Quantitative Molecular Imaging – A Mathematical Challenge (?)

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## Quantitative Molecular Imaging – A Mathematical Challenge (?)

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**Quantitative Molecular Imaging – A Mathematical Challenge**(?)**Molecular Imaging**Mathematical Imaging@WWU Christoph Brune Alex Sawatzky Frank Wübbeling Thomas Kösters Martin Benning Marzena Franek Christina Stöcker Mary Wolfram (Linz) Thomas Grosser Jahn Müller**Molecular Imaging**Major Cooperation Partners: SFB 656 /EIMI • Otmar Schober (Nuclear Medicine) • Klaus Schäfers (Medical Physics, EIMI) • Florian Büther (EIMI)Funding: Regularization with Singular Energies (DFG), SFB 656 (DFG), European Institute for Molecular Imaging (WWU + SIEMENS Medical Solutions)**Molecular Imaging**Major Cooperation Partners: Nanoscopy • Andreas Schönle, Stefan Hell (MPI Göttingen) • Thorsten Hohage, Axel Munk (Univ Göttingen) • Nico Bissantz (Bochum)Funding: „Verbundprojekt INVERS“ (BMBF )**Molecular Imaging**21st Century Imaging • Imaging nowadays mainly separates into two steps • Image Reconstruction: computation of an image from (indirectly) measured data – solution of inverse problems • Image Processing: improvement of given images / image sequences – filtering, variational problems • Mathematical issues and approaches (as well as communities) are very separated • Images are passed on from step 1 to step 2 • Is this an optimal approach ?**Molecular Imaging**Image reconstruction and inverse problems • Inverse Problems consist in reconstruction of the cause of an observed effect (via a mathematical model relating them) • Diagnosis in medicine is a prototypical example (non-invase approaches always need indirect measurements) • Crime is another one … • "The grand thing is to be able to reason backwards." Arthur Conan Doyle (A study in scarlet)**Molecular Imaging**Medical Imaging: CT • Classical image reconstruction example: • computerized tomography (CT) • Mathematical Problem: • Reconstruction of a density • function from its line integrals • Inversion of the Radon transformcf. Natterer 86, Natterer-Wübbeling 02**Molecular Imaging**Medical Imaging: CT • + Low noise level • + High spatial resolution • + Exact reconstruction • + Reasonable Costs • Restricted to few seconds (radiation exposure, 20 mSiewert) • - No functional information • - Few mathematical challenges left Soret, Bacharach, Buvat 07 Schäfers et al 07**Molecular Imaging**Medical Imaging: MR • + Low noise level • + High spatial resolution • + Reconstruction by Fourier inversion • + No radiation exposure • + Good contrast in soft matter • Low tracer sensitivity • - Limited functional information • - Expensive • - Few mathematical challenges left Courtesy Carsten Wolters, University Hospital Münster**Molecular Imaging**Molecular Imaging: PET (Human / Small animal) • Positron-Emission-Tomography • Data: detecting decay events of an radioactive tracer • Decay events are random, but their rate is proportional to the tracer uptake (Radon transform with random directions) • Imaging of molecular properties**Molecular Imaging**Medical Imaging: PET • + High sensitivity • + Long time (mins ~ 1 hour, radiation exposure 8-12 mSiewert)+ Functional information • + Many open mathematical questions • - Few anatomical information • High noise level and disturbing effects (damping, scattering, … ) • Low spatial resolution Soret, Bacharach, Buvat 07 Schäfers et al 07**Molecular Imaging**Image reconstruction in PET • Stochastic models needed: typically measurements drawn from Poisson model • „Image“ u equals density function (uptake) of tracer • Linear Operator K equals Radon-transform • Possibly additional (Gaussian) measurement noise b**Molecular Imaging**Data model • Image Data • Otmar Schober • Klaus Schäfers**Molecular Imaging**Image reconstruction in PET • Same model with different K can be used for imaging with photons (microscopy, CCD cameras, ..) • Typically the Poisson statistic is good (many photon counts), measurement noise dominates • In PET (and modern nanoscopy) the opposite is true !**Molecular Imaging**Maximum Likelihood / Bayes • Reconstruct maximum-likelihood estimate • Model of posterior probability (Bayes)**Molecular Imaging**EM-Algorithm: A fixed point iteration • Continuum limit (relative entropy)Optimality condition leads to fixed point equation**Molecular Imaging**EM-Algorithm: A fixed point iteration • Fixed point iterationConvergence analysis for exact data: descent in objective functional in Kullback-Leibler divergence (relative entropy) between to consecutive iterations (images) • Regularizing properties for ill-posed problems not completely clear, partial results Resmerita-Iusem-Engl 07**Molecular Imaging**PET Reconstruction: Small Animal PET • Reconstruction with optimized EM-Algorithm, Good statistics**f**u k ( ) ¤ K u = k 1 + K K 1 ¤ u k Molecular Imaging EM-Algorithm: A fixed point iteration • Fixed point iterationConvergence analysis for exact data: descent in objective functional in Kullback-Leibler divergence (relative entropy) between to consecutive iterations (images) • Regularizing properties for ill-posed problems not completely clear, partial results Resmerita-Iusem-Engl 07**Molecular Imaging**EM-Algorithm at the limit • Bad statistics arising due to lower radioactive activity or isotopes decaying fast (e.g. H2O15)Desireable for patientsDesireable for certain quantitative investigations (H2O15 is useful tracer for blood flow) ~10.000 Events ~600 Events**Molecular Imaging**PET at the resolution limit • How can we get reasonable answers in the case of bad data ? • Need additional (a-priori) information about: • known structures in the image • desired structures to be investigated • dynamics (4D imaging)**Z**[ ( ) ] ( ) f l K K R ¡ + u o g u ® u ( ) R ¡ ( ) u P u e » Molecular Imaging Back to Bayes • EM algorithm uses uniform prior probability distribution, any image explains data is considered of equal relevance • Prior probability can be related to regularization functional (such as energy in statistical mechanics) • Same analysis yields regularized log-likelihood**Z**[ ( ) ] ( ) f l K K R ¡ + u o g u ® u Molecular Imaging Minimization of penalized log-likelihood • Minimization of • subject to nonnegativity is a difficult task • Combines nonlocal part (including K ) with local regularization • functional (typically dependent on u and 5u ) • Ideallyingredients of EM-step should be used (Implementations • of K and K* including several corrections)**f**( ) ( ) ¤ ¤ @ K K R 1 0 ¡ + 2 ® p p u = ; K u Molecular Imaging Minimization of penalized log-likelihood • Assume K is convex, but not necessarily differentiable • Optimality condition for a positive solution • For simplicity assume K*1 = 1 in the following (standard • operator scaling)**f**( ) ( ) ¤ @ K R 1 0 ¡ + 2 ® p p u = k k k 1 1 1 + + + ; K u k Molecular Imaging Minimization of penalized log-likelihood • Simplest idea: gradient-type method • Not robust if J nonsmooth, possibly extreme damping needed for gradient-dependent J • Better: evaluate nonlocal term at last iterate and subgradient at new iterationNo preservation of positivity (even with damping)**u**u f f = k k 1 + 1 2 + u ( ) k 1 @ + R 0 ¡ + 2 ( ) ( ) ( ) ¤ ¤ ® p p u @ K K R 0 = k k k ¡ + 1 1 1 + + + 2 u ® p u p u ; = = = k k k k k 1 1 1 + + + 1 2 + ; u K K k u u u k k k Molecular Imaging Minimization of penalized log-likelihood • Improved: approximate also first termRealized via two-step method**f**f u k ( ) ( ) ( ) ¤ ¤ ¤ @ K K K R 1 0 ¡ + 2 u ® p p u = = = k 1 2 + ; K K K 1 ¤ u u k Molecular Imaging Minimization of penalized log-likelihood • Assume K is convex, but not necessarily differentiable • Optimality condition for a positive solution • subject to nonnegativity of u**Molecular Imaging**Hybrid Imaging: PET-CT (PET-MR) • Hybrid imaging becomes increasingly popular. Combine • Anatomical information (CT or MR) • Functional information (PET) • Anatomical information yields a-priori knowledge • about structures, e.g. • exact tumor location and size Soret, Bacharach, Buvat 07**Z**( ) ( ( ) ) d R P u x u x x = ; Molecular Imaging Regularization and Constraints • Anatomical priors (CT images) can be incorporated into the reconstruction process as constraints or as regularization: • constraints: uptake equals zero in certain tissues • - regularization: penalization of (high) uptake in certain tissues • Both cases can be unified into a penalization functional of the form • with P possibly infinite in the constrained case**Molecular Imaging**TV-Methods • Penalization of total VariationFormalExactROF-Model for denoising g: minimize total variation subject to • Rudin-Osher-Fatemi 89,92**Molecular Imaging**Why TV-Methods ? • Cartooning Linear Filter TV-Method**Molecular Imaging**Why TV-Methods ? • Cartooning • ROF Model with increasing allowed variance**u**u = k k 1 + 1 2 + ( ) @ J 1 0 ¡ ¡ + 2 ® p p u = k k k 1 1 1 + + + ; ( ) @ J 0 + 2 u u g ® p p u ® = = k ; ; ¸ Molecular Imaging TV-Methods • There exists Lagrange Parameter, such that ROF is equivalent to • Optimality condition • Compare with**2**f ( ) Z 1 u u = ¡ k 1 2 + ( ) ¤ K ( ) J i u u + = = k k 1 2 + ® u m n ! K 2 u k B V 2 u u k Molecular Imaging EM-TV Methods • EM-step • TV-correction step by minimizing • in order to obtain next iterate**f**2 ( ) 2 ( ) Z Z 1 u u ¿ u u = ¡ ¡ k 1 2 k + ( ) ¤ K ( ) J i u u + + = = k k 1 2 + ® u m n ! K 2 2 u k B V 2 u u u k k Molecular Imaging Damped EM-TV Methods • EM-step • Damped TV-correction step by minimizing • in order to obtain next iterate**Molecular Imaging**EM-TV: Analysis • Iterates well-defined in BV (existence, uniqueness) • Preservation of positivity (as usual for EM-step, maximum principle for TV minimization) • Descent of the objective functional with damping (yields uniform bound in BV and hence stability) • Remaining issue: • - Second derivative of logarithmic likelihood term is not uniformly bounded in general (related to lower bound for density)**Molecular Imaging**Computational Issues in TV-minimization • - Regularization term not differentiable, not strictly convex • Degenerate differential operator • No strong convergence in BV • Discontinuous solutions expected • Large data sets (3D / 4D Imaging, future 4 D / 5D with different regularization in dimensions > 3 ) • Solution approaches: • Dual or primal-dual schemes • Parallel implementations based on dual domain decomposition**Molecular Imaging**Primal dual discretization • Use characterization of subgradients as elements of the convex setOptimality condition for ROF can be reformulated as a primal-dual (or dual) variational inequality**Molecular Imaging**Primal dual discretization • Discretize variational inequality by finite elements, usually on square / cubical elements - piecewise constant for u (discontinuous anyway) • - Raviart-Thomas for p (stability) Or higher-order alternatives**Molecular Imaging**Error estimation • Error estimates need appropriate distance measure, • generalized Bregman-distancemb-Osher 04 • mb 08 DFG funding, „Regularisierung mit Singulären Energien“, 2008-2011**Molecular Imaging**Parallel Implementations • Diploma thesis Jahn Müller, jointly supervised with Sergej Gorlatch (Computer Science, WWU)**Molecular Imaging**~600 Events EM EM-TV**Molecular Imaging**EM-TV reconstruction from synthetic data • Bild Daten EM EM-TV**Molecular Imaging**H2O15 Data – Right Ventricular EM EM-Gauss EM-TV**Molecular Imaging**H2O15 Data – Left Ventricular EM EM-Gauss EM-TV**Molecular Imaging**Quantification • Results can be used as input for quantification • Standard approach: Rough region partition in PET images • Computation of mean physiological parameters (e.g. perfusion) in each region (parameter fit in ordinary differential equations • Needs 4D PET reconstructions • DFG-Funding SFB 656, Subproject PM6 (mb/Klaus Schäfers)**Molecular Imaging**Quantification • Remaining problem: systematic error for TV-Methods • Variation reduced too strongly, quantitative Values can differ in particular in small structures • Problems in quantitative evaluations**1**^ u 1 1 1 ( ) ( ( [ ) ( ) ( ( ) ( ) h ) i ] ) ^ ^ ^ J J J ¡ + ¡ + ¡ p u e x p p u u e x p u u p u u » » ; Molecular Imaging Quantitative PET • Contrast correction via iterative Regularization • Prior probability centered at zero • Adaptation: maximum likelihood estimater of Poisson-TV model. Second step with shifted prior probability Iterative algorithm, EM-TV can be used for each substep Osher-mb-Goldfarb-Xu-Yin 05, mb-Gilboa-Osher-Xu 06**Molecular Imaging**Quantitative PET • Contrast correction via iterative Regularization • Significant improvement of quantitative densities • Often not visible in images • Directly visible for smallstructures, e.g. for analogousproblem in nanoscopy (4-Pi, STED) Operator K is a convolution**Molecular Imaging**Nanoscopy – STED & 4Pi • Stimulated Emission Depletion (Stefan Hell, MPI Göttingen)BMBF funded, „INVERS“, Göttingen(MPI+Univ)-Münster-Bochum, Leica