COMPARISON OF BRAIN NETWORKS WITH UNKNOWN CORRESPONDENCES SI Ktena, S Parisot, J Passerat-Palmbach and D Rueckert
The brain from a network perspective • cognition is a network phenomenon [ Sporns, Dial. Clin. Neurosc. (2013) ] Brain parcellation • no physical trace of certain diseases, only changes in Timeseries Imaging the physical wiring and data data strength of connections • given two brain graphs Functional Structural brain network brain network representing connectivity, how similar are they? (w ithin/between subjects, Network analysis between modalities etc. ) 2
Inference: Naive approach Model training Graph and inference construction Graph embedding { { Richiardi et al., IEEE Signal Processing Magazine (2013) class class Varoquaux and Craddock, Neuroimage (2013) 3
Why individual parcellations? • Standard anatomical atlases subdivide the brain based on cytoarchitecture (e.g. Brodmann) or anatomical landmarks (e.g. Desikan-Killiany) • Individual variability in terms of anatomy or function due to maturation or brain injury are not accounted for [ Timofiyeva, PLoS One (2014) ] • Data-driven single subject parcellations capture this variability 4
Inexact graph matching •Evaluate how much two graphs share Graph embedding Graph kernels s e t c n n e e m d e n l o e p s e e s d e c i r n v r e o o d r c p n e o t d o p o n s e n o r r s d o e r c i u q e r Craddock et al., Nature Methods (2013) Jie et al. Human Brain Mapping (2014) 5
Graph edit distance • Measure of dissimilarity between graphs, defined directly in their domain as a nonnegative function this is a long lon. d : G × G → R + G Able to model structural variation in a very intuitive and illustrative way. G G1 G2 1 1 5 edge-delete (2,3) label-change (1,5) 2 3 2 3 2 3 node-add (6) 4 4 4 5 5 7 5 edge-add(2,6) node-add(7) 2 3 2 3 2 3 4 6 4 6 4 6 G4 G3 G’ 6
GED computation • The Hungarian algorithm provides a fast approximate solution to the GED computation [ Riesen & Bunke, Img & Vis. Comp. , (2009) ] • Given two labeled graphs , with and G 1 G 2 |V ( G 1 ) | = n his is a long l a square cost matrix C of order is |V ( G 2 ) | = m n + m defined, which encodes all the possible edit operation costs c 1 , 1 . . . c 1 ,m c 1 , ε . . . ∞ . . . . ... ... substitutions . . . deletions . . . . . c n, 1 . . . c n,m ∞ . . . c n, ε C = 0 0 c ε , 1 . . . ∞ . . . . . . . ... ... . . . . insertions . . . . 0 0 ∞ . . . c ε ,m . . . 7
Node features • Spatial information (coordinates in standard brain space) • Feature information (network measures, egonet based) network features f 1 v 1 f 2 v 1 f 3 v 1 f 4 v 1 f 5 v 1 f 6 v 1 spatial coordinates v 1 x v 1 y v 1 z v 1 standard mean deviation v 3 v 2 d v 3 s v 3 c v 3 egonet v 4 d v 4 s v 4 c v 4 8
Node distance • Spatial distance • Feature distance Egonet l v = ( x, y, z ) f v = ( ¯ d u , ¯ s u , ¯ c u ) d | f v 1 i − f v 2 i | X d l = d eucl ( v 1 , v 2 ) = k l v 1 � l v 2 k d f = d canb ( v 1 , v 2 ) = d ( v 1 , v 2 ) = α ∗ d eucl ( v 1 , v 2 ) + (1 − α ) ∗ d canb ( v 1 , v 2 ) | f v 1 i | + | f v 2 i | i =1 9
Tailoring GED for brain graphs • In order to achieve better approximation of the true edit distance, edge operations need to be involved. • Constrain substitutions to the nearest N nodes - set cost to Inf for the rest of the substitutions if then u j ∈ neigh ( v i ) c i,j = α ∗ d euclidean ( v i , u j ) + (1 − α ) ∗ d canberra ( v i , u j ) + d edge ( v i , u j ) else c i,j = ∞ • Take into account node betweenness centrality g for the cost of node insertion/deletion c ✏ ,j = α + (1 − α ) ∗ g ( u j ) c i, ✏ = α + (1 − α ) ∗ g ( v i ) 10
Summary Graph Edit Distance Brain connectivity Labeled graphs (Hungarian algorithm) networks . . . . . . 11
Evaluation • Diffusion and functional MRI data from the Human Connectome Project • 30 healthy unrelated subjects as well as 20 monozygotic and 20 dizygotic female twin pairs (MZ twins share 100% of genetic information, while DZ share only 50%) • Connectivity driven single-subject parcellations [ Parisot et al., MICCAI , (2016) ] • Structural networks derived with probabilistic tractography • Functional networks estimated using partial correlation 12
Single-subject parcellations same subject different subjects GED(same subject) < GED(different subjects) 13
Structural networks * p<0.05 ** p<0.001 ns non-significant 14
Functional networks * p<0.05 ** p<0.001 ns non-significant 15
Monozygotic vs. unrelated pair 16
Monozygotic vs. unrelated pair 17
Conclusions • Novel way of evaluating graph similarity between brain networks based on graph edit distance • Enforces spatial constraints and incorporates feature information • Applied on healthy unrelated subjects and twin pairs and was able to reflect similarities between corresponding networks • Future steps: ‣ Predicting phenotype using GED distance matrix ‣ Network dynamics (brain development, disease progression, brain plasticity) 18
Acknowledgements: Daniel Rueckert Sarah Parisot Salim Arslan J. Passerat-Palmbach Emma Robinson Funding sources: EPSRC Foundation for Education and European Culture
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