Introduc)on to PET Image Reconstruc)on Adam Alessio http://faculty.washington.edu/aalessio/ Nuclear Medicine Lectures Imaging Research Laboratory Division of Nuclear Medicine University of Washington Fall 2011 https://www.rad.washington.edu/research/research/groups/irl/ Summary of PET (SUV) Lecture from Robert Doot, PhD Quantitative analysis of FDG uptake is important in tumor imaging, especially for research Standard uptake values (SUV) are clinically feasible and require no extra effort But SUVs require attention to detail (i.e. weigh pts every time) SUV is less precise than more complex quantitative analysis methods Protocol standardization improves quantitative precision Projec)on Imaging overlay of all information (non quantitative) Tomographic Imaging Tomo + graphy = Greek: slice + picture source z y x z detector x p(x,z) = f(x,y,z) dy orbiting source + detector data for all angles true cross-sectional image 3 4 Types of imaging systems From photon detec)on to data in form of Sinograms The number of events detected along an (LOR) is proportional to the integral of activity (i.e. FDG concentration) along that line. s Projection: collection of parallel LORs (a single view) source Transmission (TX) Emission (EM) but same mathematics of tomography 511 kev photon detections Patient FDG distribution point source 180 o 0 o s s Sinogram (all views) single projection sine wave traced out by point source 5 6 Adam Alessio, aalessio@u.washington.edu 1
Sinogram Example Data Correc)ons Sinogram P(s, θ) Order of corrections (common application): Start with Raw Data: Prompt Events = Trues + Randoms + Scatter Scanner B Source Objects D C A The sinogram is P(s, θ) organized as a 2D histogram - Radon Transform of the object A D S C B 1. Randoms correction (Yr = Prompt-Randoms) 2. Detection efficiency normalization (Yn = Yr * Norm) 3. Deadtime (Yd = Yn * Dead) 4. Scatter (Ys = Yd - Scat) 5. Attenuation ( Ya = Ys * ACF) attenuation correction factors 6. Image Reconstruction 7 9 Sinograms reconstructed into images Analy)cal Methods: Data Forma)on? y s Projection domain x Image domain Reconstruction Problem All modalities that collect line-integral data (sinograms) can employ the same methods for reconstruction: 1. Analytical Methods - most popular: FBP- Filtered Backprojection 2. Iterative Methods - most popular: OSEM - ordered subset expectation maximization 10 Point Source - Forward Projection to 3 projections Can get estimate of point source with Back-projection 11 Analy)cal Methods: simple back- projec)on of point source Image & Sinogram Space Angle Instead of getting a point source, end up with: 1/r = Distance Backprojected Image No filtering ==> Result = true 1/r Need to do Filtered Back-Projection FBP To undo this, filter projections with ramp filter before backprojecting 12 13 Adam Alessio, aalessio@u.washington.edu 2
Analy)cal Method: Filtered back- projec)on Ramp filter then backprojection provides the exact solution in the absence of noise Ramp filter accentuates high frequency - Not good for noise High Frequency 0 frequency High Frequency Either clip ramp filter or often use filters to clip ramp to reduce noise: ex: Hanning filter R.H. Bracewell and A.C. Riddle, Inversion of fan beam scans in radio astronomy, Astrophysics Journal, vol. 150, pp. 427-434, 1967. 14 FBP Characteris)cs PROS: Analytic method ( inverse Radon transform ) FBP is exact IF: No noise -??Which filter is exact?? No attenuation Complete, continuously sampled data Uniform spatial resolution Easy to implement Computationally Fast Linear, other properties well understood (2x uptake = 2x intensity in image) Can Adjust filter window to trade off bias vs. variance CONS: cannot model noise in data, cannot model non-idealities of system, (resolution recovery methods) does not easily work with unusual geometries, cannot include knowledge about the image (like non-negative activity) 15 The Reconstruc)on Problem: An Inverse Problem Observed PET data system matrix x is N x 1 image (typically N ~ 128 x 128) y is M x 1 data (typically M ~ 280 x 336) Unknown image Error in observations P is M x N system matrix (provides probability entry j from x will be placed in entry i of y) If we could just invert P, we could solve this. But P is 16,000 x 95,000 entries Need to solve iteratively. Itera)ve Reconstruc)on Characteris)cs Pros General results in reduced variance (noise) for a given level of accuracy (bias, resolution, etc.) Reduce or eliminate streaks Incorporate (model) physical effects Counting statistics (noise) Confidence weighting Distance dependent resolution Scatter, attenuation, detector efficiency, deadtime, randoms a priori information (non-negativity, anatomical information, etc.) Cons Slow (computationally intensive) Non-linear -- hard to analyze A lot of knobs to adjust: smoothing parameters, number iterations, etc. Streaks replaced with different noise character (e.g. blobs ) Adam Alessio - 16 17 Itera)ve Reconstruc)on: Basic Components 1. Description of the form of the image ( pixels, voxels, blobs ) 2. System model relating unknown image to each detector measurement : relates image to data (Can include detector response, corrections for attenuation, efficiencies, etc ) 3. Statistical Model describing how each measurement behaves around its mean (Poisson, Gaussian, ) 4. Objective Function defining the best image estimate (Log-likelihood, WLS,MAP ) 5. Method for maximizing the objective function (EM) Main Point: Lots of options, Not all EM algorithms the EM for NM Reconstruc)on in a Nutshell Expectation Maximization is a general method for solving all kinds of statistical estimation problems, in tomography results in easy maximization step Back project Image NEW = Image OLD!BP Measured_Data % # FP(Image OLD ) & Forward project Tomography EM: Method for maximizing the Poisson Likelihood Function (ML-EM) same 18 19 EM Forward projection takes image into data space Back projection takes data into image space Adam Alessio, aalessio@u.washington.edu 3
Common Methods 1.EM (ML-EM - maximum likelihood method) 2.OSEM (Ordered Subset Expectation Maximization) Subset A Subset B Variant of EM (still Maximum Likelihood) - Pro: Fast Con: Does not converge Uses subsets of the data to compute each estimate Subset A then B then then repeat Image NEW = Image OLD!BP Measured_Data % A # FP A (Image OLD ) & Then Image NEW = Image OLD!BP Measured_Data % B # FP B (Image OLD ) & OSEM Example Image NEW = Image OLD!BP Measured_Data % # FP(Image OLD ) & Don t Know This True Image 5 10 15 15 20 35 Data (noiseless) Know This 25 20 30 25 1st Guess First Image First Subset Second Subset 10 10 10.00 1.00 15.00 1.50 25.00 0.60 6.00 0.60 9.00 0.60 0.60 10 10 10.00 1.00 15.00 1.50 25.00 1.40 14.00 1.40 21.00 1.40 1.40 20 20 1.00 1.50 23 27 1.00 1.50 1.09 0.93 Third Subset 5.56 0.93 9.78 1.09 End of Iteration 1 15.22 1.09 19.44 0.93 20 1.09 0.93 Slide courtesy of Tom Lewellen 21 3. Regularized Methods Common Methods MAP(maximum a posteriori), PWLS(penalized weighted least squares), GEM (generalized EM) All consist of variation in objective function x ˆ REGULARIZED = arg max ( Plain_ Objective(x)+ Regularizer(x) ) x Iteration # OSEM Convergence (Implicit Smoothing- Stop algorithm before convergence) Assume have some knowledge about the image before we even get the data (a priori knowledge) PROS: Can enforce noise/resolution properties in final image (don t need to post-smooth), Can include anatomical information (PET/CT?), Enforce non-negativity, Methods converge to final solution, More accurate model of data CONS: Usually takes longer, has more variables to set and understand (more knobs ), can impose odd noise structures 22 23 Iteration # 1 4 8 16 Subsets: 1 OSEM Examples (Explicit Smoothing- Post Smooth) 12 subsets, 7 itera)ons No Post Filter 7.5mm 15mm Noise free 4 16 High counts (low noise) low counts (high noise) 16 7.5mm Post Smooth 24 25 Adam Alessio, aalessio@u.washington.edu 4
No Post Filter OSEM Examples (Explicit Smoothing) 12 subsets, 7 itera)ons 7.5mm 15mm FBP Examples Noise free High counts (low noise) low counts (high noise) Reduce Variance & Increase Bias 26 (Reduce noise & Increase quantitative error) 27 Each method can tradeoff Bias and Noise FBP - can vary cutoff on filter which modifies the ramp filter OSEM - Can vary number of iterations (stop early) or vary amount of post-recon smoothing Each method can tradeoff Bias and Noise FBP - can vary cutoff on filter which modifies the ramp filter OSEM - Can vary number of iterations (stop early) or vary amount of post-recon smoothing OSEM-2 iterations FBP - Hanning OSEM-4 iterations 5mm or 10mm?? FBP - Hanning Hypothetical Tradeoff Curves-Which is better: method A or B? method C or D? 28 Hypothetical Tradeoff Curves 29 Imaging - Fully 3D vs. 2D Fully 3D Reconstruc)on Fully 3D PET data increases sensitivity of scanner (~ 8x) Drawbacks: Increased scatter Significant storage and reconstruction computation demands 2D - Direct And Cross Planes Only Fully 3D - Oblique Planes Also Direct Analytic Approach 3DRP: 3D reprojection (Kinahan and Rogers 1988) Iterative Approach Simple conceptual extension: Just need system model that relates voxel to fully 3D data (as opposed to a pixel to 2D data) System model becomes 100-2000x larger (big computational challenge. (2D:1.5 Billion entries to 3D: 1000 Billion entries) Rebinning Approach Reduce Fully 3D data to decoupled sets of 2D data, then do normal 2D reconstruction FORE (Fourier Rebinning) most common form 30 31 Adam Alessio, aalessio@u.washington.edu 5
Intro Image Reconstruction 11/2/11 Example of different post filters Some Examples: FBP, 12 mm Hann, no axial smoothing OSEM, S28, I2, LF4.3, PF6, Z0. OSEM, S28, I2, LF4.3, PF8, Z0. OSEM, S28, I2, LF4.3, PF10, Z1. OSEM, 10 mm Gaussian, no axial smoothing 32 33 Example of changing number of iterations S28, I1, LF6.3, PF8, Z1. S28, I2, LF6.3, PF8, Z1. S28, I3, LF6.3, PF8, Z1. S28, I4, LF6.3, PF8, Z1. Detectors Example of Detector Modeling with PET: Scanner bore Measuring Detector Response (Point Spread Func)on) Locations of point sources PPSF (sv,s) Each radial location has blur in radial direction counts(a.u.) Steps for measuring PSF: 1. 12 Fully-3D acquisitions of Na22 point source a different radial locations 2. Extract radial profile from each 3. Parameterize with discrete cosine transform coefficients 4. Interpolate coefficients to all radial locations to determine unique blurring kernel at each radial location: sv = 2mm sv = 348mm s s Measured profile in black, parameterized profile in red Alessio AM, Stearns CW, et al. Application and evaluation of a measured spatially variant system model for PET image reconstruction. IEEE Trans Med Imaging. 2010;29:938-949. FDG PET Example No Post Filter 3.5mm Post Filter FDG PET Example 7mm Post Filter 7mm Post Filter +PSF +PSF 109kg/240lbs patient, 10mCi injected, 5min/FOV Coronal view through images reconstructed with and proposed spatially varying system model modification. 36 Adam Alessio, aalessio@u.washington.edu 35 Observation: With this fixed resolution vs. noise setting, there is enhancement with addition of PSF (~11% higher) 37 6
Intro Image Reconstruction 11/2/11 Example of Detector Modeling with PET: 4 iterations OSEM- simple geometric projector - geometric projector with knowledge of exact locations of detectors +PSF - projector with measured spatially variant resolution response Resolution vs Iteration Line Source at 5cm from center of FOV Contrast Recovery vs. True Noise across 50 scans 8 iterations No post filter 2.5mm pixel size +PSF No post filter 2.5mm pixel size End 4 iterations Matched STD in Grey Matter Observation: As add sophistication to system model, algorithms tend to converge more slowly Inverse problem is more ill-posed when relate voxels to more locations in data space (system model is less-sparse-> harder to invert) 3mm Post Filter, standard z filter 2.5mm pixel size 38 39 Next Week: X- ray/ct with Paul Kinahan 40 Adam Alessio, aalessio@u.washington.edu 7