Five genes are commonly mutated in colorectal cancer patients, namely: APC (Adenomatosis Polyposis Coli), K-RAS (Kirsten Rat Sarcoma viral), TGF (Transforming Growth Factor), SMAD (Mothers Against Decapentaplegic) and p53 or TP53 (Tumor Protein 53). These genes belong to four specific pathways, which funnel external or internal signals that cause cell proliferation or cell death (see 15 and 1617 for more details).

The anti-growth, p53, pathway is activated in the case of DNA damage 1819. This is particularly relevant during irradiation 20. p53 pathway activation can block the cell cycle and induce apoptosis 2122. The K-RAS gene belongs to a mitogenic pathway that promotes cell proliferation in the presence of growth factors 23. Activation of the anti-growth pathways TGFβ/SMAD and WNT/APC inhibits cell proliferation. The SMAD gene is activated by hypoxia signals 2425, while APC is activated through β-catenin by loss of cell-cell contact 2627282930. Moreover, it has recently been hypothesized that overpopulation of APC mutated cells can explain the shifts of normal proliferation in early colon tumorigenesis 31.

We assume that activation of APC and SMAD is due to overpopulation and hypoxia signals respectively. Both pathways inhibit cell proliferation. In consequence, APC mutated cells promote overpopulation and SMAD or RAS mutated cells promote proliferation during hypoxia. Figure 2 shows the schematic genetic model.

We develop a Boolean model of these pathways in Figure 2. Each gene is represented by a node in the network and the interactions are encoded as the edges. The state of each node is 1 or 0, corresponding to the presence or absence of the genetic species. The state of a node can change with time according to a logical function of its state and the states of other nodes with edges incident on it 323334. The rules governing the genetic pathways are presented in Table 2.

We consider a discrete mathematical model of the cell cycle in which the cycle phase duration values were set according to the literature 35. In our model the proliferative cycle is composed of three distinct phases: S (DNA synthesis), G₁ (Gap 1) and G₂M (Mitosis). We model the 'Restriction point' R 36 at the end of G₁ where internal and external signals, i.e. cell DNA damage, overpopulation and hypoxia, are checked 37 (see Figure 3 for a schematic representation of our cell cycle model).

For each spatial position (x, y), we assume that:

- If the local concentration of oxygen is below a constant threshold Th_o and if SMAD is not mutated, hypoxia is declared and causes cells to quiesce (G₀) through SMAD gene activation (see Figure 2);

- If the local number of cells is above a constant threshold Th_t and if APC is not mutated, overpopulation is declared and leads cells to quiesce (G₀) through the APC gene (see Figure 2);

- Otherwise, if the conditions are appropriate, cells enter G₂M and divide, generating new cells at the same spatial position.

Induction of apoptosis through p53 gene activation is discussed later.

We use a fluid dynamics model to describe tissue behavior. This macroscopic-level continuous model is based on Darcy's law, which is a good model of the flow of tumor cells in the extracellular matrix 383940:

v = -k∇p     (1)

where p is the pressure field. The media permeability k is assumed to be constant.

We study the evolution of the cell densities in two dimensions. We formulate the cell densities in the tissue mathematically as advection equations, where n_φ(x, y, t) represents the density of cells with position (x, y) at time t in a given cycle phase φ. Assuming that all cells move with the same velocity given by Eq. (1) and applying the principle of mass balance, the advection equations are:

where P_φ is the cell density proliferation term in phase φ at time t, retrieved from the cell cycle model.

The global model is an age-structured model (see Section 2.7). Initial conditions for n_φ are presented in Section 2.6.

Assuming to be a constant and adding Eq. (2) for all phases, the pressure field p satisfies:

The pressure is constant on the boundary of the computational domain.

In our model, the oxygen concentration C follows a diffusion equation with Dirichlet conditions on the edge of the computation domain Ω:

C = C_max     on     Ω_bv     (5)

C_∂Ω = 0     (6)

D is the oxygen diffusion coefficient, which is constant throughout the computation domain. In this equation, Ω_bv stands for the spatial location of blood vessels, α_φ is the coefficient of oxygen uptake by cells at cell cycle phase φ and C_max is the constant oxygen concentration in blood vessels.

Cell sensitivity depends on cell cycle phase 8. We assume that only proliferative cells are sensitive to the treatment. In addition, we assume that DNA damage is proportional to the irradiation dose. This is known as the 'single hit' theory, which is governed by the expression

n_dsb = R_φd     (7)

where n_dsb is the number of double strand breaks induced by radiation dose d. As mentioned previously, the radiosensitivity R_φ has been assumed to depend on the cell cycle phase (see Table 3). Based upon radiobiological experiments found in the literature, we take the radiosensitivity as constant (2 Gy^-1) in G₁ and G₀. It decreases in S phase to 0.2 Gy^-1, and then increases to 2 Gy^-1 during G₂.

We set a constant treatment threshold Th_r such that if n_dsb due to the irradiation dose is above Th_r at any time, p53 is activated and the cells are labeled as 'DNA damaged cells'. DNA damaged cells are identified at the R point of the cell cycle and are directed to apoptosis. They die and disappear from the computational domain after T_Apoptosis, i.e. the duration of the apoptotic phase.

The standard radiotherapy protocol used in the simulations consists of a 2 Gy dose delivered each day, five days a week, and can be repeated for several weeks. The radiotherapeutic dose is assumed to be uniformly distributed over the spatial domain.

According to the radiosensitivity parameters found in the literature 414243, only a fraction of mitotic cells are assumed to be sensitive to the standard 2 Gy dose.

Cell cycle kinetic parameters were retrieved from flow cytometric analysis of human colon cancer cells 3544. Table 3 summaries the quantitative parameters used in our model.

In our two-dimensional model we study an 8 cm square tissue. We assume that the domain comprises five small circular tumor masses, the first located at the center of the computational domain and the other four towards the corners. Moreover, the domain has two sources of oxygen, to the right and left sides of the central cell cluster (see Figure 4).

The number of cells in each tumor is the same, and they are uniformly distributed. The number of cells in each phase of the cell cycle is proportional to the duration of the phase. For instance, the G₁ phase contains twice as many cells as the S phase because the G₁ phase is twice as long as the S phase. It is important to emphasize that the cell cycle phases are discrete (see Section 2.7).

The model is fully deterministic. Cell cycle phases durations τ_φ have been discretized in several elementary age intervals a ∈ {1, ..., N_φ} where N_φ is an integer such as τ_φ = dt × N_φ. Here dt is the time step of the cell cycle model. The cell density n_{a, φ} at age a in phase φ is governed by:

In this equation, φ ∈ {G₁, S, G₂M, G₀, Apoptosis} and a ∈ {1, ..., N_φ}. P_a,φ is the cell density proliferation term in phase φ at age a retrieved from the cell cycle model. In the simulations, the intracellular and extracellular conditions were identified for cells at the end of G₁ phase. These were used as initial conditions for the gene level model. The genetic model was computed until it reached steady state (this is of the order of 10 iterations).

Noting that is constant, we can sum Eqs. (8) to obtain an expression for the pressure field of the form:

The computer program starts from an initial distribution of cells in each state {a, φ}. The computations are performed using a splitting technique. First we run the cell cycle model for one time-step dt, then retrieve new values for n_a,φ and compute P_{a, φ}. Pressure is retrieved by solving Eq. (9) and velocity is computed using Darcy's law (see Eq. (1)). Since the contribution of the source term has been taken into account by the cell cycle model at the first stage of the splitting technique, Eqs. (8) are solved continuously and without second members:

which can also be written [using (9)]:

This equation is then solved using a splitting technique. The advection parts of Eq. (11) are solved by sub-cycling finite different scheme computations, with time-step dt_adv being smaller than dt (for stability reasons). We set n_a,φ = 0 on the part of the boundary where v·υ < 0, υ denoting the outgoing normal to the boundary. For the pressure p, we set p = 0 on the boundary.

All simulations (except the ones shown in Figure 7) were run for 320 h with time step dt = 1 h in a discrete computational domain composed by 100 × 100 elementary spatial units.

Figure 5 shows the simulated growth of cell colonies. According to the model settings, the colony of normal cells grows up to 10⁶ cells and is then regulated through activation of gene APC owing to overpopulation. APC mutated tumor cells are not sensitive to overpopulation and reproduce exponentially until late regulation because of hypoxia, through SMAD gene activation. Finally, according to the model parameters, APC and SMAD/RAS mutated tumor cells cannot be regulated at all and thus induce an exponential growth profile.

The simulation results reproduce the evolution of colorectal cancer 1645. Indeed, APC has been shown to promote shifts in pattern of the normal cell population in early colorectal tumorigenesis, and SMAD/RAS mutations promote evolution from early adenoma to adenocarcinoma.

We study the potential influence of the choice of parameters values on the model's results. The most critical parameters to account for include:

• cell-specific radiosensitivity parameters (α_φ);

• anti-growth signals, i.e. hypoxia and overpopulation, activation thresholds above which cells go into quiescence (Th_o and Th_t);

• initial conditions, i.e. initial number of cells and spatial configurations of oxygen sources.

Treatment protocol efficacy depends directly on cell-specific radiosensitivity parameters. Figure 13 compares the evolution of total cell number over time when the standard treatment protocol is applied. Model simulations show that the standard treatment is efficient when the parameters make cells in G₁ phase become radiosensitive. APC and SMAD/RAS activation, which leads cells to become quiescent, is controlled by the two threshold parameters Th_t and Th_o. Increasing Th_t results in delay of the overpopulation signal, while increasing Th_o speeds hypoxia activation.

Decreasing the initial number of cells has the same effect as increasing Th_t, while decreasing the number or the initial strength of the oxygen sources has the same effect as increasing Th_o. The initial configuration of tumor cells and oxygen sources is important for spatial propagation of the hypoxia signal. Indeed, Figure 9 shows a particular hypoxia propagation in the tumor cell masses that is correlated with the locations of the oxygen sources. Since Th_t and Th_o are merely constants, it seems that we may change the spatial configuration and size of the initial cell population and vary the oxygen sources and yet produce the same qualitative results.

Finally, Figure 14 shows the difference in evolution of the overpopulation signal over time if the initial distribution of cells in the clusters is uniform or random. The step by step evolution of overpopulation activation is softened but still exists when the cells are randomly distributed within the initial tumor masses.