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Quantitative Molecular Imaging – A Mathematical Challenge (?) Mathematical Imaging@WWU Christoph Brune Alex Sawatzky Frank Wübbeling Thomas Kösters Martin Benning Marzena Franek Christina Stöcker Mary Wolfram (Linz) Thomas Grosser Jahn Müller

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mathematical imaging@wwu
Molecular ImagingMathematical Imaging@WWU

Christoph Brune Alex Sawatzky Frank Wübbeling Thomas Kösters Martin Benning

Marzena Franek Christina Stöcker Mary Wolfram (Linz) Thomas Grosser Jahn Müller

major cooperation partners sfb 656 eimi
Molecular ImagingMajor 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)
major cooperation partners nanoscopy
Molecular ImagingMajor Cooperation Partners: Nanoscopy
  • Andreas Schönle, Stefan Hell (MPI Göttingen)
  • Thorsten Hohage, Axel Munk (Univ Göttingen)
  • Nico Bissantz (Bochum)Funding: „Verbundprojekt INVERS“ (BMBF )
21st century imaging
Molecular Imaging21st 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 ?
image reconstruction and inverse problems
Molecular ImagingImage 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)
medical imaging ct
Molecular ImagingMedical 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
medical imaging ct8
Molecular ImagingMedical 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

medical imaging mr
Molecular ImagingMedical 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 pet human small animal
Molecular ImagingMolecular 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
medical imaging pet
Molecular ImagingMedical 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

image reconstruction in pet
Molecular ImagingImage 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
data model
Molecular ImagingData model
  • Image Data
  • Otmar Schober
  • Klaus Schäfers
image reconstruction in pet14
Molecular ImagingImage 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 !
maximum likelihood bayes
Molecular ImagingMaximum Likelihood / Bayes
  • Reconstruct maximum-likelihood estimate
  • Model of posterior probability (Bayes)
em algorithm a fixed point iteration
Molecular ImagingEM-Algorithm: A fixed point iteration
  • Continuum limit (relative entropy)Optimality condition leads to fixed point equation
em algorithm a fixed point iteration17
Molecular ImagingEM-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
pet reconstruction small animal pet
Molecular ImagingPET Reconstruction: Small Animal PET
  • Reconstruction with optimized EM-Algorithm, Good statistics
em algorithm a fixed point iteration19

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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
em algorithm at the limit
Molecular ImagingEM-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

pet at the resolution limit
Molecular ImagingPET 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)
back to bayes

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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
minimization of penalized log likelihood

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Molecular Imaging

Minimization of penalized log-likelihood
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  • Combines nonlocal part (including K ) with local regularization
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  • Ideallyingredients of EM-step should be used (Implementations
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minimization of penalized log likelihood24

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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)
minimization of penalized log likelihood25

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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)
minimization of penalized log likelihood26

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Molecular Imaging

Minimization of penalized log-likelihood
  • Improved: approximate also first termRealized via two-step method
minimization of penalized log likelihood27

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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
hybrid imaging pet ct pet mr
Molecular ImagingHybrid 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

regularization and constraints

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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
tv methods
Molecular ImagingTV-Methods
  • Penalization of total VariationFormalExactROF-Model for denoising g: minimize total variation subject to
  • Rudin-Osher-Fatemi 89,92
why tv methods
Molecular ImagingWhy TV-Methods ?
  • Cartooning Linear Filter TV-Method
why tv methods32
Molecular ImagingWhy TV-Methods ?
  • Cartooning
  • ROF Model with increasing allowed variance
tv methods33

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Molecular Imaging

TV-Methods
  • There exists Lagrange Parameter, such that ROF is equivalent to
  • Optimality condition
  • Compare with
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Molecular Imaging

EM-TV Methods
  • EM-step
  • TV-correction step by minimizing
  • in order to obtain next iterate
damped em tv methods

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Molecular Imaging

Damped EM-TV Methods
  • EM-step
  • Damped TV-correction step by minimizing
  • in order to obtain next iterate
em tv analysis
Molecular ImagingEM-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)
computational issues in tv minimization
Molecular ImagingComputational 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
primal dual discretization
Molecular ImagingPrimal 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
primal dual discretization39
Molecular ImagingPrimal 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
error estimation
Molecular ImagingError estimation
  • Error estimates need appropriate distance measure,
  • generalized Bregman-distancemb-Osher 04
  • mb 08 DFG funding, „Regularisierung mit Singulären Energien“, 2008-2011
parallel implementations
Molecular ImagingParallel Implementations
  • Diploma thesis Jahn Müller, jointly supervised with Sergej Gorlatch (Computer Science, WWU)
quantification
Molecular ImagingQuantification
  • 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)
quantification47
Molecular ImagingQuantification
  • Remaining problem: systematic error for TV-Methods
  • Variation reduced too strongly, quantitative Values can differ in particular in small structures
  • Problems in quantitative evaluations
quantitative pet

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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
quantitative pet49
Molecular ImagingQuantitative 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
nanoscopy sted 4pi
Molecular ImagingNanoscopy – STED & 4Pi
  • Stimulated Emission Depletion (Stefan Hell, MPI Göttingen)BMBF funded, „INVERS“, Göttingen(MPI+Univ)-Münster-Bochum, Leica
nanoscopy at the limit
Molecular ImagingNanoscopy at the limit
  • Syntaxin PC 12, 53 nm
  • EM EM-TV Iterated EM-TV
  • Christoph Brune
3d cell structure
Molecular Imaging3D Cell Structure
  • Christoph Brune
  • EM-TV Iterated EM-TV
quantification of blood flow
Molecular ImagingQuantification of Blood Flow
  • Current quantification with radioactive water has limited resolution, due to poor quality of reconstructions at each time step
outlook imaging of physiological quantities

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Molecular Imaging

Outlook: Imaging of Physiological Quantities
  • Instead of reconstruction images with bad statistics, use direct model based inversion from 4D dataSchematic data model:
  • Physiological Activation Images PET dataquantities,3D+1D curves, 4D 4D 4DF, r, CA CT u f
  • Nonlinear physiological models, lead to nonlinear inverse problems
myocardial blood flow

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Molecular Imaging

Myocardial Blood Flow
  • Two-compartment model: computation of flow into tissue CT from arterial blood flow CA aus
  • Nonlinearity in the ODE, exponential dependence of CT on F
myocardial blood flow56
Molecular ImagingMyocardial Blood Flow
  • Left ventricular image computed from the ODE solution via
  • Indicator functions obtained from segmentation (EM-TV). Corrections by spillover terms s1, s2
quantification of myocardial blood flow
Molecular ImagingQuantification of Myocardial Blood Flow
  • Solve nonlinear inverse problem again by two-step procedure,
  • i.e. EM alternated with parameter identification in coupled ODEs
  • A-priori knowledge on parameters in regularization and constraints
quantification of myocardial blood flow58
Molecular ImagingQuantification of Myocardial Blood Flow
  • Sequence of reconstructedimages by EM method
  • (3 D reconstruction in each time frame)
quantification of myocardial blood flow59
Molecular ImagingQuantification of Myocardial Blood Flow
  • Sequence of reconstructedimages based on blood flow
  • model
  • (4 D reconstruction)
quantification of myocardial blood flow60
Molecular ImagingQuantification of Myocardial Blood Flow
  • Sequence of reconstructedimages based on blood flow
  • model
  • (4 D reconstruction)