Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

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


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 to check access.


  1. 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

  2. 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

  3. 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

  4. 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

  5. 5

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

  6. 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

  7. 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

  8. 8

    Cheng X, Veverka V, Radhakrishnan A, et al. Human PD-L1/B7-H1/CD274 protein. Sino Biological Inc,

  9. 9

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

  10. 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

  11. 11

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

  12. 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

  13. 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

  14. 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

  15. 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

  16. 16

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

  17. 17

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

  18. 18

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

  19. 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

  20. 20

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

  21. 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

  22. 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

  23. 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

  24. 24

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

  25. 25

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

  26. 26

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

  27. 27

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

  28. 28

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

  29. 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

  30. 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

  31. 31

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

  32. 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

  33. 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

  34. 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

  35. 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

  36. 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

  37. 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

  38. 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

  39. 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

  40. 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

  41. 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

  42. 42

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

  43. 43

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

  44. 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

  45. 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

  46. 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

  47. 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

  48. 48

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

  49. 49

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

  50. 50

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

  51. 51

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

  52. 52

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

  53. 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

  54. 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

  55. 55


  56. 56


  57. 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

  58. 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

  59. 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

  60. 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

  61. 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

  62. 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

  63. 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

  64. 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

  65. 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

  66. 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

  67. 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

  68. 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

  69. 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

  70. 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

  71. 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

  72. 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

  73. 73

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

  74. 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

  75. 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

  76. 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

  77. 77

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

  78. 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

  79. 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

  80. 80

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

Download references


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

Correspondence to Avner Friedman.

Rights and permissions

Reprints and Permissions

About this article

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. (2020).

Download citation


  • mathematical model
  • anti-PD-L1
  • radiotherapy
  • combination therapy
  • scheduling


  • 35R35
  • 35Q92
  • 92C50
  • 35R37
  • 35K55
  • 92B05