Reflection Matrix Approach for Quantitative Imaging of Scattering Media

We present a physically intuitive matrix approach for wave imaging and characterization in scattering media. The experimental proof of concept is performed with ultrasonic waves, but this approach can be applied to any field of wave physics for which multielement technology is available. The concept is that focused beam forming enables the synthesis, in transmit and receive, of an array of virtual transducers which map the entire medium to be imaged. The interelement responses of this virtual array form a focused reflection matrix from which spatial maps of various characteristics of the propagating wave can be retrieved. Here we demonstrate (i) a local focusing criterion that enables the image quality and the wave velocity to be evaluated everywhere inside the medium, including in random speckle, and (ii) a highly resolved spatial mapping of the prevalence of multiple scattering, which constitutes a new and unique contrast for ultrasonic imaging. The approach is demonstrated for a controllable phantom system and for in vivo imaging of the human abdomen. More generally, this matrix approach opens an original and powerful route for quantitative imaging in wave physics.

Popular Summary

In medical ultrasound imaging, transducer arrays (collections of small independent sources and receivers) are placed on the outside of the body, with the aim of gathering information about the structure deep inside. Ideally, one would place the sources and receivers closer or inside the organ of interest. However, this is hardly possible for noninvasive imaging. Here, we show that data collected by a commercial medical imaging apparatus can be manipulated to create tiny “virtual” acoustic sources and receivers anywhere inside the human body.

This technique enables us to observe how acoustic waves travel between various areas in a human organ, which we demonstrate for in vivo measurements on the liver. Because the path and speed of acoustic waves is directly related to the structure through which they travel, these observations enable the creation of new 2D images of various organ properties related to their density and composition. These images, in turn, can give valuable insight into the health of the organ tissue.

More broadly, our approach can be extended to any kind of wave that can be controlled by a multielement array. Applications range from biomedical diagnosis in optical imaging to crack detection in industrial materials or monitoring volcanoes and fault zones in geophysics.

Figure 1

Principle of the focused reflection matrix approach. (a) Sketch of the experimental setup for the acquisition of ${\mathbf{R}}_{\mathbf{u}\theta }(t)$. An ultrasonic transducer array is in direct contact with a layer of bovine tissue placed on the top of a tissue-mimicking phantom. ${\mathbf{R}}_{\mathbf{u}\theta }(t)$ is acquired by recording the time-dependent reflected field at each transducer element ${\mathbf{u}}_{\mathbf{out}}$, for each plane-wave illumination ${\theta }_{\mathrm{in}}$. (b) Each pixel of a conventional image results from confocal beam forming applied to ${\mathbf{R}}_{\mathbf{u}\theta }$ in emission and reception. (c) Matrix imaging consists of performing focused beam forming to probe distinct points ${\mathbf{r}}_{\mathbf{in}}$ and ${\mathbf{r}}_{\mathbf{out}}$ in emission and reception. The set of impulse responses between such virtual transducers form a focused reflection matrix ${\mathbf{R}}_{xx}$ at each depth.

Figure 2

Matrix approach applied to wave velocity mapping of the bovine tissue and phantom system described in Fig. 1. (a),(b) The matrix ${\mathbf{R}}_{xx}$ is displayed at depth $z=18$ and 30 mm, respectively, assuming a homogeneous wave velocity model ($c=1542\mathrm{m}{\mathrm{s}}^{-1}$). The local image resolution $w$ is extracted from each antidiagonal of ${\mathbf{R}}_{xx}$. (c) Corresponding ultrasound image built from the confocal elements of ${\mathbf{R}}_{xx}$. (d) The optimized wave velocities are displayed versus depth for the bovine tissue, found using a homogeneous model (blue open symbols), and the phantom, found using a bilayer model (red solid symbols). (e) The focusing criterion $F$, averaged over the depth ranges [17.4, 19.4] (blue circles) and [30, 32] mm (red disks), is displayed versus the wave velocity hypothesis $c$. (f) Ultrasound image built from the confocal elements of ${\mathbf{R}}_{xx}$ using the optimized wave velocity model ($c_t=1573\mathrm{m}/\mathrm{s}$, $c_p=1546\mathrm{m}/\mathrm{s}$). (g),(h) Corresponding reflection matrices ${\mathbf{R}}_{xx}$ are shown for depths $z=18\mathrm{mm}$ and $z=30\mathrm{mm}$.

Figure 3

Maps of local focusing parameter $F(\mathbf{r})$ for the bovine tissue and phantom system, superimposed over the echographic image in Fig. 2. (a) The homogeneous model with a constant speed of sound ($c=1542\mathrm{m}/\mathrm{s}$) results in a poor quality of focus in some areas. (b) The two-layer model used to construct ${\mathbf{R}}_{\mathbf{r}\mathbf{r}}$ results in close to ideal focus quality throughout the image.

Figure 4

(a) The modulus of matrix ${\mathbf{R}}_{xx}$ is shown at depth $z=44.5\mathrm{mm}$. Signals from multiple scattering can be seen at elements far from the diagonal. (b) Sketch of multiple-scattering paths (red or blue path) involved in the matrix ${\mathbf{R}}_{xx}$. The constructive interference between reciprocal paths (red and blue paths) occurs only when $|{\mathbf{r}}_{\mathbf{out}}-{\mathbf{r}}_{\mathbf{in}}|<\delta x$ (CBS).

Figure 5

Local mapping of multiple scattering. Normalized mean intensity profiles for (a) the focused basis $I_{\mathrm{av}}(\mathbf{r},\mathrm{\Delta }x)$ and (b) the far field $I_{\mathrm{av}}(\mathbf{r},\mathrm{\Delta }\theta )$ are displayed for the different areas highlighted in (c) the corresponding ultrasound image (formed using the two-layer model in Sec. 3a). Maps of multiple-scattering rates (d) $\rho (\mathbf{r})$ [Eq. (22)] and (e) $\epsilon (\mathbf{r})$ [Eq. (23)] are shown, superimposed on the ultrasound image. Some structures in the bovine tissue layer (indicated by white arrows) cause a significant amount of multiple scattering to occur behind them (outlined by dashed lines). The solid lines outline the image area that suffers from artifacts due to the double-reflection event between the probe and the bovine tissue–phantom interface.

Figure 6

In vivo imaging of a human liver using the focused reflection matrix approach. (a) Ultrasound image calculated using a four-layer model [Eq. (10)]. The four layers can be identified as skin, fat, muscle, and liver tissue. (b) The speed of sound is calculated for each depth via the optimization of $F$ (Sec. 3a). (c) The ratio quantity $\epsilon (\mathbf{r})$ is shown in the same region of interest as (a).