Mt oc tD

which is valid for all times. Obviously, in this regime, the equation for the tumor becomes independent of the equation for circulating EPCs. The numbers of the BM EPCs and circulating EPCs remain at their equilibrium level, x(t) = x0, y(t) = y0-The very simple behavior of this system is presented in Figure 10.2.

Vasculogenesis-driven dynamics. Next, let us consider the opposite regime by taking 7 = 0. Now all the equations are coupled. As time goes by, the tumor size will increase, and this will lead to an increase in the number of circulating EPCs. Consequently, the number of BM EPCs will drop. Qualitatively, the dynamics can be described as follows. In the beginning, y(t) increases. Then it reaches a maximum, after which it will decrease to zero. x(t) drops to low numbers as y(t) reaches its maximum, and then it continues to decrease to zero. The dynamics of M(t) also has two stages. It grows faster than linear in the beginning, and then saturates at a linear growth with slope bo, as y —> oo.

To understand the long-term behavior, let us suppose that d2 = 0, since it does not change the behavior qualitatively. We note that initially, the balance in the equation for x is defined by the b0 and -(Ao + d0)x terms. As time goes by and M increases, we have the balance bo ~ AMx, which means that x —> 0. Also, this expression defines the dynamics of y; we have bo ~ rSy, where S —> oo, and y —> 0.

A typical outcome of a numerical simulation of system (10.8-10.10) is given in Figure 10.3. A note of caution: the numerical values of the functions should not be taken literally. The point of Figures 10.2 and 10.3 is to depict the qualitative behavior of the angiogenesis- and vasculogenesis-driven systems, which turns out to be very different.

Fig. 10.3 Typical vasculogenesis-driven tumor dynamics: the quantities x(t), y(t) and M(t) are plotted as a function of time. The parameter values are as in Figure 10.2, except A = 0.1, r = 0.1 and 7 = 0.

Main qualitative differences. From this analysis we conclude that the behavior of the system driven by angiogenesis and vasculogenesis is different. The two main points of difference are as follows:

• For angiogenesis driven systems, the amount of BM EPCs and circulating EPCs stays constant in time. For vasculogenesis driven systems, the amount of BM EPCs steadily decreases, and the amount of circulating EPCs experiences a sharp peak in the beginning and then also decreases.

• The tumor mass in angiogenesis driven systems grows as a cube of time, that is, the diameter of the tumor grows linearly with time. For vasculogenesis driven systems, tumor growth has two stages: at first, the tumor mass grows faster than linear, and later, once the BM is depleted of EPCs, the tumor mass grows linearly with time, which means that the diameter of the tumor grows as a cubic root of time.

Another mathematical model of tumor vasculogenesis has been recently proposed [Stoll et al. (2003)]. In this paper, the emphasis is on the geometry of the tumor and its growth dynamics. However, this model does not take account of the independent dynamics of BM and circulating EPCs. Our model concentrates on the description of the fine balance between the three compartments: the BM, the circulatory system and the tumor.

10.5 Applications

We have presented a mathematical model of tumor growth driven by angiogenesis and vasculogenesis and found that the dynamics are quite different in the two cases, as predicted by our equations. Indeed, if angiogenesis was the entire story, then we would expect a cubic (third power) growth of the tumor mass, and constant levels of BM EPCs and circulating EPCs. On the other hand, if the tumor growth is driven by vasculogenesis, then the dynamics will go through two stages. First, the level of circulating EPCs will increase and the tumor will grow faster than linear, and then, when the BM is depleted of EPCs, the level of circulating EPCs will also go down and tumor growth will slow down to linear, which means that the tumor diameter will only grow as a cubic root of time.

10.5.1 Dynamics of BM-derived EPCs

Even though the exact role the BM-derived EPCs play in the formation of de novo blood vessels in the process of tumorigenesis is heavily debated, there is growing evidence of their importance for the formation of tumor vasculature [Bolontrade et al. (2002); Davidoff et al. (2001)]. A paper by [Lyden et al. (2001)] suggests that "recruitment of VEGF-responsive BM-derived precursors is necessary and sufficient for tumor angiogenesis". Schuch et al. [Schuch et al. (2003)] propose that EPCs can be a novel target for endostatin and suggest that their relative numbers can serve as a surrogate marker for the biological activity of antiangiogenic treatment.

In this chapter, we developed a model with a predictive power regarding the dynamics of BM-derived EPCs. This is a first attempt to quantify the level of EPCs in relation to carcinogenesis.

10.5.2 Re-evaluation of apparently contradictory experimental data

The two-stage dynamics characteristic of vasculogenesis-driven tumor growth is consistent with some experimental data published recently. In particular, our model can help resolve some contradicting reports on the levels of circulating EPCs in cancer patients. In the paper [Beerepoot et al. (2004)] it is found that the level of circulating endothelial cells in peripheral blood of cancer patients is increased compared with healthy subjects. More specifically, cancer patients with progressive disease had on average 3.6-fold more circulating EPCs than healthy subjects. Patients with stable disease had circulating EPC numbers equal to that in healthy subjects. On the other hand, [Kim et al. (2003)] reports that the number of circulating EPCs was not found to be increased in cancer patients, although the plasma levels of VEGF were elevated. It was further concluded by the authors that VEGF, at concentrations typical of those observed in the blood of cancer patients, does not mobilize EPCs into the peripheral blood.

With our model, it is possible to resolve this apparent discrepancy in measurements of the level of circulating EPCs. If we look at Figure 10.3, we can see that the level of EPCs first increases and then drops even below the level corresponding to the equilibrium in healthy subjects. Therefore, the timing of measurements becomes crucially important. The level of circulating EPCs will depend on the stage of cancer development. It experiences a peak and drops considerably afterward. It would be very interesting to perform systematic measurements to find out the exact timing of this process.

Application for diagnostics. Dynamic analysis of the number of circulated EPCs in blood opens up a clinically important avenue of research. Circulated stem cells can be used as a surrogate marker of tumor vas-

culogenesis. Development of assays which would allow us to monitor the recruitment of labeled EPCs could eventually be transformed into a clinical diagnostic test for estimating the intensity of tumor vasculogenesis.

10.5.3 Tumor growth kinetics

The kinetics of tumor growth is a complicated question, and no universal answer has been given as to how exactly tumors develop in time. There may be a good reason for this: different tumors may grow according to different scenarios, and these scenarios may be very complicated. The Gom-pertzian law of tumor growth has been extensively discussed over the last four decades, see e.g. [Laird (1969); Lazareff et al. (1999); Norton (1988)]. This "sigmoidal" empirical law models an exponential growth of a tumor at the initial stages followed by saturation at a constant level. There are many papers which suggest that this law does not hold [Retsky et al. (1990)], and propose different models [Ferreira et al. (2002); Gatenby and Gawlin-ski (1996); Guiot et al. (2003); Kansal et al. (2000); Sherratt and Chaplain (2001)]. Many of the mathematical models seem to be concerned with the avascular stage in tumor growth. While this may be of theoretical interest, it is believed that most of the observed tumors are dependent on blood supply. Therefore, the formation of new blood vessels should be a part of a realistic model.

In general, there is a curious situation regarding the state of affairs with the studies of tumor growth kinetics. It seems that many modelers come up with different theoretical constructs predicting various modes of growth, while there is very little directed research in this area on the part of experimental cancer biologists. One possible explanation is of course the impossibility of studying tumor growth kinetics in humans, without treatment. Another factor is the difficulty of precise measurements: with only a limited number of data points and a large error of measurement, it is impossible to make out subtleties of the growth dynamics. Finally, the very complexity of multistage tumorigenesis poses a problem when trying to identify any "universal" behavior, thus leaving theorists with their theories unchecked, and making experimental biologists concentrate on issues of "survival", "treatment success" etc., which present more possibilities for direct applications in treatment.

It is probably safe to say that different factors affect the rate of tumor growth during different stages of tumorigenesis. In the very beginning (the avascular stage), a mutation (or a set of mutations) throws the growth and death regulation out of balance, which may lead to an exponential accumulation of such (pre-)malignant cells. Then, for some reason, the growth slows down. This can be related to space/density control or the lack of specific growth factors. The growth of the lesion plateaus, until the next mutation breaks out of homeostatic regulation, leading to another increase in cell number. At some point, the lesion reaches the size where it is impossible to keep up the functioning of cells unless additional blood supply is provided. At this stage, the rate-limiting factor becomes the making of the new blood vessels.

It is this stage of the growth that we concentrated on in our model. We assumed that new blood vessels are formed near the surface of the existing tumor, thus making the cells near the surface divide more often than the core. A similar assumption was made in the interesting paper by [Bru et al. (2003)]. There, a linear growth of the diameter with time was observed in colonies of tumor cell lines in vitro. The authors went on to develop a model which takes account of the fractal structure of a tumor. Most of the growth activity (i.e. mitosis) was assumed to be concentrated on the boundary of the colony/tumor, which leads to a linear growth law for the colony diameter.

Our model is similar to this in the assumption that tumor growth happens mostly on the surface. However, it makes more explicit statements on the kinetics of de-novo vascularization. If we assume that new blood vessels are formed locally, that is, if the tumor dynamics are dominated by the process of angiogenesis, then we find that tumor mass grows as a third power of time (this means that the tumor diameter grows linearly, like in the model by [Bru et al. (2003)]). On the other hand, if circulating EPCs are recruited from the blood stream, that is, vasculogenesis is the dominant process, the growth of the tumor mass (after some transition period) will be linear in time.

The point of our model is to address a specific question, namely, whether new blood vessels are formed "locally" or "globally". It is incomplete without an experimental validation. An experimental test can be performed to find out which of the processes contributes more to tumor growth. If a linear growth is found, then we can conclude that vasculogenesis is more important. If the growth is cubic in time, then angiogenesis wins. Several studies have reported the kinetics of tumor growth which can be used to test our predictions. For instance, in the paper by Schuch et al. [Schuch et al. (2002)], the law of tumor volume growth for pancreatic cancers resembles linear, which suggests the prevalence of vasculogenesis. The paper [Hah-

nfeldt et al. (1999b)] contains data on Lewis lung carcinoma implanted in mice, where a linear growth of three-dimensional tumor size was observed. This is again consistent with our vasculogenesis-driven dynamics. On the other hand, [Mandonnet et al. (2003)] report the linear growth of the tumor diameter for gliomas, which is consistent with the dominance of angiogenesis. However, statistically it may be difficult to distinguish between a linear and a cubic growth curve unless we have many experimental points. A conscious experiment with this specific question in mind would be very desirable to address this issue.

Chapter 11

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