A two-compartment model of angiogenesis and angiogenic inhibition.

Many generalizations involve time-evolution analyzed with systems of first-order ODEs. For example, two ODEs are usually needed when considering two cell populations (or, more generally, two ``compartments'') - e.g. hypoxic and normoxic tumor cells, or cells in the S and other phases of the cell cycle, etc. This section discusses one recent example.

Quantifying the dynamics of tumor cells interacting with an endothelial cell compartment is currently of major interest. A sufficiently small tumor may persist in a dormant, prevascular state. But it is now know that due to diffusion limitations tumors depend on neovascularization for their growth to larger sizes [10, 11]. Endothelial cells supply the neovascularization. They influence the tumor by, roughly speaking, increasing carrying capacity. The tumor influences them via long-range inhibitors and short-range stimulators [12]. Attempts to capture the essential features of tumor/vasculature dynamics by using a pair of ODEs were made by Liotta and coworkers [13, 14] and related models were considered by Michelson and Leith, e.g. [15, 16]. We next describe a "dynamic carrying capacity" model [17] which is based on extensions of the ideas behind these models.

One assumes a tumor cell population with $N(t)$ cells and considers a carrying capacity $K(t)$, regarded as a dynamical variable in its own right, proportional to the amount of neovascularization. The tumor is assumed so large that neovascularization dominates the availability of oxygen and nutrients. The dynamic equations are taken to be the following. For the tumor compartment:

$$dN/dt= N F(N/K)$$ (6)

where $F(x)$ is a smooth, decreasing function on $(0, \infty)$ with $F(1)=0$, as exemplified in Eq. 3. For the endothelial compartment:

$$dK/dt= - \alpha c(t)K + \omega N - \gamma N^{2/3} K, \quad \qquad 0 \leq \alpha , \omega, \gamma .$$ (7)

Here the term $\alpha c K$ was discussed above, the term $\omega N$ corresponds to short range stimulation by the tumor, and the term $\gamma N^{2/3}K$ corresponds to long-range inhibition. Eq. 7 is a simple, representative example. It incorporates in minimal form the essential features of angiogenic response to externally administered angiogenic agents (the term involving $\alpha$), tumor-generated stimulators, and tumor-generated inhibitors. Substituting minor variations (e.g. adding a term $- \lambda' K$ with $0 <\lambda' < \omega$ for apoptosis of endothelial cells, and/or replacing $\omega N$ by a term proportional to $N K^{1/2}$, and/or replacing $\gamma N^{2/3}K$ by a term proportional to $N K$) does not change the main conclusions which can be drawn from Eqs. 6 and 7.

Standard arguments show that in the absence of external treatment, i.e. for $c(t) \equiv 0$, Eqs. 6 and 7 specify one global attractor in the positive $N,K$ quadrant, with $N_0 =K_0$ and $N_0 = (\omega / \gamma )^{3/2}$.

In the absence of treatment, Eqs. 6 and 7 give a more mechanistic explanation of tumor growth deceleration than do the one-ODE models of the preceding section, as follows: after leaving a prevascular stage (not modelled here) a small tumor can recruit neovascularization via endothelial stimulation so that the tumor can grow; as the tumor grows larger, the long-range inhibition term gradually becomes more important and neovascularization is slowed; if death or treatment do not intervene, the carrying capacity approaches a limit; the tumor cannot grow beyond the limiting carrying capacity, and the whole system approaches the global attractor at the finite number $N_0$. Thus the two near-universal features of tumor deceleration and tumor dependence on angiogenesis are linked in a causal and quantitative way. This scenario represents an important shift in thinking compared to the thinking behind the one-ODE models. Endothelial cells are much less diverse and subject to phenotypical changes than are tumor cells, so the former are more susceptible to effective control, and a scenario where the endothelial cells determine growth has far-reaching implications for treatment strategies.

Indeed an important application for Eqs. 6 and 7 is in describing various anti-angiogenic treatment regimens [17]. When an inhibitor such as angiostatin or endostatin is administered, the result is to slow or reverse neovascularization, as modelled by the term in Eq. 7 involving $c(t)$. The tumor thereafter decelerates or shrinks. Figure 1 gives an example. The parameters and initial values in the figure are assigned in a way believed to be roughly appropriate for endostatin treatment of a murine tumor [18] at 20 mg/kgm daily. The use of the ODE pair 6 and 7 is a practical and flexible way to interrelate various current results on angiogenic treatment.