Invasive cancer is an emergent phenomenon resulting from non-linear, stochastic interactions between evolving cellular phenotypes and multiple environmental selection factors. A clinically detectable cancer is the end result of this complex system dynamics. Through quantitative models, the critical parameters that control the interactions at the tumor-host interface can be identified as the environmental carrying capacity for the tumor cells and the interaction of the tumor cells with normal cellular populations. Both of these are lumped phenomenologic terms. The former encompasses response to normal tissue growth inhibitions such as cellular interactions with other cells, the extracellular matrix, and various soluble growth factors and substrate-mediated growth controls such as angiogenesis. The latter includes the negative effects of host immune response on cancer cells as well as the negative effects of cancer cells on normal tissue due, for example, to excess production and excretion of H+ ions as detailed above.
Although conclusions from these simple models must be drawn with some caution, the population models clearly raise doubts that cancer therapy based solely on systemic cytotoxic drugs will ever successfully eradicate a broad range of tumors. While a therapeutic strategy that relies solely on killing tumor cells (without altering critical system parameters) may be sufficiently effective to reduce the tumor size, the models clearly demonstrate that dNT/dt will remain >0 as long as NT is >0. That is, the tumor population will inevitably rebound unless all proliferative cells within the population are eliminated. Two fundamental barriers to this requirement of total eradication of the tumor cells emerge readily from evolutionary game theory. First, the heterogeneity of tumor phenotypes (defined by the strategy parameter u) related to the stochastic nature of the random underlying mutations and the non-linear interactions with microenvironmental selection parameters (which are also variable due to spatial and temporal variations in, e.g., blood flow) will likely produce a broad range of cellular sensitivity to cytotoxic drugs. Second, a cytotoxic therapy that "harvests" a large number (but not all) of the tumor cells simply becomes an additional environmental selection parameter. Tumor cells, because of their ability to evolve, adapt to this new factor and resistant populations readily emerge. Ultimately, the tumor regrows as the resistant populations proliferate rendering the therapy ineffective.
In summary, any therapy relying solely on tumor cytotoxic effects will be curative only if it is sufficiently effective to overcome the tumor phenotypic diversity such that it kills all proliferative cells in a time period sufficiently short to prevent evolution of resistance.
On the other hand, the models support the current trend toward therapeutic strategies focused on disrupting the interaction of tumor cells with their environment such as blocking EGFRs or anti-angiogenesis drugs because they alter the fundamental system parameters in addition to directly killing tumor cells. By altering the critical parameters in the state equations, these strategies have the potential advantage of rendering the tumor solution unstable, driving the system to a new steady state in which tumor cells are absent or at least remain at equilibrium with normal cells. This effect will likely be more durable because under these conditions, dNT/dt will be 0 for NT > 0 or NT > NTmax depending on whether the solution admits the presence of any tumor cells.
Finally, integrative models may allow more rational therapeutic design. It is apparent from the models that several critical parameters govern system dynamics in invasive cancer. Therapy directed toward only one of these parameters may be ineffective because each term in the inequalities required for stability of the tumor solution (
NTKT/KN > 1 and TNKN/KTthree lumped parameters each of which, in turn, may consist of several biological processes. Furthermore, any therapy is subject to long-term failure due to evolving tumor phenotypes that adapt to and overcome proliferation constraints. It is likely that the most effective therapies will be rationally designed combinations targeting more than one parameter or more than one component of each parameter. For example, KT could be reduced through the simultaneous administration of growth inhibitors and anti-angiogenesis drugs (37, 38). This may explain the surpraadditive (37) growth inhibition of anti-EGFR combined with C225 (a chimerized version of the anti-EGFR antibody MAb225 that inhibits expression of VEGF and vascular permeability factor, thus decreasing angiogenesis). Other valuable combinations predicted by the models include the addition of cytotoxic drugs to certain biological modifiers. Reduction of tumor population density below the carrying capacity (KT) allows the tumor solution to be destabilized with reversal either NTKT/KN > 1 or TNKN/KT when KT tumor cells are present. Thus, immune therapy which increases TN will likely not be effective as initial tumor therapy but may strongly affect tumor dynamics after debulking with chemotherapy. In fact, studies with trastuzumab, an antibody against the HER-2/NEU receptor, demonstrated 16% response rate to the antibody alone (39) but 52% response to the antibody combined with an anthracycline (40) and 42% combined with a taxane regimen. Similarly, the addition of the antibody increased survival when compared to cytotoxic therapy alone by 16% at 1 year (41) and 25% at 29 months (42) as predicted by the mathematical models.
Appendix A
The G-function (25–27, 31, 32, 34) is used to examine the evolutionary stability characteristics of this model. In the above model, it is assumed that the tumor populations can exhibit a range of strategies as given by specifying the virtual variable v.
 |
(8) |
 |
(9) |
where the
variables denote variances due to genetic diversity. We have previously shown (
25–27) that, by varying the environmental parameter
k, the dynamical system can have equilibrium solutions composed of one or more cell types. That is, given a constant strategy vector
u, there exists at least one non-zero equilibrium solution
N* (
i.e., not every component of
N* is zero). There are two ways on which we can find an equilibrium solution for
N*. One way is to solve for
N* from the system of equations (see
Eq. E)
Note that, in general, a solution to this system of equations will require a numerical procedure. If more than one equilibrium solution exists, then the particular solution obtained will depend on the initial guess made for the solution. A second method for determining an equilibrium solution (when such a solution is asymptotically stable), is to choose
u and an initial condition
N (0) and simply let the solution to equations determine the equilibrium point by integrating the differential equations.
At equilibrium, one or more components of N* must be positive and non-zero for there to be a viable solution. The corresponding strategies are those that can coexist in the population of cells. However, the equilibrium solution to the above system of equations only considers the outcome for those strategies already resident in the population and does not consider, nor can it consider, the potentially infinite number of feasible strategies that may occur in the future via selection and/or mutation.
For our model, consider the following parameter values
 |
(10) |
In this situation, for a given
u, there is only one non-zero equilibrium solution to the system
(Eq. E). Choosing different strategies will result in different equilibrium values. For example
ui = 0 results in
N1* = 100.0. However, this solution is not evolutionarily stable because it can be displaced by introducing another cell population (even at small numbers) using the strategy
u2 = 1. We seek an evolutionarily stable strategy (ESS) that has the property that it cannot be displaced by introducing a mutant strategy. We can obtain such a strategy for this set of parameters by simply picking any strategy (
e.g.,
u1 = 0) and letting it evolve according to the system of
Eqs. E and
F. The equilibrium solution to this set of equations is
u1* = 1.213 and
N1* = 83.2 and it is an ESS solution. We view this as normal tissue. This stability is maintained in spite of a low baseline mutation rate. That is, a normal cellular population will remain evolutionarily stable so long as conditions do not change.
With this model, if we increase k from the value given above to 
(e.g., due to damage or changes in surrounding tissue), we have previously shown (25) that there exist two equilibrium solutions to Eq. E. As a consequence, if we introduce a cancer cell at some strategy other than 1.213, it can coexist. In fact if we allow both the normal cells and cancer cells to evolve according to Eqs. E and F, we would arrive at an ESS composed of two strategies (the normal cells would no longer be at 1.213). However, normal cells are limited in their ability to evolve significantly within the lifetime of the host, whereas, tumor cells have no such limitation and can evolve to an equilibrium condition. This was incorporated into the integration of Eqs. E and F by setting 1 = 0 (no evolution of normal cells) and 2 > 0 (evolution allowed for the cancer cells). The results are depicted in Fig. 3.
Whether a population of cancer cells will evolve depends on the nature of the environmental growth constraints. For example, if the host-immune system is capable of eliminating tumor cells that express certain cell-surface antigens, the system dynamics will favor strategies that do not express these antigens and, over time, these cells will proliferate and the tumor will successfully evade the host response. As illustrated in the text, application of any tumor therapy similarly introduces new environmental selection forces that will, over time, favor evolution or resistant phenotypes.
Footnotes
The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked advertisement in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 3/17/03; revised 6/17/03; accepted 6/19/03.
Answers to all your Biology Questions
