Predicting treatment response or survival of cancer patients remains challenging in immuno-oncology. Efforts to overcome these challenges focus, among others, on the discovery of new biomarkers. Despite advances in cellular and molecular approaches, only a limited number of candidate biomarkers eventually enter clinical practice.

A computational modeling approach based on ordinary differential equations was used to simulate the fundamental mechanisms that dictate tumor-immune dynamics and to investigate its implications on responses to immune checkpoint inhibition (ICI) and patient survival. Using in silico biomarker discovery trials, we revealed fundamental principles that explain the diverging success rates of biomarker discovery programs.

Our model shows that a tipping point—a sharp state transition between immune control and immune evasion—induces a strongly non-linear relationship between patient survival and both immunological and tumor-related parameters. In patients close to the tipping point, ICI therapy may lead to long-lasting survival benefits, whereas patients far from the tipping point may fail to benefit from these potent treatments.

These findings have two important implications for clinical oncology. First, the apparent conundrum that ICI induces substantial benefits in some patients yet completely fails in others could be, to a large extent, explained by the presence of a tipping point. Second, predictive biomarkers for immunotherapy should ideally combine both immunological and tumor-related markers, as a patient’s distance from the tipping point can typically not be reliably determined from solely one of these. The notion of a tipping point in cancer-immune dynamics helps to devise more accurate strategies to select appropriate treatments for patients with cancer.

Immunotherapies are revolutionizing clinical care for cancer patients. The most widely used approach, immune checkpoint inhibition (ICI), can lead to long-term survival benefits in patients with advanced melanoma,

To unravel the complexities of cancers and their treatments, researchers have adopted mathematical and computational approaches to complement laboratory research. A plethora of modeling approaches are available, ranging from simple one-variable equations to complex spatial agent-based simulation models. In silico modeling has contributed to fundamental insights into tumor growth and cancer progression,

In this study, we investigate the consequences of tumor-immune dynamics on patients’ responses to ICI and survival in an ODE model. Our model reveals a tipping point within tumor-immune dynamics—a critical threshold for survival culminating in an all-or-nothing principle—that has profound implications for a patient’s disease course and outcome. We show how the presence of a tipping point alone robustly induces heterogeneous immunotherapy treatment outcomes, and how this complicates the search for both prognostic and predictive biomarkers.

We constructed a mathematical model consisting of a system of ODEs to capture essential interactions between cancer cells and lymphocytes during tumor formation. Our model represents tumorigenesis in patients, starting with the malignant transformation of a single cell.

The model consists of five equations that describe essential processes in the tumor microenvironment and the lymphatic organs (

An in silico model of the tumor microenvironment generates realistic and modifiable disease courses of cancer patients. (A) The ODE model describes fundamental processes in the tumor microenvironment. Parameters: α=naive T cell priming rate, δ=effector T cell death rate, ξ=effector T cell killing rate, ρ=tumor growth rate, p_{s}=effector T cell proliferation rate, and m_{s}=effector T cell migration rate. (B) An effective anti-tumor immune response can eradicate tumor cells before the clinical manifestation of a tumor. (C) After an initial state in which the tumor outpaces the immune system, the immune system can suppress tumor growth and controls it in a subclinical state. (D) The natural course of disease for a clinically apparent tumor. An initial malignant transformation is followed by tumor growth until clinical diagnosis. Despite the activation of adaptive immunity, the tumor prevails. A stage of progressive disease follows, ultimately culminating in cancer-related death. The horizontal gray lines indicate (from bottom to top): the tumor burden at diagnosis and the tumor burden at death, respectively. Simulation parameters are added in

We modeled tumor growth—that is, the formation of tumor cells during carcinogenesis—with the generalized exponential model proposed by Mendelsohn, in which

where

The killing rate expression is derived from the conventional Michaelis-Menten kinetics for enzyme-substrate interaction

in which E, S, and P are the enzyme, substrate, and product, respectively. k_{1}, k_{2}, and ξ represent the enzyme-substrate complex formation rate, the complex dissociation rate, and the catalytic rate. Given that complex formation and dissociation occur at a rate that is at least an order of magnitude faster than tumor growth, Borghans

in which I is the number of immune cells in the tumor microenvironment, ξ is the T cell killing rate, h_{I} is the saturation constant of the effector T cells, and h_{T} is the tumor cells’ saturation constant. Here, we consider T cells to follow a ‘monogamous killing’ strategy, meaning that one T cell interacts with one tumor cell at a time.

Combining T cell-mediated tumor cell killing (Equation 1b) and tumor growth (Equation 1a), we obtain the complete differential equation that describes the tumor burden over time:

Subsequently, the immunogenicity of the tumor triggers an anti-tumor immune response. Lymph node-resident T cells (S) migrate at rate m_{s} from the lymph nodes to the tumor microenvironment. The number of intratumoral T cells over time is determined by migration and death. Therefore, by combining a migration term with a death term at rate δ, we obtain the following equation for the evolution of intratumoral T cells over time:

Intratumoral T cells migrate from the lymph nodes where they are produced. This process starts with converting lymph node-resident naive T cells (ie, not activated antigen-specific; N) into antigen-specific effector T cells (S) at priming rate α. The priming rate α is scaled by the tumor size (ie, a smaller tumor will cause less T cell priming than a larger tumor) with a scaling term ^{7} cells (ie, a sphere with a radius of 0.29 cm). Effector T cells expand clonally at proliferation rate _{s}

The simulations used the following initial conditions: T(0)=1, I(0)=0, S(0)=0, and N(0)=10^{6}.

The simulation parameters are listed in

Simulation parameters in the ODE model

Symbol | Parameter (dimension) | Default value (range*) |

| Tumor growth rate (cells/day) | 1 (0–7) |

| Relative killing rate (cells/day) | 0.001 (0–0.05) |

| Michaelis constant (cells) | 571 |

| Death rate of immune cells (cells/day) | 0.019 |

α | Conversion rate of naive T cells into specific T cells (cells/day) | 0.0025 |

_{S}
| Total production rate of effector T cells from lymph nodes (cells/day) | 1 |

_{S}
| Migration rate from lymph node to tumor microenvironment (cells/day) | 1 |

*If not fixed.

ODE, ordinary differential equation.

The parameters were chosen to mimic realistic in vivo intercellular behavior. The rationale for the choice of each parameter is explained below.

In a human adult, an estimated repertoire of approximately 10^{10}–10^{11} naive CD8^{+} T cells is present.^{+} T cells need to be primed to become activated effector T cells. The CD8^{+} T cell precursor frequency—the frequency at which any given peptide-MHC complex is recognized by naive antigen-specific CD8^{+} T cells—is on the order of 1: 100 000.^{+} T cells in one of the tumor areas draining lymph nodes. A human body contains ±600 lymph nodes. At a steady state, roughly 40% of all lymphocytes reside in lymph nodes, meaning that 40 000 naive T cells (≈70 naive CD8^{+} T cells per lymph node) can be primed.^{6} antigen-specific CD8^{+} T cells per day via clonal expansion. Next, we assume that all antigen-specific effector T cells migrate into the tumor microenvironment to interact with tumor cells (ie, complex formation).

Complex formation and dissociation rates are described by the ‘Michaelis constant’, which we derived from the literature.

The killing rate of effector T cells has been investigated mainly in the context of infectious disease. In their review, Halle

For the simulations involving dynamic patient trajectories, we varied the tumor growth rate

We simulated tumor development in patients up to a maximum of 5 years. Note that depending on emergent tumor-immune dynamics, simulated patients may not reach the overall survival endpoint during this interval. Each time step in the simulation corresponded to 1 day. At baseline, one tumor cell and a pool of 10^{6} naïve tumor-specific T cells are present in a patient. Activated effector T cells are absent. We defined the time of diagnosis as the time at which the tumor exceeded 65×10^{8} cells and became clinically apparent. This cut-off corresponds to the assumption that a tumor with a volume of 1 cm^{3} contains 10^{8} tumor cells^{12} cells, corresponding to a total tumor mass of approximately 22×22×22 cm.

Model findings related to biomarker discovery programs were validated in a cohort of 58 patients with metastatic cutaneous melanoma that were treated with dendritic cell vaccination. Full details of this cohort, including baseline characteristics, were published previously.

We implemented our ODE model in C++. The Boost library ‘odeint’ was used to solve the system of ODEs.

To investigate the consequences of tumor-immune dynamics on the survival kinetics of patients, we used a computational modeling approach. We aimed to capture the interplay between tumor- and immune cells in the tumor microenvironment and simulate tumor growth in patients (see the Methods section). Our ODE model captured essential processes in antitumor immunity: priming of naive antigen-specific CD8^{+} T cells, clonal expansion of effector T cells in lymph nodes, tumor growth leading to effector T cell attraction into the tumor microenvironment, and formation of tumor-immune cell complexes to enable tumor cell killing (

We simulated tumor development from malignant transformation of a single cell, via clinical detection of a tumor, to advanced disease and possibly death. Depending on the tumor growth and the cytotoxic capacity of effector T cells, the ‘time to clinical manifestation’ and overall survival varied. Despite this variation, our simulations consistently showed three possible outcomes: (1) effector T cells inhibited tumor cell outgrowth and eradicated the tumor before clinical manifestation (

To better characterize these dichotomous survival kinetics, we examined how tumor-immune dynamics influenced patient survival by varying the tumor growth rate and the T cell killing rate over a broad range of possible values.

First, we focused solely on the tumor-component by varying the tumor growth rate. An increase in tumor growth did not gradually shorten overall survival in patients (

A tipping point in the tumor-immune interaction determines a patient’s outcome. (A) A gradual increase in tumor growth reveals a tipping point, where long-term survival (immune control; inset 1) abruptly changes to short-term survival (immune evasion; inset 2). (B) A similar analysis reveals a tipping point along the immune axis, again differentiating short-term survival (immune evasion; inset 1) from long-term control (immune control; inset 2). (C) The tipping point is present across the entire range of parameters examined. Cure and progressive disease are the dominant states, whereas subclinical tumor control only occurs within a limited parameter range (inset). Simulation parameters are shown in

Second, we investigated the influence of the T cell killing rate on overall survival. As for the death rate, a gradual increase in the cytotoxic capacity of effector T cells did not induce a gradual change in survival times. Instead, a sharp state transition that differentiated short from long survival was observed again (

To visualize this sudden state transition or ‘tipping point’ in tumor-immune dynamics as a function of both tumor proliferation and cytotoxic killing at the same time, we visualized the joint influence of the tumor growth rate and T cell killing rate on survival in a heatmap (^{3} tumor cells;

Next, we expanded these analyses to characterize the tipping point in different tumor types. A fundamental distinction between tumors is the rate at which they induce T cell priming, for instance, through tumor-specific immunogenicity or by specific characteristics of the immunosuppressive microenvironment. To this end, we simulated four tumor types: a tumor without T cell priming and three tumors in which T cell priming was varied from low to high. Without T cell priming, survival was only determined by the tumor growth rate—logically, no tipping point exists in the absence of T cells (

In general, the presence of a tipping point indicates that small perturbations in either tumor growth rate or T cell killing rate in the vicinity of a tipping point may result in substantial overall survival differences in patients. In contrast, much larger perturbations far away from the tipping point would have far less effect.

So far, we have described tumor-immune interactions during the natural course of malignant disease. In a clinical setting, however, therapeutic interventions are available to steer disease courses. Dependent on the treatment of choice, a specific effect is exerted on the tumor microenvironment. Treatment effects vary from constraining the proliferative capacity of tumor cells (eg, chemotherapy or targeted therapy) to increasing the T cell pool (eg, CAR T cells) or expanding the proliferative capacity of T cells (eg, cancer vaccines;

Tipping points induce dichotomous clinical outcomes in heterogeneous patient populations. (A) Treatments target processes or cell populations in the tumor microenvironment. (B, C) Two criteria need to be met to induce long-term survival: (B) ICI need to augment T cell killing sufficiently and (C) the treatment effect needs to be retained for a prolonged time. An inadequate treatment effect or limited treatment duration led at maximum to a temporary survival benefit. (D, E) In patient populations with variation in only (D) the tumor (ie, growth rate), or (E) the immune system (ie, T cell killing rate), the distance to a tipping point determines the clinical benefit. Without treatment, survival was limited (gray bars). In contrast, ICI induced long-term survival solely in patients close to a tipping point (green bars). See also

In the presence of a tipping point, ICI could induce a long-term survival benefit under two conditions: (1) the effect of treatment needs to be potent enough to shift a patient over a tipping point (

Thus far, our simulations considered tipping points generated in patients with fixed characteristics. However, disease courses in patients are certainly not fixed and are, to a certain extent, subject to (possibly random) variation. We hypothesized that interpatient variability in clinical outcomes could be partially attributable to this dynamic behavior of cancers and the interaction with the immune system. Such variation might reflect biological processes (eg, accumulating mutations, the expression of checkpoint molecules, and the availability of nutrients) that alter antitumor immunity and promote or hamper tumor development. We reasoned that the subsequent dynamics could drive patients towards and ultimately over a tipping point—or move patients away from it, which would limit the survival benefit of these treatments. To verify this hypothesis, we simulated the effect of dynamically evolving tumors (

Survival outcomes are strongly affected by evolving patient dynamics. (A, B) Examples of dynamic disease courses in patients with identical tumors and immune systems at baseline, respectively. (A) Evolving tumors (ie, random variation in tumor growth rate over time) and (B) continuous variation in the potency of the immune system (ie, killing rate) lead to divergent survival outcomes. The gray dotted lines indicate the baseline values for the growth rate and killing rate, respectively. (C) Dynamic trajectories in a heterogeneous patient population can move patients towards or away from a tipping point. The gray box indicates patients in which dynamic trajectories (blue) strongly alter survival outcomes compared to static trajectories (red). See also

Biomarker discovery studies aim to improve the prediction of patient survival on treatment. We observed that tipping points are crucial in shaping survival kinetics. Therefore, accurate survival predictions would require the consideration of tipping points. Ideally, a prognostic biomarker (or biomarker panel) would consistently distinguish long-term survivors from their counterparts. Since the non-linear survival dynamics following a tipping point weaken the correlation between a single biomarker and survival, the question is: how can we screen for biomarkers in a more efficient manner that takes this tipping point into account?

At first, we approached this question with an in silico biomarker discovery study. We measured the value of two potential biomarkers at baseline in simulated patients (n=100) that were subsequently treated with ICI (cohort characteristics are specified in

Non-linear tumor-immune dynamics complicate biomarker discovery. (A) An in silico biomarker discovery study in a ‘fixed’ patient cohort: while a single biomarker—either a tumor or an immune marker—can predict survival to some extent (the first and second columns), information from both markers in a biomarker panel enhances the predictive capacity greatly (third column). (B) Dynamic disease trajectories challenge survival prediction with ‘baseline’ biomarkers. In dynamic disease courses, the predictive value of single ‘baseline’ biomarkers is limited (the first and second columns; compare to (A). A biomarker panel improves survival predictions in this cohort (the third column) but is still defied by evolving dynamics. See also

In clinical practice, the assumption of a ‘fixed’ patient trajectory does not hold. Therefore, we simulated this cohort again with dynamic trajectories. Due to the dynamics, a subgroup of patients did not develop clinical tumors and was excluded from the analysis. The prediction of a patient’s prognosis with a single biomarker, either from the tumor or the immune system, in a dynamic cohort became increasingly challenging (the first and second columns of

These in silico experiments suggest that biomarker discovery efforts benefit from considering tumor and immune markers in concert rather than alone. To test this hypothesis, we retrospectively analyzed clinical data derived from previous trials in patients with metastatic melanoma (n=58; see baseline characteristics in ^{−7}) and I/P ratio (p=3.6×10^{−7}) explained survival better than chance on their own, but a bivariable model (p=9.3×10^{−10};

A composite biomarker consisting of a tumor and an immune component outperforms single markers in a retrospective analysis of metastatic melanoma patients. (A) A linear classifier based on LDH level at baseline, a surrogate marker for tumor growth, classified 71% of all patients correctly as short survivors (<9 months) or long survivors (>9 months). With an accuracy of 78%, the I/P ratio—an immune marker—performs better in this cohort. A linear combination of both markers leads to an even better classification (86% accuracy) than either one alone. (B) A Cox proportional hazard model based on both markers fits the data better than models based on either marker as measured by the Bayesian Information Criterion (BIC) (the lower, the better, and differences above 10 are considered strongly favoring one model over another). I/P, intratumoral versus peritumoral; LDH, lactate dehydrogenase; OS, overall survival.

Two important findings are derived from these observations: First, due to the non-linear tumor-immune dynamics with respect to survival, it can be complicated for a single biomarker to predict a patients’ prognosis accurately. Since survival kinetics emerge from the interplay between a cancer and the immune system, biomarkers from both systems need to be incorporated simultaneously into a biomarker panel to improve the predictive value. Second, biomarker measurements at baseline are merely a situational snapshot of the disease conditions at a specific point in time. Depending on the magnitude of the dynamics, it might become challenging or even impossible to predict the prognosis of patients from these biomarkers correctly.

This study investigated how tumor-immune dynamics relate to ICI-induced treatment responses and survival kinetics of patients. We predict that a tipping point is present in the tumor-immune interaction. This finding implies that underneath the intricate interplay between a developing malignancy and the immune system, two contrasting disease states determine disease outcome: a state where the immune system controls tumor outgrowth and a state in which a tumor escapes immune defense. A stable ‘steady state’ in which tumor growth and the immune response perfectly balance each other for extended periods seems only plausible in a subclinical setting. We show that treatment with ICI can induce a survival benefit by shifting a patient over a tipping point, thereby tipping the balance in tumor-immune dynamics in favor of survival. In line with clinical observations of interpatient variability in disease courses, we found that dynamics in patient trajectories pose major challenges for treatment response prediction. Moreover, we showed how a tipping point in dynamic patient trajectories defies simple strategies for outcome prediction in biomarker discovery studies. In particular, when facing highly dynamic disease courses, adaptive treatment strategies based on continuous monitoring might be more promising than simple patient stratification at baseline.

Tipping points are well known in complex systems such as financial markets and ecosystems but are also present in medicine.

Tipping points within tumor-immune dynamics have important implications for biomarker discovery. Biomarkers are developed to predict prognosis and steer clinical decision-making. Disease outcomes in cancer patients are essentially determined by the interplay between two complex systems: the tumor and the immune system. Our model predicts that factors from both systems should be considered to improve the predictive power of biomarkers. However, in contrast with this seemingly straightforward prediction, current research mainly focuses on factors derived from one of the two complex systems. Expression of PD-L1 on tumor tissue illustrates this: while 45% of patients with PD-L1 positive tumors show objective responses to anti-PD(L)1 immunotherapy, 15% of patients with PD-L1 negative tumors also show objective responses.

Our approach has to be interpreted in light of some limitations. Although the ‘coarse-grained’ nature of ODE models allows focusing on the major common underlying mechanisms in many cancers, it is also a potential pitfall. For example, metabolic processes such as hypoxia, immune-suppressive characteristics of the tumor microenvironment such as the presence of FoxP3^{+} regulatory T cells or expression of transforming growth factor β, the presence of other relevant effector cells such as natural killer cells, and the availability of nutrients are only implicitly represented by our model in a single killing efficacy parameter. This simplification also holds for treatments. In this study, ICI was limited to its main mode of action: the augmentation of the T cell killing rate. While the ‘true’ mechanistic effects might be more widespread, sufficient data to correctly parameterize more complex models remains scarce. Furthermore, it should be emphasized that an ODE model contains limited spatial information; while we distinguish between lymphatic tissue and the tumor microenvironment, all cells within the microenvironment are identical, and all processes affect cells in the same manner. Although we do not expect that explicit incorporation of these processes or translation of the model into a spatial variant alters our central finding of a tipping point, it could nevertheless be of interest to verify these hypotheses in future research using more complex, spatial agent-based models.

In conclusion, we used computational modeling to show that the clinical outcome of cancer patients is determined by tipping points in tumor-immune dynamics. A tipping point influences not only treatment response but also the prognosis of patients and has major implications for future biomarker research.

JHAC and JT conceived this study. JHAC performed the experiments and wrote the manuscript under the supervision of JT. All authors provided feedback on the manuscript and reviewed the manuscript prior to submission.

JHAC was funded by the Radboudumc. WJL was supported by Fellowships from the NHMRC, the Simon Lee Foundation and the Cancer Council Western Australia. CGF received an ERC Adv Grant ARTimmune (834618) and an NWO Spinoza grant. IJMdV received an NWO-Vici grant (918.14.655). JT was supported by a Young Investigator Grant (10620) from the Dutch Cancer Society and an NWO grant (VI.Vidi.192.084).

WJL reports consultancy activities for Douglas Pharmaceuticals and MSD; research funding from Douglas Pharmaceuticals, AstraZeneca, and ENA therapeutics; patents PCT/AU2019/050259 and PCT/AU2015/000458 (all outside this work). NM reports personal fees from Bayer and Merck Sharp & Dohme; grants and personal fees from Jansen-Cilag, Roche, Astellas, and Sanofi (all outside this work). WRG reports consultancy activities for Bristol-Myers Squibb, IMS Health, Janssen-Cilag, Sanofi, and MSD; speaker fees from ESMO and MSD; and research funding from Bayer, Astellas, Janssen-Cilag, and Sanofi (all outside this work).

Not commissioned; externally peer reviewed.

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The code of the ODE model and the data shown in Figure 6 are available at GitHub: