Analyzing Brain Topography Via Directional Derivatives of Connectivity.

2177 

Abstract Submission 

Simona Leserri^1,2, Dogu Baran Aydogan^1,3

¹AIVI Virtanen Institute for Molecular Sciences, University of Eastern Finland, Kuopio, Finland, ²Department of Psychiatry, University of Helsinki and Helsinki University Hospital, Helsinki, Finland, ³Department of Neuroscience and Biomedical Engineering, Aalto University School of Science, Espoo, Finland

Simona Leserri  
AIVI Virtanen Institute for Molecular Sciences, University of Eastern Finland|Department of Psychiatry, University of Helsinki and Helsinki University Hospital
Kuopio, Finland|Helsinki, Finland

Dogu Baran Aydogan  
AIVI Virtanen Institute for Molecular Sciences, University of Eastern Finland|Department of Neuroscience and Biomedical Engineering, Aalto University School of Science
Kuopio, Finland|Espoo, Finland

Topographic organization stands as an important characteristic of the brain [6, 10]. Tractography studies have highlighted this principle in the sensory and motor systems [1,4].

Gradients have been used to describe spatial changes in connectivity patterns [5]. Since gradients are rooted in dimensionality reduction [9], these informative methods may not comprehensively encapsulate the complexity of the brain's structural organization.

Here, we introduce a novel approach founded on the mathematical concept of directional derivatives. This method effectively quantifies variations in structural connectivity across multiple directions, presenting a robust tool for the detailed exploration of the brain's topography.

Figure 1 illustrates how we compute the structural connectivity and its derivatives starting from dMRI-based tractography. Consider the tractogram T composed of streamlines s_i. We define T_S(x,r) as the subset of streamlines in T that intersect a sphere S(x,r), centered at x ∈ R³, with radius r. Each streamline in T_S(x,r) contributes to structural connectivity at x, and the contribution is determined by a weighted sum w(x,i). This weight is calculated by integrating over the segments of s_i that intersect S(x,r). For each segment, we consider its length as well as its distance to point x. Each streamline is connected to the brain surface, Ω, that can be obtained through FreeSurfer [3]. This surface is composed of cells and vertices. We then assign the weight of each streamline to the vertices surrounding the intersecting cell to obtain f(x) ∈ R² that is our metric of structural connectivity.

We can evaluate the same function at slightly shifted position around x, with a displacement h. Finally, the numerical approximation of the directional derivative of structural connectivity along a direction d on the unit sphere can be obtained with finite difference as in ∇_d f(x) = (f(x+hd) − f(x))/h. The sum of ∇_d f(x) over the surface is the scalar that encapsulates the structural changes along the specified direction, and is stored in the output image at position x.

To test our method, we conducted experiments with subject 100307 from the Human Connectome Project [8]. The entire brain tractogram comprised 100 million streamlines, generated through parallel transport tractography [2] with anatomically constrained tractography [7], so to ensure streamline termination on Ω.

Figure 2A depicts the gradual changes in connectivity on the surface due to small displacements in the neighborhood of a point in the corpus callosum. It highlights how our metric of structural connectivity f(x) provides insights into the topographic organization of connections along various directions.
Figure 2B displays RGB-encoded directional derivatives. The image is formed by using the absolute values of the derivatives along the directions [1 0 0], [0 1 0], and [0 0 1], represented by red, green, and blue colors, respectively. The images are computed on a 0.5 mm isotropic image grid, employing h=0.0125 mm.

This study proposes an innovative computational tool for investigating the brain and its topographic organization. While our present focus is on structural connectivity, the concept of directional derivatives in connectivity stands for other types of connectivity and could be extended to functional measurements such as fMRI and EEG. Beyond contributing to our understanding of the brain's organization principles, we anticipate the relevance of our method in all contexts where precision and personalization are crucial, including surgical planning and brain stimulation studies.

Connectivity (eg. functional, effective, structural) ²

White Matter Anatomy, Fiber Pathways and Connectivity ¹

Tractography

White Matter

Other - Topographic organization; Connectivity