# Detecting perfusion abnormalities in Arterial Spin Labelling Random-effects and Mixed-effects GLM

Camille Maumet

<figure><img src="image_1.png" /></figure>

## To cite this version:

Camille Maumet. Detecting perfusion abnormalities in Arterial Spin Labelling Random-effects and Mixed-effects GLM. NeuroImaging seminars, Institute of Psychiatry King's College London, Feb 2014, London, United Kingdom. hal-04091257

HAL Id: hal-04091257

https://inria.hal.science/hal-04091257

Submitted on 7 May 2023

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<figure><img src="image_2.png" /></figure>
# Detecting perfusion abnormalities in Arterial Spin Labelling

## Random-effects and Mixed-effects GLM

Camille Maumet ¹,²

¹Neuroimaging Statistics, University of Warwick, UK.

²Formerly affiliated to University of Rennes 1 & Inria, VisAGeS, France.

NeuroImaging seminars, Institute of Psychiatry
King's College London - February 14th, 2014

<figure><img src="image_3.png" /></figure>
# Outline

## Detecting perfusion abnormalities

## Random-Effects and Mixed-Effects General Linear Model

## Detections in ASL

## Conclusions
# Outline

## Detecting perfusion abnormalities

- Arterial Spin Labelling

- Group analysis

- Single-subject analysis

## Random-Effects and Mixed-Effects General Linear Model

- Hypothesis testing

- Random-effects and Mixed-effects

## Detections in ASL

- Methods

- Experiment

- Results

## Conclusions
# Outline

## Detecting perfusion abnormalities
### Arterial Spin Labelling

* Group analysis
* Single-subject analysis

## Random-Effects and Mixed-Effects General Linear Model
### Hypothesis testing
### Random-effects and Mixed-effects

## Detections in ASL
### Methods
### Experiment
### Results

## Conclusions
# Brain perfusion

Detecting
perfusion
abnormalities in
Arterial Spin
Labelling

Camille Maumet

Detecting
perfusion
abnormalities

ASL

Group
Single-subject

General Linear
Model: MFX and
RFX

Hypothesis
testing
RFX and MFX

Detections in ASL
Methods
Experiment
Results

Conclusions

<figure><img src="image_4.png" /></figure>

Blood supply to the brain
## Brain perfusion

Detecting

perfusion

abnormalities in

Arterial Spin

Labelling

Camille Maumet

Detecting

perfusion

abnormalities

ASL

Group

Single-subject

General Linear

Model: MFX and

RFX

Hypothesis

testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions

<figure><img src="image_5.png" /></figure>

Blood supply to the brain

**Brain perfusion** is the biological process that ensures the delivery of oxygen and nutrients to the cerebral tissues by means of microcirculation.

<figure><img src="image_6.png" /></figure>

Example of perfusion parameters
# Arterial Spin Labelling (ASL)

<figure><img src="image_7.png" /><figcaption>Labelling</figcaption></figure>

Detecting
perfusion
abnormalities in
Arterial Spin
Labelling

Camille Maumet

Detecting
perfusion
abnormalities

ASL

Group
Single-subject

General Linear
Model: MFX and
RFX

Hypothesis
testing
RFX and MFX

Detections in ASL

Methods
Experiment
Results

Conclusions

[Detre et al., MRM 1992]
# Arterial Spin Labelling (ASL)

<figure><img src="image_8.png" /><figcaption>Labelling</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Delay</figcaption></figure>

Detecting
perfusion
abnormalities in
Arterial Spin
Labelling

Camille Maumet

Detecting
perfusion
abnormalities

ASL

Group

Single-subject

General Linear
Model: MFX and
RFX

Hypothesis
testing
RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions

[Detre et al., MRM 1992]
Arterial Spin Labelling (ASL)

<figure><img src="image_10.png" /></figure>

Labelling

<figure><img src="image_11.png" /></figure>

Delay

<figure><img src="image_12.png" /></figure>

Readout

Detecting
perfusion

abnormalities in
Arterial Spin
Labelling

Camille Maumet

Detecting
perfusion
abnormalities

ASL

Group

Single-subject

General Linear
Model: MFX and

RFX

Hypothesis
testing
RFX and MFX

Detections in ASL

Methods
Experiment
Results

Conclusions

[Detre et al., MRM 1992]
Arterial Spin Labelling (ASL)

Detecting
perfusion
abnormalities in
Arterial Spin
Labelling

Camille Maumet

Detecting
perfusion
abnormalities

ASL

Group
Single-subject

General Linear
Model: MFX and

RFX

Hypothesis
testing
RFX and MFX

Detections in ASL

Methods
Experiment
Results

Conclusions

<figure><img src="image_13.png" /></figure>

Labelling

<figure><img src="image_14.png" /></figure>

Delay

<figure><img src="image_15.png" /></figure>

Readout

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /></figure>

Label

[Detre et al., MRM 1992]
Arterial Spin Labelling (ASL)

<figure><img src="image_18.png" /></figure>

Detecting
perfusion
abnormalities in
Arterial Spin
Labelling

<figure><img src="image_19.png" /></figure>

Camille Maumet

perfusion
abnormalities

ASL

Group

Single-subject

General Linear
Model: MFX and
RFX

Hypothesis
testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions

[Detre et al., MRM 1992]
Arterial Spin Labelling (ASL)

Detecting

perfusion

abnormalities in

Arterial Spin

Labelling

Camille Maumet

Detecting

perfusion

abnormalities

ASL

Group

Single-subject

General Linear

Model: MFX and

RFX

Hypothesis

testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions

<figure><img src="image_20.png" /></figure>

Labelling

<figure><img src="image_21.png" /></figure>

Delay

<figure><img src="image_22.png" /></figure>

Readout

<figure><img src="image_23.png" /></figure>

<figure><img src="image_24.png" /></figure>

Label

<figure><img src="image_25.png" /></figure>

Readout

<figure><img src="image_26.png" /></figure>

<figure><img src="image_27.png" /></figure>

Control

[Detre et al., MRM 1992]
Arterial Spin Labelling (ASL)

Detecting

perfusion

abnormalities in

Arterial Spin

Labelling

Camille Maumet

Detecting

perfusion

abnormalities

ASL

Group

Single-subject

General Linear

Model: MFX and

RFX

Hypothesis

testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions

<figure><img src="image_28.png" /></figure>

Labelling

<figure><img src="image_29.png" /></figure>

Delay

<figure><img src="image_30.png" /></figure>

Readout

<figure><img src="image_31.png" /></figure>

<figure><img src="image_32.png" /></figure>

Label

<figure><img src="image_33.png" /></figure>

Readout

<figure><img src="image_34.png" /></figure>

<figure><img src="image_35.png" /></figure>

Control

<figure><img src="image_36.png" /></figure>

Control

<figure><img src="image_37.png" /></figure>

<figure><img src="image_38.png" /></figure>

Label

<figure><img src="image_39.png" /></figure>

<figure><img src="image_40.png" /></figure>

Difference
1 pair

[Detre et al., MRM 1992]
Arterial Spin Labelling (ASL)

Detecting

perfusion

abnormalities in

Arterial Spin

Labelling

Camille Maumet

Detecting

perfusion

abnormalities

ASL

Group

Single-subject

General Linear

Model: MFX and

RFX

Hypothesis

testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions

<figure><img src="image_41.png" /></figure>

Labelling

<figure><img src="image_42.png" /></figure>

Delay

<figure><img src="image_43.png" /></figure>

Readout

<figure><img src="image_44.png" /></figure>

<figure><img src="image_45.png" /></figure>

Label

<figure><img src="image_46.png" /></figure>

Readout

<figure><img src="image_47.png" /></figure>

<figure><img src="image_48.png" /></figure>

Control

<figure><img src="image_49.png" /></figure>

Control

<figure><img src="image_50.png" /></figure>

<figure><img src="image_51.png" /></figure>

Label

<figure><img src="image_52.png" /></figure>

<figure><img src="image_53.png" /></figure>

Difference
1 pair

<figure><img src="image_54.png" /></figure>

Difference
60 pairs

[Detre et al., MRM 1992]
# Outline

## Detecting perfusion abnormalities

Arterial Spin Labelling

**Group analysis**

Single-subject analysis

## Random-Effects and Mixed-Effects General Linear Model

Hypothesis testing

Random-effects and Mixed-effects

## Detections in ASL

Methods

Experiment

Results

## Conclusions
<figure><img src="image_55.png" /></figure>

Detecting

perfusion

abnormalities in

Arterial Spin

Labelling

Camille Maumet

Detecting

perfusion

abnormalities

ASL

Group

Single-subject

General Linear

Model: MFX and

RFX

Hypothesis

testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions
Detecting
perfusion

abnormalities in
Arterial Spin
Labelling

Camille Maumet

Detecting
perfusion
abnormalities

ASL

Group

Single-subject

General Linear
Model: MFX and

RFX

Hypothesis
testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions

Within-group analysis

Identify common patterns across a group of subjects.

<figure><img src="image_56.png" /></figure>
# Within-group analysis

Identify common patterns across a group of subjects.

<figure><img src="image_57.png" /></figure>

## Examples

<figure><img src="image_58.png" /></figure>

Group cerebral blood flow.

<figure><img src="image_59.png" /></figure>

Group activation for a language task.

Detecting

perfusion

abnormalities in

Arterial Spin

Labelling

Camille Maumet

Detecting

perfusion

abnormalities

ASL

Group

Single-subject

General Linear

Model: MFX and

RFX

Hypothesis

testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions
Between-group analysis

<figure><img src="image_60.png" /></figure>

<figure><img src="image_61.png" /></figure>

...

<figure><img src="image_62.png" /></figure>

<figure><img src="image_63.png" /></figure>

<figure><img src="image_64.png" /></figure>

...

<figure><img src="image_65.png" /></figure>

Detecting
perfusion

abnormalities in
Arterial Spin
Labelling

Camille Maumet

Detecting
perfusion
abnormalities

ASL

Group

Single-subject

General Linear
Model: MFX and
RFX

Hypothesis
testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions
Between-group analysis

Identify differences at the group level.

<figure><img src="image_66.png" /></figure>

Detecting
perfusion
abnormalities in
Arterial Spin
Labelling

Camille Maumet

Detecting
perfusion
abnormalities
ASL

Group

Single-subject

General Linear
Model: MFX and
RFX

Hypothesis
testing
RFX and MFX

Detections in ASL
Methods
Experiment
Results

Conclusions
# Between-group analysis

Identify differences at the group level.

<figure><img src="image_67.png" /></figure>

Example

<figure><img src="image_68.png" /></figure>

Detecting
perfusion
abnormalities in
Arterial Spin
Labelling

Camille Maumet

Detecting
perfusion
abnormalities
ASL

Group

Single-subject

General Linear Model: MFX and RFX

Hypothesis
testing
RFX and MFX

Detections in ASL
Methods
Experiment
Results

Conclusions
# Outline

## Detecting perfusion abnormalities

Arterial Spin Labelling

Group analysis

Single-subject analysis

## Random-Effects and Mixed-Effects General Linear Model

Hypothesis testing

Random-effects and Mixed-effects

## Detections in ASL

Methods

Experiment

Results

## Conclusions
Within-subject analysis

<figure><img src="image_69.png" /></figure>

Detecting
perfusion
abnormalities in
Arterial Spin
Labelling

Camille Maumet

Detecting
perfusion
abnormalities

ASL
Group

Single-subject

General Linear
Model: MFX and
RFX
Hypothesis
testing
RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions
Within-subject analysis

Identify patterns of perfusion (or activation) in a single
subject.

<figure><img src="image_70.png" /></figure>

Detecting
perfusion

abnormalities in
Arterial Spin
Labelling

Camille Maumet

Detecting

perfusion

abnormalities

ASL

Group

Single-subject

General Linear

Model: MFX and

RFX

Hypothesis

testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions
Within-subject analysis

Identify patterns of perfusion (or activation) in a single
subject.

<figure><img src="image_71.png" /></figure>

Examples

<figure><img src="image_72.png" /><figcaption>Cerebral blood flow.</figcaption></figure>

<figure><img src="image_73.png" /><figcaption>Subject activation for a language task.</figcaption></figure>

Detecting
perfusion

abnormalities in

Arterial Spin

Labelling

Camille Maumet

Detecting

perfusion

abnormalities

ASL

Group

Single-subject

General Linear

Model: MFX and

RFX

Hypothesis

testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions
Between-group individual analysis

Detecting

perfusion

abnormalities in

Arterial Spin

Labelling

Camille Maumet

Detecting

perfusion

abnormalities

ASL

Group

Single-subject

General Linear

Model: MFX and

RFX

Hypothesis

testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions

<figure><img src="image_74.png" /></figure>

<figure><img src="image_75.png" /></figure>

<figure><img src="image_76.png" /></figure>

<figure><img src="image_77.png" /></figure>
Between-group individual analysis

Identify deviation from normality in a single subject.

<figure><img src="image_78.png" /></figure>

Detecting
perfusion
abnormalities in
Arterial Spin
Labelling

Camille Maumet

Detecting
perfusion
abnormalities

ASL
Group

Single-subject

General Linear
Model: MFX and
RFX

Hypothesis
testing
RFX and MFX

Detections in ASL
Methods
Experiment
Results

Conclusions
# Between-group individual analysis

Identify deviation from normality in a single subject.

<figure><img src="image_79.png" /></figure>

## Example

<figure><img src="image_80.png" /></figure>

Hyper- and hypo-perfusions in a patient diagnosed with brain tumour.
# Within-group and between-group analyses

## Group analyses

<figure><img src="image_81.png" /></figure>

## Individual analyses

<figure><img src="image_82.png" /></figure>

<figure><img src="image_83.png" /></figure>

* Study of typical brain perfusion.

* Provide a better understanding of **brain dysfunction** associated with a pathology.

* Study of **brain perfusion** in a particular subject.

* Outline **deviation from normality** (or from a reference group).
## Outline

Detecting
perfusion
abnormalities in
Arterial Spin
Labelling

Camille Maumet

Detecting
perfusion
abnormalities

ASL
Group
Single-subject

General Linear
Model: MFX and
RFX

Hypothesis
testing
RFX and MFX

Detections in ASL
Methods
Experiment
Results

Conclusions

Detecting perfusion abnormalities
Arterial Spin Labelling
Group analysis
Single-subject analysis

## Random-Effects and Mixed-Effects General Linear Model

Hypothesis testing

Random-effects and Mixed-effects

## Detections in ASL

Methods
Experiment
Results

## Conclusions
# Outline

Detecting perfusion abnormalities

Arterial Spin Labelling

Group analysis

Single-subject analysis

## Random-Effects and Mixed-Effects General Linear Model

### Hypothesis testing

Random-effects and Mixed-effects

## Detections in ASL

Methods
Experiment
Results

## Conclusions
A massively univariate approach

<figure><img src="image_84.png" /></figure>

Detecting
perfusion
abnormalities in
Arterial Spin
Labelling

<figure><img src="image_85.png" /></figure>

Camille Maumet

Detecting
perfusion
abnormalities

ASL

Group

Single-subject

General Linear
Model: MFX and
RFX

Hypothesis
testing

RFX and MFX

Detections in ASL

Methods
Experiment
Results

Conclusions
A massively univariate approach

<figure><img src="image_86.png" /></figure>

Detecting
perfusion

abnormalities in
Arterial Spin

Labelling

Camille Maumet

Detecting

perfusion
abnormalities

ASL

Group

Single-subject

General Linear

Model: MFX and

RFX

Hypothesis
testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions
<figure><img src="image_87.png" /></figure>
<figure><img src="image_88.png" /></figure>
<figure><img src="image_89.png" /></figure>
<figure><img src="image_90.png" /></figure>
# Hypothesis testing: overview

Detecting

perfusion

abnormalities in

Arterial Spin

Labelling

Camille Maumet

Detecting

perfusion

abnormalities

ASL

Group

Single-subject

General Linear

Model: MFX and

RFX

Hypothesis
testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions

<figure><img src="image_91.png" /></figure>
# Hypothesis testing: overview

Detecting

perfusion

abnormalities in

Arterial Spin

Labelling

Camille Maumet

Detecting

perfusion

abnormalities

ASL

Group

Single-subject

General Linear

Model: MFX and

RFX

Hypothesis
testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions

<figure><img src="image_92.png" /></figure>
Model fitting

Using the GLM, the dataset of interest is modelled as a
linear combination of pre-defined parameters.

<figure><img src="image_93.png" /></figure>

Source: "The General Linear Model for fMRI analyses" by FIL Methods Group, SPM Course, 2013.

Detecting
perfusion

abnormalities in

Arterial Spin

Labelling

Camille Maumet

Detecting

perfusion

abnormalities

ASL

Group

Single-subject

General Linear

Model: MFX and

RFX

Hypothesis
testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions
Model fitting

Using the GLM, the dataset of interest is modelled as a
linear combination of pre-defined parameters.

<figure><img src="image_94.png" /></figure>

Source: "The General Linear Model for fMRI analyses" by FIL Methods Group, SPM Course, 2013.

Independent and identically distributed errors, i.e. given
$e \sim N(0, \sigma^2 I)$, we can use Ordinary Least Squares:

$$
\hat{\beta}_{OLS} = (X^T X)^{-1} X^T Y, \quad \widehat{\text{Var}}(\hat{\beta}_{OLS}) = \hat{\sigma}^2 (X^T X)^{-1}. \quad (1)
$$

Detecting
perfusion

abnormalities in

Arterial Spin

Labelling

Camille Maumet

Detecting

perfusion

abnormalities

ASL

Group

Single-subject

General Linear

Model: MFX and

RFX

Hypothesis
testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions
## Model fitting

Using the GLM, the dataset of interest is modelled as a linear combination of pre-defined parameters.

Detecting
perfusion

abnormalities in
Arterial Spin
Labelling

Camille Maumet

Detecting

perfusion
abnormalities

ASL

Group

Single-subject

General Linear
Model: MFX and
RFX

Hypothesis
testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions

<figure><img src="image_95.png" /></figure>

Source: "The General Linear Model for fMRI analyses" by FIL Methods Group, SPM Course, 2013.

Independent and identically distributed errors, i.e. given $e \sim \mathcal{N}(0, \sigma^2I)$, we can use Ordinary Least Squares:

$$ \hat{\beta}_{oLS} = (X^T X)^{-1} X^T Y, \quad \widehat{\text{Var}}(\hat{\beta}_{oLS}) = \hat{\sigma}^2 (X^T X)^{-1}. \quad (1) $$

Otherwise, given $e \sim \mathcal{N}(0, V)$, Weighed Least Squares:

$$ \hat{\beta}_{wLS} = (X^T V^{-1} X)^{-1} X^T V^{-1} Y, \quad \widehat{\text{Var}}(\hat{\beta}_{wLS}) = (X^T V^{-1} X)^{-1}. \quad (2) $$
# Hypothesis testing: overview

Detecting

perfusion

abnormalities in

Arterial Spin

Labelling

Camille Maumet

Detecting

perfusion

abnormalities

ASL

Group

Single-subject

General Linear

Model: MFX and

RFX

Hypothesis
testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions

<figure><img src="image_96.png" /></figure>
# Hypothesis testing

Under the null hypothesis:

$$H_0: c\beta(v) = 0.$$

Assuming the normality of the error, the t-statistic at voxel v is defined by:

$$ \frac{c\hat{\beta}(v)}{\sqrt{\widehat{\mathrm{Var}}(c\hat{\beta}(v))}} \sim \mathcal{T}_{N-p}. \quad (3) $$
# Hypothesis testing: overview

Detecting

perfusion

abnormalities in

Arterial Spin

Labelling

Camille Maumet

Detecting

perfusion

abnormalities

ASL

Group

Single-subject

General Linear

Model: MFX and

RFX

Hypothesis
testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions

<figure><img src="image_97.png" /></figure>
# Outline

Detecting perfusion abnormalities

Arterial Spin Labelling

Group analysis

Single-subject analysis

## Random-Effects and Mixed-Effects General Linear Model

Hypothesis testing

## Random-effects and Mixed-effects

### Detections in ASL

Methods

Experiment

Results

## Conclusions
<figure><img src="image_98.png" /><figcaption>Repetition 1</figcaption></figure>

<figure><img src="image_99.png" /><figcaption>Repetition 2</figcaption></figure>

<figure><img src="image_100.png" /><figcaption>Repetition r</figcaption></figure>
<figure><img src="image_101.png" /></figure>

Repetition 1

<figure><img src="image_102.png" /></figure>

Repetition 2

...

<figure><img src="image_103.png" /></figure>

Repetition r

<figure><img src="image_104.png" /></figure>

<figure><img src="image_105.png" /></figure>

Perfusion
estimate in
subject 1: $\beta_1$
Modelling: subject level

Detecting
perfusion
abnormalities in
Arterial Spin
Labelling

Camille Maumet

Detecting
perfusion
abnormalities
ASL
Group
Single-subject

General Linear
Model: MFX and
RFX
Hypothesis
testing
RFX and MFX

Detections in ASL
Methods
Experiment
Results

Conclusions

Control Group

<figure><img src="image_106.png" /></figure>

Repetition 1

<figure><img src="image_107.png" /></figure>

Repetition 1

<figure><img src="image_108.png" /></figure>

Repetition 2

<figure><img src="image_109.png" /></figure>

Repetition 2

...

<figure><img src="image_110.png" /></figure>

Repetition r

<figure><img src="image_111.png" /></figure>

Repetition r

<figure><img src="image_112.png" /></figure>

<figure><img src="image_113.png" /></figure>

Perfusion
estimate in
subject 1: $\beta_1$

<figure><img src="image_114.png" /></figure>

<figure><img src="image_115.png" /></figure>

$\beta_{n1}$
Modelling: subject level

Detecting

perfusion

abnormalities in

Arterial Spin

Labelling

Camille Maumet

Detecting

perfusion

abnormalities

ASL

Group

Single-subject

General Linear

Model: MFX and

RFX

Hypothesis

testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions

<figure><img src="image_116.png" /></figure>

Repetition 1

<figure><img src="image_117.png" /></figure>

Repetition 2

...

<figure><img src="image_118.png" /></figure>

Repetition r

<figure><img src="image_119.png" /></figure>

<figure><img src="image_120.png" /></figure>

Perfusion
estimate in
subject 1: β₁

<figure><img src="image_121.png" /></figure>

Repetition 1

<figure><img src="image_122.png" /></figure>

Repetition 2

...

<figure><img src="image_123.png" /></figure>

Repetition r

<figure><img src="image_124.png" /></figure>

<figure><img src="image_125.png" /></figure>

βₙ₁

<figure><img src="image_126.png" /></figure>

Repetition 1

<figure><img src="image_127.png" /></figure>

Repetition 2

...

<figure><img src="image_128.png" /></figure>

Repetition r

<figure><img src="image_129.png" /></figure>

<figure><img src="image_130.png" /></figure>

βₙ₁+₁

<figure><img src="image_131.png" /></figure>

Repetition 1

<figure><img src="image_132.png" /></figure>

Repetition 2

...

<figure><img src="image_133.png" /></figure>

Repetition r

<figure><img src="image_134.png" /></figure>

<figure><img src="image_135.png" /></figure>

βₙ₁+ₙ₂
Modelling: group level

Detecting

perfusion

abnormalities in

Arterial Spin

Labelling

Camille Maumet

Control Group

<figure><img src="image_136.png" /></figure>

...

<figure><img src="image_137.png" /></figure>

Repetition 1

Repetition 2

Repetition r

r₁

<figure><img src="image_138.png" /></figure>

Repetition 1

Repetition 2

Repetition r

r₁ + 1

Patient Group

...

<figure><img src="image_139.png" /></figure>

Repetition 1

Repetition 2

Repetition r

r₁ + r₂

Detecting

perfusion

abnormalities

ASL

Group

Single-subject

General Linear

Model: MFX and

RFX

Hypothesis

testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions
Modelling: group level

Detecting

perfusion

abnormalities in

Arterial Spin

Labelling

Camille Maumet

Detecting

perfusion

abnormalities

ASL

Group

Single-subject

General Linear

Model: MFX and

RFX

Hypothesis
testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions

<figure><img src="image_140.png" /></figure>
Modelling: group level

Detecting

perfusion

abnormalities in

Arterial Spin

Labelling

Camille Maumet

Detecting

perfusion

abnormalities

ASL

Group

Single-subject

General Linear

Model: MFX and

RFX

Hypothesis

testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions

Control Group

<figure><img src="image_141.png" /></figure>

<figure><img src="image_142.png" /></figure>

<figure><img src="image_143.png" /></figure>

<figure><img src="image_144.png" /></figure>

<figure><img src="image_145.png" /></figure>

<figure><img src="image_146.png" /></figure>

<figure><img src="image_147.png" /></figure>

Patient Group

<figure><img src="image_148.png" /></figure>

<figure><img src="image_149.png" /></figure>

<figure><img src="image_150.png" /></figure>
<figure><img src="image_151.png" /></figure>

Random-effects (RFX) analysis.
Modelling: group level

Detecting
perfusion
abnormalities in
Arterial Spin
Labelling

Camille Maumet

<figure><img src="image_152.png" /></figure>

Detecting
perfusion
abnormalities

ASL

Group

Single-subject

General Linear
Model: MFX and
RFX

Hypothesis
testing

RFX and MFX

<figure><img src="image_153.png" /></figure>

Detections in ASL

Methods

Experiment

Results

Conclusions

Mixed-effects (MFX) analysis.
# Random-effects assumptions

RFX analyses assume that the **within-subject variance** is:

* **negligible** by comparison to the between-subject variance; or

* **roughly constant** across subjects.
# Random-effects or Mixed-effects analyses

In functional MRI there is no consensus:

* Superiority of MFX, [Beckmann 2003, Mumford 2006, Thirion 2007].

* Validity of RFX for one-sample t-tests in BOLD fMRI, [Mumford 2009].

* Invalidity of RFX [Chen 2012].

Both approaches are in use in the neuroimaging community:

* Random-effects analyses (SPM<fnref n="1" />)

* Mixed-effects analyses (FSL<fnref n="2" />, AFNI<fnref n="3" />).

What about ASL?

<footnote n="1">¹www.fil.ion.ucl.ac.uk/spm/</footnote>

<footnote n="2">²fsl.fmrib.ox.ac.uk/fsl/fslwiki/</footnote>

<footnote n="3">³afni.nimh.nih.gov/afni/</footnote>
# Outline

Detecting perfusion abnormalities
Arterial Spin Labelling
Group analysis
Single-subject analysis

Random-Effects and Mixed-Effects General Linear Model
Hypothesis testing
Random-effects and Mixed-effects

## Detections in ASL
* Methods
* Experiment
* Results

## Conclusions
# Outline

Detecting perfusion abnormalities

Arterial Spin Labelling

Group analysis

Single-subject analysis

Random-Effects and Mixed-Effects General Linear Model

Hypothesis testing

Random-effects and Mixed-effects

Detections in ASL

**Methods**

Experiment

Results

Conclusions
# Detecting perfusion abnormalities using the GLM

Detecting

perfusion

abnormalities in

Arterial Spin

Labelling

Camille Maumet

Detecting

perfusion

abnormalities

ASL

Group

Single-subject

General Linear

Model: MFX and

RFX

Hypothesis

testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions

Between-group analyses

<figure><img src="image_154.png" /></figure>

↔

<figure><img src="image_155.png" /></figure>

Individual analyses

<figure><img src="image_156.png" /></figure>

↔

<figure><img src="image_157.png" /></figure>

* Modelling and estimation using the GLM.

* Difference between random-effects and mixed-effects analyses.

Note: For ease of calculation, the models will be presented in the following without covariates.
# Detecting perfusion abnormalities using the GLM

Detecting

perfusion

abnormalities in

Arterial Spin

Labelling

Camille Maumet

Detecting

perfusion

abnormalities

ASL

Group

Single-subject

General Linear

Model: MFX and

RFX

Hypothesis

testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions

Between-group analyses

<figure><img src="image_158.png" /></figure>

Individual analyses

<figure><img src="image_159.png" /></figure>

* Modelling and estimation using the GLM.

* Difference between random-effects and mixed-effects analyses.

Note: For ease of calculation, the models will be presented in the following without covariates.
# GLM: subject level

<figure><img src="image_160.png" /></figure>

Repetition 1

<figure><img src="image_161.png" /></figure>

Repetition 2

<figure><img src="image_162.png" /></figure>

Repetition r

...

Detecting

perfusion

abnormalities in

Arterial Spin

Labelling

Camille Maumet

Detecting

perfusion

abnormalities

ASL

Group

Single-subject

General Linear

Model: MFX and

RFX

Hypothesis

testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions
# GLM: subject level

<figure><img src="image_163.png" /></figure>

Detecting

perfusion

abnormalities in

Arterial Spin

Labelling

Camille Maumet

Detecting

perfusion

abnormalities

ASL

Group

Single-subject

General Linear

Model: MFX and

RFX

Hypothesis

testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions
GLM: subject level

Detecting

perfusion

abnormalities in

Arterial Spin

Labelling

Camille Maumet

Detecting

perfusion

abnormalities

ASL

Group

Single-subject

General Linear

Model: MFX and

RFX

Hypothesis

testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions

<figure><img src="image_164.png" /></figure>

Repetition 1

<figure><img src="image_165.png" /></figure>

Repetition 2

...

<figure><img src="image_166.png" /></figure>

Repetition r

<figure><img src="image_167.png" /></figure>

<figure><img src="image_168.png" /></figure>

$\beta_s$

<figure><img src="image_169.png" /></figure>

$\text{Var}(\beta_s)$
GLM: subject level

Detecting

perfusion

abnormalities in

Arterial Spin

Labelling

Camille Maumet

Detecting

perfusion

abnormalities

ASL

Group

Single-subject

General Linear

Model: MFX and

RFX

Hypothesis

testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions

<figure><img src="image_170.png" /></figure>

Repetition 1

<figure><img src="image_171.png" /></figure>

Repetition 2

...

<figure><img src="image_172.png" /></figure>

Repetition r

<figure><img src="image_173.png" /></figure>

<figure><img src="image_174.png" /></figure>

$\beta_s$

<figure><img src="image_175.png" /></figure>

$\text{Var}(\beta_s)$

Given a voxel, for each subject $s$ we have:

$$Y_s = X_s \beta_s + \epsilon_s, \quad (4)$$

where

* $Y_s$ vector of observations;

* $X_s$ subject-level design matrix;

* $\beta_s$ parameters to be estimated;

* $\epsilon_s$ residual error
GLM: subject level

<figure><img src="image_176.png" /></figure>

Detecting
perfusion
abnormalities in
Arterial Spin
Labelling

Camille Maumet

Detecting
perfusion
abnormalities

ASL

Group

Single-subject

General Linear
Model: MFX and

RFX

Hypothesis
testing
RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions

Given a voxel, for each subject *s* we have:

$$ Y_s = \begin{bmatrix} 1 \\ \vdots \\ 1 \end{bmatrix} \beta_s + \epsilon_s. \quad (5) $$

where

* $Y_s$ vector of observations;

* $X_s$ subject-level design matrix;

* $\beta_s$ parameters to be estimated;

* $\epsilon_s$ residual error, $\epsilon_s \sim N(0, \sigma_s^2)$. [Aguirre 2002]
GLM: subject level

<figure><img src="image_177.png" /></figure>

Assuming $\epsilon_s \sim N(0, \sigma_s^2)$, by ordinary least squares we have:

$$ \hat{\beta}_s = \frac{1}{r} \sum_{i=1}^{r} y_{s,i}, \quad \text{and} \quad \widehat{\text{Var}}(\hat{\beta}_s) = \frac{\hat{\sigma}_s^2}{r} $$

where

* $y_{s,i}$ is the $i^{th}$ element of vector $Y_s$

* $\hat{\sigma}_s^2$ the estimated within-subjet variance.

Detecting
perfusion

abnormalities in
Arterial Spin
Labelling

Camille Maumet

Detecting
perfusion
abnormalities

ASL

Group

Single-subject

General Linear
Model: MFX and
RFX

Hypothesis
testing
RFX and MFX

Detections in ASL

Methods

Experiment
Results

Conclusions
GLM: between-group

<figure><img src="image_178.png" /></figure>

Camille Maumet

Detecting
perfusion
abnormalities
Arterial Spin
Labelling

ASL

Group

Single-subject

General Linear
Model: MFX and
RFX

Hypothesis
testing
RFX and MFX

Detections in ASL

Methods

Experiment
Results

Conclusions

The subject parameters ($\beta_s$)$_{1\le s\le n_1+1}$ can be combined using:

$$ \begin{bmatrix} \beta_1 \\ \vdots \\ \beta_{n_1+n_2} \end{bmatrix} = X_G \beta_G + \gamma_G, \quad (6) $$

where

* $X_G$ is the group-level design matrix;

* $\beta_G$ the group parameters;

* $\gamma_G^s$ the residual error term.
GLM: between-group

<figure><img src="image_179.png" /><figcaption>Camille Maumet</figcaption></figure>

The subject parameters $(\beta_s)_{1\le s \le n_1+1}$ can be combined using:

$$ \begin{bmatrix} \beta_1 \\ \vdots \\ \beta_{n_1} \\ \beta_{n_1+1} \\ \vdots \\ \beta_{n_1+n_2} \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ \vdots & \vdots \\ 1 & 0 \\ 0 & 1 \\ \vdots & \vdots \\ 0 & 1 \end{bmatrix} \begin{bmatrix} \beta_{\text{controls}} \\ \beta_{\text{patients}} \end{bmatrix} + \gamma_G. \quad (7) $$

where

* $\gamma_G$ the residual error term, $\gamma_G^s \sim \mathcal{N}(0, \sigma_{G,i}^2)$.

Detecting

perfusion

abnormalities in

Arterial Spin

Detecting

perfusion

abnormalities

ASL

Group

Single-subject

General Linear

Model: MFX and

RFX

Hypothesis

testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions
GLM: between-group

<figure><img src="image_180.png" /><figcaption>Camille Maumet</figcaption></figure>

The subject parameters $(\beta_s)_{1\le s\le n_1+1}$ can be combined using:

$$ \begin{bmatrix} \hat{\beta}_1 \\ \vdots \\ \hat{\beta}_{n_1} \\ \hat{\beta}_{n_1+1} \\ \vdots \\ \hat{\beta}_{n_1+n_2} \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ \vdots & \vdots \\ 1 & 0 \\ 0 & 1 \\ \vdots & \vdots \\ 0 & 1 \end{bmatrix} \begin{bmatrix} \beta_{\text{controls}} \\ \beta_{\text{patients}} \end{bmatrix} + \gamma_{G_c} \quad (8) $$

where

* $\gamma_{G_c}$ the residual error term, $\gamma_{G_c} \sim \mathcal{N}(0, \sigma_{G,i}^2 + \frac{\sigma_s^2}{r})$.

Detecting perfusion abnormalities in Arterial Spin Labelling

Detecting perfusion abnormalities
ASL

Group
Single-subject

General Linear
Model: MFX and
RFX

Hypothesis
testing

RFX and MFX

Detections in ASL

Methods

Experiment
Results

Conclusions
Contrast of interest

<figure><img src="image_181.png" /></figure>

Detecting
perfusion
abnormalities in
Arterial Spin
Labelling

Camille Maumet

Detecting
perfusion
abnormalities

ASL

Group

Single-subject

General Linear
Model: MFX and
RFX

Hypothesis
testing
RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions

We are interested in the null hypothesis:

$$H0: \beta_{controls} = \beta_{patients}. \qquad (9)$$

$$H0: c \beta_G = 0. \qquad (10)$$
Contrast of interest

<figure><img src="image_182.png" /></figure>

Camille Maumet

Detecting
perfusion
abnormalities

ASL

Group

Single-subject

General Linear
Model: MFX and

RFX

Hypothesis
testing
RFX and MFX

Detections in ASL

Methods

Experiment
Results

Conclusions

We are interested in the null hypothesis:

$$
H0 : \beta_{\text{controls}} = \beta_{\text{patients}}. \tag{9}
$$

$$
H0 : c \beta_G = 0. \tag{10}
$$

Corresponding to the patient versus control group
contrast:

$$
c = [1 - 1] \tag{11}
$$
Random-effects (RFX) between-group
analysis

<figure><img src="image_183.png" /></figure>

Assuming $\hat{\gamma}_{G_C, i}^s \sim \mathcal{N}(0, \sigma_{G_C, i}^2)$, by weighted least squares we have:

$$ \hat{\beta}_{\text{controls}}^{\text{RFX}} = \frac{1}{n_1} \sum_{s=1}^{n_1} \hat{\beta}_s, \quad \hat{\beta}_{\text{patients}}^{\text{RFX}} = \frac{1}{n_2} \sum_{s=n_1+1}^{n_1+n_2} \hat{\beta}_s, \quad (12) $$

The associated sampling variances are:

$$ \widehat{\mathrm{Var}}(\hat{\beta}_{\text{controls}}^{\text{RFX}}) = \frac{\hat{\sigma}_{G_C, 1}^2}{n_1}, \quad \widehat{\mathrm{Var}}(\hat{\beta}_{\text{patients}}^{\text{RFX}}) = \frac{\hat{\sigma}_{G_C, 2}^2}{n_2} \quad (13) $$
Mixed-effects (MFX) between-group analysis

<figure><img src="image_184.png" /></figure>

Assuming $\gamma_{G,C,i}^s \sim \mathcal{N}(0, \sigma_{G,i}^2 + \frac{\sigma_s^2}{r})$, by weighted least squares we have:

$$ 
\begin{aligned}
\hat{\beta}_{\text{controls}}^{\text{MFX}} &= \frac{1}{\sum_{j=1}^{n_1} w_{j,1}} \sum_{s=1}^{n_1} w_{s,1} \hat{\beta}_s, && \text{where } w_{s,i} = \frac{1}{\hat{\sigma}_{G,i}^2 + \frac{\hat{\sigma}_s^2}{r}} \\
\hat{\beta}_{\text{patients}}^{\text{MFX}} &= \frac{1}{\sum_{j=n_1+1}^{n_1+n_2} w_{j,2}} \sum_{s=n_1+1}^{n_1+n_2} w_{s,2} \hat{\beta}_s.
\end{aligned}
\quad (14) $$

The associated sampling variances are:

$$ \widehat{\mathrm{Var}}(\hat{\beta}_{\text{controls}}^{\text{MFX}}) = \frac{1}{\sum_{s=1}^{n_1} w_{s,1}}, \quad \widehat{\mathrm{Var}}(\hat{\beta}_{\text{patients}}^{\text{MFX}}) = \frac{1}{\sum_{s=n_1+1}^{n_1+n_2} w_{s,2}} \quad (15) $$
# Detecting perfusion abnormalities using the GLM

Detecting
perfusion

abnormalities in
Arterial Spin
Labelling

Camille Maumet

Detecting
perfusion

abnormalities

ASL

Group

Single-subject

General Linear
Model: MFX and
RFX

Hypothesis
testing
RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions

Between-group analyses

<figure><img src="image_185.png" /></figure>

Individual analyses

<figure><img src="image_186.png" /></figure>
Random-effects (RFX) analysis

<figure><img src="image_187.png" /></figure>

Detecting
perfusion
abnormalities in
Arterial Spin
Labelling

Camille Maumet

Detecting
perfusion
abnormalities

ASL

Group

Single-subject

General Linear
Model: MFX and
RFX

Hypothesis
testing
RFX and MFX

Detections in ASL

Methods

Experiment
Results

Conclusions

Assuming $\gamma_{G_C}^s \sim \mathcal{N}(0, \sigma_{G_C}^2)$, by ordinary least squares we
have:

$$
\hat{\beta}_{\text{controls}}^{\text{RFX}} = \frac{1}{n_1} \sum_{s=1}^{n_1} \hat{\beta}_s, \quad \hat{\beta}_{\text{patient}}^{\text{RFX}} = \hat{\beta}_{n+1}, \quad (16)
$$

The associated sampling variances are:

$$
\widehat{\mathrm{Var}}(\hat{\beta}_{\mathrm{controls}}^{\mathrm{RFX}}) = \frac{\hat{\sigma}_{G_C}^2}{n_1}, \quad \widehat{\mathrm{Var}}(\hat{\beta}_{\mathrm{patient}}^{\mathrm{RFX}}) = \hat{\sigma}_{G_C}^2 \quad (17)
$$
Mixed-effects (MFX) analysis

<figure><img src="image_188.png" /><figcaption>Camille Maumet</figcaption></figure>

Detecting
perfusion
abnormalities

ASL

Group

Single-subject

General Linear
Model: MFX and

RFX

Hypothesis
testing
RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions

Assuming $\gamma_{G_C}^s \sim \mathcal{N}(0, \sigma_G^2 + \frac{\sigma_s^2}{r})$, by weighted least squares we get:

$$ \hat{\beta}_{controls}^{MFX} = \frac{1}{\sum_{j=1}^{n_1} w_{j,1}} \sum_{s=1}^{n_1} w_{s,1} \hat{\beta}_s, \quad \hat{\beta}_{patient}^{MFX} = \hat{\beta}_{n_1+1} \quad (18) $$

$$ \text{where } w_s = \frac{1}{\hat{\sigma}_G^2 + \frac{\hat{\sigma}_s^2}{r}}. $$

The associated sampling variances are:

$$ \widehat{\mathrm{Var}}(\hat{\beta}_{controls}^{MFX}) = \frac{1}{\sum_{s=1}^{n_1} w_s}, \quad \widehat{\mathrm{Var}}(\hat{\beta}_{patients}^{MFX}) = \hat{\sigma}_G^2 + \frac{\hat{\sigma}_{n_1+1}^2}{r} \quad (19) $$
# Random-effects and Mixed-effects analyses

Detecting
perfusion
abnormalities in
Arterial Spin
Labelling

Camille Maumet

Detecting
perfusion
abnormalities
ASL

Group
Single-subject

General Linear
Model: MFX and
RFX

Hypothesis
testing
RFX and MFX

Detections in ASL

Methods

Experiment
Results

Conclusions

<table><caption>Control group and patient estimates and sampling variances with RFX and MFX.</caption><thead><tr><th></th><th>β<sub>controls</sub></th><th>β<sub>patient</sub></th><th>Var (β<sub>controls</sub>)</th><th>Var (β<sub>patient</sub>)</th></tr></thead><tbody><tr><td>RFX</td><td><sup>1</sup>/<sub>n<sub>1</sub></sub> Σ<sub>s=1</sub><sup>n<sub>1</sub></sup> β<sub>s</sub></td><td>β<sub>n<sub>1</sub>+1</sub></td><td><sup>σ<sup>2</sup><sub>G<sub>C</sub></sub></sup>/<sub>n<sub>1</sub></sub></td><td>σ<sup>2</sup><sub>G<sub>C</sub></sub></td></tr><tr><td>MXF</td><td><sup>1</sup>/<sub>Σ<sub>j=1</sub><sup>n<sub>1</sub></sup> <sup>1</sup>/<sub>σ<sup>2</sup><sub>G</sub> + <sup>1</sup>/<sub>r</sub></sub></sub> Σ<sub>s=1</sub><sup>n<sub>1</sub></sup> <sup>1</sup>/<sub>σ<sup>2</sup><sub>G</sub> + <sup>1</sup>/<sub>r</sub></sub> β<sub>s</sub></td><td>β<sub>n<sub>1</sub>+1</sub></td><td><sup>1</sup>/<sub>Σ<sub>s=1</sub><sup>n<sub>1</sub></sup> <sup>1</sup>/<sub>σ<sup>2</sup><sub>G</sub> + <sup>1</sup>/<sub>r</sub></sub></sub></td><td>σ<sup>2</sup><sub>G</sub> + <sup>σ<sup>2</sup><sub>n<sub>1</sub>+1</sub></sup>/<sub>r</sub></td></tr></tbody></table>
Detecting
perfusion

abnormalities in
Arterial Spin

Labelling

Camille Maumet

Detecting
perfusion
abnormalities

ASL

Group

Single-subject

General Linear
Model: MFX and

RFX

Hypothesis
testing

RFX and MFX

Detections in ASL
Methods

Experiment

Results

Conclusions

# Outline

Detecting perfusion abnormalities

Arterial Spin Labelling

Group analysis

Single-subject analysis

Random-Effects and Mixed-Effects General Linear Model

Hypothesis testing

Random-effects and Mixed-effects

Detections in ASL

Methods

Experiment

Results

Conclusions
# Subjects and imaging protocol

25 patients diagnosed with brain tumours and 61 control subjects participated in this study.

## Imaging protocol:

* PICORE Q2TIPS Pulsed ASL, 60 repetitions

* MPRAGE T1 3D

* T2 FLAIR

## For the patients only:

* T1 3D Gadolinium

* Dynamic Susceptibility Contrast imaging (DSC)
Validation: Ground Truth

Detecting

perfusion

abnormalities in

Arterial Spin

Labelling

Camille Maumet

Detecting

perfusion

abnormalities

ASL

Group

Single-subject

General Linear

Model: MFX and

RFX

Hypothesis

testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions
# Validation: Ground Truth

## 1. Segmentation of the tumour:

<figure><img src="image_189.png" /></figure>

Detecting perfusion abnormalities in Arterial Spin Labelling

Camille Maumet

Detecting perfusion abnormalities

ASL

Group

Single-subject

General Linear Model: MFX and RFX

Hypothesis testing
RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions
# Validation: Ground Truth

## 1. Segmentation of the tumour:

Detecting
perfusion

abnormalities in
Arterial Spin

Labelling

Camille Maumet

Detecting
perfusion

abnormalities
ASL

Group

Single-subject

General Linear
Model: MFX and

RFX

Hypothesis
testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions

<figure><img src="image_190.png" /></figure>

T1 Gadolinium

<figure><img src="image_191.png" /></figure>

T2 FLAIR

<figure><img src="image_192.png" /></figure>

<figure><img src="image_193.png" /></figure>

Tumour and
oedema
segmentation

## 2. Combination with T2 perfusion information:

<figure><img src="image_194.png" /></figure>

Tumour
segmentation

<figure><img src="image_195.png" /></figure>

DSC CBF

<figure><img src="image_196.png" /></figure>

<figure><img src="image_197.png" /></figure>

“Ground truth”
## 1. Segmentation of the tumour:

<figure><img src="image_198.png" /></figure>

## 2. Combination with T2 perfusion information:

<figure><img src="image_199.png" /></figure>

## 3. Visual assessment and manual corrections by a clinician.
# Outline

Detecting perfusion abnormalities

Arterial Spin Labelling

Group analysis

Single-subject analysis

Random-Effects and Mixed-Effects General Linear Model

Hypothesis testing

Random-effects and Mixed-effects

Detections in ASL

Methods
Experiment

Results

Conclusions
# Assumptions of RFX analyses

* Assumption 1: Within-subject variance negligible by comparison to the between-subject variance.

Detecting
perfusion
abnormalities in
Arterial Spin
Labelling

Camille Maumet

Detecting
perfusion
abnormalities

ASL

Group

Single-subject

General Linear
Model: MFX and
RFX

Hypothesis
testing
RFX and MFX

Detections in ASL
Methods
Experiment

Results

Conclusions
Assumptions of RFX analyses

<figure><img src="image_200.png" /><figcaption>► Assumption 1: Within-subject variance negligible by comparison to the between-subject variance.</figcaption></figure>

Detecting

perfusion

abnormalities in

Arterial Spin

Labelling

Camille Maumet

Detecting

perfusion

abnormalities

ASL

Group

Single-subject

General Linear

Model: MFX and

RFX

Hypothesis

testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions
Assumptions of RFX analyses

► Assumption 1: Within-subject variance negligible by comparison to the between-subject variance.

<figure><img src="image_201.png" /></figure>

× not verified

Detecting

perfusion

abnormalities in

Arterial Spin

Labelling

Camille Maumet

Detecting

perfusion

abnormalities

ASL

Group

Single-subject

General Linear

Model: MFX and

RFX

Hypothesis

testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions
# Assumptions of RFX analyses

* Assumption 2: Within-subject variance roughly constant across subjects.

Detecting
perfusion
abnormalities in
Arterial Spin
Labelling

Camille Maumet

Detecting
perfusion
abnormalities

ASL

Group

Single-subject

General Linear
Model: MFX and
RFX

Hypothesis
testing
RFX and MFX

Detections in ASL
Methods
Experiment

Results

Conclusions
Assumptions of RFX analyses

► **Assumption 2: Within-subject variance roughly constant across subjects.**

<figure><img src="image_202.png" /><figcaption>Within-subject standard deviation in two control subjects:</figcaption></figure>

Detecting perfusion abnormalities in Arterial Spin Labelling

Camille Maumet

Detecting perfusion abnormalities

ASL
Group

Single-subject

General Linear Model: MFX and RFX

Hypothesis testing
RFX and MFX

Detections in ASL
Methods
Experiment
Results

Conclusions
Assumptions of RFX analyses

► **Assumption 2: Within-subject variance roughly constant across subjects.**

<figure><img src="image_203.png" /></figure>

Detecting
perfusion
abnormalities in
Arterial Spin
Labelling

Camille Maumet

Detecting
perfusion
abnormalities

ASL

Group

Single-subject

General Linear
Model: MFX and
RFX

Hypothesis
testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions
Assumptions of RFX analyses

► **Assumption 2: Within-subject variance roughly constant across subjects.**

<figure><img src="image_204.png" /></figure>

× not verified

Detecting
perfusion
abnormalities in
Arterial Spin
Labelling

Camille Maumet

Detecting
perfusion
abnormalities

ASL

Group

Single-subject

General Linear
Model: MFX and
RFX

Hypothesis
testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions
# Results: RFX versus MFX

Detecting

perfusion

abnormalities in

Arterial Spin

Labelling

Camille Maumet

Detecting

perfusion

abnormalities

ASL

Group

Single-subject

General Linear

Model: MFX and

RFX

Hypothesis

testing

RFX and MFX

Detections in ASL

Methods

Experiment

Results

Conclusions

Ground Truth

Patient 15, TPR = 50%

<figure><img src="image_205.png" /></figure>

Homoscedastic model

Heteroscedastic model

Patient 16, FPR = 1‰

<figure><img src="image_206.png" /></figure>
# Results: RFX versus MFX

Detecting
perfusion
abnormalities in
Arterial Spin
Labelling

Camille Maumet

Detecting
perfusion
abnormalities

ASL

Group

Single-subject

General Linear
Model: MFX and
RFX

Hypothesis
testing
RFX and MFX

Detections in ASL

Methods
Experiment

Results

Conclusions

<figure><img src="image_207.png" /><figcaption>ROC curves of the homoscedastic and heteroscedastic models</figcaption></figure>

<table><thead><tr><th></th><th colspan="5">Random-effects analysis</th></tr><tr><th>FWHM (mm³)</th><th>0</th><th>4</th><th>6</th><th>8</th><th>10</th></tr></thead><tbody><tr><th>ROC Area</th><td>0.46</td><td><strong>0.49</strong></td><td><strong>0.49</strong></td><td><strong>0.49</strong></td><td>0.48</td></tr></tbody></table>

<table><thead><tr><th></th><th colspan="5">Mixed-effects analysis</th></tr><tr><th>FWHM (mm³)</th><th>0</th><th>4</th><th>6</th><th>8</th><th>10</th></tr></thead><tbody><tr><th>ROC Area</th><td>0.63</td><td>0.70</td><td><strong>0.72</strong></td><td><strong>0.72</strong></td><td>0.69</td></tr></tbody></table>

Area under the ROC curve.

ROC curves for perfusion abnormality detections.

[Maumet et al., NeuroImage 2013]
## Outline

- Detecting perfusion abnormalities

- Arterial Spin Labelling

- Group analysis

- Single-subject analysis

- Random-Effects and Mixed-Effects General Linear Model

- Hypothesis testing

- Random-effects and Mixed-effects

- Detections in ASL

- Methods

- Experiment

- Results

## Conclusions

- Detecting perfusion abnormalities in Arterial Spin Labelling

- Camille Maumet

- Detecting perfusion abnormalities
Detecting
perfusion
abnormalities in
Arterial Spin
Labelling

Camille Maumet

Detecting
perfusion
abnormalities

ASL

Group

Single-subject

General Linear
Model: MFX and
RFX

Hypothesis
testing
RFX and MFX

Detections in ASL

Methods
Experiment
Results

Conclusions

Conclusion

We demonstrated on a dataset of PICORE Q2TIPS ASL images that:

► the assumptions of RFX analyses were violated.

► using an **MFX analysis** was essential in the detection of perfusion abnormalities at the patient level.
Detecting
perfusion
abnormalities in
Arterial Spin
Labelling

Camille Maumet

Detecting
perfusion
abnormalities

ASL

Group

Single-subject

General Linear
Model: MFX and
RFX

Hypothesis
testing
RFX and MFX

Detections in ASL

Methods
Experiment
Results

Conclusions

Conclusion

We demonstrated on a dataset of PICORE Q2TIPS ASL
images that:

► the assumptions of RFX analyses were violated.

► using an **MX** **a**nalysis was essential in the
detection of perfusion abnormalities at the patient
level.

► According to the literature, the difference between RFX and MFX is less pronounced when comparing 2 groups.
Conclusion

We demonstrated on a dataset of PICORE Q2TIPS ASL images that:

► the assumptions of RFX analyses were violated.

► using an **MXF analysis** was essential in the detection of perfusion abnormalities at the patient level.

► According to the literature, the difference between RFX and MFX is less pronounced when comparing 2 groups.

► Further investigation using other ASL sequences are needed.

► Software packages: **RFX (SPM)**, **MXF (FSL)**, **AFNI**).
# Conclusion

We demonstrated on a dataset of PICORE Q2TIPS ASL images that:

* the assumptions of RFX analyses were violated.

* using an **MXF analysis** was essential in the detection of perfusion abnormalities at the patient level.

* According to the literature, the difference between RFX and MFX is less pronounced when comparing 2 groups.

* Further investigation using other ASL sequences are needed.

* Software packages: **RFX (SPM)**, **MXF (FSL)**, **AFNI**.

* More details on RFX and MFX [Beckmann 2003, Mumford 2006, Mumford 2009].

Detecting perfusion abnormalities in Arterial Spin Labelling

Camille Maumet

Detecting perfusion abnormalities

ASL

Group

Single-subject

General Linear Model: MFX and RFX

Hypothesis testing
RFX and MFX

Detections in ASL

Methods
Experiment
Results

Conclusions
Thank you

VisAGeS team:

► Christian Barillot.

► Pierre Maurel.

► Jean-Christophe Ferré.

► Béatrice Carsin.

C. Maumet, P. Maurel, J-C. Ferré, B. Carsin, C. Barillot. *Patient-specific detection of perfusion abnormalities combining within-subject and between-subject variances in Arterial Spin Labeling*. NeuroImage, 2013, 81C, pp. 121-130. Freely available online.
## Outline

Detecting

perfusion

abnormalities in

Arterial Spin

Labelling

Camille Maumet

Appendix

## Appendix
A model of normal perfusion

Perfusion estimate $\hat{\beta}_{controls}$

Mean perfusion estimate

<figure><img src="image_208.png" /></figure>
# A model of normal perfusion

## Perfusion estimate $\hat{\beta}_{controls}$

Mean perfusion estimate

<figure><img src="image_209.png" /></figure>

## Standard deviation estimates

RFX Combined within- and between-subject standard deviation

<figure><img src="image_210.png" /></figure>

MFX Between-subject standard deviation

<figure><img src="image_211.png" /></figure>

Root-square of the average within-subject variance

<figure><img src="image_212.png" /></figure>
# A model of normal perfusion

Perfusion estimate $\hat{\beta}_{controls}$

Mean perfusion estimate

<figure><img src="image_213.png" /></figure>

Standard deviation estimates

RFX Combined within- and between-subject standard deviation

<figure><img src="image_214.png" /></figure>

MFX

Between-subject standard deviation

<figure><img src="image_215.png" /></figure>

Root-square of the average within-subject variance

<figure><img src="image_216.png" /></figure>

<figure><img src="image_217.png" /></figure>

Brain arteries

<figure><img src="image_218.png" /></figure>

Brain veins