Statistical Methods for Reconstruction of Parametric Images from Dynamic PET Data

Transcription

1 Statistical Methods for Reconstruction of Parametric Images from Dynamic PET Data Mustafa E. Kamasak Purdue University School of Electrical and Computer Engineering March 25, 25

2 Outline Introduction and Problem Statement Two-tissue Compartment Model Image Domain Parameter Estimation Methods Direct Parameter Estimation Simulations HR+ Monkey Data Future Work and Conclusions 2

3 Dynamic PET PET accumulates/averages the emissions of voxels. Time resolution can be achieved by dividing the data into time frames. Used in imaging heart perfusion, brain activation, glucose metabolizing rate, receptor availability Time response of voxels are governed by ODEs Parameters of these ODEs are physiologically relevant 3

4 2-tissue Compartment Model Used in; Glucose metabolism imaging (FDG) Receptor availability imaging ( C-raclopride, 8 F-fallypride) C P k C F C B k 2 k 4 C P (nci/ml): Molar tracer concentration in plasma C F (nci/ml): Molar concentration of unbound tracer C B (nci/ml): Molar concentration of metabolized or bound tracer k 3 4

5 Physiologically Important Parameters For receptor-ligand imaging binding potential (BP) and volume distribution (VD) are physiologically important parameters. BP = k 3 k 4 VD = k k 2 ( + k ) 3 k 4 BP available receptor sites VD steady state distribution of tracer between plasma and tissue 5

6 2-tissue Compartment Model Equations C P is assumed to be known (arterial sampling, estimation), same for all voxels Kinetic parameters vary for each voxel location Time variation of molar tracer concentrations at voxel s dc F (t) dt dc B (t) dt PET signal at voxel s, = k s C P (t) (k 2s + k 3s )C F (t)+k 4s C B (t) = k 3s C F (t) k 4s C B (t) f(ϕ s,t) = ( V B )(C F (t)+c B (t))s A e λt + V B C WB (t) 6

7 Image Domain Dense Parameter Estimation Methods Require reconstructed image Pixelwise weighted least squares (PWLS): ˆϕ s =argmin ϕ s x s f(ϕ s ) 2 W s no a priori information, high spatial variance Pixelwise weighted least squares with spatial regularization: ˆϕ =argmin ϕ N s= x s f(ϕ s ) 2 W s + S(ϕ) PWLSZ - constrain using smoothed PWLS estimates (Zhou et al.) PWLSR - GMRF 7

8 Our Approach: Direct Parametric Image Reconstruction t=.. K Parametric Image Reconstruction t ϕ = k k 2 k 3 k 4 k k 2 k 3 k k (N ) k 2(N ) k 3(N ) k 4(N ) Scanner Geometry parametric image PET DATA Advantages: Directly reconstructs parameters from sinogram data Dimensionality reduction Improves SNR Produces a single full image of parameter vector 8

9 Parametric Reconstruction Model t ϕ = k k 2 k 3 k 4 k 2 k 3 k 4 k k (N ) k 2(N ) k 3(N ) k 4(N ) 2 tissue Compartment Model F( ϕ) ϕ LL(Y ) parametric image Scanner Geometry ϕ s parameter vector of voxel s f(t,ϕ s ) f(ϕ s ) =. time response at voxel s f(t K,ϕ s ) F (ϕ) =[f(ϕ ),f(ϕ 2 ),...,f(ϕ N )] time response of all voxels Y is the sinogram data 9

10 MAP Estimate of Parametric Image C(Y ϕ) = LL(Y ϕ)+s(ϕ) ˆϕ = argmin ϕ C(Y ϕ) How do we efficiently compute ˆϕ?

11 Log Likelihood (data) Term Y sinogram matrix; Y mk independent Poisson distributed A system matrix; A ij detection probability of an event from voxel j by detector pair i E[Y F (ϕ)] = AF (ϕ)+µ. Log likelihood LL(Y ϕ) = K k= M m= Y mk log(a m F (ϕ, t k )+µ) (A m F (ϕ, t k )+µ) log(y mk!).

12 Stabilizing (regularization/prior) Term Model the distribution of parametric image with Markov Random Field (MRF) Gibbs distribution p(ϕ) = z exp{ {s,r} N g s r ϕ s ϕ r q W } Negative logarithm of distribution is used as stabilizing function S(ϕ) = g s r T (ϕ s ) T (ϕ r ) 2 W. {s,r} N T ( ), let you regularize physiologically relevant parameters. 2

13 PICD - Parametric Iterative Coordinate Descent Efficient implementation of ICD for reconstruction with kinetic models Sequentially update parameter ϕ s vector at each voxel C( ϕ s,ϕ s ) f( ϕ s,ϕ s )θ + 2 f( ϕ s,ϕ s ) 2 θ 2 + r sg s r T ( ϕ s ) T (ϕ r ) 2 W where f( ϕ s,ϕ s )=[f( ϕ s,t ) f(ϕ s,t ),,f( ϕ s,t K ) f(ϕ s,t K )] Update ϕ s ϕ s arg min ϕ s C( ϕ s,ϕ s ) LL(y ϕ)+s(ϕ) will decrease with each PICD iteration 3

14 PICD - Update Strategy We re-parametrize using ϕ s =[a s,b s,c s,d s ] Then the time response is f(t, ϕ s )=( V B )[(ae ct +be dt )u(t) C P (t)]s A e λt +V B C WB (t) Two-stage nested optimization. { (c s,d s ) arg min c s d s c s,d s line search a s,b s gradient search arg min ã s, b s { C([ã s, b s, c s, d s ],ϕ s )} }. 4

15 Computational Complexity N number of voxels M average number of projections per voxel K number of time frames K c number of time points in the time-convolution kernel L ab number iterations required for each update of (ã, b) L cd number iterations required for each update of ( c, d) Algorithm Per Iteration Complexity PICD KN(L cd L ab + K c L cd + M ) PWLS KN(L cd L ab + K c L cd ) PWLSR/PWLSZ KN(L cd L ab + K c L cd ) ICD KN(M ) 5

16 Multiresolution Reconstruction Multiresolution reconstruction Coarsest scale initialized to constant value Coarse scale solutions are used to initialize fine scale solutions Used 3 scales (32 32, and 28 28) Speeds up convergence at low frequency components. 6

17 Simulations - Phantom Rat phantom with seven separate regions is used to assess the estimation methods nonbrain nonspecific gray matter striatum cortex white matter Activity (nci/ml) nonbrain nonspecific gray matter striatum cortex white matter time (min) Regions are obtained by segmenting MRI scans of a rat 7

18 Simulations - Phantom Kinetic parameters of regions are obtained from literature Region k k 2 k 3 k 4 Background CSF Nonbrain (NB) Whole brain (WB) Straitum (STR) Cortex (COR) White matter (WM) Total scan time is 6 min., divided into 8 time frames: 4.5 min, 4 2minand 5min 8

19 Simulations - Assumptions Raclopride with C is used as tracer. The blood function, C P (t) was generated as described in [Wong et. al. ] Activity scaled to are scaled M counts 8 projection angles each with 2 projection (.875 mm spacing) Used 4 mm. wide triangular PSF Poisson noise model with accidental coincidences Image domain methods use MAP reconstruction with a single regularization parameter for all time frames 9

20 Kinetic Parameter Estimates k k 2 k 3 k 4 (a) original (b) PWLS (c) PWLZ

21 Kinetic Parameter Estimates k k 2 k 3 k 4 (d) PWLSR (e) PICD (f) PICD

22 Normalized RMSE of the kinetic parameter estimates Normalized RMSE.5.5 PWLS Zhou PWLSR PICD PICD k k2 k3 k4 22

23 Estimates of Physiologically Important Parameters BP V D BP V D (a) Original (b) PWLS (c) PWLSZ (d) PWLSR (e) Logan (f) PICD 23

24 Normalized RMSE of BP and VD Normalized RMSE PWLS PWLSZ PWLSR Logan PICD.2 BP VD 24

25 Emission Images Frame 5 Frame Frame 5 Frame 5 Frame Frame 5 (a) Original (b) FBP (c) MAP (d) PICD 25

26 Normalized RMSE of Emission Images Normalized RMSE of emission images Frame number FBP MAP PICD FBP MAP PICD (a) (b) 26

27 CPU Time per Iteration Method time for iteration (sec.) PWLS 474 PWLSZ 487 PWLSR 526 PICD

28 RMSE % of the final parameter estimate Convergence fixed resolution PICD PWLS PWLSZ PWLSR multiresolution PICD 5 5 CPU time (min) 28

29 EXACT HR+ Dataset Monkey head imaged with 8 F-fallypride scanner: Siemens EXACT HR+ supplied by Dr. Brad Christian, Kettering Medical Center, OH. 29

30 FORE + Corrections for scatter, attenuation, and detector normalization a 2D + no corrections FORE + corrections a Thanks to Dr. Christian and CTI 3

31 What Does Data Look Like 3

32 k, k 2, k 3, k 4 Images for Monkey Data k k 2 k 3 k PWLS PWLSR k.9 k2 k3.8 k PICD

33 BP and VD Images for Monkey Data PWLS PWLSR PICD 2 2 BP BP VD 5 VD

34 PWLS Fit for voxel (7,6) ϕ (7,6) =[.422,.794,.697,.37] T, BP=8.869, VD= Measured Fit. activity (nci/ml) residual activity (nci/ml) time (min) time (min) 34

35 PWLS Fit for voxel (99,56) ϕ (99,56) =[.495,.353,.78,.] T, {BP,V D} = Measured Fit.25.2 activity (nci/ml) residual activity (nci/ml) time (min) time (min) 35

36 PWLS Fit for voxel (68,7) ϕ (68,7) =[.258,.22,.2,.74] T, BP=.55,VD= Measured Fit.2.5 activity (nci/ml) residual activity (nci/ml) time (min) time (min) 36

37 PICD Fit for sinogram voxel (6,49) counts/min measured sinogram forward projected activity residual counts/min time (min) time (min) 37

38 PICD Fit for sinogram voxel (56,59) counts/min measured sinogram forward projected activity residual counts/min time (min) time (min) 38

39 PICD Fit for sinogram voxel (6,67) counts/min measured sinogram forward projected activity residual counts/min time (min) time (min) 39

40 Limitations of PICD Interframe image registration and motion compensation Time response of some voxels do not follow the model Goodness-of-fit measure Model validation for new tracers Some of these can be solved future work 4

41 Conclusions Propose direct reconstruction of parametric image Advantages Dimensionality reduction Higher SNR Dense parameter estimates Propose computationally efficient optimization algorithm Propose regularization on physiologically relevant domain(s) Demonstrated improved quality, lower RMSE estimations and fast convergence on realistic simulation data Demonstrated increased resolution on real data 4

42 Future Work More accurate scanner model Extension of the work to nonlinear models (with more parameters) Goodness-of-fit analysis on sinogram domain Model selection Parameter estimation reliability Requires estimation of ϕ s f(ϕ s,t)= k s f([k s,k 2s,k 3s,k 4s ],t) k 2s f([k s,k 2s,k 3s,k 4s ],t) k 3s f([k s,k 2s,k 3s,k 4s ],t) k 4s f([k s,k 2s,k 3s,k 4s ],t) for each voxel s. 42

43 Even More Future Work Parameter identifiability Requires estimation of Hessian H = [ ][ ] T f(ϕ s,t) f(ϕ s,t) ϕ s ϕ s for each voxel s. Better optimization algorithms ie. multigrid method Motion estimation and compensation Higher order spline implementation for list-mode data 43

44 Acknowledgements Prof. Bouman and Prof. Morris Prof. Christian and Prof. Sauer Prof. Allebach, Prof. Delp, and Prof. Lucier EISL colleagues Cristian Constantinescu, Chunzhi Wang, Dr. Karmen Yoder, Marc Normandin, and Dr. Ti-Qiang Li 44

45 Thank You! 45