Comparing the growth kinetics of cell populations in two and three dimensions

We study the kinetics of growing cell populations by means of a kinetic Monte Carlo method. By applying the same growth mechanism to a two-dimensional (2D) and a three-dimensional (3D) model, and making direct comparison with experimental studies, we show that both models exhibit similar behavior. Based on this we propose a method for establishment of a mapping between the 2D and 3D results. Additionally, we present an analytic approach to obtain the time evolution, and show in case of the 3D model how synchronization effects can influence the growth kinetics. Finally, we compare the results of our models to experimental data of the growth kinetics of 2D monolayers and 3D NIH3T3 xenografts in mice.

Figure 1

(Color online) (a) 2D lattice: based on a square lattice (white lines) the Voronoi points (black) are distributed and connected by the Delaunay triangulation (red lines). The corresponding polygon (blue) for the shape of one cell (white) is shown. The radius of the green circle denotes the proliferation length $\Delta L$ up to which a dividing cell (white) can proliferate. (b) The dual graph of the analog 3D lattice: the polyhedrons correspond to the cells. Here cells inside the range $\Delta L$ of a proliferating cell (orange) are colored blue instead of drawing a sphere surrounding them.

Figure 2

(Color online) Examples of the irregular lattice and corresponding cell structure in two and three dimensions. (a) The Voronoi structure, i.e., the cell geometry, in the 2D model: inactive cells (light blue) and proliferating cells (dark blue), with an example path for a cell to proliferate inside this rim (light green line), and the involved cells (dark red). (b) The Delaunay lattice structure corresponding to (a): inactive points (light blue), proliferating points (dark blue), path for proliferation (light green line), and involved lattice points (dark red). (c) The Voronoi structure and the cell structure in the 3D model. Cells are depicted as spheres. (d) The Delaunay lattice structure corresponding to (c).

Figure 3

(Color online) Lattice artifacts in two and three dimensions. (a) Simulation with the individual-cell 2D model on a simple square lattice with $\Delta L=1$, $m=10000$, and $\varphi =0$. (b) Simulation with the individual-cell 2D model on the Voronoi structure with the same parameters as (a). (c) Simulation with the individual-cell 3D model on a simple cubic lattice with $\Delta L=1$, $m=10000$, and $\varphi =0$, $n_{\text{max}}=1$. (d) Simulation with the individual-cell 3D model on the Voronoi structure with the same parameters as (c). The color represents the distance from the center of mass in the range between overall minimum and maximum distance. The lattice artifacts in lattice models on regular lattices become the more pronounced the larger $m$ is. With $m\to 1$ (assumption of Poissonian cycle time distribution), increasing $\Delta L$ (thick proliferating rim) or if free migration of cells on the lattice (detaching cells that move freely) is considered, the lattice artifacts disappear.

Figure 4

(Color online) Cell-cycle time distribution of clusters in two (light blue shading) and three dimensions (coarse-grained model; red squares) in comparison with the Erlang distribution (black dash-dotted line). Parameters: $\Delta L=9$, $m=60$, and $\varphi =0$ for the 2D model, and $\Delta L=2.21$ with a coarse-grained factor $s=3$ in the 3D model with compounded cells; hence $n_{\text{max}}=27$.

Figure 5

(Color online) Scheme of the development of the mean radius and the corresponding growth velocity from a 3D simulation with parameters $\Delta L=4$, $\varphi =2$, and $m=5$. (a) Radius $\overline{R}$ vs time $t$ divided into three phases of development: an exponential phase, a linear phase, and a transient in between. (b) Growth velocity $v=d\overline{R}/dt$ vs time $t$ averaged over 20 neighboring data points to suppress large fluctuations. The initial $t^{1/2}$ phase is very short due to $\varphi$ being very small; hence it cannot be seen.

Figure 6

(Color online) Asymptotic expansion velocity $v$ vs Erlang number $m$ ($\Delta L=1$, $\varphi =0$) in two (red) and three (green) dimensions, obtained by single-cell-based kinetic Monte Carlo simulations. The lines are fits using Eq. (4).

Figure 7

(Color online) Square of the asymptotic expansion velocity $v^2$ of the single-cell-based kinetic Monte Carlo simulations vs square of proliferation length $\Delta L^2$. (a) The migration rate is kept constant with values $\varphi =0$ (circles), $\varphi =10$ (squares), and $\varphi =20$ (diamonds) in two dimensions. (b) In three dimensions the migration rate is chosen to be $\varphi =0$ (circles), $\varphi =0.5$ (squares), and $\varphi =2$ (diamonds). In both cases the Erlang number is set to $m=1$. The continuous curves (same color) depict the result of Eq. (4) for the corresponding parameters.

Figure 8

(Color online) Square of the asymptotic expansion velocity $v^2$ of the single-cell-based kinetic Monte Carlo simulations vs migration rate $\varphi$. (a) The proliferation length is kept constant with values $\Delta L=1$ (circles), $\Delta L=3$ (squares), and $\Delta L=6$ (diamonds) in two dimensions. (b) In three dimensions the proliferation length is chosen to be $\Delta L=2$ (circles), $\Delta L=3$ (squares), and $\Delta L=5$ (diamonds). In both cases the Erlang number is set to $m=1$. The continuous curves (same color) depict the result of Eq. (4) for the corresponding parameters.

Figure 9

(Color online) Time evolution of cell colonies with parameters $\Delta L=6$, $\varphi =0$, and $m=1$ (single-cell-based model). The colonies are depicted for times $t=8,12,20$ (in units of average cell-cycle time). The active proliferation rim is labeled by color. (a) 2D case: proliferating cells in dark blue and inactive cells in light blue. (b) 3D case: surface cells in light blue, proliferating cells in medium blue, and inactive cells in black. Here one-eighth of the spheroid is cut off, so that the interior becomes visible.

Figure 10

(Color online) Comparison of the analytical models for the growth kinetics in two and three dimensions. The parameters were chosen to be $\Delta L_{\text{eff}}=5$, $\varphi =0$, and $N_0=1$. (a) Long-time development and (b) crossover between exponential and linear regimes. Thick orange (gray) line: result of Eq. (10) $\left(d=2\right)$; thick red (black) line: result of Eq. (11) $\left(d=3\right)$. The thin lines are results of the phenomenological growth law presented in [46] using the Hill function $H\left(t\right)=t_c^n/\left(t_c^n+t^n\right)$—thin green (gray) line: $d=2$, $n=8$; dashed green line: $d=2$, $n=16$; thin blue (black) line: $d=3$, $n=4$; dashed blue line: $d=3$, $n=8$.

Figure 11

(Color online) Mean radius $R$ of the cell aggregate vs time $t$. (a) In two dimensions—blue crosses: experiment for C6 rat astrocyte glioma cells [15, 16]; red pluses: kinetic Monte Carlo simulation of the 2D model with the same parameters as in Fig. 4; green line: analytical result of Eq. (10) with $\Delta L_{\text{eff}}=B{L}^{\prime }\approx 8.725$ and $N_0=1$, corresponding approximately to $\Delta L=9$. We chose $l=10\mu \text{m}$ for the cell diameter and $\tau =19\text{h}$ for the cell-cycle duration as in Ref. [14]. (b) Three dimensions—blue circles, green diamonds, violet squares: in vivo NIH3T3 mouse-fibroblast xenografts; solid blue line: coarse-grained kinetic Monte Carlo simulations with parameters $\Delta L=2.85$, $\varphi =0$, $m=1$, and $n_{\text{max}}=1000$; dashed red line: analytical result of Eq. (10) with $\Delta L_{\text{eff}}=28.5$ and $N_0=1$. We have chosen $l=17\mu \text{m}$ [51] and $\tau =24\text{h}$ [52].

Figure 12

(Color online) Comparison of numerical simulation and analytical result of the population size $n\left(t\right)$ for high Erlang numbers in the exponential-growth phase. Red crosses: individual-cell-based model with $\Delta L=8$, $\varphi =0$, and $m=100$. Blue pluses: Analytical result of Eq. (14) for $m=100$. The population was synchronized at time $t\approx 6.3\overline{\tau }$.

Figure 13

(Color online) Population sizes vs time for different Erlang numbers $m=10,20,50,100$ in a half-logarithmic diagram. The 3D individual-cell-based model was used with the parameters $\Delta L=8$ and $\varphi =0$. The vertical blue line represents the time of total synchronization. An ordinate offset between the curves was added for convenience so as to not let them overlap.