Response of patients with melanoma to immune checkpoint blockade – insights gleaned from analysis of a new mathematical mechanistic model
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).
References (65)
Programmed death-1 ligand 1 interacts specifically with the B7-1 costimulatory molecule to inhibit T cell responses
Immunity
(2007)Progressive upregulation of PD-1 in primary and metastatic melanomas associated with blunted TCR signaling in infiltrating T lymphocytes
J. Invest. Dermatol.
(2011)Delay-induced oscillatory dynamics of tumour–immune system interaction
Math. Comput. Modell.
(2010)A general framework for modeling tumor-immune system competition and immunotherapy: mathematical analysis and biomedical inferences
Physica D
(2005)Metamodeling tumor–immune system interaction, tumor evasion and immunotherapy
Math. Comput. Modell.
(2008)New response evaluation criteria in solid tumours: revised RECIST guideline (version 1.1)
Eur. J. Cancer
(2009)Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis
Bull. Math. Biol.
(1994)Criterion of Hopf bifurcations without using eigenvalues
J. Math. Anal. Appl.
(1994)- et al.
Melanoma early detection
Hematol./Oncol. Clin.
(2009) Generalized form of Hurwitz-Routh criterion and Hopf bifurcation of higher order
Appl. Math. Lett.
(2002)
A Survey of Models for Tumor-Immune System Dynamics
Employing dynamical computational models for personalizing cancer immunotherapy
Expert Opin. Biol. Ther.
Human macrophages and dendritic cells can equally present MART-1 antigen to CD8+ T cells after phagocytosis of gamma-irradiated melanoma cells
PLoS One
Control of NK cell activation by immune checkpoint molecules
Int. J. Mol. Sci.
Examining the presentation of tumor-associated antigens on peptide-pulsed T2 cells
Oncoimmunology
The Qualitative Theory of Ordinary Differential Equations: an Introduction
Tumor doubling time of cutaneous melanoma and its metastasis
Am. J. Dermatopathol.
Tumor immunogenicity determines the effect of B7 costimulation on T cell-mediated tumor immunity
J. Exp. Med.
Tumor-infiltrating lymphocytes: apparently good for melanoma patients. But why?
Cancer Immunol. Immunother.
CRP identifies homeostatic immune oscillations in cancer patients: a potential treatment targeting tool?
J. Transl. Med.
Abstract LB-116: Myeloid derived suppressor cells and NK cells are correlated with clinical benefit and survival in advanced melanoma patients treated with PD-1 blocking antibodies nivolumab and pembrolizumab
In: Proceedings of the American Association for Cancer Research Annual Meeting 2018; 2018 Apr 14-18; Chicago, IL. Philadelphia (PA): AACR; Cancer Res 2018;78(13 Suppl): Abstract nr LB-116
Effective migration of antigen-pulsed dendritic cells to lymph nodes in melanoma patients is determined by their maturation state
Cancer Res.
Magnetic resonance tracking of dendritic cells in melanoma patients for monitoring of cellular therapy
Nat. Biotechnol.
Does the cell number 109 still really fit one gram of tumor tissue?
Cell Cycle
Complete coefficient criteria for five-dimensional hopf bifurcations, with an application to economic dynamics
J. Nonlinear Dyn. vol. 2015, Article ID 278234, 11 pages, 2015. https://doi.org/10.1155/2015/278234
The immunological synapse
Arthritis Res.
The immunological synapse
Cancer Immunol. Res.
Mathematical models for immunology: current state of the art and future research directions
Bull. Math. Biol.
Adjuvant Pembrolizumab versus placebo in resected stage III melanoma
N. Engl. J. Med.
Modeling cancer-immune responses to therapy
J. Pharmacokinet. Pharmacodyn.
Spontaneous regression of cancer
Ann. N. Y. Acad. Sci.
Asymptotic dynamics of some t-periodic one-dimensional model with application to prostate cancer immunotherapy
J. Math. Biol.
Cited by (17)
A mathematical model for tumor-immune competitive system with multiple time delays
2024, Chaos, Solitons and FractalsA tumor–immune interaction model with the effect of impulse therapy
2023, Communications in Nonlinear Science and Numerical SimulationTreatment of melanoma with dendritic cell vaccines and immune checkpoint inhibitors: A mathematical modeling study
2023, Journal of Theoretical BiologyStability and Hopf bifurcation of a tumor–immune system interaction model with an immune checkpoint inhibitor
2023, Communications in Nonlinear Science and Numerical SimulationCitation Excerpt :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.
Modeling the dynamics of mixed immunotherapy and chemotherapy for the treatment of immunogenic tumor
2024, European Physical Journal Plus