Time reversal for elastic scatterer location from acoustic recording

Highlights

• The data, recorded in the acoustic part of the medium (basically scalar quantities) contain enough information to identify the properties and locations of the inclusions located in the elastic part. This was made possible by introducing criteria allowing us construct images of these inclusions.

• Using the same procedure, one can differentiate between benign and malignant inclusions.

• This approach is fairly insensitive to noise in the data.

Abstract

The aim of this paper is to study the feasibility of time-reversal methods in a non homogeneous elastic medium, from data recorded in an acoustic medium. We aim to determine, from partial aperture boundary measurements, the presence and some physical properties of elastic unknown “inclusions”, i.e. not observable solid objects, located in the elastic medium. We first derive a variational formulation of the acousto-elastic problem, from which one constructs a time-dependent finite element method to solve the forward, and then, the time reversed problem. Several criteria, derived from the reverse time migration framework, are then proposed to construct images of the inclusions, and to determine their locations. The dependence of the approach to several parameters (aperture, number of sources, etc.) is also investigated. In particular, it is shown that one can differentiate between a benign and malignant close inclusions. This technique is fairly insensitive to noise in the data.

Introduction

Time reversal (TR) is a subject of very active research for over two decades, and was experimentally developed by M. Fink in 1992 in acoustics, showing very interesting features [34]. It remains an active subject of research, in the theoretical, physical and numerical points of view, with many applications such as medical imaging [41] or earthquake prevention, see [42] and references therein.

TR is a procedure based on the reversibility property of wave propagation phenomena in non-dissipative media, like in acoustic, electromagnetic or elastodynamics. A consequence of this property is that one can “time-reverse” developed signals, by letting them propagate back in time to the location of the source (or scatterers) that emitted them originally. This remarkable property allows proposing innovative methods, for instance in medical imaging [52], [51], [13], [28], seismic inversion [50] and active or passive detection problems [19]. The outstanding aspect of this experiment is the possibility to refocus in a very precise way the signal, without knowing the details of the source that emitted it. The first mathematical analysis can be found in [10] for a homogeneous medium and in [25], [16] for a random layered medium.

TR can also be used to identify scatterers, using that a scatterer illuminated by an incident wave behaves as a source of the reflected field. More generally, scatterers behave like secondary sources, so that TR causes reflected signals to refocus at the scatterer location.

TR can also be considered as an inverse problem, for which the solution is obtained without the need of an iterative process. Theoretically, solving a time reversed problem under ideal circumstances, should yield the exact solution. However, there is always the possibility that under some (realistic!) conditions, the time reverse process will fail. This may happen due to several reasons: measurement noise, availability of only partial information in space or in time, and lack of knowledge about the medium properties or the source.

TR is also related to imaging methods. Regarding elasticity imaging, we reader the reader to the works of Ammari and co-workers, see for instance [1], and more recently [2]. The authors consider TR approach, and explore elasticity imaging based on topological derivative-based imaging framework for detecting elastic inclusions, working in the time-harmonic regime.

As far as ultrasonic imaging is concerned, TR can be also related to ultrasonic testing, a family of non-destructive testing techniques, based on the propagation of ultrasonic waves in the examined object. On this vast topic, we refer the reader for instance to the total focusing method, an ultrasonic array technique used to synthetically focus at every point of a region of interest. But unlike TR, these techniques are mainly based on the extraction of the time-of-flight corresponding to the reflected echoes generated by the embedded defects. Some of these methods use TR [48], but not all of them, which are based on Fourier transform [43] for example, or more recently on wavelet transform [49].

Finally, one can also cite works on electromagnetic inverse problem: a first attempt to perform an electromagnetic TR experiment was reported in [44]. More recently, let us mention the recent book of Chen [20] and the references therein, that deals with theory, algorithms and applications of electromagnetic inverse scattering problems. Among the classical methods of inverse scattering, the author proposes a chapter that deals with TR imaging, which can be useful to link the different domains of this technique.

From a computational point of view, TR involves advancing the numerical solution of the relevant wave problem “backward in time”. The most direct use of the refocusing property allows one to solve the inverse problem of finding the location of a source when remote measurements of the wave field are given. Frequent applications are in seismology, for locating the epicenter of an earthquake from measurements taken on the ground [35], [45].

Here, we are concerned with the refocusing on scatterers of the diffracted wave to obtain elastic properties of these scatterers. Detection of elastic properties of objects inside a given “noisy” medium, by ultrasound waves can have a lot of applications, such as medical imaging, material detection inside a rock, see [38], [21] and references therein.

As usual for inverse problems, it will require to define criteria, like cost functions in inverse problems [35], to measure or at least evaluate the quality of the final result, namely the refocusing of wave on the diffracting object we aim to recover.

The main goal of this article is to derive and analyze the TR approach for the time-dependent acousto-elastodynamics configuration. As described in [13], [31], the recording of reflected waves is carried out in the fluid part of the domain, which means that only acoustic pressure is recorded. Our goal here is to deal with this partial data (acoustic pressure without knowledge of elastic displacements). The main applications of linear elastodynamics, or acousto-elastodynamics are generally structural engineering [15], where the main purpose is to examine beams that support buildings and bridges, or seismology, where the waves can be excited naturally by earthquakes or artificially using man made explosions, to explore the subsoil properties. In this paper, we will focus on another application: instead of waves that are excited on earth, we will consider waves that propagate through a tissue, for example, through breast tissue. The applications we have in mind are for instance the detection of tumors.

As far as applications are concerned, this approach is supposed to determine the elastic properties of obstacles in media that are assumed to mimic breast tissue, benign and malignant tumors [47], [31]. Applications that we have in mind are to help in the identification of breast tumors, using efficient ultrasound wave simulations. Such a technique could be used instead of intrusive diagnosis in cases where the usual diagnosis is not guaranteed (e.g. Mammography, human palpation, Ultrasound, etc.).

For these reasons, mathematical models and numerical simulations will mimic the process of sending ultrasound waves through a breast tissue according to (partially) known mechanical properties supplemented with noise. Then, they will try to detect cancerous tumors by the reflected waves, in case of partial information. Hence in this article,¹ in order to be closer to what happens in many real cases as in medical imaging and other fields, we do not assume that the line of receivers encloses the area of interest Ω. We rather consider that the aperture is reduced. Then, we will apply the method to identify an “inclusion” or to differentiate between a single inclusion and two close inclusions with different elastic properties, that could correspond to benign and malignant breast tumors. Typically, a benign tumor corresponds to normal breast tissues, with a Young modulus between 1 and 70 KPa, whereas malignant tumors have a Young modulus varying from 15 to 500 KPa (see for instance [33]). Note also that, as pointed out for instance in [5], the method does not require a priori knowledge of the physical properties of the inclusions.

The paper is organized as follows: we present in Section 1, the forward problem, which is practically used only for numerically generating the measurement data (to be inverted). In Section 2, we describe the process of time reversal and the way it is derived for the acousto-elastic problem. In Section 3, we describe the cost functions used to evaluate the quality of identification. We then present numerical experiments and results, with a particular focus on breast cancer detection applications. A conclusion is drawn at the end of the paper.