Promotion

The development of cancer is regulated on many levels. So far, we have concentrated on the development of the cancerous phenotype itself. That is we investigated the processes which lead to the generation of a malignant cell and examined conditions under which genetically unstable cells can emerge. However, even if cancer cells have been generated and can in principle evolve to accumulate more mutations, these cancer cells might not be able to grow beyond a very small size. The reason is that the body is characterized by specific defenses which try to prevent the growth and pathogenicity of selfish transformed cells once they have been generated. In particular, the microenvironment in which the cancer emerges is thought to play a pivotal role in deciding whether the cancer will succeed at growing to high levels or not [Tlsty (2001); Tlsty and Hein (2001)]. Indeed, the development of cancer may require a conspiracy between tumor cells and their microenvironment [Hsu et al. (2002)].

One of the most important players in this respect is the blood supply which provides cancer cells with oxygen, the necessary nutrients, and factors required for replication and survival. A given tissue or organ must have a sufficient blood supply in order to function. No extra blood supply is available though, which will hinder any potential abnormal growth. Cancer cells have to induce the generation of new blood supply in order to sustain their growth. This process is called angiogenesis (as explained already in Chapter 1). Research on the role of angiogenesis for cancer progression has been pioneered by Judah Folkman in the 1960s and 70s [Folkman (1971)], and work from his laboratory has been dominating the literature up to now (e.g. [Folkman (1995a); Folkman (2002)]). In early experiments, Folk-man and colleagues placed a small number of rabbit melanoma cells on the surface of the rabbit thyroid gland. They observed that the tumor cells initially grew but subsequently stopped growing once they reached a relatively small size comparable to that of a pea. The reason is that the tumor cells run out of blood supply.

It is now clear that growth to larger sizes requires the emergence of so-called angiogenic tumor cells. The ability of the cancer to grow depends on the balance between so-called angiogenesis inhibitors, and angiogen-esis promoters. Examples of inhibitors are thrombospondin, tumstatin, canstatin, endostatin, angiostatin and interferons. Examples of promoters are growth factors such as FGF, VEGF, IL-8, and PDGF. Normal tissue produces mostly angiogenesis inhibitors. So do cancer cells. This serves as a preventative measure against abnormal growth. Angiogenic cancer cells, on the other hand, have mutations which allow the balance between inhibitors and promoters to be shifted away from inhibition, and towards promotion. This is done by activating the production of angiogenesis promoters, or by inactivating genes which encode inhibitors. Once such angiogenic cells have evolved, it is possible for the cancer to recruit new blood vessels and hence to grow to larger sizes. Folkman's research has also given rise to exciting new avenues of therapies [Hahnfeldt et al. (1999b)]: Administration of angiogenesis inhibitors can destroy blood supply and result in remission of cancers. While encouraging results have been obtained in laboratory animals, our understanding is far less complete in the context of human pathologies.

This chapter reviews mathematical models which have examined the dynamics of angiogenesis-dependent tumor growth. The absence of blood supply can affect tumor cells in two basic ways [Folkman (2002)]. On the one hand, it can increase the death rate of tumor cells. In the absence of blood supply, apoptosis can be triggered as a result of hypoxia. On the other hand, the absence of blood supply can prevent cell division and growth. In this case the cells are dormant; that is, they do not divide or die. Both types of scenarios have been modeled and will be discussed. The models have similar properties, and we will discuss the requirements for the evolution of angiogenic cell lines and for the transition from a small and non-pathogenic tumor to a tumor with malignant potential. We will then take one of the models and incorporate the spread of the tumor across space into the equations. We will discuss how the dynamics between tumor promotion and inhibition influence more advanced tumor growth within a tissue. Finally, we will discuss clinical implications of the modeling results.

9.1 Model 1: Angiogenesis inhibition induces cell death

We describe and analyze a model for the evolution of angiogenic tumor cell lines [Wodarz and Krakauer (2001)]. The model consists of three basic variables (Figure 9.1).

Balance in favor of inhibition

Balance in favor of inhibition

Balance in favor of inhibition

Fig. 9.1 Schematic diagram illustrating the central assumptions underlying the mathematical model.

Healthy host tissue, xo; a first transformed cell line, Xi, which is non-angiogenic and cannot grow above a given threshold size; an angiogenic tumor cell line which has the potential to progress, x2. It is thought that the formation of new blood vessels depends on a balance of angiogenesis inhibitors and promoters. If the balance is in favor of the inhibitors, new blood vessels are not formed. On the other hand, if it is in favor of the promoters, angiogenesis can proceed. Hence, the model assumes that healthy tissue, xo, and stage one tumor cells, Xi, produce a ratio of inhibitors and promoters that is in favor of angiogenesis inhibition. On the other hand, it is assumed that angiogenic tumor cell lines have the ability to shift the balance in favor of angiogenesis promotion. We first consider progression from the wildtype cells to a first transformed cell line. The basic model is given by the following pair of differential equations, x0 =

r0x0 - (1 ~ ^o) - d0x0, x\ = f¿oroxo ( 1 - Y^j + TlXl ~ jrj i1 ~~ Vi) ~ dlXl•

Healthy cells are assumed to replicate at a density dependent rate i*0X0(l — x0/k0). The value of ko represents the maximum size this population of cells can achieve, or the carrying capacity. The cells die at a rate dijXQ. We assume that the rate of mutation is proportional to the rate of replication of the cells, and is thus given by fiwr0x0(l — xo/ko). The mutations give rise to the first stage of tumor progression, x\, i.e. to a tumor cell line that is not angiogenic. This cell line will depend on the blood supply of the healthy tissue and will not be able to grow beyond a small size. These cells replicate at a density dependent rate riX\{l — x\/ki), where the carrying capacity k\ is assumed to be relatively small (k\ « ko). They die at a rate dixi, and mutate to give rise to an angiogenic tumor cell line, x2, at a rate ^xTix\{l — xi/ki). In the model, the population of healthy cells attains a homeostatic setpoint given by Xq = ko{ro — do)/ro- The mutation rate Ho can be assumed to be very small, since healthy tissue has intact repair mechanisms that ensure faithful replication of the genome. Once mutation gives rise to the first tumor cell line, it will grow to its small homeostatic set point level defined by = k\(r\ — d\)/r\.

The wildtype cell population and the small population of first stage tumors are assumed to reach constant levels in a relatively short time. In other words, they reach an equilibrium abundance. Further tumor growth requires the emergence of the angiogenic cell line, x2. In the following we investigate the conditions required for angiogenic tumor cell lines to evolve assuming a constant background abundance of xq and x\.

The angiogenic cell line replicates at a density dependent rate r2x2(1 — x2/k2). As these cells can potentially influence the balance of inhibitors and promoters in favor of promoters, we have to take these dynamics into account. The death rate of these cells is determined by two components. The angiogenic tumor cells are characterized by a composite background death rate d2x2, as with x0 and x j. In addition, the model assumes that the death rate can be increased if the balance between angiogenesis inhibition and promotion is in favor of inhibition. Hence, this death rate is expressed as (p0X0 + P1X1 +p2x2)/(qx2 + 1). Thus all three cell types lead to the inhibition of angiogenesis, whereas inhibition of angiogenesis can only be overcome by cell line x2.

As we have assumed that xo and X\ are at equilibrium, we start our analysis by ignoring mutation and simply looking at the dynamics of the angiogenic cell line, x2. These dynamics are described by the equation,

where x*{) and x* are defined above. Two outcomes are possible, (i) The cell line X2 cannot invade, resulting in equilibrium EO where xip = 0. (ii) The cell line x2 can invade and converges to equilibrium El described by

2r2?2

where Q = kq% (d2 — 7*2) + r2 + kp2 and subscript I refers to the inhibitory cell lines: piXi = PoXq + pix\.

In the following we examine the stability properties of these two equilibria which are summarized in Figure 9.2. If pjxj < r2 — d2, then the equilibrium describing the extinction of the angiogenic tumor, is not stable. The equilibrium describing the invasion of the angiogenic tumor cell line, a;is stable. In other words, if the above condition is fulfilled, then the degree of angiogenesis inhibition is too weak, and the angiogenic tumor cell line can emerge, marking progression of the disease.

(a) Emergence of <b> »btabUIty: angiogenic cells both outcome» possible

(c) No Emergence of angiogenic cells

Level of angiogenesis inhibition

(a) Emergence of <b> »btabUIty: angiogenic cells both outcome» possible

(c) No Emergence of angiogenic cells

Level of angiogenesis inhibition

Extinction equilibrium stable

Fig. 9.2 Graph showing the stability properties and the outcome of the model. Parameters were chosen as follows: ri = 1; fa = 1; = 0.1; P2 = 1; <? = 10.

Extinction equilibrium stable

Fig. 9.2 Graph showing the stability properties and the outcome of the model. Parameters were chosen as follows: ri = 1; fa = 1; = 0.1; P2 = 1; <? = 10.

On the other hand, if pixj > r2 — d2, the degree of inhibition is stronger and the situation is more complicated (Figure 9.2). The equilibrium describing the extinction of the angiogenic cell population, becomes stable. However, equilibrium describing the emergence of angiogenic tumor cells, may or may not be stable (Figure 9.2).

(1) If the degree of angiogenesis inhibition lies above a certain threshold, equilibrium x^ is unstable and the angiogenic cell line cannot invade. It was not possible to define this threshold in a meaningful way.

(2) If the degree of angiogenesis inhibition lies below this threshold, equilibrium x^ remains stable. Now, both the extinction and the emergence equilibria are stable (Figure 9.2). This means that two outcomes are possible and that the outcome depends on the initial conditions. Either the angiogenic cell line fails to emerge, or the angiogenic cell line does emerge, resulting in tumor progression. As shown in Figure 9.3, a low initial abundance of angiogenic tumor cells results in failure of growth. On the other hand, a high initial number of angiogenic tumor cells results in growth of the tumor and progression (Figure 9.3).

To summarize, the model shows the existence of three parameter regions (Figure 9.2). If the degree of angiogenesis inhibition by healthy tissue and stage one tumor cells lies below a threshold, angiogenic tumor cell lines always invade resulting in progression of the disease. If the degree of inhibition lies above a threshold, the angiogenic cell lines can never emerge and pathology is prevented. Between these two thresholds, both outcomes are possible depending on the initial conditions. A high initial number of angiogenic tumor cells results in growth of this cell line and progression of the disease.

What does the initial number of angiogenic cells mean in biological terms? The dependence of growth on the initial number of angiogenic tumor cells presents an effective barrier against pathologic tumor growth. Given that a small number of non-angiogenic tumor cells exists, it will be difficult to create a sufficiently large number of angiogenic mutants to overcome the blood supply barrier. This difficulty could explain why, upon autopsy, people tend to show small tumors which have failed to grow to larger sizes. The initial number of the angiogenic cells could be determined by the mutation rate which gives rise to the angiogenic cells. If the mutation rate is high, the initial number of angiogenic cells will be high. On the other hand, if the mutation rate is low, the initial number of the c <u 00 o '5ö

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