Efficient coarse simulation of a growing avascular tumor

The subject of this work is the development and implementation of algorithms which accelerate the simulation of early stage tumor growth models. Among the different computational approaches used for the simulation of tumor progression, discrete stochastic models (e.g., cellular automata) have been widely used to describe processes occurring at the cell and subcell scales (e.g., cell-cell interactions and signaling processes). To describe macroscopic characteristics (e.g., morphology) of growing tumors, large numbers of interacting cells must be simulated. However, the high computational demands of stochastic models make the simulation of large-scale systems impractical. Alternatively, continuum models, which can describe behavior at the tumor scale, often rely on phenomenological assumptions in place of rigorous upscaling of microscopic models. This limits their predictive power. In this work, we circumvent the derivation of closed macroscopic equations for the growing cancer cell populations; instead, we construct, based on the so-called “equation-free” framework, a computational superstructure, which wraps around the individual-based cell-level simulator and accelerates the computations required for the study of the long-time behavior of systems involving many interacting cells. The microscopic model, e.g., a cellular automaton, which simulates the evolution of cancer cell populations, is executed for relatively short time intervals, at the end of which coarse-scale information is obtained. These coarse variables evolve on slower time scales than each individual cell in the population, enabling the application of forward projection schemes, which extrapolate their values at later times. This technique is referred to as coarse projective integration. Increasing the ratio of projection times to microscopic simulator execution times enhances the computational savings. Crucial accuracy issues arising for growing tumors with radial symmetry are addressed by applying the coarse projective integration scheme in a cotraveling (cogrowing) frame. As a proof of principle, we demonstrate that the application of this scheme yields highly accurate solutions, while preserving the computational savings of coarse projective integration.

Figure 1

Discretized two-dimensional host tissue. Blood vessels provide the tissue, at its borders, with proliferation and survival nutrients at a constant rate.

Figure 2

Results obtained from direct simulation of the cellular automaton when $L_x=L_y=L=1001$, ${\theta }_{\mathrm{nec}}=0.03$, ${\theta }_{\mathrm{div}}=0.3$, ${\theta }_{\mathrm{mig}}=1000$, $a=1/L$, ${\lambda }_N=25$, and ${\lambda }_M=25$. Here, we show snapshots of the tumor cell distribution at dimensionless times: (a) $t=100$, (b) $t=500$, and (c) $t=1000$. The growing tumor exhibits a (coarsely) radially symmetric shape on the coarse scale. See text for additional details.

Figure 3

Evolution of the total number of tumor cells of the growing tumor shown in Fig. 2.

Figure 4

Tumor front evolution (red line) for the growing tumor shown in Fig. 2. The black line corresponds to a circular front with radius equal to the mean radius of the tumor front.

Figure 5

Mean radial tumor cell distribution at time $t=400$, when the domain is divided into (a) two sectors ($-\pi \le \theta \le 0,0\le \theta \le \pi$) and (b) four sectors ($-\pi \le \theta \le -\pi /2,-\pi /2\le \theta \le 0,0\le \theta \le \pi /2,\pi /2\le \theta \le \pi$). The mean radial distributions computed at different sectors appear “coarsely” the same, indicating the radial symmetry of the developing tumor.

Figure 6

Schematic of a coarse time stepper for the model of a growing avascular tumor with radial symmetry.

Figure 7

(a) Initial mean radial tumor cell density $f_0$ given by the expression in Eq.(10). (b) One microscopic cell distribution consistent with $f_0$. (c) $f_{\text{lift}}$ (red broken line) is the restriction of the distribution in figure (b), lifted from $f_0$. Comparing $f_{\text{lift}}$ with the initial distribution $f_0$ (blue solid line) demonstrates that the restriction operation is the (coarse) approximate inverse of the lifting step.

Figure 8

Comparison of mean radial density of tumor cells at time $t=450$, obtained by direct simulations (blue line) and coarse projective integration (red line). The (clearly erroneous) result of simple (non-comoving frame) coarse projective integration is obtained using $t_1=300$, $t_2=350$, and $t_{\text{project}}=450$ with $K=50$ Fourier coefficients.

Figure 9

Mean radial density of tumor cells (blue line) at $t=500$ and the corresponding distribution produced by a 50 Fourier mode truncation (red broken line). The two distributions visually coincide.

Figure 10

Evolution of the mean radial distribution of tumor cells $f\left(r\right)$. We observe that the evolution is characteristic of a moving front.

Figure 11

Comparison of mean radial density of tumor cells at times $t=550$, 750, and 950. The blue line corresponds to the results obtained from direct simulation, and with red broken lines we depict the results of the coarse projective integration application in a comoving frame with $t_2-t_1=50$ and $t_{\text{project}}-t_2=100$ using $K=50$ Fourier coefficients. Good visual agreement is observed.

Figure 12

Comparison of mean radial density of tumor cells at times $t=600$, 750, and 900. The blue lines correspond to the results obtained from direct simulation, and with red broken lines we depict the results of the application of coarse projective integration in a comoving frame with $t_2-t_1=25$ and $t_{\text{project}}-t_2=100$ using $K=50$ Fourier coefficients.