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Proceedings of Spie PROCEEDINGS OF SPIE SPIEDigitalLibrary.org/conference-proceedings-of-spie Random field interferometry for medical ultrasound Ulrich, Ines Elisa, Boehm, Christian, Fichtner, Andreas Ines Elisa Ulrich, Christian Boehm, Andreas Fichtner, "Random field interferometry for medical ultrasound," Proc. SPIE 11319, Medical Imaging 2020: Ultrasonic Imaging and Tomography, 1131912 (16 March 2020); doi: 10.1117/12.2559852 Event: SPIE Medical Imaging, 2020, Houston, Texas, United States Downloaded From: https://www.spiedigitallibrary.org/conference-proceedings-of-spie on 25 Mar 2020 Terms of Use: https://www.spiedigitallibrary.org/terms-of-use Random field interferometry for medical ultrasound Ines Elisa Ulrich, Christian Boehm and Andreas Fichtner Department of Earth Sciences, ETH Z¨urich, Sonneggstrasse 5, CH-8092 Z¨urich, Switzerland ABSTRACT We present a novel approach to obtain time-of-flight measurements between transducer pairs in an Ultrasound computed tomography (USCT) scanner by applying the interferometry principle, which has been used success- fully in seismic imaging to recover the subsurface velocity structure from ambient noise recordings. To apply this approach to a USCT aperture, random wavefields are generated by activating the emitting transducers in a random sequence. By correlating the random signals recorded by the receiving transducers, we obtain an approximation of the Green's functions between all receiver pairs, where one is acting as a virtual source. This eliminates specific source imprints, and thus avoids the need for reference measurements and calibration. The retrieved Green's functions between any two measurement locations can then be used as new data to invert the sound speed map. On the basis of the cross-correlation travel times a ray-based time-of-flight tomography is developed and solved with an iterative least-squares method. As a proof of concept, the algorithm is tested on numerical breast phantoms in a synthetic 2D study. Keywords: Random field interferometry, ultrasound computed tomography, time-of-flight inversion 1. INTRODUCTION Ultrasound computed tomography (USCT) is frequently used for medical purposes to image soft tissue body parts, as for instance the breast. The property of interest is the speed of sound of the breast tissue, tissue density or attenuation. Commonly, malignant cell regions are denser and stiffer than benign regions, thus speed of sound maps image breast tissue.1{3 Breast cancer detection using USCT usually works with a collection of ultrasound scans that measure the pressure wavefield emitted by individual transducers. A state-of-the-art USCT acquisition process often requires a large number of emitter-receiver pairs to obtain a good coverage of the domain of interest and careful calibration of the emitting transducers using reference measurements in water.4{6 This is a time-consuming process during which patient movements can degrade the quality of the aquired data. Furthermore, calibration of emitters using reference measurements in water is often subject to strong assumptions, for instance, that the temperature of the water is uniform and constant, which are difficult to establish during real-life aquisition. In recent years, scientific contributions from geophysical research have proven to be a fruitful addition to the medical imaging community. Although the imaged medium in the geophysical application compared to the medical application seems considerably different, Pratt et al.7,8 have shown the possibilities arising from applying geophysical imaging methods to breast cancer screening. Further studies, extending the research to image other human body parts such as the brain9 or the limbs,10 have demonstrated the potential of ultrasound imaging. With this work, we propose to circumvent the aforementioned challenges of standard USCT data acquisition techniques by transferring yet another theory widely applied in seismic imaging, to breast imaging with USCT. This theory allows us to eliminate the need for calibration and individual source wavefields by implementing the random field interferometry principle in which active sources are substituted by virtual ones without specific imprints. A similar approach of evaluating tissue parameters has been studied by Gallot et al.11 and by Sebra et al.12 for elastography, where shear waves are used in contrast to pressure waves as considered in this study. In Earth science, random field interferometry has spawned new possibilities to image the interior of the Earth Further author information: (Send correspondence to I.E.U.) I.E.U.: E-mail: ines.ulrich@erdw.ethz.ch C.B.: E-mail: christian.boehm@erdw.ethz.ch A.F.: E-mail: andreas.fichtner@erdw.ethz.ch Medical Imaging 2020: Ultrasonic Imaging and Tomography, edited by Brett C. Byram, Nicole V. Ruiter, Proc. of SPIE Vol. 11319, 1131912 · © 2020 SPIE · CCC code: 1605-7422/20/$21 · doi: 10.1117/12.2559852 Proc. of SPIE Vol. 11319 1131912-1 Downloaded From: https://www.spiedigitallibrary.org/conference-proceedings-of-spie on 25 Mar 2020 Terms of Use: https://www.spiedigitallibrary.org/terms-of-use by using passive measurements of quasi-random wavefields,13, 14 which has inspired the term ambient noise interferometry. Instead of using the direct transmission data from one source to one receiver, an approximation of the Green's function between any two points can be obtained by cross-correlating recordings of a random wavefield at these two positions,15 resulting in deterministic signals that constrain the medium. Cross-correlation is a statistical method to extract the similarities of recordings at two locations as a function of time, suppressing all non-coherent parts in the two signals. Therefore, through constructive interference, the time of flight between two receivers is marked by a peak in the cross-correlation function of two displacement recordings of a random wavefield. The description of seismic and medical imaging problems is based on the same mathematical relations, which allows us to transfer ideas developed in seismic research and apply them in a medical context. In seismic imaging, one relies on ambient random wavefields, generated for example by ocean waves. Since there is no ambient random wavefield in the medical application, we generate a random wavefield actively by a random superposition of individual sources. As a proof of concept, we develop an algorithm that enables an active generation of a random wavefield and implements the cross-correlation method on the scale of the USCT setup to extract times of flight. We test the algorithm on a 2D numerical breast phantom, representing a coronal slice through the breast. Variations in the speed of sound inside the phantom are included to model regions with malignant cells. The implemented speed of sound values are based on real human soft tissue values reported in the literature1{3 with variations on the order of 5% with respect to the speed of sound of water. To evaluate the quality of the extracted cross-correlation travel times, a ray-based time-of-flight tomography is developed and solved with an iterative least-squares scheme. We show that the travel time shifts between the obtained cross-correlation travel times and the analytical solution for water are accurate enough such that the speed of sound distribution within the phantom can be reconstructed. 2. METHOD 2.1 Random field interferometry The propagation of acoustic energy in an inhomogeneous medium is described by the acoustic wave equation that is given in the frequency domain by !2 1 − p(x;!) − @ @ p(x;!) = f(x;!); (1) ρ(x)c2 xi ρ(x) xi where ! is the angular frequency, ρ is the density, c is the acoustic velocity and p(x) and f(x) are the space- frequency domain representations of the acoustic pressure and the source distribution. We use Einstein's sum- mation convention to sum over repeated lower case subscripts. Equation (1) states that the acoustic pressure at any point x inside the medium is directly connected to the speed of sound of the propagation medium. This is the basic argument for the following analysis on the cross-correlation function of pressure measurements from a random wavefield. Just as in medical imaging, seismology is often concerned with finding the velocity of a wave traveling through an unknown medium in order to deduce its properties. For convenience, we use the frequency-domain representation of the wavefields and the frequency-domain Green's function G(x; x0;!) in the following and omit the dependency on !. The Green's function is the solution to the acoustic wave equation (1) for a point source at location x0, therefore satisfying LG(x; x0) = δ(x − x0); (2) !2 1 where L is the forward wave operator collecting the terms − ρ(x)c2 (·)−@xi ρ(x) @xi (·) in equation (1). A relation between the pressure wavefield and its sources can be established by invoking the acoustic frequency-domain representation theorem16, 17 of the form Z p(x) = G(x; x0)f(x0)dx0: (3) D Proc. of SPIE Vol. 11319 1131912-2 Downloaded From: https://www.spiedigitallibrary.org/conference-proceedings-of-spie on 25 Mar 2020 Terms of Use: https://www.spiedigitallibrary.org/terms-of-use With the help of the Green's function G(x; x0), the pressure p recorded at position x can be computed for arbitrary sources f located inside the domain D of interest. Two sources at neighbouring positions x0 and x00 are assumed to be uncorrelated in time, such that f ∗(x0)f(x00) = S(x0)δ(x0 − x00); (4) where S(x0) is the spectral density of the sources and ∗ denotes complex conjugation.15 In the following, we refer to a wavefield that is generated by uncorrelated sources in the sense of equation (4) as a random wavefield. Using the assumption that the sources are uncorrelated, a representation theorem for the cross-correlation function between two pressure measurements p(xA) and p(xB) is given by Z ∗ ∗ C(xA; xB) = p(xA)p (xB) = G(xA; x)G (xB; x)S(x)dx: (5) D Here, the cross-correlation function C(xA; xB) models the forward wavefield between two measurement locations.
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