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Quantitative Molecular Imaging –
A Mathematical Challenge (?)
Martin Burger
Institut für Numerische und Angewandte Mathematik
CeNoS
European Institute for Molecular Imaging
Molecular Imaging
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Mathematical Imaging@WWU
Christoph Brune
Benning
Alex Sawatzky
Frank Wübbeling
Thomas Kösters Martin
Marzena Franek Christina Stöcker Mary Wolfram (Linz) Thomas Grosser
Müller
Martin Burger
Stuttgart, 27.5.2008
Jahn
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)
Martin Burger
Stuttgart, 27.5.2008
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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 )
Martin Burger
Stuttgart, 27.5.2008
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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 ?
Martin Burger
Stuttgart, 27.5.2008
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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 (noninvase 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)
Martin Burger
Stuttgart, 27.5.2008
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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 transform
cf. Natterer 86, Natterer-Wübbeling 02
Martin Burger
Stuttgart, 27.5.2008
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Molecular Imaging
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Medical Imaging: CT
+ Low noise level
+ High spatial resolution
+ Exact reconstruction
+ Reasonable Costs
Soret, Bacharach, Buvat 07
CT
- Restricted to few seconds
(radiation exposure, 20 mSiewert)
- No functional information
- Few mathematical challenges left
Martin Burger
Stuttgart, 27.5.2008
Schäfers et al 07
Molecular Imaging
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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
Martin Burger
Stuttgart, 27.5.2008
Courtesy Carsten Wolters,
University Hospital Münster
Molecular Imaging
Molecular Imaging: PET (Human / Small
animal)
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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
Martin Burger
Stuttgart, 27.5.2008
Molecular Imaging
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Medical Imaging: PET
+ High sensitivity
+ Long time (mins ~ 1 hour,
radiation exposure 8-12 mSiewert)
+ Functional information
+ Many open mathematical questions
Martin Burger
Stuttgart, 27.5.2008
PET
- Few anatomical information
- High noise level and disturbing
effects (damping, scattering, … )
- Low spatial resolution
Soret, Bacharach, Buvat 07
Schäfers et al 07
Molecular Imaging
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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
Martin Burger
Stuttgart, 27.5.2008
Molecular Imaging
Data model
Image
Otmar Schober
Klaus Schäfers
Martin Burger
Stuttgart, 27.5.2008
Data
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Molecular Imaging
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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 !
Martin Burger
Stuttgart, 27.5.2008
Molecular Imaging
Maximum Likelihood / Bayes
Reconstruct maximum-likelihood estimate
Model of posterior probability (Bayes)
Martin Burger
Stuttgart, 27.5.2008
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Molecular Imaging
EM-Algorithm: A fixed point iteration
Continuum limit (relative entropy)
Optimality condition leads to fixed point equation
Martin Burger
Stuttgart, 27.5.2008
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Molecular Imaging
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EM-Algorithm: A fixed point iteration
Fixed point iteration
Convergence 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
Martin Burger
Stuttgart, 27.5.2008
Molecular Imaging
PET Reconstruction: Small Animal PET
Reconstruction with
optimized EM-Algorithm,
Good statistics
Martin Burger
Stuttgart, 27.5.2008
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Molecular Imaging
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EM-Algorithm: A fixed point iteration
Fixed point iteration
uk + 1
uk
f
=
K ¤(
)
K ¤1
K uk
Convergence 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
Martin Burger
Stuttgart, 27.5.2008
Molecular Imaging
EM-Algorithm at the limit
Bad statistics arising due to
lower
radioactive activity or isotopes
decaying fast (e.g. H2O15)
~10.000
Events
Desireable for patients
Desireable for certain
~600
15
quantitative investigations (H2O Events
is useful tracer for blood flow)
Martin Burger
Stuttgart, 27.5.2008
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Molecular Imaging
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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)
Martin Burger
Stuttgart, 27.5.2008
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)
P (u) » e¡ R ( u )
Z yields regularized log-likelihood
Same analysis
[K u ¡ f log(K u)] + ®R(u)
Martin Burger
Stuttgart, 27.5.2008
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Molecular Imaging
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Minimization of penalized log-likelihood
Z
Minimization of
[K u ¡ f log(K u)] + ®R(u)
subject to nonnegativity is a difficult task
Combines nonlocal part (including K ) with local regularization
functional (typically dependent on u and 5u )
Ideally ingredients of EM-step should be used
(Implementations
of K and K* including several corrections)
Martin Burger
Stuttgart, 27.5.2008
Molecular Imaging
Minimization of penalized log-likelihood
Assume K is convex, but not necessarily differentiable
Optimality condition for a positive solution
f
K ¤1 ¡ K ¤(
) + ®p = 0;
Ku
p 2 @R(u)
For simplicity assume K*1 = 1 in the following (standard
operator scaling)
Martin Burger
Stuttgart, 27.5.2008
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Molecular Imaging
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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 iteration f
1 ¡ K ¤(
K uk
) + ®pk + 1 = 0;
pk + 1 2 @R(uk + 1 )
No preservation of positivity (even with damping)
Martin Burger
Stuttgart, 27.5.2008
Molecular Imaging
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Minimization of penalized log-likelihood
Improved: approximate also first term
uk + 1
f
¡ K ¤(
) + ®pk + 1 = 0;
uk
K uk
pk + 1 2 @R(uk + 1 )
Realized via two-step method
uk + 1=2
f
= uk K ¤ (
)
K uk
uk + 1 ¡ uk + 1=2
+ ®pk + 1 = 0;
uk
Martin Burger
Stuttgart, 27.5.2008
pk + 1 2 @R(uk + 1 )
Molecular Imaging
Minimization of penalized log-likelihood
Assume K is convex, but not necessarily differentiable
Optimality condition for a positive solution
f
K ¤1 ¡ K ¤(
) + ®p = 0;
Ku
subject to nonnegativity of u
uk + 1=2
Martin Burger
uk
f
=
K ¤(
)
K ¤1
K uk
Stuttgart, 27.5.2008
p 2 @R(u)
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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
Martin Burger
Stuttgart, 27.5.2008
Soret, Bacharach, Buvat 07
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Molecular Imaging
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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
Z into a penalization functional of the
Both cases can be unified
R(u) =
P (x; u(x)) dx
form
with P possibly infinite in the constrained case
Martin Burger
Stuttgart, 27.5.2008
Molecular Imaging
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TV-Methods
Penalization of total Variation
Formal
Exact
ROF-Model for denoising g : minimize total variation subject to
Rudin-Osher-Fatemi 89,92
Martin Burger
Stuttgart, 27.5.2008
Molecular Imaging
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Why TV-Methods ?
Cartooning
Linear Filter
Martin Burger
Stuttgart, 27.5.2008
TV-Method
Molecular Imaging
Why TV-Methods ?
Cartooning
ROF Model with increasing allowed
variance
Martin Burger
Stuttgart, 27.5.2008
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Molecular Imaging
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TV-Methods
There exists Lagrange Parameter, such that ROF is equivalent
to
u ¡ g + ®p
= 0;
Optimality
condition
p 2 @J (u);
uk + 1 ¡ uk + 1=2
+ ®pk + 1 = 0;
Compare with
uk
Martin Burger
Stuttgart, 27.5.2008
1
®=
¸
pk + 1 2 @J (uk + 1 )
Molecular Imaging
EM-TV Methods
EM-step
uk + 1=2 = uk K
¤(
f
)
K uk
TV-correction step by minimizing
1
2
Z
(u ¡ uk + 1=2 ) 2
+ ®J (u) !
uk
in order to obtain next iterate
Martin Burger
Stuttgart, 27.5.2008
min
u2 B V
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Molecular Imaging
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Damped EM-TV Methods
EM-step
uk + 1=2 = uk K
¤(
f
)
K uk
Damped
TV-correctionZstep by minimizing
Z
¿
2
(u ¡ uk ) 2
1
+
uk
2
(u ¡ uk + 1=2 ) 2
+ ®J (u) !
uk
in order to obtain next iterate
Martin Burger
Stuttgart, 27.5.2008
min
u2 B V
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)
Martin Burger
Stuttgart, 27.5.2008
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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
Martin Burger
Stuttgart, 27.5.2008
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Molecular Imaging
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Primal dual discretization
Use characterization of subgradients as elements of the
convex set
Optimality condition for ROF can be reformulated as a primaldual (or dual) variational inequality
Martin Burger
Stuttgart, 27.5.2008
Molecular Imaging
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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
Martin Burger
Stuttgart, 27.5.2008
Molecular Imaging
Error estimation
Error estimates need appropriate distance measure,
generalized Bregman-distance
mb-Osher 04
mb 08
DFG funding, „Regularisierung mit Singulären Energien“, 2008-2011
Martin Burger
Stuttgart, 27.5.2008
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Molecular Imaging
Parallel Implementations
Diploma thesis Jahn Müller, jointly supervised with Sergej
Gorlatch (Computer Science, WWU)
Martin Burger
Stuttgart, 27.5.2008
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Molecular Imaging
~600 Events
EM
EM-TV
Martin Burger
Stuttgart, 27.5.2008
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Molecular Imaging
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EM-TV reconstruction from synthetic data
Bild
Martin Burger
Daten
Stuttgart, 27.5.2008
EM
EM-TV
Molecular Imaging
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H2O15 Data – Right Ventricular
EM
Martin Burger
EM-Gauss
Stuttgart, 27.5.2008
EM-TV
Molecular Imaging
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H2O15 Data – Left Ventricular
EM
Martin Burger
EM-Gauss
Stuttgart, 27.5.2008
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)
Martin Burger
Stuttgart, 27.5.2008
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Molecular Imaging
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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
Martin Burger
Stuttgart, 27.5.2008
Molecular Imaging
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Quantitative PET
Contrast correction via iterative Regularization
p(u) » exp(¡ J (u))
Prior probability
centered at zero
u^ 1
Adaptation:
maximum likelihood estimater of Poisson-TV
model. Second step with shifted prior probability
p(u) » exp(¡ [J (u) + J ( u^ 1 ) + h^
p1 ; u ¡ u^ 1 i ])
Iterative algorithm, EM-TV can be used for each substep
Osher-mb-Goldfarb-Xu-Yin 05, mb-Gilboa-Osher-Xu 06
Martin Burger
Stuttgart, 27.5.2008
Molecular Imaging
Quantitative PET
Contrast correction via iterative Regularization
Significant improvement of quantitative densities
Often not visible in images
Directly visible for small
structures, e.g. for analogous
problem in nanoscopy
(4-Pi, STED)
Operator K is a convolution
Martin Burger
Stuttgart, 27.5.2008
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Molecular Imaging
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Nanoscopy – STED & 4Pi
Stimulated Emission Depletion (Stefan Hell, MPI Göttingen)
BMBF funded, „INVERS“, Göttingen(MPI+Univ)-Münster-Bochum, Leica
Martin Burger
Stuttgart, 27.5.2008
Molecular Imaging
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Nanoscopy at the limit
Syntaxin PC 12, 53 nm
EM
EM-TV
EM-TV
Christoph Brune
Martin Burger
Stuttgart, 27.5.2008
Iterated
Molecular Imaging
3D Cell Structure
EM-TV
Iterated EM-TV
Christoph Brune
Martin Burger
Stuttgart, 27.5.2008
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Molecular Imaging
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Quantification of Blood Flow
Current quantification with radioactive water has limited
resolution, due to poor quality of reconstructions at each time
step
Martin Burger
Stuttgart, 27.5.2008
Molecular Imaging
Outlook: Imaging of Physiological
Quantities
54
Instead of reconstruction images with bad statistics, use direct
model based inversion from 4D data
Schematic data model:
!
!
Physiological
Activation
quantities,3D+1D curves, 4D
F, r, CA
CT
!
Images
4D
u
PET data
4D
f
Nonlinear physiological models, lead to nonlinear inverse
problems
Martin Burger
Stuttgart, 27.5.2008
Molecular Imaging
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Myocardial Blood Flow
Two-compartment model: computation of flow into tissue CT
from arterial blood flow CA aus
@CT (x; t)
CT (x; t)
= F (x)(CA (t) ¡
)
@t
VD
Nonlinearity in the ODE, exponential dependence of CT on F
Martin Burger
Stuttgart, 27.5.2008
Molecular Imaging
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Myocardial Blood Flow
Left ventricular image computed
from the ODE solution via
Indicator functions
obtained from
segmentation (EM-TV). Corrections by spillover terms s1, s2
Martin Burger
Stuttgart, 27.5.2008
Molecular Imaging
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Quantification 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
Martin Burger
Stuttgart, 27.5.2008
Molecular Imaging
Quantification of Myocardial Blood Flow
Sequence of reconstructed
images by EM method
(3 D reconstruction in each
time frame)
Martin Burger
Stuttgart, 27.5.2008
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Molecular Imaging
Quantification of Myocardial Blood Flow
Sequence of reconstructed
images based on blood flow
model
(4 D reconstruction)
Martin Burger
Stuttgart, 27.5.2008
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Molecular Imaging
Quantification of Myocardial Blood Flow
Sequence of reconstructed
images based on blood flow
model
(4 D reconstruction)
Martin Burger
Stuttgart, 27.5.2008
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