Interactions between particles

We understand the particles as isolated cancer cells or cancer-cell aggregates of similar sizes and masses. To maintain similar masses, we split and merge the particles according to their mass and position. In the particular case that a particle represents an isolated cancer cell, we set mref to be the reference cell mass, cf. Table 6.1. Moreover, because

three dimensions, i.e. the mass of the cancer cell is modelled to be distributed evenly throughout the cubeK0. We then proceed as follows:

Splitting. A large particle (xp, mp) with mass mp > 4₃mref is split into two smaller

particles (x1

p, m1p), (x2p, m2p) of the same position x1p = x2p = xp and each of mass

m1_p = m2_p = 1₂mp. From that moment onwards, these two particles are considered

to be distinct from each other.

Merging. Asmall particle (xp, mp)with mass mp < 2₃mref ismerged with another small

particle(xq, mq)if they are close to each other, i.e. if

kxp−xqk<diam(K0),

where k · k describes the two-dimensional Euclidean norm. The resulting particle

is set to have the cumulative mass of the two particles and to be located at their

centre of mass mpxp+mqxq mp+mq , mp+mq . (6.1.26)

If more than two small particles are found in merging distance at the same time, they are merged pair-wise in the order they have been created.

Given that the distance between the particles is sufficiently small, iterations of themerging and splitting processes lead to particles with masses mp ∈

₂ 3mref, 4 3mref , i.e. particles with masses that are close to the reference cell mass mref.

Besides the merging and splitting procedures, we do not consider other processes that alter the masses of the particles. Moreover, we do not consider any further interactions between the particles in this work (such as competition for free space or development of collision forces) as we try to be consistent with the dynamics that are usually assumed by macroscopic deterministic models similar to that in equations (6.1.1)–(6.1.3). If two or more particles occupy the same position in this model, this can be understood as a comparatively high local cell density.

6.2

Implementation and model calibration

We choose our domain to be of size 8500µm×8500µm×8500µm following the experi- mental settings of Nurmenniemi et al. (2009) described in Section 2.5.3.

The construction of the initial ECM density distribution is based on discrete prin- ciples. For this, we first created a random landscape of a predefined size—e.g. an 8×8

random matrix for two-dimensional experiments or an 8×8×8 random matrix for three-

dimensional experiments. These random values are chosen from a normal distribution between the predefined minimum and maximum ECM density value and, respectively, will serve as the ‘heights of the hills’ and ‘depths of the valleys’ of the final ECM landscape. With subsequent refinements and periodic interpolations, we increased the dimensionality of the matrix. This way, from the 8×8 matrix in two dimensions we deduced a 16×16

matrix, from which we deduced a 32×32 matrix and so on—as shown in Figure 6.3. At

8 ×8 16 ×16 32 ×32

64 ×64 128 ×128 256 ×256

Figure 6.3: Construction of a sample initial ECM density distribution in two dimensions. The initial ECM density distribution that we ultimately use in our model is a result of multiple refinements. Here, we show the process in two dimensions starting with an 8×8 random matrix (top left panel). This matrix is progressively refined to a 256×256 matrix (bottom right panel) via repeated periodic interpolations. The result is a sample two-dimensional initial ECM density distribution with values between 0.9×wmax and wmax, where wmax = 1.06 g−1cm3 (ICRP, 2009). The corresponding process was applied in three dimensions up to a refinement of 64×64×64 for our simulations.

of the values. The process in two dimensions up to the deduction of a 256×256 matrix is

described visually in Figure 6.3.

In what follows, we additionally provide a pseudo-algorithm for the construction of the ECM to explain how the construction of the ECM is based on discrete principles. For the sake of simplicity of presentation, we here consider the one-dimensional case and the domain[0,1).

• An initial approximation of the ECM is set by deciding on the number of the major

‘hills’ and ‘valleys’ per direction along the grid. In our one-dimensional example, we choose an initial rough structure of the ECM with 8 ‘hills’ and ‘valleys’, i.e. we approximate the ECM as

8 X i=1 c(8)_i X_C(8) i (x), x∈[0,1), where C_i(8) = [x(8)_i−1/2, x(8)_i+1/2), x(8)_i−1/2 = (i−1)∆x(8)_{, i = 1, ...,8, ∆x}(8) ₌ 1 8, and where the coefficientsc(8)_i are uniformly distributed random numbers chosen within

the interval [0,1).

• Subsequently, a refinement takes place. The domain [0,1) is now discretised by

C_i(16) = [x(16)_i−1/2, x(16)_i+1/2), ∆x(16) = ₁₆1, i = 1, ...,16 , x(16)_i−1/2 = (i−1)∆x(16). Accord-

ingly, the ECM is then approximated by the simple function 16 X i=1 c(16)_i X_C(16) i (x), x∈[0,1).

The new coefficients c(16)_i interpolate—with some random noise—between the pre-

vious values: c(16)_i =1 + 0.002r(16)_i −0.5c (8) bi/2c+c (8) bi/2c+1 2 , i= 1, . . . ,16,

where b·c represents the floor function, and where r(16)_i are uniformly distributed

random numbers within[0,1). We note that the first and last coefficients,c(16)₁ and c(16)₁₆ , are computed periodically with respect to thec(8)· -values. The rescaling factor

0.002 is chosen to that the multiplicative randomness/noise is adjusted to 0.1% of the interpolated value.

• In a similar manner, the resolution of the ECM increases further so that the ECM

is approximated by 32 X i=1 c(32)_i X_C(32) i (x), x∈[0,1], where C_i(32) = [x(32)_i−1/2, x(32)_i+1/2),x(32)_i−1/2 = (i−1)∆x(32)_{, ∆x}(32)₌ 1 32, i= 1, ...,32, and c(32)_i =1 + 0.002r_i(32)−0.5 c (16) bi/2c+c (16) bi/2c+1 2 , i= 1. . .32.

• The refinements are repeated until the desired resolution is reached.

• At a final stage when the desired resolution of the ECM is reached, the values of

the density of the matrix are rescaled between the biological range of a minimum and maximum ECM density.

We initially considered a single layer of epithelial-like cancer cells to reside on the upper non-uniform matrix surface. This way, our initial conditions correspond to those in the experiments by Nurmenniemi et al. (2009), for which 7×105 epithelial-like cancer cells and no mesenchymal-like cancer cells were placed on top of each myoma disc. For the three-dimensional model withx= (x, y, z)∈Ω, this single layer of epithelial-like cells is translated to cell density as

c(t,x) =c(t, x, y, z) =

(

1, z > zmax−1.5×10−3 0, else.

Here, 1.5×10−3 is the non-dimensional diameter of a single cell corresponding to the the average diameter of 15 to 20 µm of a HSC-3 cell (Japanese Collection of Research

Table 6.1: Parameter settings for the simulations. Epithelial-like HSC-3 cells and mesenchymal-like HSC-3 cells are abbreviated ECC and MCC, respectively. At this instance, we only model the activity of membrane-bound, non-diffusive MMPs like MT1-MMP following the experimental results by Sabeh et al. (2009) shown in Figure 4.13. We achieve this by setting the coefficients Dm, ρEm, ρMm andλm in equation (6.1.2)to zero.

We also set the initial conditions for the diffusible MMP to be m(x,0) = 1. This allows for possible extensions of the model that include diffusible MMPs, while continuing to ac- count for the effects of membrane-bound MMPs in our model via equation (6.1.3). Note that ν_E and ν_M have different units as the former refers to a density of cancer cells and the latter to individual cancer cells.

Description Value Range Reference

DE ECC density diffusion coefficient 8.64×10

−8_cm3 _1×10−9_−1×10−12_cm2_s−1 _{Chaplain and Lolas (2005)}

Brú et al. (2003)

ρE

c ECC density proliferation 1.2 d

−1 _{1.2 d}−1 _{Fujinaga et al. (2014)}

coefficient

σ MCC particle diffusion coefficient 3.3675cm d−1

2 Parameter estimation

µ MCC particle drift coefficient 7.4595×10−2_d−1 _{Parameter estimation}

s Maximum MCC particle speed 2.16cm d−1 _1.83×10−5_−3.83×10−5_{cm s}−1 _{Butler et al. (2010)} mref MCC particle reference mass 2.3×10

−9_{g cell}−1 _2.3×10−9_−3.3×10−9_{g cell}−1 _{Park et al. (2008)} |V0| MCC particle reference volume 2.3×10−9cm3 2.2×10−9−5.2×10−9cm3 Puck et al. (1956) νE EMT rate 7.502×10

−2_{M cm}−3_d−1 _{Parameter estimation} ν_M MET rate 4.7697×10−1_d−1 _{Parameter estimation} wmax Maximum (initial) ECM density 1.06 g cm−3 1.02−1.05g cm−3 ICRP (2009)

λw ECM degradation rate 1.8383×10−4M cm−3d−1 Parameter estimation

by ECCs & MCCs

Bioresources Cell Bank, 2015). z describes the height of the three-dimensional domain

and zmax is the height of the upper surface of the myoma assay without the upper initial cell layer placed on it. Figure 6.5 shows this domain.

Throughout, we implement the model with the zero Neumann boundary conditions for the densities in the model in equations (6.1.1)–(6.1.3). However, since the MMPs accounted for in this set of simulations are membrane-bound and hence follow the spa- tiotemporal evolution of the cancer cells and since the ECM is immovable, we practically only need to enforce the boundary conditions for the cancer cells modelled through equa- tion (6.1.1). Furthermore, to represent the experimental conditions of Nurmenniemi et al. (2009) through our simulations, we do not allow particles to escape the domain. If a par- ticle would have escaped the domain, we force it to remain on the boundary of the domain instead, retracted along its linear movement trajectory. During the next time step, such a particle again moves according to the model, i.e. according to equation (6.1.7).