Skip to main content
SpringerLink
Log in

Mathematical modeling of cancer treatment with radiation and PD-L1 inhibitor

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

Radiation therapy is a longstanding cancer treatment. More recently, it has been demonstrated that radiation therapy (RT) elicits anti-cancer immune response. For this reason, there is a growing interest in combining RT with immunotherapy, specifically with checkpoint inhibitors such as anti-CTLA-4 and anti-PD-L1. In the present paper, we develop a mathematical model of combination therapy with RT and anti-PD-L1. The model is used to compare different schedules in clinical trials. Simulations of the model show that applying both RT and anti-PD-L1 at the same week has more benefits than applying them in separate adjacent weeks. Furthermore, applying anti-PD-L1 before RT has more benefits than applying RT before anti-PD-L1.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agata Y, Kawasaki A, Nishimura H, et al. Expression of the PD-1 antigen on the surface of stimulated mouse T and B lymphocytes. Int Immunol, 1996, 124: 765–772

    Google Scholar 

  2. Azad A, Lim S Y, D’Costa Z, et al. PD-L1 blockade enhances response of pancreatic ductal adenocarcinoma to radiotherapy. EMBO Mol Med, 2017, 124: 167–180

    Google Scholar 

  3. Brahmer J R, Tykodi S S, Chow L Q, et al. Safety and activity of anti-PD-L1 antibody in patients with advanced cancer. N Engl J Med, 2012, 124: 2455–2465

    Google Scholar 

  4. Buttea M J, Pena-Cruzc V, Kima M-J, et al. Interaction of human PD-L1 and B7-1. Mol Immunol, 2008, 124: 3567–3572

    Google Scholar 

  5. Cantelli G, Crosas-Molist E, Georgouli M, et al. TGFB-induced transcription in cancer. Semin Cancer Biol, 2017, 124: 60–69

    Google Scholar 

  6. Chen D, Bobko A A, Gross A C, et al. Involvement of tumor macrophage HIFs in chemotherapy effectiveness: Mathematical modeling of oxygen, pH, and glutathione. PLoS One, 2014, 124: e107511

    Google Scholar 

  7. Cheng X, Veverka V, Radhakrishnan A, et al. Structure and interactions of the human programmed cell death 1 receptor. J Biol Chem, 2013, 124: 11771–11785

    Google Scholar 

  8. Cheng X, Veverka V, Radhakrishnan A, et al. Human PD-L1/B7-H1/CD274 protein. Sino Biological Inc, http://www.sinobiological.com/PD-L1-B7-H1-CD274-Protein-g-533.html

  9. Condamine T, Gabrilovich D I. Molecular mechanisms regulating myeloid-derived suppressor cell differentiation and function. Trends Immunol, 2011, 124: 19–25

    Google Scholar 

  10. Czarkowska-Paczek B, Bartlomiejczyk I, Przybylski J. The serum levels of growth factors: PDGF, TGF-beta and VEGF are increased after strenuous physical exercise. J Physiol Pharmacol, 2006, 124: 189–189

    Google Scholar 

  11. D’Acunto B. Computational Methods for PDE in Mechanics. Series on Advances in Mathematics for Applied Sciences, vol. 67. Singapore: Word Scientific, 2004

    Google Scholar 

  12. Daly M E, Monjazeb A M, Kelly K. Clinical trials integrating immunotherapy and radiation for non-small-cell lung cancer. J Thorac Oncol, 2015, 124: 1685–1693

    Google Scholar 

  13. Deng L, Liang H, Burnette B, et al. Irradiation and anti-PD-L1 treatment synergistically promote antitumor immunity in mice. J Clin Invest, 2014, 124: 687–695

    Google Scholar 

  14. Dovedi S J, Adlard A L, Lipowska-Bhalla G, et al. Acquired resistance to fractionated radiotherapy can be overcome by concurrent PD-L1 blockade. Cancer Res, 2014, 124: 5458–5468

    Google Scholar 

  15. Eckert F, Gaipl U, Niedermann G, et al. Beyond checkpoint inhibition Immunotherapeutical strategies in combination with radiation. Clin Transl Radiat Oncol, 2017, 124: 29–35

    Google Scholar 

  16. Enderling H, Chaplain M A J, Hahnfeldt P. Quantitative modeling of tumor dynamics and radiotherapy. Acta Biotheor, 2010, 124: 341–353

    Google Scholar 

  17. Escorcia F E, Postow M A, Barker C A. Radiotherapy and immune checkpoint blockade for melanoma. Cancer J, 2017, 124: 32–39

    Google Scholar 

  18. Friedman A, Hao W. The role of exosomes in pancreatic cancer microenvironment. Bull Math Biol, 2017, 124: 1111–1133

    MathSciNet  MATH  Google Scholar 

  19. Hamza T, Barnett J B, Li B. Interleukin 12 a key immunoregulatory cytokine in infection applications. Int J Mol Sci, 2010, 124: 789–806

    Google Scholar 

  20. Hao W, Friedman A. Mathematical model on Alzheimer’s disease. BMC Syst Biol, 2016, 124: 1–18

    Google Scholar 

  21. Hu Z I, Ho A Y, Mcarthur H L. Combined radiation therapy and immune checkpoint blockade therapy for breast cancer. Int J Radiat Oncol, 2017, 124: 153–164

    Google Scholar 

  22. Itakura E, Huang R-R, Wen D-R, et al. IL-10 expression by primary tumor cells correlates with melanoma progression from radial to vertical growth phase and development of metastatic competence. Mod Pathol, 2011, 124: 801–809

    Google Scholar 

  23. Jafarzadeh A, Minaee K, Farsinejad A-R, et al. Evaluation of the circulating levels of IL-12 and IL-33 in patients with breast cancer: Influences of the tumor stages and cytokine gene polymorphisms. Iran J Basic Med Sci, 2015, 124: 1189–1198

    Google Scholar 

  24. Janco J M T, Lamichhane P, Karyampudi L, et al. Tumor-infiltrating dendritic cells in cancer pathogenesis. J Immunol, 2015, 124: 2985–2991

    Google Scholar 

  25. Kaminska B, Wesolowska A, Danilkiewicz M. TGF beta signalling and its role in tumour pathogenesis. Acta Biochim Pol, 2005, 124: 329–337

    Google Scholar 

  26. Kang J, Demaria S, Formenti S. Current clinical trials testing the combination of immunotherapy with radiotherapy. J Immunother Cancer, 2016, 124: 1–20

    Google Scholar 

  27. Kaur P, Asea A. Radiation-induced effects and the immune system in cancer. Front Onco, 2012, 124: 191

    Google Scholar 

  28. Krüger-Krasagakes S, Krasagakis K, Garbe C, et al. Expression of interleukin 10 in human melanoma. Brit J Cancer, 1994, 124: 1182–1185

    Google Scholar 

  29. Lai X, Friedman A. Combination therapy of cancer with BRAF inhibitor and immune checkpoint inhibitor: A mathematical model. BMC Syst Biol, 2017, 124: 70

    Google Scholar 

  30. Lai X, Stiff A, Duggan M, et al. Modeling combination therapy for breast cancer with BET and immune checkpoint inhibitors. Proc Natl Acad Sci USA, 2018, 124: 5534–5539

    Google Scholar 

  31. Lawrence Y R, Dicker A P. Radiation therapy and the immune system: Learning to live together. Future Oncol, 2014, 124: 777–780

    Google Scholar 

  32. Lee E-J, Lee S J, Kim J-H, et al. Radiation inhibits interleukin-12 production via inhibition of C-rel through the interleukin-6/signal transducer and activator of transcription 3 signaling pathway in dendritic cells. PLoS One, 2016, 124: e0146463

    Google Scholar 

  33. Li H-H, Wang Y-W, Chen R, et al. Ionizing radiation impairs T cell activation by affecting metabolic reprogramming. Int J Biol Sci, 2015, 124: 726–736

    Google Scholar 

  34. Liao K L, Bai X F, Friedman A. Mathematical modeling of interleukin-27 induction of anti-tumor T cells response. PLoS One, 2014, 124: e91844

    Google Scholar 

  35. Liniker E, Menzies A, Kong B, et al. Activity and safety of radiotherapy with anti-PD-1 drug therapy in patients with metastatic melanoma. Oncoimmunology, 2016, 124: e1214788

    Google Scholar 

  36. Lisiero D N, Soto H, Liau L M, et al. Enhanced sensitivity to IL-2 signaling regulates the clinical responsiveness of IL-12Cprimed CD8 T cells in a melanoma model. J Immunol, 2011, 124: 5068–5077

    Google Scholar 

  37. Liu S, Sun X, Luo J, et al. Effects of radiation on T regulatory cells in normal states and cancer: Mechanisms and clinical implications. Am J Cancer Res, 2015, 124: 3276–3285

    Google Scholar 

  38. Lo W-C, Arsenescu V, Arsenescu R I, et al. Inflammatory Bowel disease: How effective is TNF-alpha suppression? PLoS One, 2016, 124: e0165782

    Google Scholar 

  39. Lonergan D M, Mikulec A A, Hanasono M M, et al. Growth factor profile of irradiated human dermal fibroblasts using a serum-free method. Plast Reconstr Surg, 2003, 124: 1960–1968

    Google Scholar 

  40. Longoria T C, Tewari K S. Evaluation of the pharmacokinetics and metabolism of pembrolizumab in the treatment of melanoma. Expert Opin Drug Metab Toxicol, 2016, 124: 1247–1253

    Google Scholar 

  41. Lowther D E, Goods B A, Lucca L E, et al. PD-1 marks dysfunctional regulatory T cells in malignant gliomas. JCI Insight, 2016, 124: e85935

    Google Scholar 

  42. Ma Y, Shurin1 G V, Peiyuan Z, et al. Dendritic Cells in the Cancer Microenvironment. J Cancer, 2013, 124: 36–44

    Google Scholar 

  43. Maggio F D, Minafra L, Forte G, et al. Portrait of inflammatory response to ionizing radiation treatment. J Inflamm, 2015, 124: 14

    Google Scholar 

  44. Manda K, Glasow A, Paape D, et al. Effects of ionizing radiation on the immune system with special emphasis on the interaction of dendritic and T cells. Front Onco, 2012, 124: 102

    Google Scholar 

  45. Marino S, Hogue I, Ray C, et al. A methodology for performing global uncertainty and sensitivity analysis in systems biology. J Theo Biol, 2008, 124: 178–196

    MathSciNet  MATH  Google Scholar 

  46. Mautea R L, Gordona S R, Mayere A T, et al. Engineering high-affinity PD-1 variants for optimized immunotherapy and immuno-PET imaging. Proc Natl Acad Sci USA, 2015, 124: E6506–14

    Google Scholar 

  47. Merrick A, Errington F, Milward K, et al. Immunosuppressive effects of radiation on human dendritic cells: Reduced IL-12 production on activation and impairment of näive T-cell priming. Brit J Cancer, 2005, 124: 1450–1458

    Google Scholar 

  48. Meziani L, Deutsch E, Mondini M. Macrophages in radiation injury: A new therapeutic target. Oncoimmunology, 2018, 124: e1494488

    Google Scholar 

  49. Munn D H, Mellor A L. IDO in the tumor microenvironment: Inflammation, counter-regulation, and tolerance. Trends Immunol, 2016, 124: 193–207

    Google Scholar 

  50. Muppidi M R, George S. Immune checkpoint inhibitors in renal cell carcinoma. J Target Ther Cancer, 2015, 124: 47–52

    Google Scholar 

  51. Ott P A, Hodi F S, Kaufman H L, et al. Combination immunotherapy: A road map. J Immunother Cancer, 2017, 124: 16

    Google Scholar 

  52. Palucka J, Banchereau J. Cancer immunotherapy via dendritic cells. Nat Rev Cancer, 2012, 124: 265–277

    Google Scholar 

  53. Perrot C Y, Javelaud D, Mauviel A. Insights into the transforming growth factor-beta signaling pathway in cutaneous melanoma. Ann Dermatol, 2013, 124: 135–144

    Google Scholar 

  54. Persa E, Balogh A, Safrany G, et al. The effect of ionizing radiation on regulatory T cells in health and disease. Cancer Lett, 2015, 124: 252–261

    Google Scholar 

  55. PhosphoSitePlus/IL2(human). http://www.phosphosite.org/proteinAction?id=14691&showAllSites=true

  56. PhosphoSitePlus/IL10(human). http://www.phosphosite.org/proteinAction.action?id=2473887

  57. Pinto A T, Pinto M L, Cardoso A P, et al. Ionizing radiation modulates human macrophages towards a pro-inflammatory phenotype preserving their pro-invasive and pro-angiogenic capacities. Sci Rep, 2016, 124: 18765

    Google Scholar 

  58. Poniatowski L A, Wojdasiewicz P, Gasik R, et al. Transforming growth factor beta family: Insight into the role of growth factors in regulation of fracture healing biology and potential clinical applications. Mediat Inflamm, 2015, 124: 137823

    Google Scholar 

  59. Rockne R, Alvord C E, Rockhill J K, et al. A mathematical model for brain tumor response to radiation therapy. J Math Biol, 2009, 124: 561–578

    MathSciNet  MATH  Google Scholar 

  60. Roses R E, Datta J, Czerniecki B J. Radiation as immunomodulator: Implications for dendritic cell-based immunother-apy. Radiat Res, 2014, 124: 211–218

    Google Scholar 

  61. Sachs K, Hahnfeld P, Bre D J. The link between low-LET dose-response relations and the underlying kinetics of damage production/repair/misrepair. Int J Radiat Biol, 1997, 124: 351–374

    Google Scholar 

  62. Saenz R, Futalan D, Leutenez L, et al. TLR4-dependent activation of dendritic cells by an HMGB1-derived peptide adjuvant. J Transl Med, 2014, 124: 1–11

    Google Scholar 

  63. Safarzadeh E, Hashemzadeh S, Duijf P H, et al. Circulating myeloid-derived suppressor cells: An independent prognostic factor in patients with breast cancer. J Cell Physiol, 2018, 124: 3515–3525

    Google Scholar 

  64. Santibanez J F, Quintanilla M, Bernabeu C. TGF-beta/TGF-beta receptor system and its role in physiological and pathological conditions. Clin Sci, 2011, 124: 233–251

    Google Scholar 

  65. Sharabi A B, Lim M, DeWeese T L, et al. Radiation and checkpoint blockade immunotherapy: Radiosensitisation and potential mechanisms of synergy. Lancet Oncol, 2015, 124: e498–e509

    Google Scholar 

  66. Shi L, Chen S, Yang L, et al. The role of PD-1 and PD-L1 in T-cell immune suppression in patients with hematological malignancies. J Hematol Oncol, 2013, 124: 74

    Google Scholar 

  67. Shimizu T, Seto T, Hirai F, et al. Phase 1 study of pembrolizumab (MK-3475; anti-PD-1 monoclonal antibody) in Japanese patients with advanced solid tumors. Invest New Drug, 2016, 124: 347–354

    Google Scholar 

  68. Shui Y B, Wang X, Hu J S, et al. Vascular endothelial growth factor expression and signaling in the lens. Invest Ophthalmol Vis Sci, 2003, 124: 3911–3919

    Google Scholar 

  69. Sindoni A, Minutoli F, Ascenti G, et al. Combination of immune checkpoint inhibitors and radiotherapy: Review of the literature. Crit Rev Oncol Hemat, 2017, 124: 63–70

    Google Scholar 

  70. Teng F, Kong L, Meng X, et al. Radiotherapy combined with immune checkpoint blockade immunotherapy: Achievements and challenges. Cancer Lett, 2015, 124: 23–29

    Google Scholar 

  71. Umansky V, Blattner C, Gebhardt C, et al. The role of myeloid-derived suppressor cells (MDSC) in cancer progression. Vaccines, 2016, 124: 1–16

    Google Scholar 

  72. Vescovi R, Monti M, Moratto D, et al. Collapse of the plasmacytoid dendritic cell compartment in advanced cutaneous melanomas by components of the tumor cell secretome. Cancer Immunol Res, 2018, 124: 12–28

    Google Scholar 

  73. Wang J-S, Wang H-J, Qian H-L. Biological effects of radiation on cancer cells. Mil Med Res, 2018, 124: 1–10

    Google Scholar 

  74. Wang W, Green M, Liu J R, et al. CD8+ T cells in immunotherapy, radiotherapy, and chemotherapy. In: Zitvogel L, Kroemer G, eds. Oncoimmunology. Cham: Springer, 2018, 23–39

    Google Scholar 

  75. Watanabe Y, Dahlman E L, Leder K Z, et al. A mathematical model of tumor growth and its response to single irradiation. Theor Biol Med Model, 2016, 124: 6

    Google Scholar 

  76. Whitehouse G, Gray E, Mastoridis S, et al. IL-2 therapy restores regulatory T-cell dysfunction induced by calcineurin inhibitors. Proc Natl Acad Sci USA, 2017, 124: 7083–7088

    Google Scholar 

  77. Whiteside T L. The role of regulatory T cells in cancer immunology. Immunotargets Ther, 2015, 124: 159–171

    Google Scholar 

  78. Wu Q, Allouch A, Martins I, et al. Macrophage biology plays a central role during ionizing radiation-elicited tumor response. Biomed J, 2017, 124: 200–211

    Google Scholar 

  79. Young K H, Baird J R, Savage T, et al. Optimizing timing of immunotherapy improves control of tumors by hypofrac-tionated radiation therapy. PLoS One, 2016, 124: e0157164

    Google Scholar 

  80. Young M E. Estimation of diffusion coefficients of proteins. Biotechnol Bioeng, 1980, 124: 947–955

    Google Scholar 

Download references

Acknowledgements

The first author was supported by the Fundamental Research Funds for the Central Universities (Grant No. 19XNLG14), the Research Funds of Renmin University of China, and National Natural Science Foundation of China (Grant Nos. 11501568 and 11571364). The authors thank Mathematical Biosciences Institute for the support of this collaboration.

Author information

Authors and Affiliations

  1. Institute for Mathematical Sciences, Renmin University of China, Beijing, 100872, China

    Xiulan Lai

  2. Mathematical Bioscience Institute & Department of Mathematics, Ohio State University, Columbus, OH, 43210, USA

    Avner Friedman

Authors
  1. Xiulan Lai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Avner Friedman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Avner Friedman.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lai, X., Friedman, A. Mathematical modeling of cancer treatment with radiation and PD-L1 inhibitor. Sci. China Math. 63, 465–484 (2020). https://doi.org/10.1007/s11425-019-1648-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-019-1648-6

Keywords

MSC(2010)

Search

Navigation