Statistical Parametric Mapping (SPM) Talk I: Spatial Pre-processing & Morphometry Talk II: General Linear Model Talk III: Experimental Design & Efficiency Talk IV: EEG/MEG

Spatial Pre-processing & Morphometry Rik Henson With thanks to: John Ashburner, Matthew Brett, Karl Friston

Overview fMRI time-series Statistical Parametric Map Smoothing kernel General Linear Model Design matrix Motion correction Parameter Estimates Spatial normalisation Standard template

Overview 1. Realignment (within-modality) 2. Between-modality Coregistration 3. Normalisation (to stereotactic space) 4. Smoothing 5. Unified Segmentation & Normalisation 6. Morphometry (VBM/DBM/TBM)

Overview 1. Realignment (within-modality) - Spatial (movement correction) - Unwarping (movement-by-distortion) - Temporal (slice-timing correction) 2. Between-modality Coregistration 3. Normalisation (to stereotactic space) 4. Smoothing 5. Unified Segmentation & Normalisation 6. Morphometry (VBM/DBM/TBM)

Spatial Realignment: Reasons for Motion Correction Subjects will always move in the scanner The sensitivity of the analysis depends on the residual noise in the image series, so movement that is unrelated to the subject’s task will add to this noise and hence realignment will increase the sensitivity However, subject movement may also correlate with the task… …in which case realignment may reduce sensitivity (and it may not be possible to discount artefacts that owe to motion) Realignment (of same-modality images from same subject) involves two stages: 1. Registration - estimate the 6 parameters that describe the rigid body transformation between each image and a reference image 2. Reslicing - re-sample each image according to the determined transformation parameters

Rigid body transformations parameterised by: 1. Registration Determine the rigid body transformation that minimises the sum of squared difference between images Rigid body transformation is defined by: 3 translations - in X, Y & Z directions 3 rotations - about X, Y & Z axes Operations can be represented as affine transformation matrices: x1 = m1,1x0 + m1,2y0 + m1,3z0 + m1,4 y1 = m2,1x0 + m2,2y0 + m2,3z0 + m2,4 z1 = m3,1x0 + m3,2y0 + m3,3z0 + m3,4 Squared Error Translations Pitch Roll Yaw Rigid body transformations parameterised by:

1. Registration Iterative procedure (Gauss-Newton ascent) Additional scaling parameter Nx6 matrix of realignment parameters written to file (N is number of scans) Orientation matrices in header of image file (data not changed until reslicing)

2. Reslicing Nearest Neighbour Linear Application of registration parameters involves re-sampling the image to create new voxels by interpolation from existing voxels Interpolation can be nearest neighbour (0-order), tri-linear (1st-order), (windowed) fourier/sinc, or nth-order “b-splines” Full sinc (no alias) Windowed sinc

Residual Errors after Spatial Realignment Interpolation errors, especially with tri-linear interpolation and small-window sinc Ghosts (and other artefacts) in the image (which do not move as a rigid body) Rapid movements within a scan (which cause non-rigid image deformation) Spin excitation history effects (residual magnetisation effects of previous scans) Interaction between movement and local field inhomogeniety, giving non-rigid distortion… Before realignment After realignment

Overview 1. Realignment (within-modality) - Spatial (movement correction) - Unwarping (movement-by-distortion) - Temporal (slice-timing correction) 2. Between-modality Coregistration 3. Normalisation (to stereotactic space) 4. Smoothing 5. Unified Segmentation & Normalisation 6. Morphometry (VBM/DBM/TBM)

Unwarping EPIs contain distortions owing to field inhomogenieties (susceptibility artifacts, specifically in phase-encoding direction) They can be “undistorted” by use of a field-map (available in the “FieldMap” SPM toolbox) (Note that susceptibility artifacts that cause drop- out are more difficult to correct) However, movement interacts with the field inhomogeniety (presence of object affects B0), ie distortions change with position of object in field This movement-by-distortion can be accommodated during realignment using “unwarp” Field-map Distorted image Corrected image

Unwarping Pitch  +  B0  Roll Estimated derivative fields One could include the movement parameters as confounds in the statistical model of activations However, this may remove activations of interest if they are correlated with the movement Better is to incorporate physics knowledge, eg to model how field changes as function of pitch and roll (assuming phase-encoding is in y-direction)… … using Taylor expansion (about mean realigned image): Iterate: 1) estimate movement parameters (, ), 2) estimate deformation fields, 1) re-estimate movement … Fields expressed by spatial basis functions (3D discrete cosine set)… Pitch Estimated derivative fields  +  B0  Roll

Unwarping tmax=13.38 tmax=5.06 tmax=9.57 Example: Movement correlated with design tmax=13.38 No correction tmax=5.06 Correction by covariation tmax=9.57 Correction by Unwarp

Overview 1. Realignment (within-modality) - Spatial (movement correction) - Unwarping (movement-by-distortion) - Temporal (slice-timing correction) 2. Between-modality Coregistration 3. Normalisation (to stereotactic space) 4. Smoothing 5. Unified Segmentation & Normalisation 6. Morphometry (VBM/DBM/TBM)

Temporal Realignment (Slice-Timing Correction) Most functional MRI uses Echo-Planar Imaging (EPI) Each plane (slice) is typically acquired every 3mm normally axial… … requiring ~32 slices to cover cortex (40 to cover cerebellum too) (actually consists of slice-thickness, eg 2mm, and interslice gap, eg 1mm, sometimes expressed in terms of “distance factor”) (slices can be acquired contiguously, eg [1 2 3 4 5 6], or interleaved, eg [1 3 5 2 4 6]) Each plane (slice) takes about ~60ms to acquire… …entailing a typical TR for whole volume of 2-3s Volumes normally acquired continuously (though sometimes gap so that TR>TA) 2-3s between sampling the BOLD response in the first slice and the last slice (a problem for transient neural activity; less so for sustained neural activity)

Temporal Realignment (Slice-Timing Correction) Bottom Slice Top Slice SPM{t} TR=3s …the “Slice-timing Problem” (Henson et al, 1999) Slices acquired at different times, yet model is the same for all slices (slice-specific models would have problems when reslice, eg normalise) Two main solutions: Temporal interpolation of data (to a specified reference slice) … but less good for longer TRs 2. More general basis set (e.g., with temporal derivatives) … but inference more difficult Interpolated SPM{t} Derivative SPM{F}

Temporal Interpolation 10 slice volume Voxel in slice 1 Voxel in slice 5 Interpolated time course Estimated value at time of first slice

Temporal Realignment (Slice-Timing Correction) Interpolation artefacts often greater the more the interpolation, so choose middle slice in time as reference slice (but don’t forget to synchronise model with the middle of the scan!) Should any slice-time correction (temporal realignment) be done before or after motion correction (spatial realignment)? Well, depends on slice order (sequential or interleaved), and amount of movement… Movement of more than one slice (eg >3mm) would cause temporal interpolation of data from different tissue-types (markedly different values, eg brain boundary), increasing interpolation artefacts…

Interpolation Artifacts Synch Interpolation can spread artefact across time (scans)

Temporal Realignment (Slice-Timing Correction) Interpolation artefacts often greater the more the interpolation, so choose middle slice in time as reference slice (but don’t forget to synchronise model with the middle of the scan!) Should any slice-time correction (temporal realignment) be done before or after motion correction (spatial realignment)? Well, depends on slice order (sequential or interleaved), and amount of movement… Movement of more than one slice (eg >3mm) would cause temporal interpolation of data from different tissue-types (markedly different values, eg brain boundary), increasing interpolation artefacts… …which may be than greater timing errors resulting from movement correction first (e.g, only ~TR/N for contiguous, or TR/2 for interleaved) Probably best to perform slice-time correction AFTER movement correction, particularly for contiguous acquisition

Overview 1. Realignment (within-modality) - Spatial (movement correction) - Unwarping (movement-by-distortion) - Temporal (slice-timing correction) 2. Between-modality Coregistration 3. Normalisation (to stereotactic space) 4. Smoothing 5. Unified Segmentation & Normalisation 6. Morphometry (VBM/DBM/TBM)

Between Modality Co-registration Useful, for example, to display functional results (EPI) onto high resolution structural image (T1)… …indeed, necessary if spatial normalisation is determined by T1 image Because different modality images have different properties (e.g., relative intensity of gray/white matter), cannot simply minimise difference between images Therefore, use Mutual Information as cost function, rather than squared differences… EPI T2 T1 Transm PD PET

Between Modality Co-registration

Between Modality Coregistration: Mutual Information Mutual Information apparent from 2D histogram (plot of one image against other) For histograms normalised to integrate to unity, the Mutual Information is: SiSj hij log (hij) Sk hik Sl hlj PET T1 MRI

Overview 1. Realignment (within-modality) - Temporal (slice-timing correction) - Spatial (movement correction) - Unwarping (movement-by-distortion) 2. Between-modality Coregistration 3. Normalisation (to stereotactic space) 4. Smoothing 5. Unified Segmentation & Normalisation 6. Morphometry (VBM/DBM/TBM)

Reasons for Normalisation Inter-subject averaging extrapolate findings to the population as a whole increase statistical power above that obtained from single subject Reporting of activations as co-ordinates within a standard stereotactic space e.g. the space described by Talairach & Tournoux Label-based approaches: Warp the images such that defined landmarks (points/lines/surfaces) are aligned but few readily identifiable landmarks (and manually defined?) sulcal matching may help for cortex, but not deep-brain structures Intensity-based approaches: Warp to images to maximise some voxel-wise similarity measure eg, squared error, assuming intensity correspondence (within-modality) Normalisation constrained to correct for only gross differences; residual variability accommodated by subsequent spatial smoothing

Spatial Normalisation Summary Original image Spatially normalised Determine transformation that minimises the sum of squared difference between an image and a (combination of) template image(s) Two stages: 1. affine registration to match size and position of the images 2. non-linear warping to match the overall brain shape Uses a Bayesian framework to constrain affine and warps Spatial Normalisation Template image Deformation field

Stage 1. Full Affine Transformation The first part of normalisation is a 12 parameter affine transformation 3 translations 3 rotations 3 zooms 3 shears Better if template image in same modality (eg because of image distortions in EPI but not T1) Rotation Translation Zoom Shear Rigid body

Six affine registered images Insufficiency of Affine-only normalisation Six affine registered images Six affine + nonlinear registered

Stage 2. Nonlinear Warps Deformations consist of a linear combination of smooth basis images These are the lowest frequency basis images of a 3-D discrete cosine transform Brain masks can be applied (eg for lesions)

Bayesian Constraints Template image Affine Registration (2 = 472.1) Non-linear registration with regularisation (2 = 302.7) Non-linear registration without regularisation (2 = 287.3) Without the Bayesian formulation, the non-linear spatial normalisation can introduce unnecessary warping into the spatially normalised images

Bayesian Constraints Using Bayes rule, we can constrain (“regularise”) the nonlinear fit by incorporating prior knowledge of the likely extent of deformations: p(p|e)  p(e|p) p(p) (Bayes Rule) p(p|e) is the a posteriori probability of parameters p given errors e p(e|p) is the likelihood of observing errors e given parameters p p(p) is the a priori probability of parameters p For Maximum a posteriori (MAP) estimate, we minimise (taking logs): H(p|e)  H(e|p) + H(p) H(e|p) (-log p(e|p)) is the squared difference between the images (error) H(p) (-log p(p)) constrains parameters (penalises unlikely deformations)  is “regularisation” hyperparameter, weighting effect of “priors”

Bayesian Constraints Algorithm simultaneously minimises: Sum of squared difference between template and object Squared distance between the parameters and their expectation Bayesian constraints applied to both: 1) affine transformations based on empirical prior ranges 2) nonlinear deformations based on smoothness constraint (minimising membrane energy) Empirically generated priors

DARTEL: Diffeomorphic Registration (SPM8) Nonlinear warps do not guarantee a 1-to-1 mapping between object and target image... ...not “diffeomorphic”... ...so inverting this mapping (eg un-normalising a brain) is not precise Better approach is to construct a flow field, so one image can slowly “flow” into another => DARTEL

DARTEL: Diffeomorphic Registration (SPM8) Grey matter average of 452 subjects Affine Grey matter average of 471 subjects DARTEL

Overview 1. Realignment (within-modality) - Temporal (slice-timing correction) - Spatial (movement correction) - Unwarping (movement-by-distortion) 2. Between-modality Coregistration 3. Normalisation (to stereotactic space) 4. Smoothing 5. Unified Segmentation & Normalisation 6. Morphometry (VBM/DBM/TBM)

Reasons for Smoothing Potentially increase signal to noise (matched filter theorem) Inter-subject averaging (allowing for residual differences after normalisation) Increase validity of statistics (more likely that errors distributed normally) Kernel defined in terms of FWHM (full width at half maximum) of filter - usually ~16-20mm (PET) or ~6-8mm (fMRI) of a Gaussian Ultimate smoothness is function of applied smoothing and intrinsic image smoothness (sometimes expressed as “resels” - RESolvable Elements) Gaussian smoothing kernel FWHM

Overview 1. Realignment (within-modality) - Temporal (slice-timing correction) - Spatial (movement correction) - Unwarping (movement-by-distortion) 2. Between-modality Coregistration 3. Normalisation (to stereotactic space) 4. Smoothing 5. Unified Segmentation & Normalisation 6. Morphometry (VBM/DBM/TBM)

Image Segmentation (in general) Partition into gray matter (GM), white matter (WM) and cerebrospinal fluid (CSF) ‘Mixture Model’ cluster analysis used, which assumes each voxel is one of a number of distinct tissue types (clusters), each with a (multivariate) normal distribution Further Bayesian constraints from prior probability images, which are overlaid Additional correction for (low-spatial frequency) intensity inhomogeniety… . Intensity histogram fit by multi-Gaussians Priors: Image: Brain/skull CSF WM GM

Image Segmentation (in practice (SPM5)) Intensity-based partitioning of image is complicated by spatial image intensity inhomogenieties… …but correction of image inhomogeniety is complicated by different spatial distributions of different tissue-types Original Corrected

Image Segmentation (in practice (SPM5)) Intensity-based partitioning of image is aided by coregistration with prior tissue-type probabilities in template space…. …but coregistration of an image with template space (ie normalisation) is aided by separating different tissue-types (eg, might want normalisation to be based on gray-matter only)

Image Segmentation (in practice (SPM5)) These interdependencies prevent traditional serial-stage processing (e.g, inhomogeniety-correction -> normalisation -> segmentation) Need a single, unified generative model (solved iteratively)…

Image Segmentation (in practice (SPM5)) Initial coregistration with templates using (4x256) Mutual Information Then Repeat until convergence: 1. Estimate field inhomogenieties (using discrete cosine set) 2. Estimate tissue classifications (using Mixture of Gaussians) 3. Estimate new warps for coregistration with templates Result is not only segmentation, but superior normalisation to standard normalisation… …and more robust to lesions (Though aided by manual repositioning close to template space)

Overview 1. Realignment (within-modality) - Temporal (slice-timing correction) - Spatial (movement correction) - Unwarping (movement-by-distortion) 2. Between-modality Coregistration 3. Normalisation (to stereotactic space) 4. Smoothing 5. Unified Segmentation & Normalisation 6. Morphometry (VBM/DBM/TBM)

Morphometry (Computational Neuroanatomy) Voxel-by-voxel: where are the differences between populations? Univariate: e.g, Voxel-Based Morphometry (VBM) Multivariate: e.g, Tensor-Based Morphometry (TBM) Volume-based: is there a difference between populations? Multivariate: e.g, Deformation-Based Morphometry (DBM) Continuum: perfect normalisation => all information in Deformation field (no VBM differences) no normalisation => all in VBM Spatial Normalisation Original Template Normalised Deformation field VBM TBM DBM

Voxel-Based Morphometry (VBM) A voxel by voxel statistical analysis is used to detect regional differences in the amount of grey matter between populations Original image Spatially normalised Segmented grey matter Smoothed SPM Voxel intensities can be further modulated by determinant of Jacobean of warps: i.e, whether want to compare graymatter density (unmodulated) or volume (modulated)

cingulate/parafalcine VBM Examples: Aging Grey matter volume loss with age superior parietal pre and post central insula cingulate/parafalcine

VBM Examples: Brain Asymmetries Right frontal and left occipital petalia

Morphometry on deformation fields: DBM/TBM Deformation-based Morphometry looks at absolute displacements Tensor-based Morphometry looks at local shapes Vector field Tensor field

Deformation-based Morphometry (DBM) fields ... Remove positional and size information - leave shape Parameter reduction using principal component analysis (SVD) Multivariate analysis of covariance used to identify differences between groups Canonical correlation analysis used to characterise differences between groups

DBM Example: Sex Differences Non-linear warps of sex differences characterised by canonical variates analysis Mean differences (mapping from an average female to male brain)

Tensor-based morphometry Template Warped Original If the original Jacobian matrix is donated by A, then this can be decomposed into: A = RU, where R is an orthonormal rotation matrix, and U is a symmetric matrix containing only zooms and shears. Relative volumes Strain tensor Strain tensors are defined that model the amount of distortion. If there is no strain, then tensors are all zero. Generically, the family of Lagrangean strain tensors are given by: (Um-I)/m when m~=0, and log(U) if m==0.

References Friston et al (1995): Spatial registration and normalisation of images. Human Brain Mapping 3(3):165-189 Ashburner & Friston (1997): Multimodal image coregistration and partitioning - a unified framework. NeuroImage 6(3):209-217 Collignon et al (1995): Automated multi-modality image registration based on information theory. IPMI’95 pp 263-274 Ashburner et al (1997): Incorporating prior knowledge into image registration. NeuroImage 6(4):344-352 Ashburner et al (1999): Nonlinear spatial normalisation using basis functions. Human Brain Mapping 7(4):254-266 Ashburner & Friston (2000): Voxel-based morphometry - the methods. NeuroImage 11:805-821

SPM5 requirements… Workstation Matlab 6.5 - 7.1 developed on Sun Solaris UNIX Solaris, Linux, Mac & Windows supported other UNIX disk & memory… Matlab 6.5 - 7.1 no special “toolboxes” required ANSII C Compiler to compile external C–mex routines ready for Solaris, Linux, & Windows NIfTI/Analyze format images conversion program extend SPM Internet access …for SPMweb & the email discussion list Plenty of time!

SPM documentation… peer reviewed literature SPM course notes, Human Brain Function & SPM manual algorithm descriptions, code annotations, pseudo-code online help & function descriptions & SPM5 Manual

SPM Online Bibliography

some SPM internet resources… SPMweb site http://www.fil.ion.ucl.ac.uk/spm/ SPM email discussion list http://www.fil.ion.ucl.ac.uk/spm/support/ <spm@jiscmail.ac.uk> FIL neuroscience resources links http://www.fil.ion.ucl.ac.uk/resources.html SPM Wiki http://en.wikibooks.org/wiki/SPM MRC-CBU imagers http://www.mrc-cbu.cam.ac.uk/Imaging/Common/

SPMweb… Introduction to SPM SPM distribution: SPM99, SPM2, SPM5 Documentation & Bibliography SPM email discussion list SPM short course Example data sets SPM extensions http://www.fil.ion.ucl.ac.uk/spm/

SPM – email discussion list spm@jiscmail.ac.uk Web home page http://www.fil.ion.ucl.ac.uk/spm/support/ Archives, archive searches, membership lists, instructions Subscribe http://www.jiscmail.ac.uk/ email jiscmail@jiscmail.ac.uk join spm Firstname Lastname Participate & learn email spm@jiscmail.ac.uk Monitored by SPMauthors Usage queries, theoretical discussions, bug reports, patches, techniques, &c… http://www.fil.ion.ucl.ac.uk/spm/support/ spm@jiscmail.ac.uk

(0th-order term can be determined from fieldmap) Unwarping (0th-order term can be determined from fieldmap) B0{i} B0 = +  +  + error - f1 fi  1 +2 + ... +5 + ... i 