Response of patients with melanoma to immune checkpoint blockade – insights gleaned from analysis of a new mathematical mechanistic model

https://doi.org/10.1016/j.jtbi.2019.110033Get rights and content

Highlights

  • Smaller tumors better respond to immunotherapy.

  • Ratio of activation-exhaustion rates of T cells modifies disease fate.

  • Large activation rate leads to stable disease.

  • Low exhaustion rate can cause oscillations in tumor size.

  • Analysis enables to identify sensitive personal parameters.

Abstract

Immune checkpoint inhibitors (ICI) are becoming widely used in the treatment of metastatic melanoma. However, the ability to predict the patient's benefit from these therapeutics remains an unmet clinical need. Mathematical models that predict melanoma patients’ response to ICI can contribute to better informed clinical decisions. Here, we developed a simple mathematical population model for pembrolizumab-treated advanced melanoma patients, and analyzed the local and global dynamics of the system. Our results show that zero, one, or two steady states of the mathematical system exist in the phase plane, depending on the parameter values of individual patients. Without treatment, the simulated tumors grew uncontrollably. At increased efficacy of the immune system, e.g., due to immunotherapy, two steady states were found, one leading to uncontrollable tumor growth, and the other resulting in tumor size stabilization. Model analysis indicates that a sufficient increase in the activation of CD8+ T cells results in stable disease, whereas a significant reduction in T-cell exhaustion, another process contributing CD8+ T cell activity, temporarily reduces the tumor mass, but fails to control disease progression in the long run. Importantly, the initial tumor burden influences the response to treatment: small tumors respond better to treatment than larger tumors. In conclusion, our model suggests that disease progression and response to ICI depend on the ratio between activation and exhaustion rates of CD8+ T cells. The analysis of the model provides a foundation for the use of computational methods to personalize immunotherapy.

Introduction

Advanced melanoma is the most deadly skin cancer. In 2015, 351,880 new cases were diagnosed worldwide, and 59,782 deaths were reported (Karimkhani et al., 2017). For 2018, a total of 91,279 new cases, and 9,320 deaths are expected in the United States (Siegel et al., 2018). Most early-detected melanomas are curable by resection (Terushkin and Halpern, 2009), whereas metastatic disease requires systemic treatment. Melanoma cells can stimulate host immunity by their high mutation burden, enabling recognition as non-self-antigens and activation of antigen presenting cells (APCs). The latter stimulate CD8+ T cells to differentiate into memory and cytotoxic effector CD8+ T cells (Dustin, 2014; Gattinoni et al., 2012). Melanoma can also suppress host immunity by expressing ligands which bind to regulatory receptors on activated immune cells - cytotoxic T lymphocyte-associated protein 4 (CTLA-4) and programmed cell death 1 (PD-1) - and inhibit immune activity (Butte et al., 2007; Chapon et al., 2011; Walker and Sansom, 2011).

Treatment of melanoma has been revolutionized with the approval of ICIs. For example, treatments based on pembrolizumab and nivolumab, inhibitors of the immune checkpoint receptor, programmed cell death 1 (PD-1), greatly improved prognosis in metastatic disease (Robert et al., 2015). However, even though ICIs can induce durable response in some patients (Ott et al., 2013; Prieto et al., 2012), the overall response rate to these drugs is still modest (Hamid et al., 2013; Hodi et al., 2010) and reliable markers to predict treatment efficacy are still under development (Wang et al., 2012; Weide et al., 2016)

The complex interplay of cancer cells and the host immune system, as affected by immunotherapy, renders the reasoning of treatment causality difficult. Having the capacity to succinctly integrate this interplay in one coherent framework, and enable its analysis, mathematical models may become instrumental in predicting the interactive dynamics of host immunity, cancer progression and immunotherapy, thereby providing a new powerful tool for treatment personalization. Mathematical models have been previously employed for studying the interactions of cancer with the immune system, for investigating the response to different immunotherapies, and for patient-specific regimen personalization (Adam and Bellomo, 2012; Agur et al., 2016; d'Onofrio, 2005; d'Onofrio, 2008; Eftimie et al., 2016; Eladdadi and Radunskaya, 2014; Foryś et al., 2016; Kogan et al., 2012; Kronik et al., 2010). While suited to their specific aims, none of these models included cellular immunity in a way that enables analysis of treatment by ICI. In particular, previous models did not address the recently discovered effects of Effector T cell exhaustion on the treatment. The elucidation of these effects are crucial for evaluating the efficacy of ICI and, therefore, are introduced into the mathematical model developed and analyzed in this work.

In this study, we developed a mathematical mechanistic model for the interactions of melanoma cells with the host immune system, and analyzed the effects ICIs have on this interplay. Our study indicates that different potential immunotherapy strategies, which are expected to enhance the efficacy of CD8+ T cells, result in distinct tumor dynamics and disease fates.

Section snippets

Mathematical model

Our mathematical model simplifies the overall system to its main driving forces, namely, melanoma cells, antigen-presenting cells (APCs), and effector CD8+ T cells. It takes into account the following assumptions about the involved dynamics:

  • 1.

    Mutated tumor cells express non-self-antigens and activate APCs. The number of activated functional APCs, denoted A, depends on the tumor immunogenicity (Chen et al., 1994; Rizvi et al., 2015; Schumacher and Schreiber, 2015; Snyder et al., 2014), which is

Defining a biologically relevant domain for model analysis

Asymptotic solutions for the system defined in Eq. (1) are of interest, since they indicate the potential fates of the system. For example, a positive, steadily growing number of cancer cells over time suggests inability to cure the disease. Additional potential solutions include, for instance, a decrease in the number of tumor cells down to a constant amount, indicating shrinkage and tumor stabilization thereafter. Further solutions may present oscillations that might indicate alternating

Discussion

In this article we showed by mathematical modeling and analysis within the bio-medically relevant parameter ranges, that the ratio between activation and exhaustion rates of CD8+ T cells can determine the outcomes of melanoma immunotherapy. Based on our results, we suggest to evaluate T cell activation and exhaustion rates in individual patients for improving the prediction accuracy of their response to treatment.

Our theoretical and numerical analyses suggest that under realistic assumptions on

Acknowledgments

We thank Dr. Moran Elishmereni from Optimata Ltd. for sharing her knowledge about parameter estimation techniques. This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No. 642295 (MEL-PLEX).

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      Several mathematical modeling efforts of immune checkpoint inhibitors anti-PD-1 and anti-PD-L1 have been made over the last few years. Some models focus on monotherapies [16–20], others concentrate on combinations with other therapies, such as radiotherapy (RT), oncolytic virus therapy, and DC vaccines [21–24]. Serre et al. [24] first proposed a discrete time pharmacokinetic model, investigated the effects of combining ICIs (anti-PD-1, anti-PD-L1, anti-CTLA-4) with radiotherapy, showed that synergistic effects occur between the combination treatments, and validated results of various experiments.

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