The Contribution of Evolutionary Game Theory to Understanding and Treating Cancer

Cancer is a disease of unregulated proliferation, caused by abnormal function of genes responsible for regulating cell division. The genesis of cancer has strong ties to human life history [6, 70, 93, 94, 179], and its progression is driven by natural selection, characterized by cancer cells exhibiting the following three conditions [58]: This Darwinian view of cancer aligns with the premises of evolutionary game theory (EGT), which assumes that evolution tests heritable traits in an ongoing competition for survival [38, 100, 127, 128]. EGT is a branch of mathematics that has helped to conceptualize and understand the behavior of real-world biological systems, including several counter-intuitive biological phenomena [91, 92, 128, 162, 182, 201], and is being increasingly recognized as an important tool for mathematical oncologists [18, 154, 165].

EGT deals with situations where organisms using different strategies and/or possessing different traits interact with each other. Unlike in classical game theory [139, 190, 191], these organisms do not need to be overtly rational, i.e., their strategies (often referred to as “types”) are inherited rather than rationally chosen [37, 100] (although a rational population-level interpretation of the dynamics is also possible [109]). Some strategies might confer higher fitness and the individuals using these strategies will in the long run dominate the population. Thus, if we see cancer as a Darwinian process, it can be described as an evolutionary game, where cancer cells are the players, their heritable traits correspond to the strategies, and the payoffs are represented in terms of survival and proliferation (fitness) [38, 129]. This is a dynamic game, as one can analyze how frequencies of different strategies and numbers of individuals corresponding to these different strategies change in time. We refer to those changes as evolutionary and ecological dynamics, respectively. Both together are called eco-evolutionary dynamics.

Compared to other fields of applied mathematics, EGT of cancer is a relatively new field, just a few decades old [123, 181]. Tomlinson was first to explicitly frame cancer as an evolutionary game [181]. Since then, at least 60 publications on cancer have called their research game-theoretic. This body of literature has grown into diverse and different groupings. Given that cancer is an evolutionary process, cancer treatment could benefit from insights from evolutionary theory, giving rise to Evolutionary or Darwinian medicine [77, 79, 83]. The increasing interest in this field is reflected in the recent update of medical curricula to include evolutionary reasoning [140]. Clearly, EGT can only improve cancer treatment if there is something gained from these evolutionary insights. Standard of Care (SoC) in treating cancer typically applies therapy at Maximum Tolerable Dose (MTD), to remove as many tumor cells as fast as possible. For some aggressive cancers, such as advanced Non-Small Cell Lung Cancer, no better treatment than MTD has been found so far [9, 19, 20]. Yet, unless the patient is cured, the MTD strategy promotes evolution of treatment-induced resistance which leads to treatment failure [76, 164, 206].

The fact that even personalized therapies tailored to the cancer’s genetic signature and to the individual’s genetic disposition fail can be attributed to the extensive adaptive potential of the human genome. As MTD can only eradicate therapy-sensitive tumor cells, it benefits therapy-resistant cells [82, 146]. Subsequently, growth-limiting constraints due to competition may temporarily vanish and increase the per capita growth rate of the resistant types (competitive release [50, 68, 204]). In turn, some experiments show that when treatment is stalled (drug holiday), resistant types are typically at a disadvantage (cost of resistance [168], although this is not universal [113]). This evidence suggests that MTD might be evolutionarily unwise when it promotes treatment-induced resistance in cancer cells. Additionally, there is evidence for selection for evolvability in tumor cells, e.g., hyper-mutators [44]. Recent works showed that a game-theoretic approach may help to provide an alternative to MTD, based on anticipating and steering the cancer eco-evolutionary dynamics in response to the treatment [88, 165].

The aims of this paper are: (1) to discuss the achievements of the existing works on game theory of cancer and (2) to show the future potential of game theory to understand cancer mechanisms, inspire novel research, and design better treatment protocols.

In the remainder of this paper, we first introduce models where the interaction among cells is explicitly framed as an evolutionary game, with either no or fixed treatment (Sect. 2). Second, we will review cancer models where the physician, as a rational player optimizing their own objective(s), enters the evolutionary game (Sect. 3). Third, we will focus on the clinical aspects of EGT therapy models (Sect. 4). We close with a discussion on limitations and future steps in game theory of cancer and its treatment (Sect. 5).

In the literature of EGT models of cancer with no or predefined treatment \({\mathbf {m}}\), the authors either focus on finding the ESS resistance strategy \({\mathbf {U}}^*\) at the ecological equilibrium \({\mathbf {x}}^*\), or they analyze transient dynamics toward \(\left( {\mathbf {x}}^*,{\mathbf {U}}^*\right) \) for particular (predefined) choices of \({\mathbf {m}},\) to see what choices of \({\mathbf {m}}\) are better than others in terms of some prespecified metrics, such as progression-free or overall survival.

We will first present research that utilizes both Eqs. (3) and (5), followed by models that somewhat simplify the two equations by using a fitness and a competition matrix, respectively, and spatial models.

From a game-theoretic perspective, the physician is not a real player when the treatment protocol is decided a priori. This is the case for continuous MTD, metronomic therapies, and adaptive therapy when the therapy switching rules are decided beforehand. This was the case in the models introduced in Sect. 2.

Here, we consider the case where the physician becomes a true player in the game. When viewing cancer as an evolutionary game between the physician and the cancer cells, a natural question arises: Can we drive cancer into a stable state, corresponding to either a cure or a chronic disease, which is not too harmful for the patient and can be maintained at a stable tumor burden? This concept of stability corresponds to the Evolutionarily Stable Strategies introduced in Sect. 2. Alternatively, if cure or stable tumor burden cannot be achieved, a relevant question is whether we can maximally delay undesirable states (e.g., too high tumor burden or too high level of resistance), by more dynamical treatment protocols than currently used as SoC. To this aim, we introduce an objective function to be optimized by the physician, \(Q\left( {\mathbf {U}}(\cdot ),{\mathbf {x}}(\cdot ),{\mathbf {m}}(\cdot )\right) ,\) which varies with \({\mathbf {m}}(\cdot ) {\mathop {=}\limits ^\mathrm{{def}}}\left[ {\mathbf {m}}(t)\right] _{t\in [0,T]},\) \({\mathbf {U}}(\cdot ) {\mathop {=}\limits ^\mathrm{{def}}}\left[ {\mathbf {U}}(t)\right] _{t\in [0,T]},\) \({\mathbf {x}}(\cdot ) {\mathop {=}\limits ^\mathrm{{def}}}\left[ {\mathbf {x}}(t)\right] _{t\in [0,T]}\). We can refer to this function as the Quality of Life function of the patient. The physician’s goal is to find the optimal \({\mathbf {m}}^*(\cdot )\) which optimizes such an objective, i.e., findwhere Q has been decided by the physician and patient a priori. In such a situation, cancer cells are playing an evolutionary game with each other and their eco-evolutionary dynamics can still be described by Eqs. (3) and (5). However, they become followers in a Stackelberg (i.e., leader-follower) game, with the physician as a rational leader [164]. Since the followers are evolutionary players, we call these types of games Stackelberg evolutionary games (SEGs), in accordance with recent research on this topic [156, 158]. It is noteworthy that the physician, as the only rational player in this SEG, can anticipate and steer the eco-evolutionary response of the cancer cells defined by (3)–(5), while the cancer cells can only adapt to the actions already taken by the physician. The theory of Stackelberg games was originally devised in economics to conceptualize interactions with an imbalance in control or power, e.g., the competition between a market leader and follower [27, 97]. Its extension into SEGs not only applies to cancer treatment, but also to other problems involving a rational player interacting with an evolutionary system such as pest management, fisheries management, or the control of infectious diseases [39, 40, 96, 158].

Here, we divide existing literature into two categories: When it comes to the objective of the leader, we identify two important categories of literature: In Table 1, we summarize these options and indicate the sections where each is discussed.

Application of EGT principles in therapy, in order to anticipate and steer cancer eco-evolutionary response, is a powerful tool, but relies on our ability to estimate tumor size and composition prior to treatment. The intra-tumoral evolutionary process leads to sub-clonal diversification and generates the genetic and phenotypic intra-tumor heterogeneity, which determines the tumor composition and therefore the evolutionary state [124]. In order to optimize the model parameters, determined by the tumor composition, monitoring of the tumor’s behavior during therapy is required. At best, this encompasses continuous surveillance of the total number of tumor cells and their cell type composition. In clinic, the personalized therapeutic strategy then needs to be optimized after every measurement, i.e., after each clinical visit. Kaznatcheev et al. recently showed how to assess the game played by different cell types of non-small cell lung cancer cells in vitro [113]. This game changes in response to different treatment regimens. Due to the in vitro setup, the experiments could be monitored with relative ease by performing time-lapse microscopy. However, in a clinical setting the key constraint is the low amount of information available about intra-tumoral evolution and the speed of evolution during treatment. It is still challenging to identify, quantify and monitor the evolving strategy distribution in heterogeneous tumors. A sufficient technology for this is yet unavailable; however, several techniques can be proposed which we discuss in the following paragraphs.

Firstly, tissue biopsies of the primary tumor and of metastases can be sampled to reveal genetic and phenotypic differences between cancer cell types. Genetic differences are revealed by genome sequencing, while phenotypic heterogeneity is typically assessed with histology techniques and proteomics [32]. Nevertheless, to monitor the cancer cells’ response to treatment, tissues need to be isolated at the time of initial diagnosis as well as successively sampled throughout treatment. In the clinic, such repeated biopsies are not easily acceptable, due to their invasive nature and expense. Such is the case in taking biopsies of disseminated bone disease in mCRCP patients [72]. Furthermore, often only a fraction of the tumor is isolated, which does not represent the complete genomic and phenotypic landscape, and the detection of small lesions and deriving biopsies from them is a major challenge [104, 145].

Secondly, an alternative approach is based on liquid biopsies. They consist of several sources of tumor material including circulating tumor DNA (ctDNA) and circulating tumor cells (CTCs). The ctDNA is a DNA released by malignant cancer cells, with diagnostic genetic and epigenetic alterations. Several studies have shown that exome-wide analysis of ctDNA may contribute to monitoring the evolution of acquired drug resistance and track the outgrowth of resistant cell types [42, 135, 136, 143]. To be able to use the genotypic information obtained from ctDNA, we need to know the relationship between mutations and their phenotypic impact, i.e., the genotype-phenotype map [4, 141]. Predicting what genotypes will eventually evolve to drive phenotypic resistance remains a significant challenge [63].

From CTCs, besides genotypic information, phenotypic information about the strategy distribution can directly be obtained for use in the EGT models. CTCs represent intact, viable non-hematological cells with malignant features. The resistant CTC populations may be phenotypically distinct from their precursors in physical size, shape and surface marker expression. For instance, Tsao et al. detected tumor progression and proliferation of resistant melanoma cell types by observing surface marker up-regulation from CTCs [183]. They saw how a widening of the signal distribution detected by spectroscopy, reflected a more heterogeneous CTC population. In mCRPC, Zhang et al. detected testosterone producing cells by the presence of CTCs expressing CYP17A1, which is a key enzyme for androgen synthesis [206, 207]. Androgen receptors (AR) can also be detected and monitored in real time from mCRPC CTCs. The AR splice variant 7 was proved to be predictive of resistance to anti-AR treatment, such as ADT therapy and treatment with both abiraterone and enzalutamide [8, 131, 132, 166, 178]. Additionally, CTCs can be assayed for human epidermal growth factor receptor 2 (HER2) in breast cancer, which contributes to treatment resistance [51, 152]. This technique may also be applied to display the strategy distribution in other cancer types, when specific up-regulation or down-regulation of specific surface markers in resistant cell types occurs.

Taking liquid biopsies and isolating CTCs has advantages over conventional tissue biopsies since they are less invasive to the patient. Additionally, it may reflect the heterogeneity of the tumor more appropriately and it allows continuously monitoring of a patient’s tumor composition [119]. Nevertheless, the liquid biopsies provide neither spatial information nor information on the composition of individual metastatic lesions, since the primary tumor and its metastases are not measured individually. Accordingly, liquid biopsies may contain a mixture of tumor cells originating from multiple independent lesions. Analyses of primary and disseminated tumor cells show large differences in genetic variation [167], and CTCs are unlikely to represent the full spectrum of mutations and differences in protein expression in tumor lesions since CTC biopsies might only show the ‘tip of the iceberg’ [205]. While it is better to have this aggregated information as a proxy for the cancer’s evolutionary dynamics than no information at all, the information found this way can be used to measure the evolutionary states of different metastatic lesions if multiple metastatic lesions located at different sites shed CTCs homogeneously or if the variation in the composition of these lesions is low. This is shown in BRAF status concordance in primary and metastatic melanoma [35, 36] and colorectal carcinoma and KRAS mutation status in colorectal adenocarcinoma [60]. Alternatively, one needs to identify the tissue of origin of CTCs by using expression profiling of organ-specific metastatic features. Studies have shown that certain methylation patterns are tissue specific, which may serve to determine the source of tumor cells or ctDNA [117, 169].

Thirdly, another approach is blood sampling to measure blood serum markers. These are biomarkers produced by specific tumor cell types. In studies by Zhang et al., prostate cancer volume is determined by assessing PSA levels in the blood, while in the latter research testosterone blood levels under androgen deprivation are measured [206, 207]. The testosterone levels are used as a proxy for the amount of testosterone-producing cancer cells. Nevertheless, whether each tumor cell produces the same amount of PSA might depend on the sensitivity of the tumor cells to androgen stimulation for the expression of PSA. Some cell types have been shown to lose sensitivity to androgen and produce even more PSA than androgen sensitive cell types [59, 106]. This feature that differs between prostate cancer cell types might provide ways to measure androgen-independent and dependent types of cancer cells. However, this is again aggregated information combining all metastatic lesions. A study in melanoma showed a higher expression of BRAF (V600E) oncoprotein in vemurafenib-resistant tumor cells compared to sensitive cells. This difference might be used for parameterizing EGT models of melanoma [174]. For other cancer types investigated in adaptive therapy studies, there is a lack of reliable biomarkers presented yet.

Fourthly, modern imaging techniques are another emerging approach for gaining tumor and intra-tumoral metrics. Imaging can provide a holistic view of the entire tumor and since it is noninvasive, it is suitable for repeated monitoring. Magnetic resonance imaging (MRI) and computed tomography (CT) can be used to track spatial and temporal patterns of heterogeneity. For example, these techniques may reveal tumor habitats such as necrosis, hypoxia and vascular permeability. Such habitats may select for different cells with varying levels of responsiveness to therapy [176, 177].

Radiomics provide images of tumor habitats which seek correlations between cell phenotypes and their visual appearance. Quantitative imaging features can include shape, edge to volume ratio, texture or tissue environment. Such features can be built into predictive models relating image features to tumor cell types [3, 81]. It has already been demonstrated in studies of patients with glioblastoma multiforme that differences in cancer cell protein expression within a tumor correlate with regions of varying contrast-enhancement from MRI images [62, 99]. However, before quantitative imaging features can be used for clinical monitoring of cancer cell strategies, it needs to be ensured that specific imaging features can be linked to the underlying composition of cell types that differ in their response to treatment.

Positron Emission Tomography (PET), which can be performed along with CT or MRI scans, provides additional anatomic and spatial information. PET scans can show differential amounts and patterns of uptake of radiotracers by cells within a tumor. This might provide the ability to label and quantify the resistant as well as the sensitive cells. For example, the variability of tumor glycolytic metabolism within the same lesion can be assessed with the use of 2-flouro-2-deoxy-D-glucose F 18 ([18F]-FDG) PET imaging [28]. Uptake patterns influence, thus patients’ outcome and thus provide insights into the prevalence of resistant cancer cells within the tumor. Additionally, PET imaging using fluorodihydrotestosterone F 18 ([18F]-FDHT) permits labelling and detection of androgen receptors [202]. Accordingly, a combination of [18F]-FDG and [18F]-FDHT PET imaging can identify AR positive and negative lesions, and therefore the ability to discriminate sensitive and resistant prostate cancer cell types [71]. A radiotracer to label prostate-specific membrane antibody (PSMA), a cell surface protein with high expression in prostate cancer cells, is also available for PET imaging. PSMA is expressed on nearly all prostate cancer cells, and therefore accessible to labelling [126]. Furthermore, it is under research whether the radiotracer N-succinimidyl-4-[18F]fluorobenzoate ([18F]-SFB) is suitable for labeling HER2 overexpressing cells in breast cancer [198].

Modern imaging techniques may hold more promise than tissue and liquid biopsies. This is because it can reveal relevant information about both the location of the lesions and the tumor cell types within these lesions, to reveal both tumor eco-evolutionary dynamics and spatial characteristics. Furthermore, it is noninvasive and overcomes sampling errors of biopsies. In particular, we propose PET imaging because it can provide insight in both total tumor mass and the tumor’s cell type composition. Therefore, the discovery of radiotracers, which are able to classify different tumor cell types, is of uppermost clinical importance. Nevertheless, current modern imaging studies mostly focus on how tumor metrics and not cell type composition can be used as a prognostic marker for overall survival, malignancy or therapy response. For example, Aerts et al. used radiomic data from CT images of patients with early stage NSCLC and used a response phenotype that can predict a patient’s sensitivity toward Gefitinib therapy [2]. In order to parameterize the EGT models, all different tumor cell types in a tumor need to be identified and monitored.

It may be worthwhile to use newly developed techniques such as organoids [155, 184] and xenografts [41] to measure cell type compositions and protein expression to monitor tumor evolution and improve our understanding of the eco-evolutionary dynamics. Early preclinical in vivo studies of adaptive therapy included ovarian cancer cell line xenografts treated with carboplatin, and MDA-MB-231/luc triple-negative and MCF7 ER+ breast cancer cell lines treated with paclitaxel. In all cases, adaptive therapy could stabilize tumor volume, though the underlying sub-populations were not explicitly measured [67, 83]. In both of these studies, once initial tumor volume control was achieved, it could be maintained with constant or even progressively smaller drug doses, suggestive of stable eco-evolutionary equilibria.

Once patient-specific data of tumor cell types are available and monitored, it can be used to parameterize and optimize the EGT models to guide adaptive therapy protocols [98]. Subsequent measurements to inform patient specific parameters would then greatly improve modeling and predictions regarding tumor characteristics [206, 207]. After every measurement, the optimal next step in the adaptive therapeutic protocol could be calculated and used to stabilize the tumor burden, or may even be steered to create a pathway toward cure. To compare different mathematical models and seek the optimal cancer treatment, an optimal control theory approach may suffice [7, 47, 54, 57, 133, 134, 160]. Additionally, model predictive control (MPC) can use real-time monitored data to update the optimal cancer treatment. MPC involves model-based control techniques which can update the model and the optimal treatment schedule with each new clinical measure [134].

Critically, a model for tumor treatment can only be as effective as its associated empirical methods allow, i.e., in order to parameterize and validate it. Data may be retrospective (histologies, radiographies, biopsies, etc.) as well as derived from mouse or cell culture studies. For mapping genotypic or phenotypic data to treatment strategies, traditional statistical approaches can be used, but opportunities for machine learning and/or artificial intelligence are evident [43, 107]. Furthermore, the in vitro and in vivo competition assay has been shown to be well suited to feed EGT models. Such experiments have already shown that the success of cancer lineages depends on its frequency and the frequency of all other lineages with other strategies [23, 102, 103]. General models, when augmented by measurements, will permit EGT to inform clinical practice [89].

We considered evolutionary approaches to treatment and anticipated increased life expectancy from evolutionary therapy as compared to the traditional therapy, when resistant types are either pre-existing or evolve in response to therapy.

The biggest obstacle to applying EGT treatment methods to clinics remains the difficulty of estimating the tumor composition which currently can be done only for some types of cancers. Therapy for such cancers is particularly suited for our approach. They have discrete cell types and can therefore be understood via simpler EGT models. We reviewed some approaches for estimating the tumor composition in Sect. 4.

There is a clear gap between the complexity of the models that we introduced by (3) and (5), with a resistance matrix \({\mathbf {U}}\), and existing models, where either scalar or vector-valued traits are considered. We could find no research where one actually considers the resistance level of each known cancer type to each possible treatment. In all research we reviewed, resistance was either a single evolving trait of a monomorphic cancer population, a strategy within a polymorphic cancer population with one treatment, or multiple strategies of a polymorphic population with multiple treatments, where the resistance to these treatments does not evolve according to (5) but represents discrete and fixed strategies. The effects of these modeling assumptions (monomorphic or polymorphic population) and how they impact the superiority of adaptive therapy over continuous therapy with MTD have been recently investigated by Pressley et al., where time to progression for monomorphic and polymorphic models was compared between adaptive therapy and MTD [151]. The most general form of the cancer model, given by (3) and (5), has recently been used by Reed et al. to model pediatric sarcomas, where tumor growth is suppressed by multiple drugs, toward which resistance is evolving (e.g., vinorelbine, dactinomycin, cyclophosphamide) [153].

Most commonly used replicator dynamics and LV equations describe only one of the Eqs. (3) and (5). There is a rich theory for both, presumably as these models are simpler than the most general ones.

Some topics are not addressed in this paper that may become relevant for the future use of game theory for cancer and its treatment. For example, we did not specify whether the cancer cells’ types correspond to genetic or non-genetic traits (e.g., epigenetics; [161]). Generally, we believe that this does not influence the conceptualization of the game-theoretic models. Furthermore, whether strategies are genetic, epigenetic or phenotypically plastic will, at times, influence evolutionary speed. It may be that the tumor micro-environment can influence the epigenetics of a cell and thereby change its type, while this would not be the case if the type is genetically determined. Future models may need to pay close attention to the role of the micro-environment on the capacity for cancer cells to switch strategies. This switching may also happen in cancer stem cells (CSCs) and have consequences for tumor heterogeneity and the composition of cancer cell types. This may be interesting to study in the context of EGT modeling and cancer therapy.

Many common cancer types are shown to be propagated by small populations of CSCs. Genetic and epigenetic alterations can lead to CSCs emerging from non-stem cells endowed with stem cell properties. Therefore, stem cell identity may not be strictly a property of that cell, but may also depend on extrinsic cues provided by the adjacent cells and microenvironment. If stemness is not an intrinsic property, the malignant cells will regenerate new cancer stem cells, even if those with stem-like properties have been eliminated. Accordingly, the stem state of a cell is a dual phenotype. Therefore, in modeling CSCs a choice must be made on whether stemness is an intrinsic property or whether cell type switching takes place or not. Analysis of the evolution of stemness can help to identify whether these different types of stemness evolve according to different selective pressures, such as tissue maintenance and repair. This phenomenon also poses the question as to how non-genetically encoded plasticity will affect EGT modeling.

To conclude, many game theoretical models hypothesize on cancer behavior that yet has to be validated with real data [29, 31, 151, 188]. Close communication and collaboration between theoretical and empirical scientists will be of the utmost importance in advancing evolutionary therapies based on evolutionary game theory, to improve treatment results and patients’ quality of life.