Front instabilities and invasiveness of simulated 3D avascular tumors

Abstract

We use the Glazier-Graner-Hogeweg model to simulate three-dimensional (3D), single-phenotype, avascular tumors growing in an homogeneous tissue matrix (TM) supplying a single limiting nutrient. We study the effects of two parameters on tumor morphology: a diffusion-limitation parameter defined as the ratio of the tumor-substrate consumption rate to the substrate-transport rate, and the tumor-TM surface tension. This initial model omits necrosis and oxidative/hypoxic metabolism effects, which can further influence tumor morphology, but our simplified model still shows significant parameter dependencies. The diffusion-limitation parameter determines whether the growing solid tumor develops a smooth (noninvasive) or fingered (invasive) interface, as in our earlier two-dimensional (2D) simulations. The sensitivity of 3D tumor morphology to tumor-TM surface tension increases with the size of the diffusion-limitation parameter, as in 2D. The 3D results are unexpectedly close to those in 2D. Our results therefore may justify using simpler 2D simulations of tumor growth, instead of more realistic but more computationally expensive 3D simulations. While geometrical artifacts mean that 2D sections of connected 3D tumors may be disconnected, the morphologies of 3D simulated tumors nevertheless correlate with the morphologies of their 2D sections, especially for low-surface-tension tumors, allowing the use of 2D sections to partially reconstruct medically-important 3D-tumor structures.

Figure 1. Simulated growing tumors with .

(a) . 2D sections of a 3D simulation along the XY (first row), XZ (second row) and YZ plane (third row). The developing tumor is initially spherical; it then becomes slightly irregular. Fourth row: 3D visualization of the same simulation. (b) . 2D sections of a 3D simulation along the XY (first row), XZ (second row) and YZ plane (third row). The developing tumor is initially spherical; it then becomes grooved. Fourth row: 3D visualization of the same simulation. (c) . 2D sections of a 3D simulation along the XY (first row), XZ (second row) and YZ plane (third row). The developing tumor is initially spherical; it then becomes grooved. Fourth row: 3D visualization of the same simulation. (d) . 2D sections of a 3D simulation along the XY (first row), XZ (second row) and YZ plane (third row). The developing tumor is initially spherical; it then becomes grooved with a rough surface. Fourth row: 3D visualization of the same simulation. The simulation time is indicated in days beneath each column, where 1 day = 400 MCS.

Figure 2. Simulated growing tumors with .

(a) . 2D sections of a 3D simulation along the XY (first row), XZ (second row) and YZ plane (third row). The developing tumor is initially compact; it then becomes dendritic. The disconnected parts in the last image connect to the backbone of the tumor out of the section plate. Fourth row: 3D visualization of the same simulation. (b) . 2D sections of a 3D simulation along the XY (first row), XZ (second row) and YZ plane (third row). The developing tumor is initially compact; it then becomes dendritic. The disconnected parts in the last two images connect to the backbone of the tumor out of the section plate. Fourth row: 3D visualization of the same simulation. (c) . 2D sections of a 3D simulation along the XY (first row), XZ (second row) and YZ plane (third row). The developing tumor is initially compact with a rough surface; it then becomes seaweed-like. The disconnected parts in the last two images connect to the backbone of the tumor out of the section plate. Fourth row: 3D visualization of the same simulation. (d) . 2D sections of a 3D simulation along the XY (first row), XZ (second row) and YZ plane (third row). The developing tumor is seaweed-like with a rough surface. The disconnected parts in the images connect to the backbone of the tumor out of the section plate. Fourth row: 3D visualization of the same simulation. The simulation time is indicated in days beneath each column, where 1 day = 400 MCS.

Figure 3. Simulated growing tumors with .

(a) . 2D sections of a 3D simulation along the XY (first row), XZ (second row) and YZ plane (third row). The developing tumor is initially compact; it then becomes dendritic. The disconnected parts in the last two images connect to the backbone of the tumor out of the section plate. Fourth row: 3D visualization of the same simulation. (b) . 2D sections of a 3D simulation along the XY (first row), XZ (second row) and YZ plane (third row). The developing tumor becomes dendritic. The disconnected parts in the last two images connect to the backbone of the tumor out of the section plate. Fourth row: 3D visualization of the same simulation. (c) . 2D sections of a 3D simulation along the XY (first row), XZ (second row) and YZ plane (third row). The developing tumor has a form intermediate between dendrite and seaweed. The disconnected parts in the images connect to the backbone of the tumor out of the section plate. Fourth row: 3D visualization of the same simulation. (d) . 2D sections of a 3D simulation along the XY (first row), XZ (second row) and YZ plane (third row). The developing tumor is seaweed-like. The disconnected parts in the images connect to the backbone of the tumor out of the section plate. Fourth row: 3D visualization of the same simulation. The simulation time is indicated in days beneath each column, where 1 day = 400 MCS.

Figure 4. Simulated growing tumors with .

(a) . 2D sections of a 3D simulation along the XY plane. The developing tumor remains compact and ceases proliferating. We do not show 2D sections along the XZ and YZ planes because they are essentially indistinguishable from those along the XY plane. We do not show 3D visualization of the same simulation because it does not provide any new information about the simulated tumor. (b) . 2D sections of a 3D simulation along the XY (first row), XZ (second row) and YZ plane (third row). The developing tumor forms a truncated dendrite. The disconnected parts in the last image connect to the backbone of the tumor out of the section plate. Fourth row: 3D visualization of the same simulation. (c) . 2D sections of a 3D simulation along the XY (first row), XZ (second row) and YZ plane (third row). The developing tumor has a form intermediate between dendrite and seaweed, with thinner fingers. The disconnected parts in the images connect to the backbone of the tumor out of the section plate. Fourth row: 3D visualization of the same simulation. (d) . 2D sections of a 3D simulation along the XY (first row), XZ (second row) and YZ plane (third row). The developing tumor forms a seaweed. The disconnected parts in the images connect to the backbone of the tumor out of the section plate. Fourth row: 3D visualization of the same simulation. The simulation time is indicated in days beneath each column, where 1 day = 400 MCS.

Figure 5. Morphologies of 3D tumors visualized in 3D and sphericity as a function of and , observed when the simulated tumor reaches the boundaries of the simulation domain (6 mm).

The standard deviation for is less than 0.02. The panel for and is blank because the corresponding tumor never grows to this size.

Figure 6. Sphericity of simulated tumors as a function of for different .

(a) When they reach the boundary of the simulation domain, (b) with 1000 Generalized Cells.

Figure 7. Mean circularity as a function of time for 2D sections of 3D simulations of tumor growth.

(a) , (b) , (c) , and (d) .

Figure 8. Sphericity as a function of time for 3D simulations of tumor growth.

(a) , (b) , (c) , and (d) .

Figure 9. as a function of time for 3D simulations of tumor growth.

(a) , (b) , (c) , and (d) .

Figure 10. Simulated growing tumors with quiescence or necrosis for .

(a) 2D sections of 3D simulations with quiescence with . (b) 2D sections of 3D simulations with quiescence with . Green - proliferating cells, blue - quiescent cells. (c) 2D sections of 3D simulations with necrosis with . (d) 2D sections of 3D simulations with necrosis with . Green - proliferating cells, red - necrotic cells.