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Integrative System Approaches to Medical Imaging and Image Computing
Physiological Modeling In Situ ObservationRobust Integration
Motivations Observing in situ living systems across temporal
and spatial scales, analyzing and understanding the related structural and functional segregation and integration mechanisms through model-based strategies and data fusion, recognizing and classifying pathological extents and degrees
Biomedical imaging Biomedical image computing and intervention Biological and physiological modeling
Perspectives 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 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 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 Integrative system approaches to biomedical
imaging and image computing: 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 Biomedical imaging:
PET: activity and parametric reconstruction low-count and dynamic PET pharmacokinetics
SPECT: activity and attenuation reconstruction Medical image computing:
Computational cardiac information recovery: electrical propagation, electro-mechanical coupling, material
elasticity, kinematics, geometry fMRI analysis and applications:
biophysical model based analysis Fundamental medical image analysis problems
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 EngineeringHong Kong University of Science and Technology
Outline Background and review
Introduce tracer kinetics into reconstruction, to incorporate information of physiological processes
Tracer kinetics modeling and imaging model for dynamic PET
State-space formulation of dynamic PET reconstruction problem
Sampled-data H∞ filtering for reconstruction
Experiments
Background Dynamic PET imaging
Measures the spatiotemporal distribution of metabolically active compounds in living tissue
A sinogram sequence from contiguous acquisitions Two types of reconstruction problems
Activity reconstruction: estimate the spatial distribution of radioactivity over time
Parametric reconstruction: estimate physiological parameters that indicate functional state of the imaged tissue
Activity imageof human brain
Parametric imageof rat brain phantom
Dynamic PET Reconstruction — Review on existing methods
Frame-by-frame reconstruction 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 Spatial priors: smoothness constrain, shape prior Temporal priors: But information of the physiological
process is not taken into account
Introduce Tracer Kinetics into Reconstruction
Motivation Incorporate knowledge of physiological modeling Go beyond limits imposed by statistical quality of data
Tracer kinetic modeling 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
Tracer kinetics as continuous state equation Sinogram sequence in discrete measurement equation
Biological Process
Observations
Reconstruction Framework Formulated as a state estimation problem in a hybrid paradigm
Sampled-data H∞ filter for estimation
Described by tracer kinetic models
Represented byPET data
Tracer Kinetics Guided Dynamic PET Reconstruction — Overview
Main contributions
Physiological information included
Temporal information of data is explored
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
Compartment: a form of tracer that behaves in a kinetically equivalent manner. Interconnection: fluxes of material and biochemical conversions : 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
Governing kinetic equation for each voxel i:
Compact notation:
(1)
(2)
Two-Tissue Compartment Modeling for PET Tracer Kinetics
Total radioactivity concentration in tissue:
Directly generate PET measurements via positron emission
Neglect contribution of blood to PET activity
(3)
Typical time activity curves
Imaging Model for Dynamic PET Data
Measure the accumulation of total concentration of radioactivity on the scanning time interval
Activity image of kth scan AC-corrected measurements:
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
(4)
(5)
State-Space Formulation for Dynamic PET Reconstruction
Time integration of Eq.(2)
where , System kinetic equation for all voxels:
where , system noise A: block diagonal with blocks , Activity image expressed as
Let , construct measurement equation:
(6)
(7)
(8)
(9)
State-Space Formulation for Dynamic PET Reconstruction
Standard state-space representation Continuous tracer kinetics in Eq.(7) Discrete measurements in Eq.(9)
State estimation problem in a hybrid paradigm Estimate given , and obtain activity
reconstruction using Eq.(8)
(7)
(8)
(9)
Sampled-Data H∞ Filtering for Dynamic PET Reconstruction
Mini-max H∞ criterion 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) Continuous kinetics, discrete measurements Sampled-data filter to solve incompatibility of system
and measurements
Mini-max H∞ Criterion
Performance measure (relative estimation error)
, S(t), Q(t), V(t), Po: weightings
Given noise attenuation level , the optimal estimate should satisfy
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
(11)
(10)
Sampled-Data H∞ Filter
Prediction stage
Predict state and on time interval with and as initial conditions
Eq.(13) is Riccati differential equation Update stage
At , the new measurement is used to update the estimate with filter gain
(12)
(13)
(14)
(15)
System Complexity & Numerical Issues
Large degree of freedom In PET reconstruction with N voxels (128*128)
Numerical Issues 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.
Number of elements in
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
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
Activity image sequence Sinogram sequence
Experiments — Setup
Different data sets 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
Kinetic parameters unknown a priori for a specific subject, may have model mismatch 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 — ResultsPerfect model recovery under 30% noise
Truth ML-EM H∞
Frame #4
Frame #8
Frame #12
ML-EM H∞
Low count High count
Experiments — ResultsDisturbed model recovery for low counts data
Truth ML-EM H∞
Frame #4
Frame #8
Frame #12
ML-EM H∞
30% noise 50% noise
Experiments — Results
Quantitative analysis of estimated activity images, with each cell representing the estimation error in terms of bias ± variance.
Quantitative results for different data sets
Future Work
Current efforts Monte Carlo simulations Real data experiments
Planned future work Reduce filtering complexity Parametric reconstruction: using system ID/joint
estimation strategies