrasound diffraction tomography. 16th Congress of the French Acoustical Society. Marseille; 2022.

Doveri E, Sabatier L, Long V, Lasaygues P. Reflection-Mode Ultrasound Computed Tomography Based on Wavelet Processing for High-Contrast Anatomical and Morphometric Imaging. Appl. Sci. Oct. 9, 2021;11(20):9368.

Lasaygues P, Espinosa L, Bernard S, Petit P, Guillermin R. Ultrasound Computed Tomography. In: Laugier P, Grimal Q, eds. Bone Quant. Ultrasound: New Horizons. P. Laugier & Q. Grimal. Springer

International Publishing; 2022. pp. 227–50.
Lasaygues P, Tanne D, Mensah S, Lefebvre JP. Circular Antenna for Breast Ultrasonic Diffraction

Tomography. Ultrasound. Imaging. July 2002;24(3):177–89.

Lefebvre, J.-P. Progress in Linear Inverse Scattering Imaging: NDE Applications of Ultrasonic Reflection

Tomography. Inverse Problems in Engineering Mechanics. Rotterdam/Brookfield: A.A. Balkema; 1994. pp. 371–5.

Lefebvre J-P, Lasaygues P, Mensah S. Acoustic Tomography, Ultrasonic Tomography. In: Bruneau M,

Potel C, eds. Mater. Acoust. Handb. [Internet]. London, UK: ISTE; 2009 [cited Apr. 3, 2016]. p. 887–

906. Available from: http://doi.wiley.com/10.1002/9780470611609.ch35

Mensah S, Rouyer J, Lasaygues P, Franceschini E. Ultrasound mammography for breast lobe examination.

IEEE; 2011 [cited Dec. 9, 2015]. pp. 1399–402. Available from:

http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6293561