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

2. 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

3. 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

4. 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)

5. 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

6. 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

7. 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

8. 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

9. 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 image of human brain Parametric image of rat brain phantom

10. 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

11. 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

12. 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 by PET data

13. 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

14. 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

15. Two-Tissue Compartment Modeling for PET Tracer Kinetics  Governing kinetic equation for each voxel i:  Compact notation: (1) (2)

16. 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

17. 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)

18. 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)

19. 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)

20. 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

21. 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)

22. 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)

23. 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

24. 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

25. 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

26. Experiments — Setup  Different data sets  Different noise levels: 30% and 50% AC events of the total counts per scan  High and low count cases: 10 7 and 10 5 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

27. Experiments — Results Perfect model recovery under 30% noise TruthML-EMH∞ Frame #4 Frame #8 Frame #12 ML-EMH∞ Low countHigh count

28. Experiments — Results Disturbed model recovery for low counts data TruthML-EMH∞ Frame #4 Frame #8 Frame #12 ML-EMH∞ 30% noise50% noise

29. 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

30. Future Work  Current efforts  Monte Carlo simulations  Real data experiments  Planned future work  Reduce filtering complexity  Parametric reconstruction: using system ID/joint estimation strategies