Analysis of Functional MRI Timeseries Data Using Signal …bouman/publications/pdf/... ·...

Analysis of Functional MRI Timeseries DataUsing Signal Processing Techniques

Sea Chen

Department of Biomedical Engineering

Advisors: Dr. Charles A. Bouman and Dr. Mark J. Lowe

S. Chen – Final Exam – October 7, 2002 – p.1/39

Overview

� Introduction

� Update: Supertemporal Resolution Analysis

� Review

� New simulations

� New data

� Clustered Components Analysis

� Motivation

� Theory

� Methods

� Results

� Conclusions


Goals

We would like to aid in the understanding of theblood-oxygenation-level-dependent (BOLD) contrastmechanisms used in functional magnetic resonanceimaging (fMRI) through

� achieving a high signal-to-noise (SNR) estimate ofthe BOLD response.

� achieving a high temporal resolution estimate ofthe BOLD response.


fMRI: The basic idea

�

Experimental paradigm designed to activate neuronal metabolism

�

Changes in blood oxygenation during activation � changes in physicalparameters affecting MR signal

�

Contrast produced by difference between active and control states

�

Data set is volume of pixels repeated over time

Considerthis pixel Time

0 168 24 32 40

ResponseSignal

One pixel of one slice through time

... ... ... ... ... ...

StimulusSignal


Supertemporal Resolution:Motivation

� Short TR

� Better time resolution

� Lower SNR due to saturation effects

� BOLD signal is distorted by blood inflow effects

� Long TR

� Poorer time resolution

� BOLD effect more dominant in activation signal


Supertemporal Resolution: Review

�

Assumption: Voxels exhibiting the same generating activationsignal span different slices in a 2D acquisition

�

Method exploits the timing characteristics of the 2D acquisition

�

Bayesian prior used to implement temporal regularization


Supertemporal Resolution: Review

� MAP estimate for Supertemporal Resoution (STR)

��

�

��

� � ��

� � � � � � � � �

��

��

! � � � " � # �

where

� � �$ % &

�('$ �' �$

� Optimization performed using conjugate gradient

� Regularization parameter ) found by crossvalidationstrategy


Supertemporal Resolution: Updates

� Reduction in computation time

� Minor software revisions

� New hardware

� New simulations

� Introduced amplitude amplification factors � (1x,2x, 4x, 6x, 8x simulating increase in

�� -field)

� Generated multiple (20 / �) datasets

� New human visual system data

� Three inch surface coil

� Multiple runs (3 TR = 2.0s, 3 TR = 0.5) of 3.5cycles


Performance comparison

�

Simple averaging (SA) method

�

Alignment of slices into timeframe of first slice

�

NO regularization

�

Closed form solution

�

0.5 s time resolution estimate from TR = 0.5 s dataset

�

Interpolation with regularization (IWR) method

�

Alignment of slices into timeframe of first slice

�

Regularization applied and chosen with crossvalidation

�

Numerical optimization with conjugate gradient

�


�

Supertemporal regularization (STR) method

�

Slice timing considered in data model

�

Regularization applied and chose with crossvalidation

�

Numerical optimization with conjugate gradient

�



Simulation results: Performance

1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

Amplitude Amplification Factor

Mea

n S

quar

e E

rror

(A

U)

Mean square error of estimates versus synthetic amplitude amplification

mean SA errormean IWR errormean STR errorindividual SA errorsindividual IWR errorsindividual STR errorse

0.5 error

e2.0

error

Mean square error of simulation results for different analysis methods plotted againstamplitude amplification factor �


Simulation results: Examples

40 50 60 70 80 90 100 110 120−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Time (s)

Inte

nsity

(A

U)

IWR estimate on synthetic dataset at 4x template amplitudes

normalized IWR estimateinjected BOLD signal

40 50 60 70 80 90 100 110 120−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Time (s)

Inte

nsity

(A

U)

STR estimate on synthetic dataset at 4x template amplitudes

normalized STR estimateinjected BOLD signal

IWR estimate on TR=0.5s data STR estimate on TR=2.0s dataExamples of 0.5 second estimates at � � �


Human data results:Simple averaging method

40 50 60 70 80 90 100 110 120−0.15

−0.1

−0.05

0

0.05

0.1

0.15

SA estimates for V(r#)0.5

data series

Nor

mal

ized

inte

nsity

(A

U)

Time (s)

V(r1)0.5

V(r2)0.5

V(r3)0.5

40 50 60 70 80 90 100 110 120−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Time (s)

Nor

mal

ized

inte

nsity

(A

U)

Statistics on SA estimates for V(r#)0.5

data series

meanmean ± std

SA estimates Mean and std. dev. of SA estimatesSimple averaging estimates on the TR=0.5 second dataset (3 experiments)


Human data results: Interpolationwith regularization method

40 50 60 70 80 90 100 110 120−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Time (s)

Nor

mal

ized

inte

nsity

(A

U)

IWR estimates for V(r#)0.5

data series

V(r1)0.5

V(r2)0.5

V(r3)0.5

40 50 60 70 80 90 100 110 120−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Time (s)

Nor

mal

ized

inte

nsity

(A

U)

Statistics on IWR estimates for V(r#)0.5

data series

meanmean ± std

IWR estimates Mean and std. dev. of IWR estimatesInterpolation with regularization estimates on the TR=0.5 second dataset (3 experiments)


Human data results:Supertemporal Resolution method

40 50 60 70 80 90 100 110 120−0.15

−0.1

−0.05

0

0.05

0.1

0.15

STR estimates for V(r#)2.0

data series

Time (s)

Nor

mal

ized

inte

nsity

(A

U)

V(r1)2.0

V(r2)2.0

V(r3)2.0

40 50 60 70 80 90 100 110 120−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Time (s)

Nor

mal

ized

inte

nsity

(A

U)

Statistics on STR estimates for V(r#)2.0

data series

meanmean ± std

STR estimates Mean and std. dev. of STR estimatesSupertemporal resolution estimates on the TR=2.0 second dataset (3 experiments)


Discussion and conclusions

� Simulated data

� Initially at low SNR, STR performs worse thanIWR because small features masked by noise

� At increasing SNR, STR performs better thanIWR/SA as inherent physical advantangebecomes apparent

� In human data, STR estimates qualitativelydifferent from IWR/SA estimates

� Conclusion: STR may be a valuable tool incharacterizing small features in the BOLD signal athigher static field strengths or higher SNR


Clustered components analysis:Objectives

Hypothesis: Activation by specific functional tasks � Distinct neuralresponses in different parts of the brainTherefore, we propose the following goals:

�

Design and run fMRI experiment activating visual, auditory, andmotor cortices.

�

Estimate number of distinct neural responses (# ofclasses/clusters)

�

Extract an estimate for each response

�

Determine voxel memberships


Existing approaches to signalestimation

�

Principle component analysis (PCA)

�

Extracts orthogonal signals

�

Disadvantage: Signals not usually orthogonal

�

Independent component analysis (ICA)

�

Extracts spatially independent signals

�

Disadvantage: Signals may not be independent

�

Conventional Clustering

�

Groups signal vectors as spheres about a mean

�

Disadvantage: Signals may not form spherical clusters

�

General Comment: None of these methods start with an explicitmodel of the data. All go about estimating the distinct signals inan ad hoc way.


Analysis framework

�

Dimensionality Reduction

�

Signal subspace is orthogonal to noise subspace

�

Noise can be accurately modeled in fMRI

�

Separate signal subspace (dim�

) from noise subspace (dim��

)

�

Clustered Components Analysis

�

Useful information is in shape of signal, amplitude unimportant

�

Component direction is found instead of mean

�

Amplitude can vary in cluster so long as shape preserved

�

Clusters found in cylinders instead of spheres


Interpretation of ClusteredComponents Analysis

�

Because amplitude of the voxel signal is not important, themethod clusters around component directions, not componentmeans.

�

This means the clusters can be thought of as cylinders rather thanthe traditional spheres.


Dimensionality reduction:Harmonic decomposition

Data model for harmonic decomposition

� � � ��

� �

:

� �

detrended voxel timecourse matrix (

=# of timepoints,�

=# of voxels)

� �

:

�

matrix of sampled sines and cosines (

�

harmoniccomponents)

�

��

:

��

harmonic image

� � :

�

maxtrix of residuals from the least squares fit


Dimensionality reduction: Signalsubspace estimation

�

Signal + noise covariance:

��

��

�

Noise covariance:

�� trace

� � � � � � ��

�

Signal covariance:

��

�

Eigen decomposition

��

�

Only the columns of the

� �eigenvector matrix

��

corresponding to the

�positive eigenvalues of

�� are retained,

yielding the

� �

modified eigenvector matrix

� �� .

� � �

reduced dimensionality feature vector matrix:

� � � � � �

��

(

�

is a whitening vector matrix derived from� )


Data Model for Clustered ComponentAnalysis

�

Assumptions

�

Only shape of the response important

�

Amplitude is NOT important

�

Noise independent in space and time (time-independence can be relaxed)

�

Our Model

� �$ is

�

-dimensional column vector representation of the � � �

timecourse

�$ � �$ �� $

e1

e2

e3

an Xnewhere

= 1.5, X = 1an n

an Xn ne + W

� �$ is the unknown scalar amplitude for pixel �,

� � � � �� are the

�

component directions

� �� $ � �is class of the pixel

� � $ is a Gaussian noise vector


Clustered components approach

�

Goal: Minimize minimum description length (MDL) criterion

MDL � � loglikelihood

��

��

# of parameters � � � �# of datapoints

�

Unknown model parameters

� �

is the model order (number of clusters)

� � � is the amplitude of each pixel

� ��

is the set of distinct neural responses

� � � � ��

are the prior probabilities for each class

�

Use maximum likelihood (ML) estimate

� � � implicitly

�

Find ML estimates

� �� and

� � � using theExpectation-Maximization (EM) algorithm for each model order

�

�

Estimate model order

� �

by cluster merging and minimizing theMDL criterion


Voxel likelihood function

�

Likelihood for each voxel

� '$ � �$��

�

�� ! � � � #� ��

��

�

ML estimate of the scalar amplitude

� � � � ��

�

Voxel log-likelihood

�� '$ � �$��

��

�

� � � � ��

��

� � � � ��


Maximum likelihood estimate

�

Log-likelihood of the entire dataset

� � � �� ! ' � �� # ��

$ � ��

��

�� ! '$ � � � �� $ # �

��

$ � ��

� ��

� �

! # � � � ��

�

� ' �$ '$ � � �� '$ '$ � � ��

�

ML estimate of the parameters

� � �� "! # �� $ % �'& ( ��


Expectation-maximization equations

�

Posterior probability

$ �� %�� ( & ��

$ %� � ��'& �( ��

�� $ %� � ��

� & �( ��

�

E-step

� � ��

��

$ �� %�� ( & ��

� � � � � � � � �

��

��

� � �

��

& � & � $ �� %�

� � ( & ��

�

M-step � � �

� � ��

� � � �

� � � �

� � � � � � � � � � �


Order Estimation through ClusterMerging

1. Start with large number of clusters (

�

) and initialize

2. Run EM algorithm to convergence

3. Choose the two clusters that minimize the distance function (which is the upperbound on the change in MDL)

� !� � � # � � �� ! �� # � � � � ! �� # � � � � � ! �� #

4. Merge clusters using

��

� ��

5. Decrement

�

and initialize next iteration with new clusters

6. Repeat 2 through 5 until

�= 1

7. Choose number of components minimizing the MDL criterion

�� ! �� # � � � � � �� ! ' � �� # � � � � � � !� � #


Synthetic fMRI Images

�

Synthetic data

�

Baseline control images created at each sample point

�

During periods of activation, 3 different realistic signals with varyingamplitudes were injected

�

Gaussian white noise added to each voxel at each timepoint time

�

Verification and comparison using different analysis methods applied before andafter signal subspace estimation (SSE)

�

PCA, using 3 components corresponding to the 3 largest variances

�

Spatial ICA constrained to yield 3 components

�

Spatial ICA unconstrained, using 3 best components

�

Fuzzy C-means (FCM) clustering constrained to yield 3 clusters

�

CCA


Paradigm Design

�

For our dimensionality reduction, activation must be periodic

�

Block activation scheme

�

1 cycle = 32 seconds control (rest state), 32 seconds

�

1 scan = 16 seconds lead in - 4.5 paradigm cycles - 16 seconds lead out (onlyuse samples during paradigm)

�

0.5 Hz sample rate (TR = 2 seconds)

�

To illustrate the power of the clustering method, many different types of functionalcortex must be activated

�

Visual: Flashing 8Hz checkerboard

�

Auditory: Forward vs. Backward speech (backwards is the control)

�

Motor: Self paced finger tapping

On OnOn On OnOff Off Off Off Off

Lead-in Lead-out

0:00 0:16 0:48 1:20 1:52 2:24 2:56 3:28 4:00 4:32 5:04 5:20


Simulated data: Hard classifications

Results for CCA applied to synthetic data


Simulated data: Qualitative results

10 20 30 40 50 60 70 80 90 100 110

−0.5

0

0.5

10 20 30 40 50 60 70 80 90 100 110

−0.5

0

0.5

10 20 30 40 50 60 70 80 90 100 110

−0.5

0

0.5

10 20 30 40 50 60 70 80 90 100 110

−0.5

0

0.5

10 20 30 40 50 60 70 80 90 100 110

−0.5

0

0.5

10 20 30 40 50 60 70 80 90 100 110

−0.5

0

0.5

10 20 30 40 50 60 70 80 90 100 110

−0.5

0

0.5

10 20 30 40 50 60 70 80 90 100 110

−0.5

0

0.5

10 20 30 40 50 60 70 80 90 100 110

−0.5

0

0.5

PCA FCM constrained ICA

10 20 30 40 50 60 70 80 90 100 110

−0.5

0

0.5

10 20 30 40 50 60 70 80 90 100 110

−0.5

0

0.5

10 20 30 40 50 60 70 80 90 100 110

−0.5

0

0.5

10 20 30 40 50 60 70 80 90 100 110

−0.5

0

0.5

10 20 30 40 50 60 70 80 90 100 110

−0.5

0

0.5

10 20 30 40 50 60 70 80 90 100 110

−0.5

0

0.5

unconstrained ICA CCAEstimates after application of SSE


Simulated data: Quantitative results

Mean squared error for analyses on synthetic data before and after signal subspaceestimation (SSE)

PCA FCM ICA (c) ICA (u) CCA

Before SSE

��

��

� � � ��

� � ��

� � ��

� � ��

After SSE

��

��

� � ��

� � ��

� � � ��

� � ��

Number of voxels classified correctly on synthetic data before and after signal subspaceestimation (SSE) out of 192 total voxels

PCA ICA (c) ICA (u) FCM CCA

Before SSE 61 113 38 95 167

After SSE 111 162 77 151 169


Human data:Timesequence realizations

20 40 60 80 100 120 140 160 180 200 220−2

−1.5

−1

−0.5

0

0.5

1

1.5

2Class 1Class 2Class 3Class 4Class 5

First 5 clusters


Human data:Timesequence realizations

20 40 60 80 100 120 140 160 180 200 220−2

−1.5

−1

−0.5

0

0.5

1

1.5

2Class 6Class 7Class 8Class 9

Clusters 6-9


Human data: Hard classifications

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

Motor cortex (first 5 clusters): (L) Upper slice, (R) Lower slice



50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

Auditory cortex (first 5 clusters): (L) Upper slice, (R) Lower slice



50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

Visual cortex (first 5 clusters): (L) Upper slice, (R) Lower slice


Conclusions

�

Clustered component analysis is a new method of extracting

signals where only shape, not amplitude, is important

�

CCA has been shown to perform well on simulated data

�

The experimental results show the following:

�

The distinct neuronal signals do not correlate strongly

with the known functional cortices

�

The clusters tend to lie along sulcal-gyral boundaries,

possibly correlated with vasculature

�

CCA can be used with dimensionality reduction strategies

other than the ones used in our experiments

�

CCA may also be adapted for use with applications other

than fMRIS. Chen – Final Exam – October 7, 2002 – p.38/39

Acknowledgements

� Major Professors: Dr. Mark J. Lowe and Dr.Charles A. Bouman

� Committee Members: Dr. Peter C. Doerschuk andDr. Edward J. Delp

� Department of Biomedical Engineering andDivision of Imaging Sciences


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