Figure 1

Experimental data described by Barrandon and Green [2] illustrate the complicated relationship between cell motility and cell proliferation. Three colonies are shown after 24 days’ incubation. The top colony (black circles) was supplied with $\text{TGF-}\alpha$, the middle colony (open circles) was supplied with EGF and the bottom colony (open triangles) was supplied with neither chemical. These chemicals upregulate cell motility and we see that the increase in motility leads to dramatic population growth over a period of 24 days. Reprinted from [2] with permission from Elsevier.Reuse & Permissions

Figure 2

Experiments performed by Tremel et al. [18] were used to estimate the proliferation rate of a population of fibroblast cells in vitro. Motile and proliferative fibroblast cells were placed uniformly in two-dimensional culture and the number of cells in the system was observed at regular intervals of time, for example, at (a) 54–60 and (b) 96–102 h after initial seeding. The growth in cell numbers was fitted to a logistic growth model (c). Reprinted from [18] with permission from Elsevier.Reuse & Permissions

Figure 3

(Color online) The influence of motility on proliferation patterns in two (left) and three (right) spatial dimensions for different initial conditions. The top plots show the evolution of density over time with (a) and (b) showing 2D results and (c) and (d) showing 3D results. The red (gray) lines show the solution of the logistic equation as a function of rescaled time, $\overline{t}=\left(P_p-P_d\right)t$, and the black lines the discrete results averaged over 40 realizations for $P_m=1.0$, $P_d=0.0$ and $P_p=0.005$, 0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1.0, again as a function of rescaled time. Agreement with the logistic equation becomes worse as $P_p$ is increased relative to $P_m$. The bottom plots show snapshots of the density at different time points for $P_p=0.5$ and illustrate the spatial correlations (clusters) that arise: (e) 2D simulation at rescaled time $t=4$; (f) 2D simulation at rescaled time $t=7$; (g) 3D simulation at rescaled time $t=2$; (g) 3D simulation at rescaled time $t=3$.Reuse & Permissions

Figure 4

(Color online) The influence of motility on proliferation patterns in 2D (top row) and 3D (bottom row). In each case, the red (dark gray) line shows the solution of the logistic equation, the black line the averaged discrete results and the green (light gray) line the results given by the inclusion of correlations. In most cases, the KSA results (green/light gray) are visually indistinguishable from the simulation results (black).Reuse & Permissions

Figure 5

(Color online) The influence of motility on proliferation patterns with the inclusion of agent death in 2D (top row) and 3D (bottom row). In each case, the red (dark gray) line shows the solution of the logistic equation, the black line the averaged discrete results and the green (light gray) line the results given by the inclusion of correlations.Reuse & Permissions

Figure 6

Investigation of the correlation at different distances in 2D (top row) and 3D (bottom row). The results show $F_{A,A}\left(1\right),F_{A,A}\left(2\right),\dots ,F_{A,A}\left(5\right)$ (decaying as the distance increases).Reuse & Permissions

Figure 7

(Color online) Investigating truncation of the correlation equations at different distances in 2D (top row) and 3D (bottom row). Mean field (red/dark gray) and simulation (black) results are presented together with a suite of solutions of the correlation equations (green/light gray). In each case, four solutions of the correlation equations are given with the most accurate representing the full system of correlations. The remaining three solutions of the correlation equations correspond to truncating the system at $r=1,2,3$ with the accuracy decreasing as $r$ decreasing.Reuse & Permissions

Figure 8

Calculating the coefficient ${ϵ}_1$ for the correction of the KSA with birth only and a 1% and 5% initial seedings. Note that a significantly larger correction is required for 1% seeding compared to 5% seeding.Reuse & Permissions

Figure 9

(Color online) Estimation of parameter values with and without inclusion of the correlations. The top row shows the results of fitting with $P_d=0.0$. The left-hand figure shows the results of fitting density data only for $P_p=0.8$, $P_d=0.0$ with agent density shown in black, results from fitting with the inclusion of nearest neighbor correlations in green (light gray), and fitting the logistic curve in red (dark gray). The predicted values of $P_p$, $P_d$ are $P_p=0.76$, $P_d=0.02$ (correlations included) and $P_p=0.60$, $P_d=0.02$ (correlations not included). The center and right-hand figures compare the predicted birth rates given by fitting with the logistic curve (red/dark gray) and with the inclusion of nearest neighbor correlations (green/light gray). The center plot shows results from fitting density data only and the right-hand plot fitting with both density and correlation data. The bottom row shows results of fitting using both the density and nearest neighbor correlation data when death is nonzero. Details are as above, with the center plot showing the predicted birth rate and the right-hand plot the predicted death rate. The predicted values of $P_p$ and $P_d$ on the left-hand plot are $P_p=0.64$, $P_d=0.33$ (correlations included) and $P_p=0.07$, $P_d=0.04$ (correlations not included).Reuse & Permissions