Model Selection. Results. Anatomically Informed. Bayesian Model Selection for fMRI Group Data Analysis. M. Keller. CEA, NeuroSpin. University of Paris Sud.
Introduction
Spatial Uncertainty
Model Selection
Anatomically Informed Bayesian Model Selection for fMRI Group Data Analysis M. Keller CEA, NeuroSpin University of Paris Sud
Select-NeuroSpin meetings, may 2009
Results
Introduction
Spatial Uncertainty
Model Selection
fMRI Group Data Analysis
Goal : Identify brain structures consistently involved in a certain cognitive task accross individuals Example : Activation maps of n subjects during a number processing task ,→ Infer regions of positive mean group activation Strong between-subject variability, both functional and anatomical
Results
Introduction
Spatial Uncertainty
Model Selection
The Mass-Univariate Approach
Normalize individual images to a common brain template Compute t-statistic map Identify clusters above a given threshold Clusters whose size exceed a critical value are reported [Hayasaka and Nichols, 2003] Clusters related to anatomical structures using digital atlases [Tzourio-Mazoyer et al., 2002, Perrot et al., 2009]
Results
Introduction
Spatial Uncertainty
Model Selection
Limits of The Mass-Univariate Approach
Simple, widely applicable, the above method [Friston, 1997] suffers from the following drawbacks : Arbitrary cluster-forming threshold May result in merging or missing activation regions Exclusive false positive control Their is no guarantee that the whole functional network can be recovered Assumption of perfect match between individual brains Neglects registration errors ; may result in blurring the activation map, and creating false positives
Results
Introduction
Spatial Uncertainty
Model Selection
Dependence on the cluster-forming threshold
t-map thresholded at FPR = 10−2 , 10−3 and 10−4 Clusters detected at FWER = 0.05
Results
Introduction
Spatial Uncertainty
Model Selection
Existing Alternatives
Extension of the mass-univariate approach to control false negatives using Bayesian inference [Friston et al., 2002] Threshold-Free Cluster Enhancement (TFCE) [Smith and Nichols, 2009] Spatial normalization errors addressed through feature-based approaches [Xu et al., 2007, Thirion et al., 2007, Tucholka et al., 2008], or by extending the mass univariate model [Keller et al., 2008]
Results
Introduction
Spatial Uncertainty
Model Selection
Proposed Approach
Address spatial normalization errors by extending the mass univariate model, as in [Keller et al., 2008] Define candidate regions by pre-defined parcellations, such as the atlas in [Perrot et al., 2009] Select functional network associated to target task using Bayesian model selection
Results
Introduction
Spatial Uncertainty
Model Selection
Results
Observation Model For each subject i = 1, . . . , n, at voxel k = 1, . . . , p : Yi (vk ) εik
= ind.
∼
µ(vk + uik ) + εik ;
(1)
N (0, σI2 + sik2 ),
(2)
Yi (vk ) estimated effect (observation) µ ∈ Rp map of mean group effects uik subject-to-atlas registration error sik2 ∈ Rp within-subject variance (known) σI2 between-subject variance Generalizes [Mériaux et al., 2006] (uik ≡ 0), [Friston, 1997] (sik2 ≡ 0)
Introduction
Spatial Uncertainty
Model Selection
Results
Deformation Field Model The registration error for subject i at voxel k is given by : uik
=
B X
K(vk , vkb )wib
(3)
b=1
K(vk , vkb ) wi
= i.i.d
∼
exp −{kvk − vkb k2 /γ 2 }
(4)
N (0, σS2 I3B ) 1{∀k , vk + uik ∈ V}
(5)
{vkb , b = 1, . . . , B} fixed (user-chosen) control points wib elementary displacements γ displacement field regularity σS standard registration error
Introduction
Spatial Uncertainty
Model Selection
Results
A Partial Justification Theorem ([Girosi et al., 1995]) For i = 1, . . . , n, given (wib , vkb )1≤b≤B , the function f : R3 → R3 defined by f (v ) =
B X
K(v , vkb )wib
(6)
(f (vkb ) − wib )2 + λφ[f ],
(7)
b=1
minimizes the functional H[f ] =
B X b=1
for a certain regularization parameter λ and a certain smoothness functional φ[f ].
Introduction
Spatial Uncertainty
Model Selection
Results
Illustration on Toy Data
Original Dignal
Simulated Data
No Spatial Uncertainty
Spatial Uncertainty
n = 30 images simulated under full model (≈ 3 000 control points ; γ = 3 voxels) θ = (µ, σI2 , σS2 ) estimated by the MAP θˆ = arg maxθ p(θ|Y ) using a MCMC-SAEM algorithm [Kuhn and Lavielle, 2004] (≈ 100 control points, γ = 3 voxels)
Introduction
Spatial Uncertainty
Model Selection
Sensitivity Study
Loss
1 p
P
k (µk
−µ ˆk )2 over 100 trials, against noise level
Field regularity in voxels (left to right) : γ = 2.0, 3.0, 4.0. Model estimated with γ = 3.0
Results
Introduction
Spatial Uncertainty
Model Selection
Illustration on Real Data
n = 24 subjects, number processing task (subtraction), Localizer Database [Pinel et al., 2007]
No Spatial Uncertainty
Spatial Uncertainty
Results
Introduction
Spatial Uncertainty
Model Selection
Results
Regionalized Group Activation Model The search volume V = {vk , 1 ≤ k ≤ p} ⊂ R3 is divided into N fixed regions of interest, assumed functionally homogeneous : V = V1 ∪ . . . ∪ VN
(8)
In region Vj , mean group effects {µk , vk ∈ Vj } are modeled according to : µk
i.i.d.
∼
N (ηj , σj2 ),
with : ηj the regional mean effect, σj2 the regional variance
(9)
Introduction
Spatial Uncertainty
Model Selection
Results
Most Probable Functional Network Region Vj is said to be involved in the task if ηj 6= 0 inactive if ηj = 0 The functional network we wish to recover is represented by the unknown binary label vector Γ? ∈ {0, 1}N , such that Γ?j = 1 if ηj 6= 0 Γ?j = 0 if ηj = 0. Goal : estimate Γ? by the a posteriori most probable network : ˆ? = arg max p(Γ|Y ) Γ Γ
(10)
= arg max p{∀j, Γj = 1 : ηj 6= 0; ∀j, Γj = 0 : ηj = 0|Y }. Γ
Introduction
Spatial Uncertainty
Model Selection
Results
Independence Approximation Computing p(Γ|Y ) for all 2N possible choices of Γ is intractable. Instead, we use the following independance approximation : p(Γ|Y ) ≈
Y Γj =1
p(ηj 6= 0|Y )
Y
p(ηj = 0|Y ).
(11)
Γj =0
(11) would be exact conditional on σI2 , if σS2 = 0, uik ≡ 0. ˆ? = 1 for all j such that p(ηj 6= 0|Y ) > 0.5. Using (11), Γ j
Introduction
Spatial Uncertainty
Model Selection
Results
Posterior Probability Computation p(ηj 6= 0|Y ) = Pj and p(ηj = 0|Y ) = 1 − Pj computed through their Bayes’ factor : Pj 1 − Pj
=
=
p(Y |ηj 6= 0) p(Y |ηj = 0) Z p(Y |ηj , σj2 ) π(ηj , σj2 ) dηj dσj2 Z p(Y |ηj = 0, σj2 ) π(σj2 ) dσj2
assuming a uniform prior π(ηj 6= 0) = π(ηj = 0) = 0.5 π(ηj , σj2 ) and π(σj2 ) are normal-inverse-Gamma and inverse-Gamma priors, as in [Keller et al., 2008].
(12)
Introduction
Spatial Uncertainty
Model Selection
Results
Marginal Likelihood Computation Computing p(Y ) = p(Y |ηj 6= 0) or p(Y |ηj = 0) is difficult (integral on high-dimensional space) Basic Marginal Equality (BME) can be used [Chib, 1995] : p(Y ) =
p(Y |θ) π(θ) , π(θ|Y )
(13)
Valid for any θ = (σI2 , σS2 , ηj , σj2 ) or (σI2 , σS2 , σj2 ) We use the MAP θˆ = arg maxθ p(Y |θ), estimated using a MCMC-SAEM algorithm [Kuhn and Lavielle, 2004] ˆ and π(θ|Y ˆ )! Still have to evaluate p(Y |θ)
Introduction
Spatial Uncertainty
Model Selection
Results
Maximum Likelihood Estimation Likelihood obtained by integrating out elementary displacements w : Z ˆ ˆ p(w|θ) ˆ dw, p(Y |θ) = p(Y |w, θ)
(14)
Cannot be computed explicitely ˆ 1, except “near” w ˆ ˆ = arg max p(w|Y , θ) p(Y |w, θ) w
ˆ by the upper bound : ,→ Estimate p(Y |θ) ˆ = p(Y |w, ˆ ˆ (Y |θ) ˆ θ), p ˆ estimated by simulated annealing. w
(15)
Introduction
Spatial Uncertainty
Model Selection
Results
Posterior Density Estimation Posterior density obtained by integrating out hidden variables Z = (w, µ) Z ˆ ˆ Y ) p(Z|Y ) dZ. p(θ|Y ) = p(θ|Z, (16)
Cannot be computed explicitely Upper bound can be estimated using the MCMC-SAEM output Z1 , . . . , ZG : ˆ ) / G−1 log p(θ|Y
G X g=1
ˆ g, Y ) log p(θ|Z
(17)
Introduction
Spatial Uncertainty
Model Selection
Simulations
True µ, CSA atlas
t-score map
ˆ E(µ|Y , θ)
Pj histogram
Regions j with ηj ≥ 0.04 all reported active (Pj > 0.5) P Average relative error ε = N1 j |ˆ ηj − ηj |/ηj ≈ 7% Classical approach merged neighboring activated regions
Results
Introduction
Spatial Uncertainty
Model Selection
Number Processing Task
t-score map
ˆ E(µ|Y , θ)
Pj map
n = 38 subjects Candidate regions defined by CSA atlas
Pj histogram
Results
Introduction
Spatial Uncertainty
Model Selection
Results
Detected Regions Frontal lobe Parietal lobe Sulcus/Fissure
Pj
η ˆj
Left middle frontal
1.00
2.94
Right middle frontal
1.00
3.17
Left superior frontal
0.94
2.11
Right superior frontal
1.00
1.93
Left middle precentral
1.00
4.11
Right middle precentral
1.00
Sulcus/Fissure
Pj
η ˆj
Left intra-parietal
1.00
4.87
Right intra-parietal
1.00
3.03
Left precuneus
0.98
5.58
Right precuneus
0.85
3.84
Right inferior postcentral
0.88
2.30
Right parieto-occipital
0.83
1.30
Pj
η ˆj
0.79
1.99
1.98
Left inferior precentral
1.00
5.30
Right inferior precentral
0.97
2.69
Left anterior cingular
1.00
3.39
Right anterior cingular
1.00
4.15
Left inferior frontal
1.00
4.38
Other Sulcus/Fissure Right callosal
Introduction
Spatial Uncertainty
Model Selection
Perspectives
Short Term : Use spatial normalization in [Perrot et al., 2009] instead of standard SPM normalization for individual data processing Apply to larger cohorts (IMAGEN database)
Middle Term : Generalize to probabilistic region labels to account for their variability accross subjects Prove consistency of MAP estimate (Proof in [Allassonnière et al., 2007] in similar settings) Improve estimation of marginal likelihood
Results
Introduction
Spatial Uncertainty
Model Selection
Allassonnière, S., Amit, Y., and A., T. (2007). Towards a coherent statistical framework for dense deformable template estimation. Journal of the Royal Statistical Society : Series B (Statistical Methodology), 69(1) :3–29. Chib, S. (1995). Marginal likelihood from the Gibbs output. J. Amer. Statist. Assoc., 90 :1313–1321. Friston, K., Glaser, D. E., Henson, R. N. A., Kiebel, S., Phillips, C., and Ashburner, J. (2002). Classical and Bayesian inference in neuroimaging : Applications. Neuroimage, 16(2) :484–512. Friston, K. J. (1997). Human Brain Function, chapter 2, pages 25–42. Academic Press. Girosi, F., Jones, M., and Poggio, T. (1995). Regularization Theory and Neural Networks Architectures. Neural Computation, 7 :219–269. Hayasaka, S. and Nichols, T. (2003). Validating Cluster Size Inference : Random Field and Permutation Methods. Neuroimage, 20(4) :2343–2356. Keller, M., Roche, A., Tucholka, A., and B.Thirion (2008). Dealing with spatial normalization errors in fMRI group inference using hierarchical modeling. Statistica Sinica, 18(4) :1357–1374. Kuhn, E. and Lavielle, M. (2004). Coupling a stochastic approximation of EM with a MCMC procedure. ESAIM P&S, 8 :115–131. Mériaux, S., Roche, A., Dehaene-Lambertz, G., Thirion, B., and Poline, J.-B. (2006). Combined permutation test and mixed-effect model for group average analysis in fMRI.
Results
Introduction
Spatial Uncertainty
Model Selection
Results
Hum. Brain Mapp., 27(5) :402–410. Perrot, M., Rivière, D., Tucholka, A., and Mangin, J.-F. (2009). Joint Bayesian Cortical Sulci Recognition and Spatial Normalization. In 21st. IPMI, pages xx–xx, Williamsburg, U.S.A. Pinel, P., Thirion, B., Mériaux, S., Jobert, A., Serres, J., Le Bihan, D., Poline, J.-B., and Dehaene, S. (2007). Fast reproducible identification and large-scale databasing of individual functional cognitive networks. BMC Neurosci, 8(1) :91. Smith, S. M. and Nichols, T. E. (2009). Threshold-free cluster enhancement : addressing problems of smoothing, threshold dependence and localisation in cluster inference. Neuroimage, 44(1) :83–98. Thirion, B., Tucholka, A., Keller, M., Pinel, P., Roche, A., Mangin, J.-F., and Poline, J.-B. (2007). High level group analysis of FMRI data based on Dirichlet process mixture models. In IPMI, volume 4584 of LNCS, pages 482–494. Tucholka, A., Thirion, B., Pinel, P., Poline, J.-B., and Mangin, J.-F. (2008). Triangulating cortical functional networks with anatomical landmarks. In 5th Proc. IEEE ISBI, pages 612–615, Paris, France. Tzourio-Mazoyer, N., Landeau, B., Papathanassiou, D., Crivello, F., Etard, O., Delcroix, N., Mazoyer, B., and Joliot, M. (2002). Automated anatomical labeling of activations in SPM using a macroscopic anatomical parcellation of the MNI MRI single-subject brain. Neuroimage, 15(1) :273–89. Xu, L., Johnson, T., and Nichols, T. (2007). Bayesian spatial modeling of fMRI data : A multiple-subject analysis. Technical report, The University of Michigan Department of Biostatistics.
