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Methods for Forward and Inverse Scattering in Ultrasound Tomography

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Quantitative Ultrasound in Soft Tissues

Abstract

Ultrasonic computed tomography (UCT) is a potentially useful technique that has been explored for decades in the context of medical imaging. UCT can provide quantitative images of acoustical parameters such as speed of sound, attenuation, and density from measurements of pressure fields. Throughout the years, several algorithms that rely on different wave propagation models have been developed. In this chapter, the fundamentals of forward and inverse solvers for ultrasonic tomography will be described.

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Notes

  1. 1.

    The eikonal equation can alternatively be derived from Fermat’s principle, and therefore this principle can also be used for ray tracing. The interested reader can refer to acoustics textbooks such as Pierce (1989).

  2. 2.

    Equation 14.13 can actually be derived from direct first-order simplifications of the wave equation which are not presented here for brevity. The interested reader may refer to Jensen (1990).

  3. 3.

    This approach is in fact a synthetic aperture reconstruction method (Soumekh 1999) equivalent to the delay-and-sum algorithm. A commonly used variation is to perform the backpropagation operation using envelope-detected data, which results in a spatial compounding reconstruction method (Trahey et al. 1986).

  4. 4.

    Density information from dispersive media can also be isolated from \(O_\rho (k,\mathbf {r})\) profiles at different frequencies if the dispersion can be properly modeled as a function of \(\omega \).

  5. 5.

    In general, distortions may arise when reconstructing cross-sections of non-cylindrical objects using 2D data. The interested reader may refer to (Lavarello and Oelze 2009; Duncan et al. 2009).

  6. 6.

    See Sect. 14.5 for a discussion on the discretization of the wave equation.

  7. 7.

    Direct algebraic manipulation reveals that, for the linear-operator case (i.e., \(R_m(f) = \mathbf{R }_m \cdot \bar{\mathbf{f }}\)), (14.56) reduces to (14.52).

  8. 8.

    Like in the case of sound speed imaging with DBIM, frequency hopping can be used to improve the spatial resolution of density tomograms constructed with the T-matrix formulation.

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Authors and Affiliations

  1. Departamento de Ingeniería, Pontificia Universidad Católica del Perú, Av. Universitaria 1801, San Miguel Lima 32, Lima, Perú

    Roberto J. Lavarello

  2. Department of Electrical and Computer Engineering, University of Rochester, 601 Elmwood Avenue, UR Medical Center, Box 648, Rochester, NY, 14642-8648, USA

    Andrew J. Hesford

Authors
  1. Roberto J. Lavarello
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  2. Andrew J. Hesford
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Correspondence to Roberto J. Lavarello .

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Editors and Affiliations

  1. Biomedical Engineering, Riverside Research Institute F.L. Lizzi Center, New York, New York, USA

    Jonathan Mamou

  2. Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA

    Michael L. Oelze

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Lavarello, R.J., Hesford, A.J. (2013). Methods for Forward and Inverse Scattering in Ultrasound Tomography. In: Mamou, J., Oelze, M. (eds) Quantitative Ultrasound in Soft Tissues. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6952-6_14

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