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 (figure 1A). In the tumor microenvironment, tumor growth (Equation 1a) and T cell-mediated killing of tumor cells (Equation 1b) determine the evolution of the tumor burden (the numbers of the equations correspond to those used in figure 1A). Tumor-infiltrating lymphocytes migrate from the lymph nodes to the microenvironment (equation 2). Before migration, T cells expand clonally in the lymph nodes’ T cell zones (equation 3) after conversion of naive T cells into antigen-specific effector T cells (equation 4). Below, we provide in-depth descriptions of each model equation.

We modeled tumor growth—that is, the formation of tumor cells during carcinogenesis—with the generalized exponential model proposed by Mendelsohn, in which ρ represents a tumor growth rate constant.27 Essentially, this means that at each time interval, a fraction of tumor cells divide. The dividing fraction decreases as the tumor burden increases since substantial parts of a larger tumor mass, such as the necrotic core, are no longer able to proliferate. Since the tumor burden (T) is determined by the combination of tumor growth and tumor cell killing, the first part of equation 1—describing the tumor burden over time – will be:

(1a) dTdt=ρT45

where dTdt=ρT45=ρT1/5T=f(T)T , resulting in f(T)=ρT1/5 as the fraction of dividing cells per time interval, which scales inversely with the tumor burden T.

The killing rate expression is derived from the conventional Michaelis-Menten kinetics for enzyme-substrate interaction28 29:

E+S⇌k2k1ES→ξE+P

in which E, S, and P are the enzyme, substrate, and product, respectively. k₁, k₂, 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 et al 28 argued that the Michaelis-Menten kinetics could be simplified using a quasi-steady-state assumption. Simplification using a Padé approximation and subsequent rearrangement leads to a conventional Double Saturation (DS) model that describes effector T cell-mediated killing28 29:

(1b) dTdt=−ξIT1+IhI+ThT

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.28 29

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:

(1) dTdt=ρT45−ξIT1+IhI+ThT

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:

(2) dIdt=msS−δI

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 (T107+T) , meaning that the priming rate is at half of its maximum rate and starts to saturate in tumors larger than 10⁷ cells (ie, a sphere with a radius of 0.29 cm). Effector T cells expand clonally at proliferation rate p_s and migrate into the tumor microenvironment. Combining these processes, we arrive at the final two differential equations that describe the evolution of naïve and primed T cells in the lymph nodes:

(3) dSdt=α(T107+T)N+psS−msS

(4) dNdt=−α(T107+T)N

The simulations used the following initial conditions: T(0)=1, I(0)=0, S(0)=0, and N(0)=10⁶.

The simulation parameters are listed in table 1.

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 10¹⁰–10¹¹ naive CD8⁺ T cells is present.30 31 Naive CD8⁺ 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: 100 000.30 Priming should be limited primarily to naive CD8⁺ 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.32 33 We assume that priming occurs primarily in the tumor-draining lymph node station (per station harboring around 20 lymph nodes34). Then, 1400 T cells would be available for priming at any given time, and this pool would be refreshed approximately once per day by T cell recirculation. Considering that dendritic cells might present multiple epitopes and antigens, and that T cell priming in vivo might occur suboptimally, we set a priming rate of at most 2500 cells per day. The order of magnitude of these priming rates corresponds to priming rates found in chronic infectious diseases.35 Due to evasive mechanisms, antitumor immunity is a more dormant process than an immune response to infections.36 Therefore, we scaled the priming rate with tumor size, which translates into a maximum production of 10⁶ 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.29 The Michaelis constant describes the ratio between complex formation and dissociation.

The killing rate of effector T cells has been investigated mainly in the context of infectious disease. In their review, Halle et al 37 discuss discrepancies between in vitro and in vivo killing rates of effector T cells. Depending on the context, killing rates of effector T cells vary from 1 target per 5 min to 0–10 targets per day,37 but tumor cells are considered difficult to kill. Extensive variation in experimental in vivo per capita killing rates (ie, the number of cells killed by an effector T cell per unit of time) complicates the selection of a default fixed killing parameter. Therefore, we investigated T cell dynamics over a wide range of killing rates as described using the monogamous killing regime in a DS model by Gadhamsetty et al.29 The DS model ensures that the killing rate saturates with respect to the tumor cell and the effector cell densities. Consequently, our model’s maximum per capita killing rate is 2.5: one T cell can kill at most 2.5 tumor cells per day, provided there are abundant target cells available, and there is no competition with other T cells. The default tumor growth rate is 1 cell/day, but we varied this parameter extensively in our simulations. Taken together, our default parameter values led to simulations of disease courses with realistic survival times in patients with malignancies and matched the order of magnitude of tumor growth rates as reported by others.38

For the simulations involving dynamic patient trajectories, we varied the tumor growth rate ρ and the T cell killing rate ξ in a stochastic manner over time. Briefly, we set one value per month of simulated time by multiplying the baseline parameter value with a random number drawn from a normal distribution with a fixed SD. The values used for the SD are given in online supplemental tables 4 and 5 (‘stochasticity’). From these monthly reference values, we generated time-dependent functions using cubic B-spline interpolation. For details, see our simulation code (link given below).

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 10⁶ 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 65×10⁸ cells and became clinically apparent. This cut-off corresponds to the assumption that a tumor with a volume of 1 cm³ contains 10⁸ tumor cells39 and that several primary tumors (eg, lung cancer, colon carcinoma, and renal cell carcinoma) are diagnosed as spherical structures with a median diameter of approximately 5 cm.40–42 The ‘lethal’ tumor burden of patients in these simulations is estimated at 10¹² 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.43 None of the patients received prior or subsequent immunotherapy. The serum lactate dehydrogenase levels at baseline (ie, before therapy) were analyzed as a surrogate marker for tumor growth. The ratio of intratumoral versus peritumoral T cell densities, obtained by immunohistochemical staining of the primary tumor, was selected as a surrogate marker for the T cell killing rate. Overall survival data were available for all patients.

We implemented our ODE model in C++. The Boost library ‘odeint’ was used to solve the system of ODEs.44 The code is available at GitHub: https://github.com/jeroencreemers/tipping-point-cancer-immune-dynamics. Analyses and visualizations were performed in R.

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 (figure 1A).

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 (figure 1B); (2) effector T cells were initially unable to inhibit tumor cell outgrowth but caught up and suppressed tumor growth to a balanced subclinical state (figure 1C); or (3) exponential tumor growth outpaced the immune system’s control and gave rise to a clinically detectable tumor (figure 1D). These three scenarios only led to two clinically different outcomes in patients: either a tumor became clinically evident, or the immune system could suppress or eradicate a tumor at an early stage (ie, before the tumor could reach a clinically detectable size). A balanced equilibrium state, in which the immune system keeps a clinically evident tumor under persistent control, does not exist in this deterministic version of our model.

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 (figure 2A). On the contrary, a critical threshold was present. Once the threshold was exceeded, the kinetics ‘flipped’ from a state of immune control (figure 2A, inset 1) to a state in which the tumor could evade immune control (figure 2A, inset 2).

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 (figure 2B). This coincided with the phenotypes ‘immune evasion’ (figure 2B, inset 1) and ‘immune control’ (figure 2B, inset 2).

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 (figure 2C). This ‘phase diagram’ shows that the tipping point is not only present for specific parameter values but is a fundamental property in our model. By contrast, the state of subclinical tumor control was not universally present around the tipping point (figure 2C, inset) but manifested itself only in a narrow range of parameters. Within both the ‘cure’ and ‘control’ domain (figure 2, inset), the immune system prevented tumors from reaching a detectable size, precluding the clinical classification as ‘patient’. The difference between individuals in the ‘cure’ and ‘control’ domains was that all tumor cells were eradicated in the former, while in the latter, the immune system kept the tumor in an undetectable subclinical state (ie, a tumor size of around 10³ tumor cells; figure 1B,C).

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 (online supplemental figure 1A). With T cell priming, tipping points became apparent. The location of the tipping point was affected by the priming rate. A higher priming rate facilitated improved tumor eradication through an increased influx of cytotoxic T cells into the tumor microenvironment (online supplemental figure 1B–D).

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; figure 3A). Given the unparalleled responses of advanced malignancies to immunotherapy, we focused on the consequences of a tipping point for responses to ICI, but these findings could be extended to other therapies as well. In this study, we limited the treatment effect of ICI to their primary mode of action: the augmentation of the T cell killing rate (figure 3A).

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 (figure 3B), and (2) the treatment effect needs to be sustained long enough for a patient to benefit from the treatment (figure 3C). The treatment effect was defined as the multiplication factor of the T cell killing rate. When both criteria were satisfied, ICI were able to induce a long-term survival benefit. However, if the treatment effect (anti-PD1 effect <12.6) or duration (less than ±5 months) proved inadequate, any survival benefit was only temporary, and inevitable tumor progression would ultimately limit overall survival (figure 3B,C; insets). These survival kinetics depend not solely on therapeutic features of ICI but rather on the interplay between patient and ICI characteristics. To illustrate this, we simulated twenty patients with identical immune systems (ie, identical T cell killing rates). In the absence of ICI therapy, variation in the tumor growth rate—that is, variation in the distance to a tipping point—led to a limited variation in survival (figure 3D; gray bars). When these same patients were treated with ICI, a survival benefit is induced in all patients. However, the extent of this benefit differs and depends on the distance to a tipping point. Following clinical observations, long-term survival is only induced in the subset of patients close to a tipping point (figure 3D; green bars). Similar findings were obtained in a population of patients with identical tumors but different immune systems. Without treatment, hardly any survival variation is present (figure 3E; gray bars). Again, treatment with ICI induced dichotomous clinical outcomes: a small survival benefit in most patients, with long-term survival in a subset (figure 3E; green bars). Hence, the mere presence of a tipping point yields heterogeneity in treatment outcomes.

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 (figure 4A) or immune systems (figure 4B) in identical patients compared with a static reference patient. Specifically, we varied the tumor growth rate and the T cell killing rate randomly over time (parameter values are included in online supplemental table 4). On reaching a diagnosable tumor volume, all patients in these examples were treated with ICI. As expected, stochastic dynamics prompted survival differences and induced a survival benefit in a subset of patients. In a heterogeneous patient population, this led to an interesting finding: the initial distance to a tipping point, along with the dynamics itself, determined the clinical outcome of patients treated with ICI (figure 4C, online supplemental figure 2). At population level, this led to a distinction between three subsets of patients: (1) patients far away from a tipping point with an unmodifiable bad prognosis (non-responders), (2) patients close to a tipping point with a favorable prognosis (responders), and most importantly, (3) patients in between these groups (potential responders). In the last subset, tumor dynamics ultimately determined the treatment response, and thereby the clinical outcome (figure 4C; gray box). A clinically important ramification of dynamic trajectories is that even if the subset to which a patient belongs is known at baseline, dynamics could alter the distance to a tipping point and, thereby, the prognosis of a patient. Therefore, it might be impossible to predict the prognosis solely based on characteristics measured at diagnosis. Dynamic trajectories can significantly diversify patient outcomes, meaning that continuous variation in the tumor growth rate (figure 4D) or T cell killing rate (figure 4E) leads to an entire spectrum of patient outcomes.

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 online supplemental table 5). We simplified the cohort by fixing the tumor and immune characteristics of these patients over time and assumed to have access to an entirely accurate biomarker (ie, no measurement error; figure 5A). Within this cohort, we predicted the prognosis of patients based on either the tumor or immune marker (the first and second columns of figure 5A, respectively). As is common in practice (though from a statistical point of view far from ideal), we dichotomized the biomarker using its median as a cut-off. Although survival differentiation based on these biomarkers alone was partially possible, it remained far from optimal. However, when we constructed a biomarker panel including both biomarkers, it highly accurately discriminated short-term from long-term survivors (third column of figure 5A). Note that despite variability in time from diagnosis, the initial plateau in the survival curves was caused by the fact that all tumors were diagnosed with identical sizes and immediately treated.

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 figure 5B). The combination of both markers in a biomarker panel increased the predictive capacity slightly, enabling the prediction of prognosis to some extent. However, in line with the notion of personalized medicine, the accurate and individualized prediction of prognosis based on baseline characteristics was not feasible in a significant subgroup of patients due to dynamic tumor-immune interactions (third column of figure 5B).

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 online supplemental table 6).43 We assessed whether a combination of two biomarkers would provide more information on a patient’s survival than either marker alone. Baseline lactate dehydrogenase (LDH) was selected as a surrogate marker for tumor growth, and the ratio between immunohistochemically determined intratumoral versus peritumoral (I/P) immune cells on the primary tumor was selected as an immune marker. We then used two different methods to measure the amount of information these markers provide on patient survival. First, we applied linear discriminant analysis to determine marker cut-off values that distinguish ‘short survivors’ (<9 months) from ‘long survivors’ (>9 months, corresponding to the median survival in the cohort). A cut-off based on the tumor marker LDH alone correctly classifies 71% of patients (figure 6A), which increased to 78% when using the I/P ratio as an immune marker instead. A combination of both markers achieves 86% accuracy, with the discrimination line following a roughly diagonal slope akin to the tipping point in our ‘in silico’ cohort (figure 2). Second, we compared Cox proportional hazard models based on LDH alone and I/P ratio alone to a model including both markers. Both LDH (likelihood ratio test: p=4×10⁻⁷) 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; online supplemental table 7) provided the best fit to the data as measured by a Bayesian Information Criterion, which was lower by 11.6 compared with the LDH-only model and by 20.7 compared with the I/P ratio-only model. The Kaplan-Meier plots shown in figure 6B illustrate the performance of each model by comparing the patients with the highest 50% estimated relative hazard to the lowest 50%. These results support our in silico-generated hypothesis that a combination of tumor and immune markers form a better basis for patient stratification than either marker on its own.

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.