Fig. 9.5 Emergence of Turing instability. As ai increases and through its critical value, the function F(uj) (equation (9.17)) crosses zero. Negative regions of F{uf) correspond to unstable wave-numbers. The wave-number which becomes unstable first is denoted by u>c• The parameters are as follows: r = 1; <5 = 0.1; ap = 5\bp = 0.1; 6/ = 0.01; Dq = 0.00001; Dj = 0.001.

solution is stable. As a/ increases, the function F(u>) crosses the line F = 0. The critical value of a,/, a/iC, for which F(ojc) = 0, is determined from where a and (3 both depend on a/. We solved this equation numerically to find the critical value of a/iC, see Figure 9.5.

The applicability of the above analysis depends on the parameters of the system. First of all, we need conditions (9.15-9.16) to be satisfied. They mean that without diffusion, a positive, spatially uniform solution is stable. Next, we need to be in a weakly nonlinear regime, where the function F(u>) has only very narrow regions of uj corresponding to negative values. More precisely, Au ~ L1, where L is the spatial dimension of the system. In terms of parameter aj, we require that it is sufficiently close to a/jC. Then, we can calculate the "most unstable" wave-number, that is, u>c defined by equation (9.18), with ljcj. This value will determine the spatial period of the solution,

9.3.2 Stationary periodic solutions

Let us start from the value aj below the critical, a/ < a/)C. The system exhibits bistability. If we start in the vicinity of a (0,0) solution, then cancer will not grow and decay to zero. If we start from a point (C, I) in the domain of attraction of the solution (C, I), then the system will develop towards this positive spatially homogeneous stationary solution.

Next, let us suppose we have a/ > a/jC, but make sure that it is sufficiently close to a/jC (the exact meaning of "close" is specified in the analysis above). Again, if the initial conditions are close to the zero solution, then the zero state will be the state that the system will attain. However, if we start in the vicinity of the (C, I) state, we will observe interesting behavior. Solution (C, I) is now unstable, and we will see "ripples" developing on top of this solution. This is Turing instability. The spatial period of the ripple was calculated in the previous section. Long-time evolution of this state is of course not in the realm of linear stability analysis, but we can predict that the spatial scale of the resulting solution will be given by (9.19).

Finally, let us assume that aj is much higher than critical. Now, solution (C, I) is unstable even in the system of ODEs. However, a periodic solution will develop, unless the initial condition is in the domain of attraction of the zero solution. The spatial scale of the periodic solution is determined intrinsically by the parameters of the system, and it grows with a/. Intuitively this is easy to understand, because higher values of aj correspond to higher levels of inhibition, so the distance between regions of large C will become larger. Note that the exact period of the periodic solution is adjusted to fit the boundary conditions of the system. For instance, with the Neumann boundary conditions, the boundary points are forced to be troughs of the wave-like pattern. In other words, the period of the solution must be an integer fraction of L.

9.3.3 Biological implications and numerical simulations

We start with a scenario where the degree of inhibition is much larger than the degree of promotion (ai/bj » ap/bp). This corresponds to the early stages when the tumor is generated. We then investigate how tumor growth changes as the degree of inhibition is reduced relative to the level of promotion (i.e. the value of aifbi is reduced). We consider the following parameter regions (Figure 9.6).

(1) If the degree of inhibition is strong and lies above a threshold, growth of the cancer cells to higher levels does not occur (not shown). Only a small number of cells which do not require promotion for survival would remain.

(2) If the degree of inhibition is weaker, the cancer cells can grow. The spread across space is, however, self-limited (Figure 9.6a). The cancer cells migrate across space. The inhibitors produced by the cancer cells also spread across space, while the promoters do not. Therefore, as the cancer cells migrate, they enter regions of the tissue where the balance of inhibitors to promoters is heavily in favor of inhibitors. Consequently, these cells cannot grow within the space. They remain dormant and may eventually die. In biological terms, this corresponds to a single coherent but self-limited lesion (uni-focal). Note that this does not mean that it is in principle impossible to generate more lesions. It means that the space between lesions is bigger than the space provided for cancer growth within the tissue.

(3) As the production of inhibitors is further reduced, we enter another parameter region. Now fewer inhibitors diffuse across space. We observe that multiple lesions or foci are formed (Figure 9.6b). They are separated by tissue space which does not contain any tumor cells. The separate lesions produce some inhibitors, and they diffuse across space. This explains the absence of tumor cells between lesions. Because the production of inhibitors is weakened, however, tumor growth is only inhibited in a certain area around the lesion, and not across the whole space. How many lesions are found within a tissue depends on the parameters in the model, in particular on the relative strength of inhibition and promotion (Figures 9.6b and c). The stronger the degree of inhibition, the larger the space between lesions, and the fewer lesions we expect. The weaker the degree of inhibition, the smaller the space between lesions, and the larger the expected number of lesions. Analytical expressions for the space between lesions are given below. In biological terms, the occurrence of multiple lesions within a tissue which arise from a single tumor is often referred to as multi-focal cancers.

(4) If the degree of inhibition is further reduced and lies below a threshold, spread of inhibitors is sufficiently diminished such that the tumor cells can invade the entire space and tissue (Figure 9.6d). In biological terms, this corresponds to the most extensive tumor growth possible within a tissue.

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