Fig 1.
(A) Fluorescence microscopy image of a human mesenchymal stem cell (hMSC), scale bar represents 100 μm.
(B) Fluorescence microscopy image of the corresponding nucleus. (C) Plot of the corresponding contour of that cell with the centre of the cell shown as a red dot.
Fig 2.
A) The input data: a cell contour and the centre of its nucleus marked; the latter serves as the base point for the radial distance function.
B) The radial distance function: The complete cell contour forms a graph G. The edges of this graph are measured relative to the cell centre by computing the largest Euclidean distance between the centre and the endpoints of the edge: the corresponding measure is the radial distance function with respect to the centre. Edges whose radial distance function is below a given cut-off value (or ‘time step’), illustrated as concentric circles around the centre, define a sub-graph of the whole contour. C) Graph filtration: Examples of sub-graphs for five different time steps. The different graphs obtained at increasing values of time form a filtration of the graph G. D) The persistence diagram captures the topological properties of the graph filtration. The points marked as ‘’ and ‘
’ indicate that the corresponding points have multiplicity 2 and 3, respectively, in the persistence diagram.
Fig 3.
Heat map of the distance matrix for X1.
There is a cell that has distinctly higher than average distances to the other cells, indicated by the row/column of mostly bright yellow. Generated with [34].
Fig 4.
Image of the unusual cell shape (X1-015) identified in Fig 3 (image processed using [35]).
Fig 5.
Dendrograms for X1, the colours correspond to 4 clusters obtained using average linkage.
(A) Average linkage. (B) Complete linkage. (C) Single linkage. (D) Ward linkage. In each of these, there is an outlier, with the corresponding leaf coloured purple. Generated with [36].
Table 1.
Purity score of the 4 clusters obtained with the average linkage for X1 without X1-015, see Fig 5.
The colour and size of each cluster is in parentheses.
Fig 6.
Example cells from each cluster of the set X1 without X1-015.
Those clusters are identified with HCA and average linkage scheme (see Fig 5). All cell images are shown at the same magnification level. Images were processed using [35].
Fig 7.
Heat map of the distance matrix for Y1.
Generated with [34].
Fig 8.
Dendrograms for Y1, the colours correspond to 4 clusters obtained using average linkage.
(A) Average linkage. (B) Complete linkage. (C) Single linkage. (D) Ward linkage. In each of these, there is a consistent sub-population, coloured purple. Generated using [36].
Table 2.
Purity score of the 4 clusters obtained with the average linkage for Y1, see Fig 8.
The colour and size of each cluster is in parentheses.
Fig 9.
Example cells from each cluster of the set Y1.
Those clusters are identified with HCA and average linkage scheme (see Fig 8). All cell images are shown at the same magnification level. Images were processed using [35].
Fig 10.
The MDS embedding of the combined dataset X1Y1.
Y1 cells are represented in dark yellow; the region containing those cells is magnified in the insert. X1 cells are coloured according to the cluster they belong to, where the clusters have been defined in Sect 4.1.1: cluster A in red, cluster B in blue, cluster C in green, and cluster D in purple. The outlier X1–015 (see Sect 4.1.1) is shown in black. The explained variance of each principal component is provided in parenthesis.
Fig 11.
Example cells from the set X1Y1.
Those cells are identified in the MDS of X1Y1 in Fig 10. All four cells are at the same scale. Images were processed using [35].
Fig 12.
The MDS embeddings of the combined dataset X1Y1 for 4 different distances between cell contours, (A) aspect ratio distance, (B) elastic shape distance, (C) Fourier descriptors with M = 10, and (D) Fourier descriptors with M = 50 (see text for details on each of the distance).
We label the position of X1–015, the known outlier among X1 cells, on all four panels. The explained variance of each principal component is provided in parenthesis. Colours are described in the caption of Fig 10.
Fig 13.
ROC analyses of four measures of cell geometry similarity.
We compare the effectiveness of a the aspect ratio distance (in red), the elastic shape distance (in blue) the Fourier distance (with M=10) (in purple), and PH (in black) to measure cell contour similarities in the combined dataset X1Y1. “True” relationships are defined either with a binary separation of X1 and Y1 cells, Level 1 (panel A), or with a finer set of clusters involving the four clusters of X1, the outlier X1–015 as its own cluster, and Y1 (6 clusters, Level2, panel B). Curves close to the first diagonal (such as the ROC curve for the aspect ratio distance) indicate poor performance, while the upper most curve (such as the PH-based curve) indicates good performance.
Table 3.
ROC analyses of the effectiveness of four measures of cell contour similarity in classifying cells in the combined X1Y1 dataset.
Fig 14.
Sensitivity of the PH distance to the position of the nucleus centre used as a focal point to the radial function.
Pairwise distances between cell contours of the X1Y1 dataset were computed after perturbing the position of the nucleus randomly between [–s,s], where s is a noise level in pixel. The distances are then assessed using a ROC analysis and measured with an AUC value (see text for details). Those experiments are repeated 20 times at each noise level. Panel (A) reports the mean (dashed line) and standard deviation (shaded area) of the relative RMS change of the individual distance as a function of s, while panel (B) shows the changes in AUC values (both means, dashed lines, and standard deviations, shaded areas) at two level of granularity versus s (see text for details).