Flow Correlated Percolation during Vascular Remodeling in Growing Tumors

A theoretical model based on the molecular interactions between a growing tumor and a dynamically evolving blood vessel network describes the transformation of the regular vasculature in normal tissues into a highly inhomogeneous tumor specific capillary network. The emerging morphology, characterized by the compartmentalization of the tumor into several regions differing in vessel density, diameter, and necrosis, is in accordance with experimental data for human melanoma. Vessel collapse due to a combination of severely reduced blood flow and solid stress exerted by the tumor leads to a correlated percolation process that is driven towards criticality by the mechanism of hydrodynamic vessel stabilization.

Figure 1

Schematic illustration of the model: (a) TC proliferation, (b) TC death, (c) vessel growth, (d) vessel dilatation, (e) vessel collapse due to low shear force, and (f) collapse of uncirculated vessels.

Figure 2

The time evolution of the tumor or vessel system demonstrated by 3 snapshots at time $t=0$, 200, and 400. (a)–(c) Only the tumor is presented (note the necrotic regions inside), (d)–(f) only the vessel network is presented (note the increased MVD at the tumor periphery, and the reduced MVD and dilated vessels in the tumor center), and (g)–(i) shows an equatorial cross section of the whole system in the $xy$ plane at $z=L/2$. The parameter values are given in the text. The color code of the TCs represents the age scaled to $[0,1]$ and the color code of the vessel indicates the scaled blood flow, $q(e)/q_0$.

Figure 3

(a) Tumor density ${\rho }_{\mathrm{TC}}$, (b) MVD, (c) vessel diameter $d$, (d) blood pressure gradient $\nabla P$, and (f) shear force $f$ as a function of the distance to the center $R=|\mathbf{r}-{\mathbf{r}}_{\mathrm{c}}|$ for different times $t$ [see (a)]. Panel (e) shows $\nabla P$ as a function of the azimuthal angle $\theta$. The data are averaged over all sites with the same $R$ (or $\theta$). Except for ${\rho }_{\mathrm{TC}}$, all quantities are normalized to their (constant) values in the original network.

Figure 4

Log-log plot of the box count $N_{\epsilon }$ (see text) vs $\epsilon$ (in units of $a$) for vessel networks at $t=400$ for different values of $t_{\mathrm{collapse}}$ and $t_{\max }$. The straight line is the best fit $N_{\epsilon }\sim {\epsilon }^{-D_f}$, with $D_f=2.52(5)$ being its slope. The local slope of the data increases monotonically from 1, the fractal dimension of an individual vessel, to its asymptotic value (cf. 19). The inset shows $N_{\epsilon }$ of different concentric shells of thickness 20 for $t_{\mathrm{collapse}}=50$ and $t_{\max }=20$. The slopes of the dashed upper line and solid lower line are $-2.24$ and $-1.68$, respectively.