that is, this equilibrium is always stable. For the points (C±, J±), we get the following Jacobian,

(-¿(e-7-W±r) ¿(c-7-W-r) \ 2 w 2-yw i ba —hi J '

where we denote for convenience, T = y/(e + 7 — W)2 — 4ej. It is easy to show that the eigenvalues of this matrix for the solution (C_, /_) are given by

and for the solution (¿7+, J+) we have eigenvalues

where K± = + —7-W±T). We can see that solution (C_,/_) is always unstable and we will not consider it any longer. Solution (C+,/+), which we call for simplicity (C, I) from now on, is stable as long as r+ > 0. (9.6)

9.2.2 Conclusions from the linear analysis

As we can see this model has very similar properties compared to the last one, and they are summarized as follows. There are two outcomes. (i) The cancer cells cannot grow and consequently go extinct. That is, C = 0, P = 0 and I = The cancer goes extinct in the model because we only consider cells which require the presence of promoters for division. If the level of promoters is not sufficient, the rate of cell death is larger than the rate of cell division. In reality, however, it is possible that a small population of non-angiogenic tumor cells survives. This was modeled in more detail in the previous section. Here, we omit this for simplicity, (ii) The population of cancer cells grows to significant levels, that is, C = C.

How do the parameter values influence the outcome of cancer growth? The cancer extinction outcome is always stable. The reason is as follows. The cancer cells require promoters to grow. The promoters, however, are produced by the cancer cells themselves. If we start with a relatively low initial number of cancer cells, this small population cannot produce enough promoters to overcome the presence of inhibitors. Consequently, the cancer fails to grow and goes extinct. This outcome is always a possibility, regardless of the parameter values. Significant cancer growth can be observed if the intrinsic growth rate, r, lies above a threshold relative to the death rate of the cells, 8, and degree of tumor cell inhibition (ap and bp relative to aj and bi). The exact condition is given by (9.6). In this case, the outcome is either failure of cancer growth, or successful growth to large numbers. Which outcome is achieved depends on the initial conditions. Successful growth is only observed if the initial number of cancer cells lies above a threshold. Then, enough promoters are initially produced to overcome inhibition. This is the same result as presented in the previous section; in biological terms this may mean that mutant cells which produce promoters must be generated frequently (e.g. by mutator phenotypes) in order to initiate tumor growth to higher levels [Wodarz and Krakauer (2001)].

9.3 Spread of tumors across space

In this section, we introduce space into the above described model. We consider a one-dimensional space along which tumor cells can migrate. The model is formulated as a set of partial differential equations and is written as follows, dC ( rC \ ( P \ . _ _ dC2 fn dT \eC + lJ \I + 1J dx2

The model assumes that tumor cells can migrate, and this is described by the diffusion coefficient Dc. Inhibitors can also diffuse across space, and this is described by the diffusion coefficient Dj. It is generally thought that inhibitors act over a longer range, while promoters act locally [Folk-man (2002); Ramanujan et al. (2000)]. Therefore, we make the extreme assumption that promoters do not diffuse. For simplicity we assume that inhibitors are only produced by cancer cells and ignore the production by normal tissue (that is, £ = 0). This simplification is justified because this model concentrates on the tumor dynamics, and numerical simulations show that the results considered here are not altered by this simplification. As mentioned above, the model considers tumor spread across space. It is im portant to point out that we do not consider long-range metastatic spread. Instead, we consider local spread of a tumor within a tissue, such as the breast, liver, brain, or esophagus.

These equations must be equipped with appropriate initial and boundary conditions. In the simulations we used the following (Neumann) boundary conditions:


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