Brain Morphometrics from MRI Scans John Ashburner The Wellcome Trust Centre for Neuroimaging 12 Queen Square, London, UK.

Contents Introduction Voxel-based Morphometry Diffeomorphic Shape Modelling

Neuroscientists currently model a fraction of the available information in MRI data. –Models are geared for questions with relatively simple answers. Findings are easier to explain and visualize on the printed page. –Two main approaches commonly used: 1.Examine whole brain and assume independence among regions. 2.Focus attention on a few specific brain regions. –Findings from whole brain multivariate analyses are too complicated to add to human understanding of the brain. See what data mining can achieve from it. Scavenging for data is likely to become easier. Some journals require primary data to be made available. Funding bodies now promote data sharing. Introduction

Example data management and sharing policies UK Wellcome Trust –“The Trust considers that the benefits gained from research data will be maximised when they are made widely available to the research community as soon as feasible, so that they can be verified, built upon and used to advance knowledge.” – UK Medical Research Council –“The MRC Data Sharing and Preservation Initiative aims to maximise opportunities for enabling wider and better use of this data for further high quality, ethical research.” – USA NIH –“Data should be made as widely and freely available as possible while safeguarding the privacy of participants, and protecting confidential and proprietary data.” –

Some currently available datasets IXI : Brain MR images from 550 normal subjects between 20 and 80 years. – OASIS : Cross-sectional MRI Data in young, middle aged, nondemented and demented older adults. 416 subjects, aged 18 to 96. – ADNI : 200 elderly controls, 400 subjects with mild cognitive impairment and 200 subjects with Alzheimer’s. – MIRAID & ELUDE : Late life depression data. – The Extensible Neuroimaging Archive Toolkit ( XNAT ) is an open source software platform designed to facilitate management and exploration of neuroimaging and related data. –

Typical anatomical MRI data T1- weighted MRA T2- weighted Proton Density weighted Grey matter White matter Grey and white matter segmented from original scans. Many other types of MRI can be collected. Eg. Functional MRI (fMRI). Diffusion Weighted Images (DWI). Anatomical scans are volumetric, typically of about 1mm isotropic resolution. Image dimensions are about 256x256x150 voxels.

Contents Introduction Voxel-based Morphometry Diffeomorphic Shape Modeling

Voxel-Based Morphometry Produce a map of statistically significant differences among populations of subjects. –e.g. compare a patient group with a control group. –or identify correlations with age, test-score etc. The data are pre-processed to sensitise the tests to regional tissue volumes. –Usually grey or white matter.

Volumetry T1-Weighted MRI Grey Matter Probably the easiest approach to understand and describe.

Images are blurred Before convolution Convolved with a circle Convolved with a Gaussian Each voxel after blurring effectively becomes the result of applying a weighted region of interest (ROI).

Statistical Parametric Mapping… group 1group 2 voxel by voxel modelling –  parameter estimate standard error = statistic image or SPM Statistics are corrected for multiple dependent comparisons using Gaussian random field theory.

Possible Explanations for Findings Thickening Thinning Folding Mis-classify Mis-register

Contents Introduction Voxel-based Morphometry Diffeomorphic Shape Modelling

D’Arcy Thompson (1917). GROWTH AND FORM. The morphologist, when comparing one organism with another, describes the differences between them point by point, and “character” by “character”. If he is from time to time constrained to admit the existence of “correlation” between characters (as a hundred years ago Cuvier first showed the way), yet all the while he recognises this fact of correlation somewhat vaguely, as a phenomenon due to causes which, except in rare instances, he cannot hope to trace; and he falls readily into the habit of thinking and talking of evolution as though it had proceeded on the lines of his own descriptions, point by point and character by character. But if, on the other hand, diverse and dissimilar [fish/brains] can be referred as a whole to identical functions of very different coordinate systems, this fact will of itself constitute a proof that a comprehensive “law of growth” has pervaded the whole structure in its integrity, and that some more or less simple and recognizable system of forces has been at work.

A shape-based generative model Grey matter average of 471 subjects White matter average of 471 subjects μ t1t1 ϕ1ϕ1 t2t2 ϕ2ϕ2 t3t3 ϕ3ϕ3 t4t4 ϕ4ϕ4 t5t5 ϕ5ϕ5 t – individual’s data (diverse and dissimilar brains) μ – template (identical function) Φ – deformation (very different coordinate systems) Template data

Individual Warped IndividualTemplate

Deformations and Jacobian Deformation FieldJacobians determinants (should be positive)

Displacements don’t add linearly Forward Inverse Composed Subtracted

Diffeomorphisms: a more sophisticated shape modelling framework. Diffeomorphisms are a key component of Pattern Theory. Smooth, continuous one-to-one mappings. Provide parsimonious representations of relative shapes. –Metrics describing similarities between shapes. –Shapes can be encoded by initial velocity/momentum. Image registration usually by a variational approach based on the principle of stationary action (LDDMM):

Geodesic Shooting: Generating warps from initial velocities Magnitude of momentum Velocity Forward Deformation Inverse Deformation Jacobians

Diffeomorphic Eigenwarps 550 scans from the IXI dataset, registered to a common average using a diffeomorphic registration model. Eigen-decomposition of initial velocities/momenta. Exaggerated warps generated by a geodesic shooting method. - Preserves one-to-one mapping. 1st 3rd 10th - +

Predictions Brain shapes can be used to make predictions about individuals. This figure shows predictions of subjects ages made from IXI dataset using Tipping’s Relevance Vector Regression (a kernel method). More useful predictions may be possible from other data. Cross-validation results for age predictions.

Some References on Diffeomorphisms M. F. Beg, M. I. Miller, A.Trouve & L. Younes. "Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms". International Journal Of Computer Vision (2003). M. I. Miller, A. Trouve, & L. Younes. "Geodesic Shooting for Computational Anatomy". Journal of Mathematical Imaging and Vision (2004) L. Wang, M. F. Beg, J. T. Ratnanather, C. Ceritoglu, L. Younes, J. C. Morris, J. G. Csernansky & M. I. Miller. "Large Deformation Diffeomorphism and Momentum Based Hippocampal Shape Discrimination in Dementia of the Alzheimer Type". IEEE Trans. Med Imaging (2006) L. Younes, F. Arrate & M. I. Miller. "Evolution Equations in Computational Anatomy". Neuroimage (2008) And my own approximation to this is: J. Ashburner. “A fast diffeomorphic image registration algorithm”. NeuroImage (2007).

Contents Introduction Voxel-based Morphometry Segmentation Diffeomorphic Shape Modelling

Image Intensity Distributions (T1-weighted MRI)

Tissue Probability Maps Tissue probability maps (TPMs) are used to represent the prior probabilities of different tissues at each location. Note the similarity with some non-negative matrix factorization models. Factors are: Tissue intensity distributions Tissue probability maps

Deforming the Tissue Probability Maps Tissue probability maps are deformed so that they can be overlaid on top of the image to segment.

Modelling intensity non- uniformity artifact A bias field can by modelled by a linear combination of spatial basis functions. Corrupted image Corrected image Bias Field

Principle of Stationary Action Lagrangian = Kinetic Energy – Potential Energy Principle of stationary action: LDDMM is a variational method based on this principle. Position Velocity

Hamiltonian Mechanics Introduce momentum: The Hamiltonian formalism gives: The dynamical system equations are then and

Hamiltonian for diffeomorphisms In the landmark-based framework So update equations are t=0 t=1 See e.g. the work of Steve Marsland et al