Algorithms for Polyenergetic Breast Tomosynthesis Image - PowerPoint PPT Presentation

Motivation Mathematical Formulation Reconstruction and Results Final Remarks Algorithms for Polyenergetic Breast Tomosynthesis Image Reconstruction Julianne Chung ∗ , James Nagy † , Ioannis Sechopoulos ‡ ∗ Department of Computer Science University of Maryland, College Park, MD † Department of Mathematics and Computer Science Emory University, Atlanta, GA ‡ Department of Radiology and Winship Cancer Institute Emory University, Atlanta, GA Julianne Chung, James Nagy, Ioannis Sechopoulos Polyenergetic Tomosynthesis Reconstruction

Motivation Mathematical Formulation Reconstruction and Results Final Remarks Outline Motivation 1 Mathematical Formulation 2 Reconstruction and Results 3 Final Remarks 4 Julianne Chung, James Nagy, Ioannis Sechopoulos Polyenergetic Tomosynthesis Reconstruction

Motivation Mathematical Formulation An Important Application Reconstruction and Results Polyenergetic Model, Monoenergetic Algorithm Final Remarks Outline Motivation 1 Mathematical Formulation 2 Reconstruction and Results 3 Final Remarks 4 Julianne Chung, James Nagy, Ioannis Sechopoulos Polyenergetic Tomosynthesis Reconstruction

Motivation Mathematical Formulation An Important Application Reconstruction and Results Polyenergetic Model, Monoenergetic Algorithm Final Remarks Breast Cancer Statistics 2nd most common type of cancer worldwide > 1 million women each year diagnosed with breast cancer ∼ 465,000 die each year from the disease Julianne Chung, James Nagy, Ioannis Sechopoulos Polyenergetic Tomosynthesis Reconstruction

Motivation Mathematical Formulation An Important Application Reconstruction and Results Polyenergetic Model, Monoenergetic Algorithm Final Remarks Digital Tomosynthesis X-ray Mammography Digital Tomosynthesis Computed Tomography Julianne Chung, James Nagy, Ioannis Sechopoulos Polyenergetic Tomosynthesis Reconstruction

Motivation Mathematical Formulation An Important Application Reconstruction and Results Polyenergetic Model, Monoenergetic Algorithm Final Remarks An Inverse Problem True Images Given: 2D projection images Goal: Reconstruct a 3D volume Julianne Chung, James Nagy, Ioannis Sechopoulos Polyenergetic Tomosynthesis Reconstruction

Motivation Mathematical Formulation An Important Application Reconstruction and Results Polyenergetic Model, Monoenergetic Algorithm Final Remarks Simulated Problem 21 projection images: Original object: Reconstruction: 200 × 300 pixels 300 × 300 × 200 voxels 150 × 150 × 50 voxels (10 × 15 cm) (7 . 5 × 7 . 5 × 5 cm) (7 . 5 × 7 . 5 × 5 cm) − 30 ◦ to 30 ◦ , every 3 ◦ X-ray Tube X-ray Tube Center of Rotation Chest Wall Compression Plate Compressed Breast Support Plate Detector Detector Front view Side view with X-ray tube at 0 ◦ Julianne Chung, James Nagy, Ioannis Sechopoulos Polyenergetic Tomosynthesis Reconstruction

Motivation Mathematical Formulation An Important Application Reconstruction and Results Polyenergetic Model, Monoenergetic Algorithm Final Remarks Polyenergetic Model Incident X-ray has a distribution of energies Rh/Rh 28kVp: 47 energy levels, 5keV - 28keV Consequences: Beam Hardening: Low energy photons preferentially absorbed Unnecessary radiation Linear attenuation coefficient depends on energy Julianne Chung, James Nagy, Ioannis Sechopoulos Polyenergetic Tomosynthesis Reconstruction

Motivation Mathematical Formulation An Important Application Reconstruction and Results Polyenergetic Model, Monoenergetic Algorithm Final Remarks Polyenergetic Model Incident X-ray has a distribution of energies Rh/Rh 28kVp: 47 energy levels, 5keV - 28keV Consequences: Beam Hardening: Low energy photons preferentially absorbed Unnecessary radiation Linear attenuation coefficient depends on energy Julianne Chung, James Nagy, Ioannis Sechopoulos Polyenergetic Tomosynthesis Reconstruction

Motivation Mathematical Formulation An Important Application Reconstruction and Results Polyenergetic Model, Monoenergetic Algorithm Final Remarks Monoenergetic Algorithm Monoenergetic Reconstruction Lange and Fessler’s Convex MLEM Algorithm Beam hardening artifacts Julianne Chung, James Nagy, Ioannis Sechopoulos Polyenergetic Tomosynthesis Reconstruction

Motivation Mathematical Formulation An Important Application Reconstruction and Results Polyenergetic Model, Monoenergetic Algorithm Final Remarks Previous Methods Methods for eliminating beam hardening artifacts: Dual Energy Methods Alvarez and Macovski (1976), Fessler et al (2002) FBP + Segmentation Joseph and Spital (1978) Filter function based on density De Man et al (2001), Elbakri and Fessler (2003) Julianne Chung, James Nagy, Ioannis Sechopoulos Polyenergetic Tomosynthesis Reconstruction

Motivation Mathematical Formulation Problem Set-up Reconstruction and Results Statistical Assumptions Final Remarks Outline Motivation 1 Mathematical Formulation 2 Reconstruction and Results 3 Final Remarks 4 Julianne Chung, James Nagy, Ioannis Sechopoulos Polyenergetic Tomosynthesis Reconstruction

Motivation Mathematical Formulation Problem Set-up Reconstruction and Results Statistical Assumptions Final Remarks A Polyenergetic Mathematical Representation Voxel j Energy-dependent Attenuation Coefficient: µ ( e ) ( j ) = s ( e ) x ( j ) + z ( e ) where x ( j ) represents unknown glandular fraction of j th voxel s ( e ) and z ( e ) are known linear fit coefficients Julianne Chung, James Nagy, Ioannis Sechopoulos Polyenergetic Tomosynthesis Reconstruction

Motivation Mathematical Formulation Problem Set-up Reconstruction and Results Statistical Assumptions Final Remarks Computing Image Projections Ray Trace: L i 60 N � 50 � µ ( e ) ( j ) a ( ij ) µ ( e ) dl ≈ 40 L i j = 1 30 20 10 6 Vector Notation 4 0 0 1 2 2 3 4 5 0 6 µ ( e ) = s ( e ) x + z ( e ) ⇒ s ( e ) A θ x + z ( e ) A θ 1 Julianne Chung, James Nagy, Ioannis Sechopoulos Polyenergetic Tomosynthesis Reconstruction

Motivation Mathematical Formulation Problem Set-up Reconstruction and Results Statistical Assumptions Final Remarks Computing Image Projections Ray Trace: 60 N 50 � � µ ( e ) ( j ) a ( ij ) µ ( e ) dl ≈ 40 L i j = 1 30 20 10 6 Vector Notation 4 0 0 1 2 2 3 4 5 0 6 µ ( e ) = s ( e ) x + z ( e ) ⇒ s ( e ) A θ x + z ( e ) A θ 1 Julianne Chung, James Nagy, Ioannis Sechopoulos Polyenergetic Tomosynthesis Reconstruction

Motivation Mathematical Formulation Problem Set-up Reconstruction and Results Statistical Assumptions Final Remarks Computing Image Projections Ray Trace: 60 N 50 � � µ ( e ) ( j ) a ( ij ) µ ( e ) dl ≈ 40 L i j = 1 30 20 10 6 Vector Notation 4 0 0 1 2 2 3 4 5 0 6 µ ( e ) = s ( e ) x + z ( e ) ⇒ s ( e ) A θ x + z ( e ) A θ 1 Polyenergetic Projection: n e � ̺ ( e ) exp ( − [ s ( e ) A θ x true + z ( e ) A θ 1 ]) e = 1 Julianne Chung, James Nagy, Ioannis Sechopoulos Polyenergetic Tomosynthesis Reconstruction

Motivation Mathematical Formulation Problem Set-up Reconstruction and Results Statistical Assumptions Final Remarks Statistical Model Given x , define the expected value, ¯ b ( i ) θ , as the i th entry of n e � ̺ ( e ) exp ( − [ s ( e ) A θ x + z ( e ) A θ 1 ]) e = 1 η ( i ) be the statistical mean of the noise. Let ¯ η ( i ) ∈ R is the expected or average observation. Then ¯ b ( i ) θ + ¯ b ( i ) ∼ Poisson (¯ b ( i ) η ( i ) ) θ + ¯ Observed Data: θ Julianne Chung, James Nagy, Ioannis Sechopoulos Polyenergetic Tomosynthesis Reconstruction

Motivation Mathematical Formulation Problem Set-up Reconstruction and Results Statistical Assumptions Final Remarks Statistical Model Likelihood Function: e − (¯ b ( i ) η ( i ) ) b ( i ) M η ( i ) ) (¯ b ( i ) θ +¯ θ + ¯ θ � p ( b θ , x ) = b ( i ) θ ! i = 1 Negative Log Likelihood Function: − L θ ( x ) = − log p ( b θ , x ) M b ( i ) η ( i ) ) − b ( i ) b ( i ) � (¯ θ log (¯ η ( i ) ) = θ + ¯ θ + ¯ i = 1 Julianne Chung, James Nagy, Ioannis Sechopoulos Polyenergetic Tomosynthesis Reconstruction

Motivation Reconstruction Algorithms Mathematical Formulation Numerical Results Reconstruction and Results Some Considerations Final Remarks Outline Motivation 1 Mathematical Formulation 2 Reconstruction and Results 3 Final Remarks 4 Julianne Chung, James Nagy, Ioannis Sechopoulos Polyenergetic Tomosynthesis Reconstruction

Motivation Reconstruction Algorithms Mathematical Formulation Numerical Results Reconstruction and Results Some Considerations Final Remarks Volume Reconstruction Maximum Likelihood Estimate: � n θ � � x MLE = argmin − L θ ( x ) x θ = 1 Numerical Optimization: Gradient Descent: x k + 1 = x k − α k ∇ L ( x k ) Newton Approach: x k + 1 = x k − α k H − 1 k ∇ L ( x k ) Julianne Chung, James Nagy, Ioannis Sechopoulos Polyenergetic Tomosynthesis Reconstruction

Motivation Reconstruction Algorithms Mathematical Formulation Numerical Results Reconstruction and Results Some Considerations Final Remarks Optimization Tools Gradient: ∇ L ( x k ) = A T v k where � n e b ( i ) � v ( i ) = � s ( e ) a T i x k + z ( e ) a T � � �� η ( i ) − 1 ̺ ( e ) s ( e ) exp − i 1 b ( i ) + ¯ ¯ e = 1 Hessian: H k = A T W k A 1 − 1 H k s k = −∇ L ( x k ) ⇔ min s k || W k As k − W 2 2 v || 2 k Julianne Chung, James Nagy, Ioannis Sechopoulos Polyenergetic Tomosynthesis Reconstruction