Spatial preprocessing of fMRI data

Methods & models for fMRI data analysis in neuroeconomics17 April 2010

Klaas Enno Stephan Laboratory for Social and Neural Systrems ResearchInstitute for Empirical Research in EconomicsUniversity of Zurich

Functional Imaging Laboratory (FIL)Wellcome Trust Centre for NeuroimagingUniversity College London

With many thanks for helpful slides to:John AshburnerMeike GrolGed Ridgway

Overview of SPM

Realignment Smoothing

Normalisation

General linear model

Statistical parametric map (SPM)Image time-series

Parameter estimates

Design matrix

Template

Kernel

Gaussian field theory

p <0.05

Statisticalinference

Functional MRI (fMRI)• Uses echo planar imaging (EPI) for

fast acquisition of T2*-weighted images.

• Spatial resolution:– 3 mm (standard 1.5 T scanner)– < 200 μm (high-field systems)

• Sampling speed:– 1 slice: 50-100 ms

• Requires spatial pre-processing and statistical analysis.

EPI(T2*)

T1

dropout

subjects

sessions

runs

single run

volume

slices

Terminology of fMRI

TR = repetition timetime required to scan one volume

voxel

Terminology of fMRI

Slice thicknesse.g., 3 mm

Scan Volume:Field of View

(FOV),e.g. 192 mm

Axial slices

3 mm

3 mm

3 mm

Voxel Size(volumetric pixel)

Matrix Sizee.g., 64 x 64

In-plane resolution192 mm / 64

= 3 mm

Standard space

The Talairach Atlas The MNI/ICBM AVG152 Template

Why does fMRI require spatial preprocessing?

• Head motion artefacts during scanning

• Problems of EPI acquisition: distortion and signal dropouts

• Brains are quite different across subjects

→ Realignment

→ “Unwarping”

→ Normalisation (“Warping”)→ Smoothing

Realignment or “motion correction”• Even small head movements can be a major problem:

– increase in residual variance– data may get completely lost if sudden movements

occur during a single volume– movements may be correlated with the task

performed• Therefore:

– always constrain the volunteer’s head– instruct him/her explicitly to remain as calm as

possible– do not scan for too long – everyone will move after

while !

minimising movements is one of the most important factors for ensuring good data quality!

Realignment = rigid-body registration

• Assumes that all movements are those of a rigid body, i.e. the shape of the brain does not change

• Two steps:Registration: optimising six parameters that

describe a rigid body transformation between the source and a reference image

Transformation: re-sampling according to the determined transformation

Linear (affine) transformations• Rigid-body transformations are a subset• Parallel lines remain parallel • Operations can be represented by:

x1 = m11x0 + m12y0 + m13z0 + m14

y1 = m21x0 + m22y0 + m23z0 + m24

z1 = m31x0 + m32y0 + m33z0 + m34

• Or as matrices:

1zyx

1000mmmmmmmmmmmm

1zyx

0

0

0

34333231

24232221

14131211

1

1

1

2D affine transforms• Translations by tx and ty

x1 = 1 x0 + 0 y0 + tx

y1 = 0 x0 + 1 y0 + ty

• Rotation around the origin by radiansx1 = cos() x0 + sin() y0 + 0y1 = -sin() x0 + cos() y0 + 0

• Zooms by sx and sy:x1 = sx x0 + 0 y0 + 0y1 = 0 x0 + sy y0 + 0

Shearx1 = 1 x0 + h y0 + 0y1 = 0 x0 + 1 y0 + 0

3D rigid-body transformations• A 3D rigid body transform is defined by:

– 3 translations - in X, Y & Z directions– 3 rotations - about X, Y & Z axes

• Non-commutative: the order of the operations matters

1000010000cossin00sincos

10000cos0sin00100sin0cos

10000cossin00sincos00001

1000Zt100Y010X001

rans

trans

trans

ΩΩΩΩ

ΘΘ

ΘΘ

ΦΦΦΦ

Translations Pitchabout x axis

Rollabout y axis

Yawabout z axis

Realignment

• Goal: minimise squared differences between source and reference image

• Other methods available (e.g. mutual information)

A special case ...

• If a subject remained perfectly still during a fMRI study, would realignment still be a good idea to perform?

• When could this issue be of practical relevance?

Coregistration

• also affine registration (like realignment)

• used to register a structural image to a (mean) functional one– allows more accurate anatomical localisation of activations– must be done before spatial normalisation (if warping

parameters are estimated from the T1 image)

• examples in SPM8 Manual, Chapters 28, 29

Practical demonstration: realignment & coregistration

Joint and marginal histograms

intensityfrequency

intensity

frequency

• Nearest neighbour– Take the value of the

closest voxel

• linear (2D: bilinear; 3D: trilinear)– Just a weighted average

of the neighbouring voxels– f5 = f1 x2 + f2 x1

– f6 = f3 x2 + f4 x1

– f7 = f5 y2 + f6 y1

Interpolation

B-spline interpolation

B-splines are piecewise polynomials

A continuous function is represented by a linear combination of basis functions

2D B-spline basis functions of degrees 0, 1, 2 and 3

Nearest neighbour and trilinear interpolation are the same as B-spline interpolation with degrees 0 and 1.

Why does fMRI require spatial preprocessing?

• Head motion artefacts during scanning

• Problems of EPI acquisition: distortion and signal dropouts

• Brains are quite different across subjects

→ Realignment

→ “Unwarping”

→ Normalisation (“Warping”)→ Smoothing

Residual errors after realignment

• Resampling can introduce interpolation errors• Slices are not acquired simultaneously

– rapid movements not accounted for by rigid body model

• Image artefacts may not move according to a rigid body model– image distortion– image dropout– Nyquist ghost

• Functions of the estimated motion parameters can be included as confound regressors in subsequent statistical analyses.

Movement by distortion interactions

• Subject disrupts B0 field, rendering it inhomogeneous

→ distortions in phase-encode direction

• Subject moves during EPI time series

→ distortions vary with subject orientation→ shape of imaged brain

varies

Andersson et al. 2001, NeuroImage

Movement by distortion interaction

Movement by distortion interactions

after head rotationoriginal deformations

deformations after realignment mismatch in deformationsAndersson et al. 2001, NeuroImage

Different strategies for correcting movement artefacts

• liberal control:realignment only

• moderate control:realignment + “unwarping”

• strict control:realignment + inclusion of realignment parameters in statistical model

Why does fMRI require spatial preprocessing?

• Head motion artefacts during scanning

• Problems of EPI acquisition: distortion and signal dropouts

• Brains are quite different across subjects

→ Realignment

→ “Unwarping”

→ Normalisation (“Warping”)→ Smoothing

Individual brains differ in size, shape and folding

Spatial normalisation: why necessary?

• Inter-subject averaging– Increase sensitivity with more subjects

• Fixed-effects analysis– Extrapolate findings to the population as a

whole• Random / mixed-effects analysis

• Make results from different studies comparable by bringing them into a standard coordinate system– e.g. MNI space

Spatial normalisation: objective• Warp the images such that functionally

corresponding regions from different subjects are as close together as possible

• Problems:– Not always exact match between structure and

function– Different brains are organised differently– Computational problems (local minima, not enough

information in the images, computationally expensive)

• Compromise by correcting gross differences followed by smoothing of normalised images

Spatial normalisation: affine step

• The first part is a 12 parameter affine transform– 3 translations– 3 rotations– 3 zooms– 3 shears

• Fits overall shape and size

Spatial normalisation: non-linear step

Deformations consist of a linear combination of smooth basis functions.

These basis functions result from a 3D discrete cosine transform (DCT).

Spatial normalisation: Bayesian regularisation

Deformations consist of a linear combination of smooth basis functions set of frequencies from a 3D

discrete cosine transform.Find maximum a posteriori (MAP) estimates: simultaneously minimise

– squared difference between template and source image – squared difference between parameters and their priors

)(log)(log)|(log)|(log yppypyp MAP:

Deformation parameters

“Difference” between template and source image

Squared distance between parameters and their expected values

(regularisation)

Templateimage

Affine registration.(2 = 472.1)

Non-linearregistration

withoutregularisation.(2 = 287.3)

Non-linearregistration

usingregularisation.(2 = 302.7)

Without regularisation, the non-linear spatial normalisation can introduce unnecessary warps.

Spatial normalisation: overfitting

Segmentation

GM and WM segmentations overlaid on original images

Structural image, GM and WM segments, and brain-mask (sum of GM and WM)

Segmentation & normalisation• Circular relationship between segmentation & normalisation:

– Knowing which tissue type a voxel belongs to helps normalisation.– Knowing where a voxel is (in standard space) helps segmentation.

• Build a joint generative model:– model how voxel intensities result from mixture of tissue type distributions– model how tissue types of one brain have to be spatially deformed to

match those of another brain

• Using a priori knowledge about the parameters: adopt Bayesian approach and maximise the posterior probability

Ashburner & Friston 2005, NeuroImage

Unified segmentation with tissue class priors

• Goal: for each voxel, compute probability that it belongs to a particular tissue type, given its intensity

• Likelihood model: Intensities are modelled by a mixture of Gaussian distributions representing different tissue classes (e.g. GM, WM, CSF).

• Priors are obtained from tissue probability maps (segmented images of 151 subjects).

Ashburner & Friston 2005, NeuroImage

p (tissue | intensity) p (intensity | tissue) ∙ p (tissue)

Normalisation options in practice

• Conventional normalisation:– either warp functional scans to EPI template directly– or coregister structural scan to functional scans and then

warp structural scan to T1 template; then apply these parameters to functional scans (“Normalise: Write”)

• Unified segmentation:– coregister structural scan to functional scans– unified segmentation provides normalisation parameters– apply these parameters to functional scans (“Normalise:

Write”)

Normalising structural image to T1 template

Estimate normalization parameters (T1 -> T1 template)

Apply the normalization parameters to the T1 image

And apply the normalization parameters to the (coregistered) functional images

T1 image

EPI time series

Normalising EPI images to EPI template

Calculate mean of EPI time series

Estimate normalization parameters (EPI -> EPI template)

EPI time seriesApply the normalization parameters to all EPI images

T1 image

Apply the normalization parameters to the (coregistered) T1 image

Smoothing• Why smooth?

– increase signal to noise– inter-subject averaging– increase validity of Gaussian Random Field theory

• In SPM, smoothing is a convolution with a Gaussian kernel.• Kernel defined in terms of FWHM (full width at half maximum).

Gaussian convolution is separable Gaussian smoothing kernel

Smoothing

Before convolution Convolved with a circleConvolved with a Gaussian

Smoothing is done by convolving with a 3D Gaussian which is defined by its full width at half maximum (FWHM).Each voxel after smoothing effectively becomes the result of applying a weighted region of interest.

Summary: spatial preprocessing steps

• Head motion artefacts during scanning

• Problems of EPI acquisition: distortion and signal dropouts

• Brains are quite different across subjects

→ Realignment

→ “Unwarping”

→ Normalisation (“Warping”)→ Smoothing

Thank you!

Supplementary slides

World coords(2, 4, -4) mm

(4, 4, -2)

(2, 2, -4)

Voxel index(45, 66, 35)

(44, 66, 36)

(45, 65, 35)

x 2 92y 2 -128z 2 -74

W V

W V

W V

xyz

World space & voxel space

7412892

zyx

200020002

zyx

V

V

V

W

W

W

1zyx

100074200

12802092002

1zyx

V

V

V

W

W

WBy moving to 4D, one can includetranslations within a single matrix multiplication

These 4-by-4 “homogeneous matrices” are the currency of voxel-world mappings, affine coreg. and realignment. In SPM5 & SPM8, they are stored in the .hdr files.  In SPM2, they are stored in .mat files. 

To find inverse mappings, or results of concatenating multiple transformations, we simply follow the rules of matrix algebra

Changing coordinate systems

74-2zz128-2y

922x

VW

VW

VW

yx

Representing rotations

0

0

1

1

yx

)θcos(θ)sin(θ)sin()θcos(

yx

0

0

0

1

1

1

zyx

1000)θcos()θsin(0)θsin()θcos(

zyx