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