Quantifying spatial structure in experimental observations and agent-based simulations using pair-correlation functions

We define a pair-correlation function that can be used to characterize spatiotemporal patterning in experimental images and snapshots from discrete simulations. Unlike previous pair-correlation functions, the pair-correlation functions developed here depend on the location and size of objects. The pair-correlation function can be used to indicate complete spatial randomness, aggregation, or segregation over a range of length scales, and quantifies spatial structures such as the shape, size, and distribution of clusters. Comparing pair-correlation data for various experimental and simulation images illustrates their potential use as a summary statistic for calibrating discrete models of various physical processes.

Figure 1

Images of spatial patterning. (a),(b) An initially uniform coculture of cells forms multicellular aggregates (reprinted from Thomas et al. [1], Fig. 3, with permission from Eur. Cells Mater.). (c),(d) A segregated mixture of steel and glass beads evolves to a less segregated mixture when vibrated (reprinted from Yang [2], Fig. 2, with permission from Powder Technol.). (e),(f) Aggregates of dye on a viscous fluid evolve to a more regular pattern when the fluid is stirred [reprinted from Kobayashi et al. [3], picture 4.1(1a and 1d), with permission from Topol. Appl.].

Figure 2

(a) Gray squares represent sites that are occupied by unit sized area agents on a lattice with $X=4$ and $Y=2$. The counts of the pair distances between area agents in the $x$ direction are indicated with arrows. There is one pair of agents separated by a distance of one [$c_x\left(1\right)=1$], one pair of agents separated by a distance of two [$c_x\left(2\right)=1$], one pair of agents separated by a distance of three [$c_x\left(3\right)=1$], and no pairs of agents separated by a distance of four [$c_x\left(4\right)=0$]. (b)–(d) Counts of pairs of sites on a lattice with $X=4$ and $Y=1$. In (b) we have three pairs of sites separated by a distance of one [$r_x\left(1\right)=3$]. In (c) we have two pairs of sites separated by a distance of two [$r_x\left(2\right)=2$]. In (d) we have one pair of sites separated by a distance of three [$r_x\left(3\right)=1$]. (e)–(g) Counts of pairs of lattice sites on a lattice with $X=4$ and $Y=2$. In these subfigures we have used a combination of solid and dashed arrows to make the counts of pairs of sites clear. In (e) we have six pairs of sites separated by a distance of one. In (f) we have four pairs of sites separated by a distance of two. In (g) we have two pairs of sites separated by a distance of three.

Figure 3

Domain populated uniformly at random with (black) unit square area agents. (a)–(c) $XY=1200$, $\rho =0.25$, and ${\Delta }_x={\Delta }_y=1$. (a) Typical realization. (b) Counts of pair distances $C_x$ and $C_y$ (solid and dashed curves). The dotted lines are for the normalization values ${\widehat{C}}_x$ and ${\widehat{C}}_y$. (c) Average pair-correlation function $\overline{P}$, $N=10000$, $XY=100$, and $\Delta =1$. The dotted, dashed, and solid curves are for mean field densities $\rho =\{0.10,0.25,0.5\}$. The upper three curves are for $\tilde{\rho }$, given by Eq. (10). The lower three curves are for $\tilde{\rho }=\rho$.

Figure 4

Cellular-agent aggregates generated by the discrete proliferation mechanism, $XY=100000$, ${\mathbb{P}}_p=1.0$, ${\mathbb{P}}_m=0$, and $s=1$. (a) Initial condition. (b) Typical realization at $t=10$. (c) Pair-correlation functions $P_x$ (solid) and $P_y$ (dashed) for (b), at $t=10$ and ${\Delta }_x={\Delta }_y=1$. The dotted curve is the average pair-correlation function $\overline{P}$, $N=1$, and $\Delta =1$.

Figure 5

Cellular-agent aggregates generated by the discrete proliferation mechanism, $XY=100000$, ${\mathbb{P}}_p=1.0$, ${\mathbb{P}}_m=0$, and $s=1$. Initially the domain was populated uniformly at random with unit square cellular agents and $\rho \left(0\right)=0.001$. (a),(b) Typical snapshots at $t=10$ and 20, respectively. (c) $\overline{P}$ at $t=10$ (dashed) and $t=20$ (dotted) with $N=100$ and $\Delta =1$. The solid curve is for the initial distribution of agents with $N=1000$ and $\Delta =1$.

Figure 6

Aggregation patterns generated by the discrete proliferation and motility mechanism, $XY=100000$, ${\mathbb{P}}_p=0.1$, and ${\mathbb{P}}_m=1.0$. Initially the domain was populated uniformly at random with unit square cellular agents at $\rho \left(0\right)=0.01$. (a) Typical realization at $t=20$. (b) Average pair-correlation functions $\overline{P}$ at $t=10$ and $\Delta =1$. The dotted curve is for all the agents within the domain, $N=100$. The solid curve is for the agents that initially seeded the domain, $N=200$. (c) Average index $\overline{I}$ (solid curve) for $B=400$ and $N=100$. The dashed curve represents the average ECSR limiting value ${\overline{I}}_{ecsr}$.

Figure 7

Domain populated uniformly at random with square cellular agents, $s=169$, $XY=10000$, and $\rho =0.1$. (a) Typical realization. (b),(c) Average pair-correlation functions $\overline{P}$. The solid curves are for all the area agents in the domain, $N=100$. The dotted curves are for the central area agents within each of the cellular agents, $N=10000$. (b) $\Delta =1$. (c) $\Delta =10$.

Figure 8

Aggregation patterns generated by the discrete proliferation mechanism, $XY=100000$, ${\mathbb{P}}_p=1.0$, and ${\mathbb{P}}_m=0$. (a) Snapshot of a realization at $t=5$ for a domain initially populated uniformly at random with cellular agents of size $s=49$ and $\rho \left(0\right)=0.01$. (b) Average pair-correlation functions $\overline{P}$ for (a). The dotted curve is for the central area agents within each of the cellular agents, $N=10000$ and $\Delta =1$. The solid curve is for all the area agents in the domain, $N=100$ and $\Delta =1$.

Figure 9

Simulation of a scrape wound assay, using the discrete proliferation and motility mechanism, $s=1$, $XY=1200$, ${\mathbb{P}}_p=0.1$, and ${\mathbb{P}}_m=1.0$. (a) Initial condition at $t=0$. The uniform density in each populated strip is ${\rho }_s=0.5$. (c) and (e) Snapshots of the simulation at $t=20$ and $t=40$. (b), (d), and (f) Pair-correlation functions $P_x$ (solid curves) and $P_y$ (dashed curves) for (a), (c), and (e), $N=1$ and ${\Delta }_x={\Delta }_y=1$.

Figure 10

Spatial analysis of a scrape wound assay, $XY=281190$. (a) Experimental image [reprinted from Glab et al. [33], Fig. 1(a), with permission from Pattern Recogn.]. The black markers indicate the location of each cell, $n=554$. (b) Overestimate of the total area (gray) of the domain that is occupied by cells, $S=205000$ [reprinted from Glab et al. [33], Fig. 4(c), with permission from Pattern Recogn.]. (c) $P_x$ and $P_y$ with ${\Delta }_x={\Delta }_y=1$ (solid curves) and ${\Delta }_x={\Delta }_y=20\approx \sqrt{S/n}$ (dashed curves). The curves that fluctuate around unity are for $P_x$, while the curves that deviate from unity are for $P_y$.

Figure 11

Spatial analysis of multicellular aggregates, $XY=90000$. (a) Experimental image [reprinted from Shimoyama et al. [27], Fig. 4(L9-30), with permission from Biochem. J.]. (b) Image agents that approximate the area occupied with cells in (a), $S=31950$, $\widehat{n}=1278$, and $s=25$. (c) Average pair-correlation function $\overline{P}$ for (b), $\Delta =5$ and $N=1$.

Figure 12

Spatial analysis of evenly distributed cells, $XY=250000$. (a) Experimental image used in the analysis [38]. The superimposed markers represent the location of each cell, $n=163$. (b) Image agents that approximate the area occupied with cells in (a), $S=67050$, $\widehat{n}=2682$, and $s=25$. (c) Average pair-correlation functions $\overline{P}$, $N=10$ and $\Delta =5$. The solid curve is for image agents that approximate the area occupied with cells [e.g., (b)]. The dotted curve is for the locations of the cells, for example, the superimposed markers in (a).