Modelling Strategies in Functional Magnetic Resonce Imaging

1
Modelling Strategies in Functional Magnetic
Resonce Imaging
14th of October 2009
Kristoffer Hougaard Madsen
Danish Research Centre for Magnetic
Resonance Copenhagen University Hospital Hvidovre
DTU Informatics Technical University of Denmark
2
Overview

• Part I Preliminaries
• Functional Magnetic Resonance Imaging
• Pre-processing
• Nuisance Variable Regression
• Part II Statistical Parametric Mapping
• Part III Unsupervised analysis
• Conclusion

3
Magnetic Resonance Imaging

• Hydrogen has spin property
• Strong magnetic field causes partial alignment
• Perturbed by RF waves at resonance frequency
• Emits RF during the return to equilibrium
• Spatially varying gradients allow images to be
  recorded

4
Functional MRI

• Typically based on haemodynamics
• Indirect measure of neural activity
• Neuronal activity requires oxygen
• Brain is well perfused
• Increased activity -gt local increase in blood flow

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Blood Oxygenation Level Dependent (BOLD) signal

• Increased neuronal activity gives rise to
  increased oxygen metabolism (CMRO2)
• This causes increased blood flow (CBF) and
  therefore increased blood volume (CBV)
• Presence of deoxygenated blood (dHb) causes
  signal loss in T2 weighted MRI (transverse
  magnetization lost more quickly)

CMRO2
CBF
-


-
CBV
dHb

Buxton et al. (Balloon model), 1998 MRM
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Data acqusition

• Measure volumes repeatedly
  (repetition time 0.1-3 s)

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Noise

• Scanner instability
• Typically low frequency (lt0.01 Hz)
• Subject movement
• Rigid body
• Spin history
• Movement by field inhomogeniety
• Physiological effects
• Cardiac cycle (0.5-2 Hz, often appears aliased)
• Respiratory cycle ( 0.1-0.5 Hz)
• End tidal CO2 volume / respiration volume over
  time
• Secondary and interaction effects

8
Typical pre-processing steps

• Retrospective rigid body realignment
• Co-registering/Normalisation
• Slice-timing (time interpolation)
• Spatial smoothing
• Modelling nuisance effects
• High-pass filter
• Cardiac/respiratory regressors
• Movement (residual effects)

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Nuisance Variable Regression (NVR)

• Cardiac effects
• Effects of the cardiac cycle will appear aliased
  due to the low sampling rate
• Can appear at any frequency
• Measure cardiac cycle using ECG or pulse-oximeter
• Model effect by aliased sine and cosine functions
  representing the cardiac cycle (model any phase)
• Respiration effects
• Gross head movement
• Blood oxygenation changes
• Field changes due to movement of organs in the
  abdomen
• Measure respiration using pneumatic belt
• Model effect by Fourier expansion of the
  respiration cycle
• Residual motion
• Most prominent near edges
• Model by Volterra expansion of the movement
  parameters from the realignment procedure

Lund et al., Neuroimage 2006 Glover et al.,
MRM 2000 Friston et al. 1996, MRM, 35,
346-55.
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Detecting fMRI signals
High-pass filter
Residual movement
Paradigm
Note Bayesian model selection No
thresholding Madsen and Hansen, 2008
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Constructing the fMRI Matrix
TR
time
data array 646440
data array 646440
data array 646440
...
time
space
YT
spatial column vector 163840x1
...

data matrix (X) of spacetime
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Overview

• Part I Basic Principles
• Part II Statistical Parametric Mapping
• Stimulation Protocol
• General Linear Model (GLM)
• Inference
• Part III Unsupervised analysis
• Part IV Experiments
• Conclusion

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Paradigm

• Control what kind of neural processing goes on
  simple visual example
• Construct reference time series of expected
  activation by convolving with impulse response
  function
• Assuming LTI system
• In the analysis compare signal to this

time
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SPM and the General Linear Model (GLM)

• Model (M) (univariate - single voxel time series)
• Likelihood (multivariate Gaussian noise)

Design matrix (TK)
Residual noise (model inadequacy)
Parameters (estimated)
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Hypothesis testing

• Estimate the scale of the covariance from data
  (assuming rest of covariance is known)
• Null distribution
• How likely is that? (p-value small means
  unlikely meaning we will accept the alternative
  hypothesis)

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Overlaying activity

• Threshold
• Multiple comparisons
• Familywise error (random field theory)
• False discovery rate
• Overlay thresholded SPM on anatomical image

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Overview

• Part I Basic Principles
• Part II Statistical Parametric Mapping
• Part III Unsupervised analysis
• Linear latent variable model
• Uniqueness, how many components?
• Multisubject analysis and multiway decomposition

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Decomposition as a cocktail party problem
Instantaneous mixing
A1,1
A1,2
A2,1

Y
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The fMRI Party problem
Mixing matrix (estimated)
Residual noise (model inadequacy)
Sources (estimated)

• Likelihood (matrix-variate Gaussian noise)

Where are diagonal (residual is
independent over space and time)
Assumption Data instantaneous mixture of
temporal signatures. (PCA/ICA/NMF)
?
Flaw
X?AS(AQ-1)(QS)ÂS
Representation not unique!
Additional information needed (independence,
sparseness, non-negativity)
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Spatial FA model without the math
matrix of sources (S) aka. spatial components
mixing matrix (A) aka. component time series
data matrix (Y) timespace
space
time
?
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Temporal FA model without the math
data matrix (Y T) spacetime
mixing matrix (A) voxel maps
matrix of sources (S) aka. temporal components
time
space
?
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Singular Value Decomposition

• SVD
• Unique (up to permutation of components)
• Equivalent to PCA
• Convex optimization problem
  (one global
  solution easy to find)
• Sort components according to singular values
• Truncate to obtain approximate model
• The orthogonality constraint is often not
  appropriate
• Spatial/temporal versions are equivalent

diagonal
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Infomax-ICA

• Start with truncated SVD solution ( - k
  sources)
• Determine unmixing matrix
• Maximize probability of sources thereby
  determining
• Assumes independent sources (in this case
  spatially)
• Sources are assumed non-gaussian (prior)
• Feasible because mixing causes more Gaussian
  distributions (central-limit theorem)
• In general a non-convex optimasation problem
• Must select non-linearity/prior for sources

Bell Sejnowski, 1995
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Independent Component Analysis
(Example of single subject analysis)
Stimuli full-checkerboard (8Hz), each trial
consist of 10 seconds pause 10 seconds stimuli
and 10 seconds pause. Data acquired at 3 Hz.
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Sparse coding

• Laplace prior for sources (S)
• Negative log posterior proportional to
• Avoid trivial solution where S 0 and A 8
• Normalise A
• Gaussian prior for A
• When performing MAP estimation S tends to become
  sparse (many zeros for high l)
• Estimate alternating between A and S

Olshausen and Fields, Nature 1996
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How many components?

• Look at singular values of data
• Find the corner (broken-stick method)

Heuristic Very difficult to do in practice the
curve is often very smooth Threshold can be
regarded a noise variance

• Large sample approximations to the model evidence
• Laplace approximation to the model evidence
• Bayesian Information Criterion (BIC) / Minimum
  Description Lenght (MDL)
• Akaikes Information Criterion (AIC)
• Final Prediction Error (FPE)
• Automatic Relevance Determination

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Cross-validation
Split data into training and test set, learn
model parameters on training set and use these
parameters to predict
Problem facing unsupervised learning diffuculties
in selecting training and test set so they are
independent due to correlation in residual (i.e.
noise correlated, rendering missing values not
truly missing in the training set)
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From 2-way to multi-way analysis
Space
Trial/Condition/Subject
Time
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Multi-subject analysis

• At least four possibilities
• Pre-average data
• Separate analysis
• Data concatenation
• Tensor models

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Concatenation

• Temporal concatenation
• Common spatial map
• Separate temporal profile

• Spatial concatenation
• Common temporal profile
• Separate spatial maps

space
time
space
time
Subjects can have different spatial maps Same
time series
Subjects can have different time profiles Often
combined with data reduction by SVD for each
subject Spatial ICA, GIFT Toolbox
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Multilinear modelling

• Bilinear Model

Assumption Data instantaneous mixture of
temporal signatures.
(PCA/ICA/NMF)
Trilinear Model
Assumption Data instantaneous mixture of
temporal signatures that are expressed
to various degree over the trials
(Canoncial Decomposition, Parallel Factor
(CP))
(weighted averages over the trials)
"A surprising fact is that the nonrotatability
characteristic can hold even when the number of
factors extracted is greater than every dimension
of the three-way array. - Kruskal 1976
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Unfortunately, multi-linear models are often to
restrictive

• Trilinear model can encompass
• Variability in strength over repeats
• However, other common causes of variation are
• Delay Variability
• Shape Variability

Trial 1 Trial 2
Trial 1 Trial 2
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Violation of multi-linearity causes degeneracy
Space
Trial
Time
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Decomposition with Invariance
Shift Invariance
?1,1
A1,1
A1,2
?2,1
?1,2
A2,1
?3,1
?2,2
?3,2

Y
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Modelling Delay Variability
Shifted CP
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(Mørup et al., NeuroImage 2008)
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Delay modelling of fMRI data fromretinotopic
mapping paradigm
B
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(No Transcript)
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Reverberation/Convolution

Y
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Modeling Delay and Shape Variability
convolutive CP

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CP, ShiftCP and ConvCP
ConvCP Can model arbitrary number of component
delays within the trials and account for shape
variation within the convolutional model
representation
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Convolutive Multi-linear decomposition
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Analysis of fMRI data
Degeneracy
Degeneracy
Each trial consists of a visual stimulus
delivered as an annular full-field checkerboard
reversing at 8 Hz.
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Shift Invariant multiway decompostion
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Overview

• Part I Basic Principles
• Part II Bayesian model comparison
• Part III Unsupervised analysis
• Part IV Experiments
• Conclusion

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Conclusions

• Functional MRI data is noise
• Modelling nuisances can help supress known noise
  sources
• Unsupervised learning is an important framework
  for multivariate analysis of neuroimaging data
  such as fMRI
• Explores pattern in data
• Identifies noise source
• Drives new hypothesis
• Bi-linear analysis ambiguous requiring additional
  assumption such as independence or sparsity
  (forming ICA and Sparse coding

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Conclusions

• Multi-linear modeling offers the ability to
  extract the consistent activity of neuroimaging
  data over repeats/subjects/conditions etc.
• However, violation of multi-linearity due to
  variability causes degeneracy
• Common causes of variability in neuroimaging data
  are delay and shape variation
• Advancing the CP model to ShiftCP and ConvCP
  enables to address these types of variability.
• Modelling delay and shape changes is also
  relevant for bi-linear modelling and open
  doorways to address latent causal relations.

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