Topological analysis of hemodynamic response to cardiac resynchronization therapy

The Mapper algorithm is a qualitative method in topological data analysis that constructs graphs from point clouds by combining dimensionality reduction and clustering. We apply Mapper, together with quantitative graph indices proposed in this work, to compare the effects of left ventricular epicardial and endocardial biventricular pacing in a swine model of pacing-induced non-ischemic cardiomyopathy. Distributions of four hemodynamic variables from a previous study were analyzed using Mapper, complemented by numerical indices quantifying self-connectivity, scattering, and homogeneity of the resulting graphs. Statistically significant differences were observed among basal, mid, and apical pacing regions, with self-connectivity index values of 0.62 (basal), 0.29 (mid), and 0.21 (apical), and with p < 0.05 for both basal-versus-mid and basal-vs.-apical comparisons. Moreover, endocardial stimulation at lateral sites increased the contrast between basal and mid or apical distributions compared with epicardial stimulation. These findings indicate that the proposed graph indices extend Mapper from a qualitative visualization tool to a quantitative framework for comparing pacing-induced hemodynamic distributions.

Topological methods for data analysis (TDA) have been widely applied across a variety of disciplines since the decade of 2010. While numerical descriptors based on persistent homology are the most frequently used [1], the Mapper algorithm is a qualitative method in topological data analysis, introduced in [2], that transforms point clouds into graphs by combining dimensionality reduction and clustering techniques. The Mapper algorithm has been applied in fields such as industry, biomedicine, finance, and many others [3–5]. It has also been used in artificial intelligence to analyze the structure of neural network activations and to assess their generalization capacity [6]. Mapper is most often used in the context of unsupervised learning —common applications include visualization, clustering, and feature discovery [7–9]. It has proved particularly useful in medical studies, owing to its ability to capture the global structure of datasets even when sample sizes are limited. A further advantage of Mapper is its interpretability: it produces graphs that highlight

and summarize the relevant features of the data in a visually accessible way. In parallel with these applications, parameter selection and stability of Mapper have been discussed in several works, including confidence regions for topological features and stability analyses [10, 11]. These approaches are complementary to the objective of the present work: rather than performing inference on the topological features of Mapper itself, we use Mapper as an interpretable graph representation and quantify the organization of predefined experimental labels on this graph.

Even though Mapper has traditionally been applied as a qualitative exploratory tool, relying on visualization of the resulting graphs, there has been significant progress toward defining quantitative measures for analyzing its resulting graphs. In [12], a patient-patient similarity network was constructed using Mapper and enrichment analysis was applied to statistically assess the overrepresentation of type 2 diabetes within nodes, as well as to identify comorbidity patterns. In [13], spatial analysis of functional enrichment was used to compute enrichment scores across nodes and their neighborhoods, thus highlighting localized associations in microbiome data. More generally, a mathematical framework was proposed in [14] for quantifying interesting paths in Mapper graphs, providing optimization formulations to systematically identify substructures of interest. The enhancement of Mapper graphs with statistical methods has also been explored in [15]. Related quantitative approaches have also used graph-level or node-level descriptors of Mapper outputs, for instance to assess purity or to compare structures across Mapper graphs [6, 16].

In this article, we introduce a quantitative procedure for obtaining statistically interpretable results from colored Mapper graphs and apply it to data from cardiac resynchronization studies reported in [17, 18]. Mapper and related TDA-based approaches have previously been used in cardiovascular applications, for instance for heart-disease classification and risk stratification after acute myocardial infarction [19, 20]. However, the aim of the present work is different: we use Mapper to represent acute hemodynamic responses to cardiac resynchronization therapy and quantify how predefined experimental labels are organized across the resulting graphs. Our approach is related to graph-theoretic notions such as assortative mixing, which describes the tendency of connected nodes to share categorical attributes [21–23]. The indices proposed here are adapted to the specific structure of Mapper graphs, where each node represents a cluster of data points and may contain a mixture of experimental labels. Our main methodological contribution is the use of label-aware quantitative indices on colored Mapper graphs to compare treatment-response patterns associated with clinically relevant labels. In the present study, these labels encode the pacing site and the stimulation modality (endocardial or epicardial).

The main objective of the study reported in [17] was to compare the effects of biventricular pacing from left ventricular epicardium versus the endocardium in a swine model of pacing-induced non-ischemic cardiomyopathy. The results presented here are biologically plausible and consistent with those obtained in [17], achieving greater statistical power. Thus, our work provides a 3R enhancement (Replacement, Reduction, and Refinement), in the framework of animal research, of the experimental approach used in [17]. Specifically, integration of several configurations of the cardiac resynchronization device yields an expanded dataset, facilitating the attainment of statistical significance through our topology-based method.

The paper is organized as follows. Section 2 describes the dataset, reviews the Mapper algorithm, and introduces the quantitative indices proposed in this work for the analysis of colored Mapper graphs. Section 3 presents the results obtained with these indices, including comparisons of different left ventricular pacing locations, endocardial versus epicardial stimulation, and the assessment of potential outlier effects. Finally, Section 4 discusses a biological interpretation of the results, methodological implications of the proposed indices, and some limitations of the study.

Dataset

This study was performed using data from [17], where six female domestic swine with pacing-induced non-ischemic cardiomyopathy were submitted to endocardial or epicardial biventricular pacing. Different electrodes were used to pace the right atrium, the right ventricle, and the left ventricle epicardium or endocardium. With the aim of comparing the effect of the pacing site, electrodes were placed at three basal regions (anterior, lateral and posterior), three mid regions (anterior, lateral and posterior) and two apical regions (anterior and lateral), both from the epicardium and from the endocardium. In addition, at each site, six different configurations of the resynchronization therapy (CRT) device were performed (two atrioventricular delays: 80 and 110 ms and three interventricular intervals: 0, 30, and −30 ms). Electrocardiograms and intra-cavitary pressure probes were registered and signals were digitized for subsequent analysis. Values obtained after biventricular pacing were compared with baseline values obtained during previous dyssynchronous pacing, and differences were expressed as percentages of change.

The hemodynamic variables considered in this study are left-ventricular peak pressure (LVP), the maximum and minimum time derivatives of LVP (DPDT+ and DPDT–), and mean aortic blood flow (ABF). However, the minimum derivative of LVP (denoted DPDT–) reflects different physiological features compared to the other three variables. For this reason, we repeated our analyses both including and excluding DPDT–, and compared the corresponding self-connectivity indices of Mapper graphs. We also assessed the effect of potential outliers by recalculating our results while excluding one individual at a time.

Thus, our study dataset is a point cloud consisting of 576 points in a 4-dimensional space, whose coordinates are percentages of change of the four study variables with respect to their baseline values. For each of the six study individuals (labeled as FIS06, FIS13, FIS14, FIS15, FIS18, and FIS19), the point cloud contains 36 basal points, 36 mid points and 24 apical points, half of which are endocardial and the rest are epicardial, obtained with six CRT device configurations.

Topological data analysis

The Mapper algorithm was introduced in [2] for qualitative analyses based on shape features of point clouds. Mapper is a combination of dimensionality reduction and clustering techniques, yielding graph representations of dataset structures. Its main ingredients are a filter function and a clustering procedure. The filter function f is defined on the dataset and takes values in the real numbers ℝ or in a multidimensional space ℝ d. The image of f is covered with a finite number of overlapping open sets { U i ∣ i ∈ I }, where the cardinality of I and the size of the covering sets U i are parameters to be chosen in the algorithm. For each i ∈ I, the subset f - 1 (U i) is split into clusters C i 1, …, C i n i. The output of the algorithm is an undirected graph whose set of nodes is in bijective correspondence with the set of clusters and there is an edge between nodes i r and j s if the subsets C i r and C j s have nonempty intersection. Hence the nodes of the graph are weighted by the number of data points contained in the corresponding cluster. In this study, we have chosen principal component analysis (PCA) as filter function —by projecting the data point cloud onto a variance-maximizing 2-dimensional parameter space— and we have used k -means as clustering algorithm. Since k -means includes a random seed, for reproducibility purposes the chosen random seeds are specified in each case.

It should be emphasized that Mapper graphs are purely relational structures. Thus, while adjacency is crucial, distances between nodes are meaningless and location of nodes in graph pictures is irrelevant. In this paper, Figures 1 and 2, as well as those in the Supplementary material, have been produced using the Python library NetworkX [24]. The graph layout algorithm also depends on a random seed. For easier reproducibility, all graph visualizations in this article were generated using layout random seed 3.

Although Mapper is a descriptive tool that yields graphs as outcomes, in this study we have defined three indices for quantitative analysis of colored Mapper graphs, as described in Subsection 2.2.1. We suppose that points in the dataset are labeled with a set of colors —basal points are red, mid points are yellow, and apical points are blue— and have customized Mapper (retrieved from [25]) so that nodes of graphs are also labeled with the same set of colors. Since each node of the graph corresponds to a cluster of points in the dataset, we assign to a node the dominant color among the points in the corresponding cluster (Figure 1). In case of ties, the dominant color among adjacent nodes is chosen, and if this does not break a tie then the dominant color in the graph is selected.

Numerical descriptors

Let G = (V, E) be a colored Mapper graph with colors { c 1, …, c k }, where V is its set of nodes (or vertices) and E is its set of edges. Then V = V (c 1)∪⋯∪ V (c k), where V (c i) denotes the set of nodes of color c i. A node v ∈ V (c i) may contain data points of several colors, with a majority of c i -colored points. The number of elements of V (c i) is denoted by N i in what follows.

Self-connectivity A node v ∈ V (c i) is called insular if there is no edge from v to a node w ∈ V (c j) with c j ≠ c i (see Figure 3). Nodes with no incident edges are also considered insular. For each i ∈{1, …, k }, let r i be the number of insular nodes of color c i. The self-connectivity index of color c i is defined as the ratio r i / N i between the number of c i -colored insular nodes and the total number of c i -colored nodes:

To assess whether the observed differences in self-connectivity between two colors are larger than expected by chance, we use a permutation test. If S ^ (c) denotes the empirical self-connectivity index associated with color c, then, for two colors c 1 and c 2, the observed test statistic is defined as

Since the Mapper graph is constructed independently of the experimental labels, these labels are exchangeable under the null hypothesis that the true self-connectivity indices S (c 1) and S (c 2) are equal, and labels can therefore be permuted while keeping the graph fixed. We generate a null distribution by randomly permuting the color labels of the points, recoloring the Mapper graph according to the permuted point labels, and recomputing self-connectivity values. For each permutation b = 1, …, B, we compute

where S ^ b (c) is the estimated self-connectivity index obtained after the b -th permutation. The two-sided empirical p -value is then computed as

This procedure preserves the Mapper graph structure and the number of points assigned to each color exactly, while testing whether the observed organization of colors on the graph is compatible with random relabeling.

In graph theory there is a similar measure, called homophily index, which is a ratio of the probability of linkages between nodes belonging to different groups by the probability of linkages between nodes within the same group [23]. Network homophily (or attribute assortativity) refers to the phenomenon that nodes tend to connect with similar nodes; it has alternatively been defined as the fraction of edges that connect nodes with the same labels [26].

Homogeneity For each node v ∈ V (c i) and j = {1, …, k }, let m j be the number of c j -colored data points within v. The ratio

is the proportion of c i -colored points within the node v, where k is the number of colors. The homogeneity index of a colored Mapper graph G = (V, E) for color c i, which we denote by h i (G), is the average value of h i (v) over all nodes v ∈ V (c i). Hence, for each color c i, the homogeneity index is the average percentage of c i -colored data points within c i -colored nodes. This indicator is used to detect a possible bias due to an unbalanced distribution of colors within the nodes of the graph. In the context of neural network analysis, there is a similar notion called node-wise purity, which is defined for each node i as 1/ c i, where c i is the number of class labels in node i; see [6].

Scattering For each color c i, the scattering index of a colored Mapper graph G = (V, E) is defined as

where D i is the number of c i -colored data points. For each color c i, the scattering index is the ratio between the number of c i -colored nodes and the total number of nodes, divided by the ratio between the number of c i -colored points in the dataset and the total number of points in the dataset:

This ratio measures the dominance of color c i in the collection of clusters obtained within the Mapper algorithm. Hence, a high scattering index value (greater than 1) for color c i indicates that the proportion of c i -colored graph nodes among all nodes is higher than expected based on the distribution of color c i in the original point cloud. The most plausible interpretation is that c i -colored data points tend to be more scattered throughout the dataset, thus forming smaller clusters, compared to other colors.

Interactive visualization

A relevant feature of Mapper graphs that makes them especially useful as analytical devices is the possibility of enriching them with metadata that can be consulted interactively. The graphs depicted in this article can be navigated through the following link: https://ainafem.github.io/mapper_cardiac_analysis. Information given on nodes includes cluster labels and the number of data points of each color contained in the corresponding cluster.

Comparison between left ventricle locations

Figure 1 shows the joint distribution of percentage changes of four hemodynamic variables. Graph nodes correspond to clusters of points in 4-dimensional space (LVP, DPDT+, DPDT–, ABF), and each node is colored with the dominant color among the points in the corresponding cluster. Graph edges indicate overlapping between neighboring clusters. Red-colored nodes, corresponding to study individuals paced at the basal site, are more scattered than yellow-colored nodes (mid site) and blue-colored (apical site) nodes. Moreover, the self-connectivity index is higher for red-colored nodes, indicating a higher proportion of insular nodes with respect to the total number of red-colored nodes.

Values of self-connectivity, scattering and homogeneity indices obtained from the graph in Figure 1 are shown in Table 1, where self-connectivity and scattering are seen to be higher for the basal site, while node homogeneity is similar for all sites. Corresponding p -values of comparison tests for self-connectivity indices are shown in Table 2.

The p -values in Table 2 indicate a significant distinction between pacing at basal locations versus mid or apical locations, with a higher self-connectivity index value in case of the basal location. The scattering index is also much higher for the basal region, suggesting that basal points are more dispersed in the original dataset, while homogeneity values indicate that there is no relevant bias attributable to an uneven distribution of colors across graph nodes.

The clustering step in Mapper was performed using k -means, which depends on a random initialization. To assess whether this stochastic component affected the results, we carried out a sensitivity analysis with respect to the random seed. Specifically, we recomputed the Mapper graph using 12 different random initializations, while keeping all other Mapper parameters fixed, and recorded the corresponding quantitative indices. Supplementary Table S1 reports the mean and standard deviation of these indices across the 12 runs. The observed variability is small, indicating that the conclusions are robust with respect to the k -means initialization.

Since the minimum rising rate of left-ventricular peak pressure (DPDT–) reflects different physiological features than the other study variables, we repeated all calculations using only LVP, DPDT+ and ABF, for the sake of comparison. The results are shown in Supplementary Tables S3, S4, and lead to similar conclusions as Tables 1 and 2.

Endocardial location versus epicardial location

The effect of endocardial pacing was compared with the effect of epicardial pacing by drawing new Mapper graphs and computing the corresponding numerical indices. Although the dataset is the same, there are only two colors now (red: epicardial, blue: endocardial), and therefore the coloring of nodes in the resulting graphs is different, as shown in Figure 2. In this graph, self-connectivity is higher in blue-colored nodes, suggesting an increased departure from average values in the case of endocardial pacing.

A significant difference (0.60 versus 0.29, p < 0.05) was found in a comparison test for self-connectivity indices in endocardial sites versus epicardial sites. Scattering is also higher for endocardial sites (1.25 versus 0.75), while homogeneity is similar in both sites. The corresponding summaries and p -values of comparison of self-connectivity indices are given in Tables 3 and 4.

A stochastic sensitivity analysis was also performed for the comparison between endocardial and epicardial stimulation. As before, the Mapper parameters were kept fixed and the graph was recomputed using 12 different random initializations of k -means. Supplementary Table S2 reports the corresponding mean and standard deviation values, showing that the quantitative indices remain stable under changes in the random seed.

In order to assess the importance of anterior regions versus lateral or posterior regions at each location in the distinction between endocardial and epicardial sites, self-connectivity indices were calculated by grouping regions (anterior, lateral and posterior) in all possible combinations. This was first done for all locations together (basal, mid and apical), and then grouping only basal and mid zones to estimate the relative contribution of the apical zone. Corresponding p -values are shown in Table 5. These results suggest that the lateral region consistently discriminates between endocardial and epicardial pacing, while the posterior region shows lowest discrimination capacity.

Seeking for further evidence of the distinction between endocardial pacing and epicardial pacing, Mapper graphs were generated separately for each subsample (endocardial and epicardial) of the study dataset. The Mapper graph corresponding to epicardial points is shown in Supplementary Figure S1, and the one corresponding to endocardial points is shown in Supplementary Figure S2. The results of comparison of self-connectivity indices of basal versus mid and basal versus apical are shown in Table 6 for each of the two graphs (Supplementary Figures S1, S2). Statistical significance is consistently higher in the case of endocardial pacing.

Hence, our results support the claim that the self-connectivity index of colored Mapper graphs is a valid numerical indicator for statistical comparisons.

Effect of outliers

Outlier individuals could potentially be influencing the above results. To account for this effect, p -values were computed anew by excluding each individual, one at a time. Results are shown in Supplementary Table S9 and compared with the corresponding results by excluding minimum rising rate of left-ventricular peak pressure (DPDT–) from the analyses (Supplementary Table S10).

When taking into account intra-individual variability, it should be reported that one individual, labeled as FIS14, was treated with norepinephrine during the original study [17]. However, no anomalies were detected involving FIS14. While the results in Supplementary Tables S9, S10 tend to be consistent, variability in statistical significance arises from the inherent randomness of the Mapper algorithm and the small sample size.

Mapper parameter sensitivity

The Mapper algorithm is known to be sensitive to the choice of parameters, in particular to the resolution and overlap of the cover and to the clustering procedure [10, 11]. To assess the effect of this sensitivity in our study, we recomputed the Mapper graph using all four hemodynamic variables under a range of parameter choices, as follows:

Number of intervals in the cover, from 5 to 8.

Percentage of overlap, taking values 25%, 35%, 40%, and 50%.

Number of clusters in the k -means step, from 2 to 4.

For each combination of parameters, we obtained Mapper graphs colored by heart stimulation region, and computed the corresponding quantitative indices. Supplementary Table S11 summarizes the overall variability of the indices across all parameter combinations.

In the main analysis, stimulation at the the basal region showed the clearest distinction with respect to mid and apical stimulation, as expressed by its larger self-connectivity index. Our subsequent parameter sensitivity analysis shows that this qualitative behavior is consistent across the range of parameters considered. In 47 out of 48 parameter combinations, the self-connectivity index of the basal region was larger than that of the apical region. Moreover, for all parameter combinations, the self-connectivity index of the basal region was larger than that of the mid region. Although the statistical significance of the pairwise comparisons varies with the parameter choice, the direction of the effect is highly consistent. This indicates that the main conclusion —namely, that basal stimulation exhibits a distinctive organization in the Mapper graph— is robust with respect to the Mapper parameters considered. This behavior is particularly clear for the comparison between basal and mid stimulation, as shown in Figure 4.

We also produced Mapper graphs colored according to endocardial or epicardial stimulation location. Supplementary Table S12 summarizes the variability of self-connectivity indices across all parameter combinations. Consistently with the main results, the self-connectivity index was larger for endocardial than for epicardial points in 42 out of 48 parameter combinations. Although the permutation tests did not identify this comparison as significant as uniformly as in the heart-region analysis, the direction of the effect was preserved across most parameter choices. In addition, Supplementary Figure S3 depicts the range of variation of index differences and corresponding p -values across parameter combinations.

This consistent tendency motivated a further analysis of whether endocardial or epicardial stimulation affects the organization of the Mapper graph when it is colored by heart region. The additional analyses confirmed that the distinction between the basal region and the other heart locations is better preserved when only endocardial points are considered. In particular, the difference in self-connectivity between the basal and mid regions was statistically significant in 5 out of 48 parameter combinations for the epicardial subset, compared with 29 out of 48 parameter combinations for the endocardial subset.

In summary, this work demonstrates that the distribution of hemodynamic variables within basal points in the dataset is significantly different from the distribution within mid points and apical points. The distinction is increased in the case of endocardial stimulation.

Topological data analysis

The numerical Mapper graph indices introduced in this study are novel contributions and have served as key descriptors in obtaining the results presented in the article. Higher self-connectivity values indicate that clusters dominated by a given heart region tend to connect among themselves, with few or no links to clusters dominated by other regions. This suggests that the corresponding hemodynamic responses occupy a relatively distinct part of the data space, being more similar to one another than to responses elicited from other regions. From a physiological perspective, this is consistent with a region-dependent response to pacing, distinguishing that heart region from the others. Higher values of the scattering index indicate that observations from a given heart region are partitioned into a relatively large number of clusters, suggesting that their responses are more dispersed across the data space rather than concentrated in a few compact groups. This is consistent with greater heterogeneity of the hemodynamic response to pacing within that region, with different pacing configurations producing more diverse response patterns. Finally, high homogeneity values indicate that the dominant color assigned to each Mapper node closely reflects the composition of the underlying data points, suggesting that the observed differences are unlikely to arise from bias due to mixed-color clusters.

Since Mapper is a descriptive tool for datasets, our conclusions primarily highlight distinctions. What can be asserted from our results is that the response to pacing is significantly dependent on the stimulation site, with endocardial pacing having a greater impact overall. Indeed, as previously shown in ([17], Table 3), increases in DPDT+ values are greater when pacing is applied at basal sites, which is associated with more favorable hemodynamic responses (see [17], Table 2). The observation that basal pacing may increase the likelihood of a positive response to cardiac resynchronization therapy more effectively than mid or apical pacing has also been reported in other studies [27, 28]. In ([17], Table 3), it is also shown that endocardial pacing tends to produce a stronger effect on DPDT+ in basal regions compared to epicardial pacing (16.2%±4.1 vs. 11.9%±2.1), although the differences are not conclusive. Our finding that the contrast between endocardial and epicardial stimulation is mostly localized in lateral or anterior regions could explain why the difference was not found to be statistically significant in [17].

Overall, our analysis of sensitivity of results with respect to Mapper parameters supports the qualitative conclusions of the main study. It is well-known that Mapper graphs are not stable under parameter variation [10, 11]. Rather than seeking parameter stability, one should tune the Mapper parameters for each analysis to obtain graphs that optimally discriminate the variables of interest. The main tendencies observed in the manuscript, which were obtained with a fixed initial selection of Mapper parameters (PCA filter, a cover with 7 intervals and 40% overlap, and k -means clustering with k = 3 and random seed 42) were preserved across a broad range of parameter values. This indicates that the reported effects reflect consistent structure in the data within the parameter ranges considered.

Physiological implications

The singular behavior and more favorable hemodynamic response observed during basal pacing likely stem from its ability to recruit a larger myocardial mass more efficiently [29]. This alters the mechanical activation sequence in a manner that optimizes stroke volume within this pacing-induced non-ischemic cardiomyopathy model. Furthermore, the data reveals that the topological differences between epicardial and endocardial capture are most pronounced in the lateral left ventricular wall. Physiologically, the lateral wall is typically the latest to activate in dyssynchronous failing hearts [30]. Pacing at this specific site maximizes resynchronization, which consequently magnifies the hemodynamic contrast between stimulation methods. Finally, the pronounced separability and stronger effect on DPDT+ observed with endocardial stimulation can be attributed to its alignment with the natural transmural activation sequence (endocardium to epicardium). Endocardial pacing engages more rapidly the fast-conducting Purkinje network, yielding a more rapid, physiological, and topologically distinct hemodynamic profile compared to the delayed, artificial transmural wavefronts generated by standard epicardial pacing. Importantly, these localized physiological differences were not detected using the traditional statistical framework applied in the original experimental study [17]. The standard statistic methods used in the original study such as ANOVA aggregate data by evaluating central tendencies across broad, predefined groups (e.g., all endocardial versus all epicardial sites), often treating hemodynamic variables in isolation. This conventional approach inadvertently averaged out subtle, nonlinear regional variations. In contrast, the Mapper algorithm evaluates the complete hemodynamic profile simultaneously as a high-dimensional point cloud. Mapper successfully isolates distinct phenotypic clusters by preserving local topological structures and neighborhood relationships, such as the magnified contrast specifically in the lateral wall, that remain mathematically obscured by standard univariate hypothesis testing.

Clinical implications and translational challenges

The application of topological data analysis to study cardiac hemodynamics offers promising avenues for precision medicine in cardiac resynchronization therapy (CRT). Clinically, predicting CRT response remains a significant challenge, as standard univariate metrics often fail to capture the complex, multidimensional nature of a failing heart's mechanics [31, 32]. Our findings suggest that tools like the Mapper algorithm could eventually be utilized for advanced patient stratification. By processing high-dimensional intraoperative or pre-procedural data, such as multi-site hemodynamics and electrical activation delays, TDA could help identify specific phenotypic clusters of patients. This stratification could proactively identify individuals who require targeted left ventricular endocardial pacing to achieve optimal resynchronization, rather than relying on a standardized trial-and-error approach.

Limitations

While the Mapper algorithm successfully captures the global topological structure of the dataset, this study has notable limitations. The study population comprises only six swine, subjected to multiple pacing conditions and repeated measurements. Consequently, the statistical power, robustness, and generalizability of the findings are limited accordingly. The conclusions drawn from our numerical analyses are exploratory in nature, attempting to demonstrate the usefulness of the graph indices used for comparisons in this work. Furthermore, the reliance on a constrained number of individuals entails a risk of overfitting topological graph structures to this specific sample, resulting in limited external validity. Therefore, further validation in larger, independent cohorts is required. Additional numerical analyses are also needed using quantitatively enriched Mapper graphing techniques, aiming for enhanced overall consistency, possibly by selecting different hemodynamic variables or adjusting the parameters of the Mapper algorithm with other criteria.