# Preliminary results on the graph matching of structural and functional brain networks

Yanis Aeschlimann, Anna Calissano, Andreas Abildtrup Hansen, Théodore Papadopoulos, Samuel Deslauriers-Gauthier

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Yanis Aeschlimann, Anna Calissano, Andreas Abildtrup Hansen, Théodore Papadopoulos, Samuel Deslauriers-Gauthier. Preliminary results on the graph matching of structural and functional brain networks. Neuromod meeting 2024, Jul 2024, Antibes (06), France. hal-04895290

HAL Id: hal-04895290

https://inria.hal.science/hal-04895290v1

Submitted on 17 Jan 2025

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# Preliminary results on the graph matching of structural and functional brain networks

Inria

UNIVERSITÉ
CÔTE D'AZUR

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CRONOS

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NEURO
MOD

Y. Aeschlimann¹*, A. Calissano², A. Hansen³, T. Papadopoulos¹, S. Deslauriers-Gauthier¹

¹ Inria Centre at Université Côte d'Azur, France

² Imperial College London

³ Technical University of Denmark

* *yanis.aeschlimann@inria.fr*

## CONTEXT

The brain, responsible for human behavior, can be subdivided into regions using an atlas. The cortical **atlas** proposed by Glasser et al. [1] used in this work contains 180 regions per hemisphere.

The cortex can be represented as a graph. The regions of the atlas are the nodes and the connectivity between them are the edges [2]. The strength of the connectivity between 2 regions is represented by a scalar value. In this work, we consider structural and functional connectivity. Structural connectivity is estimated from **diffusion Magnetic Resonance Imaging (dMRI)**, whereas functional connectivity is the result of **resting-state functional MRI** (rs-fMRI), that is a derived measure of brain activity.

The edges of the brain graph representing the connectivity can be represented by a square matrix, called **connectome**. Each entry (i,j) of the matrix is the connectivity between region **labeled i** and **j**.

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Brain atlas extracted from [3]

Due to inter-subject structural and functional variability, the atlas may not fit all subjects perfectly [4,5]. In this work, we correct the inter-subject structural and functional variability using concepts of graph alignment.

## MATERIALS AND METHODS

### Connectome generation pipeline

1050 subjects preprocessed dataset of the Human Connectome Project (HCP) [6] including:

- Diffusion MRI data

- Functional MRI data: resting-states and 7 different tasks

#### Diffusion MRI

We used the Tractoflow pipeline [7] to construct a tractogram [8] of ~ 2 millions of streamlines from the dMRI data of the HCP.

A streamline is a sequence of connected points supposed to follow a fiber bundle.

From the collection of streamlines, we construct the structural connectome by performing a local search to find the closest cortical region to the endpoints of the streamline [9].

Each entry (i,j) in the connectome contains the number of streamlines that connects the regions labeled i and j.

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#### Resting-state functional MRI

A Butterworth bandpass filter is used to keep only meaningful signal from HCP data.

The Pearson's coefficient is computed between every fMRI signal that is averaged over all the voxels of the region.

Functional connectomes of resting-state (run-1 and run-2) : rs-1-FC and rs-2-FC

Functional connectomes of the 7 different tasks (BOLD signals concatenated): 7-ts-FC

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### Graph alignment, FAQ algorithm

The Fast Approximate Quadratic (FAQ) algorithm [10] solves the optimization problem:

$$ \underset{P \in \mathcal{P}}{\text{minimize}} \quad \|PC_S P^T - C_S^{\text{ref}}\|_{\mathcal{F}}^2 $$

It yields a permutation matrix P that permutes the labels of the structural connectome $C_S$ in order to maximize the similarity with the connectome of reference $C_S^{\text{ref}}$.

$$ \underset{P \in \mathcal{P}}{\text{minimize}} \quad \alpha \ \|PC_S P^T - C_S^{\text{ref}}\|_{\mathcal{F}}^2 + (1 - \alpha) \|PC_F P^T - C_F^{\text{ref}}\|_{\mathcal{F}}^2 $$

Now we can align brain regions using the structural connectome $C_S$ and the functional connectome $C_F$.

The hyperparameter $\alpha$ is used to favor one modality over the other, if one is more meaningful or less noisy for instance.

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Representation of region permutation as result of graph alignment [5]

We use a regularization term to favor permutations of specific regions, for instance close regions are more likely to be permuted than remote regions.

$$ \underset{P \in \mathcal{P}}{\text{minimize}} \quad \alpha \ \|PC_S P^T - C_S^{\text{ref}}\|_{\mathcal{F}}^2 + (1 - \alpha) \|PC_F P^T - C_F^{\text{ref}}\|_{\mathcal{F}}^2 + \lambda \|P - A_{\text{reg}}\|_{\mathcal{F}}^2 $$

$A_{\text{reg}} =$

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$A_{\text{reg}}$ is a binary matrix, where entries equal to 1 when regions of row and column indices are neighbors

## RESULTS

### OPTIMIZATION OF PERMUTATION MATRICES

% of alignment on
rs-1-FC ($\alpha$ = 0) and SC ($\alpha$ = 1)

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% of alignment on
7-ts-FC ($\alpha$ = 0) and SC ($\alpha$ = 1)

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### VALIDATION OF PERMUTATION MATRICES

Generalization on rs-2-FC and 7-ts-FC

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Generalization on rs-1-FC and rs-2-FC

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## TAKE HOME MESSAGE

* It is possible to find permutation matrices that align simultaneously SC and FC

* The permutation matrices found for the alignment of the first run of resting-state FC perform well on the second run of resting-state FC

* The permutation matrices found for the alignment of the tasks FC perform well on resting-state FC, but not the converse.

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SCAN ME

**Acknowledgements:** Y. Aeschlimann is supported by a scholarship of the Université Côte d'Azur

**References:**

[1] Glasser et al. "A multi-modal parcellation of human cerebral cortex." Nature(2016)

[2] Olaf Sporns et al. The human connectome: A structural description of the human brain. 2020.

[3] Guillermo Alejandro Gallardo. Inferring and comparing structural parcellations of the human brain using diffusion MRI. Université Côte d'Azur, 2018.

[4] Laura E. Suárez et al. Linking structure and function in macroscale brain networks. Trends in Cognitive Sciences, 2020.

[5] Anna Calissano et al. Graph alignment exploiting the spatial organisation improves the similarity of brain networks. hal-03910761, 2023.

[6] Matthew F. Glasser et al. The minimal preprocessing pipelines for the humanconnectome project. NeuroImage, 2013.

[7] Guillaume Theaud et al. Tractoflow: A robust, efficient and reproducible diffusion mri pipeline leveraging nextflow singu-larity. 2020.

[8] Ben Jeurissen et al. Diffusion mri fiber tractography of the brain. Wiley NMR inbiomedicine, 2017.

[9] Chun-Hung Yeh et al. Connectomes from streamlines tractography: Assigning streamlinesto brain parcellaions is not trivial but highly consequential. NeuroImage, 2019.

[10] Vince Lyzinski et al. Fast approximate quadratic programming for graph matching. PLOS ONE, 2015.