A hierarchical group ICA model for assessing covariate effects on brain functional networks

Transcription

1 A hierarchical group ICA model for assessing covariate effects on brain functional networks Ying Guo Joint work with Ran Shi Department of Biostatistics and Bioinformatics Emory University 06/03/2013 Ying Guo (Emory University) SRCOS /03/ / 31

2 Outline Introduction of independent component analysis (ICA) for fmri data A hierarchical group PICA model for modeling covariate effects on brain functional networks Estimation methods for the proposed group ICA model Simulation studies A fmri data example Summary Ying Guo (Emory University) SRCOS /03/ / 31

3 Source Separation in fmri studies Goal: Decompose observed fmri data as a linear combination of spatio-temporal processes of underlying source signals. Ying Guo (Emory University) SRCOS /03/ / 31

4 Single-subject ICA Classic noise-free ICA Y = AS, (1) Y(T V ) is the observed fmri data. A(T q) is the mixing matrix. Each column represents the time series associated with an IC S(q V ) is statistically independent spatial source signals in its rows. The distribution of the spatial source signal is non-gaussian. Probabilistic ICA (PICA): Y = AS + E E(T V ) noise term representing residual variability in the data that are not explained by the extracted ICs. Ying Guo (Emory University) SRCOS /03/ / 31

5 Time T...1 Schematic plot for Single-subject ICA Voxels 1.. V IC 1 IC q IC 1 Y A S Statistically independent f s 1,, s q = f s 1 f(s q ) IC q Ying Guo (Emory University) SRCOS /03/ / 31

6 Existing group ICA methods Temporal-concatenation group ICA (TC-GICA) GIFT (Calhoun et al., 2001 ) Tensor PICA model (Beckmann and Smith, 2005 ) A general group PICA model (Guo and Pagnoni, 2008; Guo 2011) Assume common IC spatial maps across subjects in group ICA decomposition. A hierarchical group PICA (Guo and Tang, 2013) Incorporate subject-specific random effects in IC spatial maps in decomposing multi-subject fmri data. Ying Guo (Emory University) SRCOS /03/ / 31

7 Modeling covariate effects on brain functional networks The brain functional networks may differ depending on subjects clinical and demographical characteristics. For example, Greicius MD et al. (2007): increased default-mode network functional connectivity in subjects with major depression. Greicius MD et al. (2007). Biol Psychiatry.62(5): Ying Guo (Emory University) SRCOS /03/ / 31

8 Existing approaches for assessing covariate effects in ICA Single-subject ICA approach (e.g. Greicius MD et al., 2007) Run single-subject ICA on each subject select IC in each subject that most closely matched to the functional network of interest group analysis based on subjects best-matched IC images Existing group ICA: Estimate subject-specific spatial maps: back-construction, dual-regression, or model-based estimation group analysis based on subject-specific IC maps Ying Guo (Emory University) SRCOS /03/ / 31

9 A hierarchical group PICA regression model The first-level model The second-level model Y i (v) = A i s i (v) + γ (1) i (v), s i (v) = s(v) + β(v) x i + γ (2) i (v), Y i (v) (T 1): the observed fmri data from subject i at voxel v s i (v) (q 1): spatial source signals for subject i at voxel v A i (T q): mixing matrix for subject i γ (1) i (v): residual variability in the observed data N(0, Σ (1) ). s(v) (q 1): statistically independent population spatial source signals, x i (p 1): p covariates of subject i β(v)(p q): regression parameters of the p covariates for the q IC γ (2) i (v): subject random variability, N(0, Σ (2) ). Ying Guo (Emory University) SRCOS /03/ / 31

10 Source Distribution Model Based on the characteristics of fmri signals, we model s(v) = [s 1 (v),..., s q (v)] with a mixture of Gaussian distributions, m f (s l (v); ϕ l ) = π lj g(s l (v); µ lj, σlj) 2 j=1 where g( ) is the Gaussian pdf. Define z l (v) as latent states variable for the lth IC at voxel v with p (z l (v) = j) = π lj. Given z l (v), we have p (s l (v) z l (v) = j) = g(s l (v); µ lj, σ 2 lj), Let z(v) as q 1 latent states vector for the q ICs at voxel v. z(v) = [z 1 (v),..., z q (v)], Ying Guo (Emory University) SRCOS /03/ / 31

11 Estimation Complete Log-likelihood Function log l(θ Y, SI, S, Z) { V N [ ( = log g Yi (v); A i s i (v), Σ (1)) v=1 i=1 + log g ( s i (v); s(v) + β(v) x i, Σ (2))] } ( ) q + log g s(v); µ z(v), Σ (3) z(v) + log π lz(v) where Y = {Y i (v)}, SI = {s i (v)}, S = {s(v)} and Z = {z(v)}, θ = {{A i }, Σ (1), Σ (2), ϕ} where ϕ include parameters in the source distribution model. Ying Guo (Emory University) SRCOS /03/ / 31 l=1

12 An EM algorithm E-Step: we find the conditional expectation for the complete data log-likelihood, E (k)[l(θ Y, S, SI, Z)] S,SI,Z Y, ˆθ = Q 1 (θ ˆθ (k) ) + Q 2 (θ ˆθ (k) ) + Q 2 (θ ˆθ (k) ) + Q 4 (θ ˆθ (k) ), Ying Guo (Emory University) SRCOS /03/ / 31

13 The EM algorithm, Cont. Q 1 (θ ˆθ (k) ) = Q 2 (θ ˆθ (k) ) = Q 3 (θ ˆθ (k) ) = Q 4 (θ ˆθ (k) ) = V v=1 V v=1 N [ ( E log g S,SI,Z Y,θ (k) Yi (v); A i s i (v), Σ (1))] i=1 N [ ( E log g S,SI,Z Y,θ (k) si (v); s(v) + β(v) x i, Σ (2))] i=1 V E [log g S,SI,Z Y,θ (k) v=1 V v=1 q E S,SI,Z Y,θ (k)[log π lz(v)] l=1 ( )] s(v); µ z(v), Σ (3) z(v) Ying Guo (Emory University) SRCOS /03/ / 31

14 The EM algorithm, Cont. Derive the conditional distributions needed for Q(θ ˆθ (k) ) (p(s i y, θ), p(s y, θ), p(s, s i y, θ) and p(z y, θ)), We first derive p(s i z, y, θ), p(s z, y, θ), p(s, s i z, y, θ). Conditioning on z(v), rewrite the source distribution model as, s(v) = Gz(v)µ + γ (3) (v), Third-level Model where µ = [µ 11,..., µ qm ], Gz(v) is q mq binary indicator matrix, and γ (3) (v) MVN(0, Σ (3) z(v) ). We then derive p(z y, θ) Obtain p(s i y, θ), p(s y, θ), and p(s, s i y, θ). Ying Guo (Emory University) SRCOS /03/ / 31

15 The EM algorithm, Cont. M-Step Parameter updates for the first-level model: Â (k+1) { V i = v=1 Y i(v)e (s i (v) Y(v); ˆθ (k))} { V v=1 E (s i (v)s i (v) Y(v); ˆθ (k))} 1 { ˆΣ (1)(k+1) = 1 V N NV v=1 i=1 Y i (v)y i (v) 2Y i (v)e ( s i (v) Y(v); ˆθ (k)) Â (k+1) i ( +Â(k+1) i E s i (v)s i (v) Y(v); ˆθ (k)) } Â (k+1) i Ying Guo (Emory University) SRCOS /03/ / 31

16 The EM algorithm, Cont. For the second-level model, the covariate effects is updated by ( N ) 1 N ˆβ(v) (k+1) = x i x i x i [E (s i (v) (k)) Y(v); ˆθ i=1 i=1 ( E s(v) Y(v); ˆθ (k))], while Σ (2) = diag(ν 2 1, ν 2 2,..., ν 2 q) is updated by ˆν 2(k+1) l = 1 NV V v N i=1 { ( E s il (v) 2 Y(v); ˆθ (k)) + E (s l (v) 2 Y(v); ˆθ (k)) ( 2E s il (v)s l (v) Y(v); ˆθ (k)) + ˆβ l (v) (k+1) x i x ˆβ i l (v) (k+1) ( ( +2 E s l (v) Y(v); ˆθ (k)) ( E s il (v) Y(v); ˆθ (k))) x ˆβ i l (v) (k+1), (2) Ying Guo (Emory University) SRCOS /03/ / 31

17 An EM algorithm, Cont. Parameter updates for the source distribution model: ˆπ (k+1) lj = 1 V p(z(v) l = j Y(v); ˆθ (k) ), V ˆµ (k+1) lj = ˆσ 2(k+1) lj = v=1 V v=1 p(z(v) l = j Y(v); ˆθ (k) )E (s(v) l z(v) l = j, Y(v); ˆθ (k)) V ˆπ (k+1) lj V v=1 p(z(v) l = j Y(v); ˆθ (k) )E (s(v) 2 l z(v) l = j, Y(v); ˆθ (k)) (ˆµ (k+1) lj ) 2. V ˆπ (k+1) lj, Ying Guo (Emory University) SRCOS /03/ / 31

18 Steps for the model estimation Step 1. Start with initial parameter values ˆθ (0) = (Â (0) i, {ˆΣ (1,0) }, {ˆΣ (2,0) }, ˆϕ (0) ), β (0). Step 2. E-step: compute the conditional expectation function Q(θ ˆθ (k) ). Step 3. M-step: updating parameter estimates using the explicit solutions. ˆθ (k+1) = argmax Q(θ ˆθ (k) ) θ Step 4. Iterate between steps 2-3 until convergence. Step 5. Estimate IC spatial maps based on ˆθ for specified covariates x. ŝ (v) = E(s(v) Y, ˆθ) + ˆβ(v) x Ying Guo (Emory University) SRCOS /03/ / 31

19 Dimension Reduction and Whitening Prior to ICA, dimension reduction and whitening is performed, Ỹ i (v) = (Λ i,q σ 2 i,qi q ) 1/2 U i,qy i (v), (3) U i,q and Λ i,q : the first q eigenvectors and eigenvalues from SVD of [Y i (1),..., Y i (V )] T V. The first-level model: where Ỹ i (v) = Ãis i (v) + γ (1) i (v), Ỹ i (v) = H i Y i (v), Ãi(v) = H i A i (v), γ (1) i (v) = H i γ (1) i (v) H i = (Λ i,q σ 2 i,q I q) 1/2 U i,q: transformation matrix Ying Guo (Emory University) SRCOS /03/ / 31

20 Simulation Studies We compare the performance of our method (Covariate ICA) vs. an existing method (Dual regression) under the following simulation settings simulate fmri data from q = 3 source signals and consider sample sizes of n = 10, 20, 40 subjects for each IC: 3D population spatial maps ( ) and time course of length T = 200 simulate two covariates X = (X 1, X 2 ), X 1 U( 1, 1), X 2 Bernoulli(0.5) multi-subject fmri data with low and high between-subject variability in spatial distributions of source signals simulate subject-specific time courses for each source signal Ying Guo (Emory University) SRCOS /03/ / 31

21 Simulation Study: Group Level IC maps Group IC (True) Group IC (Cov.ICA.) Group IC (Dual.Reg.) Ying Guo (Emory University) SRCOS /03/ / 31

22 Simulation Study: Covariate Effect Maps for X 1 β(v)(true) β(v)(cov.ica.) β(v)(dual.reg.) Ying Guo (Emory University) SRCOS /03/ / 31

23 Simulation Study: Covariate Effect Maps for X 2 β(v)(true) β(v)(cov.ica.) β(v)(dual.reg.) Ying Guo (Emory University) SRCOS /03/ / 31

24 Simulation Studies Table: Simulation results: Population-level maps and covariate effects. Btw-subj Population-level spatial maps Covariate Effects Var Corr(SD) MSE(SD) Cov.ICA. Dual.Reg. Cov.ICA. Dual.Reg. Low N= (0.003) 0.953(0.017) 0.048(0.019) 0.154(0.055) N= (0.002) 0.955(0.002) 0.021(0.003) 0.127(0.044) N= (0.002) 0.952(0.003) 0.012(0.001) 0.111(0.030) MSE( ˆβ) is defined as 1 V E( V v=1 ˆβ(v) β(v) F ) Ying Guo (Emory University) SRCOS /03/ / 31

25 An Approximated EM algorithm The proposed exact EM algorithm: Pros: provide explicit solutions for E-step and M-step Cons: this exact estimation becomes computational expensive when the number of ICs increases. We propose an approximation EM algorithm for faster computation. It is derived based on the fact that fmri source signals are generally sparse and well-separated in distributed patterns. Ying Guo (Emory University) SRCOS /03/ / 31

26 Simulation Studies: the Exact EM vs. the Approximated EM Num. of Population-level spatial maps Covariate Effects IC Corr(SD) MSE(SD) Exact EM Approx EM Exact EM Approx EM q= (0.003) 0.981(0.001) 0.048(0.020) 0.048(0.019) q= (0.006) 0.980(0.006) 0.069(0.024) 0.070(0.022) q= (0.019) 0.970(0.018) 0.097(0.035) 0.103(0.035) Num. of IC Computation Time (min) Exact EM Approx EM q= q= q= Ying Guo (Emory University) SRCOS /03/ / 31

27 A Data Example A Zen meditation study 24 subjects: Zen meditation group (n=12) and control group (n=12). The two groups were matched for gender,age, and education level. A series of 520 functional MRI images were acquired with a 3.0 Tesla Siemens Magnetom Trio scanner, each containing voxels. Word or nonword items presented visually in random order; press a button with the left hand (index finger = yes, middle finger = no ); focus on breathing in other time points. We fit the proposed ICA regression model with a covariate indicating group membership: x = 1 for meditator and x = 0 for controls. Ying Guo (Emory University) SRCOS /03/ / 31

28 Figure: Model-based estimation of subgroup spatial maps for functional networks 8 Task-related network 60 Default mode network Control Temporal corr. with task time series: Meditator Temporal corr. with task time series: 0.86 Ying Guo (Emory University) SRCOS /03/ / 31

29 Summary We propose a hierarchical group PICA regression model The proposed model provides a formal statistical framework for modeling covariate effects in group ICA. Our model can provide model-based estimation of spatial distributed patterns of brain functional networks based on subjects demographic and clinical characteristics. We propose an exact EM algorithm and an approximation EM for model estimation. Simulation results show that our proposed model has better performance than the existing method. Ying Guo (Emory University) SRCOS /03/ / 31

30 Acknowledgements Giuseppe Pagnoni, PhD. Department of Biomedical Sciences and Technologies, University of Modena and Reggio Emilia, Modena, Italy Emory University URC grant and NIH grant R01-MH Ying Guo (Emory University) SRCOS /03/ / 31

31 References Beckmann, C.F. and Smith, S.M. (2005). Tensorial extensions of independent component analysis for multisubject FMRI analysis, NeuroImage 25, Calhoun, V.D., Adali, T., Pearlson, G.D. and Pekar, J.J. (2001). A method for making group inferences from functional MRI data using independent component analysis, Human Brain Mapping 14, Guo Y. and Pagnoni G. (2008). A unified framework for group independent component analysis for multi-subject fmri data. NeuroImage 42, Guo Y. (2011). A General Probabilistic Model for Group Independent Component Analysis and Its Estimation Methods. Biometrics 67, Guo Y. and Tang L. A hierarchical model for probabilistic independent component analysis of multi-subject fmri studies. Biometrics. (Accepted). Ying Guo (Emory University) SRCOS /03/ / 31