Integrative Methods for Functional and Structural Connectivity

Transcription

1 Integrative Methods for Functional and Structural Connectivity F. DuBois Bowman Department of Biostatistics Columbia University SAMSI Challenges in Functional Connectivity Modeling and Analysis Research Triangle Park, NC April 9, 2016 F. D. Bowman Integrative Neuroimaging Methods SAMSI 1 / 40

2 Multimodal Imaging Magnetic Resonance Imaging Modalities Magnetic resonance imaging (MRI) Functional MRI (fmri) Diffusion Tensor Imaging (DTI) 60 v= F. D. Bowman Integrative Neuroimaging Methods SAMSI 2 / 40

3 Multimodal Imaging Integrative Approaches There is clear interdependence between brain function and structure What are key considerations in combining information across modalities? What properties do the imaging measurements truly reflect? Are there limits on the extent to which we can model the available knowledge of brain function/structure relationships? How can we couple the measurements from multiple imaging modalities, which may have different spatial scales? F. D. Bowman Integrative Neuroimaging Methods SAMSI 3 / 40

4 Multimodal Connectivity Anatomically-weighted Functional Connectivity F. D. Bowman Integrative Neuroimaging Methods SAMSI 4 / 40

5 Multimodal Connectivity Anatomically-weighted Functional Connectivity (awfc) [Bowman et al., 2012, NeuroImage] Our main goal is to reliably detect functionally connected regions We leverage supplementary information from the SC strength between brain regions SC does not imply FC Absence of SC does not preclude FC Indirect SC (e.g. via a remote third brain region) Potential Advantages: Help determine FC in the presence of fmri noise Defaults to standard unweighted techniques when no SC is present Provides additional information about connectivity relationships F. D. Bowman Integrative Neuroimaging Methods SAMSI 5 / 40

6 Subregion Selections and Summaries Whole-brain analysis with subregions for each AAL region For fmri, Resting-state subregions selected in GM to be representative of larger regions Task-based subregions selected according to other criterion, e.g. centered on maximally activated voxel in region (across all tasks) Generate summary temporal signal for each subregion For DTI, Subregions centered in WM proximal to GM regions Probabilistic tractography used to summarize region-to-region SC fmri DTI F. D. Bowman Integrative Neuroimaging Methods SAMSI 6 / 40

7 Functional Connectivity Whole-Brain Region to Region Approaches Calculate the correlation (or partial correlation) between a all pairs of regions). Regional Associations ρ ij = max u U { T u t=1 [y i(t + u) ȳ i ][y j (t) ȳ j ] ˆσ i ˆσ j } We allow lag-u associations, u [ 2, 2] Define functional dissimilarity between regions i and j, e.g. f ij = 1 ρ ij F. D. Bowman Integrative Neuroimaging Methods SAMSI 7 / 40

8 Combined Distance Anatomically Weighted Functional Distance ( d ij = 1 π ) ij f ij λ π ij is the probability of structural connectivity We consider second-order connections Adjust for physical distances between regions λ is a parameter used to potentially attenuate the impact of anatomical weighting λ determined empirically λ [1, ) d ij f ij, as λ F. D. Bowman Integrative Neuroimaging Methods SAMSI 8 / 40

9 Determining Functional Networks Functional Clustering with DTI-based Anatomical Weighting: F. D. Bowman Integrative Neuroimaging Methods SAMSI 9 / 40

10 awfc Objective Function: 6 (a) h(λ, G) = log ( FCwi FC tot (b) ) G= λ= Σ i i (λ) (G) λ G (lag 1) (a) Empirical optimization of the attenuation parameter λ (b) Determination of the number of cluster, G, for subject 1 F. D. Bowman Integrative Neuroimaging Methods SAMSI 10 / 40

11 Data Example Auditory Task Study N=17 healthy female subjects 3 fmri runs (483 scans total for each subject) fmri Auditory Task Cue: Hear an auditory tone (2000 Hz) in one ear. Target: Subsequently hear a target tone (1000 Hz) in either ear. Instructed to click a button with right (a) index finger if the target is from the left earphone OR (b) middle finger if from right earphone. 50% of the trials have the cue and target tones in the same ear. Subjects fixated at a cross throughout the entire scanning session. DTI data also collected Objective: Study had a broad set of objectives (Mayer et al., 2009). We consider an analysis that identifies different functional networks. F. D. Bowman Integrative Neuroimaging Methods SAMSI 11 / 40

12 awfc: Analysis of Task Data Motor Motor Visual Visual Auditory F. D. Bowman Integrative Neuroimaging Methods SAMSI 12 / 40

13 awfc Results: Auditory Data (a) (b) Network Correlations median In network Cross network Network SC Proportion In network Cross network Median Correlation Proportion Connected subject subject (a) Median correlations within-networks and between-networks (b) Proportion of region pairs exhibiting SC, i.e. Pr(SC) > 0.5, within-networks and between-networks F. D. Bowman Integrative Neuroimaging Methods SAMSI 13 / 40

14 awfc Results: Auditory Data Regional Time-series from 90 Brain Regions (Subject 2) F. D. Bowman Integrative Neuroimaging Methods SAMSI 14 / 40

15 awfc Results: Auditory Data Time-series for awfc Networks (Subject 2) F. D. Bowman Integrative Neuroimaging Methods SAMSI 15 / 40

16 Integrative Methods Bayesian Multimodal Model for Joint Activation F. D. Bowman Integrative Neuroimaging Methods SAMSI 16 / 40

17 Joint FC-SC Model Figure : Histogram of joint activation counts (Z1) at a lower and higher level of SC for two subjects across region pairs based on 205 brain regions. Note that the joint activation values tend to be larger between region pairs exhibiting high SC relative to low SC. [Xue et al. (2015). Frontiers in Neuroscience] F. D. Bowman Integrative Neuroimaging Methods SAMSI 17 / 40

18 Joint FC-SC Model Joint Activation and Structural Connectivity Define: A gnt = I (R gnt > c σ gn ), R gnt = Y gnt ˆµ gn is the mean-adjusted level of neural activity for region g, subject n, and scan t c is a constant σ 2 gn is the variance of Y gnt. A gnt is an indicator of elevated regional brain activity at time t. F. D. Bowman Integrative Neuroimaging Methods SAMSI 18 / 40

19 Joint FC-SC Model Joint Activation The joint activation between two regions a and b can be expressed as: Z1 = I (A ant = 1, A bnt = 1), Z2 = I (A ant = 1, A bnt = 0) T T t=1 t=1 T T Z3 = I (A ant = 0, A bnt = 1), Z4 = I (A ant = 0, A bnt = 0). t=1 t=1 (1) Z 1 (omitting subject index) gives the number of times that both regions a and b experience an elevated fmri signal together. F. D. Bowman Integrative Neuroimaging Methods SAMSI 19 / 40

20 Joint FC-SC Model Joint Activation We assume Z = (Z 1,, Z 4 ) follows a multinomial distribution with parameters T and θ = (θ 1,, θ 4 ), where θ 1 = P (A ant = 1, A bnt = 1), θ 2 = P (A ant = 1, A bnt = 0) θ 3 = P (A ant = 0, A bnt = 1), θ 4 = P (A ant = 0, A bnt = 0). (2) F. D. Bowman Integrative Neuroimaging Methods SAMSI 20 / 40

21 Joint FC-SC Model Structural Connectivity Let S denote the DTT counts linking regions a and b for subject n (omitting subscripts for simplicity) out of M trials for probabilistic fiber tracking. We regard S to follow a binomial distribution with parameters M and π, where π is the probability of SC between regions a and b for any subject. F. D. Bowman Integrative Neuroimaging Methods SAMSI 21 / 40

22 Joint FC-SC Model Functional Coherence and Ascendancy Table : Joint activation probabilities for regions a and b. Region a Active Inactive Region b Active θ 1 θ 3 θ 1 + θ 3 Inactive θ 2 θ 4 θ 2 + θ 4 θ 1 + θ 2 θ 3 + θ 4 1 F. D. Bowman Integrative Neuroimaging Methods SAMSI 22 / 40

23 Joint FC-SC Model Functional coherence between pairs of brain regions: { θ1 +θ 4 E κ = 1 E if θ 1 θ 4 > θ 2 θ 3 0 otherwise, (3) where E = (θ 1 + θ 2 )(θ 1 + θ 3 ) + (θ 3 + θ 4 )(θ 2 + θ 4 ). Numerator measures the difference between the probability of coherence and the expected probability under independence. κ [0, 1] (we disregard cases where θ 1 + θ 4 < E) κ = 1 indicates perfect coherence (θ 1 + θ 4 = 1) κ = 0 reflects no agreement (beyond chance) Extends the agreement measure of Patel et al. (2006a), which is based on Cohen s Kappa (Cohen, 1960) F. D. Bowman Integrative Neuroimaging Methods SAMSI 23 / 40

24 Joint FC-SC Model Given that a and b are functionally connected, i.e., κ exceeds a specified threshold (say e κ ) with high probability, Ascendancy: τ ab = θ / 1 + θ 2 θ1 + θ 3. (4) θ 3 + θ 4 θ 2 + θ 4 Assess the hierarchical relationship between the functionally connected regions Odds of region a being active, P(A a = 1)/(1 P(A a = 1)), relative to the odds of region b being active, P(A b = 1)/(1 P(A b = 1)) τ ab [0, ) F. D. Bowman Integrative Neuroimaging Methods SAMSI 24 / 40

25 Joint FC-SC Model Joint Bayesian Model: Likelihood For any pair of regions a and b, the likelihood function is: p(z, S θ, π) 4 θ i=1 N Z in n=1 i π N n=1 S n(1 π) N M N n=1 S n. (5) Each repeated measure on the same region pair is regarded as independent over time Pre-whitening performed We build structure-function dependence in the distribution of [θ π] Conditional independence between S and Z, given θ and π. F. D. Bowman Integrative Neuroimaging Methods SAMSI 25 / 40

26 Joint FC-SC Model Joint Bayesian Model: Prior Distributions Define a beta prior for structural connection probabilities π: p(π) π α 0 1 (1 π) β 0 1. (6) If no prior information available for SC, can specify a flat prior for DTT data by setting α 0 = β 0 = 1 for each region pair F. D. Bowman Integrative Neuroimaging Methods SAMSI 26 / 40

27 Joint FC-SC Model Joint Bayesian Model: Prior Distributions Assume θ π Dirichlet (α(π) + α 1, α 2, α 3, α 4 ) α(π) is a function of π, reflecting relationship between FC and SC The expected value of θ 1, say η 1 = E(θ 1 ), is given by η 1 (π) = (α(π) + α 1 )/(α(π) + α 1 + α 2 + α 3 + α 4 ) (7) When α(π) is an increasing function, η 1 (π) is also an increasing function, with respect to π Matches our observation from the data F. D. Bowman Integrative Neuroimaging Methods SAMSI 27 / 40

28 Joint FC-SC Model Joint Bayesian Model: Prior Distributions A sensitivity analysis shows that our results do not change much with respect to different functions of α(π). Figure : Increasing functions of π; all have the same area under curve. F. D. Bowman Integrative Neuroimaging Methods SAMSI 28 / 40

29 Joint FC-SC Model Joint Bayesian Model: Prior Distributions Functional connectivity network and ascendacy F. D. Bowman Integrative Neuroimaging Methods SAMSI 29 / 40

30 Integrative Methods Joint Multimodal Model Prediction Models F. D. Bowman Integrative Neuroimaging Methods SAMSI 30 / 40

31 Neuroimaging Biomarkers Multimodal Imaging Biomarkers We use a multimodal approach to combine several types of imaging data, along with clinical and demographic information. Exploratory candidate biomarkers (a massive number) Hypothesis driven candidate biomarkers (imaging and clinical) Y = { 1, PD patient 0, Control F. D. Bowman Integrative Neuroimaging Methods SAMSI 31 / 40

32 Multimodal Methods Penalized likelihood approaches: l p (β, λ) = l(β) + λ[ p j=0 1 2 (1 α)β2 j + α β j ] We use a logistic model for the likelihood: l(β) = i [y ilogπ i (β) + (1 y i )log(1 π i (β))] π i (β) = exp(x i β) 1+exp(x i β) Pool predictive strength across multiple data modalities Performs variable selection or shrinkage Well-suited for high dimensional data Model development, training, testing, and validation Adapt standard implementation to encourage reproducibility and robustness [Bowman et al., 2016, Frontiers in Neuroscience] F. D. Bowman Integrative Neuroimaging Methods SAMSI 32 / 40

33 Neuroimaging Biomarkers Feature Selection: Strength and Consistency Define restricted tuning parameter space: B = {(α, λ) p 75, AUC 0.9} Repeated k-fold CV at each operating point (200 repititions) F. D. Bowman Integrative Neuroimaging Methods SAMSI 33 / 40

34 Neuroimaging Biomarkers Top features over (α, λ) Feature Upper 10% Pred. Strength Direction FC:Amygdala R x Angular R* 91.6% 1 FC: Amygdala R x Lingual L 100.0% 1 FC: Calcarine L x Thalamus L 100.0% -1 FC.Cingulum Ant R x Cingulum Post L 100.0% 1 FC: Cuneus R x Precuneus R 100.0% -1 FC: Frontal Inf Orb R x Temporal Mid R 99.6% 1 FC: Frontal Inf Orb R x Temporal Mid L 100.0% 1 FC: Frontal Inf Tri R x Temporal Pole Mid R 99.2% 1 FC: Frontal Mid Orb L x Hippocampus L 100.0% 1 FC:Frontal Sup Medial L x Cingulum Ant L 93.8% 1 FC: Frontal Sup Orb L x Insula L 100.0% 1 FC:Frontal Sup Orb L x Parietal Inf L 100.0% 1 FC: Frontal Sup Orb L x Temporal Sup R 100.0% -1 FC:Occipital Mid L x Occipital Inf R 97.9% 1 FC:Occipital Sup L x Temporal Mid R 100.0% -1 FC:Occipital Sup R x Precuneus R 100.0% -1 FC: Temporal Mid R x Temporal Pole Mid R 100.0% 1 FC: Temporal Sup R x Temporal Pole Mid L 100.0% 1 FC: Thalamus L x Temporal Pole Mid L 100.0% -1 SC: Calcarine L x Precuneus R 97.8% -1 VBM: Frontal Inf Orb R 100.0% -1 VBM: Frontal Mid R 100.0% features identified (20 FC, 2 VBM, and 1 SC). The 23 features are consistently the most predictive across a restricted tuning parameter space for (α, λ) in EN. Predictive strength, for a given (α, λ), was computed as the mean absolute coefficient (normalized) across 200 training samples. 16 features were retained at 100% of the tuning parameter values. *Two distinct FC links between these regions. F. D. Bowman Integrative Neuroimaging Methods SAMSI 34 / 40

35 Neuroimaging Biomarkers Model 1 Model 2 Model 3 Thalamic and limbic system alterations hippocampus, amygdala, orbitofrontal cortex, cingulate gyrus Three variable models achieving 100% accuracy (models adjusted for age and sex). In a bootstrap analysis, each of the models appears in all 100 bootstrap samples. F. D. Bowman Integrative Neuroimaging Methods SAMSI 35 / 40

36 Neuroimaging Biomarkers F. D. Bowman Integrative Neuroimaging Methods SAMSI 36 / 40

37 Remarks Rich information is available reflecting properties of brain function and structure There are established benefits to considering the modalities collectively Statistical modeling can be complicated by the massive amount of data generated Attention must be devoted to preprocessing details Direct correspondence between functional and structural modalities For Bayesian methods, the future holds promise for incorporating informative priors into statistical models F. D. Bowman Integrative Neuroimaging Methods SAMSI 37 / 40

38 Acknowledgements Collaborators: Daniel Drake, PhD, Columbia University Ben Cassidy, PhD, Columbia University Jaehee Kim, PhD, Duksung Women s University/Columbia Jian Kang, PhD, University of Michigan Lijun Zhang, PhD, Pennsylvania State University Wenqiong Xue, PhD, Boehringer Ingelheim Daniel Huddleston, MD, Emory University NINDS U18 NS (Bowman) and the PDBP F. D. Bowman Integrative Neuroimaging Methods SAMSI 38 / 40

39 References Bowman, F. D., Drake, D., Huddleston, D. (2016). Multimodal Imaging Signatures of Parkinson s Disease. Frontiers in Neuroscience 10: Bowman F. D., Zhang L., Derado G., Chen S. (2012). Determining functional connectivity using fmri data with diffusion-based anatomical weighting. NeuroImage 62, /j.neuroimage Xue, W., Bowman, F. D., Pileggi, A. V., Mayer, A. R. (2015). A Multimodal Approach for Determining Brain Networks by Jointly Modeling Functional and Structural Connectivity. Frontiers in Computational Neuroscience 9: F. D. Bowman Integrative Neuroimaging Methods SAMSI 39 / 40

40 Thank you! F. D. Bowman Integrative Neuroimaging Methods SAMSI 40 / 40