Introducción a la técnica y análisis de resonancia magnética funcional

Jorge L. Armony

McGill UniversityMontrealCanada

CURSO UAM 20

13

Interacciones neuronales

Kandel, Schwartz & Jessel: Principles of Neural Science

Potencial de acción

CURSO UAM 20

13

Técnicas de neuroimagen funcional: Señal electrodinámica

Electrodos de profundidad

Electroencefalografía (EEG)

Magnetoencefalografía (MEG)

CURSO UAM 20

13

Técnicas de neuroimagen funcional:Señal hemodinámica

CURSO UAM 20

13

These facts seem to us to indicate the existence of an automatic mechanism by which the blood-supply of any part of the cerebral tissue is varied in accordance with the activity of the chemical changes which underlie the functional action of that part.

J Physiol. (1890), 11: 85-158

CURSO UAM 20

13

Resonancia magnética funcional (IRMf/fMRI)

CURSO UAM 20

13

La hemoglobina

- four globin chains- each globin chain contains a heme group- at center of each heme group is an iron atom (Fe)- each heme group can attach an oxygen atom (O2)

Iron

CURSO UAM 20

13

Over ninety years ago, on November 8, 1845, Michael Faraday investigated the

magnetic properties of dried blood and made a note "Must try recent fluid blood."

If he had determined the magnetic susceptibilities of arterial and venous blood, he

would have found them to differ by a large amount (as much as twenty per cent

for completely oxygenated and completely deoxygenated blood); this discovery

without doubt would have excited much interest and would have influenced

appreciably the course of research on blood and hemoglobin.

PNAS (1936), 22: 210-216

CURSO UAM 20

13

Oxy-Hb (four O2) is diamagnetic → no ΔB effectsDeoxy-Hb is paramagnetic → if [deoxy-Hb] ↓ → local ΔB ↓

OxyhemoglobinDiamagnetic

χ ~ -0.3

DeoxyhemoglobinParamagnetic

χ ~ 1.6

B0

oxyHbdeoxyHb

Hemoglobin: Magnetic Properties

CURSO UAM 20

13

B0

oxyHbdeoxyHb

Hemoglobin: Magnetic Properties

CURSO UAM 20

13

neuronal activity ↑

tissue energy demand ↑

Glucose, O2 consumption ↑

blood flow and volume ↑

local dHb content of blood ↓

local dHb-induced magnetic field disturbance ↓

BOLD fMRI signal ↑

neurovascular coupling

fMRI relevantphysiological correlates

Source: B. Pike, MNI(Blood Oxygen Level Dependent signal) CURSO U

AM 2013

Logothetis et al. (2001). Nature 412: 150-157

Relationship between BOLD signal and neural activity

LFP: local field potentialsMUA: multiunit activitySDF: Spike density function

CURSO UAM 20

13

13

First Functional Images

Kwong et al. (1992) PNAS 89: 5675-5679CURSO U

AM 2013

14

Source: Robert Cox’s web slides

Hemodynamic response

CURSO UAM 20

13

1 25

12 2430

Resolution in x (e.g. 4mm)

Matrix in the xy plane (e.g., 64 x 64)

Res

olut

ion

in y

(e.g

. 4m

m)

Resolution in z(e.g. 4mm)

Un volumen:64 x 64 x 30 voxels

6

7 13

18

19

VOXEL(Volumetric Pixel)

4 mm

4 m

m

4 mm

122880 datos!

30 slices

CURSO UAM 20

13

time

Imagenes funcionales

CURSO UAM 20

13

Experimental Design in Neuroimaging

Most studies use a categorical, subtractive design

CATEGORICAL:Two (or more) levels of a given category (one of them usually serves as control)Examples: Happy, Fearful and Neutral Faces, Remembered and Forgotten words, Pictures and Fixation Cross

SUBTRACTIVE:Directly compares two conditions (A-B). Uses the notion of cognitive subtraction(Donders, 1868), which relies on the assumptions of pure insertion and linearity

CURSO UAM 20

13

Cogntitive subtraction = Mathematical Subtraction(principle of pure insertion: Donders, 1868)

>

= + FEARFEAR

– = + –FEARFEAR = FEARFEAR

CURSO UAM 20

13

19

Preprocessing

• Corrects for non-task-related variability in experimental data

• Usually done without consideration ofexperimental design; thus, pre-analysis

• Sometimes called post-processing, in reference to being done after acquisition

• Attempts to remove, rather than model, data variability

Source: http://www.biac.duke.eduCURSO U

AM 2013

20

Preprocessing Steps

• Realignment

• Coregistration

• Slice timing

• Normalization

• Spatial Smoothing

CURSO UAM 20

13

21

Inter-scan movement: RealignmentPeople move, even if they don’t realize!

Need for motion correction

Two steps:

1. Registration: Determine the 6 parameters that describe the rigid-body transformation between each image and a reference image (usu. first in series).

2. Transformation: Resampling each image according to the determined transformation parameters.

CURSO UAM 20

13

22

roll

pitch

yaw

x

y

z

Rigid body movement: 3 translation parameters3 rotation parameters

Same location in the grid

Same location in the brain

Inter-scan movement: Realignment

CURSO UAM 20

13

23

Small movements are corrected well

Sudden movements are more problematic (especially if correlated withexperimental paradigm)

TRASLATION

ROTATION

mm

rad

xy

z

pitch (x)

yaw (y)

roll (z)

Realignment: Registration

CURSO UAM 20

13

24

BEFORE REALIGNMENT AFTER REALIGNMENT

Difference between first and last image in one session

CURSO UAM 20

13

25

Optimisation* Optimisation involves finding some “best”

parameters according to an “objective function”, which is either minimised or maximised

* The “objective function” is often related to a probability based on some model

Value of parameter

Objective function

Most probable solution (global optimum)

Local optimumLocal optimum

Source: John AshburnerCURSO U

AM 2013

26

Objective Functions* Intra-modal

* Mean squared difference (minimise)* Normalised cross correlation (maximise)* Entropy of difference (minimise)

* Inter-modal (or intra-modal)* Mutual information (maximise)* Normalised mutual information (maximise)* Entropy correlation coefficient (maximise)* AIR cost function (minimise)

Source: John AshburnerCURSO U

AM 2013

27

Residual Errors from aligned fMRI* Re-sampling can introduce interpolation errors

* especially tri-linear interpolation

* Gaps between slices can cause aliasing artefacts* 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 modelled as confounds in subsequent analyses

Source: John AshburnerCURSO U

AM 2013

28

Unwarping

Estimate movement parameters.

Estimate new distortion fields for each image:

• estimate rate of change of field with respect to the current estimate of movement parameters in pitch and roll.

Estimate reference from mean of all scans.

Unwarp time series.

0B ϕ∂ ∂ 0B θ∂ ∂

Δϕ +Δθ

Andersson et al, 2001Source: John AshburnerCURSO U

AM 2013

29

Realignment: Transformation

Source: John AshburnerCURSO U

AM 2013

30

Inter-subject brain differences: NormalizationPeople’s brains are different!!

CURSO UAM 20

13

31

Normalization

Also useful for reporting coordinates in a standard space (e.g., Talairach and Tournoux)

CURSO UAM 20

13

32

Inter-subject brain differences: After Normalization

CURSO UAM 20

13

33

Spatial Normalisation - Procedure

Non-linear registration

* Minimise mean squared difference from template image(s)

Affine registrationSource: John Ashburner

CURSO UAM 20

13

34

Spatial Normalisation - Affine* The first part is a 12 parameter affine

transform* 3 translations* 3 rotations* 3 zooms* 3 shears

* Fits overall shape and size

* Algorithm simultaneously minimises* Mean-squared difference between template and source image* Squared distance between parameters and their expected values

(regularisation)CURSO UAM 20

13

35

Spatial Normalisation - Non-linearDeformations consist of a linear combination of smooth basis functions

These are the lowest frequencies of a 3D discrete cosine transform (DCT)

Algorithm simultaneously minimises* Mean squared difference between template and

source image * Squared distance between parameters and their

known expectationCURSO UAM 20

13

36

Templateimage

Affine registration.(χ2 = 472.1)

Non-linearregistration

withoutregularisation.(χ2 = 287.3)

Non-linearregistration

usingregularisation.(χ2 = 302.7)

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

Spatial Normalisation - Overfitting

CURSO UAM 20

13

37

CURSO UAM 20

13

38

Why smooth?

• Remove residual inter-subject brain differences

• Allow for the use of Gaussian random field theory (later…)

CURSO UAM 20

13

Crazy little thing called BOLD

Image acquisition by Sarael Alcauter, National Institute of Psychiatry “Ramón de la Fuente”, Mexico City, MexicoCURSO U

AM 2013

CURSO UAM 20

13

Mean (Queen) = 10.77 Mean (Silence) = 10.26

t(118) = 14.17 p = 2 10-27

0.000000000000000000000000002

= +

CURSO UAM 20

13

=

T-map

CURSO UAM 20

13

t-values

t0

t-values

t0

CURSO UAM 20

13

TQA: Temporal Queen Area

CURSO UAM 20

13

Respuestas cerebrales a estímulos visuales

>

Activación en la corteza fusiforme (FFA): Área involucrada en el procesamiento de rostros

Kanwisher et al. (1997)MIT, USA

Trejo & Armony (2008)HGM, México

CURSO UAM 20

13

Respuestas cerebrales a estímulos visuales

>

Activación en la corteza parahipocámpica: Área involucrada en el procesamiento de información espacial

CURSO UAM 20

13

Segregación de la respuesta en corteza temporal ventral a distintostipos de estímulos visuales

CURSO UAM 20

13

– =

Casas – Rostros

-1

0

1

2

3

4

5

6

7

8

9

10–

Mapa t

=

Error estándar de la media

Mapa t con umbral

t > 5.2 (p<0.05 FWE)

-4

-2

0

2

4

6

8

CURSO UAM 20

13

Casas – Rostros

Mapa t

=

Error estándar de la media

Mapa t con umbral

t > 5.2 (p<0.05 FWE)

-4

-2

0

2

4

6

8

Imagen T1(estructural)

Mapa de activaciones(SPM)

+

CURSO UAM 20

13

NEUTRAL

RECORDADA

NUEVA

RECORDADA

OLVIDADA

Efecto subsecuente de memoria

Subsequently Remembered > Subsequently Forgotten

Sergerie, Lepage & Armony (2005). NeuroimageCURSO U

AM 2013

0

0.03

0.06

0.09

0.12

Fear Happy Neutral

R DLPFC (48 22 16)

-0.01

0.01

0.03

0.05

Fear Happy Neutral

L DLPFC (-34 32 12)

RL

EMOTIONAL

RL

NEUTRAL

Efecto del valor emocional en la memoria

Sergerie, Lepage & Armony (2005). NeuroimageCURSO U

AM 2013

What else can we do?

• Cleaner data (e.g., high-pass filtering, scaling)

• More sophisticated averaging (modeling)

• Choose a good threshold

CURSO UAM 20

13

Modeling the expected response

Original:ON-OFF = 13.02, t(80) = 8.6, p = 2 10-13

Shifted blocks:ON-OFF = 12.45, t(80) = 17.7, p = 10-29CURSO U

AM 2013

Modeling the expected response

ASSUMPTIONS:

Experimental Paradigm• The response is solely determined by two conditions: the ON and OFF

blocks

Brain Physiology• There is a delay of about 6 sec between onset/offset of the block and the

observed signal

Brain-Design Interactions• The response is the same within a given block• The response is the same for all blocks belonging to the same condition

CURSO UAM 20

13

Modeling the expected response

0 10 20 30 40 50 60 70 80 90-15

-10

-5

0

5

10

15

20

25

0 10 20 30 40 50 60 70 80 90-15

-10

-5

0

5

10

15

20

25

CURSO UAM 20

13

A better model for the HRF

0 10 20 30 40 50 60 70 80-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 10 20 30 40 50 60 70 80-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Convolve with a model of hrf

0 10 20 30 40 50 60 70 80-15

-10

-5

0

5

10

15

20

25

CURSO UAM 20

13

Modeling the data: The General Linear Model

= β +

y x εCURSO U

AM 2013

yi = xi β + εi

data model

errorparameter

0 10 20 30 40 50 60 70 80 90-15

-10

-5

0

5

10

15

20

25

Given by the

scanner

You build it

The weight of the model

Part of the data not accounted by the model

BEFORE

DURING

AFTER

NEVER

Who’s who in the GLM

CURSO UAM 20

13

Fitting the model to the data

CURSO UAM 20

13

In search of a criterion

0 10 20 30 40 50 60 70 80 90-15

-10

-5

0

5

10

15

20

25

Try to minimize:

∑(yi – Mi)2Mi

yi

CURSO UAM 20

13

Height

Sum

of s

quar

es

CURSO UAM 20

13

Height

Sum

of s

quar

es

Choosing a height (parameter) of 8 minimizes the “distance” between the data and the model.

ORDINARY LEAST SQUARES (OLS) SOLUTION

0 10 20 30 40 50 60 70 80 90-20

-15

-10

-5

0

5

10

15

20

25

CURSO UAM 20

13

CURSO UAM 20

13

A bigger (and better) model

= β1 +

y x1 ε

β2

x2

β3

x3

+ +

= β1 + β2 β3+ +CURSO U

AM 2013

System of Linear EquationsNow we are left with a SLE of N independent equations and p unknownsThree possibilities:

1. N = pOnly one solution X = A-1b

2. N < pInfinite number of solutions (underdetermined system)

3. N > pNo solution (overdetermined system)

CURSO UAM 20

13

εβ += Xy

∑=

N

tt

1

2ε̂

Ordinary Least Squares Estimator

Given

We try to find β̂ (an estimate of the true parameters β )

such that is minimal (as small as possible)

βε ˆˆ Xy −= (residuals)

CURSO UAM 20

13

In matrix form…

=

β1

+

y X ε

β2

β3

+β= CURSO U

AM 2013

εβ += Xy

=

β

ε+X

N: number of scansp: number of regressors (explanatory variables)

N: number of scansp: number of regressors (explanatory variables)

y

General Linear Model (GLM)

N N N

1 11p

p

CURSO UAM 20

13

,εβ += XyGiven

If X was a number, then yXinvyXXy )(ˆ 1 === −β

But X is a matrix, and it typically has no inverse (it is not square)

β̂Xy =we look for

Looking for a solution…

y1 = x11β1+x12 β 2+…+x1p βp

y2 = x21β1+x22 β 2+…+x2p βp

…

yN = xN1β1+xN2 β 2+…+xNpβp

More equations (scans) than unknowns (conditions)Overdetermined System

CURSO UAM 20

13

Settling for a pseudo-solution…

yXXXyXpinv TT 1)()(ˆ −==β

So, we use the pseudo-inverse (we choose the Moore-Penrose version)

( ) TT XXXXpinv 1)( −=

But… this pseudo-solution is the OLS solution!

CURSO UAM 20

13

ye

Design space defined by X

x1

x2

PIRRye

−==

β̂ˆ Xy =

yXXX TT 1)(ˆ −=β

TT XXXXPPyy

1)(

ˆ−=

=

Residual forming matrix R

Projection matrix P

OLS estimates

A Geometric Perspective

Slide from Klaas StephanCURSO U

AM 2013

Basic Model

Linear Modulation(habituation)

Other Modulation(e.g., performance, physio)

Independent Blocks

Parametric Modeling

Categorical Modeling

CURSO UAM 20

13

Event-related Designs

CURSO UAM 20

13

Event-related Designs

CURSO UAM 20

13

CURSO UAM 20

13

GLM: The L stands for Linear

It assumes that the superposition principle holds

A BC D

A+C B+DIf thenCURSO UAM 20

13

GLM: And the M stands for Model

CURSO UAM 20

13

Introducing the Temporal Derivative

CURSO UAM 20

13

CURSO UAM 20

13

hrf alone hrf + d(hrf)/dt

CURSO UAM 20

13

HRF, temporal derivative and dispersion Gamma functions

Fourier set (sines and cosines)

Basis Functions

CURSO UAM 20

13

Basis Functions: Choosing the Right Model

Good: Easy to analyze and interpret in terms of hemodynamic activity

Bad: May fail to capture real responses that do not fit the assumed behaviour

Few, model-based functions (e.g., synthetic HRF)

Good: No a priori assumptions about the shape of the response. Can capture unexpected responses (e.g., longer delay/duration)

Bad: Difficult to interpret physiologically. They may capture non-hemodynamic responses (noise, artifacts)

Many, general basis functions (e.g., Fourier set)

All models are wrong, but some are useful – George Box

CURSO UAM 20

13

Statistical Inference

)ˆ(

ˆ

ββ

stdt =

Where in the brain is my experimental parameter (β ) significantly bigger than zero?

pNdf −=

t-map

= = p-values

-4

-2

0

2

4

6

8

10

12

14

–Why don't you say at once “it’s a miracle”?–Because it may be only chance.

F. Dostoevsky, Crime and Punishment

CURSO UAM 20

13

Statistical Inference

Where in the brain is one experimental parameter (β1 ) significantly bigger than another one (β2 ) ?

Contrast: Linear combination of parameters

Hypothesis: β1 - β2 > 0

1∗β1 + (-1)∗β2

β1 > β2

c = [1 -1] c = [1 -1 0 0] if we included mean and linear trend in the model

)ˆ(

ˆ

ββcstd

ct =

= [1 -1]β1

β2

CURSO UAM 20

13

My p-value is smaller than yours…

If P is between .1 and .9 there is certainly no reason to suspect the hypothesis tested. If it is below .02 it is strongly indicated that the hypothesis fails to account for the whole of the facts. We shall not often be astray if we draw a conventional line at .05.

R. Fisher, Statistical Methods for Research Workers (1925)

CURSO UAM 20

13

Multiple comparisons across the brainThere are ~100,000 voxels in the brain!!

• No correction (p < 0.05 uncorrected)Good: Easy, minimize Type II errorsBad: Too many false positives (Type I errors) 5% of 100,000 = 5,000 voxels!

• Bonferroni correction (p < 0.05/100,000 = 0.0000005)Good: Easy, minimize Type I errorsBad: Too stringent (worst-case scenario). Too many Type II errors

• Gaussian random field theory (spatial smoothing)Good: Works well for spatially-correlated data. Reasonable resultsBad: Still fairly stringent. Removes some specificity (due to smoothing)

• False discovery rate (FDR)Good: Fewer Type II errors while still controlling for Type I errorsBad: Significance of a voxel depends on significance of other voxels

• “Pseudo-Bonferroni” correction (p < 0.001)Good: Somewhere between 0.05 uncorrected and 0.05 correctedBad: Completely arbitraryCURSO U

AM 2013