Optimizing Cancer Treatment Using Game Theory: A Review

Abstract

Importance: While systemic therapy for disseminated cancer is often initially successful, malignant cells, using diverse adaptive strategies encoded in the human genome, almost invariably evolve resistance, leading to treatment failure. Thus, the Darwinian dynamics of resistance are formidable barriers to all forms of systemic cancer treatment but rarely integrated into clinical trial design or included within precision oncology initiatives.

Observations: We investigate cancer treatment as a game theoretic contest between the physician's therapy and the cancer cells' resistance strategies. This game has 2 critical asymmetries: (1) Only the physician can play rationally. Cancer cells, like all evolving organisms, can only adapt to current conditions; they can neither anticipate nor evolve adaptations for treatments that the physician has not yet applied. (2) It has a distinctive leader-follower (or "Stackelberg") dynamics; the "leader" oncologist plays first and the "follower" cancer cells then respond and adapt to therapy. Current treatment protocols for metastatic cancer typically exploit neither asymmetry. By repeatedly administering the same drug(s) until disease progression, the physician "plays" a fixed strategy even as the opposing cancer cells continuously evolve successful adaptive responses. Furthermore, by changing treatment only when the tumor progresses, the physician cedes leadership to the cancer cells and treatment failure becomes nearly inevitable. Without fundamental changes in strategy, standard-of-care cancer therapy typically results in "Nash solutions" in which no unilateral change in treatment can favorably alter the outcome.

Conclusions and relevance: Physicians can exploit the advantages inherent in the asymmetries of the cancer treatment game, and likely improve outcomes, by adopting more dynamic treatment protocols that integrate eco-evolutionary dynamics and modulate therapy accordingly. Implementing this approach will require new metrics of tumor response that incorporate both ecological (ie, size) and evolutionary (ie, molecular mechanisms of resistance and relative size of resistant population) changes.

Figure 1.. Mathematical Formulas for Cancer Therapy Game

An example of a cancer therapy game in which the cancer cells evolve strategies to maximize their net proliferation rates, and the physician aims to balance drug toxicity with overall tumor burden. Two approaches by the physician are considered: (1) Therapy that aims to maximize patient outcome given the current resistance strategy of the cancer cells. This treatment strategy results in a Nash equilibrium; (2) Therapy that aims to maximize patient outcome by anticipating the evolutionary and ecological response of the cancer cells. This therapy results in a Stackelberg equilibrium and a better outcome than the Nash equilibrium (see Figure 2).

Figure 2.. Nash and Stackelberg Equilibria

We assume a 2-player game in which player 1 and player 2 choose actions u₁ and u₂ to maximize their payoffs J₁(u₁, u₂) and J₂(u₁, u₂), respectively. The graph shows level curves of payoffs J₁(u₁, u₂) (solid blue curves) and J₂(u₁, u₂) (solid orange curves) in the (u₁, u₂)-space. Player 1 would like to achieve point T₁ (his or her absolute maximum, known also as team optimum in game theory), and we investigate whether he or she can get close to this outcome when playing simultaneously with player 2 and when playing first. The blue dashed line denotes the best response of player 1 to any action of player 2 and the orange dashed line denotes the best response of player 2 to any action of player 1 (both obtained by maximizing a corresponding player’s payoff for any choice of the other player). If the players play simultaneously, the outcome lies at the intersection of the 2 best-response curves, the Nash equilibrium (N). However, if the physician (player 1) applies treatment with foreknowledge of the best-response curve of tumor cells, he or she can, as the leader, play the strategy u₁ with superscript S₁ based on that information. In contrast, the nonrational, follower cancer cells can only respond with the strategy on their best-response curve. The physician can anticipate this outcome; the cancer cells cannot. By exploiting his or her leadership role, the physician can both anticipate and steer the cancer cells’ resistance evolution toward a much better patient outcome corresponding to the point S₁, the Stackelberg equilibrium.

Figure 3.. Adaptive Strategies for Metastatic Castration-Sensitive Prostate Cancer

Computer simulations of treatment outcomes using methods outlined in Gallaher et al and similar to the models used to design ongoing clinical trials. The subpopulations are color coded, and the area of each simulation represents total tumor burden. The model assumes a newly presented prostate cancer metastasis with different initial distributions of resistant and sensitive subpopulations. A, A pretreatment biopsy finds that 95% of the cancer cells express androgen receptor (AR) but not CYP17A, 3% are both AR and CYP17A1 positive, and 2% are AR negative. The frequency of the cell populations suggests that the fitness of the AR-positive phenotype is much higher than AR-negative or CYP17A phenotypes. In the top row, continuous androgen deprivation therapy (ADT) rapidly selects for resistant populations with the dominant clones overexpressing CYP17A, leading to tumor progression. An alternative approach replaces continuous ADT with the protocol used in Zhang et al. Androgen deprivation therapy is administered until the tumor burden is reduced by half (based on prostate-specific antigen measurements) and then withdrawn. In the absence of therapy, the fitness advantage of the AR-positive cells allows them to grow at the expense of the resistant populations, thus prolonging tumor control with ADT. B, The initial biopsy shows the AR-positive phenotype to be 65% of the cells, with 35% CYP17A1 and 10% AR negative. Because the relative fitness advantage of the AR-positive cells is not as great as in A (based on the higher relative fractions of AR-negative and CYP17A1 phenotypes), the adaptive strategy in A will not be as successful (simulation not shown). An alternative evolutionary strategy in the lower row alternates treatments directed against AR-positive (ADT) and CYP17A1 (abiraterone) cells, as well as treatment holidays to control the AR-positive and CYP17A1 populations while maximally reducing the growth of the AR-negative cells. Many other strategies (eg, addition of docetaxel) are available, and similar simulations can allow the treating physician to devise a patient-specific protocol that optimizes outcomes. Each arrowhead represents a treatment period. The drug used is above the arrowhead (red arrowheads indicate ADT; blue arrowheads, abiraterone). No specified drug indicates a treatment holiday (black arrowheads).

Figure 4.. Using the Bellman Theorem to Guide Brief Applications of Treatment to Estimate the Size of Resistant Populations and Their Strategies

As a simplified example of this “unmasking process,” we use the simulations in Figure 3 but assume that the metastatic prostate cancer is presenting with unknown cellular composition. Because nearly all precastration metastatic prostate cancer initially responds to androgen deprivation therapy (ADT), we assume that the androgen receptor (AR)-positive population is dominant. Here we wish to determine the size of the resistant populations by giving brief pulses of ADT and abiraterone. A, Here the ADT-resistant subpopulations are small. Initial treatment with a pulse of ADT causes a marked decrease in tumor size (measured with prostate-specific antigen [PSA]). This allows the treating physician to estimate the fraction of the AR-positive population. The physician can then briefly apply abiraterone. The smaller decrease in tumor size is used to estimate the size of the CYP17A1 phenotypes population. All other cancer cells can then be assumed to be AR negative, allowing an evolution-based treatment similar to that shown in Figure 3A. B, Here the initial combinations of ADT and abiraterone show that the population expressing CYP17A is much larger and indicate the need for combined therapy as shown in Figure 3B. Each arrowhead represents a treatment period (red arrowheads indicate ADT; blue arrowheads, abiraterone). The drug used is above the arrowhead. No specified drug indicates a treatment holiday (black arrowheads).