Fig 1.
Schematic representation of the experimental configurations.
Using the same experimental conditions, consisting of a microfluidic device in which cells were seeded in a collagen matrix, it was possible to characterize cell motility and cluster formation as two distinct processes. The results recorded at day 1 provided the data used to define locomotive forces and, consequently, cell-matrix interactions. Furthermore, the results from the 5th day of experiments were used to calibrate cell-cell interactions.
Table 1.
Reference parameter values for the model.
Fig 2.
(A)A simplified representation of the force diagram showing the cell-cell and cell-matrix interactions present in the model, as described by Eq 2. The model considers cell-generated locomotive forces, drag forces imposed by the ECM, and cell-cell adhesion and repulsion of cells that are inside an interaction radius. Although the model only accounts for the cell volume and not cell geometry, spherical geometry is assumed. (B) Representation of the locomotive forces generator function modelled as 5 through an estimation of the inverse cumulative distribution of experimental cell velocities. When fed uniformly distributed random values between 0 and 1, the represented function produced a new set of values that followed the desired force distribution, as shown by the representative boxplot showcased as the output of this function. (C) Schematic representation of the implemented workflow to model cell-generated locomotive forces (considering no cell-cell interactions). At an average of 20 simulated minutes, cells are allowed to change their velocity both in magnitude and direction. Accordingly, at those time points, we generated a new cell-generated force value through the inverse sampling method. The output of this function was subsequently incorporated into the equation of motion, given by Eq 4, producing three different velocity distributions, each corresponding to a matrix density value.
Fig 3.
Schematic representation of the simulated conditions.
(A) Representation of the single-cell migration simulation setup, which consisted of tracking the positions of an individual cell during 24 hours to calibrate cell-generated locomotive forces. Due to the short duration of the simulations, cell death and proliferation were both neglected. We used spatial data to compute the cell’s average and effective velocities, which were compared to the experimental data. Based on this analysis, we iteratively changed the parameters of the locomotive force generator function, given by Eq 5, until the results were consistent with the experimental data. (B) Representation of the second simulated setup, which aimed to define cell-cell interactions by assessing the development of multicellular clusters for seven days. Unlike Setup 1, cell death and proliferation were considered in this setup. We computed cluster area values at days 1, 3 and 5 of simulation by classifying the cells through a clustering algorithm and compared these data to the experimental results. Similar to what we did in Setup 1, we iteratively ran simulations to approximate our model to the experimental data, this time focusing on the parameters that modulate cell-cell interactions, namely, cell-cell adhesion and cell-cell repulsion.
Table 2.
Parameter values for the post-processing of results.
Fig 4.
Experimental and simulated results for the individual migration setup.
(A) Representation of relative cell trajectories for the experimental (top) and computational (bottom) results. As the density of the collagen matrix increases, cells become more confined, resulting in reduced cell movement. (B) Mean and effective cell velocities for cells seeded in matrices of varying collagen density. Both the experimental (left) and computational (right) results indicate that as the density increases, cells travel shorter distances due to the restrictions imposed by the matrix and both the mean and effective speeds decrease.
Table 3.
Statistical data of the mean and effective cell velocities for different collagen concentrations for both experimental settings and computational simulations.
Fig 5.
Representation of cell positions after five simulated days of tumour growth.
2D (top) and 3D (bottom) representations of the coordinates of cells grown in matrices of different collagen concentrations after five days. For the 2D scatter plots, we selected only cells present in a defined height of interest to remove cells that would otherwise be out of focus in microscopy images. A single replicate was chosen for each condition to produce these plots. Different colours represent different clusters, whereas black cells are those considered outliers (i.e., they do not belong to any of the cell groups). The cluster area increases with density, as cells stay closer to their original position. On the other hand, in low collagen density matrices (left), in which individual cell migration is not limited, individual cells are seen to stray away, resulting in a large number of outliers and smaller, sparser tumours.
Fig 6.
Evolution of cluster area growth over five simulated days.
Distribution of cluster areas at the end of days one, three and five for clusters grown in matrices of medium and high collagen density (4.0 and 6.0 mg/mL, respectively), for the experiment (left) and computational (right) settings. These distributions take into account data from all five replicates. Cells seeded in collagen matrices of low density (2.5 mg/mL) did not show significant multicellular cluster formation and growth and hence were not represented. An increase in tumour size through time can be seen for both the medium and high collagen concentrations, but larger densities induce the formation of clusters of larger areas.
Fig 7.
Cluster eccentricity after 7 days of growth.
Representation of the xy cell coordinates for the cells present in the entire height of the domain, overlapped by the clusters’ equivalent ellipses and eccentricity values (left). Individual cells and clusters of small areas or areas that were manually evaluated as having been misclassified by the clusterization algorithm are not represented. Clusters of cells grown in collagen matrices of high density are shown to present smaller values of eccentricity (indicating a rounder morphology), while clusters grown in matrices with a lower collagen concentration adopt slightly larger values, as cells migrate away from the cluster, producing more elongated morphologies. The distribution of eccentricity values for both densities (right) further confirms this idea.