## Ma 7 s106

We have M = Ma+Mv. Depending on the tumor geometry, the relationship between the tumor mass and its surface changes. The basic formula is S = DM~d~ , where D is the dimension of the tumor. For instance, in flat tumors, such as bladder carcinoma in situ, certain cancers of the eye or flat adenomas/adenocarcinomas of the colorectal mucosa [Hurlstone et al. (2004); Rubio et al. (1995)] we have D = 2. The two-dimensional analysis is also applicable in certain in vitro experiments. Most of the time, however, solid tumors are three-dimensional, and we will set D = 3.

10.4 Mathematical analysis

Let us rewrite the system in a closed form, x = bo — Aox — AMx - dox + d2y, (10.8)

Tumor-free equilibrium. In the absence of a tumor, we have Ma = Mv = M = S = 0, and the equilibrium level of circulating stem cells is determined by setting the right hand side of equations (10.8), (10.9) to zero, which gives

If the rate of return to BM is negligible, d2 <C di, then these equilibrium expressions become more transparent,

The expression for xo, the equilibrium number of BM EPCs in the absence of a tumor, is a balance between a constant production, tumor-independent recruitment and death. We conclude that in the absence of tumor,

Constant Recruitment + Death

This is an unstable equilibrium. Adding a small amount of M will get it out of balance and lead to a growth of tumor. It is this process that we will model next. We start from the initial condition

The advantage of the above system is that we can consider the two regimes, angiogenesis and vasculogenesis, separately. Namely, by taking 7 = 0, we can assume that the only way the tumor vasculature is built is by vasculogenesis. Alternatively, r = A = 0 means that we assume that the only mechanism is angiogenesis.

time

Fig. 10.2 Typical angiogenesis-driven tumor dynamics: the quantities x(t), y(t) and M(i) are plotted as a function of time. The parameter values are as follows: D = 3, b0 = 10, A0 = 0.01,A = 0, cio = 0.1, di = 0.1, d2 = 0, r = 0,7 = 2.

time

Fig. 10.2 Typical angiogenesis-driven tumor dynamics: the quantities x(t), y(t) and M(i) are plotted as a function of time. The parameter values are as follows: D = 3, b0 = 10, A0 = 0.01,A = 0, cio = 0.1, di = 0.1, d2 = 0, r = 0,7 = 2.

Angiogenesis-driven dynamics. We start with r = X — 0. Let us suppose that the linear size of tumor is a. Then M oc aD and S oc DaD~l. From the equation (10.10) with r = 0, we have a oc 7, and we have the law for tumor growth,

0 0