Iterative Polyenergetic Digital Tomosynthesis Reconstructions for Breast Cancer Screening Open Access

Abstract

In digital tomosynthesis imaging, multiple projections of an object are obtained along a small range of incident angles in order to reconstruct a pseudo 3D representation of the object. This technique is of relevant interest in breast cancer screening since it eliminates the problem of tissue superposition that reduces clinical performance in standard mammography. The challenge of this technique is that it is computationally and memory intensive, as it deals with millions of input pixels in order to produce a reconstruction composed of billions of voxels. Standard approaches to solve this large-scale inverse problem have relied on simplifying the physics of the image acquisition model by considering the x-ray beam to be monoenergetic, thus decreasing the number of degrees of freedom and the computational complexity of the solution. However, this approach has been shown to introduce beam hardening artifacts to the reconstructed volume. Beam hardening occurs when there is preferential absorption of low-energy photons from the x-ray by the object, thus changing the average energy of the x-ray beam.

This thesis presents an interdisciplinary collaboration to overcome the mathematical, computational, and physical constraints of standard reconstruction methods in digital tomosynthesis imaging. We begin by developing an accurate polyenergetic mathematical model for the image acquisition process and propose a stable numerical framework to iteratively solve the nonlinear inverse problem arising from this model. We provide an efficient and fast implementation of the volume reconstruction process that exploits the parallelism available on the GPU architecture. Under our framework, a full size clinical data set can be reconstructed in under five minutes. The implementation presented reduces storage and communication costs by implicitly storing operators and increasing kernel functionality. We show that our reconstructed volume has no beam hardening artifacts and has better image quality than standard reconstruction methods. Our reconstructions also provide a quantitative measure for each voxel of the volume, allowing the physician to see and measure the contrast between materials present inside the breast. The research presented in this thesis shows that large-scale medical image reconstructions can be done using physically accurate models by effectively harnessing the multi-threading power of GPUs.

Table of Contents

Contents
List of Figures xiii
List of Tables xvi
1 Introduction 2
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Inverse Problems and Regularization . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Linear Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Separable Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.3 Nonlinear Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . 15
1.3 Outline of Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 The Physics of X-ray Imaging and Digital Tomosynthesis 22
2.1 Producing a Radiograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.1 X-ray Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.1.2 Interaction Between X-rays and Matter . . . . . . . . . . . . . . . . 26
2.1.3 The Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2 Computed Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3 Digital Tomosynthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3 The Mathematics of Digital Tomosynthesis 35
3.1 Modeling the Source-Detector Interaction . . . . . . . . . . . . . . . . . . . 36
3.2 Beer's Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 The Radon Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4 Standard Reconstruction Methods . . . . . . . . . . . . . . . . . . . . . . . 42
3.4.1 Backprojection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4.2 Filtered Backprojection . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.4.3 Maximum Likelihood Estimation Maximization Method (MLEM) . . 46
3.4.4 Algebraic Reconstruction Techniques (ART) . . . . . . . . . . . . . . 48
3.5 Beam Hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4 The Linear Polyenergetic Model for Attenuation 52
4.1 Building the Forward Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 Statistical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3 The Iterative Reconstruction Framework . . . . . . . . . . . . . . . . . . . . 56
4.3.1 Gradient Descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3.2 Newton Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5 The Quadratic Polyenergetic Model for Attenuation 62
5.1 Building the Forward Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.1.1 Finding a Quadratic Fit . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.1.2 Finding an Improved Quadratic Fit . . . . . . . . . . . . . . . . . . 65
5.1.3 The Forward Projection Model . . . . . . . . . . . . . . . . . . . . . 69
5.2 Statistical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.3 The Iterative Reconstruction Framework . . . . . . . . . . . . . . . . . . . . 71
5.3.1 Gradient Descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.3.2 Newton Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6 The Polyenergetic Model for the Attenuation of Multiple Materials 80
6.1 Building the Forward Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.2 Statistical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.3 The Iterative Reconstruction Framework . . . . . . . . . . . . . . . . . . . . 86
6.3.1 Gradient Descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.3.2 Newton Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7 A Simplied Polyenergetic Multimaterial Model 94
7.1 Building the Forward Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.2 Statistical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.3 The Iterative Reconstruction Framework . . . . . . . . . . . . . . . . . . . . 99
7.3.1 Gradient Descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.3.2 Newton Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
8 Implementation Considerations 107
8.1 Computational Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
8.2 GPU Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
8.3 Programming using OpenCL . . . . . . . . . . . . . . . . . . . . . . . . . . 111
9 Trace Functions 114
9.1 Siddon's Algorithm for the Exact Radiological Path . . . . . . . . . . . . . 114
9.2 A Task Parallel Version of Siddon's Raytrace . . . . . . . . . . . . . . . . . 121
9.3 Raytracing in Digital Tomosynthesis . . . . . . . . . . . . . . . . . . . . . . 124
9.3.1 Implementation Considerations for Siddon's Algorithm . . . . . . . . 125
9.3.2 Raytracing for Digital Tomosynthesis on a GPU . . . . . . . . . . . 126
9.3.3 Our Modied Raytrace vs. a Direct Implementation of Siddon's Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
9.4 The Backprojection Operation . . . . . . . . . . . . . . . . . . . . . . . . . 133
9.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
10 Computational Design 136
10.1 Serial Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
10.2 Matrix Product Optimizations . . . . . . . . . . . . . . . . . . . . . . . . . 139
10.3 Kernel Fusion Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 144
10.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
11 Numerical Results 149
11.1 Homogeneous Phantoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
11.1.1 Homogeneous Phantom with Glandular Tissue Inserts . . . . . . . . 151
11.1.2 Homogeneous Phantom with Micro-calcication Inserts . . . . . . . 156
11.2 Heterogeneous Phantoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
11.2.1 Heterogeneous Phantom Glandular and Adipose Tissue Swirl . . . . 160
11.2.2 Heterogeneous Phantom with Glandular Tissue Inserts . . . . . . . . 162
11.2.3 Heterogeneous Phantom with Micro-calcication Inserts . . . . . . . 166
11.3 Multimaterial Model Results . . . . . . . . . . . . . . . . . . . . . . . . . . 169
11.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
12 Conclusion 173
Bibliography 176

About this Dissertation

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School	Laney Graduate School
Department	Math and Computer Science
Degree	PhD
Submission	Dissertation
Language	English
Research Field	Mathematics Health Sciences, Radiology Computer Science
Keyword	Iterative image reconstruction Polyenergetic tomosynthesis Digital tomosynthesis Tomography Gradient descent Maximum likelihood GPU Computing
Committee Chair / Thesis Advisor	Nagy, James, Emory University
Committee Members	Veneziani, Alessandro, Emory University Benzi, Michele, Emory University

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