## Tw

U) = (3pxu (1 - 17) (1 - a) H--- [1 - u + u (1 - 77) (1 - a) (1 - /?)] + 7z, px + 1

tjurw

The first term in the equation for tumor growth represents the production of tumor cells from healthy cells by mutation with a rate (3pxu{ 1 — 77)(1 — a) (see Figure 8.1). In other words, a transformed cell is generated when a healthy cell experiences a genomic alteration which is not repaired, is carcinogenic, and does not result in apoptosis. The parameter r denotes the turnover of the tumor cells. Note the absence of the parameter <fi which means that there is no constant population size. Instead, the population of tumor cells can grow exponentially. Similarly to healthy cells, tumor cells can become damaged with a rate u. With a probability 77 the damage results in cell cycle arrest and arrested cells return to a dividing state with a rate 7. With a probability 1 — 77 damage does not result in cell cycle arrest and repair. With a probability a the damage leads to cell death, for example caused by apoptosis. With a probability 1 — a, the damaged cell survives and attains an additional mutation. This mutation contributes to further cancer progression with a probability (3. In addition to these basic cell growth kinetics, the model assumes that tumor cell division is inhibited by the surrounding tissue cells with a rate p. The higher the value of p, the lower the rate of tumor cell division. If the value of p is sufficiently large, the rate of tumor cell division is less than or equal to the tumor death rate; hence, the tumor fails to grow altogether. In the model, only cells which do not repair or are not senescent can contribute to this suppressive activity. This is because repairing or senescent cells are not active. Note that the outcome of the model does not depend on the particular form by which tissue-mediated tumor cell inhibition is described. Alternatives (such as a tissue-mediated increase in the death rate of tumor cells) can also be explored, and give qualitatively identical results (see Chapter 9).

It is important to note that the model only concentrates on describing the initial growth dynamics of the tumor within a tissue environment which varies in checkpoint competence (such as p53+/+, p53+/-, and p53-/- organisms). The checkpoints under consideration (e.g. p53) can induce cell cycle arrest and apoptosis in tumor cells; they can be lost later in the course of progression [Blagosklonny (2002); Kahlem et al. (2004)]. Therefore, these initial tumor cells are assumed to have the same level of checkpoint competence as the normal tissue cells.

There are two quantities which are important for the analysis. These are the initial growth rate of the tumor and the mutation rate of the tumor cells. They will be explained in turn.

(i) The initial growth rate of the tumor is made up of two components. These are the production of tumor cells from healthy tissue, and the growth of those tumor cells (clonal expansion). While production of tumor cells by healthy tissue is important for the initiation of tumor cell growth, the growth term becomes dominant as clonal expansion takes off. Because both processes are involved in the initiation of cancer, we consider the time until a tumor reaches an arbitrary threshold size as a measure of how fast a tumor can develop. The longer this time, the slower the development of cancer. In terms of experimental data it correlates with a lower fraction of animals which show a tumor by a defined time point. It is important to note that this growth phase only corresponds to the initial expansion phase which leads to a first stage and detectable tumor. While not included in the model explicitly, this initial growth is not assumed to go on. Instead the tumor is likely to remain at a given small size until further mutations allow the cells to progress and expand further. In the model this is dealt with by stopping the simulation once the tumor has reached this critical threshold size.

(ii) Thus, once a tumor has been initiated and has grown to the threshold size, further progression requires the accumulation of additional carcinogenic mutations. This is influenced by the mutation rate of cells, /i. The higher the mutation rate, the higher the chance that an additional carcinogenic mutation is created. In the model the mutation rate of cells is given by fi = /3ru( 1 — r])( 1 — a)/(px + 1).

The following sections will examine how checkpoint competence influ ences the initial tumor cell growth and the accumulation of further mutations. We will start by examining the basic tumor cell dynamics without taking into account tissue-tumor cell interactions. Then, we will examine how tissue-mediated tumor cell inhibition influences the results. As explained above, the analysis will concentrate on the effect of cell cycle arrest and senescence, captured in the parameter 77.

### 8.5 Checkpoints and basic tumor growth

Here we consider how checkpoint competence influences initial tumor growth and the accumulation of carcinogenic mutations in the absence of tissue-mediated tumor cell inhibition (p = 0). That is, the only effect of healthy tissue is to give rise to a transformed cell by mutation. The outcome depends on what can be called the "cost of cell cycle arrest", and the "cost of cell death", and is summarized in Figure 8.3a. These concepts have been explained in detail in Chapter 7. The cost of cell cycle arrest represents the average time cells remain in an arrested state (given by I/7). The longer the duration of cell cycle arrest, the more the cell cycle becomes delayed, and the higher the fitness cost for the cells. On the other hand, the cost of cell death represents the fitness reduction of cells which is brought about by the generation of lethal mutants. Lethality can arise from necrotic cell death, but in cancer apoptosis will also play a very important role. The higher the probability that a mutant is lethal (higher value of a), the higher the cost of cell death. The behavior of the model depends on the relative magnitude of these costs.

If the cost of cell cycle arrest is significantly higher than the cost of cell death, a decrease in checkpoint competence correlates with faster initial tumor growth, and with an enhanced ability to accumulate further carcinogenic mutations (Figure 8.3a, 8.4). The reason is that cells in checkpoint deficient organisms avoid senescence, and this leads to faster cell division and a higher mutation rate. As the cost of cell cycle arrest is decreased relative to the cost of cell death (cells return to a dividing state faster and more mutations result in cell death), this picture changes (Figure 8.3a). Now a reduction in checkpoint competence can lead to a slower initial growth of tumor cells. That is, checkpoint deficient organisms are expected to show a reduced incidence of tumors. The lower the cost of cell cycle arrest, and the higher the cost of cell death, the more pronounced this trend. The reason is as follows. The advantage gained from avoiding cell cycle arrest in check-

(a) 1 (relatively long duration of cell cycle arrest)

b to

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