Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Geometrical Acoustics based Focusing Algorithm for Layered Media in Medical Ultrasound

Simon Hackl, Simon Hubmer, Ronny Ramlau

TL;DR

This work tackles ultrasound image aberrations arising from large-scale sound-speed variations within layered tissues. It introduces GOAT, a Geometrical Acoustics based Focusing Algorithm, which models layer boundaries with known $N$ layers and computes precise times of flight and focusing delays by solving a nonlinear boundary-value problem grounded in Snell's law. Theoretical contributions include existence and uniqueness results for the GOAT system and practical algorithms that outperform conventional homogeneous-medium focusing (HMFA) and compare favorably to the MINEO approach, as demonstrated by k-Wave simulations and phantom experiments. The results show that GOAT can significantly reduce ToF and delay errors, improving image quality in transducer-cover scenarios and fat-layer aberrations, with clear implications for real-time clinical ultrasound applications.

Abstract

Ultrasound imaging is a widely used, non-invasive diagnostic tool in modern medicine. A crucial assumption is a constant sound speed in the observed medium. For large scale sound speed variations, this assumption leads to blurred and distorted images. In this paper, we present a Geometrical Acoustics based Focusing Algorithm (GOAT) which is able to correct for these aberrations, given a known layered medium setting with continuously differentiable medium boundaries. Existence and uniqueness conditions for a solution to the underlying system of equations are given. Using numerical simulations, the precision of our method is evaluated. Finally, the resulting image quality improvements are demonstrated in a phantom-based experimental setup.

A Geometrical Acoustics based Focusing Algorithm for Layered Media in Medical Ultrasound

TL;DR

This work tackles ultrasound image aberrations arising from large-scale sound-speed variations within layered tissues. It introduces GOAT, a Geometrical Acoustics based Focusing Algorithm, which models layer boundaries with known layers and computes precise times of flight and focusing delays by solving a nonlinear boundary-value problem grounded in Snell's law. Theoretical contributions include existence and uniqueness results for the GOAT system and practical algorithms that outperform conventional homogeneous-medium focusing (HMFA) and compare favorably to the MINEO approach, as demonstrated by k-Wave simulations and phantom experiments. The results show that GOAT can significantly reduce ToF and delay errors, improving image quality in transducer-cover scenarios and fat-layer aberrations, with clear implications for real-time clinical ultrasound applications.

Abstract

Ultrasound imaging is a widely used, non-invasive diagnostic tool in modern medicine. A crucial assumption is a constant sound speed in the observed medium. For large scale sound speed variations, this assumption leads to blurred and distorted images. In this paper, we present a Geometrical Acoustics based Focusing Algorithm (GOAT) which is able to correct for these aberrations, given a known layered medium setting with continuously differentiable medium boundaries. Existence and uniqueness conditions for a solution to the underlying system of equations are given. Using numerical simulations, the precision of our method is evaluated. Finally, the resulting image quality improvements are demonstrated in a phantom-based experimental setup.

Paper Structure

This paper contains 29 sections, 8 theorems, 37 equations, 15 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Medical Ultrasound and Geometrical Acoustics
  3. The Basic Setting
  4. Standard Focusing Method used in Homogeneous Media
  5. Geometrical Acoustics
  6. An Adapted Geometrical Acoustics Model for Focusing in Multilayered Media
  7. Definitions and Problem Formulation
  8. Geometrical Acoustics based Focusing Algorithm (GOAT)
  9. Time of Flight calculation for given Medium Boundary Points.
  10. The Geometrical Acoustics based Focusing Algorithm (GOAT).
  11. Existence and Uniqueness Results
  12. Existence of a Solution in the GOAT System of Equations.
  13. Uniqueness Condition.
  14. Special Medium Boundary Shapes
  15. Straight medium boundary.
  16. ...and 14 more sections

Key Result

Lemma 3.4

For the coordinates of the points $P_{n-1}$, $P_n$ and $P_{n+1}$ and the angles $\alpha_n$, $\theta_n$ and $\theta'_n$, outlined in fig:mafa_geometric_analysis_of_refraction, the following equations hold:

Figures (15)

  • Figure 1.1: Left: Classic homogeneous medium focusing algorithm, Center: Aberrations in a multilayered medium, Right: Layer adapted aberration correction.
  • Figure 2.1: Left: Transmit Focused wave at multiple points in time, simulated using k-Wave. Right: The DAS-Algorithm: A single backscattered wave arrives at the transducer elements at different times, which is corrected using focusing delays before the summation of the transducer signals.
  • Figure 2.2: Geometrical acoustics is used in the following settings: Homogeneous medium (Left), Layered medium with straight boundary (Center left). Layered medium with $C^1$ medium boundary in case of transmission (Center right) or reflection (Right).
  • Figure 3.1: Left: The ToF $t_{m}$ from $P_{0,m}$ to $P_N$ is calculated as the sum of the individual ToFs $t_{1,m},\ldots,t_{N,m}$ inside the layered medium. Right: In GOAT, the medium boundary points are chosen s.t. the angles $\theta_n$ and $\theta_n'$ fulfill Snell's law of refraction (\ref{['eq:Snells_Law']}) in $P_n$.
  • Figure 3.2: Left: Geometric construction in \ref{['lemma:direct_equations_for_sin_theta']}. Right: Defining vectors $u_n$ and $t_n$ and finding their enclosed angle.
  • ...and 10 more figures

Theorems & Definitions (37)

  • Definition 2.1: Ultrasound Imaging Coordinate system szabo2013diagnostic
  • Definition 2.2: Focusing Delays
  • Remark
  • Remark
  • Remark
  • Definition 3.3
  • Remark
  • Lemma 3.4
  • proof
  • Remark
  • ...and 27 more