A Hybrid Discrete–Continuum Modelling Approach to Explore the Impact of T-Cell Infiltration on Anti-tumour Immune Response

Abstract

We present a spatial hybrid discrete–continuum modelling framework for the interaction dynamics between tumour cells and cytotoxic T cells, which play a pivotal role in the immune response against tumours. In this framework, tumour cells and T cells are modelled as individual agents while chemokines that drive the chemotactic movement of T cells towards the tumour are modelled as a continuum. We formally derive the continuum counterpart of this model, which is given by a coupled system that comprises an integro-differential equation for the density of tumour cells, a partial differential equation for the density of T cells and a partial differential equation for the concentration of chemokines. We report on computational results of the hybrid model and show that there is an excellent quantitative agreement between them and numerical solutions of the corresponding continuum model. These results shed light on the mechanisms that underlie the emergence of different levels of infiltration of T cells into the tumour and elucidate how T-cell infiltration shapes anti-tumour immune response. Moreover, to present a proof of concept for the idea that, exploiting the computational efficiency of the continuum model, extensive numerical simulations could be carried out, we investigate the impact of T-cell infiltration on the response of tumour cells to different types of anti-cancer immunotherapy.

Appendices

Appendix A Formal Derivation of the Continuum Model

Building on the methods employed in Bubba et al. (2020), we carry out a formal derivation of the deterministic continuum model given by the IDE-PDE-PDE system (3.2) for $d = 1$. Similar methods can be used in the case where $d = 2$.

1.1 A.1 Formal Derivation of the IDE for the Density of Tumour Cells n(x, t)

When tumour cell dynamics are governed by the rules described in Sects. 2.1.1 and 2.1.2, considering $i \in [0,\mathcal {N}]$, between time-steps k and $k+1$ the principle of mass balance gives the following difference equation for the tumour cell density $n_{i}^{k}$:

$$\begin{aligned} n_{i}^{k+1}=\left[ 2 \, \tau \alpha _n + 1-\tau (\alpha _n+\zeta _n K_{i}^{k}+\mu _n\rho _n^k \right] n_{i}^{k}. \end{aligned}$$

(A1)

Using the fact that the following relations hold for $\tau$ and $\chi$ sufficiently small

$$\begin{aligned}{} & {} t_k\approx t, \quad t_{k+1}\approx t+\tau , \quad x_i\approx x, \quad x_{i\pm 1}\approx x\pm \chi , \end{aligned}$$

(A2)

$$\begin{aligned}{} & {} n_{i}^{k}\approx n(x,t), \quad n_{i}^{k+1}\approx n(x,t+\tau ), \quad c_{i}^{k}\approx c(x,t), \end{aligned}$$

(A3)

$$\begin{aligned}{} & {} \rho _n^k\approx \rho _n(t):=\int _\Omega n(x,t)\,\textrm{d}x, \quad K_{i}^{k}\approx K(x,t):=\int _\Omega \eta (x,x^\prime ;\theta )c(x^\prime ,t)\,\textrm{d}x^\prime , \end{aligned}$$

(A4)

where the function $\eta$ is defined via (2.12), Eq. (A1) can be formally rewritten in the approximate form

$$\begin{aligned} n(x,t+\tau )-n(x,t)=\tau \left( \alpha _n-\zeta _n K(x,t)-\mu _n\rho _n(t) \right) n(x,t). \end{aligned}$$

(A5)

If, in addiction, the function n(x, t) is continuously differentiable with respect to the variable t, starting from Eq. (A5), and letting the time-step $\tau \rightarrow 0$, one formally obtains the following IDE for the tumour cell density n(x, t):

$$\begin{aligned} \partial _t n(x,t) =\alpha _n n(x,t)-\mu _n\rho _n(t)\, n(x,t)-\zeta _nK(x,t) n(x,t)\quad (x,t)\in \Omega \times (0,t_f].\end{aligned}$$

1.2 A.2 Formal Derivation of the PDE for the Density of T Cells c(x, t)

When T cell dynamics are governed by the rules described in Sect. 2.3, considering $i \in [1,\mathcal {N}-1]$, between time-steps k and $k+1$ the principle of mass balance gives the following difference equation for the T cell density $c_{i}^{k}$:

$$\begin{aligned} \begin{aligned} c_i^{k+1}&=c_i^k(1-\tau \,\mu _c\rho _c^k)+\tau \,\alpha _cr^k_i \\&\quad + \frac{\lambda }{2}\psi (w_i^k)\Big (c_{i+1}^k+c_{i-1}^k\Big ) \\&\quad -\frac{\lambda }{2}\Big (\psi (w_{i-1}^k)+\psi (w_{i+1}^k)\Big )c_i^k\\&\quad +\frac{\nu }{2\phi _{\max }}\psi (w_i^k)\Big [\Big (\phi _i^k-\phi _{i-1}^k\Big )_{+}c_{i-1}^k\Big ]\\&\quad +\frac{\nu }{2\phi _{\max }}\psi (w_i^k)\Big [\Big (\phi _i^k-\phi _{i+1}^k)\Big )_{+}c_{i+1}^k\Big ] \\&\quad -\frac{\nu }{2\phi _{\max }}\psi (w_{i+1}^k)\Big [\Big (\phi _{i+1}^k-\phi _i^k\Big )_{+}c_i^k\Big ]\\&\quad -\frac{\nu }{2\phi _{\max }}\psi (w_{i-1}^k)\Big [\Big (\phi _{i-1}^k-\phi _i^k\Big )_{+}c_i^k\Big ]. \end{aligned} \end{aligned}$$

(A6)

Using the fact that relations (A2A3)–(A4) and the following relations

$$\begin{aligned}{} & {} c_i^k\approx c(x,t), \quad c_{i\pm 1}^k\approx c(x\pm \chi ), \quad \rho _c^k\approx \rho _c(t):=\int _\Omega c(x,t)\,\textrm{d}x, \\{} & {} \phi _i^k\approx \phi (x,t), \quad \phi _i^{k+1}\approx \phi (x,t+\tau ), \quad \phi _{i\pm 1}^k\approx \phi (x\pm \chi ), \\{} & {} w_i^k\approx w(x,t), \text { with } w(x,t)=n(x,t)+c(x,t), \quad w_{i\pm 1}^k\approx w(x\pm \chi ), \\{} & {} r_i^k\approx r(x,t):=\phi _{tot}(t)\mathbb {1}_\omega (x), \text { with } \phi _{tot}(t):=\displaystyle \int _\Omega \phi (x,t)\,\textrm{d}x \end{aligned}$$

hold for $\tau$ and $\chi$ sufficiently small, Eq. (A6) can be formally rewritten in the approximate form

$$\begin{aligned} c(x,t+\tau )&=c(x,t)(1-\tau \,\mu _c\rho _c(t))+\tau \,\alpha _cr(x,t) \\&\quad + \frac{\lambda }{2}\psi (w(x,t))\Big (c(x+\chi ,t)+c(x-\chi ,t)\Big )\\&\quad -\frac{\lambda }{2}\Big (\psi (w(x-\chi ,t))+\psi (w(x+\chi ,t))\Big )c(x,t) \\&\quad +\frac{\nu }{2\phi _{\max }}\psi (w(x,t))\Big [\Big (\phi (x,t)-\phi (x-\chi ,t)\Big )_{+}c(x-\chi ,t)\Big ]\\&\quad +\frac{\nu }{2\phi _{\max }}\psi (w(x,t))\Big [\Big (\phi (x,t)-\phi (x+\chi ,t)\Big )_{+}c(x+\chi ,t)\Big ] \\&\quad -\frac{\nu }{2\phi _{\max }}\psi (w(x+\chi ,t))\Big [\Big (\phi (x+\chi ,t)-\phi (x,t)\Big )_{+}c(x,t)\Big ]\\&\quad -\frac{\nu }{2\phi _{\max }}\psi (w(x-\chi ,t))\Big [\Big (\phi (x-\chi ,t)-\phi (x,t)\Big )_{+}c(x,t)\Big ]. \end{aligned}$$

Building on the methods employed in Bubba et al. (2020), letting $\tau \rightarrow 0$ and $\chi \rightarrow 0$ in such a way that

$$\begin{aligned} \frac{\lambda }{2}\frac{\chi ^2}{\tau }\rightarrow \beta _c \in \mathbb {R}_*^+ \quad \text {and} \quad \frac{\nu }{2\phi _{\max }}\frac{\chi ^2}{\tau }\rightarrow \gamma _c \in \mathbb {R}_*^+ \quad \text {as } \tau \rightarrow 0, \, \chi \rightarrow 0, \end{aligned}$$

after a little algebra, considering $(x,t)\in \Omega \setminus \partial \Omega \times (0,t_f]$, we find

$$\begin{aligned} \partial _t c- \partial _x\Big [\beta _c \psi (w) \partial _x c-\gamma _c \psi (w)c\partial _x \phi -\beta _c c\psi ^\prime (w)\partial _x w\Big ]= -\mu _c\rho _c(t)c+\alpha _cr \end{aligned}$$

where $\psi$ is given by (2.22) and $w:=n+c$. Moreover, zero-flux boundary conditions easily follow from the fact that T-cell moves that require moving out of the spatial domain are not allowed.

1.3 A.3 Formal Derivation of the Balance Equation for the Chemoattractant Concentration $\phi (x,t)$

The formal derivation of the balance equation for the chemoattractant concentration $\phi (x,t)$ is obtained using the methods employed in Bubba et al. (2020).

Appendix B Details of Numerical Simulations

The numerical simulations of our hybrid and continuum models are carried out on a two-dimensional domain and are performed in Matlab.

1.1 B.1 Details of Numerical Simulations of the Hybrid Model

The flowchart in Fig. 13 illustrates the general computational procedure to carry out simulations of the hybrid model in one-dimensional settings, while the flowchart in Fig. 14 provides further details of the computational procedure to simulate cell dynamics in one-dimensional settings. Analogous strategies are used in two-dimensional settings. All random numbers mentioned in Fig. 14 are real numbers drawn from the standard uniform distribution on the interval (0, 1), which in our case are obtained using the built-in Matlab function rand.

As summarised by Fig. 14, at any time-step, each T cell undergoes a three-phase process: Phase A) undirected, random movement according to the probabilities defined via (2.26) and (2.27); Phase B) chemotaxis according to the probabilities defined via (2.23) and (2.24); Phase C) death according to the probabilities defined via (2.17) and (2.21). We let then each tumour cell proliferate with the probability defined via (2.8), die due to intra-tumour competition with the probability defined via (2.10) or die due to immune action with the probability defined via (2.13). Finally, the tumour cell density at every lattice site is computed via (2.1) and inserted into (2.14) in order to update the concentration of the chemoattractant.

In a two-dimensional setting, the positions of the single T cells are updated following a procedure analogous to that illustrated in Figs. 13 and 14, with the only differences being that: T cells are allowed to move up and down as well; the concentration of the chemoattractant is updated through the two-dimensional analogue of (2.14), where the operator $\mathcal {L}$ is defined as the finite-difference Laplacian on a two-dimensional regular lattice of step $\chi$; the tumour and T cell densities are, respectively, computed via (2.1) and (2.2).

Fig. 13

Flowchart illustrating the computational procedure to simulate the hybrid model in one-dimensional settings. A detailed summary of steps 2) and 3) is provided by the flowchart in Fig. 14. A similar procedure is used in two-dimensional settings (Color figure online)

Fig. 14

Flowchart illustrating the detailed computational procedure followed to update the positions of every T cell, as well as the fate of each tumour cell and T cell in one-dimensional settings. Analogous strategies are used in two-dimensional domains (Color figure online)

1.2 B.2 Details of Numerical Simulations of the Continuum Model

To construct numerical solutions of the IDE-PDE-PDE system (3.2), we use a uniform discretisation consisting of $N^2 = 3721$ points of the square $\Omega := [0, 1]^2$ as the computational domain of the independent variable ${\textbf {x}}\equiv (x, y)$ (i.e. $(x_{i}, y_{j}) = (i\Delta x, j\Delta x)$ with $\Delta x=0.016$ and $i, j = 0,\dots ,N)$ Moreover, we choose the time step $\Delta t= 10^{-4}$ and, unless stated otherwise, we perform numerical simulations for $15\times 10^4$ time-steps (i.e. the final time of simulations is $t_f=15$).

The method for constructing numerical solutions of the IDE-PDE-PDE system (3.2) is based on a finite difference scheme whereby the discretised dependent variables are

$$\begin{aligned}n_{i,j}^{k}:=n(x_{i},y_{j},t_k), \quad c_{i,j}^{k}:=c(x_{i},y_{j},t_k) \quad \text {and} \quad \phi _{i,j}^{k}:=\phi (x_{i},y_{j},t_k) . \end{aligned}$$

We solve numerically the IDE (3.2)$_1$ for n and the PDE (3.2)$_3$ for $\phi$ using the following schemes

$$\begin{aligned} \frac{n_{i,j}^{k+1}-n_{i,j}^{k}}{\Delta t}=\left( \alpha _n-\mu _n\rho _n^k-\zeta _nK_{i,j}^{k}\right) n_{i,j}^{k} \quad i,j=0,\dots ,N, \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \frac{\phi _{i,j}^{k+1}-\phi _{i,j}^{k}}{\Delta t}=&\beta _{\phi }\frac{\phi _{i-1,j}^k+\phi _{i+1,j}^k-2\phi _{i,j}^{k}}{(\Delta x)^2}+\beta _{\phi }\frac{\phi _{i,j-1}^k+\phi _{i,j+1}^k-2\phi _{i,j}^{k}}{(\Delta x)^2}\\&+\alpha _{\phi }n^k_{i,j}-\kappa _{\phi }\phi _{i,j}^{k}, \quad i,j=1,\dots ,N-1, \end{aligned} \end{aligned}$$

and impose zero-flux boundary conditions for $\phi$ by letting

$$\begin{aligned}{} & {} \phi _{0,j}^{k+1}=\phi _{1,j}^{k+1} \quad \text {and} \quad \phi _{N,j}^{k+1}=\phi _{N-1,j}^{k+1}, \quad j=0,\dots , N \\{} & {} \phi _{i,0}^{k+1}=\phi _{i,1}^{k+1} \quad \text {and} \quad \phi _{i,N}^{k+1}=\phi _{i,N-1}^{k+1}, \quad i=0,\dots , N \end{aligned}$$

Moreover, we solve numerically the PDE (3.2)$_2$ for c using the following explicit scheme, which is the same as the one used in Bubba et al. (2020),

$$\begin{aligned} \frac{c_{i,j}^{k+1}-c_{i,j}^{k}}{\Delta t}-\frac{F^{k}_{i+\frac{1}{2},j}+F^{k}_{i-\frac{1}{2},j}}{\Delta x}-\frac{F^{k}_{i,j+\frac{1}{2}}+F^{k}_{i,j-\frac{1}{2}}}{\Delta x}=r(\phi ^k_{i,j})-\mu _c\rho _c^kc_{i,j}^{k} \end{aligned}$$

for $i,j=0,\dots , N$, where

$$\begin{aligned} \begin{aligned}&F^{k}_{i+\frac{1}{2},j}:=\beta _c \psi \left( w^k_{i+\frac{1}{2},j}\right) \frac{c^k_{i+1,j}-c_{i,j}^{k}}{\Delta x}-\beta _c c^k_{i+\frac{1}{2},j}\psi ^\prime \left( w^k_{i+\frac{1}{2},j}\right) \frac{w^k_{i+1,j}-w_{i,j}^{k}}{\Delta x}\\&\quad -b^{k,+}_{i+\frac{1}{2},j}c_{i,j}^k\psi \left( w_{i+1,j}^k\right) +b^{k,-}_{i+\frac{1}{2},j}c_{i+1,j}^{k}\psi \left( w_{i,j}^k\right) , \quad i=0, \dots N-1, \; j=0, \dots N, \\&F^{k}_{i,j+\frac{1}{2}}:=\beta _c \psi \left( w^k_{i,j+\frac{1}{2}}\right) \frac{c^k_{i,j+1}-c_{i,j}^{k}}{\Delta x}-\beta _c c^k_{i,j+\frac{1}{2}}\psi ^\prime \left( w^k_{i,j+\frac{1}{2}}\right) \frac{w^k_{i,j+1}-w_{i,j}^{k}}{\Delta x}\\&\quad -b^{k,+}_{i,j+\frac{1}{2}}c_{i,j}^k\psi \left( w_{i,j+1}^k\right) +b^{k,-}_{i,j+\frac{1}{2}}c_{i,j+1}^{k+1}\psi \left( w_{i,j}^k\right) , \qquad i=0, \dots N, \; j=0, \dots N-1, \end{aligned} \end{aligned}$$

with

$$\begin{aligned}{} & {} w^k_{i+\frac{1}{2},j}:=\frac{w^k_{i+1,j}+w^k_{i,j}}{2}, \qquad w^k_{i,j+\frac{1}{2}}:=\frac{w^k_{i,j+1}+w^k_{i,j}}{2},\\{} & {} c^k_{i+\frac{1}{2},j}:=\frac{c^k_{i+1,j}+c^k_{i,j}}{2}, \qquad c^k_{i,j+\frac{1}{2}}:=\frac{c^k_{i,j+1}+c^k_{i,j}}{2},\\{} & {} b^{k}_{i+\frac{1}{2},j}:=\gamma _c\frac{\phi ^k_{i+1,j}-\phi ^k_{i,j}}{\Delta x}, \; b^{k,+}_{i+\frac{1}{2},j}=\max \left( 0,b^{k}_{i+\frac{1}{2},j}\right) , \; b^{k,-}_{i+\frac{1}{2},j}=\max \left( 0,-b^{k}_{i+\frac{1}{2},j}\right) \end{aligned}$$

and

$$\begin{aligned}b^{k}_{i,j+\frac{1}{2}}:=\gamma _c\frac{\phi ^k_{i,j+1}-\phi ^k_{i,j}}{\Delta x}, \; b^{k,+}_{i,j+\frac{1}{2}}=\max \left( 0,b^{k}_{i,j+\frac{1}{2}}\right) , \; b^{k,-}_{i,j+\frac{1}{2}}=\max \left( 0,-b^{k}_{i,j+\frac{1}{2}}\right) \end{aligned}$$

The discrete fluxes $F^k_{i-\frac{1}{2},j}$ for $i=1,\dots , N,\, j=0,\dots , N$ and $F^k_{i,j-\frac{1}{2}}$ for $i=0,\dots , N, \,j=1,\dots , N$ are defined in an analogous way, and we impose zero-flux boundary conditions by using the definitions

$$\begin{aligned}{} & {} F^k_{0-\frac{1}{2},j}:=0 \quad \text {and} \quad F^k_{N+\frac{1}{2},j}:=0, \quad \text {for} \quad j=0,\dots ,N,\\{} & {} F^k_{i,0-\frac{1}{2}}:=0 \quad \text {and} \quad F^k_{i,N+\frac{1}{2}}:=0, \quad \text {for} \quad i=0,\dots ,N. \end{aligned}$$