Bridging the gap between individual-based and continuum models of growing cell populations

Abstract

Continuum models for the spatial dynamics of growing cell populations have been widely used to investigate the mechanisms underpinning tissue development and tumour invasion. These models consist of nonlinear partial differential equations that describe the evolution of cellular densities in response to pressure gradients generated by population growth. Little prior work has explored the relation between such continuum models and related single-cell-based models. We present here a simple stochastic individual-based model for the spatial dynamics of multicellular systems whereby cells undergo pressure-driven movement and pressure-dependent proliferation. We show that nonlinear partial differential equations commonly used to model the spatial dynamics of growing cell populations can be formally derived from the branching random walk that underlies our discrete model. Moreover, we carry out a systematic comparison between the individual-based model and its continuum counterparts, both in the case of one single cell population and in the case of multiple cell populations with different biophysical properties. The outcomes of our comparative study demonstrate that the results of computational simulations of the individual-based model faithfully mirror the qualitative and quantitative properties of the solutions to the corresponding nonlinear partial differential equations. Ultimately, these results illustrate how the simple rules governing the dynamics of single cells in our individual-based model can lead to the emergence of complex spatial patterns of population growth observed in continuum models.

Appendices

Details of numerical simulations of the individual-based model

We use a uniform discretisation of the interval [0, 100] that consists of 1001 points as the spatial domain (i.e. the grid-step is $\chi =0.1$) and we choose the time-step $\tau =2\times 10^{-3}$. We implement zero-flux boundary conditions by letting the attempted move of a cell be aborted if it requires moving out of the spatial domain. For all simulations, we use the definition (50) of G(p) and we perform numerical computations in Matlab. Further details specific either to the case of one single cell population or to the case of two cell populations are provided in the next sections.

1.1 Setup of numerical simulations for the case of one single cell population

For consistency with Eq. (1), we set $M=1$ and we drop the index $h=1$ both from the functions and from the parameters of the individual-based model. We define the rate G in Eqs. (9)–(11) according to Eq. (50). We set the homeostatic pressure $P=120 \times 10^4$ for the simulation results reported in Figs. 2 and 3, while we choose $P=120 \times 10^5$ for the simulation results of Fig. 4. Moreover, we choose $\beta = 4 \times 10^{-6}$ for the simulation results reported in Fig. 2, $\beta = 4 \times 10^{-5}$ for the simulation results of Fig. 4, and $\beta \in \left\{ 1.5 \times 10^{-6}, 4 \times 10^{-6}, 4 \times 10^{-5} \right\}$ for the simulation results of Fig. 3. We set $\nu =0.02$ in Eqs. (13)–(15) and we define the pressure $p^k_i$ according to the following barotropic relation

$$\begin{aligned} \varPi (\rho ^k_i) = K_{\gamma } \, (\rho ^k_i)^{\gamma } \quad \text {with} \quad K_{\gamma } = \frac{\gamma +1}{\gamma } \quad \text {and} \quad \gamma >1, \end{aligned}$$

which satisfies conditions (3). We let $\gamma \in \left\{ 1.2, 1.5, 2 \right\}$ for the simulation results of Fig. 4, while we choose $\gamma = 1.2$ for the simulation results reported in Figs. 2 and 3. We use the initial cell density

$$\begin{aligned} \rho ^0_{i} = A \, \exp {\left( - b \, x_i^{2}\right) } \quad \text {with} \quad A = 2\times 10^{4} \quad \text{ and } \quad b=4\times 10^{-3}. \end{aligned}$$

Additionally, we impose compact support on the initial condition by ensuring $\rho _{i}^{0}=0$ for all $x_{i} \ge 50$.

The results presented in Figs. 2 and 3 correspond to the average over three simulations of our individual-based model, while the results in Fig. 4 correspond to one single simulation when $\gamma =1.5$ or $\gamma =2$ and the average over two simulations when $\gamma =1.2$.

1.2 Setup of numerical simulations for the case of two cell populations: Figs. 5 and 6

For consistency with the system of Eq. (7), we choose $M=2$, and we set $G_1 \equiv G$ and $G_2 \equiv 0$ in Eqs. (9)–(11), where G is defined according to Eq. (50) with the homeostatic pressure $P=10 \times 10^4$ and the factor $\beta = 4 \times 10^{-5}$. We set $\nu _1 = 0.01$ and $\nu _2 = 0.5$ in Eqs. (13)–(15) for the simulation results reported in Fig. 5, while we consider $\nu _1 = 0.5$ and $\nu _2 = 0.01$ for the simulation results of Fig. 6. We define the pressure $p^k_i$ according to the following simplified barotropic relation

$$\begin{aligned} \varPi (\rho ^k_i) = K \, \rho ^k_i \quad \text {with} \quad K = 2, \end{aligned}$$

which satisfies conditions (3). We make use of the initial cell densities

$$\begin{aligned} \rho ^0_{1 i} = A_{1} \exp {\left( -b_{1} \, x_i^{2}\right) } \quad \text{ and } \quad \rho ^0_{2 i}=\left\{ \begin{array}{ll} 0 , \quad \text {for } x_i \le 13, \\ \\ A_{2} \exp {\left( -b_{2} (x_i-14)^{2}\right) }, \quad \text {for } x_i \in (13,29), \\ \\ 0 , \quad \text {for } x_i \ge 29, \end{array} \right. \end{aligned}$$

(52)

where

$$\begin{aligned} A_{1}=1.25\times 10^{4}, \quad b_{1}=0.06, \quad A_{2}=2.5\times 10^{4} \quad \text{ and } \quad b_{2}=6\times 10^{-3}. \end{aligned}$$

The results presented in Figs. 5 and 6 correspond to one single simulation of our individual-based model.

1.3 Setup of numerical simulations for the case of two populations: Figs. 7 and 8

For consistency with the system of Eq. (7), we choose $M=2$, and we set $G_1 \equiv G$ and $G_2 \equiv 0$ in Eqs. (9)–(11), where G is defined according to Eq. (50) with the homeostatic pressure $P=10 \times 10^4$ and the factor $\beta = 4 \times 10^{-5}$. We set $\nu _1 = 0.01$ and $\nu _2 = 0.5$ in Eqs. (13)–(15) for the simulation results reported in Fig. 7, while we consider $\nu _1 = 0.5$ and $\nu _2 = 0.01$ for the simulation results of Fig. 8. We define the pressure $p^k_i$ according to the following barotropic relation

$$\begin{aligned} \varPi (\rho ^k_i) = q \, (\rho _i^k- \rho ^*)_+ \quad \text{ where } \quad q=10 \quad \text{ and } \quad \rho ^* = r \, P \; \text { with } \; r=10^{-3}, \end{aligned}$$

which satisfies conditions (51). We make use of the initial cell densities (52) with

$$\begin{aligned} A_{1}=12.5\times 10^{4}, \quad b_{1}=0.06, \quad A_{2}=25\times 10^{4} \quad \text{ and } \quad b_{2}=6\times 10^{-3}. \end{aligned}$$

The results presented in Figs. 7 and 8 correspond to one single simulation of our individual-based model.

Details of numerical simulations of the continuum models

We let $x \in [0,100]$ and we construct numerical solutions for Eq. (1) and for the system of Eq. (7) complemented with zero Neumann boundary conditions. We use a finite volume method based on a time-splitting between the conservative and nonconservative parts. For the conservative parts, transport terms are approximated through an upwind scheme whereby the cell edge states are calculated by means of a high-order extrapolation procedure (LeVeque 2002), while the forward Euler method is used to approximate the nonconservative parts. We consider a uniform discretisation of the interval [0, 100] that consists of 1001 points and we perform numerical computations in Matlab. For all simulations, we use the definition (50) of G(p). Further details specific either to the case of one cell population or to the case of two cell populations are provided in the next sections.

1.1 Setup of numerical simulations for Eq. (1)

The rate G is defined according to Eq. (50) with the homeostatic pressure $P=120 \times 10^4$ for the numerical solutions reported in Figs. 2 and 3, while we choose $P=120 \times 10^5$ for the numerical solutions of Fig. 4. Moreover, we choose $\beta = 4 \times 10^{-6}$ for the numerical solutions reported in Fig. 2, $\beta = 4 \times 10^{-5}$ for the numerical solutions of Fig. 4, and $\beta \in \left\{ 1.5 \times 10^{-6}, 4 \times 10^{-6}, 4 \times 10^{-5} \right\}$ for the numerical solutions reported in Fig. 3. We define the pressure p according to the following barotropic relation

$$\begin{aligned} \varPi (\rho ) = K_{\gamma } \, \rho ^{\gamma } \quad \text {with} \quad K_{\gamma } = \frac{\gamma +1}{\gamma } \quad \text {and} \quad \gamma >1, \end{aligned}$$

which satisfies conditions (3). We let $\gamma \in \left\{ 1.2, 1.5, 2 \right\}$ for the numerical solutions of Fig. 4, while we choose $\gamma = 1.2$ for the numerical solutions reported in Figs. 2 and 3. Given the parameter values used for the individual-based model in the case of one single cell population, we choose the mobility $\mu = 4.166\times 10^{-7}$ for the numerical solutions reported in Figs. 2 and 3, while we set $\mu = 4.166\times 10^{-8}$ for the numerical solutions of Fig. 4. In this way, both values of $\mu$ satisfy condition (21) for $h=1$. We impose the initial condition

$$\begin{aligned} \rho (0,x) = A \, \exp {\left( - b \, x^{2}\right) } \quad \text {with} \quad A = 2\times 10^{4} \quad \text{ and } \quad b=4\times 10^{-3}. \end{aligned}$$

Additionally, we impose compact support on the initial condition by ensuring $\rho (0,x)=0$ for all $x \ge 50$.

1.2 Setup of numerical simulations for the system of Eq. (7): Figs. 5 and 6

The rate G is defined according to Eq. (50) with the homeostatic pressure $P=10 \times 10^4$ and $\beta = 4 \times 10^{-5}$. Given the parameter values used for the individual-based model in the case of two cell populations, we choose the mobilities $\mu _1 = 2.5\times 10^{-7}$ and $\mu _2 = 1.25\times 10^{-5}$ for the numerical solutions reported in Fig. 5, and the mobilities $\mu _1 = 1.25\times 10^{-5}$ and $\mu _2 = 2.5\times 10^{-7}$ for the numerical solutions of Fig. 6. This ensures that conditions (21) for $h=1,2$ are satisfied. We define the pressure p according to the following simplified barotropic relation

$$\begin{aligned} \varPi (\rho ) = K \, \rho \quad \text {with} \quad K = 2, \end{aligned}$$

which satisfies conditions (3). We impose the initial conditions

$$\begin{aligned} \rho ^0_{1}(0,x) = A_{1} \exp {\left( -b_{1} \, x^{2}\right) } \quad \text{ and } \quad \rho ^0_{2}=\left\{ \begin{array}{ll} 0 , \quad \text {for } x \le 13, \\ \\ A_{2} \exp {\left( -b_{2} (x-14)^{2}\right) }, \quad \text {for } x \in (13,29), \\ \\ 0 , \quad \text {for } x \ge 29, \end{array} \right. \end{aligned}$$

(53)

where

$$\begin{aligned} A_{1}=1.25\times 10^{4}, \quad b_{1}=0.06, \quad A_{2}=2.5\times 10^{4} \quad \text{ and } \quad b_{2}=6\times 10^{-3}. \end{aligned}$$

1.3 Setup of numerical simulations for the system of Eq. (7): Figs. 7 and 8

The rate G is defined according to Eq. (50) with the homeostatic pressure $P=10 \times 10^4$ and $\beta = 4 \times 10^{-5}$. Given the parameter values used for the individual-based model in the case of two cell populations, we choose the mobilities $\mu _1 = 2.5\times 10^{-7}$ and $\mu _2 = 1.25\times 10^{-5}$ for the numerical solutions reported in Fig. 7, and the mobilities $\mu _1 = 1.25\times 10^{-5}$ and $\mu _2 = 2.5\times 10^{-7}$ for the numerical solutions of Fig. 8. This ensures that conditions (21) for $h=1,2$ are satisfied. We define the pressure p according to the following barotropic relation

$$\begin{aligned} \varPi (\rho ) = q \, (\rho - \rho ^*)_+ \quad \text{ where } \quad q=10 \quad \text{ and } \quad \rho ^* = r \, P \; \text { with } \; r=10^{-3}, \end{aligned}$$

which satisfies conditions (51). We impose the initial conditions (53) with

$$\begin{aligned} A_{1}=12.5\times 10^{4}, \quad b_{1}=0.06, \quad A_{2}=25\times 10^{4} \quad \text{ and } \quad b_{2}=6\times 10^{-3}. \end{aligned}$$