Biological test system and image acquisition

In order to show the applicability of 2CALM, we designed an artificial clot [53]. After mixing thrombin, fibrinogen and platelets, the cells coagulate on a glass slide, which yields a viscous 3D clot. For visualisation purposes, we immuno-stained platelets within the protein matrix with fluorescently labelled antibodies that target CD41 and CD62p proteins. In all clot experiments, CD41 was labelled with Alexa488-conjugated anti-CD41-antibody, and CD62p was labelled with Alexa647-conjugated anti-CD62p-antibody. For imaging, we applied direct stochastic optical reconstruction microscopy (dSTORM) [54]. We used an oxygen scavenger system (OxEA) [55], which allows for two-colour imaging without buffer exchange. To adjust the blinking rates of both fluorophores, an additional UV-laser illumination was utilized. A cylindrical lens in the optical detection pathway of the microscope introduced astigmatism and an axially dependent deformation of the point spread function (PSF) of individual emitters. The single-molecule positions were determined using customized software with fitting routines derived from rapidSTORM [56]. Automated tools for extraction, characterization and comparison of the protein distributions at the nanoscopic level were implemented.

Software features: Dataset pre-processing module

2CALM includes a collection of software tools enabling a pairwise comparative analysis of independent sets of localisation microscopy data. Datasets usually have a different number of localisation events (often gathered into 3D point-clouds), varying noise due to unspecific binding and/or localisation errors resulting from sample drift during the measurement. Our toolset enables noise filtration and drift correction. Typically, special convolution algorithms [57, 58] or fiducial markers [59, 60] are deployed to correct for sample drift; we expanded a developed drift correction algorithm [32] to support 3D position accuracy of localisation events [61]. The 3D dSTORM images contain single localisation events which are not of interest in cluster analysis and henceforth are classified as outlier points. Filtering of individual outlier points (or small group of points) is carried out using a DBSCAN (Density-Based Spatial Clustering of Applications with Noise) algorithm [62–65]. An additional modified version of DBSCAN [66] allows for an automatic determination of the ROI. It is used for spatially well-separated domains with single-molecule localisations (e.g. signals in sparsely distributed individual cells) (S11 Fig). Both tools–drift correction and outlier filtering–are combined in a dataset pre-processing module.

Software features: Equalization of data sets (bootstrapping)

Although an analysis of all points in a sample at once is possible, it is rather time-consuming and less robust to errors. When datasets have a small number of localisation events (up to 50 000), a 1^st-level statistical analysis–after equalization of both samples–can be performed. The number of points is equalized to the smaller of the pairwise compared data sets. Representative datasets are shown in S1A and S1B Fig. When larger differences in localisation events are detected, errors due to a random selection of points from the equalized sample dataset may lead to an incorrect comparison, even though the samples originate from the same biological replica. In such instances bootstrapping is used to randomly draw events from the measured datasets, which yields an equalized data set. In order to improve the robustness of the analysis, we performed multiple re-sampling (bootstrapping) [67–69] using a defined number of points. The bootstrap procedure involves choosing random samples (with replacement) from a large dataset and analysing each bootstrap sample in a similar way. Bootstrapping reduces the risk of accidental one-time errors (when there are not sufficient points in a subgroup) and allows for better parameter-estimation for subsequent comparative analyses.

Software features: Statistical analysis of cluster features (Two-level statistics)

The 1^st -level analysis relies on a direct comparison of sample features like cluster density and cluster curvature distributions for each cluster dimension. For a comparison between the protein distributions of the two samples, non-parametric statistical tests are used. The 2^nd-level analysis combines the parameters obtained from the 1^st-level analysis and compares them globally. It relies on a comparison of averaged values of localisation densities and curvatures per cluster dimension and eliminates the influence of noise (disturbance of point localisations). Additional K-/H-Ripley’s tests [70, 71] show the deviation of the distributions from a random Poisson distribution.

Software features: Classification of similarity (Two-sample comparison)

Statistical Tools.

Various statistical tests are implemented for a holistic two-sample comparison at nanoscopic and microscopic level. Non-parametric Kolmogorov-Smirnov (KS) [72–74] and Wilcoxon (WX) [75, 76] tests were used for a pairwise comparison of cluster features. In order to calculate the aggregated p-value (e.g. fusion of results of KS and WX tests) we applied the weighted t-norm-AND operator [77, 78] on the p-values of the KS-/WX-test obtained from the bootstrapping process. By introducing appropriate weights for the t-norm-AND operators the significance of a particular test increases. These tests precisely identify the cluster parameters necessary for classification of sample similarity/dissimilarity for any clustering dimension. It is not capable to automatically classify data sets.

To simplify the comparison, we determined a measure for sample similarity combining all cluster dimensions. We defined two similarity measures, sim_M and sim_L. sim_M is the rescaled aggregated p-value with an interval of [0,1]. If sim_M is < 0.5, the parameters describing clusters are dissimilar (see Fig 2C zoom-in). Sim_L measures the dependency of the critical area (spanning the cluster dimension/sizes interval and the significance level) and lower bound of the confidence interval of the aggregated p-value and delivers a value within the interval of [0,1].

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Fig 2. Two sample comparison of the CD62p and CD41 distributions in clots.

(a) and (b) show reconstructed 3D dSTORM images (approx. 100 000 data points/image). (a) shows the 3D distribution of CD41 (Alexa488-antibody). (b) represents the 3D distribution of CD62p (Alexa647-antibody). (c) shows the comparison of the cluster densities for all given cluster dimensions between the two datasets (1^st-level comparison). The blue/red lines depict the KS- and WX-test results, respectively. The aggregated p-value for KS and WX tests (dashed black line) remains within the critical p-value-area (orange) disproving the similarity hypothesis. The zoomed-in areas depict: a cluster dimension (5 nm—200 nm, sim_M, sim_L below 0.5); pairwise KS- and WX-test comparisons of cluster densities for both samples indicate dissimilarity (left) and a second cluster dimension (800 nm—1000 nm, sim_M, sim_L larger 0.5); pairwise KS- and WX-test comparisons of cluster densities for both samples show similarity (right). (d) shows the 2^nd-level comparison of the mean cluster density for all given cluster dimensions. The mean-cross p-value comparison (blue) and lower/upper confidential bounds (grey dashed line) are shown. (e) and (f) represents 600 clusters localized within the CD41/CD62p distribution respectively, displayed using the Delaunay-triangulation method (clustering dimension 390 nm). (g) shows two representative clusters from from (e) and (h) from (f).

https://doi.org/10.1371/journal.pcbi.1007902.g002

Machine learning

In order to combine all features necessary for the similarity classification, we use a machine learning method, a MLP neural network, as a classifier of sample similarity [71, 79–81] (Fig 1B). The neural net consists of three layers: an input layer, a hidden layer and an output layer. Except for the input nodes, each neuron uses a nonlinear activation function. The network requires the preparation of input data for characterization; and it applies the previously determined statistic-based features of the dataset in the training and classification mode.

We trained the MLP neural network with a data set of CD41/CD62p protein distributions in an artificial clot. Approximately 262 000 training patterns were determined using this data set. The trained MLP neural network classifier was tested on ~40 different pairs of data sets, which were not included in the training set. All of the tests showed a similarity classification with a probability between 0.75 and 0.95 for a pairwise comparison of similar samples and <0.2 for a pairwise comparison of dissimilar samples (S2 Table), i.e. the parameter extraction presented in the analysis sets is a suitable base for machine learning based pattern recognition. The results on MLP neural network classification for a representative statistical comparison of sample pairs are presented in Fig 3G. The overall values of further data sets are show in the S2 Table.

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Fig 3. Comparison of IL-1β treated and untreated clot samples.

(a) and (b) show reconstructed dSTORM images of two clots, untreated (a) and treated with IL-1β (b). In (c) zoomed images of two regions extracted from (a) and (b) are shown. In (d) two populations of clusters from both images with a cluster dimension of 145 nm are visualised. (e) shows the comparison of cluster densities for all given cluster dimensions between the two datasets (1^st-level statistic). Blue and red lines depict the results of the KS- and WX-test, respectively [75, 88]. The aggregated p-value between KS and WX tests (black dashed line) remains below the critical p-value area (orange bar) and thus proves the dissimilarity hypothesis for most of the cluster dimensions. (f) shows the aggregated p-values of averaged density and curvature distribution, determined via mean-cross analysis (blue) and the average p-value (green, including confidential intervals: black dashed lines) (sim_M = 0.07 and sim_L = 0.06). The table in (g) shows the classification table with MLP neural network classification values for dissimilar and similar samples. The output values of the trained network can be interpreted as a posteriori probability of similarity hypotheses.

https://doi.org/10.1371/journal.pcbi.1007902.g003

Proof-of-principle: Comparison of CD41 and CD62p clusters

In Fig 2, we show a typical reconstructed image of two areas in the central region of the clot, which visualises the distribution of CD62p (Fig 2A) and CD41 (Fig 2B) upon platelet activation (~ 100 000 single-molecule localisations each). The reconstructed images have been compared using the developed statistical tools, proving a dissimilarity of both datasets at a very detailed level.

1^st-level statistics: KS/WX tests of cluster density and curvature

For each pair of data sets (or regions) hierarchical spatial clustering is performed [49–52]; i.e. we performed a 1^st-level analysis for the relative density (Fig 2C) and curvature [82] (comparison data: S1A Fig) for all cluster dimensions of a given interval (typically from 5 nm to 1000 nm). Samples are typically equalized via the bootstrap method (100 curves for relative density and curvature of 100 re-samples all including 8000 points). Based on KS and WX tests, a direct comparison of the clusters density distribution and curvature distribution is performed (Fig 2C and S1D Fig). The determined p-values disprove the hypothesis of both samples being dissimilar. The aggregated p-value between KS-(blue) and WX-(red) tests = 0.04, and the similarity measures sim_M = 0.39 and sim_L = 0.45 indicate a difference in the cluster density distributions for all clustering dimensions between the two samples. The detailed density comparison proves that for clustering dimensions larger than 700 nm both samples have a significant similarity (Fig 2C). Further 1^st-level analysis results on cluster curvatures (shape) [82] are presented in S1A–S1D Fig. S1C Fig represents the aggregated data of KS-/WX-test results on the characterization of cluster curvature for all clustering dimensions. The detailed curvature analysis indicates a similarity for clustering dimensions larger than 500 nm up to 1000 nm.

For the combination of the results of various tests, the calculated p-values of the individual ones are aggregated using the t-norm-AND operator (AND operation in fuzzy logic). For the similarity measure, the p-values are further transformed into sim_M and sim_L values by averaging the aggregated p-values and taking into account their confidence intervals. The process of hierarchical aggregation of individual p-values as shown in S2 Fig.

In conclusion, the results of averaging the KS-/WX-test values for cluster densities and curvatures of the bootstrapped re-samples indicate a strong dissimilarity of the samples for the maximal clustering dimension of an interval of 5 nm-1000 nm.

2^nd-level statistics: Mean-cross comparison of cluster density and curvature distribution

2^nd-level statistics were performed solely for the bootstrap derived re-samples (Fig 2D and S1E–S1I Fig). This relies on a mean-cross p-value obtained from cross-comparison tests (KS/WX) of the averaged cluster density distributions between all bootstrap resamples. Fig 2D shows the 2^nd-level comparison of the mean cluster density for all given cluster dimensions. The mean-cross comparison for curvature distributions is presented in S1E Fig with mean-cross p-values and lower/upper confidential bounds. The mean-cross p-value (KS-/WX-test) of the density distributions equals 0.049, and sim_M = 0.49 and sim_L = 0.30 underlines the nanoscopic dissimilarity of the two samples. Similar to the results on density analysis, we determined the behaviour of the cluster curvature (S1E Fig); a mean-cross p-value for curvature of 0.23 and sim_L = 0.34 indicates a dissimilarity between the two samples. Comparable to the results of the 1^st-level statistics, only for cluster dimensions larger than 700 nm the curvature as well as density distributions indicated a stronger similarity. The mean-cross p-values of the cluster curvature show a tendency towards sample similarity for cluster dimensions ~200 nm.

Additionally, the individual clusters can be visualised three-dimensionally using the Delaunay-triangulation method [83, 84] or as maximum radius spheres packed at given locations [85] (Fig 2G and 2H). Fig 2E and 2F show 600 clusters from both datasets (cluster dimension 390 nm). In Fig 2G and 2H two randomly chosen clusters have been depicted from e and f, respectively. The software feature for cluster visualisation provides detailed information on localisation of the clusters within the clot for each clustering dimension.

For large clustering dimensions, clusters cover large parts of a platelet (size 1–5 μm, see S11E–S11G Fig) and can therefore be regarded as ‘bulk signal’ by looking at micrometer-sized clusters in the nanoscopic scale regime. Hence, the results on a nanoscopic and microscopic level for dataset comparison are provided simultaneously. Further analysis (S1F Fig) displays the aggregated p-values of a mean-cross analysis for the cluster densities and curvatures (for all cluster dimensions). These aggregated p-values remain below the boundary of the critical p-value for all clustering dimensions with a dimension smaller than 700 nm. The aggregated p-values of the mean-cross analysis for the cluster densities and curvatures again indicates a dissimilarity of the samples.

2^nd-level statistics: K-/H-Ripley’s functions

For further 2^nd-level statistics we performed a Ripley’s K-/H-analysis [70, 86, 87] (S1G–S1I Fig) on the bootstrap sample data. The K-/H-Ripley’s function values are determined for each clustering dimension and are used to additionally prove the diversity of the two samples. The comparison of the values from the KS-/WX-test applied on the 3D K-/H- Ripley’s function results for each of the cluster’s dimensions (shown in S1H Fig). The results show an average p-value of 0.087, sim_M = 0.71 and sim_L = 0.14 for the K-Ripley function (left) and an average p-value of 0.086, sim_M = 0.71 and sim_L = 0.22 for the H-Ripley function (right). Additionally, the mean-cross analysis of the K-Ripley function distributions represented in S1H Fig are shown in S1I Fig. The overall Ripley analysis (sim_L) confirms the sample dissimilarity except for a small interval of the cluster dimension (400 nm– 600 nm). For these samples (Fig 1), the comparison based on the Ripley functions confirms the analysis performed with the clustering methods.

1^st-level and 2^nd-level statistics: MLP neural network

A multilevel analysis yields a general dissimilarity between CD41 and CD62p spatial distributions within a clot for the presented data sets. Additional data sets and analysis results are shown in S3–S8 Figs. We present in detail a pairwise comparison of CD41 distributions and CD62p distributions in distinct clots. In contrast to the sample presented in Fig 2, some CD41 and CD62p distributions in clots cannot be unambiguously discriminated against by individual statistical tests. Therefore, multilevel cross-testing is explicitly required. In particular, the data presented in S7 Fig and S8 Fig show divergent results regarding cluster comparison. For these technical replicas, a significant difference between the results of the 1^st-/2^nd-level cluster-based analysis and the results on K-/H- Ripley’s analysis for all cluster distances is observable (S8 Fig (cluster analysis) and S8F–S8I Fig (Ripley analysis). The K-/H- Ripley’s analysis indicates a strong similarity for clusters larger than 400 nm, which is not the case for all other 1^st-/2^nd-level cluster-based statistics. The calculation of the mean-cross p-value for all the parameters rejects the null hypothesis of similarity, showing that both datasets are dissimilar (S8H Fig). All statistical p-values are represented in S1 Table. The data comparison confirmed a significant dissimilarity for cluster dimensions. First level analysis shows a dissimilarity (sim_M = 0.006 and sim_L = 0.5 for density and curvature comparison, respectively). A higher similarity between the curvatures occurs only for cluster < 150 nm. Second level analysis indicates a general dissimilarity (aggregated sim_M values are 0.6 and 0.3 for density/curvature comparison, respectively). The mean-cross comparison of all results indicates a strong dissimilarity (sim_M = 0.3). The Ripley’s K-/H-functions show a higher similarity sim_M = 0.67/0.69. The MLP neural network indicates a 0.74 probability for dissimilarity. A comparison of a clot sample (CD41 labelled) and simulated data is presented in S13 Fig. In the last analytical step the MLP neural network was applied for a comparison of the protein distributions of CD41 and CD62p in the platelet clots. The MLP neural network analysis showed that the samples were classified as similar with a posteriori probability of ~0.95 for a pairwise comparison of similar data sets and below 0.2 for a pairwise comparison of dissimilar ones. The a posteriori probabilities of the similarity hypotheses are depicted in S1 Table. The trained MLP neural network clearly discriminates between the two clustered protein populations under investigation within clots.

Characterization of CD62p clusters after platelet activation by the cytokine IL-1β

To quantify the effect of IL-1β on platelet activation, our software toolbox has been used to analyse dSTORM data of CD62p secretion. Various cytokine-treated and untreated samples were investigated. Due to IL-1β treatment, heterogeneities in clot formation (more sparsely distributed platelets) are observable. In order to capture the best overall picture of CD62p membrane incorporation after IL-1β treatment, images with varying cell densities were compared. We observed that the numbers of clusters changes significantly depending on clustering dimension. Herein, for cluster dimensions of 80 nm, 3620 and 9106 clusters were detected in total, whereas for cluster dimensions of 145 nm, 1849 and 5205 clusters were detected in treated and untreated samples, respectively. Furthermore, data obtained from sparse and dense clot regions were analysed. For a comparison of regions with sparsely distributed platelets, the best results were obtained for segmented images. The DBSCAN-derived segmentation is crucial for 2CALM analysis, especially in cases of sparsely distributed platelets; the unspecific signal outside the regions of interest exerts an influence on direct cluster comparison (for full image comparison). A detailed description of the differences in the comparison between segmented and full images is shown in Fig 3 and S9 Fig. Fig 3A–3D depict overviews and randomly chosen regions (image segmentation) for analysis in a cytokine-treated and untreated clot, respectively. Fig 3E depicts a direct 1^st-level comparison of the clusters density distribution (curvature distribution is shown in Supplementary S6B Fig) in the chosen regions (S9A Fig). As can be seen in Fig 3E, untreated and treated samples gain similarity for larger cluster dimensions (> 800 nm). This difference was observed reproducibly for large as well as small extracted ROIs.

The 1^st-level analysis results of cluster density and curvature indicates that both samples are in general dissimilar, whereas for clustering dimensions larger than 800 nm, both samples show a weak similarity. Overall, the sim_M- and sim_L-values determined for the 1^st-level analysis are 0.46 and 0.31, respectively. With regards to cluster density, the comparison indicates that only for cluster dimensions larger than 800 nm a similarity of these two samples can be observed. The 2^nd-level analysis of curvature confirms the dissimilarity hypothesis (sim_M = 0.46 and sim_L = 0.45) (S9B Fig). The mean-cross comparison of the aggregated KS-/WX-test values for density and curvature also shows a dissimilarity (Fig 3F).

The comparison of the averaged cluster curvatures tends to gain similarity for clustering dimensions larger than 500 nm (S9B Fig). The mean-cross analysis of the combined statistical data on 1^st- and 2^nd-level cluster density and curvature comparison is shown in S9C Fig. Results from the K-/H-Ripley-analysis support the cluster comparison data S9D and S9E Fig (K-/H-Ripley’s-function (K-left, H-right) comparison for sim_M/sim_L are 0.03/0.01 and 0.04/0.01, respectively). The aggregation values of these data sets show similar tendencies (S9E Fig). A comparison of CD62p protein distribution and clusters for IL-1β treated and untreated samples verifies the effect of this cytokine on platelet activation in cluster formation. For clustering ranges between ten and a few hundred nanometres, the data sets differ most; this effect has not been observed in such detail before (see S2 Table).

The MLP neural network was used for a comparison of IL-1β treated and untreated clots. The neural network clearly classifies the samples based on the CD62p protein distribution. The tests showed a classification with a probability of 0.9 for a pairwise comparison of similar data sets and below 0.2 for a pairwise comparison of dissimilar ones. The a posteriori probability of similarity hypotheses are depicted in S2 Table. The results of the IL-1β treated samples are remarkable: In general, a larger heterogeneity within the group of treated samples in comparison to untreated samples can be observed (similarity probability = 0.67 for MLP comparison of cytokine-treated samples; similarity probability > 0.8 for MLP neural network comparison within the group of untreated samples, Fig 3G).

Equalization and bootstrap resampling

We use inferential statistics to examine the relationships between the features of two samples based on a series of smaller samples in order to generalize how those features will relate to the larger sample [67, 36, 68, 69, 98–103]. For analysis, we chose representative subsets of two samples, which will be referred to as the bootstrap-resamples. The bootstrap procedure involves choosing random samples (with replacement) from a large dataset and analysing each bootstrap sample in a similar way. Sampling with replacement means that each point is selected randomly from the original dataset. Thus, a particular point from the original dataset may appear multiple times in a given bootstrap sample. The number of total points included in a bootstrap sample (including data from both compared samples) is equal. Let N₁, N₂ be the number of points in both samples, respectively. We perform M times random resampling of each sample with the same number of points N<min(N₁,N₂). Default values for M = 100 and N is ~2000–10 000 points. These bootstrap-samples (bs¹(k),bs²(k))(k = 1,…,M) are pairwise stored for further analysis.

Spatial clustering via hierarchical clustering

Hierarchical clustering (also called Hierarchical Cluster Analysis (HCA)) is a common algorithm, which creates a hierarchy of clusters [49–52]. The agglomerative approach at the beginning declares each point as an individual cluster. Thus, pairs of clusters merge as one-element cluster moves up the hierarchy. Hierarchical clustering creates a cluster tree or dendrogram. This tree is a multi-level hierarchy with clusters of one level being joined to clusters in the next level. Grouping of the clusters into a tree connects pairs of clusters, which are close together by using a linkage function. The linkage function uses the distance information between points to determine the proximity of clusters relative to each other. Next, newly created clusters are grouped into larger clusters until a hierarchical tree is created. We apply a complete-linkage clustering function, which uses the maximum of the pairwise distances between points for clustering [51, 52].

For data partitioning, we cut the hierarchical tree into clusters with a given maximal cluster dimension/cluster size (maximal Euclidean distance between points inside a cluster); e.g. at the level of d = dim(i), where dim(i) is a vector of cluster dimensions, by pruning off branches from the bottom of the hierarchical tree, and assigning all the clusters below each cut to a single cluster.

The use of complete-linkage hierarchical clustering guarantees that agglomerated clusters have a dimension (size^Cluster) no greater than given dim(i). In the hierarchical clustering process, cluster sizes dim(i) are ordered sequentially from size_min to size_max with a constant Δ step. In order to find features that allow comparison between two samples, both samples are clustered sequentially with a given maximum cluster size d = dim(i), where dim(i) = size_min+i∙Δ, and i = 1,…,L and L∙Δ = size_max.

For linkage, the size_max for both clustered samples is assumed to be ~ 40% to 50% of the minimum dimension of both samples. The dimension of the sample can be defined as the minimum length of the 3D box, which includes all points of the sample. We set the size_min between 5 nm-10 nm and the step Δ to be 10 nm as default values.

Cluster parameter extraction

The platform allows extraction of 6 sample-independent parameters:

• Cluster density
• Cluster curvature
• Average cluster density
• Average cluster curvature
• Ripley’s K function
• Ripley’s H function

Let Cl(d) be the set of N(d) clusters for one of the samples (1 or 2) with the cluster maximum size d = dim(i) (for sake of clarity, we will omit the i index from now on).

Cluster density

For each cluster c_k (where k = 1…N(d)) from the set Cl(d) one can calculate its density and relative density as density_k = dens_k/density(sample), where n_k is the number of points in k^th-cluster from the set Cl(d), V_k is its volume and the is the density of the whole sample.

The density of the cluster slightly depends on the method of determining its volume. The platform allows one to use three different methods to determine the cluster volume:

1. Sum of the sphere volumes with the sphere centre in each point of the cluster and the radius equal to the half-distance to the nearest neighbour (called bullet-density, S10E Fig) [84].
2. Volume of convex hull spread upon cluster points (called hull-density, S10F Fig).
3. Volume estimated as a volume of a rectangular box including cluster points (called box-density, S10G Fig).

As default, we use the bullet-density that best reflects the shape of the 3D cluster.

Cluster curvature

The value of the curvature reflects the concave-convex degree of the cluster surface. We use the method presented by He et al and Williams and Shah [82, 104] to estimate the curvature of the cluster by analysing the covariance between all cluster points. For a 3D cluster, we determine the covariance matrix of each point in a cluster based on a centroid calculation.

Let pos_k = (x,y,z)_k be a matrix of the 3D point locations in a cluster c_k from the set Cl(d) and p_k be the centroid of these points. The pos_k is a c_k x 3 matrix. The 3D covariance matrix COV_k = ∑_j(pos_k(j)−p_k)∙(pos_k(j)−p_k)^T where j = 1,…,n_k is a semi-positive definite three-order symmetric matrix. Next, the three eigenvalues of the COV_k matrix λ₁, λ₂, λ₃ and its corresponding unit eigenvectors ev₁, ev₃, ev₃ are calculated. Let λ₁ ≤ λ₂ ≤ λ₃. Eigenvalue λ₁ describes the change of the value of the surface along the normal direction. The surface variation can be expressed as 𝜏_k = λ₁/ (λ₁+ λ₂+ λ₃). The curvature curv_k of the cluster c_k can be approximated as a surface variation 𝜏_k [104].

Both features, densities and curvatures of clusters, can be used for first level comparison or detailed level sample comparison. The relative density and curvature of clusters with a given dimension d can be considered as empirical distributions with an unknown probability density function (pdf). The empirical distributions of both samples can have a different number of elements (different number of clusters).

Based on these detailed distributions, we can also specify the global features of the clusters used later in the second level analysis. For each cluster size d = dim(i), the two additional sample features can be determined:

Average density of clusters: av_density(d) = mean(density_k|k = 1,…,N(d))

Average curvature of clusters: av_curv(d) = mean(curv_k|k = 1,…,N(d))

3D Ripley’s K-function and Ripley’s H-function

Sequential clustering is simultaneously used to calculate the value of the Ripley’s K-function for each sample [70, 86, 87]. Ripley's K-function is an intuitive approach for detection of deviations of general assumptions of point distributions within cluster samples. The analysis has a non-parametric character and is therefore the first step in the characterization of spatial point patterns. The K-function can be calculated for each size d = dim(i) of clusters between given size_min and size_max.

Let Dist(l,k) denote the Euclidean 3D-distance matrix between each pair of points in the sample. An extension of the K-function from 2D to 3D is obtained by assuming 3-dimensional distance measures.

Thus, the K-function for a given distance d = dim(i) is determined as (1) where V is the volume of the sample, N is the number of sample points, e_k(d) is the edge correction term for a sphere of radius d with k point as its center and I[∙] is the indicator function [70]. The expected value of the complete spatial randomness (CSR) is E[K(d)] = 4πd³/3 as well as the function [86, 87]: (2)

Finally, for each sample, six features can be determined. The sample is sequentially divided into clusters with dimensions dim(i) = size_min+i∙Δ. For each size d = dim(i) the sample is represented by:

• Vector density(d) of the relative density of clusters for size d
• Curvature vector curv(d) of clusters for size d
• Value av_density(d) of average density of clusters
• Value av_curv(d) of average curvature of clusters
• Value K(d) of the Ripley’s function
• Value H(d) of the Ripley’s function

Using H-Ripley as the crucial dimension, representative for the largest difference (clustering caused) between the sample and complete spatial randomness can be determined.

Sample comparison based on extracted parameters

Regardless of whether comparisons are carried out on full samples or selected ROIs, we derive the above-described features for each given d = dim(i). For every d, samples are compared using the results of the Ripley functions K₁, K₂ and H₁, H₂, density and curvature distributions density₁, density₂, curv₁, curv₂ clusters of samples 1 and 2, respectively. Cluster density distribution density and the distribution of cluster curvature curv are two random variables with an unknown probability density function.

1st-level analysis

For two cluster density distributions density₁ and density₂ originating from sample 1 and 2 the null hypothesis states that both distribution density₁ and density₂ belong to the same original distribution. To resolve the validity of this hypothesis at a given significance level α, we use two-sample Kolmogorov–Smirnov test [72–74].

Kolmogorov-Smirnov (KS) test compares empirical cumulative distribution functions based on their absolute differences. The null hypothesis is rejected at a significance level α in case the difference exceeds the critical value. The KS-test returns an asymptotic p-value as a scalar value in the range [0,1]. The p-value is the probability of observing a test statistic more extreme than the observed value under an assumption of a true null hypothesis. The asymptotic p-value becomes very accurate for large sample sizes and is reasonably accurate for sample sizes N₁ and N₂, such as (N₁ N₂) / (N₁ + N₂) ≥ 4. Therefore, one can accept the p-value pv(d) = testKS(density₁(d), density₂(d)) for each size d as one comparative variable.

The second null hypothesis assumes that both distributions density₁ and density₂ are taken from continuous distributions with equal medians. To resolve the validity of this hypothesis at a given significance level 𝛼, we can use the Wilcoxon rank-sum test (WX, Wilcoxon–Mann–Whitney test) [75, 76]. This non-parametric test is used to determine if two independent samples were selected from populations with equal distributions. The test assumes that the two samples are independent. A two-sided WX-test returns the p-value as a positive scalar in the range [0,1]. Data density₁ and density₂ can have different lengths.

Therefore, in addition to the p-value of the KS-Test, one can calculate the p- value pw(d) = ranksum(density₁(d), density₂(d)) for each size d of the WX-test as the second comparative variable.

After calculating the p-values for both tests, one needs to aggregate both variables pv and pw for each size d. We apply the weighted t-norm-AND operator, which is often used in AND-aggregation in fuzzy logic [77, 78]. The weighted t-norm-AND operator calculates aggregated p-values as , where parameter w is a weight between 0 and 1 and allows to assign more significance to one of the tests. Default w is ~0.5–0.6.

The same method is applied to the clusters curvature distributions curv₁ and curv₂ with the given dimension d. As result, we obtain curves representing p-values pv,pw,pd for relative density and pcv,pcw,pc for curvature of clusters depending on the cluster size d. Additionally, the distribution-free multivariate KS-test allows for a comparison of the pair of distribution (density₁, curv₁) with the pair of distribution (density₂, curv₂) [72].

2nd-level analysis

The bootstrap technique provides M pairs of bootstrap samples (bs¹(k),bs²(k)|k = 1,…,M). For each resampled pair, features are calculated based on the sequential clustering of both bs for each cluster size d = dim(i),i = 1…L. For extracted features of each pair of bs, the above-described first level statistic tests are performed and the results are stored in a L×M matrix. The rows of the matrix are adequate to the cluster sizes and the columns represent the individual bootstrap samples. The values of the Ripley’s K- and Ripley’s H-functions of the bootstrap results are assigned to the matrices KR and HR. The p-values of the KS-test and the p-values of WX-test for relative densities and curvature distribution in clusters are included in the matrices KD and KC, WD and WC, respectively. Additionally, the bootstrapping method stores the average values of relative clusters densities av_density and of the clusters curvatures av_curv into matrices DM and CM, for each dimension d of both samples.

Next, bootstrap confidence intervals are calculated. Having multiple realizations of random variables of density distribution and curvature distribution of clusters enables the estimation of confidence intervals for p-values generated by the KS- as well as the WX-test. Alternatively, for estimation of the standard parametric confidence intervals (of known distributions) we use the semiparametric or nonparametric methods with bootstrap estimates. Hence, the bootstrapping technique estimates the standard error deviation, which is used in a normal approximation of confidence intervals for significance level α (inaccurate estimation). For α = 0.05 the normal confidence interval is approximated by pd ± 1.96 * std(pd). Alternatively, nonparametric bootstrap percentile confidence intervals (BPCI) can be calculated to infer the observed significance level of the variable. The bootstrap distributions of the p-values for each cluster size can be sorted, and α and (1 –α) percentiles of the sorted empirical distribution form the limits for the BPCI [98, 99, 102].

The comparative analysis of both samples is carried out on two levels: detailed (first level statistic) and global (second level statistic). At the detailed level, we use in the KD, KC, WD and WC matrices stored p-values of the KS-test and p-values of the Wilcoxon rank-sum test for comparison. In cases when L = 1 (whole sample), it is not possible to determine confidence intervals and the analysis is based only on the pd and pc curves and their aggregation with the t-norm AND-operator.

Having many realizations of random variables of densities and curvatures (L > 1), it is possible to carry out cross tests between the realizations as well as between their average values and Ripley functions. In addition to the previously described tests, we can directly calculate p-values (empirical p-values) between average values of the chosen variable of resamples from the first sample with all values of this variable in resamples from the second sample.

For the global level (second level statistic) comparison, we use additional matrices KR, HR (contain the values of Ripley’s K and Ripley’s H functions) and DM, CM of both series of resamples. The DM arrays (DM₁ and DM₂ –matrices of resampling series from sample 1 and sample 2, respectively) contain the average values of the relative cluster densities within a given dimension (rows) for each bootstrap sample (column). The CM (CM₁ and CM₂) matrices contain the average values of the cluster curvature with the given dimension (rows) for each resample (columns).

In order to compare the samples at this global level, we test two null hypotheses:

• Mean-Cross-Hypothesis: The average value of the mean cluster densities of all resample series determined from one original sample (mean(DM₁)) and all the resample distribution of mean densities determined from the second original sample (mean(DM₂)) originate from the same continuous distribution.
• Cross-Hypothesis: Randomly k-times chosen density distributions from one series of resamples (i^th-column of matrix DM₁) and all the mean density distributions of the second resample series (DM₂) originate from the same continuous distribution.

A similar hypothesis can be made for matrices CM₁ and CM₂ containing average values of cluster curvatures. The same hypotheses can be formulated and tested for Ripley’s function results (KR and HR matrices) for both samples. For both hypotheses, we calculated the empirical p-value with methods previously described.

Measure of similarity between samples

We determine one measure for sample similarity [105] using calculated p-value curves (for each cluster dimension). In cases of an unknown confidence interval (L = 1), we can only use the simplest method for calculation of a similarity factor between compared samples.

We use the average of the aggregated p-value (mean(p)) in significance level 𝛼 and transform, it into a similarity measure as follows: (3)

Based on confidence interval estimates, we can define an alternative measure of similarity: comparison of the intersection between the area below the critical boundary 𝛼 and the lower confidence interval boundary of aggregated p-value curve.

Let d_interval = [size_min, size_max] be the full length of the cluster sizes interval. Thus, cr = 𝛼 * d_interval spans the critical area of the significant level 𝛼. The intersec(p) is the intersection of the critical area cr and the area above the lower bound of the confidence interval of p. The measure for the similarity is defined as: (4)

The average value of the aggregated p-value and confidence intervals of p-values for each size of cluster (matrix rows) are used to calculate sim_M and sim_L values. For a sim_M and sim_L value between 0.45 and 0.55, the similarity of the samples is marginal.

Machine learning

We apply a machine learning method, namely MLP neural network as a classifier of similarity of samples [71, 79, 80] (S10H Fig). We train the MLP neural network with 11 input neurons and 20 neurons in a hidden layer with a hyperbolic tangent as activation function and two neuronal output nodes with a softmax activation function (S10H Fig). As a network training performance function, we use cross entropy measures [71]. This makes it possible to interpret the values of both outputs as a posteriori probability of samples similarity. Let us assume that we have carried out a Ripley functions and bootstrapping-based analysis of two samples and extracted features and tests on both analysis levels. We transform the results into a set of L * M vectors, where L is the number of analysed sizes of clusters in dim(i) and M the number of resampling processes. Each vector represents the result of tests and forms the input and output pattern of the MLP neural network.

Vector Input = (d,p1,…,p10) and Output = [0,1] (similar) or = [0,1] (dissimilar) where d is the cluster size, p₁ = pv(d) is the p-value derived from the KS-test for relative cluster densities, p₂ = pw(d) is the p-value derived from the WX-test for the relative cluster densities, p₃ = pcv(d) is the p-value derived from the KS-test for the curvatures of clusters, p₄ = pcw(d) is the p-value derived from the WX-test for the curvatures of clusters, p₅ = mean_cross₁ is the p-value derived from the cross-comparison of the average cluster densities between sample 1 and sample 2, p₆ = mean_cross₂ is the p-value derived from the cross comparison of average cluster densities between sample 2 and sample 1, p₇ = c_mean_cross₁ is the p-value derived from the cross comparison of the average curvature of clusters between sample 1 and sample 2, p₈ = c_mean_cross₂ is the p-value derived from the cross comparison of the average curvature of clusters between sample 2 and sample 1, p₉ = R_mean_cross₁ is the p-value derived from the cross comparison via the Ripley K-function of sample 1 with sample 2, and p₁₀ = R_mean_cross₂ is the p-value derived from the cross-comparison of Ripley K-function of sample 2 with sample 1. Output value is equal to [0,1] if the parameters describing the samples are similar or is [0,1] otherwise. For extraction of the proper network training set, K-Ripley and bootstrap tests of both sample groups are required. To get a full training set, we need to know a priori if the samples are similar or not.

Cluster visualisation

S11A and S11B Fig are showing 3D representations of platelet clusters. To obtain a graphic representation of 3D objects, the Delaunay triangulation method was used. The Delaunay triangulation allows one to visualise the selected individual clusters in two ways: as shown in S11C and S11D Fig with a set of spheres with the centre in each point of the cluster and the radius equal to the half-distance to the nearest neighbour. As the default value for the cluster dimension (size), the dimension at maximum of the H-Ripley function is chosen. The H-Ripley functions are used to detect the presence of clustering in the data [106, 107]. Since the H-Ripley function is often used to recognize clustering, a positive value of H(d) indicates clustering over that spatial range whereas a negative value indicates dispersion. The value of d that maximizes H(d) indicates the radius of maximal aggregation (i.e. the size of clusters which contains the most points per volume). Therefore, d can be chosen as a neighbourhood radius ρ in the DBSCAN clustering method. It is used to extract the main (primary) clusters from the samples. The platform enables display of these primary clusters in 3D (S11E Fig). It is also possible to determine the empirical probability distribution of densities (S11H Fig) and curvature (S11I Fig) of primary clusters for both samples and their comparison using the KS-test.

The comparative analysis was performed on a standard PC with CPU i7 Intel, 1.9 GHz, 8 logical processors and 16 GB of RAM. Total analysis time for one pair of resamples and for all cluster dimensions is dependent on the number of points in a resample: e.g. for 1000 points: 0.86 s, 5000 points: 6.61s and for 10000 points: 18.11s.

Statistics

Statistical analyses were performed using SPSS. All statistical tests are described in the Fig legends. Statistical significance value: p<0.05.