Figure 1

(Color online) (a) Schematic of the transwell apparatus, adapted from [25]. (b) A 3D cuboid region defines the geometry of the random-walk model in terms of the $\left(i,j,k\right)$ lattice indices. This 3D cuboid region is used to embed the cylindrical transwell geometry. (c) A cross section through the cuboid region where $k=1$ shows the location of the porous membrane shaded gray (light gray) and the inactive sites are shaded black (black). (d) Image from a scanning electron microscope shows several MCF-7 breast cancer cells sitting on the upper surface of a transwell membrane. The cells are white (white) and are clearly distinct from the black pores (black). This image highlights a particular cell (circled) moving through a pore (image is reproduced with kind permission from Kricker [30]).Reuse & Permissions

Figure 2

(Color online) Experimental results are shown as red squares and modeling results are shown as green, purple, and blue solid lines (light gray, medium gray, and dark gray). These results correspond to (a) the time course experiments where $N_L$ was measured at various times after placing $N_U=200000$ cells into the upper compartment with 5 ng/ml IGF-II in the lower chamber, (b) the cell seeding experiments show $N_L$ after 5 h for various $10000\le N_U\le 400000$ with 5 ng/ml IGF-II in the lower chamber, and (c) the growth factor experiments show $N_L$ after 5 h for $N_U=200000$ and various concentrations of IGF-II. Modeling results in (a)–(c) correspond to (i) $\Delta =20\mu \text{m}$, $P=0.1$, $q=0.0$, and $\tau =0.11\text{h}$ shown in blue (dark gray); (ii) $\Delta =20\mu \text{m}$, $P=0.1$, $q=0.1$, and $\tau =0.08\text{h}$ shown in green (light gray); and (iii) $\Delta =20\mu \text{m}$, $P=0.1$, $q=0.2$, and $\tau =0.06\text{h}$ shown in purple (medium gray). We note that each modeling profile in (a) has been rescaled to fit the experimental data, so that it is difficult to distinguish between the three modeling curves. Modeling results in (c) correspond to $\Delta =20\mu \text{m}$, $q=0.1$, $\tau =0.08\text{h}$, and $P\left(C\right)=0.025+\left(0.1C\right)/\left(1+C\right)$. Results in (d) are the same as (b) with different proportions of pore space, 5%, 10%, 15%, 20%, and 25%, as indicated. All simulation results are averaged over $n=20$ identically prepared realizations. The discrete model was used to predict the migration assays with the same data points as the experiments and linear interpolation was used to generate the profiles in (a)–(d).Reuse & Permissions

Figure 3

(Color online) Simulation data demonstrate the effect of agent-to-agent adhesion in an undirected population of agents. All stochastic simulations correspond to $P=\Delta =\tau =1$. Results in (a) and (b) show the distribution of a population of agents with no agent-to-agent adhesion over a period of 1000 time steps. Column-averaged densities, further averaged over $n=20$ identically prepared realizations, are shown in the solid blue (solid gray) curve in (c) and compared with a numerical solution of Eq. (A3) shown in the dotted red (dotted gray) curve with the same initial conditions and boundary conditions used in the discrete simulations. In this work all numerical solutions are obtained with $\delta x=0.01=\delta t=0.01$ and $ϵ=1\times {10}^{-6}$. Data in (d)–(f) and (g)–(i) show similar results for adhesive populations with $q=0.25$ and $q=0.50$, respectively.Reuse & Permissions

Figure 4

(Color online) Simulation data demonstrate the effect of agent-to-agent adhesion in a directed population of agents. All stochastic simulations correspond to $P=\Delta =\tau =1$. Results in (a) and (b) show the distribution of a population of agents with no agent-to-agent adhesion over a period of 1000 time steps. Column-averaged densities, further averaged over $n=20$ identically prepared realizations, are shown in the solid blue (solid gray) curve in (c) and compared with a numerical solution of Eq. (A3) shown in the dotted red (dotted gray) curve with the same initial conditions and boundary conditions used in the discrete simulations. In this work all numerical solutions are obtained with $\delta x=0.01=\delta t=0.01$ and $ϵ=1\times {10}^{-6}$. Data in (d)–(f) and (g)–(i) show similar results for adhesive populations with $q=0.25$ and $q=0.50$, respectively.Reuse & Permissions