Dynamics of tissue topology during cancer invasion and metastasis

Abstract

During tumor progression, cancer cells mix with other cell populations including epithelial and endothelial cells. Although potentially important clinically as well as for our understanding of basic tumor biology, the process of mixing is largely a mystery. Furthermore, there is no rigorous, analytical measure available for quantifying the mixing of compartments within a tumor. I present here a mathematical model of tissue repair and tumor growth based on collective cell migration that simulates a wide range of observed tumor behaviors with correct tissue compartmentalization and connectivity. The resulting dynamics are analyzed in light of the Euler characteristic number (χ), which describes key topological features such as fragmentation, looping and cavities. The analysis predicts a number of regimes in which the cancer cells can encapsulate normal tissue, form a co-interdigitating mass, or become fragmented and encapsulated by endothelial or epithelial structures. Key processes that affect the topological changes are the production of provisional matrix in the tumor, and the migration of endothelial or epithelial cells on this matrix. Furthermore, the simulations predict that topological changes during tumor invasion into blood vessels may contribute to metastasis. The topological analysis outlined here could be useful for tumor diagnosis or monitoring response to therapy and would only require high resolution, 3D image data to resolve and track the various cell compartments.

Figure 1

Topological characteristics of 3D solids. In this illustration, the Euler numbers for the orange and black objects are designated χ_E and χ_C, respectively. The Euler number for a solid object with no holes or cavities is 1, regardless of surface features (A). Each penetrating hole decreases χ by one unit (B). χ for an ensemble of objects is the sum of the individual χ values (C). When two solids interact, each has a characteristic Euler number that is determined by its own topology (D). Cavities increase χ when completely contained within the solid (E).

Figure 2

Operations of the cellular automaton model. Single 2D slices of the domain are shown, but all operations extend to the third dimension.

Figure 3

Simulated wound healing. Shown are cross-sections of the 3D domain through the center of the wound at selected time steps (A). The domain contains the surface of the skin (planar structure at top, yellow epithelium, blue basement membrane), a cylindrical blood vessel (middle, green endothelium with blue basement membrane), and a spherical gland (bottom, yellow epithelium, blue basement membrane). At t = 0 a puncture wound is simulated, leaving behind a cuboid of provisional matrix that connects these structures (B). Following the rules outlined in figure 2, the endothelium, epithelia and basement membranes close the wound and return to the correct anatomy. During this process, the Euler numbers transiently decrease due to loop formation, then increase as provisional matrix is encapsulated by the migrating endo- and epithelium. By the end of the simulation, the Euler numbers return to the baseline value of 1 ((C); green: χ_endo, yellow: χ_epi).

Figure 4

Growth of a tumor initiated by a cancer cell (black) in a plane of epithelial cells (yellow, (A)). A single y–z plane of the 3D domain is shown. A portion of a cylindrical blood vessel appears at bottom. Over 30 time steps, the tumor expands and mixes with the epithelium, but is not allowed to produce provisional matrix or cross the basement membrane. Exponential tumor growth (B) is associated with an increase in the number of epithelial objects (fragmentation; (C)), along with increases in the cancer Euler number, χ_c, and decreases in the epithelium Euler number, χ_ep (D). Plots in (B)–(D) depict the mean ± sd from 16 independent simulations.

Figure 5

Growth of a tumor initiated by a cancer cell (black) in a plane of epithelial cells (yellow, (A)). The tumor expands and produces provisional matrix (red), which provides pathways for epithelial migration. Exponential tumor growth (B) is slower than in case b because the epithelial cells are not included in the calculation of tumor volume, and their migration replaces cancer cells. The epithelium remains contiguous, but its migration partially fragments the tumor (C). Overall, this process results in Euler numbers that become increasingly negative (D), indicating that the epithelium and cancer cells interdigitate, forming two interconnected porous structures. Plots in (B)–(D) depict the mean ± sd from 16 independent simulations.

Figure 6

Illustration of two interdigitating spaces, both with negative Euler numbers.

Figure 7

Growth of a tumor with increased epithelial affinity to provisional matrix. Again, tumor growth (B) is slowed due to epithelial migration. In this case, the epithelium remains contiguous, and there is more fragmentation of the tumor (C). The positive χ_ep indicates many cancer-containing cavities within the epithelium (D); however, the decreasing, negative χ_c shows that the cancer space also contains a looping, porous structure. Plots in (B)–(D) depict the mean ± sd from 16 independent simulations.

Figure 8

Topological consequences of changing the affinity of epithelium to provisional matrix or its production rate. Parameter sets are listed in table 2; number of objects and Euler numbers are from time step = 40.

Figure 9

Intravasation of a tumor (black) into a blood vessel. The tumor is initially seeded outside the basement membrane (blue, (A)), It crosses the basement membrane and interacts with the endothelium (green) over 30 time steps. As the tumor grows (B), it encapsulates endothelial cells, causing fragmentation (C). In the process, χ_c increases, while χ_en decreases (D). Plots in (B)–(D) depict the mean ± sd from 16 independent simulations.

Figure 10

Intravasation of a tumor (black) into a blood vessel with increased production of provisional matrix (red). The cancer cells and endothelium mix, but there is minimal fragmentation of both spaces (C). Negative χ_c and χ_en values indicate interconnected looping structures (D). Plots in (B)–(D) depict the mean ± sd from 16 independent simulations.

Figure 11

Intravasation of a tumor (black) into a blood vessel with increased endothelial migration into the provisional matrix (red). The endothelial cells dominate the system, breaking up the cancer space (C). Negative χ_c indicates interconnected looping structures, although many cancer cells are encapsulated by the endothelium (positive χ_en, (D)). Plots in (B)–(D) depict the mean ± sd from 16 independent simulations.

Figure 12

Topological consequences of changing the affinity of endothelium to provisional matrix or its production rate. Parameter sets are listed in table 2; number of objects and Euler numbers are from time step = 40.