Classifying the Expansion Kinetics and Critical Surface Dynamics of Growing Cell Populations

We systematically study the growth kinetics and the critical surface dynamics of cell monolayers by a class of computationally efficient cellular automaton models avoiding lattice artifacts. Our numerically derived front velocity relationship indicates the limitations of the Fisher-Kolmogorov-Petrovskii-Piskounov equation for tumor growth simulations. The critical surface dynamics corresponds to the Kardar-Parisi-Zhang universality class, which disagrees with the interpretation by Bru et al. of their experimental observations as generic molecular-beam-epitaxy-like growth, questioning their conjecture that a successful therapy should lead away from molecular beam epitaxy.

Figure 1

(a) Construction of the CA lattice: one point (black, green) is placed in every square of a square lattice at a random position ${\underline{r}}_i$. A Voronoi tessellation is constructed from these points such that a cell $i$ consists of all points in space that are closer to point ${\underline{r}}_i$ than to any ${\underline{r}}_k$ with $k\ne i$. The shape of a biological cell (white) is identified with the corresponding Voronoi polygon (blue lines). Polygons that share a common edge are defined as neighboring and connected by red lines (Delauney triangulation). (b) Probability density distribution of the cell area for the CA lattice in (a) (brown) and for a random initial distribution of points (red). (c) Cell cluster morphology for $m={10}^4$, $\Delta L=l$ on (i) the CA lattice in (a), (ii) square, (iii) hexagonal lattice, (iv) lattice with Moore neighborhood (nearest neighbors along the axes and the diagonals), (v) off-lattice cluster [2, 20].

Figure 2

(a) Mean radius $\overline{R}$ of the cell aggregate vs time $t$. Full circles: experiment for C6 rat astrocyte glioma cells [3]. Other symbols: CA simulations for different migration rules, R5(i) (a): $\Delta L=9$, $\varphi =0$, R5(i) (b): $\Delta L=1$, $\varphi =25$, R5(ii): $\Delta L=8$, $\varphi =25$, R5(iii) $\Delta L=9$, $\varphi =25$, $\Delta E/F_T=E_0+n{E}_B$ with $E_B=10$, $n$ neighbors, $E_0=5$, R5(iv): $\Delta L=\Delta \widehat{L}=6$, $\widehat{\varphi }=1$. (b) Cell cycle time distribution $f({\tau }^{\prime })$ for off-lattice model and CA growth model in comparison with Erlang distribution. ($m=60$, $\tau =19\mathrm{h}$, $l=10\mu \mathrm{m}$ [2]. $\Delta L$ and $\varphi$ in units of $l$ and ${\tau }^{-1}$.)

Figure 3

(a) $Y=R_{gyr}^2/(\varphi +1/{\tau }_{\mathrm{eff}})$ vs $t$ for $m=1$, $\Delta L=1$ and different values for $\varphi$. (b)–(d) Growth in the linear expansion regime ($N\sim {10}^5$). (b) Square of expansion velocity, $v^2$, vs square of the proliferation zone, $\Delta L^2$ (▵: $\varphi =0$, ○: $\varphi =10$, □: $\varphi =20$; $m=1$). (c) $v^2$ vs $\varphi$ (▵: $\Delta L=1$, ○: $\Delta L=3$, □: $\Delta L=6$, ☆: $\Delta L=10$; $m=1$). (d) $v$ vs $m$ ($\Delta L=1$, $\varphi =0$). The lines are fits using Eq. (1). (Time and length in units of $\tau$ and $l$.)

Figure 4

(a) Dynamic structure function $S(k,t)$ vs $k$ for [R5(i)], $\Delta L=0$, $\varphi =0$, $m=1$. Inset: scaling function $\tilde{s}=S(k,t)k^{2\alpha +1}$ vs $k{t}^{1/z}$ ($\alpha =0.5$, $z=3/2$). (b) $S(k,t)$ vs $k$ for four alternative parameter sets: (A) ▵: $m=5$ ($\Delta L=0$, $\varphi =0$), (B) ○: $\Delta L=6$ ($m=1$, $\varphi =0$), (C) ☆: R5(ii) $\varphi =100$ ($m=1$, $\Delta L=0$), (D) □: R5(iii) $\Delta E/F_T$ as in Fig. 2a ($m=1$, $\Delta L=0$) [24]. The dashed lines are guides to the eye showing $\alpha =0.5$. (Time and length in units of $\tau$ and $l$.)