Fig 1.
Summary of key experimental and simulation results.
(A)Two experiments setups for growing spheroids considered in this study. In experiment I, the spheroid is in mechanical contact with a capsule, and the mechanical resistance is determined by the wall thickness H. In experiment II, the spheroid is immersed in a dextran polymer solution, and the mechanical resistance originates from the osmotic pressure related to the dextran concentration. (B) Radial growth curves data of the spheroids in units of R0 (= 100 μm), for experiment I and II and respective model runs. The blue full circles are the free growth data for CT26, from [26]. The thin blue line indicates theoretical pure exponential growth with doubling time of 17h. The data starts deviating from an exponential after 2 days. The other lines are simulation results. The black dashed line indicates the optimal parameter set for the stress response in experiment I, performed with the final model I. The full black line indicates the same model run for free growth in Exp.I. After re-calibration of one model parameter in model I for the Exp.II conditions in absence of dextran (full red line), the model (referred to as model II to stress the change of the parameter) predicts the stress response in experiment II (red dashed line). (C) Simulation snapshots of both experiments. The cells are colored according to their volume (cells at the border are larger than in the interior). (D-G) Model simulations for Exp.II for the cell lines BC52, AB6, FHI and HT29, respectively. Full red lines represent the same initial calibration procedure, while red dashed lines represent the predicted stress conditions. The stress conditions are p = 5 kPa for AB6, FHI and BC52, and p = 10 kPa for HT29 (see Validation of model for experiment II: same cell lines as for experiment I).
Fig 2.
Simulations were performed with a center-based model (CBM). In step 1, the contact forces in CBM were calibrated from DCM simulations with parameters (Ecor, hcor, K), yielding a variable effective contact stiffness of the CBM for each individual cell depending on the compression level. In step 2 the parameters
of the CBM for cell line CT26 were determined. Comparing simulations of the CBM with stress-free growth of multicellular CT26 spheroids in experiment I determines most parameters of
(Fig 1B, full black line). step 3: those cell-line parameters that are affected by the capsule, are specified by comparison with the data from experiment I in presence of the thin capsule. The set of experiment-specific parameters
(Young modulus and thickness of the capsule) are given by the experimental setting. For the so specified complete set of parameters the simulation reproduces the experimental data I for the thin capsule (Fig 1B, dashed black line), and, after replacement of the capsule thickness, predicts the experimental data for the thick capsule (see Fig 5B). For CT26 cells growing in experiment setting II the cell parameters remain unchanged
. The deviation of the growth dynamics of stress-free growth from an exponential in experiment II (Fig 1B, full red line) is taken into account by an experiment-specific parameter, namely the proliferative rim. Without any further fit parameter, the model then predicts the correct growth dynamics subject to dextran-mediated stress (Fig 1B, dashed red line). In order to predict the stress-affected growth kinetics of the cell lines j = {CT26, AB6, HT29, BC52, FHI}, their cell cycle duration is modified to capture the stress-free growth analogously to that of CT26 cells in experimental setting II (Fig 1D–1G, full red lines). After determining the parameters, the growth kinetics of these cell lines subject to stress could be predicted (Fig 1D–1G, dashed red lines).
Fig 3.
(A) Plot of Hill-type growth rate function as function of the volumetric strain ϵV = ϵV(p), for n = 1, 2 and a large value of n, and for a constant growth scenario ( in Eq (2)). Plot of a linear growth rate function with
such that α/α0 = 1/2. Below the pink zone indicated by αqui cells become quiescent and growth stalls. In case of a sharp threshold obtained by the choice of n → ∞, any cell with
would proliferate with maximal rate α = α0, while any cell with
would be quiescent. For finite n, there are also proliferating cells for α < α0. The points on the growth rate curves below which the cells go into quiescence are indicated by an (*). In this work we have found that the parameter set n = 1,
and αqui = 0.3 results in good fits for all cell lines. (B) simulation snapshots of a CT26 spheroid during the initial free growth, just before confinement (coloring according to cell radius), and at 48h of confinement in capsule (coloring here indicates necrotic cells (dark) and viable cells (white)).
Fig 4.
(A) Time evolution of the radius of the thin capsule for the experimental data and the simulations using Model I showing the effect of a parameter variation for n with αqui = 0.33, and n = 1 with αqui = 0.5. (B) Simulation and experimental values of the radial cell density in the spheroid at T = 0h, and T = 48h for the optimal parameters. (C) Pressure curves indicating the pressure at the transition point from free spheroid growth to spheroid growth against the thin capsule in ref. [26] and the simulation.
Fig 5.
(top) (A) Time evolution of the thick capsule radius (H = 30 μm), shown for the experimental data and the simulation with Model I, indicating the effect of the parameter n and . As the number of data sets on the thick capsule did not suffice to estimate the experimental error, the errors on the thick capsule data (gray zone) were estimated from the spreading on the thin capsule data, by determining the minimum—maximum intervals for the thin capsule data. These were then rescaled by the ratio of thin—thick capsule dilatations and shifted on to the thick capsule curve. (B) Global view of experiment I and II and respective model runs, including a model prediction for a capsule wall thickness H = 60 μm. (C) Simulated evolution of the average pressure in a capsule with H = 30 μm and H = 60 μm.
Table 1.
Nominal physical parameter values for the DCM to calibrate the CBM.
Table 2.
Reference physical parameter values for the model.
CS indicates a model choice. If CS shows up with references next to it, the value was chosen from the parameter range in the references. A reference only means the value is fixed from literature. An (*) denotes parameter variability meaning that the individual cell parameters are picked from a Gaussian distribution with ±10% on their mean value. The Gaussian distribution is clamped to 4 times the standard deviation to avoid potentially very low values or very high values. Negative values are excluded.
Fig 6.
(A) Simulation snapshot at the beginning of a free growing CT26 spheroid (R = 100 μm), indicating quiescent (dark) and proliferating cells (light). (B-D) Simulation snapshots of growing CT26 spheroids at R = 120 μm during dextran application (p = 5 kPa), indicating quiescent and proliferating cells (B), individual cell pressure (C), and volume for the cells (D).
Fig 7.
(A-B) Detail of the time evolution of radius of the CT26 and FHI spheroid relative to its initial state.
Data from [12] shown for free growth and at p = 5 kPa. Runs with Model II are for free growth and for p = 5 kPa. In the CT26 cell line an additional model run is shown assuming a linear cell cycle progression function. In the FHI cell line the vertical line indicates the presumed changes in experimental conditions for free growth over time resulting in a lower surface growth (v1 → v2). The gray zones in the plots indicate the min-max values of the data.
Fig 8.
(A) Cartoon illustrating the compression experiment using deformable cells in a capsule to calibrate the center-based model. (A, bottom) Equivalent compression experiment using the center-based model with indication of the maximal principal stress directions of the cells in the capsule during compression using Eq (11). (B) Cartoon showing the volume compartments Vi, Vint and Vcaps in a capsule with thickness H. (C) Average contact force vs. for different K values simulated using DCM (diamonds), and CBM with corrected Hertz contact force (full colored lines) replacing E by
, see Eq 14. dij is the distance between the centers of cells i and j, Rref,k the radius of a free cell k ∈ {i, j}. The modified Hertz force shows the same evolution as the force in the DCM, while an uncorrected Hertz force (gray line, Eq 6) strongly underestimates the interaction force for strong volumetric compression. (D) Pressure curves during compression of the spheroid as a function of the inter-cellular volume fraction simulated with the DCM and the CBM with modified Hertz force using here K(V). The pressure for CBM was computed using both the capsule pressure and average virial stress per cell calculated from Eq (11). A representative movie (S3 Video) of these simulations is provided).
Fig 9.
(A) Simulation snapshots of DCM cells within a scaled capsule model, for the cases of cells with a reference cortex stiffness (top) and a “stiff” cortex stiffness (bottom). The coloring is according to pressure (B) Internal cell pressure for deformable cells in a shrunk capsule for nominal cells and stiff cells, as function of distance to the capsule center. The stiff cell types show a gradient in cell pressure if moving from the spheroid center towards the edge (indicated by dashed red line), while a higher variability as compared to the softer types. Notice that like in the calibration simulations we use cells of equal volume prior to compression but the method can equally be applied to any prior volume distribution.
Fig 10.
(A) Time evolution of the radius of the thin capsule, shown for the experimental data and the simulation using Model I, with parameter variation on the individual cell compressibility (K(V) means strain hardening). (B) Time evolution of the simulated cell density. The dashed horizontal line indicates the experimentally observed cell density at 48h. (C) Pressure in the capsule versus time.
Fig 11.
(top) (A) Time evolution of the radius of the thin capsule, shown for the experimental data and simulations using Model I, showing the effect of a parameter variation for the lysis time Tlys. (B) Time evolution of the simulated cell density. (C) Cell density at 48h obtained from final model run with optimal parameters, but in which cells divide after a fixed cycle time (”timer“) instead of a fixed size.