Integrative System Approaches to Medical Imaging and Image Computing Physiological Modeling In Situ Observation Robust Integration

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Integrative System Approaches to Medical Imaging
and Image ComputingPhysiological Modeling In
Situ ObservationRobust Integration
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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

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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

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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)

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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

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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

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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

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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 H8 filtering for reconstruction
• Experiments

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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

Parametric image of rat brain phantom
Activity image of human brain
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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

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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

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Tracer Kinetics Guided Dynamic PET Reconstruction
Overview
Represented by PET data
Described by tracer kinetic models

• Tracer kinetics as continuous state equation
• Sinogram sequence in discrete measurement equation

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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

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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

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Two-Tissue Compartment Modeling for PET Tracer
Kinetics

• Governing kinetic equation for each voxel i
• Compact notation

(1)
(2)
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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
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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)
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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)
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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)
(9)
(8)
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Sampled-Data H8 Filtering for Dynamic PET
Reconstruction

• Mini-max H8 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

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Mini-max H8 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

(10)
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Sampled-Data H8 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)
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System Complexity Numerical Issues

• Large degree of freedom
• In PET reconstruction with N voxels (128128)
• Numerical Issues
• Stability issues may arise in the Riccati
  differential equation (13), Mobius schemes have
  been adopted to pass through the singularities

Number of elements in
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. 13921413, 1996.
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Experiments Setup
Time activity curves for different tissue
regions in Zubal phantom
Zubal thorax phantom
Kinetic parameters for different tissue regions
in Zubal thorax phantom
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Experiments Setup
Activity image sequence
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 Fesslers toolbox

Prof. Jeff Fessler, University of Michigan
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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
• H8 filter and ML-EM reconstruction

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Experiments Results
Perfect model recovery under 30 noise
Frame 4
Frame 8
Frame 12
Truth
ML-EM
H8
ML-EM
H8
Low count
High count
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Experiments Results
Disturbed model recovery for low counts data
Frame 4
Frame 8
Frame 12
Truth
ML-EM
H8
ML-EM
H8
30 noise
50 noise
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Experiments Results

• Quantitative results for different data sets

Quantitative analysis of estimated activity
images, with each cell representing the
estimation error in terms of bias variance.
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Future Work

• Current efforts
• Monte Carlo simulations
• Real data experiments
• Planned future work
• Reduce filtering complexity
• Parametric reconstruction using system ID/joint
  estimation strategies