A Multiscale Approach to the Migration of Cancer Stem Cells: Mathematical Modelling and Simulations

Abstract

We propose a multiscale model for the invasion of the extracellular matrix by two types of cancer cells, the differentiated cancer cells and the cancer stem cells. We investigate the epithelial mesenchymal-like transition between them being driven primarily by the epidermal growth factors. We moreover take into account the transdifferentiation program of the cancer stem cells towards the cancer-associated fibroblast cells as well as the fibroblast-driven remodelling of the extracellular matrix. The proposed haptotaxis model combines the macroscopic phenomenon of the invasion of the extracellular matrix by both types of cancer cells with the microscopic dynamics of the epidermal growth factors. We analyse our model in a component-wise manner and compare our findings with the literature. We investigate pathological situations regarding the epidermal growth factors and accordingly propose “mathematical-treatment” scenarios to control the aggressiveness of the tumour.

Appendix

1.1 Experiment Description

Here, we give technical details on the experiments that have been presented in this work. Our one-dimensional simulations have been computed on the domain $\Omega \,{=}\,[0,7.5]$ with the initial conditions (26), and the experiments in two dimensions have been conducted on $\Omega \,{=}\,[-5,5]^2$ with the initial data (27), (28). Wherever not otherwise stated, we have employed the parameters from Table 1.

The space and time domains are chosen such that we avoid interference on the wave propagation by the boundary conditions.

Experiment Ia

One-dimensional, EGF-driven EMT With parameters from Table 1 except for $\mu _0 \,{=}\, 0.55, \ k_D\,{=}\, 0.5,\ \Gamma \,{=}\, 7.5^{-1}.$ See also Figs. 2 and 7. The convergence study (see Table 2) and the sensitivity analysis, see Table 3, have been conducted on a smaller domain $\Omega _S\,{=}\,[0,3]$.

Experiment Ib

One-dimensional, adjusted EMT The parameters have been adjusted, so that the invasiveness of the CSCs coincides with the Experiment IIa: $\mu _0 \,{=}\, 0.034,\ \mu _{1/2}\,{=}\, 0.4,\ k_D\,{=}\, 0.2, \ \lambda ^D \,{=}\, 3,\ \Gamma \,{=}\, 7.5^{-1}.$ See also Fig. 5 (middle row).

Experiment Ic

One-dimensional, adjusted EMT & low EGF Same as Experiment Ib, with low EGF concentration: $\mu _0 \,{=}\, 0.034,\ \mu _{1/2}= 0.4,\ k_D\,{=}\, 0.2, \ \lambda ^D = 3,\ \Gamma \,{=}\, 10^{-3}.$ See also Fig. 5 (lower row).

Experiment Id

Adjusted EMT & low initial DCC concentration Same parameters as in Experiment Ib. Advection and diffusion terms were neglected. The initial conditions $c_0^D \,{=}\, 10^{-3},\ c_0^S \,{=}\, 0, \ c_0^F\,{=}\, 0,\ v_0\,{=}\, 1,\ m_0\,{=}\, 10^{-6}$ have been employed. See also Fig. 6 (left).

Experiment IIa

One-dimensional, constant EMT rate Our EMT model from Sect. 2.2 has been neglected, and the EMT transition coefficient in system (17) has been taken as a constant: $\mu _\text {EMT}\,{=}\, 0.017$. The invasiveness of the CSCs coincides with Experiment Ib. See Fig. 5 (upper row).

Experiment IIb

Constant EMT rate & low initial DCC concentration Same parameters as in Experiment IIa. Advection and diffusion terms were neglected. The initial conditions $c_0^D \,{=}\, 10^{-3},\ c_0^S \,{=}\, 0, \ c_0^F\,{=}\, 0,\ v_0\,{=}\, 1,\ m_0 \,{=}\, 10^{-6},$ have been employed as in Experiment Id. See also Fig. 6 (right).

Experiment IIIa

Two-dimensional, uniform ECM & fibroblast remodelling Parameters from Table 1. See Figs.  3 and 4 (lower row).

Experiment IIIb

Two-dimensional, uniform ECM & no remodelling Same as Experiment IIIa, without matrix remodelling: $\mu _v = 0$. See Fig. 4 (upper row).

Experiment IIIc

Two-dimensional, uniform ECM & self-remodelling Same as Experiment IIIa, with self-remodelling, i.e. the term $+ \mu _v\, c^\text {F}\, (1-c^\text {D}-c^\text {S}-c^\text {F}-v)^+$ for the ECM in system (17) has been replaced by $+ \mu _v\, v\, (1-c^\text {D}-c^\text {S}-c^\text {F}-v)^+$, hence the fibroblasts have been neglected. See Fig. 4 (middle row).

Experiment IIId

Two-dimensional, uniform ECM & fibroblast remodelling & smooth initial dataSame as Experiment IIIa with smooth initial data for the convergence study, see Table 2:

$$\begin{aligned} c^\text {D}_0(\mathbf x)= & {} \exp (-\Vert \mathbf x\Vert _2^2), \ c^\text {S}_0(\mathbf x) = c^\text {F}_0(\mathbf x)=0, \ v_0(\mathbf x) = 1 - c^\text {D}_0(\mathbf x), \ m_0(\mathbf x) \nonumber \\= & {} \frac{1}{20} \, c^\text {D}_0(\mathbf x), \quad \mathbf x=(x_1,x_2)\in \Omega . \end{aligned}$$

Experiment IVa

Two-dimensional, non-uniform ECM Same as Experiment IIIa with an ECM initial concentration $v_0$ different from (28). The non-uniform initial ECM $v_0$ has been constructed by a random distribution of fibroblast cell concentrations acting without other influences on the domain. We have conducted the simulation on a $100 \,{\times }\, 100$ computational grid (lower resolution as opposed to the other experiments). See Fig. 8 lower row.

Experiment IVb

Two-dimensional, non-uniform ECM Similar to Experiment IVa, the non-uniform initial ECM $v_0$ has been constructed by a random distribution of fibroblast cell concentrations acting on the domain. As a result the initial ECM concentration is different from the case in Experiment IVa. We have conducted the simulation using the standard grid resolution with 250 $\times$ 250 mesh cells. See also Fig. 9.