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W4367018901 abstract "Article Figures and data Abstract Editor's evaluation Introduction Results Discussion Methods Data availability References Decision letter Author response Article and author information Abstract The failure of cancer treatments, including immunotherapy, continues to be a major obstacle in preventing durable remission. This failure often results from tumor evolution, both genotypic and phenotypic, away from sensitive cell states. Here, we propose a mathematical framework for studying the dynamics of adaptive immune evasion that tracks the number of tumor-associated antigens available for immune targeting. We solve for the unique optimal cancer evasion strategy using stochastic dynamic programming and demonstrate that this policy results in increased cancer evasion rates compared to a passive, fixed strategy. Our foundational model relates the likelihood and temporal dynamics of cancer evasion to features of the immune microenvironment, where tumor immunogenicity reflects a balance between cancer adaptation and host recognition. In contrast with a passive strategy, optimally adaptive evaders navigating varying selective environments result in substantially heterogeneous post-escape tumor antigenicity, giving rise to immunogenically hot and cold tumors. Editor's evaluation This study presents a valuable mathematical model for the adaptive dynamics of cancer evolution in response to immune recognition. The mathematical analysis is rigorous and convincing, and overall the framework presented could be used in the future as a solid base for analytically tracking tumor evasion strategies. The work will be of interest to evolutionary cancer biologists and potentially may also have implications for the design of clinical interventions. https://doi.org/10.7554/eLife.82786.sa0 Decision letter Reviews on Sciety eLife's review process Introduction Cancer dynamics, encompassing both genotypic evolution and phenotypic progression, lies at the heart of treatment failure and disease recurrence, and therefore represents a significant and stubborn therapeutic hurdle. Prior research efforts have made substantial progress in detailing the mathematics of acquired drug resistance (Iwasa et al., 2006; Michor et al., 2004; Komarova, 2006) and the complementary roles of phenotypic and genotypic changes (Gupta et al., 2019). Recently, there has been much renewed interest in therapies that utilize the adaptive immune system to confer durable remission (Couzin-Frankel, 2013; Waldman et al., 2020). These latter breakthroughs have generated considerable interest in quantifying the cancer-immune interaction (Mayer et al., 2019; Sontag, 2017; George et al., 2017). As with targeted therapeutic resistance via compensatory evolution or adaptive rewiring (Bergholz and Zhao, 2021), tumors can similarly evade the immune system via either elimination or downregulation of tumor-associated antigens (TAAs) normally detectable by the T cell repertoire (Rosenthal et al., 2019). However, several key features distinguish immune-specific evasion from classical drug resistance (Komarova, 2006). Dynamical changes in cancer genotypes and phenotypes, while problematic for conventional therapies, create additional TAAs that may subsequently be recognized by distinct T cells (Yarchoan et al., 2017). Thus, the evolving diversity of the T cell repertoire, consisting of billions of unique clones each with a distinct T cell receptor, provides adaptive immunity and immunotherapy the unique advantage of repeated tumor recognition opportunities (George and Levine, 2021; Lakatos et al., 2020; Qi et al., 2014), making long-term evasion more challenging. Previous research efforts have investigated the diversity of evolutionary trajectories and the extent of cancer-immune co-evolution occurring in early disease progression (George and Levine, 2018; George and Levine, 2020). These works were based on increasing evidence of significant and sustained tumor evolution driven by immune surveillance (Turajlic et al., 2018; Jamal-Hanjani et al., 2017). Immunosurveillance via distinct T cell clones imposes an adaptive, stochastic recognition environment on developing cancer populations (Desponds et al., 2016) that can result either in cancer elimination, escape, or equilibrium (Schreiber et al., 2002; Dunn et al., 2004). Equilibrium results in cancer co-existence with the immune system over large time scales (Turajlic et al., 2018), thereby motivating the need for a more complete understanding of the interplay between immune recognition and cancer evolution for effective therapeutic design. In addition to parsing this complexity, the precise extent to which a cancer population may actively evade repeated immune recognition attempts is at present unknown. Previous modeling efforts have assumed that cancer adaptation occurs passively, that is, without behavior predicated on knowledge of the current immune microenvironment (IME). However, it is well known that cancer populations commonly undergo phenotypic changes capable of altering their immunogenicity (Tripathi et al., 2016); these changes could be coupled to sensing of the IME in a manner similar to cancer mechanical, chemical, and stress sensing (Lee et al., 2019; Damaghi et al., 2013; Rosenberg, 2001). Moreover, direct experimental evidence demonstrates genetic adaptation in bacterial systems capable of sensing stress and consequently varying the per-cell mutation rate (Al Mamun et al., 2012; Rosenberg and Queitsch, 2014); there appear to be similar stress pathways in cancer (Bindra et al., 2007). Therefore, an alternative to passive evolution is for cancer populations to actively sense and evade recognition in the current environment en route to metastasis in a manner that maximally benefits survival, which we refer to henceforth as the ‘optimal escape hypothesis.’ Understanding the extent and associated features of optimized tumor evasion is a crucial first step to identifying the best therapeutic approach, particularly for T cell immunotherapies that may be temporally varied. Here, we introduce a mathematical framework, which we call ‘Tumor Evasion via adaptive Antigen Loss’ (TEAL), to quantify the aggressiveness of an evolutionary strategy executed by a cancer population faced with a varying recognition environment. This framework enables a dynamical analysis of both passive and optimized evasion strategies. The TEAL model describes a discrete-time stochastic process tracking the number of targets available to a recognizing adaptive immune system. We apply dynamic programming (Bellman and Dreyfus, 1959; Ross, 2014) in order to solve the corresponding time homogeneous Bellman equation detailing the tumor optimal evasion strategy for a specific example of the assumed penalty for attempting to avoid immune detection. In doing so, we obtain an exact analytical characterization of the evasion policy that maximizes long-run population survival, which we show is the unique solution. We can then quantify the enhancement in survival for optimal threats relative to their passive counterparts under a variety of temporally varying recognition environments. Surprisingly, we find that optimized strategies exhibit substantial diversity in their dynamical behavior, distinguishing them from threats with a fixed evolutionary strategy. Notably, immune recognition efficiency and the IME microenvironment are predicted to influence the likelihood for tumors to either accumulate or lose therapeutically actionable TAAs prior to their escape. The TEAL model represents a first attempt to explicitly represent – and in the future test – the optimal escape hypothesis in order to frame cancer evasion as a dynamic and informed strategy aimed at maximizing population survival. Model development In greatest generality, our model consists of an evading clonal population that may be targeted over time by a recognizing system. We assume henceforth that the recognition-evasion pair consists of the T cell repertoire of the adaptive immune system and a cancer cell population, recognizable by a minimal collection of sn TAAs present on the surface of cancer cells in sufficient abundance for recognition to occur over some time interval n. Our focus is on a clonal population, recognizing that subclonal TAA distributions in this model may be studied by considering independent processes in parallel for each clone. Experimental evidence and prior modeling suggest that tumors may be kept in an ‘equilibrium’ state of small population size prior to either escape or elimination, with repeated epochs of recognition and evasion (Dunn et al., 2004; Turajlic et al., 2018; George and Levine, 2020). We adopt a coarse-grained strategy and assume that during each epoch, the immune system has an opportunity to independently recognize each of the sn TAAs with probability q, and also the cancer population can lose recognized TAAs, each with probability πn, which we refer to as the antigen loss rate. The antigen loss rate is either fixed or chosen by the cancer population using information available in the current period. If the immune system cannot detect any of the available TAAs in a given period, then the cancer population escapes detection. On the other hand, if rn>0 antigens are detected by the adaptive immune system in this time frame, then the cancer population is effectively targeted. This leads to cancer elimination unless the population is able to lose each of the rn recognized antigens during the same period. This loss of recognition would presumably arise in a subpopulation that would then expand at the expense of the successfully targeted cells. If evasion balances recognition and all detected antigens are lost, then equilibrium (non-escape, non-elimination) ensues, and the process repeats in the next period with a new number of target antigens given by a state transition equation (1) sn+1=sn−rn+β+fn where β represents the basal rate of new antigen accumulation, and fn represents the addition of new TAA targets dependent on the rate of escape πn in the current state. We shall refer to fn as the (intertemporal) penalty term, the idea being that changes that lead to antigen loss will out of necessity give rise to the creation of new TAAs, in the form of either overexpressed/mislocalized self-peptides or tumor neo-antigens. The model therefore defines a discrete time process that involves changes to both the tumor and the immune system. The process ends in cancer elimination if the cancer population is unable to match all of the rn recognized antigens at any period. The process ends in cancer escape if at any period the number of recognized antigens is zero (rn=0). This framework mirrors the outcomes resulting from known tumor-immune interactions, a process that leads via immunoediting to cancer escape, elimination, or equilibrium (Schreiber et al., 2002; Dunn et al., 2002; Dunn et al., 2004; Koebel et al., 2007). Here, tumor antigenicity is represented by the total number of post-escape TAAs. We do not distinguish between different types of TAA loss, which may occur through a number of mechanisms, including somatic mutation, epigenetic regulation, or phenotypic alteration. Passive evader In the passive case, the cancer population does not change its evasion rate so that πn=p is fixed and independent of any of the parameters governing the recognition landscape. For this case, we shall also use the simple assumption that the net antigen accumulation and penalty β+f is a fixed constant. Optimal evader In the optimized case, πn is chosen in order to maximize the overall evasion probability as a function of parameters realizable to the cancer at period n. We assume that sn the number of TAAs as well as rn the size of the recognized subset is knowable by the cancer prior to strategy selection. In addition, we postulate that the intertemporal penalty scales directly with πn, a reasonable assumption given, for example, the direct relationship between mutagenesis and passenger mutation accumulation (Pon and Marra, 2015; McFarland et al., 2014). While many functional forms of fn⁢(πn,rn,sn) would be reasonable, we assume in general that the penalty is πn-linear: (2) fn(sn,rn,πn)=hm(sn,rn)πn. To make our system analytically solvable, we use a specific choice in which hm scales monotonically as a function of both rn and sn and hm∝rn in the large rn limit (see ‘Methods’). Since the number of recognizable (and thus actively targeted) TAAs reflect, all else being equal, an active IME hostile to cancer, we assume that subsequent total TAA addition, β+fn, are dependent on the current level of immune detection, thereby taking into account the increased cost of surviving in, for example, an inflammatory IME. The temporal dynamics of the TEAL process are illustrated in Figure 1A and Figure 1—figure supplement 1. Figure 1 with 11 supplements see all Download asset Open asset Tumor Evasion via adaptive Antigen Loss (TEAL) model. (A) Illustration of tumor antigen detection and downregulation in the TEAL model of cancer-immune interaction. (B) The directed graph with nodes representing the states of the TEAL model and edges labeled based on the probability of their occurrence. The interaction leads to elimination, equilibrium, or escape. Both evasion and elimination are absorbing states, and the equilibrium state results in repeated interaction. (C) Plots of single-period cancer optimal antigen loss rates π* given by Equation 8 are plotted as a function of recognition rate q for various numbers of recognized antigens 0<rn≤sn with sn=5. Varying environments Using the above framework, we subject both passive and active cancer evasion tactics to temporally varying recognition profiles. We partition pre-escape dynamics into four cases based on immune recognition q and basal TAA arrival β, from which we characterize the distribution of escape time, cumulative mutational burden, and predicted post-escape tumor immunogenicity. Results The following section presents the main findings of our analysis (full mathematical details are provided in the ‘Methods’ section). For sn available and rn recognized TAAs, we have that rn∼Binom(sn,q). Conditional on recognition (rn>0), the number of downregulated antigens, ℓn, is given by ℓn∼Binom(rn,πn). Recognition therefore occurs with probability P(rn>0)=1−(1−q)sn. Similarly, non-elimination occurs following recognition with probability P(ℓn=rn)=πnrn. A decision tree for the TEAL process is illustrated in Figure 1B (passive and active decision trees used in the analysis are depicted in Figure 1—figure supplements 2–4). Passive evasion strategy For a passive evader, the TAA loss rate is fixed so that πn=p. It can be shown (see Methods Section. Distribution of lost antigens) that the dynamics governed by Equation 1 in the passive case can be represented by their mean trajectories while the cancer population is in equilibrium, given by (3) En[Sn+1]=Sn−p(1−γ)ηsn−1ηsn−γsnsn+(β+f), where η≡1−q(1−p) is the probability of equilibrium (non-escape, non-elimination) between the cancer and immune compartments for a single TAA given the existence of at least one available TAA. These dynamics may be approximated by (4) En[sn+1]≈(1−q)sn+(β+f), where En[⋅] is the conditional expectation given the information available at time n. The approximation given by Equation 4 is a lower estimate of tumor antigenicity and is accurate as long as p and q are not both small and in particular for choices that give rise to large tie probability (Figure 1—figure supplements 6 and 10). Optimal evasion strategy In contrast to the above case where πn was fixed at p, Here, the antigen loss rate is variable and selected optimally given the current state of total sn and recognized rn antigens. The use of dynamic programming to address the optimal long-term evasion policy relies on a defined value function (Bellman and Dreyfus, 1959). We shall focus on the case where the cancer population is assigned normalized values of 1 at any period resulting in escape and 0 otherwise. The corresponding stationary Bellman equation takes the form (5) Jn=maxπnEn[πnrn[(1−q)sn+1+(1−(1−q)sn+1)Jn+1]], where the value function Jn=J(sn,rn,πn) represents the maximal attainable value at period n; (Methods Section Dynamic programming solution). It can be shown that (6) Jn=Anγsn1−(1−q)sn with (7) An=δnq(1−q)β+rn/c−rn1−δnq(1−q)β+rn/c−rn satisfies Equation 5. Here, 0<δn≤1 is a free parameter that varies inversely with the risk aversion of the evader (larger values imply a bolder strategy). One advantage of the dynamic programming approach is that it reduces an infinite-period optimization problem to a sequence of single-period optimizations. The corresponding optimal policy is given by the sequence (8) πn∗=(δnq1−(1−q)sn)1/rn. Plots of πn* are given for various rn in Figure 1C and Figure 1—figure supplement 11. As expected, this closed-form strategy results in increased values for the optimal antigen loss rate πn*, which increase for increasing q and rn. We take δn=1 in subsequent analysis (so that the optimal strategy when sn=rn=1 is πn*=1). Active evasion strategies enhance population survival rates For a fixed TAA arrival, Equations 3 and 4 describe a mean-reverting process. Consequently, the mean number of TAAs approaches a stable equilibrium (9) limn→∞En[sn+1]≈(β+f)/q. as long as the cancer neither escapes nor is eliminated. In the optimal case, a similar equilibrium value s∞ may be calculated: (10) s∞=βq|1/ln⁡(1−q)−1|. In this case, stability is more complex: If immune recognition is sufficiently effective, meaning q>q∗=1−e−1, then Equation 10 is a stable equilibrium exhibiting mean reversion similar to that of the passive case. On the other hand, recognition impairment (q<q∗) gives rise to an instability, which results in a system harboring an initial number of targets s0 being driven either to escape if s0<s∞ or to large accumulations (and likely elimination) if s0>s∞ (Figure 5—figure supplement 2). We proceed by contrasting active and passive escape rates assuming no recognition impairment, and discuss the implications of immune impairment in a later section. Simulations of passive and optimized strategies with passive evasion rates matching mean optimal evasion rates (p=E[πn∗]|s∞) are compared in Figure 2. Despite identical mean TAA evolution (Figure 2A) and comparable intertemporal penalties, the optimized strategy results in substantially higher cancer escape probability (150%) compared to the passive case. Moreover, optimized strategies generate wider escape time distributions, thus illustrating an adaptive evader’s sustained effort to thwart elimination prior to escape (Figure 2B). Figure 2 Download asset Open asset Passive and optimized evasion strategies against stationary threats. (A) Comparisons of the temporal dynamics of passive (green) and active (blue) strategies with parameter selections giving equal mean behavior. In the active case, q=q*+0.1 yields stable dynamics, giving mean antigen arrival β+f=3.44. In the passive case, p=0.90 was selected to match the mean optimal evasion rate and the expected sn of the active case. Also, f=3.51 and β=0.88 both chosen so that s∞=5, and the results plotted for s0∈{2,5,8} {2, 5, 8}. (B) 106 replicates of this process were used to calculate distributions of stopping times conditioned on escape. This distribution generates passive (resp. optimized) pescape of 5.37 (resp. 8.44). Figure 2—source data 1 Source data contains a spreadsheet of data for Figure 2B. https://cdn.elifesciences.org/articles/82786/elife-82786-fig2-data1-v1.xlsx Download elife-82786-fig2-data1-v1.xlsx Arbitrary recognition landscape The above describes the dynamics of passive and optimized cancer co-evolution during adaptive immune recognition with constant governing parameters. We can more generally apply this approach to understand how an evasion strategy affects the likelihood and timing of cancer escape under a variety of temporally varying recognition landscapes. Such landscapes could, for example, be imposed by a clinician temporally modulating an immunotherapeutic intervention and are routinely proposed in the setting of traditional therapies, where attempted strategies have included a variety of cyclical burst approaches (Foo and Michor, 2009; Eigl et al., 2005). A similar approach could be taken with regard to timing and dosage of adoptive T cell immunotherapy. An advantage of our dynamic programming approach is the ability to study optimal evasion strategies for arbitrary recognition landscapes (Figure 3A). We simulate TEAL dynamics and find that optimized immune evaders are more successful in evading detection than their passive counterparts across various recognition landscapes (Figure 3B). Evasion, when it occurs in the optimized case, does so largely after a sustained interaction with the recognizing threat (Figure 3C). Collectively, our results detail the dynamics of sustained cancer-immune co-evolution via TAA loss in threats capable of adopting adaptive evasion strategies in the presence of complex treatment modulation (George and Levine, 2020; Turajlic et al., 2018). Figure 3 Download asset Open asset Passive and optimized evasion strategies for temporally varying recognition profiles. (A) Temporally varying recognition functions are selected and applied to threats employing passive (blue) and optimized (red) evasion strategies. (B) The mean and standard deviation of escape probabilities is compared across recognition profiles for each strategy (pairwise significance was assessed using two-sample t-test at significance α=0.05 with p<10-5). (C) Escape time distributions are generated for step, cyclical, increasing, and decreasing recognition environments (solid line: mean). In each case, mean total new antigen arrival β+E⁢[fn] for passive (resp. optimized) evasion were 4.39 (resp. 4.75), and 103 simulations of 103 replicates each were used for statistical comparison; all samples were aggregated for escape time violin plots (solid line denotes mean). Figure 3—source data 1 Source data contains a spreadsheet of data for Figure 3B, C. https://cdn.elifesciences.org/articles/82786/elife-82786-fig3-data1-v1.xlsx Download elife-82786-fig3-data1-v1.xlsx Optimal evaders under effective immune recognition accrue mutations at a fixed rate One consequence of mean reversion is that the rate of mutation accumulation over time, λ⁢(n), is linear in n (Methods Section Mean optimal transitions): (11) λ(n)=2βln⁡(1−q)1−ln⁡(1−q)n,q>q∗=1−e−1. The prediction of constant accumulation is consistent with empirically observed cancer mutation behavior (Lawrence et al., 2013; Alexandrov et al., 2013). This is not what holds in the impaired case (as will be discussed later), thus suggesting that early cancer progression often proceeds in an environment with effective immune recognition. Additionally, our formula shows that larger mutation rates can be caused by large evasion penalties or by reduced immune recognition. Of course, the TEAL model does not consider any specific features that determine the values of the effective parameters. Instead, its utility is in quantifying the overall effect of reducing antigen detection resulting from, for example, transitions to an immune impaired microenvironment. Post-escape tumor antigenicity determined by a balance between recognition aggressiveness and local penalties in the immune microenvironment The prior section related recognition and penalty to observed mutation rates. We now consider their combined effects on tumor immunogenicity following immune escape. The TEAL model represents immunogenicity by the number of available TAAs at the time of cancer detection, an important predictor of immunotherapeutic efficacy (Martin et al., 2016; Samstein et al., 2019; Goodman et al., 2017). We apply the TEAL model to simulate evading cancer populations, focusing exclusively on trajectories that result in tumor escape, to characterize the distribution of available TAAs. This is performed first for increasing immune recognition rates q (Figure 4A) and then for increasing penalty term β (Figure 4B). Our results demonstrate that larger penalties result in higher post-escape TAA levels, while efficient immune recognition depletes available TAAs. The presumptive reason for this latter observation is that escape in the presence of strong immune recognition biases the tumor to have low numbers of TAAs. This prediction agrees with recent empirical observations that strong immune selective pressure in early cancer development results in tumor neoantigen depletion and is prognostic of poor clinical outcome (Rosenthal et al., 2019; Lakatos et al., 2020). Figure 4 Download asset Open asset Distribution of available post-escape tumor antigens. The distribution of tumor-associated antigens (TAAs) was estimated from simulations of optimized cancer evasion resulting in escape and plotted for (A) increasing recognition probability q∈{0.6,0.65,0.7,0.75,0.8,0.85,0.9,0.95} and (B) increasing evasion penalty β∈{0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8}. For (A), β=0.59. For (B), q=0.7>q∗. In both cases, s∞=5 and n=106 simulations were performed for each histogram. Figure 4—source data 1 Source data contains a spreadsheet of data for Figure 4A, B. https://cdn.elifesciences.org/articles/82786/elife-82786-fig4-data1-v1.xlsx Download elife-82786-fig4-data1-v1.xlsx Variation in the tumor microenvironment drives the generation of immune hot vs. cold tumors under optimal evasion In the passive evader case, antigenicity fluctuates around a stable equilibrium that varies directly with penalty and inversely with recognition. The adaptive case gives rise to more complex behavior resulting from impairments in immune recognition or changes in penalty (Figure 5—figure supplements 1 and 2). These changes are important manifestations of disease progression, which may alter the immunogenic landscape via impairments in immune recognition, such as MHC downregulation, co-stimulation alteration, T cell exclusion, or the establishment of a pro-tumor IME, via. for example. M2 macrophage polarization (Liu et al., 2021; Goswami et al., 2017). Although many factors may affect recognition rates, for simplicity we shall refer to larger vs. smaller immune recognition rates q as infiltrated vs. excluded. On the other hand, the generation of new TAA targets is expected to vary substantially across tumor type, for example, due to differing somatic mutation rates. Within a given tumor subtype, variations in the hostility of the IME, resulting from a large variety of possible mechanisms (metabolic, mechanical, cytokine, environment), require cancer populations to undergo greater degrees of adaptation to survive; in our approach, this greater degree of adaptation comes with a greater penalty. Consequently, we relate large vs. small local penalty terms β to anti-tumor vs. pro-tumor IMEs. Conceptually, the baseline state (infiltrated anti-tumor IME) may give rise to three alternative states (excluded anti-tumor IME, infiltrated pro-tumor IME, or excluded pro-tumor IME), based on progression. Toward this end, we simulate the TEAL model under the above conditions and record post-escape TAA distributions. As already explained, our results predict that infiltrated (q>q∗) environments lead to an absorbing equilibrium state in the intervening period prior to escape, while exclusion (q<q∗) results in unstable equilibria. Interestingly, the sign of this equilibrium, and hence the long-term immunogenic trajectory, depends on the sign of β (Equations 88 and 89). The baseline infiltrated anti-tumor case (q>q∗, β>0) yields a positive and stable, mean-reverting TAA steady state, generating immunogenically ‘warm’ tumors. Excluded anti-tumor IMEs (q<q∗, β>0) exhibit low recognition and large TAAs arrival, resulting in a unstable TAA steady state that leads to increased immunogenicity over time, resulting in ‘hot’ tumors. Furthermore, the infiltrated pro-tumor (q>q∗, β<0) case demonstrates preserved recognition with low TAAs arrival and generates an unphysiological negative stable steady state, thereby predicting that trajectories reduce immunogenicity to zero over time, yielding ‘cold’ tumors. Lastly, excluded pro-tumor IMEs (q<q∗, β<0), having compromises in both recognition and TAA arrival rate, result in an unstable state, above which trajectories accumulate additional TAAs over time, becoming immunogenically ‘hot,’ and below which the populations are predicted to reduce the number of recognizable TAAs over time, becoming ‘cold’ (Figure 5A and B). Substantial heterogeneity in the distributions of escape time predict sustained interactions in the unimpaired case (Figure 5—figure supplement 3). Tumor exclusion leads to hot tumors so that escape, should it occur, must do so on average prior to the accumulation of many TAAs. Conversely, pro-tumor IME with immune recognition drives TAA depletion, so escape occurs relatively early. These results are summarized in Figure 5C. Figure 5 with 4 supplements see all Download asset Open asset Active Evader dynamics. Violin plots of the distribution of post-immune escape. (A) Cumulative mutation burden. (B) Post-escape immunogenicity (available tumor-associated antigens [TAAs]) as a function of time for a variety of tumor immune microenvironment (IME) conditions. (Anti-tumor-infiltrated: q=q∗+0.1, β=0.529; anti-tumor-excluded: q=q∗−0.1, β=0.505; pro-tumor-infiltrated: q=q∗+0.1, β=−0.529; pro-tumor-excluded: q=q*-0.1, β=−0.505. In all cases, β chosen to give |s∞|=3 [s∞=-3 for the pro-tumor-infiltrated case] giving strictly positive penalties. Simulatio" @default.
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W4367018901 title "Author response: Optimal cancer evasion in a dynamic immune microenvironment generates diverse post-escape tumor antigenicity profiles" @default.
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