# Linear structure unfolding: application to mouse brain in spatial transcriptomics

Morgane Fierville, Kevin Lebrigand, Pascal Barbry, Xavier Descombes

► To cite this version:

Morgane Fierville, Kevin Lebrigand, Pascal Barbry, Xavier Descombes. Linear structure unfolding: application to mouse brain in spatial transcriptomics. EMBC 2025 - 47th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, IEEE EMBS, Jul 2025, Copenhagen, Denmark. 10.1109/EMBC58623.2025.11251850. [https://doi.org/10.1109/EMBC58623.2025.11251850](https://doi.org/10.1109/EMBC58623.2025.11251850)

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# Linear structure unfolding : application to mouse brain in spatial transcriptomics

Morgane Fierville<fnref n="1" />,<fnref n="2" />

Kévin Lebrigand<fnref n="1" />

Pascal Barbry<fnref n="1" />

Xavier Descombes<fnref n="2" />

**Abstract**— Spatial transcriptomics enables serial analysis of the expression of hundreds of genes in tissue sections by combining molecular biology and imaging approaches. This technique allows the exploration of large tissue sections (up to $3\text{cm}^2$) at the single-cell resolution, allowing cell-cell interactions and subcellular RNA expression analyses. Regions of interest having particular shapes raise specific computational problems that need to be addressed. In this manuscript, we propose a method for the analysis of elongated curved shape (set of connected pixels) or point cloud spread within a shape (cell centers or RNA molecules). We develop a method to unfold linear structures and analyze its content along its principal axis, in order to study cell or transcript type proportions. The centerline of the shape is automatically computed by combining k-means clustering and salesman problem optimization. Our method allows the analysis of spatial distribution of cells or transcripts along or perpendicular to the centerline.

**Index Terms**— Spatial transcriptomics, point cloud, centerline, k-means clustering, unfolding shape

## I. INTRODUCTION

Spatial transcriptomics (ST) has been developing steadily during the last five years [1], and is now able to interrogate the expression of hundreds to thousands of transcripts (i.e. RNA molecules) in complex organs at the subcellular resolution. Different technologies are available, combining state-of-the-art genomic with histological approaches. They can be merged into two major categories either based on in-situ capture with sequencing (Visium [2], Stereo-seq [3]) or imaging (MERSCOPE [4], Xenium [5][6], CosmX [7]). Depending on the biological question to address, the computational analysis can be performed at different levels, from whole tissue section to specific regions of interest, right down to cellular or even sub-cellular resolution. The main objective of these approaches is to decipher and explore the different profiles of gene expression according to different conditions within various cell types, in specific regions of interest. The presence of certain cell types (e.g. immune cells) in a specific neighborhood can also affect gene expression in a specific resident cell type without affecting its abundance. It is particularly important to delineate the organization and structure of the tissue in a way that can be easily accessible to computation, in order to highlight differences between different conditions. Besides, computing a transformation of

the data onto a common reference space allows aggregating data from different samples of the same population, thus facilitating the computation of mean and variance statistics.

In Fig. 1a-b, we show 2 shapes in white that delimit a particular region from real spatial transcriptomics data, respectively from an epithelial region of a human lung bronchus biopsy and a region of the human colon. To facilitate the comparison between samples (individuals or conditions), we project the region into a common reference shape with a uniform geometry (Fig. 1c). We then consider linear structures (Fig. 1d-e) that can be fitted to a rectangle representation. Compartments are then obtained by splitting the rectangle along the X and/or Y axis, after projecting the centerline onto a segment [0, 1].

<figure><img src="image_2.png" /><figcaption>Fig. 1. Different shapes to illustrate the purpose of the method. a), b) show curved shapes in white, from real spatial transcriptomics data from an epithelial region of a human lung bronchus biopsy and from a region of the human colon, respectively. c) Rectangle with a uniform geometry. d), e) are the shapes of the hippocampus and cortex respectively, from a coronal section of mouse brain. Each shape has been shown in intervals [0 – 5] for c), d) and [0, 3] for e) to follow the evolution along (c, d) or perpendicular (e) to the central axis of the shape, as indicated by the different arrows.</figcaption></figure>

In computer vision, the centerline of elongated shapes can be obtained through the computation of the medial axis. Current approaches to compute the medial axis are based on morphological concepts such as the skeleton [8]. However, these approaches are sensitive to noise along edges that creates spurious branches. Effectively pruning of these

This work has been supported by the French government through the 3IA Côte d'Azur Investements ANR-19-P3IA-0002 managed by the National Research Agency.

<footnote n="1">¹Université Côte d'Azur, CNRS, INSERM, IPMC, 3IA Côte d'Azur (Sophia-Antipolis, France) mfierville@ipmc.cnrs.fr</footnote>

<footnote n="2">²Université Côte d'Azur, INRIA, CNRS, INSERM (Sophia-Antipolis, France)</footnote>
spurious branches can then be cumbersome. More importantly, considering the case of a point cloud, as given by cell or transcript data, instead of a filled image-based shape makes them poorly adapted to the problem. An alternative is to consider points on the region border and compute a Voronoi tesselation [9]. This latter approach also produce extra-branches that can hardly be discriminated from the central axis.

In this paper, we propose a method for assessing gene expression or cell type proportions along a principal axis or orthogonally, in a region that is represented as a 2D shape containing a cloud of points (i.e. cells or transcripts). Herein, we propose to first cluster the point cloud and then to link the cluster centers as in [10]. Subsequently, the centerline is defined by a proper number of clusters and the order of their centers. In addition, we propose a new strategy to link the cluster centers by using the traveling salesman algorithm, coupled with an iterative scheme to automatically compute the number of clusters by studying the local geometry of the resulting central line. We show that our method is robust with respect to noise in the data, poor density of points and local curvature of the considered elongated shape. Moreover, we exemplified the proposed centerline approach by applying it to a dataset of 3 coronal section replicates of mouse brain performed on the Xenium from 10x genomics. We specifically chose the hippocampal regions that have a curved elongated structure (Fig.1d) to describe the cell composition within the structure. We extend our method to explore the cell specificity of transcript expression in cerebral cortex layers (Fig.1e).

## II. METHODS

In this section we describe the proposed approach and show its robustness with respect to the shape geometry, the sparsity of data and edges roughness.

### A. Centerline detection

The proposed method iteratively address the three following steps: (i) assign the points (or pixels) to clusters, (ii) extract the centerline and (iii) refine the clustering by increasing the number of clusters. Finally, the obtained centerline is extrapolated to reach the shape border.

We first cluster the data with an unsupervised process. K-means is a robust and general clustering algorithm which splits a set of points into a predefined number of subsets (k in the following) by optimizing a quadratic criterion [11]. Consider a set of points $p = (p_1, p_2, ..., p_n)$ with n the number of points to be divided in k clusters $C_1, C_2, ..., C_k$. The k-means algorithm iteratively minimizes the Euclidean distance between points within clusters (see Algorithm. 1 for details):

$$ \sigma^2 = \sum_{j=1}^{k} \left[ \frac{1}{|C_j|} \sum_{i=1}^{n} \|p_i - c_j\|^2 \delta_{z_i=j} \right] \quad (1) $$

### Algorithm 1 K-means clustering

INPUT: set of point $p_i$

OUTPUT: set of clusters $C_j$

1. Initialize the cluster centers $c_j$

2. Assign a cluster $z_i$ to each point $p_i$:

$$ z_i = \arg \min_j \|p_i - c_j\|^2 $$

3. Compute the cluster centroids:

$$ C_j = \{p_i : z_i = j\} $$

$$ c_j = \frac{1}{|C_j|} \sum_{p_i \in C_j} p_i $$

4. If not converged go to 2.

where $|C_j|$ is the number of points associated to cluster j, $c_j$ is the mean of the points associated to cluster j, $z_i$ is the cluster associated to $p_i$.

We then propose to order and link the centroids from the k clusters using the traveling salesman algorithm. Note that the algorithmic complexity is counterbalanced by the low number of points (number of clusters). We define each cluster centroid as a point $c_i = (x_i, y_i)$. Then, we compute the distance matrix $D \in \mathbb{R}^{k \times k}$ of all pair-wise distances:

$$ D_{ij} = \|c_i, c_j\| = \sqrt{(x_i - x_j)^2 + (y_i - y_j)^2} \quad (2) $$

where $D_{ij}$ is the Euclidean distance between 2 centroids ($c_i, c_j$).

Moreover, if a segment $[c_i, c_j]$ between two points crosses the boundary of the shape, we assign the distance between these 2 points to infinity:

$$ D_{ij} = +\infty \quad (3) $$

From this distance matrix, we compute the shortest path (i.e. the best path) that passes through all the points exactly once, without any loop or crossing the boundary of the shape. We consider each centroid as the first point $v_0$ and compute optimal path using the traveling salesman algorithm. If the ordered path contains less than the number of cluster k, it means that the starting point is not the correct one (see details in Algorithm. 2).

### B. Optimal k estimation

The last step consists in computing the optimal number of clusters (Fig.2). More clusters provide a more precise centerline. However, too many clusters may introduce oscillations of the centerline (see Fig.2c). We therefore maximize the number of cluster while controlling the angles between consecutive segments. To do so, we calculate the angles between 3 points $\theta_i = (p_{i-1}, p_i, p_{i+1})$ in the ordered line:

$$ \theta_i = \arccos \left( \frac{\| p_i \vec{p}_{i-1} \cdot p_i \vec{p}_{i+1} \|}{\| p_i \vec{p}_{i-1} \| \| p_i \vec{p}_{i+1} \|} \right) \quad (4) $$
**Algorithm 2 Minimal path**

INPUT: set of points $c_i, i = 1, ..., k$

OUTPUT: Minimal path $P$

**for** i = 1,k **do**

$$P_i = \{v_0 = c_i\}, D_i = 0$$

**for** j = 1,k-1 **do**

$$c_n = \arg \min_{c_m \notin P_i} D(v_{j-1}, c_m)$$

**if** $D(v_{j-1}, c_n) \neq \infty$ **then**

$$v_j = c_n, P_i = P_i \cup \{v_j\}$$

$$D_i = D_i + D(v_{j-1}, v_j)$$

**else**

$$D_i = \infty$$

**BREAK**

**end if**

**end for**

**end for**

$$l = \arg \min_{u=1,...,k} D_u$$

$$P = P_l$$

where $p_i$ is the position and $k$ is the total number of point from the estimated line.

Then, we select the different angles in $\Theta = \{\theta_2, ..., \theta_{k-1}\}$ which are lower than a threshold. We continue to increase $k$, to get a more precise delineation of the centerline, until we detect an outgrowth on the line represented by two consecutive angles lower than the threshold, as exemplified on Fig.2.c and explained here :

$$id = \{j \in < 2, k-1 > | \Theta_j < threshold\}$$

**for** i = 0, $|id| - 1$ **do**

**if** $id[i] + 1 = id[i+1]$ **or** $id[i] + 2 = id[i+1]$ **then**

**STOP**

**else**

$$k \leftarrow k + 1 \text{ STOP}$$

**end if**

**end for**

where $|id|$ is the number of element in that list. The threshold parameter was determined empirically and set at 120. This single parameter of the proposed approach was robust for different shapes, point density and noise level.

<figure><img src="image_3.png" /><figcaption>Fig. 2. Generation of the centerline with different $k$ of k-means with the different steps found in sections 2.1, 2.2 and 2.4 of the method section. a) Suboptimal with $k = 3$ ; b) Optimal with $k = 5$ ; c) Over-optimal with $k = 6$.</figcaption></figure>

Lastly, we extend the line to the boundary from the first and the last segment.

Finally, with this centerline, we can define the number of intervals along this region according to the desired quantification. Each point (i.e. a cell or a transcript) is assigned to its nearest interval. Then, for each interval we can evaluate the proportion of each cell type or the gene expression for different cell types.

### C. Analysis of robustness

Firstly, we compare our method to a skeleton and to a Voronoi diagram approach. In Fig.3, we consider two noisy rectangles by applying a random variation of more or less 12.5% and 25.5% of the width for shapes 1 and 2, respectively. Next, we apply a sinus function to the two previous shapes (shapes 3 and 4). Fig.3a shows the centerline obtained with the proposed method, with the resulting splitting illustrated by five grayscale intervals. We compare the results with a skeletonization and a Voronoi diagram on points of the shape boundary for which we have removed any lines that lay outside or crossed the boundary of the shape (Fig.3c). In comparison to our proposed method, which provides a noise-insensitive centerline computation, the skeletonization and Voronoi diagram approach (Fig.3b,c) are more sensitive to noise, producing spurious branches that can be challenging to prune.

<figure><img src="image_4.png" /><figcaption>Fig. 3. Comparison of centerline on noisy shapes. The different shapes were created by adding noise to either a rectangle (1, 2) or a sinusoidal shape (3, 4). a) Application of the proposed method, with the centerline in red and five intervals in grayscale. b) Skeletonization applied to the shape image. c) Voronoi diagram approach on the shape boundary.</figcaption></figure>

In addition, to test the application of our proposed method to the case of a point cloud, we manually created a curved shape with protrusions (composed of 38,737 pixels). We generated the centerline from an image or from a point cloud of the shape using different number of points (Fig.4). The point clouds was respectively created by randomly selecting 2000, 1000, 500 and 100 pixels from the image of the shape (Fig.4b-d). We can see that our method works equally well
on a shape (Fig.4a) or on point clouds even in the case of a
sparse representation of the underlying shape (Fig.4b-d).

<figure><img src="image_5.png" /><figcaption>Fig. 4. Comparison of the centerline computed with the shape (a) or a point cloud with 2000, 1000, 500 and 100 points, respectively (b,c,d,e). The centerline obtained is represented in red.</figcaption></figure>

In Fig.5, we created a shape with a curve that has different
local curvature. After the computation of the centerline
(red line) on this shape, we were able to generate 5, 10
and 20 regular intervals along it. The proposed approach
correctly follows the main axis of the shape regardless of
local curvature.

<figure><img src="image_6.png" /><figcaption>Fig. 5. Synthetic shape of a variable curve with 3 different local curvatures. a) Representation of the shape, b-d) Representation of the shape after applying our method, with the centerline (red line) visualized using 5, 10, and 20 regular intervals, respectively.</figcaption></figure>

### III. RESULTS : APPLICATION TO TRANSCRIPTS ANALYSIS IN MOUSE BRAIN HIPPOCAMPUS AND CORTEX

#### A. *Analysis of mouse brain hippocampus replicates*

Furthermore, to test our method in a realistic biological scenario, we used a dataset of coronal sections of mouse brain with 3 replicates from 10x genomics (Fig.6a). We selected only the hippocampal region of each replicate to extract shapes, then applied our method to obtain the corresponding centerlines. Subsequently, we defined 6 intervals along these regions (Fig.6b).

The hippocampus is a symmetrical region where we observed a conserved cell type proportion according to the symmetrical axis between intervals 2 and 3 (Fig.6c). The cell proportion is also maintained between the different replicates. We can see that the immature neurons are more represented at the beginning and end of the shape, compare to telencephalon projecting neurons which are more represented in the middle of the shape (Fig.6c). However, other cell types, such as vascular cells do not appear to evolve along this region.

#### B. *Analysis of mouse brain cortex layers*

A different situation was analyzed in the cortex of the mouse brain from replicate 1 (Fig.6a,d). In the cortex, there are 6 different layers (I to VI) from the outside to the inside, each with specialized functions. They are represented by different cell types, depending on the level of annotations we have performed. Thus, we can examine the cell type proportions according to two different axis: along the central axis, or orthogonally to this one. The orthogonal axis was defined by computing the centerline, then calculating the distance of each cell from the centerline. We then normalized the distance to be 0 at the upper border and 1 at the lower border (Fig.6d).

Thanks to the centerline, we produced two different plots. The first is a box plot showing the distance from 0 to 1 for each cell type, allowing us to determine whether certain cell types are more located in the upper or lower border of the cortex (Fig.6e). The second is a histogram showing the sum of transcripts per cell as a function of the distance from 0 to 1 for genes expressed in the telencephalon projecting neurons (Fig.6f).

In Fig.6e, we can see that oligodendrocytes are more represented in the lower part of the shape, in contrast to immature neurons, which are represented in the upper part of the shape. The size of the interquartile range and whisker show the extent of cell type in the shape. Thus, we can see that the majority of cell types are represented in the middle of the shape, but the di- and mesencephale neurons are concentrated in the middle (small box size) compared with telencephalon projecting neurons which are dispersed throughout the shape (large box size).

For the gene expression, we select 3 genes expressed in telecnaphalon projecting neurons and create a histogram of the sum of transcripts per cell as a function of the distance from 0 to 1. The *Satbn2* gene appears to be constant in the
different layers compared with Calb1 and Nxph3 which are expressed in the upper and lower layers, respectively.

<figure><img src="image_7.png" /><figcaption>Fig. 6. Analysis after applying the centerline method to data from a coronal mouse brain section. a) 3 replicates from coronal mouse brain section. b) Shape of the hippocampus for the 3 replicates separated into 6 regular intervals obtained using the centerline method. c) Bar plots of the cell type proportions in each interval for the 3 replicates. d) Shape of the cortex from replicate 1 with arrow from 0 to 1 to explore the orthogonal axis of the centerline. e) Box plots of the distance from 0 to 1 by cell types. f) Histogram of the sum of transcripts per cell as a function of the distance from 0 to 1 for 3 genes expressed in the telencephalon projecting neurons.</figcaption></figure>

IV. DISCUSSION

Spatial transcriptomic data analysis include similar steps as for single cell analysis, e.g. filtering cells with low counts, cell annotation, state-dependent differential gene expression and pathway analysis. However, new challenges are raised such as neighborhood or niche analysis, based on the spatial localization of cells and RNA molecules. Defining niches or spatial domains (i.e. groups of cells that have the same local environment) are important in ST to understand what are the differences within tissue, especially in the case of diseases. Although, niches are biological structure with generally irregular shapes, so we need to define a way of analyzing them.

Our proposed method demonstrates useful functionality in the context of long, curved niches, enabling us to assess gene expression and/or cell type distribution along a region of interest. This is relevant in ST studies where we can define different regions in order to analyze genes and cell types at a high resolution. The centerline is here to define the principal axis that follows the shape. Then, the important step is to use this centerline to define the intervals to which each cell or transcript will be assigned.

An important step in our method is to find the best *k* for the k-means algorithm. We want to obtain the highest possible *k* without disturbing the central axis of the shape (Fig.2). A lower *k* provides a centerline that is approximate and may not represent the shape well (Fig.2a). In contrast, if the *k* is too high, we will obtain a noisy line preventing to properly define the intervals to be analyzed (Fig.2c). To overcome this problem, we set a threshold parameter that was determined empirically to 120. This value has proved to be robust for all the conducted experiments.

Our method can be applied to a shape or directly to a point cloud (Fig.4), yielding similar results in both cases. Our approach appears more efficient and easier to compute than skeletonization or Voronoi diagram approaches, which produce spurious branches whose can be tricky to prune (Fig.3b,c). Moreover, the Voronoi diagram approach is computed on shape contour, meaning that the method is not directly applicable to point clouds. We demonstrate our method produces a centerline without spurious branches.

The robustness of the centerline in a point cloud depends on the location and number of points in the shape. If there are not enough points to represent the structure of the shape, the shape approach is the best option. However, if the number of points is sufficient, the point cloud approach is preferable, as the calculation time is reduced. In Fig.4, the shape contains (38737 * 2) values, while the point cloud with 2000 points contains only (2000 * 2) values.

Even if our method is relatively fast (around 3 minutes to run it on the cortex of replicate 1 with 6,826,685 pixels Fig.6), the minimal path step (2.B) uses the traveling salesman algorithm which is an NP-hard problem. Despite this, the optimal number of clusters never exceeded 70, even with numerous curves (48 clusters in Fig.5).

We apply our method on a real dataset with replicates. Even in the case of just 3 replicates, we can calculate statistics to compare the cell type proportions between intervals. Additionally, with more samples under different conditions, such as diseased or healthy, we could compare cell type proportions or gene expression between intervals across conditions.

Notably, our method can also be used to follow the orthogonal axis of the centerline (see Fig.6d). This option is particular useful in case of complex region such as the cortex that is composed of layers containing different cell type compositions and functions. Our approach makes it possible to analyze the region along its secondary axis (i.e. the one that follows the layers). Both axes can also be combined to create intervals on the different layers along
the central axis. Different number of layers can be found at different developmental times or in different areas of the cortex, and additional work will be necessary to fit more precisely the intervals to the histological observation. The results presented in Fig. 6d-f are in line with the literature [12], [13], [14], [15].

Our method works on linear structure with 2 main axis. We could consider improving it to extend it to other shapes with ramifications, such as vessels or star shapes. The main problem will be at the line creation step, which will be more complex. The user will need to know in advance the number of lines and branches required.

## V. CONCLUSION

Finally, our approach is able to combine image information and transcripts, enabling downstream analyses on these intervals such as neighborhood or pseudobulk assessments. By establishing a unified framework across samples, our method enables robust comparison of niches across varying states. This is particularly valuable for analyzing regions with functional heterogeneity along spatial axes, whether in terms of cell composition, gene expression, or other relevant metadata.

## COMPLIANCE WITH ETHICAL STANDARDS

The dataset of the mouse coronal brain section is publicly available and was downloaded from 10x Genomics website (https://www.10xgenomics.com/datasets/fresh-frozen-mouse-brain-replicates-1-stand ard). This dataset is licensed under the Creative Commons Attribution license.

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