Particle-based submodel

We next give an outline of the particle-based submodel that describes the spatiotem- poral evolution of the single mesenchymal-like cancer cells. Like the remainder of the model, the methods and techniques used therein are motivated by work in Sfakianakis et al. (2018a). However, prior to their application of modelling the spatiotemporal evo- lution of mesenchymal-like cancer cells in Sfakianakis et al. (2018a), similar methods and techniques had been used in other scientific fields. An example is the classical particle- in-cell method, first proposed in Harlow (1962), which has its main application in plasma physics. Another example is the smoothed-particle hydrodynamics method used in as- trophysics and ballistics, see e.g. Gingold and Monaghan (1977). The stochastic nature of the ODEs obeyed by the particles is motivated by the seminal work in Stratonovich (1966). For the combination of the two cancer cell formulations—i.e. for the bidirectional transition between the epithelial-like and mesenchymal-like cancer cell formulations via an atomistic and a continuum formulation for both the epithelial-like and mesenchymal- like cancer cell populations—we are inspired by Blanc et al. (2007); Kitanidis (1994); Makridakis et al. (2013); Tompson and Dougherty (1992).

We consider the mesenchymal-like cancer cells to be a discrete collection of isolated mass particles that migrate through the tissue via biased random motion. We model this biased random motion strategy using SDEs. In particular, we consider a system of

N = N(t) ∈ _N mesenchymal-like cancer cells, which we index by p ∈ P = {1, . . . , N}.

We account for their positions xp(t)∈ R3 and their massesmp(t)≥ 0. Then the overall

mass distribution of the particle system {(xp, mp), p∈P} is given by

˜ ˜

c(x, t) =X

p∈P

mp(t)δ(x−xp(t)) (6.1.4)

where δ(· −xp(t))is the Dirac distribution centred at xp ∈R3. Using the characteristic

function

ζ(x) =XK0(x), x∈R

we redefine _c˜_{˜(x, t) in equation (6.1.4) as} ˜ c(x, t) = Z Ω ˜ ˜ c(x0, t)ζ(x−x0)dx0 (6.1.4)= X p∈P mp(t)ζ(x−xp(t)). (6.1.6)

Here K0 is a—for reasons of simplicity—cuboid domain that represents the volume occupied by a physical cell. Thebiased random motion of the particles that represent the mesenchymal-like cancer cells is modelled through the combination of two independent processes. Firstly, we consider directed motion that represents the haptotactic response of the cells to gradients of the ECM-bound adhesion sites. Secondly, we include persistent random motion, which we understand as Brownian motion. Both of these cell-migration processes are combined in an SDE of the form

dXp_t =µ(Xp_t, t)dt+σ(Xp_t, t)dWp_t, for p∈P, (6.1.7)

where Xp_t represents the position of the particles in physical space (here _R3_{), and W}p t is

a Wiener process with independent and normally distributed increments. The modelling assumption that the mesenchymal-like cancer cells undergo directed motion is encoded in equation (6.1.7) via thedrift coefficientµ, and their random motion via thediffusion co- efficientσ. The contribution of these coefficients to equation (6.1.7) can be understood as

follows. During a short time interval of durationδt, the changes of the stochastic process Xp_t (i.e. of the position of the particle p in physical space) follow a normal distribution

with expectation µ(Xp_t, t)δt and variance σ(Xp_t, t)2_δt.

Remark: Note that, in the special case that we consider here whereµ(x, t) =µxand σ(x, t) = σx, with µ∈ _R and σ ≥ 0, the stochastic process that solves equation (6.1.7) can be numerically computed and hence approximated by the corresponding half-step explicit Euler-Maruyama particle motion scheme

Xp_t+τ =Xp_t +µX_tpτ +σZp√τ , for p∈P . (6.1.8)

Here, τ > 0 is the time step of the scheme and Zp _{is a vector of normally distributed}

values of zero mean and unit variance, cf. Kloeden and Platen (2013).

During every time step τ, a provisional new position of each particle is computed

via equation (6.1.8). This allows estimating the speed of the respective particle. If this speed exceeds the maximal (biological) mesenchymal-like cancer cell particle speeds, the

provisional new position of the respective particle is adjusted in its magnitude to comply with s. The direction of the particle displacement is not affected.

The mesenchymal-like cancer cells participate in several dynamical processes—such as in the EMT, the MET and the proliferation of the epithelial-like cancer cells; in the production of MMPs; and in the degradation of the ECM. Yet, the particle motion scheme in equation (6.1.8) does not include any reaction processes. Instead, we account for these processes in the following way:

• As mesenchymal-like cancer cells undergo MET and become epithelial-like cancer

cells, they are transformed to density via the density-to-particle operator that will be introduced in Section 6.1.3 and the respective mesenchymal-like cancer cells are removed from the system of mesenchymal-like cancer cell particles. The additional epithelial-like cancer cell density that is created via MET augments the existing

epithelial-like cancer cell density and participates in the system of equations (6.1.1)– (6.1.3) in a regular fashion. Conversely, a part of the epithelial-like cancer cell density undergoes EMT and becomes mesenchymal-like cancer cell density. This is then transformed into particles via the particle-to-density operator defined in Section 6.1.3. These newly formed mesenchymal-like cancer cells are then added to the system of existing mesenchymal-like cancer cell particles.

• At every (time instance and) time step of the method, the full distribution of

mesenchymal-like cancer cell particles is transformed temporarily to density via the particle-to-density operator (without undergoing MET to epithelial-like cancer cells). The mesenchymal-like cancer cell density then participates in the prolifera- tion of the epithelial-like cancer cells, in the production of the MMPs, and in the degradation of the ECM, as equations (6.1.1)–(6.1.3) describe.