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Quantitative Molecular Imaging – A Mathematical Challenge (?)

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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

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)

Molecular ImagingMajor Cooperation Partners: Nanoscopy

- Andreas Schönle, Stefan Hell (MPI Göttingen)
- Thorsten Hohage, Axel Munk (Univ Göttingen)
- Nico Bissantz (Bochum)Funding: „Verbundprojekt INVERS“ (BMBF )

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 ?

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)

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

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

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 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

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

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

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 !

Molecular ImagingMaximum Likelihood / Bayes

- Reconstruct maximum-likelihood estimate
- Model of posterior probability (Bayes)

Molecular ImagingEM-Algorithm: A fixed point iteration

- Continuum limit (relative entropy)Optimality condition leads to fixed point equation

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

Molecular ImagingPET Reconstruction: Small Animal PET

- Reconstruction with optimized EM-Algorithm, Good statistics

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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

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

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)

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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

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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)

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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)

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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)

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Molecular Imaging

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

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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

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

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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

Molecular ImagingTV-Methods

- Penalization of total VariationFormalExactROF-Model for denoising g: minimize total variation subject to
- Rudin-Osher-Fatemi 89,92

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Molecular Imaging

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

EM-TV Methods- EM-step
- TV-correction step by minimizing
- in order to obtain next iterate

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Molecular Imaging

Damped EM-TV Methods- EM-step
- Damped TV-correction step by minimizing
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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)

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

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

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

Molecular ImagingError estimation

- Error estimates need appropriate distance measure,
- generalized Bregman-distancemb-Osher 04
- mb 08 DFG funding, „Regularisierung mit Singulären Energien“, 2008-2011

Molecular ImagingParallel Implementations

- Diploma thesis Jahn Müller, jointly supervised with Sergej Gorlatch (Computer Science, WWU)

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)

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

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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

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

Molecular ImagingNanoscopy – STED & 4Pi

- Stimulated Emission Depletion (Stefan Hell, MPI Göttingen)BMBF funded, „INVERS“, Göttingen(MPI+Univ)-Münster-Bochum, Leica

Molecular ImagingQuantification of Blood Flow

- Current quantification with radioactive water has limited resolution, due to poor quality of reconstructions at each time step

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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

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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

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

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

Molecular ImagingQuantification of Myocardial Blood Flow

- Sequence of reconstructedimages by EM method
- (3 D reconstruction in each time frame)

Molecular ImagingQuantification of Myocardial Blood Flow

- Sequence of reconstructedimages based on blood flow
- model
- (4 D reconstruction)

Molecular ImagingQuantification of Myocardial Blood Flow

- Sequence of reconstructedimages based on blood flow
- model
- (4 D reconstruction)

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