Fig 1.
Simulated spheroids and emerging phenotypes.
a) Cut-outs of the initial states of simulated spheroids for two ECM alignments. Each spheroid, shown in red and containing roughly 2000 cells, was placed into an (800 μm)3 volume and surrounded by either an unaligned or radially aligned ECM (green fibers). To improve visibility, the front half of the volume (dashed lines) is not shown b) Time evolution of simulated spheroids displaying four different phenotypes: “spherical”, “deformed”, “spherical with far gaslikes”, “disordered”. The ECM is radially aligned for these phenotypes, and is not shown in order to highlight the spheroid morphology. Each phenotype resulted from different combinations of parameters connected to the cell motility, the cell-cell adhesion and the interaction with the ECM (see Section 2.1 and Table 1). Each simulation lasted 250 000 Monte-Carlo (MC) steps, and shown are five snapshots for each simulation. A single MC step corresponds to roughly 1 s of real time in the context of this study. Throughout our investigation, we focused on the final configuration (orange rectangle), and used five replicates from each phenotype.
Table 1.
Differences between the four simulated phenotypes used throughout this study.
Table 2.
Simulated parameter space from which the four phenotypes were obtained.
Fig 2.
Visualization of cell based features extracted from simulation data for four different phenotypes.
a) Cell density distribution. Shown is, averaged over all replicates of each phenotype, the fraction of cells within spherical layers around the spheroid center versus the radii of these layers. The “spherical” phenotype shows a steep drop at a radius of 150 μm, while the “deformed” and “disordered” phenotypes show a long-tailed distribution. The “spherical with far gaslikes” phenotype behaves similar to the “spherical” phenotype, except for a non-zero density above 175 μm. b) Gaslike cell distribution. Shown are the average fractions of gaslike cells according to Eq 1 versus their normalized average distance to the spheroid center. The fraction of gaslikes exhibited by the “spherical”, “spherical with far gaslikes” and “deformed” phenotypes is similar, but the distance from the spheroid center is far greater for the “spherical with far gaslikes” phenotype. The “disordered” phenotype on the other hand contains many cells classified as gaslikes across the entire spheroid volume. Their normalized average distance from the center evens out to a value slightly above 1. c) Voronoi cell volume distribution. Shown are histograms of the average Voronoi cell volumes found in the four phenotypes. The “spherical” phenotype shows a sharp peak around a volume of 4000 μm3, and a smaller peak around a volume of 2000 μm3. The “spherical with far gaslikes” and “deformed” phenotypes show a similar behavior, with a slightly more pronounced tail towards larger volumes. Finally, the “disordered” phenotype shows volumes distributed over a wide range. The range between volumes of 1500 μm3 and 5000 μm3 is magnified on the right to highlight the differences between phenotypes in the two peaks.
Fig 3.
Visualization of spheroid bulk based features extracted from simulation data for four different phenotypes.
Surface information was extracted via the marching cubes algorithm [26] (see also section 2.2). a) Spheroid surface area. Shown is the average surface area found for each phenotype. The “spherical with far gaslikes” phenotype has the smallest average surface area, due to the spherical bulk containing less cells than that of the “spherical” phenotype. The larger average surfaces of the “deformed” and “disordered” phenotypes are due to their more irregular shape. b) Spheroid surface deformation. Shown are histograms of the scalar products between vertex normal vectors and vertex origin vectors, with the origin denoting the center of the spheroid. The vertices were obtained from surface triangulation of the spheroid point cloud and denote points on this surface (see also S4 Fig). The “spherical” phenotype exhibits a sharp peak at scalar products of 1, which is less pronounced for the “spherical with far gaslikes” phenotype. The remaining two phenotypes are spread more widely.
Fig 4.
Sketch of the application of Nastjapy.
An overall deviation score Di,j is calculated between two individual spheroids i and j (see also section 4.4).
Table 3.
Fitted weight factors for each feature contributing to the overall deviation score between two spheroids.
Fig 5.
Feature comparison for spheroid point clouds resulting from four different transformation functions.
Shown are the standardized metric distances between the un-transformed reference spheroid and an increasingly transformed version for each data feature. In addition, the combined deviation score is depicted in gray crosses for each transformation (see Section 2.4). Below each subfigure, we provide a top-down view snapshot of the spheroid at three levels of transformation. Blue cells are classified as non-gaslike, and red cells are classified as gaslike. a) Rotation. Except for negligible changes in the spheroid surface deformation feature, we observe no change at increasing rotation angle. This supports rotational invariance of our features. b) Noise. For each feature, the distance increases at increasing noise level. Due to the loss of a solid core at high noise levels, the spheroid surface area and deformation features are no longer sensible, and were therefore cut. The deviation score increases approximately linearly up to a noise level of 200, at which point the features related to the spheroid surface area were cut. c) Deformation. Similar behavior to b) is observed here. Above a deformation amplitude of 120 the spheroid point cloud still contains cells classified as non-gaslike but loses its solid core. Surface area and deformation values were therefore cut above this threshold. The deviation score increases approximately linearly up to a deformation amplitude of 120. d) Scaling. We observe increased distances both for scale factors below and above 1. Due to the fixed values of Dcrit and dcrit (see Eq 1), the gaslike distribution feature is scale-dependent, and also varies here. For scale factors below 1.0, no gaslikes were found, and therefore the values of this feature remained constant. The deviation score increases approximately linearly both for scale factors smaller and larger than 1.
Fig 6.
Deviation score comparison for four simulated spheroid phenotypes.
a) Shown are the deviation scores for five replicates of each phenotype on the upper triangle, and the average deviation score over all replicates of each phenotype on the lower triangle. A top-down view of the spheroid point cloud for each replicate is shown next to the respective row/column. Blue cells are classified as non-gaslike, and red cells are classified as gaslike. For better comparison, an enlarged version of each spheroid was placed at the bottom of the figure. We observe the highest deviation between the “disordered” phenotype and the others, with the maximum deviation between the “spherical” and the “disordered” phenotypes. The “spherical”, “spherical with far gaslikes” and “deformed” phenotypes, which are more similar from a visual perspective, show a smaller deviation score using our analysis, but are nonetheless distinguishable. b) Box plots of the deviation score values between the “spherical” phenotype and each other phenotype. The values used here correspond to those used for the lowest row of subfigure a). We observe that the deviation scores for the “spherical” phenotype compared with the other phenotypes consistently lie above the maximum deviation score of the “spherical” phenotype compared with itself. Significance was determined using Welch’s t-test.
Fig 7.
Deviation score comparison for in vitro MDA-MB-231 spheroids cultured in four collagen concentrations c (data provided by Kang et al [13]).
a) 2D cross-sections of 3D multiphoton microscopy image stacks depicting one replicate of each collagen concentration [13]. Spheroids were imaged at one, two or three days after embedding in collagen, and were then fixed and imaged (see Section 4.3). b) Deviation score comparison between all spheroid samples. For each day, the deviation scores for three replicates of each collagen concentration are shown on the upper triangle, and the average deviation score for each collagen concentration is shown on the lower triangle. A top-down view of the spheroid point cloud for a representative replicate of each collagen concentration is shown next to each heatmap (see S2 Fig for all replicates). Blue cells are classified as non-gaslike, and red cells are classified as gaslike. Due to matching initial conditions, we observe low deviation scores between spheroids grown for one day. These differences increase at day 2, where we observe an approximately linear increase of the average deviation score from c = 1 mg/ml to c = 4 mg/ml. Finally, at day 3, we observe the lowest deviation between c = 3 mg/ml and c = 4 mg/ml. This underlines the findings by Kang et al., who observed a transition in invasion behavior between 2 mg/ml and 3 mg/ml. c) Deviation score box plots from spheroids grown for one, two and three days respectively. The box plots for each day show the deviation score values between c = 1 mg/ml and each other concentration. These values correspond to those used for the lowest rows in b). We observe that for spheroids grown for one and two days, the deviation score values of c = 1 mg/ml compared with itself are similar to the deviation score values of c = 1 mg/ml compared with the other concentrations. The differences are nonsignificant and therefore not sufficient to clearly distinguish between the concentrations. This changes at day three, where each concentration shows a higher deviation score to c = 1 mg/ml than c = 1 mg/ml compared with itself. Significance was determined using Welch’s t-test.
Fig 8.
Deviation score comparison between in vitro spheroids grown in media at four different collagen concentrations c (data provided by Kang et al [13]), and in silico spheroids exhibiting four different phenotypes, simulated by us.
a) Shown are the deviation scores between three replicates of each collagen concentration, grown for three days, and five replicates of each simulated phenotype, simulated for 250 000 MC steps. A single MC step corresponds to roughly 1 s of real time in this context. Each individual deviation score is shown on the left, and the average within a pairing of collagen concentration and phenotype is shown on the right. A top-down view of the spheroid point cloud of a representative replicate for each collagen concentration / phenotype is shown next to both heatmaps. Blue cells are classified as non-gaslike, and red cells are classified as gaslike. See S2 Fig for an enlarged view of all experimental spheroid point clouds, and Fig 6 for an enlarged view of all simulated spheroid point clouds. We observe the highest average deviation scores between c = 2 mg/ml and the “spherical” phenotype. The lowest average deviation score is found between c = 4 mg/ml and the “deformed” phenotype b) Individual metric distances for each of the features constituting the overall deviation score. Shown are the standardized and weighted metric distances between three replicates of each collagen concentration, grown for three days, and five replicates of each simulated phenotype, simulated for 250 000 MC steps. A single MC step corresponds to roughly 1 s of real time in this context. Due to the overall deviation score being a sum of all weighted feature distances, the color range has been adjusted here. The highest deviation score observed in a) is a combination of high metric distances in all features, especially the spheroid surface deformation. On the other hand, the lowest deviation score observed in a) stems from overall low values, especially in the spheroid surface area.