Level of angiogenesis inhibition

Fig. 9.3 Direction field plot showing how the outcome of the model can depend on the initial conditions. Parameters were chosen as follows: r2 = 1; fca = 100; (¿2 = 0.1;p2 = l;g = 10. For the purpose of simplicity the populations of non-angiogenic cells were summarized in a single variable and assumed to converge towards a stable setpoint (characterized by the parameters r=0.15; d=0.1; k=10).

angiogenic cells will be low. Hence, in the parameter region where the outcome of the dynamics depends on the initial conditions, a high mutation rate promotes the emergence and growth of angiogenic tumor cells (Figure 9.4). If a high mutation rate by tumor cells defines genetic instability, then it is possible that genetic instability might be required for the invasion of angiogenic tumor cells.

9.2 Model 2: Angiogenesis inhibition prevents tumor cell division

We consider a basic mathematical model which describes the growth of a cancer cell population, assuming that the amount of blood supply does not influence cell death, but the rate of cell division [Wodarz and Iwasa (2004)]. This model will also be used to consider the effect of diffusion of cells and soluble molecules across space; this is done in the next sections. Therefore, the model will take into account explicitly the dynamics of promoters and inhibitors. This is in contrast to the last section where for the purpose

(a) genetic stability / low mutation rate

(b) genetic instability / high mutation rate a s

(a) genetic stability / low mutation rate

(b) genetic instability / high mutation rate

Time (arbitrary scale)

Fig. 9.4 Genetic instability and the emergence of angiogenic cell lines, (a) If the mutation rate is low (genetic stability), the initial number of angiogenic cells created is low. Consequently they cannot emerge, (b) On the other hand, if the mutation rate is high (genetic instability), a higher initial number of angiogenic cells is created. Hence, they emerge and become established. Parameters were chosen as follows: r0 = 0.11; fco = 10; mo = 0.001; do = 0.1; n = 0.12; fci = 2; d2 = 0.1; r2 = 2.5; k2 = 2; (¿2 = 0.1;P0 = 2;Pi = 2;p2 = 2\q = 10; for (a) m = 0.001; For (b) fn = 0.01; ¡x2 = Ml-

of simplicity inhibitors and promoters were assumed to be proportional to the number of cells which secrete them. The new model includes three variables: the population of cancer cells, C; promoters, P; and inhibitors,

I. It is assumed that both promoters and inhibitors can be produced by cancer cells. In addition, inhibitors may be produced by healthy tissue. The model is given by the following set of differential equations which describe cancer growth as a function of time,

The population of cancer cells grows with a rate r. Growth is assumed to be density dependent and saturates if the population of cancer cells becomes large (expressed in the parameter e). In addition, the growth rate of the cancer cells depends on the balance between promoters and inhibitors, expressed as P/(I+1). The higher the level of promoters relative to inhibitors, the faster the growth rate of the cancer cell population. If the level of promoters is zero, or the balance between promoters and inhibitors in heavily in favor of inhibitors, the cancer cells cannot grow and remain dormant [O'Reilly et al. (1997); O'Reilly et al. (1996); Ramanujan et al. (2000)]. Cancer cells are assumed to die at a rate 6. Promoters are produced by cancer cells at a rate ap and decay at a rate bp. Inhibitors are produced by cancer cells at a rate aj and decay at a rate 6/. In addition, the model allows for production of inhibitors by normal tissue at a rate

9.2.1 Linear stability analysis of the ODEs

Let us simplify system (9.1-9.3) by using a quasistationary approach, that is, we will assume that the level of promoters adjusts instantaneously to its steady-state value (P = Cap/bp). It is convenient to denote

Now we have a two-dimensional system, d = sc((i + Wi + D-1)' (9'4)

There can be up to three fixed points in this system,

(C, I) = (0, 0), and (C, I) = (C±, I±), where I± = 7 C±, and

It is obvious that if 7 + e-VF < 0, and (pf + e-W)2 -Ae-j > 0, then there are exactly three positive equilibria in the system. If either of these conditions is violated, the (0,0) solution is the only (biologically meaningful) stable point.

Stability analysis can be performed by the usual methods. It shows that for the (0, 0) equilibrium, the Jacobian is

0 0

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