NUMEV Seminar “Acoustic Tomography: Context, Assessment, Prospects, and Experimental Research Tools”

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
Context, assessment, prospects, and experimental research tools

Philippe Lasaygues
Aix Marseille Univ, CNRS, Centrale Marseille,
LMA UMR 7031, Marseille, France

The objective of tomography is to produce cross-sectional images of objects. X-ray tomography (or CT scanning), the gold standard in this field, achieves this result using X-rays. Acoustic tomography attempts to do the same using acoustic waves: either ultrasonic waves in the biomedical field and in non-destructive testing, or sound waves or even infrasound waves (depending on the scales of interest) in the field of geophysics (Lefebvre et al. 2009). To do this, acoustic sounding signals (usually broadband pulses) are sent to the object to be imaged from transmitters located at a number of points outside the object. the field transmitted or diffracted by the object is captured using transducers also placed outside the object, and the data is inverted using ad hoc algorithms.

But what is the difference between ultrasound (long known as echotomography) in the medical and geophysical fields, and acoustic tomography? The key difference lies in how the problem is formulated, in terms of inverse problems in tomography, and in terms of physical wave focusing in ultrasound and synthetic focusing in seismic reflection. The methods used are mathematical and numerical in inverse problems and synthetic focusing (a technique that has also been successfully transposed from geophysics to non-destructive ultrasonic testing) (Lefebvre 1994), and physical (lenses) and/or electronic (antenna processing) in ultrasound. Ultimately, it is shown that the difference between synthetic focusing and tomography is quite subtle: the former is in fact only an approximate, degraded but often sufficient version of the latter.

In tomography, a distinction is made between transmission tomography (transmitter and receiver on either side of the object) and reflection and diffraction tomography (transmitter and receiver on the same side of the object) (Lefebvre et al. 2009). Both are based, at least in their basic formalism, on the assumption of low heterogeneity of the object and low acoustic impedance contrast with respect to the host medium, so that a straight-ray approximation can be used for transmission modeling and a Born approximation for diffraction modeling. Such an approximation makes perfect sense in the biomedical field for soft tissues, which are known to consist mainly 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 ultrasonic tomography has been expanding to include musculoskeletal tissues of the lower limbs (arms) and upper limbs (legs) containing one or two long bones (Lasaygues et al. 2022). The linear approximations used until now for soft tissue examination 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 by non-linear inversion approaches, and ultrasound tomography imaging is feasible (Doveri et al. 2021), even going as far as parameterizing maps by associating a gray level 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 basics of acoustic propagation and diffraction, associated electroacoustics (transducers, acoustic field, focusing), and ultrasound and tomographic imaging. The basic mathematical formulations will be reviewed and developed to move from imaging low-contrast media, such as fibrous breast tissue, to higher-contrast media, such as the musculoskeletal tissue of children.

“Ultrasound Computed Tomography using linear and non-linear inversion schemes
From soft (breast) to hard (pediatric bones) tissue imaging”

Philippe Lasaygues
Laboratoire de Mécanique et d’Acoustique -Laboratory of Mechanics and Acoustics
LMA – CNRS / Aix Marseille Univ / Centrale Méditerranée
Aix Marseille Univ, CNRS, Centrale Marseille, LMA UMR 7031, Marseille, France

This conference presents the theoretical, numerical, and experimental frameworks for Ultrasound Computed Tomography (USCT) for soft and hard tissue imaging. The difficulties raised by both applications are somewhat different. In soft tissues, the very small fluctuations to be quantified suffer from their very low impedance contrast values. This poor echogenic index generally induces low detection probability, for example in the case of large diffuse masses. In hard imaging, the difficulties arise from the very high impedance contrast that alters the propagation of the ultrasonic waves. Solutions consist in optimally assessing these non-linear effects in an iterative approach aiming at local linearization. The linear USCT method based on the use of the first-order Born approximation, applied to the case of a homogeneous and constant background, is described. The unknown object function, which is assumed to be weakly heterogeneous, is linearly related to the field measured via a tool called the Elliptical Fourier transform. The inverse problem is related to an extension of the Filtered Back-Projection (FBP) algorithm. This technique is suitable for soft tissue inspection such as breast, 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 contrasted scatterers such as bones.

When the problem can be reduced to the study of a fluid-like cavity buried 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 experimental costs involved (multiple iterative experiments) [1]. We have therefore suggested a purely numerical non-linear inversion algorithm, and the minimization procedure between the full recorded and simulated data is solved using a conjugate gradient method mainly developed in the field of non-destructive testing or an efficient quasi-Newton technique mainly developed in seismology (full waveform imaging method).

An overview of the performance and limitations of these tomography methods applied to breast and pediatric musculoskeletal imaging problems 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 French Acoustics Congress. 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. 2021 Oct 9;11(20):9368.

Lasaygues P, Espinosa L, Bernard S, Petit P, Guillermin R. Ultrasound Computed Tomography. In: Laugier P, Grimal Q, editors. Bone Quant. Ultrasound New Horiz. 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 application of Ultrasonic Reflection

Tomography. Inverse Problems 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, editors. Mater. Acoust. Handb. [Internet]. London, UK: ISTE; 2009 [cited 2016 Apr 3]. 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 inspection.

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

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