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A Topological Data Analysis Study on Murine Pulmonary Arterial Trees with Pulmonary Hypertension

Pulmonary hypertension (PH), defined by a mean pulmonary arterial blood pressure above 20 mmHg, is a cardiovascular disease impacting the pulmonary vasculature. PH is accompanied by vascular remodeling, wherein vessels become stiffer, large vessels dilate, and smaller vessels constrict. Some types of PH, including hypoxia-induced PH (HPH), lead to microvascular rarefaction. The goal of this study is to analyze the change in pulmonary arterial network morphometry in the presence of HPH. To do so, we use novel methods from topological data analysis (TDA), employing persistent homology to quantify arterial network morphometry for control and hypertensive mice. These methods are used to characterize arterial trees extracted from micro-computed tomography (micro-CT) images. To compare results between control and hypertensive animals, we normalize generated networks using three pruning algorithms. This proof-of-concept study shows that the pruning methods effects the spatial tree statistics and complexities of the trees. Results show that HPH trees have higher depth and that the directional complexities correlate with branch number, except for trees pruned by vessel radius, where the left and anterior complexity are lower compared to control trees. While more data is required to make a conclusion about the overall effect of HPH on network topology, this study provides a framework for analyzing the topology of biological networks and is a step towards the extraction of relevant information for diagnosing and detecting HPH.


Introduction
Cardiovascular diseases are the leading cause of death in the world. The World Health Organization (WHO) [36] estimates that in 2019 17.9 million people died from cardiovascular diseases, approximately 32% of deaths. While cardiovascular diseases encompass many disease types, a common trait is that these diseases are associated with remodeling of the vasculature and heart, causing a range of functional problems, the most prominent being high blood pressure. Another important finding by the WHO is that early detection is essential to improve the burden of cardiovascular diseases [36]. Cardiovascular diseases are typically diagnosed by employing a combination of measurements, including Computed Tomography (CT) and Magnetic Resonance Imaging (MRI), dynamic heart rate and blood pressure measurements, and blood samples quantifying disease markers in the blood. Cardiovascular disease is typically diagnosed by abnormal blood pressure and blood flow values resulting, among other factors, from tissue remodeling at the local level (within specific vessels) and globally at the network level. In cardiovascular disease, tissue remodeling has two effects: changes in compliance within vessels and changes in network morphometry, impacting blood pressure and flow. This study uses techniques from topological data analysis to characterize changes in morphometry of pulmonary arteries in mice with hypoxia-induced pulmonary hypertension.
Pulmonary hypertension (PH) is characterized by high blood pressure (≥ 20 mmHg) in the main pulmonary artery caused by vascular remodeling and inflammation or increased pressure in the veins or left heart [4]. Symptoms of the disease include shortness of breath, fatigue, dizziness, chest pain, heart palpitations, and swelling of the legs and ankles. These symptoms are common in many illnesses, making PH difficult to diagnose [13]. Moreover, PH is a heterogeneous disease encompassing five subtypes, each with a separate pathophysiology [30]. However, early diagnosis and targeted treatment can improve quality of life by delaying severe complications, which is essential because all but one type of PH has no cure [22]. This study will analyze the morphometry in murine pulmonary arterial networks excised from healthy and hypoxia-induced pulmonary hypertension animals. In PH induced by hypoxia (HPH) vascular remodeling impacts the pulmonary arteries [24,35]. The disease starts in the arterioles, stiffening and constricting the vessels, then migrating to the larger arteries, which stiffen and dilate. Hopkins et al. [19] have characterized the structural changes in individual vessels, but less is known about how the network morphometry changes, the focus of this study.
Pulmonary arterial morphometry characterizes the network properties in space. Since early contributions by Murray in 1926 [25], many researchers (e.g., [5,20,24,31,35])have examined the morphometry of the pulmonary arterial network. The early studies by Murray [25], and Zamir [37] devise optimality principles, describing arterial branching that minimize pumping power and lumen volume. These studies assume that the arterial network bifurcates and that the dimensions of the two daughter vessels can be determined as functions of the parent vessel. Results from these studies include a power law defining how vessel radii change across a bifurcation, an asymmetry ratio relating the radii of the two daughter vessels, and an area ratio relating the combined cross-sectional area of the daughter vessels to that of the parent vessel. These studies were supplemented by work from Singhal et al. [31], Horsfield [20], and [27,28] that incorporate data from lung casts to devise relations between parent and daughter vessels. The casts are generated by injecting liquid resin into the arterial networks. The vessel dimensions are measured using calipers on the hardened resin cast. The data provide geometric information, but each study only examined a single lung, as human cadaver data are not easily obtained. Moreover, the casts are fragile, and there is inherent human error in using calipers to measure the vessel dimensions. More recent studies by Molthen et al. [24], Vanderpool et al. [35], Davidoiu et al. [8], and Chambers et al. [5] use medical imaging to accurately and efficiently extract geometric information from arterial networks.
Persistent homology 1 refers to a TDA technique wherein combinatorial structures are successively built from a data set, and their homology is used to define descriptors of the shape of the data. These novel techniques have been used to analyze the arterial networks in the brain [3] and bronchial networks [2], but to our knowledge, this study is the first to use persistent homology to characterize pulmonary arterial networks. Bendich et al. [3] analyze human brain arterial trees generated from a tube-tracking segmentation algorithm on magnetic resonance images (MRI). Using these trees, they compute degree-0 persistent homology and find it strongly correlated with age, while degree-1 persistent homology correlates strongly with sex. In [2], persistent homology compares airway CT images from healthy and chronic obstructive pulmonary disease (COPD) patients. This study found that degree-0 persistent homology can distinguish the patient groups, while degree-2 persistent homology only can distinguish inspiration and expiration.
Motivated by these studies, we compute persistent homology from labeled spatial trees extracted from micro-CT images. We compute the degree-0 persistent homology of these trees using a height filtration in 3 dimensions in the forward and reverse directions (resulting in 6 applications of filtration per spatial tree). We compute the total persistence in each direction, known as the directional complexity, and we compare these values between control and HPH networks.

Methods
Our objective is to identify markers that can characterize differences between control and HPH arterial networks extracted from micro-CT images. This is done in three steps. First, we generate a 3-dimensional (3D) rendered network from the micro-CT image, constructing a labeled spatial tree with edges, representing vessels, and vertices, representing junctions. Each vertex labeled by its length, spatial orientation, and radius has coordinates in R 3 . Second, we compare the spatial trees obtained from the control and HPH animals. To get biologically significant information from these spatial trees, comparing only corresponding parts of the control and HPH trees is essential. In HPH, the diameter of the large vessels increases [5], making more vessels visible in the 3D rendered network. Therefore, to compare HPH networks with controls, we generate pruning algorithms to normalize the trees. Third, we use persistent homology to characterize the normalized networks. Inspired by Belchi et al. [2], we employ height filtration in 3D (6 directions), computing the degree-0 persistent homology, and compare the directional complexities of trees from the control and HPH mice.

Imaging protocol
The micro-CT images used in this study were made available by the Chesler Lab, University of California-Irvine [35]. This study uses images from six C57BL6/J mice, age 10-12 weeks and weight 25.8 ± 2.3 grams. Of the six mice, 3 are controls and 3 have PH induced by placing them in an hypoxic environment (FiO2 reduced by half) for 10 days. The mice were mediated with (52 mg/kg body weight) pentobarbital sodium and euthanized by exsanguination before the lungs were extracted. After extraction, the lungs were imaged following the protocol described in [35]. First, a cannula (PE-90 tubing, 1.27 mm outer and 0.86 mm inner diameter) was positioned in the main pulmonary artery (MPA) well above the first arterial bifurcation. After, the lungs are ventilated in a gas mixture with (15% O 2 -6% CO 2 , balance nitrogen), rinsed with a physiological salt solution, and perfused with perfluorooctyl bromide (PFOB). The lungs were further prepared by periodically adjusting the intravascular pressure from 0 to 25 mmHg multiple. After preparation, the arterial pressure was kept constant as the lungs were rotated around an X-ray beam at 1 • increments to obtain 360 planar images. The lungs were imaged at main pulmonary arterial pressures of 6.3, 7.4, 13.0, and 17.4 mmHg ( Figure 2). The planar images for each pressure were reconstructed using the Feldkamp cone-beam algorithm and converted into a 3D volumetric dataset, saved as a Digital Imaging and Communications in Medicine (DICOM) 3.0 image (Figure 1(a)).

Segmentation and skeletonization
The micro-CT images are saved in the (DICOM) 3.0 format. Each image has the dimensions (497 × 497 × 497) voxels with a spatial resolution of 30-40 µm per voxel. Each voxel v is represented by a voxel complex with spatial coordinates (x v , y v , z v ) and intensity I(v) ∈ [0, 255] with 0 representing the intensity of a black voxel and 255 of a white voxel.
Segmentation is the process used to partition the voxels into the set of foreground voxels, which represent the arteries, and background voxels represent- ing the surrounding tissue. We segment the images using the open-source program 3D Slicer [12,21,23], Kitware, Inc., employing a combination of segmentation techniques, including global thresholding, median smoothing, and manual editing as described in detail in our previous study [5]. Global thresholding is used to identify voxels in the foreground, finding voxels with intensities I(v) ∈ [τ min , τ max ]. The pulmonary arteries are the only anatomical structures in the image, so τ max = 255 for all images. The minimum threshold τ min is adjusted ad hoc to ensure that all visible arteries are included. To reduce noise, median smoothing is used, replacing the intensity of all the voxels within a kernel of (3 × 3 × 3) voxels with the median of the adjacent voxel intensities. Since the cannula and the hypobaric cylinder can distort the image, manual editing is required to remove voxels from the foreground, which do not represent the true arterial structure. The result of segmentation is a foreground which can be visualized as a 3D rendered surface representing the pulmonary arterial tree. The constructed foreground is referred to as the segmented network (shown in Figure 1(b)). Each segmented artery is a portion of the segmented network that lies between two junctions.

Skeletonization:
The skeleton is a voxel complex comprising a thinned representation of the segmented arteries. In the skeleton, the branching structure of the segmented arteries is preserved, i.e., each branch is represented by a set of voxels that are one voxel in width and centered in the artery. The skeleton is obtained by iteratively removing voxels from the segmented arteries using Couprie and Bertrand's "Asymmetric Thinning" algorithm. For details see [5,7]. Example centered skeleton is shown in Figure 1(b).
Distance map: To obtain the dimensions for the segmented arteries, we generate a distance map, an associated voxel complex with the same spatial dimensions as the original image. For each voxel v, we compute a distance voxel v D denoting the distance from v to the nearest background voxel u computed as where u − v 2 is the Euclidean distance from v to each background voxel, u, and the distance voxel v D has the same spatial coordinates (x v , y v , z v ) and intensity This distance map, as well as the skeleton and labeled spatial tree, are obtained using the Spatial Graph Extractor Github repository (SGEXT) in the Digital Geometry Tools and Algorithms Library (DGtal) [9,15,17].

Spatial tree labels
Each skeleton is used to generate a spatial graph, a collection of vertices in R 3 connected by edges. Every voxel in the skeleton corresponds to a vertex in the spatial graph. Each vertex, denoted by a numerical ID A, has 3D coordinates (x A , y A , z A ). Each edge connected by 2 vertices A and B is denoted e AB . If deg(A) = 1, A is either at the inlet to the MPA (called A root ) or a leaf, a terminal vertex. If deg(A) = 2, A is a vertex along the length of a segmented artery. If deg(A) > 2, A is a junction point, a point where a parent artery splits into multiple daughter arteries. Typically, junction points have degree 3, as 98-99% of junctions are bifurcations [5]. All edges are oriented toward blood flow (away from A root ). With this orientation in place, we can refer to e AB as an edge from start vertex A to end vertex B.
Constructing the spatial graph is subject to errors that can be corrected using the protocol introduced in [5]. Errors include edges that do not accurately represent the segmented arteries, small cycles that arise when 3 adjacent voxels are connected in a loop, duplicated edge points, and duplicated edges that connect the same vertices. False branches must be manually identified and removed from the graph, while small cycles and duplicate edges/points can be removed automatically. Cycles are broken by removing the longest edge in the cycle. After implementing the corrections, the graph becomes a spatial tree T (Figure 1(c)).
Each vertex in T is labeled with its radius, which is determined from the distance map. Recall that the skeleton is centered in the segmented arteries. Therefore, for voxel v in the skeleton, the measurement D v calculated using equation (1) gives an estimate for the radius of the artery centered at (x v , y v , z v ).
In the 3D rendered surface, every artery is represented by a collection of edges connecting consecutive degree 2 vertices between either 2 junctions or a junction and a leaf. We refer to the collection of edges as a branch of T , i.e., each branch corresponds to an artery. Branches are labeled with an average radius and length in microns (µm). To convert the voxel measurements into µm, all dimensions, x, y, and z, are multiplied by the scaling factor λ T , obtained by dividing the cannula's radius (430 µm) by the average of the first 5 radius measurements of the MPA, the radii of vertices A root , A 1 , ..., A 4 , i.e., The radius r AB of a branch from a leaf/junction point A to a leaf/junction point B is represented by the interquartile mean (IQM) of the radius measurements from A to B, and the degree 2 vertices A 1 , ..., A m along the branch,a The length L AB of a branch is the scaled sum of the Euclidean distances between consecutive vertices along that branch,

Spatial tree statistics
To quantify the spatial trees, we counted the number of branches, leaves, and tree depth, and computed the Strahler order. We define tree depth as the number of branches in the longest direct path from A root to any leaf. The Strahler order (SO) of a branch is an indicator of its level within the tree [34]. Terminal branches all have SO = 1. If two joining daughter vessels have equal SO (i.e., SO d1 = SO d2 ), Figure 3). The Strahler order of a tree refers to the maximum Strahler order of any branch within that tree. For the trees examined here, this will always be the root branch, the MPA, denoted SO M P A . Results, reported in Table 1, show that HPH trees have significantly more branches and leaves than the control trees. While pulmonary vascular remodeling observed in HPH has been characterized by increased vessel stiffness and dilation of large arteries [5], one effect that is not typically observed is angiogenesis, the development of new blood vessels. Studies show that HPH animals have fewer arterioles (arteries with radius ≤ 50 µm) than control [35]. Since the contrast in the micro-CT images travels through dilated vessels, more vessels are captured in the segmented arteries of HPH mice compared to the control. Consequently, the increased number of branches in the HPH trees is an artifact of the imaging process and does not translate to the HPH mice having more pulmonary arteries. Additionally, the minimum radius of the vessels captured during segmentation is approximately 39 µm, with the vast majority of vessels having a radius greater than 50 µm (97.3% of vessels). Since arterioles typically have radii of 25 − 50 µm, they are not captured in the images, explaining why we do not observe microvascular rarefaction described in [35]. Figure 3: The Strahler ordering system illustrated on an example tree. All terminal branches have SO = 1. Whenever 2 branches of the same Strahler order meet, their parent has Strahler order which is 1 higher than that of its daughters. If 2 branches of different Strahler order meet, their parent is labeled with the maximum Strahler order of its daughters.

Spatial tree pruning
Our objective is to distinguish HPH mice from controls. We use tree pruning to generate trees that can be compared. If the pulmonary trees branch symmetrically, tree pruning would be simple, but pulmonary arterial networks are asymmetrical [5,25,28,37]. Consequently, a pruning approach is necessary to normalize these trees. In this study, we propose 3 pruning methods wherein terminal branches are systematically removed. Two pruning techniques rely on computing the Strahler order, and the third method uses the vessel radius.
Maximum Strahler order pruning starts at the leaves removing sister branches at the lowest Strahler order, i.e., branches for which both vessels in the juntion are of Strahler order 1 (i.e., SO d1 = SO d2 = 1). Once all branches of this type has been removed, the Strahler order of the tree is reduced by one. This pruning continues until the desired Strahler order is obtained. This pruning method is illustrated in Figure 3. Panel (a) shows the original tree and (b) the maximum Strahler order pruning removing all sister vessels with Strahler order 1 from the network. The new tree will have exactly one Strahler order lower than the original tree.
Minimum Strahler order pruning: Trees at any given Strahler vary significantly in the number of branches and tree depth. To eliminate this variation we also perform minimum Strahler order pruning. This pruning method removes all pairs of terminating sister branches for which SO d1 = SO d2 = 1. In addition, this algorithm removes any branches that will not impact the final Strahler order. Strahler orders are recomputed, and the process continues until the next iteration of branch removal gives the minimal tree with the desired Strahler order. This method is illustrated in Figure 3(c).
Radius pruning: This pruning technique involves removing pairs of terminal daughter branches from trees with a radius less than or equal to a given radius threshold τ r . This threshold is chosen such that any group of trees have the desired number of branches. After radius pruning, all bifurcations with terminal daughter branches have at least one daughter with r ≥ τ r . This method is illustrated in Figure 3(d).

Directional complexity
This study uses topological data analysis (TDA) to examine the spatial features of our labeled spatial trees. TDA provides extensive toolbox of data analysis methods for learning about the shape of data. In this study, we use persistent homology to analyze our control and HPH spatial trees [10,11,29]. Specifically, we compute the degree-0 persistent homology of a height filtration, and use this to create a barcode and compute a topological marker known as directional complexity. These techniques are robust to noise [1,26], making them useful for biological applications.
Relevant topological definitions and background are given in the Appendix. In this section, we provide a brief overview of some of those key concepts and describe the particular topological methods used in this study. In summary, computing persistent homology involves building a sequence of topological spaces upon a data set, computing the corresponding homology, and tracking a topological summary that describes the shape of the data.
Simplicial homology: A spatial tree is an example of a simplicial complex, the formal sum of building blocks called simplices. A k-simplex is a convex hull of k + 1 vertices [16] (in particular, 0-simplex is a point, a 1-simplex is edge, a 2simplex is a triangular face, and a 3-simplex is a filled tetrahedron). Each labeled spatial tree in this study amounts to a set of points (0-simplices) attached by a set of edges (1-simplices). Recall that these edges are directed away from the root of the tree to represent the direction of blood flow. Likewise, when computing simplicial homology, the k-simplices of a simplicial complex are embedded with an orientation.
For a given simplicial complex S, the chain vector space, C k (S) is the vector space whose basis is the set of oriented k-simplices in S [16]. The elements c ∈ C k are called a k-chains, c = i a i σ i , where each a i is an integer and each σ i is an oriented k-simplex from S. The boundary map, ∂ k : C k → C k−1 , sends k-dimensional chains to their k − 1-dimensional boundaries.
Let L k be the kernel of ∂ k and B k be the image of ∂ k+1 . The k th simplicial homology, H k , is the quotient group The k th Betti number, β k , is the dimension of the homology H k . It counts the number of k-dimensional "holes" in a simplicial complex [26]. We define a kfeature to be a property of a simplicial complex which is counted by β k . The Betti numbers β 0 , β 1 , and β 2 count the number of disjoint connected components, loops, and spheres S 2 , respectively.
Persistent homology: The idea behind persistent homology is to iteratively build nested simplicial complexes which connect data points based on proximity. The set of nested simplicial complexes is called a filtration and is governed by a distance function ε. For an increasing sequence of ε values ε 1 , ..., ε m , the filtration S ε 1 ⊂ S ε 2 ⊂ ... ⊂ S εm is built upon the data. Degree-k persistent homology is found by computing H k (S ε 1 ), ..., H k (S εm ) and examining the evolution of their k th Betti numbers. For any i = 1, ..., m, the Betti number β k (S ε i ) counts the number of k-features in S ε i . If a feature appears for the first time in a certain S ε i , but is no longer present in S ε j for some j > i, then the birth of that feature is ε B = ε i and the death is ε D = ε j . The persistence of a feature is P = ε D − ε B . More persistent features are ones present for a large range of ε values. The last complex created in a filtration will have the topology of a point.
Barcodes: The evolution of the homology H k (S ε i ) can be visualized by creating a degree-k barcode B k , a diagram which contains 1 bar for each new k-feature detected in the filtration. Each bar spans the length from ε B to ε D for that feature ( Figure 5). Longer bars represent features with greater persistence in the chosen filtration, while shorter bars are often interpreted to represent noise, though this depends on the application.
Height filtration: In this study, we compute the degree-0 persistent homology of spatial trees with respect to the height filtration. Given a spatial tree T with vertices V and edges E, the height filtration dictates that each S ε i includes all vertices v ∈ V and edges e ∈ E for which the value of our governing distance function ε(·) is at most ε i . The exact definition of ε(·) depends on which direction we want to examine in our filtration. There are 6 filtration directions used in this study: the positive/negative x-directions (+x/−x), the positive/negative ydirections (+y/−y), the positive/negative z-directions (+z/−z) (Figure 4). For a given filtration direction ±ξ, ε of a vertex v ∈ V with spatial coordinates where ξ T max is the maximum ξ coordinate of any v ∈ V for ξ = x, y, or z. For an edge e ∈ E between two vertices For example, a height filtration in the +z direction indicates that the filtration moves along the z-axis in the positive direction, and each S ε i includes all vertices with z v ≤ ε i and all edges with max {z v 1 , z v 2 } ≤ ε i . Conversely, a direction of −z indicates that the filtration moves along the z-axis in the negative direction, and each S ε i includes all vertices with z T max − z v ≤ ε i and all edges with An illustration of the −z height filtration is shown in Figure 5.
Directional complexity: For each tree T and filtration direction ±ξ, the degree-0 barcode is computed by tracking the number of disjoint connected components in the nested set of subgraphs at each ε i ( Figure 5). The directional complexity (DC) of T in each direction is the total persistence of all 0-features in the filtration, computed by summing the lengths of the bars in the barcode, Directional complexities are named relative to murine anatomy and captures the length and occurrence of branches in the named direction ( Figure 4). For example, the −z filtration yields anterior complexity which refers to branches pointing toward the head of the mouse ( Figure 5).  Table 1 contains spatial tree statistics of the original and pruned trees in this study. These include number of branches, number of leaves, tree depth, and Strahler order. Table 2 shows the trees' directional complexities generated from the degree-0 barcode from the height filtrations in 6 directions. Arteries were imaged at the 4 different contrast pressures, 6.3, 7.4, 13.0, and 17.4 mmHg, with results for trees from all pressures shown. All values in these tables are reported as mean ± standard deviation unless otherwise stated.

Spatial tree statistics
The number of branches, number of leaves, and tree depth are all lower in the original control trees than in the original HPH trees. All original HPH trees have Strahler order SO M P A = 7, while control trees had SO M P A = 6 at all pressures except 17.2 mmHg, for which they had SO M P A = 7. This can be explained by the fact that HPH causes vessels to dilate [5] (indicated by a higher length-to-radius ratio), so more vessels are detected in the micro-CT images, leading to more branches in the spatial trees.
To generate normalized trees (shown in Figure 6), we apply maximum Strahler order pruning to the HPH trees at all pressures except 17.2 mmHg, since the control and HPH trees already had the same Strahler order (SO M P A = 6) at that pressure. At the two lowest pressures (6.3 and 7.4 mmHg) these pruned HPH trees still have more branches, leaves, and a higher tree depth than the control trees. At 13.0 mmHg, the pruned HPH trees had fewer branches and leaves than the control trees, which could be attributed to increased vessel stiffness in HPH [5]. Because the control vessels are more compliant, they expand more at higher pressure than HPH vessels, allowing the imaging contrast to perfuse more vessels. At the highest pressure (17.2 mmHg), maximum Strahler order pruning did not affect the HPH trees, since they already had the same Strahler order as the controls.  Minimum Strahler order pruning is applied to both control and HPH trees, generating trees of Strahler order SO M P A = 6 for pressures 6.3, 7.4, and 13.0 mmHg, and Strahler order SO M P A = 7 for the highest pressure, 17.2 mmHg. At the two lowest pressures (6.3 and 7.4 mmHg), the pruned control trees have more branches and leaves, but lower tree depth than the pruned HPH trees. At 13.0 mmHg all statistics are higher in the pruned control trees, and at the highest pressure 17.2 mmHg (pruned to Strahler order SO M P A = 7) these values were higher in the pruned HPH trees.
Radius pruning is applied to HPH trees only, bringing the number of branches and leaves close to those of control trees. On average, the radius-pruned HPH trees have slightly more branches than the control trees at 6.3 and 17.2 mmHg, fewer branches at 13.0 mmHg, and the same number of branches at 7.4 mmHg. The radius-pruned HPH trees have a few more leaves than the control trees at all pressures. The depth of radius-pruned HPH trees is higher than that of control trees at all pressures. The Strahler order of all radius-pruned HPH trees are approximately the same as the control trees.

Directional complexity
All directional complexities are higher in the original HPH trees than in the original control trees. In the maximum Strahler order pruned HPH trees, all directional complexities are higher than controls at 6.3 mmHg. At 7.4 mmHg, right and posterior complexities are higher in control trees than in the radius-pruned HPH trees; all other directional complexities are higher in the radius-pruned HPH trees at this pressure. At 13.0 mmHg, all directional complexities are lower in the pruned HPH trees than controls. Again, this may be due to higher vessel stiffness in HPH mice [5]. The control vessels are more compliant, so they dilate more at higher pressure than HPH vessels, resulting in more branches in the spatial trees. When more branches point in all directions, directional complexities are higher.
For the minimum Strahler order pruned trees, all directional complexities are higher in pruned control trees than in pruned HPH trees for pressures 6.3, 7.4, and 13.0 mmHg. At 17.2 mmHg, right, ventral, and posterior complexities are lower in pruned HPH trees than controls, while left, dorsal, and anterior complexities are higher in pruned HPH trees.
In original, maximum, and minimum Strahler pruned trees, the group (control or HPH) with the more branches at each pressure generally has higher directional complexities, with few exceptions mentioned above. The radius-pruned HPH trees, by design, have similar branch counts to the controls. At all pressures, the radius-pruned HPH trees have lower right, ventral, and posterior complexities and higher left and anterior complexities than controls. This indicates that more (and/or longer) branches point in the left and anterior directions in HPH, possibly due to the arterial network adapting to distribute blood more effectively these regions. Since mice are quadripeds, dorsal complexity captures branches pointing in the opposite direction of gravity. Dorsal complexity is lower in radius-pruned HPH trees than controls at all pressures except 17.2 mmHg. This coincides with the finding in Belchi et al. [2] that upwards (toward the head, also opposite of Table 1: Spatial tree statistics from micro-CT images at 4 different contrast pressures. Abbrevia- Table 2: Directional complexities of trees from micro-CT images at 4 different contrast pressures in 6 different directions. Abbreviations used are "C" for control trees, "HPH" for unpruned HPH trees, "HPH R " for radius-pruned HPH trees, "HPH M " for maximal Strahler-order pruned HPH trees, and "C m " and "HPH m " for the minimal Strahler-order pruned control and HPH trees, respectively. All values are reported as mean ± standard deviation. Data for HPH M at pressure 17.2 mmHg is missing because the control and original HPH trees at this pressure already have the same Strahler order. gravity) complexity in human bronchial networks decreases with COPD, a disease commonly associated with HPH.

Discussion
This study characterizes control and HPH spatial trees, quantifying the number of branches and leaves, tree depth, Strahler order, and directional complexities. We found that the tree depth is higher in HPH trees compared to control, a phenomenon which is preserved after pruning the trees. This difference can be explained by vascular remodeling in HPH, causing the large arteries to stiffen and dilate [35,5].
Tree Pruning: The radius pruned HPH trees have higher left and anterior complexity compared to the controls. This means that in HPH, more vessels point toward the head and the left. This may indicate that the network is adapting to effectively transport blood to these regions. Overall, the directional complexities correlate with number of branches. However, the HPH spatial trees have more branches due to increased vessel radius, which allows the contrast to perfuse more arteries. By design, the radius-pruning algorithm makes the number of branches in HPH trees close to that of controls. Therefore, the directional complexity results from these trees are more significant, as they will be less influenced by branch count. Moreover, the control trees have higher right, ventral, dorsal, and posterior complexities than the radius-pruned HPH trees, with the exception of dorsal complexity at the highest pressure (17.2 mmHg). This exception reflects our previous finding that higher pressure trees are more difficult to distinguish.
CT Image Pressure: At higher pressures, it is more difficult to distinguish control and hypertensive trees. However, at higher contrast pressures (13.0 and 17.2 mmHg) the control and HPH trees become more similar. This can be explained by the fact that in control animals, the vessels are more compliant and expand more when perfused at higher pressures. The latter is also observed with the minimal Strahler order pruning at the second highest pressure (13.0 mmHg), where the control trees' depth exceeds that of the HPH trees.
Directional complexity: Our use of degree-0 persistent homology via the height filtration is inspired by Belchi et al. [2], who found that humans with mild to moderate COPD had lower upwards complexity of the bronchial network compared to healthy subjects. In this context, "upwards" complexity refers to bronchi pointing toward the head. Since HPH is the type of PH most commonly associated with COPD [13] and the pulmonary arteries branch in a manner similar to the bronchi, we implemented the same filtration as [2] on the murine spatial trees. Belchi et al. [2] found that upwards complexity of the bronchi is lower in disease; this is contrary to our findings that anterior complexity is higher in HPH. However, mice are quadrupeds and their lungs are rotated compared to humans. Thus, dorsal complexity might be a more apt comparison to human upwards complexity because it is the direction which counters gravity. We found that the dorsal complexity is lower in the radius pruned HPH trees compared to control trees at the three lowest pressures. Finally, Belchi et al. [2] found that the choice of filtration direction do not influence results, reporting upwards complexity only. In our study, we did detect a difference in some of the directional complexities in the pruned trees, which could be reflective of the differences between human and mouse lung structures. Moreover, Belchi et al. [2] examined the airways, which could explain the differences in our results. Thet had more scans, but the graphs had fewer generations. We had fewer scans but generated larger trees.

Limitations and future work
The major limitation in this study is a lack of data. We analyzed data from three control and HPH subjects. TDA methods are designed for analysis of large datasets, yet for physiological studies it can be difficult to conduct experiments with large number of animals. One way to circumvent this is by using machine learning to generate surrogate networks representative of the actual data, an approach we plan to pursue in future studies. This approach was used in the persistent homology study by Bendich et al. [3] analyzing brain arterial networks. Using machine learning they generated 98 trees from repeated iterations of a tubetracking algorithm. We have had some success in generating simulated networks by attaching self-similar structured trees to principal branches of our labeled spatial trees [5], though these trees did not capture the three-dimensional nature of arterial branching. Using angles extracted from our spatial trees, we plan to generate networks with more similar branching properties to the trees found in this study and implement our methods to see if stronger conclusions can be drawn.
It is also critical to test our methods on trees extracted from human lung images. Such images can be difficult to obtain due to privacy concerns of patients. Moreover, segmenting these images is challenging because clinical CT images have lower resolution and in vivo images also contain airways and veins, making it difficult to isolate the arteries. The mouse images analyzed here were ideal. The CT images were gathered at high resolution and the pulmonary arteries were excised, i.e., the arterial network was the anatomical objects in the images.

Conclusion
In this study, we extracted spatial trees from micro-CT images representing the pulmonary arterial networks in control and HPH mice. There is a large discrepancy between branch counts between the two groups, which we hypothesize is an imaging consequence rather than evidence of angiogenesis, as microvascular rarefaction has been observed in mice with HPH [35]. We devised pruning algorithms based on vessel radius and Strahler order, each with their own advantages and disadvantages. Guided by a study conducted on similar networks [2], we computed the directional complexities using a degree-0 persistent homology of a height filtration. Overall, we found that tree depth is larger in the HPH trees, directional complexity correlates with the branch count, and that the left and dorsal complexities are lower in HPH. Finally, we noticed that results at low perfusion pressures reveal more differences, likely because at higher pressure increased elasticity in healthy animals cause vessels to dilate more making them appear similar to HPH animals. This study serves as a proof-of-concept for the use of TDA to detect differences in diseased vascular networks.