Transcript Cardiac Motion Recovery: Continuous Dynamics, Discrete

Integrative System Approaches to
Medical Imaging and Image Computing
Physiological Modeling
In Situ Observation
Robust Integration
Motivations
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Observing in situ living systems across temporal
and spatial scales, analyzing and understanding
the related structural and functional segregation
and integration mechanisms through modelbased strategies and data fusion, recognizing
and classifying pathological extents and degrees
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Biomedical imaging
Biomedical image computing and intervention
Biological and physiological modeling
Perspectives
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Recent biological & technological breakthroughs,
such as genomics and medical imaging, have
made it possible to make objective and
quantitative observations across temporal and
spatial scales on population and on individuals
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At the population level, such rich information facilitates the
development of a hierarchy of computational models dealing
with (normal and pathological) biophysics at various scales
but all linked so that parameters in one model are the
inputs/outputs of models at a different spatial or temporal
scale
At the individual level, the challenge is to integrate
complementary observation data, together with the
computational modeling tailored to the anatomy, physiology
and genetics of that individual, for diagnosis or treatment of
that individual
Perspectives
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In order to quantitatively understand specific
human pathologies in terms of the altered model
structures and/or parameters from normal
physiology, the data-driven information recovery
tasks must be properly addressed within the
content of physiological plausibility and
computational feasibility (for such inverse
problems)
Philosophy
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Integrative system approaches to biomedical
imaging and image computing:
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System modeling of the biological/physiological
phenomena and/or imaging processes: physical
appropriateness, computational feasibility, and model
uncertainties
Observations on the phenomena: imaging and other
medical data, typically corrupted by noises of various
types and levels
Robust integration of the models and measurements:
patient-specific model structure and/or parameter
identification, optimal estimation of measurements
Validation: accuracy, robustness, efficiency, clinical
relevance
Current Research Topics
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Biomedical imaging:
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PET: activity and parametric reconstruction
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low-count and dynamic PET
pharmacokinetics
SPECT: activity and attenuation reconstruction
Medical image computing:
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Computational cardiac information recovery:
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fMRI analysis and applications:
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electrical propagation, electro-mechanical coupling, material
elasticity, kinematics, geometry
biophysical model based analysis
Fundamental medical image analysis problems
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Efficient representation and computation platform
Robust image segmentation:
 Level set on point cloud
 Local weakform active contour
Inverse-consistent image registration
Tracer Kinetics Guided
Dynamic PET Reconstruction
Shan Tong, Huafeng Liu, Pengcheng Shi
Department of Electronic and Computer Engineering
Hong Kong University of Science and Technology
Outline
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Background and review
Introduce tracer kinetics into reconstruction,
to incorporate information of physiological
processes
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Tracer kinetics modeling and imaging model
for dynamic PET
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State-space formulation of dynamic PET
reconstruction problem
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Sampled-data H∞ filtering for reconstruction
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Experiments
Background
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Dynamic PET imaging
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Measures the spatiotemporal distribution of
metabolically active compounds in living tissue
A sinogram sequence from contiguous acquisitions
Two types of reconstruction problems
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Activity reconstruction: estimate the spatial
distribution of radioactivity over time
Parametric reconstruction: estimate physiological
parameters that indicate functional state of the
imaged tissue
Activity image
of human brain
Parametric image
of rat brain phantom
Dynamic PET Reconstruction —
Review on existing methods
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Frame-by-frame reconstruction
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Reconstruct a sequence of activity images independently
at each measurement time
Analytical (FBP) and statistical (ML-EM,OSEM) methods
from static reconstruction
Suffer from low SNR (sacrificed for temporal resolution)
and lack of temporal information of data
Statistical methods assume data distribution
that may not be valid (Poisson or Shifted Poisson)
Prior knowledge to constrain the problem
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Spatial priors: smoothness constrain, shape prior
Temporal priors: But information of the physiological
process is not taken into account
Introduce Tracer Kinetics into
Reconstruction
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Motivation
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Incorporate knowledge of physiological modeling
Go beyond limits imposed by statistical quality of data
Tracer kinetic modeling
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Kinetics: spatial and temporal distributions of a
substance in a biological system
Provide quantitative description of physiological
processes that generate the PET measurements
Used as physiology-based priors
Tracer Kinetics Guided Dynamic PET
Reconstruction — Overview
Biological
Process
Reconstruction
Framework
Represented by
PET data
Formulated as a
state estimation
problem in a hybrid
paradigm
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Described by
tracer kinetic models
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Sampled-data H∞
filter for estimation
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Observations
Tracer kinetics as continuous state equation
Sinogram sequence in discrete measurement equation
Tracer Kinetics Guided Dynamic PET
Reconstruction — Overview
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Main contributions
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Physiological information included
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Temporal information of data is explored
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No assumptions on system and data statistics, robust
reconstruction
General framework for incorporating prior knowledge
to guide reconstruction
Two-Tissue Compartment Modeling for
PET Tracer Kinetics
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Compartment: a form of tracer that behaves in
a kinetically equivalent manner.
Interconnection: fluxes of material and
biochemical conversions
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: arterial concentration of nonmetabolized tracer in
plasma
: concentration of nonmetabolized tracer in tissue
: concentration of isotope-labeled metabolic products
in tissue
: first-order rate constants specifying the tracer
exchange rates
Two-Tissue Compartment Modeling for
PET Tracer Kinetics
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Governing kinetic equation for each voxel i:
(1)
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Compact notation:
(2)
Two-Tissue Compartment Modeling for
PET Tracer Kinetics
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Total radioactivity concentration in tissue:
(3)
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Directly generate PET measurements via positron
emission
Neglect contribution of blood to PET activity
Typical time
activity curves
Imaging Model for Dynamic PET Data
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Measure the accumulation of total concentration
of radioactivity on the scanning time interval
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Activity image of kth scan
AC-corrected measurements:
(5)
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Imaging matrix D: contain probabilities of detecting an
emission from one voxel at a particular detector pair
Complicated data statistics due to SC events, scanner
sensitivity and dead time, violating assumptions in
statistical reconstruction
State-Space Formulation for Dynamic
PET Reconstruction
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Time integration of Eq.(2)
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where
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System kinetic equation for all voxels:
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where
A: block diagonal with blocks
Activity image expressed as
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, system noise
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Let
, construct measurement equation:
(9)
State-Space Formulation for Dynamic
PET Reconstruction
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Standard state-space representation
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Continuous tracer kinetics in Eq.(7)
Discrete measurements in Eq.(9)
(7)
(9)
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State estimation problem in a hybrid paradigm
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Estimate
given
, and obtain activity
reconstruction
using Eq.(8)
(8)
Sampled-Data H∞ Filtering for Dynamic
PET Reconstruction
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Mini-max H∞ criterion
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Requires no prior knowledge of noise statistics
Suited for the complicated statistics of PET data
Robust reconstruction
Sampled-data filtering for the hybrid paradigm
of Eq.(7)(9)
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Continuous kinetics, discrete measurements
Sampled-data filter to solve incompatibility of system
and measurements
Mini-max H∞ Criterion
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Performance measure (relative estimation error)
(10)
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, S(t), Q(t), V(t), Po: weightings
Given noise attenuation level
estimate should satisfy
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, the optimal
(11)
Supremum taken over all possible disturbances and initial
states
Minimize the estimation error under the worst possible
disturbances
Guarantee bounded estimation error over all disturbances of
finite energy, regardless of noise statistics
Sampled-Data H∞ Filter
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Prediction stage
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(13)
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Predict state
and
on time interval
with
and
as initial conditions
Eq.(13) is Riccati differential equation
Update stage
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At , the new measurement is used to update
the estimate with filter gain
System Complexity & Numerical Issues
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Large degree of freedom
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In PET reconstruction with N voxels (128*128)
Number of
elements in
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Numerical Issues
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Stability issues may arise in the Riccati differential
equation (13), Mobius schemes have been adopted
to pass through the singularities*
*J. Schiff and S. Shnider, “A natural approach to the numerical integration of Riccati
differential equations,” SIAM Journal on Numerical Analysis, vol. 36(5), pp. 1392–1413, 1996.
Experiments — Setup
Zubal thorax phantom
Time activity curves for different
tissue regions in Zubal phantom
Kinetic parameters for different tissue regions in Zubal thorax phantom
Experiments — Setup
Activity image sequence
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Sinogram sequence
Total scan: 60min, 18 frames with 4 × 0.5min, 4 ×
2min, and 10 × 5min
Input function
Project activity images to a sinogram sequence,
simulate AC-corrected data with imaging matrix
modeled by Fessler’s toolbox*
*Prof. Jeff Fessler, University of Michigan
Experiments — Setup
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Different data sets
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Kinetic parameters unknown a priori for a
specific subject, may have model mismatch
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Different noise levels: 30% and 50% AC events
of the total counts per scan
High and low count cases: 107 and 105 counts for
the entire sinogram sequence
Perfect model recovery: same parameters in data
generation and recovery
Disturbed model recovery: 10% parameter
disturbance added in data generation
H∞ filter and ML-EM reconstruction
Experiments — Results
Perfect model recovery under 30% noise
Frame #4
Frame #8
Frame #12
Truth
ML-EM
H∞
Low count
ML-EM
H∞
High count
Experiments — Results
Disturbed model recovery for low counts data
Frame #4
Frame #8
Frame #12
Truth
ML-EM
H∞
30% noise
ML-EM
H∞
50% noise
Experiments — Results
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Quantitative results for different data sets
Quantitative analysis of estimated activity images, with each cell
representing the estimation error in terms of bias ± variance.
Future Work
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Current efforts
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Monte Carlo simulations
Real data experiments
Planned future work
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Reduce filtering complexity
Parametric reconstruction: using system ID/joint
estimation strategies