| 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 |
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.
| [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. |
| [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. |
| [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. |
| [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. |
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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. |
| [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. |
| [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. |
| [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. |
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Y. Kuang, J. D. Nagy and S. E. Eikenberry, Introduction to Mathematical Oncology, CRC Press, Boca Raton, FL, 2016.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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/. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [21] |
S. Shi, J. Huang and Y. Kuang, Global dynamics in a tumor-immune model with an immune checkpoint inhibitor, DCDS-B, in review. |
| [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/. |
| [23] |
A. Stéphanou, P. Ballet and G. Powathil, Hybrid modelling in oncology: Success, challenges and hopes, (2019), https://arXiv.org/pdf/1901.05652.pdf. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
show all 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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [9] |
Y. Kuang, J. D. Nagy and S. E. Eikenberry, Introduction to Mathematical Oncology, CRC Press, Boca Raton, FL, 2016.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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/. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [21] |
S. Shi, J. Huang and Y. Kuang, Global dynamics in a tumor-immune model with an immune checkpoint inhibitor, DCDS-B, in review. |
| [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/. |
| [23] |
A. Stéphanou, P. Ballet and G. Powathil, Hybrid modelling in oncology: Success, challenges and hopes, (2019), https://arXiv.org/pdf/1901.05652.pdf. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| 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$^3$ |
| 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$^3$ |
| 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 |
| 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 |
| 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 | $75^{-1}10^{-7}$ |
| 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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