OPTIMIZED BEAMFORMING AND LIMITED ANGLE TOMOGRAPHY OF THE COMPRESSED BREAST by Fong Ming Hooi A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Biomedical Engineering) in The University of Michigan 2012 Doctoral Committee: Professor Paul L. Carson, Chair Associate Professor Jeff Fessler Assistant Research Professor Oliver Kripfgans Assistant Professor Zhen Xu Chief Technologist Kai Thomenius, General Electric Co. Brick walls are there for a reason. The brick walls are not there to keep us out. The brick walls are there to give us a chance to show how badly we want something. Because the brick walls are there to stop the people who don’t want it badly enough. Randy Pausch (1960-2008) (cid:13)c Fong Ming Hooi All rights reserved 2012 Dedicated to my family, especially Mom and Dad, who have offered me unconditional love and support, and for the one who stands by me every step of the way, Dan. ii ACKNOWLEDGMENTS I would like to first acknowledge my advisor, Paul Carson, for all his advice and guidance along my graduate school career. Without him, this dissertation would not have been possible, in more ways than one. I would also like to thank my committee members, Jeff Fessler, Oliver Kripfgans, Zhen Xu, and Kai Thomenius, for their valuable input on my research topic and insight into improving my dissertation. I thank the members of BRS, old and new, for their help in setting up various experiments and hunting down lab equipment, which was a monumental task in itself at times. Special thanks goes out to Fouzaan Zafar, who has spent countless hours debugging hardware with me to get speed of sound data acquisitions going, Oliver again, in so many ways, from debugging experimental setups to generously providing access to hausboot to run my massive computations. I owe Sumedha Sinha and Takashi Kozai my sanity, for their friendship kept me going and reminded me to relax and leave the apartment (from time to time). Last, but not least, I have to thank Sacha Verweij for his tremendous help and support within the last year, offering relevant advice that affects not only, but transcends beyond, this thesis. iii TABLE OF CONTENTS DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . iii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix CHAPTER I. Breast Cancer Imaging . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Background and Significance . . . . . . . . . . . . . . . . . . 1 1.2 Specific Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 II. Optimized Beamforming with 2D Reconfigurable Arrays . . 20 2.1 Benefits of Annular Arrays . . . . . . . . . . . . . . . . . . . 20 2.2 Constructing Beam with Extended Focus . . . . . . . . . . . 21 2.2.1 Transmitted Beam Optimization . . . . . . . . . . . 21 2.2.2 Comparison with purely spherical beams . . . . . . 26 2.2.3 Beam Steering . . . . . . . . . . . . . . . . . . . . . 26 2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3.1 Beam Optimization . . . . . . . . . . . . . . . . . . 29 2.3.2 Comparison with Purely Spherical Beams . . . . . . 33 2.3.3 Beam Steering . . . . . . . . . . . . . . . . . . . . . 33 2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.4.1 Beam Optimization . . . . . . . . . . . . . . . . . . 36 2.4.2 Comparison with Purely Spherical Beams . . . . . . 38 2.4.3 Beam Steering . . . . . . . . . . . . . . . . . . . . . 42 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 III. Speed of Sound Imaging using First Arrival Traveltimes . . . 48 3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 Inverse Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.3.1 Eikonal Forward Model . . . . . . . . . . . . . . . . 54 3.3.2 Cost Function Minimization via Conjugate Gradient Updates . . . . . . . . . . . . . . . . . . . . . . . . 55 iv 3.3.3 Choice of Covariance Matrices . . . . . . . . . . . . 55 3.3.4 Experimental Setup . . . . . . . . . . . . . . . . . . 57 3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.4.1 Simulated Speed of Sound Reconstructions . . . . . 60 3.4.2 Speed of Sound Reconstructions with Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 IV. Attenuation Imaging . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.2 Inverse Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.2.1 Generating Attenuation Data . . . . . . . . . . . . . 82 4.2.2 Covariance Matrices . . . . . . . . . . . . . . . . . . 83 4.2.3 Full Wave Model . . . . . . . . . . . . . . . . . . . 84 4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.3.1 Forward Wave Propagation Model . . . . . . . . . . 85 4.3.2 Experimental Data . . . . . . . . . . . . . . . . . . 85 4.3.3 Reconstruction . . . . . . . . . . . . . . . . . . . . . 87 4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 V. Reconstruction Algorithm Refinements for Grayscale, Speed of Sound, and Attenuation Imaging . . . . . . . . . . . . . . . . 101 5.1 Summary of Developed Algorithms . . . . . . . . . . . . . . . 101 5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 v LIST OF FIGURES Figure 1.1 Acoustic properties of normal tissues and masses [56]. . . . . . . . . 3 1.2 (Left) Original reflection image and (right) refraction corrected imag- ing using SOS [56]. Image boundaries are less distorted and image has overall improved resolution. . . . . . . . . . . . . . . . . . . . . 4 1.3 Schematic diagram of iterative process of inverse problems. The num- ber of iterations required depends on the initial guess of the medium. A reasonable guess is required for convergence lest the algorithm falls into a local minimum. . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 Diagramofhybridfocusingandparameters. Across-sectionofthe2D reconfigurable array aperture implementing hybrid annular focusing is shown containing the central axis of the array. . . . . . . . . . . . 23 2.2 To illustrate aperture ring selection, a 128-ring aperture steered 45◦ is shown. Each ring is linked to a different channel input delay. Ring shape was determined based on quantization of time delays in order to minimize artifacts due to phase error. . . . . . . . . . . . . . . . 28 2.3 The axicon beam produced by the outer axicon ring aperture. The axicon central focus is defined as the spatial peak intensity of the beam, which is located in between the intersection with the central axis of the beams normal to the axicon aperture from its inner and outer edges. Changing r , r , and angle φ will affect the location of 1 2 the axicon central focus. . . . . . . . . . . . . . . . . . . . . . . . . 30 2.4 Minimization function and first energy moment averaged over beam and at focus for r =2 mm and r =3 mm aperture over various axicon 1 2 angles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.5 Minimization function varying axicon fraction A over different total aperture radii, r . The best value found for A was approximately 0.25. 32 2 2.6 PSF with 40 dB dynamic range and axial amplitude comparison for closest and farthest aperture. The corresponding central A-line is plotted on each side. The axial amplitude of the hybrid apertures shows that they outperform the spherical at the lower depths for each aperture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 vi 2.7 C-scan display on a 40 dB scale of the PSF of a beam steered at 45◦ using traditional rings (top) and rings with subelements selected by quantization (bottom). The large artifact on the top image due to phase delay error is removed. . . . . . . . . . . . . . . . . . . . . . . 35 2.8 Surface of minimization function for aperture at deepest focal zone is shown. The chosen aperture falls within the valley signifying at least a local minimum of our cost functional. . . . . . . . . . . . . . . . . 38 2.9 The A-line plots of the hybrid and spherical apertures are overlaid for the aperture of the first focal zone. At the lower depths, we see that the hybrid aperture is more sensitive when compared to the spherical aperture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.10 Point spread function and spherical void simulation comparison of spherical (left) and hybrid (right) apertures for a composite of the four focal zones over the entire 4.5 cm depth. The depth extremes of each of the four transmit focal zones of the spherical void simulations show higher sensitivity for the hybrid apertures. . . . . . . . . . . . 41 3.1 Schematic of inverse problem algorithm is shown. The iteration starts with an initial guess using a homogeneous medium, and ends when the problem has converged when the misfit error satisfies a chosen criterion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.2 Experimental setup with two ATL L7-4 linear arrays. The distance between both transducers is approximately 5 cm apart, which is the average thickness of a compressed breast. A worm rubber contrast is placed in the center and the tank is filled with water for experimental data collection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.3 Reconstructed image with regularization based on a priori informa- tion on data and model space. There are some striations noted in the image resulting from undersampling with respect to the reconstructed grid. The reconstructed SOS value reached approximately 1470 m/s. 62 3.4 Reconstructed image from simulation includes regularization via cor- relation between assumed homogeneous pixels. The reconstructed image of cylinder reaches approximately 1440 m/s. . . . . . . . . . 63 3.5 Images reconstructed from simulation with correlated homogeneous region with different correlation coefficient values. As the value in- creases, the reconstruction improves and limited aperture smearing artifact is diminished. . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.6 Cross-section through the simulated object when using different cor- relation coefficients. The speed of sound value obtained in the object monotonically approaches the correct value as we increase the value of the correlation coefficient in the values of Fig. 3.5 (0.00001 - Blue, 0.0001 - Magenta, 0.001 Black, 0.01 - Red). . . . . . . . . . . . . . . 65 3.7 Two simulated 6 mm diameter cylinders placed close to each other to demonstrate the PSF overlap. The speed of sound dips only to 1470 m/s and the cylinders can barely be distinguished from each other. 66 vii 3.8 Reconstructed image of the same data as Fig. 3.7 but with pixels correlated together as assumed homogeneous regions. The two cylin- ders can now be differentiated from each other, and the reconstructed value reaches approximately 1440 m/s. . . . . . . . . . . . . . . . . . 67 3.9 Experimental data was taken with 128 transmitters and 128 receivers placed in an opposed array geometry with a cylinder of 1406 m/s placed in the center. The contrast is recovered, and the diamond shaped point spread function is noticeable. Similar to simulation, the value within the contrasting cylinder dips to approximately 1440 m/s due to the smearing artifact. . . . . . . . . . . . . . . . . . . . . . . 70 3.10 A thick ray implementation of the path length matrix is demon- strated. The speed of sound image is smoother when compared to the pencil ray implementation. The value within the contrasting cylinder is similar, but the microvariations throughout the image are largely decreased. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.11 The homogeneous region was correlated together to aid the inversion. The reconstructed speed of sound value reached approximately 1400 m/s, which is more accurate than the inversion without regularization via correlated pixels. . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.1 RF data recorded when transmitting with the 64th element on one array. The experimental and simulated shots are very similar to each other. The most important data to extract from the model are the bandsoutsidethecenterportion(correspondingtothebeamtraveling by the edge of the cylinder) as the recorded energy is very small and greatly distort attenuation image reconstructions. . . . . . . . . . . 86 4.2 Raw signal energy data is shown, uncorrected. The large bands of low energy for the object data illustrate the diffraction and multipath phase cancellation of the wave at the edge of the low SOS cylinder. . 88 4.3 The energy ratio across the receivers when transmitting with the 64th element is shown. The dips in energy shown in experimental data (green) align closely with those found from simulation (blue). . . . 89 4.4 Thecorrectedattenuationdata(lowerimage)tobefedintotherecon- struction algorithm is illustrated. The low energy bands correspond- ing to the edge of the cylinder are removed, while the beams that traverse the center of the cylinder indicate a much higher attenuation than the uncorrected data. . . . . . . . . . . . . . . . . . . . . . . . 90 4.5 The resulting attenuation image using a weighted least squares model with a priori information is depicted for the uncorrected (top image) and corrected dataset (bottom). The cylinder is recovered with a high relative attenuation compared to the background of water when the dataset is corrected. When uncorrected, the resulting image shows only a strong silhouette of the contrasting cylinder with faint attenu- ation within the cylinder. . . . . . . . . . . . . . . . . . . . . . . . . 91 viii