Quantitative models from population biology and evolutionary game theory frame the tumor-host …

• Abstract
• Introduction
• Mathematical Models
• Discussion
• References
• Figures

Biology Articles » Biomathematics » Application of quantitative models from population biology and evolutionary game theory to tumor therapeutic strategies » Mathematical Models

Mathematical Models

- Application of quantitative models from population biology and evolutionary game theory to tumor therapeutic strategies

 

Non-Evolutionary Population Interactions
We have previously demonstrated that the tumor-host interface can be explored using the principles of population ecology (4, 5). In this paradigm, tumor populations are viewed as invading species disrupting the well-ordered, cooperative cellular societies that ordinarily form functioning tissue in multicellular organisms. Each tumor "species" arising within or entering this somatic microecology may experience one of three outcomes: (a) it may undergo unconstrained proliferation driving normal cellular populations to extinction thus forming an invasive cancer; (b) it may develop a stable coexistence with normal cells forming a "benign" tumor; and (c) it may be eliminated by the host tissue and suffer extinction. We first examine the dynamics of the tumor-host interface to determine the critical parameters that control these outcomes.

The simplest and most widely used model describing the interaction of multiple populations is of the Lotka-Volterra type (6). There are a variety of other mathematical models of tumor growth kinetics (see, e.g., Chapter 3 of 6) for a survey of models). However, because all such models, including Lotka-Volterra, predict the same qualitative behavior, we have employed the one that is most simple to present and analyze mathematically (7–9). Here, we write Lotka-Volterra equations with one dominant, phenotypically stable (see below for populations that are mutagenic and, hence, evolutionary) tumor population (N_T) interacting with normal cells (N_N).

(1)

(2)

where R_N and R_T are maximum growth rates of normal and tumor cells, respectively (i.e., the net result of tumor cell doubling minus tumor cell loss from apoptosis or necrosis); K_N and K_T denote the maximal normal and tumor cell densities; and _NT and _TN are the competition coefficients. These parameters represent a lumped phenomenologic terms with the following biological significance at the tumor-host interface. _TN encompasses a variety of host defenses including the immune response that serve to decrease growth of the tumor populations, _NT is the negative effect of tumor on normal tissue such as tumor-induced extracellular matrix breakdown and microenvironmental changes. The carrying capacity term, K, represents the summation of growth promotion and constraint within the tissue. This includes growth promoters such as the concentration of epidermal growth factor in the environment and the number of receptors (EGFR) expressed on the cell surface, negative growth factors such as ß-catenin (including its regulators such as the APC gene product), and the availability of adequate substrate to allow synthesis of macromolecules for new cells. This is further discussed below.

In the absence of tumor, this system achieves a stable, nonzero steady state of normal cells but also exhibits solutions in which, under some circumstances, one population (i.e., invasive cancer) may invade and destroy the other. If initial conditions specify normal cells at their carrying capacity (K_N) when a small number of tumor cells emerge in or migrate to the region, the following steady states [i.e., (dN_N)/(dt) = (dN_T)/(dt) = 0] may result.

N_N = 0, N_T = 0. This is the trivial solution and is not biologically relevant.

N_N = K_N, N_T = 0. This corresponds to normal tissue with no tumor cell present. That is, the tumor regresses completely. Regardless of the starting point, the system will always arrive at this state if both _TNK_N/K_T > 1 and _NTK_T/K_Npoint is sufficiently close to N_N = K_N, N_T = 0 (as we suppose the initial conditions will be in early tumor development), only the former condition need be satisfied.

N_N = (K_N - _NTK_T)/(1 - _NTTN), N_T = (K_T - _TNK_N)/(1 - _NTTN). This corresponds to a stable coexistence of tumor and normal cells that we interpret as benign, non-invasive tumor. The system will always arrive at this state if both _NTK_T/K_NTNK_N/K_T

N_N = 0, N_T = K_T. This corresponds to an invasive cancer with complete overgrowth of the normal tissue by the tumor cells. Regardless of the starting point, the system will always arrive at this state if both _NTK_T/K_N > 1 and _TNK_N/K_Tthe starting point is sufficiently close to N_N = 0, N_T = K_T (as would occur when tumor treatment is initiated), only the former condition need be satisfied.

For any given set of parameters, there are either five or six possible equilibrium points which lie in the non-negative orthant. These are determined by the intersection of isoclines defined by lines along which derivatives are zero. There are four isoclines defined by

One equilibrium point is the origin defined by the intersection of the first two isoclines. Four equilibrium points are defined by the intersection of the last two isoclines with the first two and the sixth possible equilibrium point defined by the intersection of the last two isoclines, provided that such an intersection exists. Fig. 1 illustrates possible situations. The three panels illustrate steady states 2, 3, and 4 as discussed above using the following parameter values:
In the first panel, the only stable equilibrium point is N_N = 10, N_T = 0. There are six equilibrium points in the middle panel with only the positive one stable at N_N = N_T = 6(2/3). In the last panel, the equilibrium point is given by N_N = N_T = 10. Note, in this example, the transition from an equilibrium point that does not allow tumor growth to one that allows tumor-normal cell coexistence (i.e., benign tumor growth) to one in which the tumor cells drive normal populations to extinction (i.e., invasive cancer) is dependent only on changes in the cellular interaction parameter . Using the parameters cited above, we see in Fig. 2 how the system moves from the same initial starting condition to an equilibrium solution.

Given the initial conditions expected in carcinogenesis (i.e., a small number of cancer cells numerically dominated by normal cell populations), an invasive cancer must possess the parameter values specified under steady state solution 2 (_NTK_T/K_N > 1 and _TNK_N/K_Tis favored if K_T > K_N. Biologically, the carrying capacity represents the restrictions on cell numbers due to: (a) normal tissue controls mediated by positive and negative growth factors (e.g., oncogenes and tumor suppressor genes) encompassing interactions with other cells, the extracellular matrix, and soluble growth factors; and (b) substrate availability (the cell must be able to accumulate substrate in excess of basal metabolic demands to synthesize necessary components for mitosis). Thus, oncogene and tumor suppressor gene mutations that characteristically accumulate during carcinogenesis (10, 11) will tend to increase K_T. However, substrate availability due to lack of vascularity will tend to decrease K_T as demonstrated by the transition from non-invasive growth to invasive tumor growth coincident with the onset of angiogenesis (12).

The invasive cancer solution is also favored if _NT is large. Recall that the term encompasses the negative effects of one population on the other. Thus, the system dynamics dictating that development of an invasive cancer requires tumor cells exert a substantial negative effect on the normal cells. One clear mechanism for this is expression of the glycolytic phenotype. We have previously demonstrated that the consequent acidification of the extracellular tumor microenvironment will produce an acid gradient extending into adjacent normal tissue (4) resulting in normal cell death mediated (13) by p53-dependent apoptosis pathways [induced by increased caspase activity (14)] as well as degradation of extracellular matrix [through acid-induced release of Cathepsin B and other proteolytic enzymes (15)], inhibition of tumor immune response (16) and induction of angiogenesis [mediated by acid-induced release of VEGF and IL8 (17, 18)].

Therapeutic Strategies in Non-Evolving Tumor Cells
To evaluate potential cancer therapies, we can examine the system after it has achieved the invasive cancer stable state of N_T = K_T and N_N = 0. The goal of therapy is to destabilize this solution so that the system will evolve to one of the other solutions in which the tumor cells are eliminated entirely or at least remain in stable equilibrium with normal tissue (i.e., solutions 2 and 3 above).

Classical cytotoxic therapies seek to eliminate the tumor by directly killing as many individual tumor cells as possible. The fundamental flaw in this strategy is apparent from simple inspection of the system equations and Figs. 1 and 2. Reducing the population of tumor cells does not alter the basic system dynamics. That is, given the parameter values necessary for invasive cancer, the system will always tend to the state N_N = 0, N_T = K_T provided N_T > 0. In other words, cytotoxic therapies may eliminate a large number of malignant cells reducing the tumor size but, unless all of the malignant cells are eliminated, invasive cancer remains the stable steady state solution to the state equations so even a small surviving tumor population will inevitably repopulate the tissue so that the invasive cancer recurs. This is illustrated in Fig. 2. If we assume that cytotoxic therapy has reduced the tumor population from K_T to that shown at the starting point, system dynamics dictate (i.e., case 4) return to the original state in which N_T = K_T (where K_T = 10 in our example) and N_N = 0.

Therapeutic strategies more consistent with the quantitative models would typically focus on altering the key parameters that confer stability on the invasive tumor solution. That is, alter the underlying system dynamics as shown in Fig. 2 to produce case 3 or, better, case 2. Assuming the initial conditions are N_T = K_T and N_N = 0, the tumor state will be destabilized only if both of the above inequalities are reversed so that _NTK_T/K_{N TN}K_N/K_T > 1. Assuming the carrying capacity of normal cells remain constant, three general strategies are apparent: decrease the value of K_T, increase _TN, and decrease _NT.

Translating these parameter changes to conventional tumor strategies, we note that anti-angiogenesis drugs will diminish the carrying capacity of the environment for tumor cells thus decreasing the value of K_T. Strategies that increase immune response to tumor antigens will increase the value of _TN (the negative effects on the cancer cells due to the presence of normal host cells). Strategies that reduce activity of metaloproteinases will decrease the value of _NT (the negative effects on normal cells generated by tumor cells). Note, however, that the latter two strategies will each affect only one of the two inequalities. Therefore, to fully destabilize the tumor solution, these approaches should ideally be combined or added to the anti-angiogenesis therapy. Interestingly, if the number of tumor cells is near K_T, then only the first inequality needs to be satisfied to insure stability of the tumor solution. Thus, immunotherapy, by increasing the value of _TN, will typically exhibit a biological effect only in tumors that have already undergone a significant population decline due to chemotherapy, radiation therapy, or surgery. This appears consistent with clinical trials using the 17-1A antibody in colorectal cancer which demonstrated that treatment was more effective in eliminating microscopic metastatic disease than bulkier local recurrence (19–21). Similarly, anti-EGFR antibodies appear more effective when added to cisplatin (22) or doxorubicin (23).

Two caveats are also apparent from this simple analysis: (a) Threshold effects should be expected. Thus, for example, anti-angiogenesis therapy may result in some changes in tumor growth but it will not fundamentally change the dynamics and result in complete tumor regression unless the change in K_T is sufficient to satisfy the above inequalities. (b) Unexpected effects may confound therapeutic expectations. Again using anti-angiogenesis strategies, for example, note that if this approach also restricts the vascularity of normal tissue at the tumor-host interface, the resulting reduction in K_N will tend to stabilize the tumor solution.

In addition, we have thus far assumed the tumor phenotype to be stable. In fact, the mutagenic phenotype found in cancer cells confers the ability to evolve and adapt to changing environmental conditions. The potential effects of this property on therapeutic strategies are presented next.

Cellular Heterogeneity and Evolution
In the previous section, we assumed that the tumor cells possess a homogeneous and static phenotype. This is, of course, biologically unrealistic because cellular populations in both malignant and preneoplastic lesions exhibit substantial spatial and temporal heterogeneity. In part this cellular heterogeneity is due to genetic instability induced by the mutator phenotype or a mutagenic environment such as chronic inflammation or acidosis. Because of this increased mutation rate, tumor cells possess the ability to evolve at a greater rate than non-transformed cells. That is, each mutation produces an "experimental" phenotype that interacts with environmental selection factors. Those mutations that confer a survival advantage are rewarded by proliferation and clonal expansion.

This evolutionary capacity is of enormous importance in developing therapeutic strategies because, as pointed out by Coldman and Goldie (24) "much experimental evidence has accrued that [cancer] cells which display inheritable resistance are the cause of treatment failure." Several mathematical models describing the development of drug resistance in cancer therapy have been developed using the Lotka-Volterra competition model as presented above (25–31). The reader may want to consult the literature for other modelling approaches (26). Here we directly incorporate cellular evolution into the cellular and microenvironmental dynamics during cancer treatment.

This model, based on evolutionary game theory, requires some additional concepts and terms. The phenotypic properties of each cellular population of size N_i is identified by a "strategy" u_i that defines its interaction with environmental factors controlling cellular proliferation such as growth promoters or inhibitors (including chemotherapeutic agents as outlined below) and substrate delivery. Typically a range of possible strategies is available depending on the number of viable phenotypes that can be generated through mutations of the genome. Each mutation in the genome produces a new cellular strategy which, in turn, confers some fitness function H_i on the cellular population as defined by its ability to proliferate in the context of extant strategies employed by the current populations. The Lotka-Volterra equations, in terms of n different cellular populations currently in the tissue, are formulated in terms of fitness function as follows:

where

(3)

and R_i is the maximum proliferation rate, k(u_i) is the carrying capacity for population i, and (u_i, u_j) is the competitive effect of population j using strategy u_j on the fitness of individuals of species i using strategy u_i. If the fitness function of a mutant cell results in increased proliferation (i.e., the cell is more fit than its ancestors), clonal expansion results. Cells that are less fit do not proliferate and that strategy is eliminated. Neutral mutations may also be maintained in the population. Through this combination of random mutations interacting with microenvironmental selection parameters, progressively fitter populations emerge over time. This constitutes the somatic evolution of cancer so that the observed cellular strategy varies over time as new, fitter populations emerge sequentially during the transition from normal tissue to premalignant lesions to invasive cancer. This ability to evolve also confer an ability to adapt to environmental changes and is, thus, fundamentally important to the emergence of tumor resistance to any host response or therapeutic strategy.

This cellular fitness may be expressed using a fitness generating function, generally called a G-function(31, 34). A full exposition of this mathematical technique is beyond the scope of this article. A brief discussion for the mathematically inclined reader is included in Appendix A. Here, the G-function for a cellular community that includes tumor populations and has a range of possible strategies is defined by:

(4)

where v is a virtual variable. Note that by setting the virtual variable equal to the strategy used by any cellular population results in the fitness function for that population. That is
In this context, the G-function describes the evolutionary potential for all evolutionarily identical individuals.

In terms of the G-function, the population dynamics equations are written as

(5)

Given the fact that variability exists in the strategies u_i, it can be shown (25) that strategies evolve according to

(6)

where _i is related to the variance in strategies. In the simulations below, we assume that normal cells cannot evolve by setting ₁ = 0.

Consider again the simple two-population normal-cancer cell situation. By integrating Eqs. 6 and 7 we find that, over time, the tumor cells evolve, reach a maximum steady state of cellular fitness. This represent the state of an invasive cancer within the context of unmodified (i.e., untreated) tissue dynamics. The top frame of Fig. 3 illustrates this situation. This figure shows the changes in population number of both the normal and cancer cells with time keeping the normal cell strategy constant, but allowing the cancer cells to evolve as illustrated in the bottom frame. The cancer cells are introduced in small numbers at a strategy different than the normal cell strategy with the normal cells at their carrying capacity (in the absence of cancer cells).

 
Fig. 4 illustrates the fact that, at equilibrium, the cancer cells are at a local maximum on an adaptive fitness landscape as defined by the G-function. This maximum represents a local evolutionary stability for the cancer cells, which makes it impossible for other cells with similar strategy values to invade. The slight rise in the number of normal cells initially is due to the fact that _k has changed that in turn changes the equilibrium solution.

We can now model institution of treatment of a cancer population that has reached steady state values shown in Fig. 3. Using cell-specific drugs to eliminate the cancer by adding an appropriate "harvesting" term so that the G-function becomes

(7)

where k_h is a term expressing the level of drug dosage, is the cancer cell strategy at which the drug is most effective, and _h is the variance in effectiveness. Starting with the equilibrium conditions above and integrating Eqs. E and F with the G-function defined by Eq. G (see 34 for details of parameter values), it is found that cytotoxic chemotherapy is effective initially, but the cancer cells ultimately recover because they can evolve and a new tumor equilibrium state is obtained as illustrated in the first frame of Fig. 5 . The net effect is that rather than curing the cancer, the cell-specific drug caused the cancer to evolve to a new form (second frame of Fig. 5) that is now highly resistant to the current and any similar therapeutic strategies. This is illustrated in Fig. 6 by the fact that the cancer cells are again sitting at a local maximum. Note that the normal cells are at a fitness less than zero resulting in a zero equilibrium population. These results are essentially identical to evolution of multi-drug resistance observed in treated human tumors (35, 36).