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doi: 10.3934/dcdsb.2020138

Mathematical modeling of an immune checkpoint inhibitor and its synergy with an immunostimulant

Elpiniki Nikolopoulou 1, , Steffen E. Eikenberry 1, , Jana L. Gevertz 2, and  Yang Kuang 3,,

1. 

School of Mathematical and Statistical Sciences, Arizona state University, Tempe, AZ 85287-1804, USA
2. 

Department of Mathematics and Statistics, The College of New Jersey, Ewing, NJ, USA
3. 

School of Mathematical and Statistical Sciences, Arizona state University, Tempe, AZ 85287-1804, USA

* Corresponding author: Yang Kuang

Received  July 2019 Revised  December 2019 Published  April 2020

Immune checkpoint inhibitors (ICIs) are a novel cancer therapy that may induce tumor regression across multiple types of cancer. There has recently been interest in combining the ICIs with other forms of treatments, as not all patients benefit from monotherapy. We propose a mathematical model consisting of ordinary differential equations to investigate the combination treatments of the ICI avelumab and the immunostimulant NHS-muIL12. We validated the model using the average tumor volume curves provided in Xu et al. (2017). We initially analyzed a simple generic model without the use of any drug, which provided us with mathematical conditions for local stability for both the tumorous and tumor-free equilibrium. This enabled us to adapt these conditions for special cases of the model. Additionally, we conducted systematic mathematical analysis for the case that both drugs are applied continuously. Numerical simulations suggest that the two drugs act synergistically, such that, compared to monotherapy, only about one-third the dose of both drugs is required in combination for tumor control.

Keywords: Immune checkpoint inhibitors, PD-L1, immunostimulant, synergy, combination treatment, mathematical analysis, mathematical model.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Elpiniki Nikolopoulou, Steffen E. Eikenberry, Jana L. Gevertz, Yang Kuang. Mathematical modeling of an immune checkpoint inhibitor and its synergy with an immunostimulant. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020138
References:
[1]

H. O. Alsaab, S. Sau, R. Alzhrani, K. Tatiparti, K. Bhise, S. K. Kashaw and A. K. Iyer, PD-1 and PD-L1 checkpoint signaling inhibition for cancer immunotherapy: Mechanism, combinations, and clinical outcome, Frontiers in Pharmacology, 8 (2017), https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5572324/. doi: 10.3389/fphar.2017.00561.  Google Scholar

[2]

R. H. Blair, D. L. Trichler and D. P. Gaille, Mathematical and statistical modeling in cancer systems biology, Frontiers in Physiology, 3 (2012), https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3385354/. doi: 10.3389/fphys.2012.00227.  Google Scholar

[3]

R. H. Byrd, J. C. Gilbert and J. Nocedal, A trust region method based on interior point techniques for nonlinear programming, Mathematical Programming, 89 (2000), 149-185.  doi: 10.1007/PL00011391.  Google Scholar

[4]

K. Chin, V. K. Chand and D. S. A. Nuyten, Avelumab: Clinical trial innovation and collaboration to advance anti-PD-L1 immunotherapy, Annals of Oncology: Official Journal of the European Society for Medical Oncology, 28 (2017), 1658–1666, https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5834034/. doi: 10.1093/annonc/mdx170.  Google Scholar

[5]

S. Halle, K. A. Keyser, F. R. Stahl, A. Busche, A. Marquardt, X. Zheng, M. Galla, V. Heissmeyer, K. Heller, J. Boelter, K. Wagner, Y. Bischoff, R. Martens, A. Braun, K. Werth, A. Uvarovskii, H. Kempf, M. Meyer-Hermann, R. Arens, M. Kremer, G. Sutter, M. Messerle and Reinhold Förster, In vivo killing capacity of cytotoxic T Cells is limited and involves dynamic interactions and T Cell cooperativity, \emphImmunity, 44 (2016), 233-245, https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4846978/. doi: 10.1016/j.immuni.2016.01.010.  Google Scholar

[6]

L. F. Han, S. Eikenberry, C. H. He, L. Johnson, M. C. Preul, E. J. Kostelich and Y. Kuang, Patient-specific parameter estimates of glioblastoma multiforme growth dynamics from a model with explicit birth and death rates, Mathematical Biosciences and Engineering, 16 (2019), 5307–5323, https://www.aimspress.com/fileOther/PDF/MBE/mbe-16-05-265.pdf. doi: 10.3934/mbe.2019265.  Google Scholar

[7]

V. R. Juneja, K. A. McGuire, R. T. Manguso, M. W. LaFleur, N. Collins, W. N. Haining, Gordon J. Freeman and Arlene H. Sharpe, PD-L1 on tumor cells is sufficient for immune evasion in immunogenic tumors and inhibits CD8 T cell cytotoxicity, The Journal of Experimental Medicine, 214 (2017), 895–904, https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5379970/. doi: 10.1084/jem.20160801.  Google Scholar

[8]

J. Kang, S. Demaria and S. Formenti, Current clinical trials testing the combination of immunotherapy with radiotherapy, Journal for Immunotherapy of Cancer, 4 (2016), https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5028964/. doi: 10.1186/s40425-016-0156-7.  Google Scholar

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V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis, Bulletin of Mathematical Biology, 56 (1994), 295–321, https://www.sciencedirect.com/science/article/pii/S0092824005802605. Google Scholar

[11]

X. L. Lai and A. Friedman, Combination therapy of cancer with cancer vaccine and immune checkpoint inhibitors: A mathematical model, PLoS ONE, 12 (2017), 1-24.  doi: 10.1371/journal.pone.0178479.  Google Scholar

[12]

T. List and D. Neri, Immunocytokines: A review of molecules in clinical development for cancer therapy, Clinical Pharmacology, 5 (2013), 29–45, https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3753206/. doi: 10.2147/CPAA.S49231.  Google Scholar

[13]

X.-Y. Meng, N.-N. Qin and H.-F. Huo, Dynamics analysis of a predator-prey system with harvesting prey and disease in prey species, Journal of Biological Dynamics, 12 (2018), 342-374.  doi: 10.1080/17513758.2018.1454515.  Google Scholar

[14]

E. Nikolopoulou, L. R. Johnson, D. Harris, J. D. Nagy, E. C. Stites and Y. Kuang, Tumour-immune dynamics with an immune checkpoint inhibitor, Letters in Biomathematics, 5 (2018), S137–S159. doi: 10.1080/23737867.2018.1440978.  Google Scholar

[15]

Z. P. Parra-Guillen, A. Janda, P. Alzuguren, P. Berraondo, R. Hernandez-Alcoceba and I. F. Troconiz, Target-mediated disposition model describing the dynamics of IL12 and IFN$\gamma$ after administration of a mifepristone-inducible adenoviral vector for IL-12 expression in mice, AAPS Journal, 15 (2013), 183–194, https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3535095/. Google Scholar

[16]

A. Radunskaya, R. Kim and T. Woods II, Mathematical modeling of tumor immune interactions: A closer look at the role of a PD-L1 inhibitor in cancer immunotherapy, Spora: A Journal of Biomathematics, 4 (2018), 25–41, https://ir.library.illinoisstate.edu/cgi/viewcontent.cgi?article=1022&context=spora. doi: 10.30707/SPORA4.1Radunskaya.  Google Scholar

[17]

A. Rao and M. R. Patel, A review of avelumab in locally advanced and metastatic bladder cancer, Therapeutic Advances in Urology, 11 (2019), https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6354303/. doi: 10.1177/1756287218823485.  Google Scholar

[18]

E. A. Saparata and L. G. Pillis, A Comparison and catalog of intrinsic tumor growth models, Bulletin of Mathematical Biology, 76 (2014), 2010–2024, https://www.ncbi.nlm.nih.gov/pubmed/25081547. doi: 10.1007/s11538-014-9986-y.  Google Scholar

[19]

R. Serre, S. Benzekry, L. Padovani, C. Meille, N. André, J. Ciccolini, F. Barlesi, X. Muracciole and D. Barbolosi, Mathematical modeling of cancer immunotherapy and its synergy with radiotherapy, Cancer Research, 76 (2016), 4931–4940, http://cancerres.aacrjournals.org/content/76/17/4931. doi: 10.1158/0008-5472.CAN-15-3567.  Google Scholar

[20]

L. Shi, S. Chen, L. Yang and Y. Li, The role of PD-1 and PD-L1 in T-cell immune suppression in patients with hematological malignancies., Journal of Hematology and Oncology, 6 (2013), https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3851976/. doi: 10.1186/1756-8722-6-74.  Google Scholar

[21]

S. Shi, J. Huang and Y. Kuang, Global dynamics in a tumor-immune model with an immune checkpoint inhibitor, DCDS-B, in review. Google Scholar

[22]

S. Simon and N. Labarriere, PD-1 expression on tumor-specific T cells: Friend or foe for immunotherapy?, \emphOncoimmunology, 7 (2017), https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5739549/. Google Scholar

[23]

A. Stéphanou, P. Ballet and G. Powathil, Hybrid modelling in oncology: Success, challenges and hopes, (2019), https://arXiv.org/pdf/1901.05652.pdf. Google Scholar

[24]

A. Talkington and R. Durett, Estimating tumor growth rates in vivo, Bulletin of Mathematical Biology, 77 (2015), 1934–1954, https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4764475/. doi: 10.1007/s11538-015-0110-8.  Google Scholar

[25]

S. G. Tan, K. F. Liu, Y. Chai, C. W.-H. Zhang, S. Gao, G. F. Gao and J. X. Qi, Distinct PD-L1 binding characteristics of therapeutic monoclonal antibody durvalumab, Protein and Cell, 9 (2018), 135-139.  doi: 10.1007/s13238-017-0412-8.  Google Scholar

[26]

J. Tang, J. X. Yu, V. M. Hubbard-Lucey, S. T. Neftelinov, J. P. Hodge and Y. Q. Lin, The clinical trial landscape for PD1/PDL1 immune checkpoint inhibitors, Nature Reviews Drug Discovery, 17 (2018), 854-855.  doi: 10.1038/nrd.2018.210.  Google Scholar

[27]

J. R. Wares, J. J. Crivelli, C. Yun, I. Choi, J. L. Gevertz and P. S. Kim, Treatment strategies for combining immunostimulatory oncolytic virus therapeutics with dendritic cell injections, Mathematical Biosciences and Engineering, 12 (2015), 1237-125.  doi: 10.3934/mbe.2015.12.1237.  Google Scholar

[28]

A. B. Warner and M. A. Postow, Combination controversies: Checkpoint inhibition alone or in combination for the treatment of melanoma?, \emphOncology, 32 (2018), 228–34, https://www.ncbi.nlm.nih.gov/pubmed/29847853. Google Scholar

[29]

C. X. Xu, Y. P. Zhang, P. A. Rolfe, V. M. Hernández, W. Guzman, G. Kradjian, B. Marelli, G. Qin, J. Qi, H. Wang, H. Yu, R. Tighe, K. Lo, J. M. English, L. Radvanyi and Y. Lan, Combination therapy with NHS-muIL12 and avelumab (anti-PD-L1) enhances antitumor efficacy in preclinical cancer models, Clinical Cancer Research, 23 (2017), 5869–5880, http://clincancerres.aacrjournals.org/content/23/19/5869. doi: 10.1158/1078-0432.CCR-17-0483.  Google Scholar

[30]

Y. Y. Yan, A. B. Kumar, H. Finnes, S. N. Markovic, S. Park, R. S. Dronca and H. Dong, Combining immune checkpoint inhibitors with conventional cancer therapy, Frontiers in Immunology, 9 (2018), https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6072836/. doi: 10.3389/fimmu.2018.01739.  Google Scholar

[31]

D. Zamarin and M. A. Postow, Immune checkpoint modulation: Rational design of combination strategies, Pharmacology and Therapeutics, 150 (2015), 23–32, https://www.sciencedirect.com/science/article/pii/S0163725815000042. doi: 10.1016/j.pharmthera.2015.01.003.  Google Scholar

show all references

References:
[1]

H. O. Alsaab, S. Sau, R. Alzhrani, K. Tatiparti, K. Bhise, S. K. Kashaw and A. K. Iyer, PD-1 and PD-L1 checkpoint signaling inhibition for cancer immunotherapy: Mechanism, combinations, and clinical outcome, Frontiers in Pharmacology, 8 (2017), https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5572324/. doi: 10.3389/fphar.2017.00561.  Google Scholar

[2]

R. H. Blair, D. L. Trichler and D. P. Gaille, Mathematical and statistical modeling in cancer systems biology, Frontiers in Physiology, 3 (2012), https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3385354/. doi: 10.3389/fphys.2012.00227.  Google Scholar

[3]

R. H. Byrd, J. C. Gilbert and J. Nocedal, A trust region method based on interior point techniques for nonlinear programming, Mathematical Programming, 89 (2000), 149-185.  doi: 10.1007/PL00011391.  Google Scholar

[4]

K. Chin, V. K. Chand and D. S. A. Nuyten, Avelumab: Clinical trial innovation and collaboration to advance anti-PD-L1 immunotherapy, Annals of Oncology: Official Journal of the European Society for Medical Oncology, 28 (2017), 1658–1666, https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5834034/. doi: 10.1093/annonc/mdx170.  Google Scholar

[5]

S. Halle, K. A. Keyser, F. R. Stahl, A. Busche, A. Marquardt, X. Zheng, M. Galla, V. Heissmeyer, K. Heller, J. Boelter, K. Wagner, Y. Bischoff, R. Martens, A. Braun, K. Werth, A. Uvarovskii, H. Kempf, M. Meyer-Hermann, R. Arens, M. Kremer, G. Sutter, M. Messerle and Reinhold Förster, In vivo killing capacity of cytotoxic T Cells is limited and involves dynamic interactions and T Cell cooperativity, \emphImmunity, 44 (2016), 233-245, https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4846978/. doi: 10.1016/j.immuni.2016.01.010.  Google Scholar

[6]

L. F. Han, S. Eikenberry, C. H. He, L. Johnson, M. C. Preul, E. J. Kostelich and Y. Kuang, Patient-specific parameter estimates of glioblastoma multiforme growth dynamics from a model with explicit birth and death rates, Mathematical Biosciences and Engineering, 16 (2019), 5307–5323, https://www.aimspress.com/fileOther/PDF/MBE/mbe-16-05-265.pdf. doi: 10.3934/mbe.2019265.  Google Scholar

[7]

V. R. Juneja, K. A. McGuire, R. T. Manguso, M. W. LaFleur, N. Collins, W. N. Haining, Gordon J. Freeman and Arlene H. Sharpe, PD-L1 on tumor cells is sufficient for immune evasion in immunogenic tumors and inhibits CD8 T cell cytotoxicity, The Journal of Experimental Medicine, 214 (2017), 895–904, https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5379970/. doi: 10.1084/jem.20160801.  Google Scholar

[8]

J. Kang, S. Demaria and S. Formenti, Current clinical trials testing the combination of immunotherapy with radiotherapy, Journal for Immunotherapy of Cancer, 4 (2016), https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5028964/. doi: 10.1186/s40425-016-0156-7.  Google Scholar

[9] Y. Kuang, J. D. Nagy and S. E. Eikenberry, Introduction to Mathematical Oncology, CRC Press, Boca Raton, FL, 2016.   Google Scholar
[10]

V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis, Bulletin of Mathematical Biology, 56 (1994), 295–321, https://www.sciencedirect.com/science/article/pii/S0092824005802605. Google Scholar

[11]

X. L. Lai and A. Friedman, Combination therapy of cancer with cancer vaccine and immune checkpoint inhibitors: A mathematical model, PLoS ONE, 12 (2017), 1-24.  doi: 10.1371/journal.pone.0178479.  Google Scholar

[12]

T. List and D. Neri, Immunocytokines: A review of molecules in clinical development for cancer therapy, Clinical Pharmacology, 5 (2013), 29–45, https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3753206/. doi: 10.2147/CPAA.S49231.  Google Scholar

[13]

X.-Y. Meng, N.-N. Qin and H.-F. Huo, Dynamics analysis of a predator-prey system with harvesting prey and disease in prey species, Journal of Biological Dynamics, 12 (2018), 342-374.  doi: 10.1080/17513758.2018.1454515.  Google Scholar

[14]

E. Nikolopoulou, L. R. Johnson, D. Harris, J. D. Nagy, E. C. Stites and Y. Kuang, Tumour-immune dynamics with an immune checkpoint inhibitor, Letters in Biomathematics, 5 (2018), S137–S159. doi: 10.1080/23737867.2018.1440978.  Google Scholar

[15]

Z. P. Parra-Guillen, A. Janda, P. Alzuguren, P. Berraondo, R. Hernandez-Alcoceba and I. F. Troconiz, Target-mediated disposition model describing the dynamics of IL12 and IFN$\gamma$ after administration of a mifepristone-inducible adenoviral vector for IL-12 expression in mice, AAPS Journal, 15 (2013), 183–194, https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3535095/. Google Scholar

[16]

A. Radunskaya, R. Kim and T. Woods II, Mathematical modeling of tumor immune interactions: A closer look at the role of a PD-L1 inhibitor in cancer immunotherapy, Spora: A Journal of Biomathematics, 4 (2018), 25–41, https://ir.library.illinoisstate.edu/cgi/viewcontent.cgi?article=1022&context=spora. doi: 10.30707/SPORA4.1Radunskaya.  Google Scholar

[17]

A. Rao and M. R. Patel, A review of avelumab in locally advanced and metastatic bladder cancer, Therapeutic Advances in Urology, 11 (2019), https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6354303/. doi: 10.1177/1756287218823485.  Google Scholar

[18]

E. A. Saparata and L. G. Pillis, A Comparison and catalog of intrinsic tumor growth models, Bulletin of Mathematical Biology, 76 (2014), 2010–2024, https://www.ncbi.nlm.nih.gov/pubmed/25081547. doi: 10.1007/s11538-014-9986-y.  Google Scholar

[19]

R. Serre, S. Benzekry, L. Padovani, C. Meille, N. André, J. Ciccolini, F. Barlesi, X. Muracciole and D. Barbolosi, Mathematical modeling of cancer immunotherapy and its synergy with radiotherapy, Cancer Research, 76 (2016), 4931–4940, http://cancerres.aacrjournals.org/content/76/17/4931. doi: 10.1158/0008-5472.CAN-15-3567.  Google Scholar

[20]

L. Shi, S. Chen, L. Yang and Y. Li, The role of PD-1 and PD-L1 in T-cell immune suppression in patients with hematological malignancies., Journal of Hematology and Oncology, 6 (2013), https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3851976/. doi: 10.1186/1756-8722-6-74.  Google Scholar

[21]

S. Shi, J. Huang and Y. Kuang, Global dynamics in a tumor-immune model with an immune checkpoint inhibitor, DCDS-B, in review. Google Scholar

[22]

S. Simon and N. Labarriere, PD-1 expression on tumor-specific T cells: Friend or foe for immunotherapy?, \emphOncoimmunology, 7 (2017), https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5739549/. Google Scholar

[23]

A. Stéphanou, P. Ballet and G. Powathil, Hybrid modelling in oncology: Success, challenges and hopes, (2019), https://arXiv.org/pdf/1901.05652.pdf. Google Scholar

[24]

A. Talkington and R. Durett, Estimating tumor growth rates in vivo, Bulletin of Mathematical Biology, 77 (2015), 1934–1954, https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4764475/. doi: 10.1007/s11538-015-0110-8.  Google Scholar

[25]

S. G. Tan, K. F. Liu, Y. Chai, C. W.-H. Zhang, S. Gao, G. F. Gao and J. X. Qi, Distinct PD-L1 binding characteristics of therapeutic monoclonal antibody durvalumab, Protein and Cell, 9 (2018), 135-139.  doi: 10.1007/s13238-017-0412-8.  Google Scholar

[26]

J. Tang, J. X. Yu, V. M. Hubbard-Lucey, S. T. Neftelinov, J. P. Hodge and Y. Q. Lin, The clinical trial landscape for PD1/PDL1 immune checkpoint inhibitors, Nature Reviews Drug Discovery, 17 (2018), 854-855.  doi: 10.1038/nrd.2018.210.  Google Scholar

[27]

J. R. Wares, J. J. Crivelli, C. Yun, I. Choi, J. L. Gevertz and P. S. Kim, Treatment strategies for combining immunostimulatory oncolytic virus therapeutics with dendritic cell injections, Mathematical Biosciences and Engineering, 12 (2015), 1237-125.  doi: 10.3934/mbe.2015.12.1237.  Google Scholar

[28]

A. B. Warner and M. A. Postow, Combination controversies: Checkpoint inhibition alone or in combination for the treatment of melanoma?, \emphOncology, 32 (2018), 228–34, https://www.ncbi.nlm.nih.gov/pubmed/29847853. Google Scholar

[29]

C. X. Xu, Y. P. Zhang, P. A. Rolfe, V. M. Hernández, W. Guzman, G. Kradjian, B. Marelli, G. Qin, J. Qi, H. Wang, H. Yu, R. Tighe, K. Lo, J. M. English, L. Radvanyi and Y. Lan, Combination therapy with NHS-muIL12 and avelumab (anti-PD-L1) enhances antitumor efficacy in preclinical cancer models, Clinical Cancer Research, 23 (2017), 5869–5880, http://clincancerres.aacrjournals.org/content/23/19/5869. doi: 10.1158/1078-0432.CCR-17-0483.  Google Scholar

[30]

Y. Y. Yan, A. B. Kumar, H. Finnes, S. N. Markovic, S. Park, R. S. Dronca and H. Dong, Combining immune checkpoint inhibitors with conventional cancer therapy, Frontiers in Immunology, 9 (2018), https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6072836/. doi: 10.3389/fimmu.2018.01739.  Google Scholar

[31]

D. Zamarin and M. A. Postow, Immune checkpoint modulation: Rational design of combination strategies, Pharmacology and Therapeutics, 150 (2015), 23–32, https://www.sciencedirect.com/science/article/pii/S0163725815000042. doi: 10.1016/j.pharmthera.2015.01.003.  Google Scholar

Figure 1.  Schematic model representation for Tumor-Immune Interactions. Sharp arrows indicate proliferation/activation. Blocked arrows indicate killing/blocking. Dashed lines indicate proteins on the tumour (V) or T cells (T). IL-12 stimulates the proliferation of activated T-cells. NHS-muIL12's role is to administrate IL-12 to damaged intratumoral regions. PD-L1 is mainly expressed on tumor cells. Anti-PD-L1 binds to PD-L1 to prevent the formation of the PD-1-PD-L1 complex
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Figure 2.  Tumor volume data and simulations using model (3)-(6) for each single-agent and combination therapy case. Case (a) administration of no drug. Case (b), (c) treatment with NHS-muIL12 $ 2\mu g $ and $ 10 \mu g $ respectively on day $ 0 $. Case (d) administration of avelumab $ 200 \mu g $ on days 0, 3 and 6. Case (e) treatment with both avelumab ($ 200 \mu g $) on days 0, 3, 6 and NHS-muIL12 ($ 2\mu g $) on day 0
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Figure 3.  First graph depicts the model simulation for the combination treatment case of avelumab ($ 200 \mu g $) and NHS-muIL12 ($ 10\mu g $). The remaining two graphs correspond to each of the drugs administrated with respect to time
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Figure 4.  Surface plot depicting the asymptotic behavior of systems (3)-(6) and (18)-(19) for combinations of $ \gamma_1 $ (avelumab) and $ \gamma_2 $ (NHS-muIL12) drugs, where both are applied continuously. Based on a threshold, three regions, namely, 'tumor control', 'tumor escape' and 'intermediate' are identified, which are separated by hyperbolic curves. The hyperbolic curve closer to the origin corresponds to the system (18)-(19) and the other one to (3)-(6). Six representative points indicated by different markers and colors, are chosen and shown on the plot. Their respective tumor volume behavior is plotted in Figure 5 employing the same marker and color notation. Note that the red point falls in the intermediate region, where the behavior is dependent on the system being used
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Figure 5.  A subset of tumor volume behaviors of both the full system (3)-(6) and the limiting system (18)-(19) for continuous dosage with $ \gamma_1 $ (avelumab) and $ \gamma_2 $ (NHS-muIL12) respectively. The threshold chosen is shown on the plot with a dashed line. We observe that the qualitative tumor volume behaviors between the systems are comparable with the exception of the red point, which lies in the intermediate region as shown in Figure 4
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Figure 6.  Surface plot depicting the asymptotic behavior of the full system (3)-(6) for combinations of $ \gamma_1 $ (avelumab) and $ \gamma_2 $ (NHS-muIL12) drugs, where both are applied continuously. Based on a threshold, two regions, namely, 'Tumor Control' and 'Tumor Escape' are identified, which are separated by a hyperbolic curve. Six representative points indicated by different markers and colors, are chosen and shown on the plot
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Figure 7.  Surface plot depicting the asymptotic behavior of the limit case system (18)-(19) for combinations of $ \gamma_1 $ (avelumab) and $ \gamma_2 $ (NHS-muIL12) drugs, where both are applied continuously. Based on a threshold, two regions, namely, 'Tumor Control' and 'Tumor Escape' are identified, which are separated by a hyperbolic curve. Six representative points indicated by different markers and colors, are chosen and shown on the plot
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Table 1.  State Variables of the model system (1)-(2) and (3)-(6).
Variable Meaning Unit
$V$ tumor cell volume mm$^3$
$T$ volume of activated T cells mm$^3$
$L$ free PD-L1 volume mm$^3$
$P$ free PD-1 volume mm$^3$
$A_{1}$ anti-PD-L1 concentration g
$A_{2}$ NHS-muIL12 concentration g
$Q$ PD-1-PD-L1 volume mm$^3$
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Variable Meaning Unit
$V$ tumor cell volume mm$^3$
$T$ volume of activated T cells mm$^3$
$L$ free PD-L1 volume mm$^3$
$P$ free PD-1 volume mm$^3$
$A_{1}$ anti-PD-L1 concentration g
$A_{2}$ NHS-muIL12 concentration g
$Q$ PD-1-PD-L1 volume mm$^3$
Table 2.  Parameters and variables (Var.) of the model system (3)-(6)
Var. Meaning Value Reference
$r$ Tumor cell growth rate $0.213\text{ day}^{-1}$ fitted
$\eta$ Kill rate of tumor cells by T cells $1$ mm$^{-3}$ $\cdot$ day$^{-1}$ fitted
$\delta$ Source of T cell activation 0.02 mm$^{3}$ /day estimated
$\lambda_{TI_{12}}$ Activation rate of T cells by IL-12 8.81 day$^{-1}$ [14]
$K_{A_{2}}$ Dissociation constant of $A_{2}$ $7 \cdot 10^{-14}$ moles/liter estimated
$K_{TQ}$ Inhibition of function of T cells by PD-1-PD-L1 $10^{-13}$ mm$^{6}$ estimated
$d_{T}$ Death rate of T cells $0-0.5\text{ day}^{-1}$ [14]
$d_{A_{1}}$ Degradation rate of Anti-PD-L1 $ 0.1136\text{ day}^{-1}$ [17]
$d_{A_{2}}$ Degradation rate of NHS-muIL12 0.69 day$^{-1}$ [12]
$\rho_{p}$ Expression level of PD-1 $3.19\cdot10^{-7}$ - $8.49\cdot 10^{-7}$ [11]
$\rho_{L}$ Expression level of PD-L1 $3.56\cdot10^{-7}$ - $1.967\cdot 10^{-6}$ [11]
$K_{A_{1}}$ Dissociation constant of free PD-L1 with anti-PD-L1 10$^{-13}$ mol/liter estimated
$\epsilon_{v}$ Expression of PD-L1 in tumor cells vs. T cells 1-100 [14]
$\sigma$ fraction of complex association and dissociation 0.01mm$^{-3}$ estimated
$\gamma_1$ continuous infusion rate of avelumab $10^{-7}-9\cdot 10^{-5}$ g/day estimated
$\gamma_2$ continuous infusion rate of NHS-muIL12 $10^{-9}-2\cdot 10^{-6}$ g/day estimated
$c_1$ conversion constant for $A_1$ drug $55^{-1}10^{-7}-55^{-1}10^{-6}$ estimated
$c_2$ conversion constant for $A_2$ drug $75^{-1}10^{-7}- 75^{-1}10^{-6}$ estimated
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Var. Meaning Value Reference
$r$ Tumor cell growth rate $0.213\text{ day}^{-1}$ fitted
$\eta$ Kill rate of tumor cells by T cells $1$ mm$^{-3}$ $\cdot$ day$^{-1}$ fitted
$\delta$ Source of T cell activation 0.02 mm$^{3}$ /day estimated
$\lambda_{TI_{12}}$ Activation rate of T cells by IL-12 8.81 day$^{-1}$ [14]
$K_{A_{2}}$ Dissociation constant of $A_{2}$ $7 \cdot 10^{-14}$ moles/liter estimated
$K_{TQ}$ Inhibition of function of T cells by PD-1-PD-L1 $10^{-13}$ mm$^{6}$ estimated
$d_{T}$ Death rate of T cells $0-0.5\text{ day}^{-1}$ [14]
$d_{A_{1}}$ Degradation rate of Anti-PD-L1 $ 0.1136\text{ day}^{-1}$ [17]
$d_{A_{2}}$ Degradation rate of NHS-muIL12 0.69 day$^{-1}$ [12]
$\rho_{p}$ Expression level of PD-1 $3.19\cdot10^{-7}$ - $8.49\cdot 10^{-7}$ [11]
$\rho_{L}$ Expression level of PD-L1 $3.56\cdot10^{-7}$ - $1.967\cdot 10^{-6}$ [11]
$K_{A_{1}}$ Dissociation constant of free PD-L1 with anti-PD-L1 10$^{-13}$ mol/liter estimated
$\epsilon_{v}$ Expression of PD-L1 in tumor cells vs. T cells 1-100 [14]
$\sigma$ fraction of complex association and dissociation 0.01mm$^{-3}$ estimated
$\gamma_1$ continuous infusion rate of avelumab $10^{-7}-9\cdot 10^{-5}$ g/day estimated
$\gamma_2$ continuous infusion rate of NHS-muIL12 $10^{-9}-2\cdot 10^{-6}$ g/day estimated
$c_1$ conversion constant for $A_1$ drug $55^{-1}10^{-7}-55^{-1}10^{-6}$ estimated
$c_2$ conversion constant for $A_2$ drug $75^{-1}10^{-7}- 75^{-1}10^{-6}$ estimated
Table 3.  Parameter values used in simulations for equations (3)-(6)
Variable Meaning Value
$r$ Tumor cell growth rate $0.213\text{ day}^{-1}$
$\eta$ Kill rate of tumor cells by T cells $1$ mm$^{-3}$ $\cdot$ day$^{-1}$
$\delta$ Source of activation 0.02 mm$^{3}$ /day
$\lambda_{TI_{12}}$ Activation rate of T cells by IL-12 8.81 day$^{-1}$
$K_{A_{2}}$ Dissociation constant of $A_{2}$ $7 \cdot 10^{-14}$ moles/liter
$K_{TQ}$ Inhibition of function of T cells by PD-1-PD-L1 $10^{-13}$ mm$^{6}$
$d_{T}$ Death rate of T cells $0.05\text{ day}^{-1}$
$d_{A_{1}}$ Degradation rate of Anti-PD-L1 $ 0.1136\text{ day}^{-1}$
$d_{A_{2}}$ Degradation rate of NHS-muIL12 0.69 day$^{-1}$
$\rho_{p}$ Expression level of PD-1 $5.84\cdot 10^{-7}$
$\rho_{L}$ Expression level of PD-L1 $2.7635\cdot 10^{-7}$
$K_{A_{1}}$ Dissociation constant of PD-L1 with anti-PD-L1 10$^{-13}$ mol/liter
$\epsilon_{v}$ Expression of PD-L1 in tumor cells vs. T cells 50
$\sigma$ fraction of complex association and dissociation 0.001 mm$^{-3}$
$\gamma_1$ prescribed infusion rate of avelumab $10^{-7}-9\cdot 10^{-5}$ g/day
$\gamma_2$ prescribed infusion rate of NHS-muIL12 $10^{-9}-2\cdot 10^{-6}$ g/day
$c_1$ conversion constant for $A_1$ drug $55^{-1}10^{-7}$
$c_2$ conversion constant for $A_2$ drug $75^{-1}10^{-7}$
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Variable Meaning Value
$r$ Tumor cell growth rate $0.213\text{ day}^{-1}$
$\eta$ Kill rate of tumor cells by T cells $1$ mm$^{-3}$ $\cdot$ day$^{-1}$
$\delta$ Source of activation 0.02 mm$^{3}$ /day
$\lambda_{TI_{12}}$ Activation rate of T cells by IL-12 8.81 day$^{-1}$
$K_{A_{2}}$ Dissociation constant of $A_{2}$ $7 \cdot 10^{-14}$ moles/liter
$K_{TQ}$ Inhibition of function of T cells by PD-1-PD-L1 $10^{-13}$ mm$^{6}$
$d_{T}$ Death rate of T cells $0.05\text{ day}^{-1}$
$d_{A_{1}}$ Degradation rate of Anti-PD-L1 $ 0.1136\text{ day}^{-1}$
$d_{A_{2}}$ Degradation rate of NHS-muIL12 0.69 day$^{-1}$
$\rho_{p}$ Expression level of PD-1 $5.84\cdot 10^{-7}$
$\rho_{L}$ Expression level of PD-L1 $2.7635\cdot 10^{-7}$
$K_{A_{1}}$ Dissociation constant of PD-L1 with anti-PD-L1 10$^{-13}$ mol/liter
$\epsilon_{v}$ Expression of PD-L1 in tumor cells vs. T cells 50
$\sigma$ fraction of complex association and dissociation 0.001 mm$^{-3}$
$\gamma_1$ prescribed infusion rate of avelumab $10^{-7}-9\cdot 10^{-5}$ g/day
$\gamma_2$ prescribed infusion rate of NHS-muIL12 $10^{-9}-2\cdot 10^{-6}$ g/day
$c_1$ conversion constant for $A_1$ drug $55^{-1}10^{-7}$
$c_2$ conversion constant for $A_2$ drug $75^{-1}10^{-7}$
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