Cancer and immune responses

As pointed out in the previous chapter, the body is characterized by defense systems which can limit the growth and pathogenicity of selfish tumor cells once they have arisen by a series of mutations. The previous chapter explored how the limitation of blood supply can prevent cancers from growing beyond a very small size and from progressing. This is a mechanism which is supported very well by experimental and clinical data, and which is also studied from a therapeutic point of view. Another mechanism which can potentially counter the growth of cancer cells is the immune system. As will become apparent in this chapter, however, the role of the immune system in cancer is highly debated and uncertain.

The immune system defends human beings from intruders such as pathogens which would otherwise kill them. It does so by specifically recognizing proteins derived from the pathogens (for example, viruses, bacteria, or parasites). Through complicated mechanisms which will be discussed briefly later on, the immune system knows that these proteins are foreign and that they are not derived from the organism that it is supposed to protect. What about cancer? As discussed throughout this book, carcinogenesis involves the accumulation of multiple mutations and in general often exhibits genetic instability. This means that many mutated proteins are produced which are different from the organism's own proteins and should thus appear foreign. In principle, these should be visible to the immune system which could potentially remove tumor cells and prevent the development of cancer.

A role of the immune system in the fight against cancer was first suggested in 1909 by Paul Ehrlich [Ehrlich (1909)]. It was not, however, until the 1950s, when the idea was pursued more vigorously and the immune surveillance hypothesis was formulated by Burnet [Burnet (1957)]. It stated that while cancers continuously arise, they are eliminated by specific immune responses. The successful establishment of cancer was thought to come about by the occasional escape of cancer cells from the immune responses. In support of this hypothesis, experimental data indicated that cancer cells show many characteristics which prevent the immune system from recognizing the mutated proteins and from killing the cells successfully.

Following a lot of enthusiastic research in this context, clinical and experimental data cast doubt on the immune surveillance hypothesis [Dunn et al. (2002)]. Patients characterized by impaired immune systems showed no significant increase in the incidence of cancers which are not induced by viruses. Similarly, nude mice - which lack adaptive immune responses - have a similar incidence of cancers compared to normal mice. Moreover, experimental studies tracked immune cells specific for cancer proteins and found that they did not react successfully in the first place. This in fact contradicted the hypothesis that the immune system can play any surveillance role in the context of cancers. Since then, many papers have investigated the relevance of immune responses in cancer [Dunn et al. (2002)]. While our understanding is still rudimentary, it seems that the truth lies somewhere in between these two extreme views.

This chapter will review mathematical work which tried to account for some of the experimental data on immunity and cancer. We will start with a brief overview of immunity which gives the necessary background before discussing the model and equations.

11.1 Some facts about immune responses

This section will briefly review some basic immunological principles which form the basis for the rest of the chapter. More extensive descriptions of the immune system can be found in any standard immunology text book, for example [Janeway et al. (1999)]. We can distinguish between two basic types of immune responses. Innate immune responses provide a first line of defense. They do not recognize foreign proteins specifically. They provide environments which generally inhibit the spread of intruders. While they may be important to limit the initial growth of a pathogen, they are usually not sufficient to resolve diseases. They will not be considered further here. On the other hand, adaptive immune responses can specifically recognize foreign proteins and can resolve diseases. We concentrate on this type of response here. Adaptive immunity can be subdivided broadly into two types of responses: antibodies and killer cells. Antibodies recognize proteins outside the cells, such as free virus particles, extracellular bacteria or parasites. Killer cells recognize foreign proteins which are displayed on cells. For example, viruses replicate inside cells. During this process, the cell captures some viral proteins and displays them on the cell surface. When the killer cells recognize the foreign proteins on the cell surface, they release substances which kill. Mutated cancer proteins are displayed on the surface of cancer cells. Therefore, killer cells are the most important branch of the immune system in the fight against cancer. The rest of this chapter will discuss only the role of killer cell responses. The scientific term is cytotoxic T lymphocyte, abbreviated as CTL. They can also be referred to as CD8-I- T cells because they are characterized by the expression of the CD8 molecule on the cell surface.

The CTL are able to recognize the cell which displays a foreign protein in the following way. When proteins inside the cell are captured for display on the cell surface, they are presented in conjunction with so-called major histocompatibility complex (MHC) molecules. The MHC genes are highly variable, and different MHC genotypes present different proteins. This accounts for the variability between different people in immune responses against the same pathogen. The CTL carries the so-called T cell receptor or TCR. The TCR recognizes the protein-MHC complexes. This triggers the release of specific molecules such as perforin or FAS, which induce apoptosis in the cell that displays the foreign proteins. In immunology, the foreign protein which is recognized by the immune cells is also referred to as antigen.

It is important to note that all proteins of a cell are processed and displayed in this way, not just the ones which are supposed to be recognized by the immune system. With this in mind, how can the immune cells distinguish between self and foreign? After all, the vast majority of proteins produced by a cell are normal self proteins. Immune cells usually do not react against self proteins. This is called tolerance. They do, however, react against most pathogens. This is called reactivity. In the context of cancers it appears that the immune response can remain tolerant to mutated cancer proteins during the natural history of progression, but that reactivity can be induced by certain vaccination approaches. What determines whether we observe tolerance or reactivity? This question is still debated. Different hypotheses have been put forward. According to one hypothesis, immune cells can distinguish between "self" and "foreign" [Janeway (2002)]. Neg ative selection early during the development of immune cells can result in the deletion of cells which react to self proteins. This hypothesis has difficulties to explain immunological tolerance in the context of tumors because the mutated tumor cell proteins should appear foreign to the immune system. Another idea is the danger signal hypothesis [Fuchs and Matzinger (1996)]. This states that immune responses only react if they sense so-called "danger signals". They can be released as a result of tissue injury, necrosis, or virus infections. Since tumor cells die predominantly by apop-tosis, such danger signals are not released. This could explain why the immune system remains tolerant in the context of cancers, while it reacts to infectious agents. The problem with the danger signal hypothesis is that such signals have yet to be identified in terms of chemical compounds.

Recently, the phenomenon of cross-priming and cross-presentation has received attention in the context of CTL regulation [Albert et al. (2001); Albert et al. (1998); Belz et al. (2002); Blankenstein and Schuler (2002); den Haan and Bevan (2001); den Haan et al. (2000); Heath and Carbone (2001a); Heath and Carbone (2001b)]. This relates to the initiation of immune responses. Before the antigen is seen, very few specific immune cells exist. When antigen is recognized the specific CTL start to divide and undergo clonal expansion. If clonal expansion is successful, we observe reactivity. If it is not successful, we observe tolerance. This initiation of the response does not occur at the site where the aberrant target cells are located, but in the lymph nodes. Cross priming means that the initiation of the CTL response is not mediated by antigen which is displayed on the target cells themselves. Instead, it occurs when antigen is recognized on so-called antigen presenting cells (APCs). The job of these cells is to take up antigen, transport it to the lymph nodes, and present it to the CTL. This activates the CTL and induces clonal CTL expansion. Once expanded, the CTL can migrate to the site where the aberrant cells are located. Once there, they perform their effector function; that is, they kill the cells.

In summary, the indirect recognition of antigen by CTL on antigen presenting cells is called cross-priming or cross-presentation. On the other hand, the direct recognition of antigen on the target cells which are supposed to be killed is called direct presentation. These concepts are explained in more detail schematically in Figure 11.1.

Because cross-presentation is thought to play a role in deciding whether we observe reactivity or tolerance, this process is the subject of mathematical models in this chapter. We start by introducing a basic mathematical model of CTL response regulation and then discusses basic mathematical results. These mathematical results are subsequently applied to aspects of cancer evolution, progression and treatment.

Cross-presentation Direct presentation

Cross-presentation Direct presentation

Fig. 11.1 Schematic representation of the concept of cross-priming, which is central to this chapter. So called "antigen presenting cells" can take up antigen (proteins derived from pathogens or cells) and display them on their surface. Before the APCs can function, they need to be activated. This is achieved by so called helper T cells (Th) which can recognize the antigen on the APC. The activated APC subsequently can interact with CTL. CTL can also specifically recognize the antigen on the APC. This interaction activates the CTL which can then turn into effector cells and kill the troubled target cells which display the antigen. These target cells are different from APCs and can be for example virus-infected cells or tumor cells. This process is called cross-priming because the CTL do not get activated directly by the troubled cells which need to be killed, but indirectly by the APCs which can take up and display the antigen.

Fig. 11.1 Schematic representation of the concept of cross-priming, which is central to this chapter. So called "antigen presenting cells" can take up antigen (proteins derived from pathogens or cells) and display them on their surface. Before the APCs can function, they need to be activated. This is achieved by so called helper T cells (Th) which can recognize the antigen on the APC. The activated APC subsequently can interact with CTL. CTL can also specifically recognize the antigen on the APC. This interaction activates the CTL which can then turn into effector cells and kill the troubled target cells which display the antigen. These target cells are different from APCs and can be for example virus-infected cells or tumor cells. This process is called cross-priming because the CTL do not get activated directly by the troubled cells which need to be killed, but indirectly by the APCs which can take up and display the antigen.

11.2 The model

We describe a model containing four variables: cells directly displaying antigen such as infected cells or tumor cells, T (we will refer to these cells as "target cells"); non-activated APCs which do not present the antigen, A; loaded and activated APCs which have taken up antigen and display it, A*\ CTL, C. The model is given by the following system of differential equations which describe the development of these populations over time,

The infected or tumor cells grow at a density dependent rate rT(l-T/k). In case of virus infections, this represents viral replication, where virus load is limited by the availability of susceptible cells, captured in the parameter k. In case of tumors, this corresponds to division of the tumor cells, and the parameter k denotes the maximum size the tumor can achieve, limited for example by spatial constraints. The cells die at a rate dT, and are in addition killed by CTL at a rate 7 TC. APCs, A, are produced at a constant rate A and die at a rate 8\A. They take up antigen and become activated at a rate a AT. The parameter a summarizes several processes: the rate at which antigen is released from the cells, T, and the rate at which this antigen is taken up by APCs and processed for display and cross-presentation. Loaded APCs, A*, are lost at a rate S2A*. This corresponds either to death of the loaded APC, or to loss of the antigen-MHC complexes on the APC. Upon cross-presentation, CTL expand at a rate r]A*C/(eC+l). The saturation term, eC+1, has been included to account for the limited expansion of CTL in the presence of strong cross-stimulation [De Boer and Perelson (1995)]. The activated and expanding population of CTL can kill the infected cells upon direct presentation. In addition, it is assumed that direct presentation can result in removal of CTL at a rate qTC. This can be brought about, for example, by antigen-induced cell death, or overdifferentiation into effectors followed by death. Finally, CTL die at a rate HC.

Thus a central assumption of the model is that cross-presentation can induce CTL expansion, while direct presentation does not have that effect; instead it can result in the decline of the CTL population. This assumption implies that the magnitude of cross-presentation relative to direct presentation could be a decisive factor which determines the outcome of a CTL response: activation or tolerance. In the model, the ratio of cross-presentation to direct presentation is given by cA*/qT.

We assume that r > a. That is, the rate of increase of the target cells, T, is greater than their death rate. This ensures that this population of cells can grow and remain present. If this is fulfilled, the system can converge to a number of different equilibria (Figure 11.2). The expressions for the equilibria will not be written out here since most of them involve complicated expressions.

(1) The CTL response fails to expand, i.e. C — 0. The population of target cells grows to a high equilibrium level, unchecked by the CTL. The populations of unloaded and loaded APCs, A and A*, also equilibrate.

(2) The CTL response expands, i.e. C > 0. In this case, the system can converge to one of two different outcomes, (a) The number of

(b)
(c)

Time (arbitrary units)

Fig. 11.2 Different outcomes of the model shown as time series, (a) Tolerance; CTL go extinct, (b) Tolerance outcome where CTL do not go extinct but are maintained at very low levels, (c) Immunity outcome. Parameters were chosen as follows: r=0.5; k=10; d=0.1; 7=1; A=1; 8^=0.1; S2=1.S; r]=2; e=l; q=0.5; n=0.1. ce=0.2 for (i) and a=0.1 for (ii). For (iii) a=0.05; r=10; r)=10.

Time (arbitrary units)

Fig. 11.2 Different outcomes of the model shown as time series, (a) Tolerance; CTL go extinct, (b) Tolerance outcome where CTL do not go extinct but are maintained at very low levels, (c) Immunity outcome. Parameters were chosen as follows: r=0.5; k=10; d=0.1; 7=1; A=1; 8^=0.1; S2=1.S; r]=2; e=l; q=0.5; n=0.1. ce=0.2 for (i) and a=0.1 for (ii). For (iii) a=0.05; r=10; r)=10.

CTL is low and the number of target cells is high. This outcome is qualitatively similar to (i), because the CTL population does not fully expand, and the population of target remains high, (b) The number of CTL is high and the number of target cells is low. This can be considered the immune control equilibrium. If the population of target cells is reduced to very low levels, this can be considered equivalent to extinction (number of cells below one).

These equilibria therefore fall into two basic categories: (a) Tolerance; this is described by two equilibria. Either the immune response goes extinct, or it exists at low and ineffective levels. (b) Reactivity; this is described by only one equilibrium. The immune response expands to higher levels and exerts significant levels of effector activity. The following sections will examine which outcomes are achieved under which circumstances.

11.3 Method of model analysis

The equilibrium outcomes of the model describe the states to which the system can converge: reactivity or tolerance. These equilibria are roots of polynomials of degree larger than two. Consequently, the stability analysis of the equilibria was performed numerically, and will be presented as bifurcation plots below. The numerical analysis was carried out with a program called Content. It allows tracking the position of the equilibria as we vary parameters of the system. It also determines their eigenvalues as a function of parameters, thus giving full information on their stability properties.

11.4 Properties of the model

The two most important parameters in the present context are the rate of antigen uptake by APCs, a, and the growth rate of the target cells, r. This is because variation in these parameters can significantly influence the ratio of cross-presentation to direct presentation which is the subject of investigation. Hence, in the following sections we will examine the behavior of the model in dependence of these two parameters.

The rate of antigen uptake by APCs. The rate of antigen uptake by APCs comprises two processes: (i) the degree to which the antigen is made available for uptake; this can be determined for example by the amount of antigen released from the target cell, or the amount of apoptosis going on [Albert et al. (1998)]. (ii) The rate at which the APCs take up the available antigen and process it for presentation. As the rate of antigen uptake by APCs, a, decreases, the ratio of cross-presentation to direct presentation decreases (Figure 11.3a). When the value of a is high, the outcome is immunity. If the value of a is decreased and crosses a threshold, we enter a region of bistability (Figure 11.3a): both the immunity and the tolerance equilibria are stable. Which outcome is achieved depends on the initial conditions. If the value of a is further decreased and crosses another threshold, the immune control equilibrium loses stability. The only stable

tolerance equilibrium

tolerance equilibrium

Rate of antigen presentation by APCs, cc

Growth rate of target cells, r.

Fig. 11.3 Bifurcation diagram showing the outcome of the model as a function of (a) the rate of antigen presentation by APCs, a, and (b) the growth rate of target cells, r. Virus load and the ratio of cross-presentation to direct presentation at equilibrium are shown. Parameters were chosen as follows: r=0.5; k=10; d=0.1; 7=1; A=1; 8i=0.1; a=0.5; ¿2=1-5; r)=2; s=l; q=0.5; fi=0.1.

Rate of antigen presentation by APCs, cc

Growth rate of target cells, r.

Fig. 11.3 Bifurcation diagram showing the outcome of the model as a function of (a) the rate of antigen presentation by APCs, a, and (b) the growth rate of target cells, r. Virus load and the ratio of cross-presentation to direct presentation at equilibrium are shown. Parameters were chosen as follows: r=0.5; k=10; d=0.1; 7=1; A=1; 8i=0.1; a=0.5; ¿2=1-5; r)=2; s=l; q=0.5; fi=0.1.

outcome is tolerance (Figure 11.3a).

In the region of bistability, the dependence on initial conditions is as follows. Convergence to the immune control equilibrium is promoted by low initial numbers of target cells, high initial numbers of presenting APCs, and high initial numbers of CTL. This is because under these initial conditions, the dynamics start out with a high ratio of cross-presentation to direct presentation and this promotes the expansion of the CTL. On the other hand, if the initial number of target cells is high and the initial number of presenting APCs and CTL is low, then the initial ratio of cross-presentation to direct presentation is low and this promotes tolerance. There are some slight variations to this general picture. As they do not alter the basic results, however, the reader is referred to [Wodarz and Jansen (2001)] for details.

In summary, as the rate of antigen uptake by APCs is decreased, the ratio of cross-presentation to direct presentation decreases, and this shifts the dynamics of the CTL response in the direction of tolerance. This can include a parameter region in which both the tolerance and the immunity outcome are stable, depending on the initial conditions. If the CTL responsiveness to cross-presentation is very strong, tolerance becomes an unlikely outcome.

The growth rate of target cells. An increase in the growth rate of target cells, r, results in a decrease in the ratio of cross-presentation to direct presentation in the model. Hence an increase in the growth rate of target cells shifts the dynamics of the CTL from a responsive state towards tolerance. The dependence of the dynamics on the parameter r is shown in Figure 11.3b. The growth rate of target cells needs to lie above a threshold to enable the CTL to potentially react. This is because for very low values of r, the number of target cells is very low, not sufficient to trigger immunity. If the growth rate of target cells is sufficiently high to potentially induce immunity, we observe the following behavior (Figure 11.3b). If the value of r lies below a threshold, the only stable outcome is immunity. If the value of r is increased and crosses a threshold, we enter a region of bistability. That is, both the immunity and the tolerance outcomes are possible, depending on the initial conditions. The dependence on the initial conditions is the same as explained in the last section. If the value of r is further increased and crosses another threshold, the immunity equilibrium loses stability and the only possible outcome is tolerance. Again, there are some slight variations to this general picture. As they do not alter the basic results, however, the reader is referred to [Wodarz and Jansen (2001)] for details.

In summary, an increase in the growth rate of target cells has a similar effect as a decrease in the rate of antigen uptake by APCs: the ratio of cross-presentation to direct presentation becomes smaller, and the outcome of the dynamics is driven from immunity towards tolerance. Again, this includes a parameter region where both the immunity and tolerance outcomes are stable and where the outcome depends on the initial conditions. The higher the overall responsiveness of the CTL to cross-stimulation, the less likely it is that a high growth rate of target cells can induce tolerance.

11.5 Immunity versus tolerance

The models have investigated the topic of CTL regulation from a dynamical point of view. We showed that the immune system can switch between two states: tolerance and activation. Which state is reached need not depend on the presence or absence of signals, but on the relative magnitude of cross-presentation to direct presentation. This shows that regulation can be accomplished without signals but in response to a continuously varying parameter. Thus, the regulation of CTL responses could be implicit in the dynamics. This relies on the assumption that there is a difference in the effect of cross-presentation and direct presentation. The mathematical model assumes that while cross-presentation results in CTL expansion, direct presentation results in lysis followed by removal of the CTL. Some mechanisms described in the literature support this notion. The simplest mechanism resulting in CTL removal could be antigen-induced cell death [Baumann et al. (2002); Budd (2001); Hildeman et al. (2002)]. That is, exposure to large amounts of antigen by direct presentation can trigger apoptosis in the T cells. Another mechanisms could be that exposure to direct presentation of antigen on the target cells induces the generation of short lived effectors which are destined for death [Guilloux et al. (2001)]. Since CTL effectors are thought to die shortly after killing target cells, exposure to large amounts of direct presentation can result in over-differentiation and an overall loss of CTL.

With the assumptions explained above, we find a very simple rule that determines whether CTL responses expand and react, or whether they remain silent and tolerant. CTL expansion and immunity is promoted if the ratio of cross-presentation to direct presentation is relatively high. This is because the amount of CTL expansion upon cross-presentation outweighs the degree of CTL loss upon direct presentation. On the other hand, tolerance is promoted if the ratio cross-presentation to direct presentation is relatively low. This is because the amount of CTL loss upon direct presentation outweighs the amount of CTL expansion upon cross-presentation.

For self antigen displayed on cells of the body, the ratio of cross-presentation to direct presentation is normally low. This is because these cells do not die at a high enough rate or release the antigen at a high enough rate for the amount of cross presentation to be strong. On the other hand, large amounts of this antigen can be available on the surface of the cells expressing them (direct presentation). In terms of our model, this situation can best be described by a low value of a. Hence, in our model,

CTL responses are not predicted to become established against self antigens. Instead, the outcome is tolerance. In addition, the initial conditions favor tolerance in this scenario. When immune cells with specificity for self are created and try to react, the number of these immune cells is very low and the number of target cells (tissue) is relatively high. This promotes failure of the CTL response to expand and to become established. On the other hand, with infectious agents, antigen is abundantly available. For example, virus particles are released from infected cells, ready to be taken up by APCs for cross-presentation. Therefore, the immune responses react and become fully established. In contrast, tumors may fail to induce CTL responses because tumor antigens are largely displayed on the surface of the tumor cells, but relatively little tumor antigen is made available for uptake by dendritic cells and hence for cross-presentation. The relative amount of cross-presentation, however, is influenced by parameters such as the growth rate of the target cells, and we observe a parameter region where the outcome of the CTL dynamics can depend on the initial conditions. Since cancer cells continuously evolve towards less inhibited growth, these results have implications for the role of CTL in tumor progression and cancer therapy. This is explored in the following sections.

11.6 Cancer initiation

A tumor cell is characterized by mutations which enable it to escape growth control mechanisms which keep healthy cells in check. According to the model, the generation of a tumor cell can lead to three different scenarios (Figure 11.4):

(i) A CTL response is induced which clears the cancer, (ii) A CTL response develops which is weaker; it controls the cancer at low levels, but does not eradicate it. (iii) A CTL response fails to develop; tolerance is achieved and the cancer can grow uncontrolled. Which outcome is attained depends on the characteristics of the cancer cells. In particular it depends on how fast the cancer cells can grow (r in the model), and how resistant they are against death and apoptosis. Cell death, and in particular apop-tosis, is thought to increase the amount of cross-presentation [Albert et al. (1998)]. Resistance to apoptosis thus corresponds to a reduction in the parameter a in the model. Three parameter regions can be distinguished (Figure 11.5).

(i) If the cancer cells replicate slowly and/or still retain the ability to

(b)

• ¡1 <i I' «I it „ , , . -i yyyuv'^v'vvvv'.'w.^ ;

0 50 100 ISO 200 250 X0

Time (arbitrary scale)

Fig. 11.4 Time series plots showing the different possible outcomes when a tumor is generated, (a) Clearance, (b) Immune control but failure to clear the target cells, (c) Tolerance. Parameters were chosen as follows: k=10; d=0.1; 7 =1; A=1; 5\=0.1; a=0.5; 52=1.5; V =2; e=l; q=0.5; fj. =0.1. (a) r = 0.13. (b, c) r = 1. The difference between graphs (b) and (c) lies in the initial number of CTL, 2.

undergo apoptosis, the cancer will be cleared, because strong CTL responses are induced, (ii) If the cancer cells replicate faster and/or the degree of apoptosis is weaker, the ratio of cross-presentation to direct presentation is reduced. This can shift the dynamics into the bistable parameter region. That is, the outcome depends on the initial conditions. If the initial size of the tumor is relatively small, it is likely that CTL responses will be develop successfully. This will result in control of the tumor. Because the response is less efficient, however, clearance is not likely. If the size of the tumor is already relatively large when the CTL response is activated, the likely outcome is tolerance, (iii) If the growth rate of the tumor cells is

CTL-mediated clearance

(cancer goes extinct)

CTL-mediated clearance

(cancer goes extinct)

CTL-mediated control

(cancer persists at low levels)

CTL collapse

(uncontrolled cancer growth)

CTL-mediated control

(cancer persists at low levels)

CTL collapse

(uncontrolled cancer growth)

/ uslcr rule "/ i um tri clt replication and/or less apoptosis

Fig. 11.5 Three parameter regions of the model: CTL mediated clearance, CTL-mediated control with tumor persistence at low levels, and CTL collapse leading to uncontrolled tumor growth. Which outcome is observed depends on the rate of cancer cell replication and on the ability of cells to undergo apoptosis.

still higher and/or the degree of apoptosis is still lower, then the ratio of cross-presentation to direct presentation falls below a threshold; now the only possible outcome is tolerance and uncontrolled tumor growth.

11.7 Tumor dormancy, evolution, and progression

Here, we investigate in more detail the scenario where the growth rate of the tumor is intermediate, and both the tolerance and the CTL control outcomes are possible, depending on the initial conditions. Assume the CTL control equilibrium is attained because the initial tumor size is small. The number of tumor cells is kept at low levels, but the tumor is unlikely to be cleared because in this bistable parameter region the ratio of cross-presentation to direct presentation is already reduced. If the tumor persists at low levels, the cells can keep evolving over time. They can evolve, through selection and accumulation of mutations, either towards a higher growth rate, r, or towards a reduced rate of apoptosis which leads to reduced levels of antigen uptake by dendritic cells, a. Both cases result in similar evolutionary dynamics. This is illustrated in Figure 11.6 assuming that the cancer cells evolve towards faster growth rates (higher values of r).

Growth rate of the tumor, r

Fig. 11.6 Equilibrium tumor load (a) and the number of tumor specific CTL (b) as a function of the growth rate of tumor cells, r. This graph can by interpreted to show the effect of tumor evolution towards faster growth rates over time. As evolution increase the value of r over time, the tumor population and the CTL attain a new equilibrium. Parameters were chosen as follows: r=0.5; k=10; d=0.1; 7 =1; A=1; 5\=0.1; a=0.5; 62=1-5; r/ =2; e=l; q=0.5; p, =0.1.

Growth rate of the tumor, r

Fig. 11.6 Equilibrium tumor load (a) and the number of tumor specific CTL (b) as a function of the growth rate of tumor cells, r. This graph can by interpreted to show the effect of tumor evolution towards faster growth rates over time. As evolution increase the value of r over time, the tumor population and the CTL attain a new equilibrium. Parameters were chosen as follows: r=0.5; k=10; d=0.1; 7 =1; A=1; 5\=0.1; a=0.5; 62=1-5; r/ =2; e=l; q=0.5; p, =0.1.

An increase in the growth rate of tumor cells does not lead to a significant increase in tumor load. At the same time, it results in an increase in the number of tumor-specific CTL. The reason is that a faster growth rate of tumor cells stimulates more CTL which counter this growth and keep the number of tumor cells at low levels. When the growth rate of the tumor cells evolves beyond a threshold, the equilibrium describing CTL-mediated control of the cancer becomes unstable. Consequently, the CTL response collapses and the tumor can grow to high levels.

The dynamics of tumor growth and progression can include a phase called "dormancy". During this phase the tumor size remains steady at a low level over a prolonged period of time before breaking out of dormancy and progressing further. Several mechanisms could account for this phe nomenon. The limitation of blood supply, or inhibition of angiogenesis, can prevent a tumor from growing above a certain size threshold [Folkman (1995b)]. When angiogenic tumor cell lines evolve, the cancer can progress further. Other mechanisms that have been suggested to account for dormancy are immune mediated, although a precise nature of this regulation remains elusive [Uhr and Marches (2001)]. As shown in this section, the model presented here can account for a dormancy phase in tumor progression. If the overall growth rate of the cancer cells evolves beyond this threshold, dormancy is broken: the CTL response collapses and the tumor progresses.

11.8 Immunotherapy against cancers

Assuming that the CTL response has failed and the cancer can grow unchecked, we investigate how immunotherapy can be used to restore CTL mediated control or to eradicate the tumor. In the context of the model, the aim of immunotherapy should be to increase the ratio of cross-presentation to direct presentation. The most straightforward way to do this is dendritic cell vaccination. In the model, this corresponds to an increase in the number of activated and presenting dendritic cells, A*. We have to distinguish between two scenarios: (i) The tumor cells have evolved sufficiently so that the CTL control equilibrium is not stable anymore, and the only stable outcome is tolerance, (ii) The tumor has evolved and progressed less; the equilibrium describing CTL mediated control is still stable.

First we consider the situation where the tumor has progressed far enough so that the CTL control equilibrium is not stable anymore. Upon dendritic cell vaccination, tolerance is temporarily broken (Figure 11.7). That is, the CTL expand and reduce the tumor cell population. This CTL expansion is, however, not sustained and tumor growth relapses (Figure 11.7). The reason is as follows. Upon dendritic cell vaccination, the ratio of cross-presentation to direct presentation is increased sufficiently, enabling the CTL to expand. However, this boost in the level of cross-presentation subsequently declines, allowing the tumor to get the upper hand and re-grow. The model suggests, however, that the tumor can be eradicated if the level of cross-presentation is continuously maintained at high levels. This can be achieved by repeated vaccination events (Figure 11.7). The next vaccination event has to occur before the level of cross-presentation has significantly declined. This will drive tumor load below a threshold level

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