# American Institute of Mathematical Sciences

May  2019, 24(5): 2237-2250. doi: 10.3934/dcdsb.2019093

## Antagonism and negative side-effects in combination therapy for cancer

 1 Mathematical Bioscience Institute & Department of Mathematics, Ohio State University, Columbus, OH, USA 2 Institute for Mathematical Sciences, Renmin University of China, Beijing, China

* Corresponding author: xiulanlai@ruc.edu.cn

Received  January 2018 Revised  January 2019 Published  May 2019 Early access  March 2019

Fund Project: The first author is supported by the Mathematical Biosciences Institute and the National Science Foundation under Grant DMS 0931642.

Most clinical trials with combination therapy fail. One of the reasons is that not enough forethought is given to the interaction between the different agents, as well as the potential negative side-effects that may arise in the combined therapy. In the present paper we consider a generic cancer model with combination therapy consisting of chemotherapy agent $X$ and checkpoint inhibitor $A$. We use a mathematical model to investigate the results of injecting different amounts $\gamma_X$ of $X$ and $\gamma_A$ of $A$. We show that there are some regions in the $(\gamma_A,\gamma_X)$-plane where as increase in $\gamma_X$ or $\gamma_A$ actually decreases the tumor volume; such 'regions of antagonism' should be avoided in clinical trials. We also show how to achieve the same level of tumor volume reduction with least negative-side effects, where the side-effects are represented by the level of inflammation of the tumor microenvironment.

Citation: Avner Friedman, Xiulan Lai. Antagonism and negative side-effects in combination therapy for cancer. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2237-2250. doi: 10.3934/dcdsb.2019093
##### References:
 [1] G. Cantelli, E. Crosas-Molist1, M. Georgouli and V. Sanz-Moreno, TGFB-induced transcription in cancer, Seminars in Cancer Biology, 42 (2017), 60-69.  doi: 10.1016/j.semcancer.2016.08.009. [2] T. Chanmee, P. Ontong, K. Konno and N. Itano, Tumor-associated macrophages as major players in the tumor microenvironment, Cancers, 6 (2014), 1670-1690. [3] X. Chen and J. J. Oppenheim, Contrasting effects of TNF and anti-TNF on the activation of effector T cells and regulatory T cells in autoimmunity, FEBS Lett., 585 (2011), 3611-3618. [4] D. Chen, A. A. Bobko, A. C. Gross, R. Evans, C. B. Marsh and V. V. Khramtsov, et al., Involvement of tumor macrophage HIFs in chemotherapy effectiveness: mathematical modeling of oxygen, pH, and glutathione., PLoS ONE, 9 (2014), e107511. [5] X. Cheng, V. Veverka, A. Radhakrishnan, L. C. Waters, F. W. Muskett and S. H. Morgan, 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. [6] T. Condamine and D. I. Gabrilovich, Molecular mechanisms regulating myeloid-derived suppressor cell differentiation and function, Trends Immunol., 32 (2011), 19-25. [7] A. Friedman and X. Lai, Combination therapy for cancer with oncolytic virus and checkpoint inhibitor: A mathematical model, PLoS ONE, 2 (2018), e0192449. [8] W. Hao and A. Friedman, The role of exosomes in pancreatic cancer microenvironment, Bull. Math. Biol., 80 (2018), 1111-1133.  doi: 10.1007/s11538-017-0254-9. [9] W. Hao and A. Friedman, Mathematical model on Alzheimer's disease, BMC Syst. Biol., 10 (2016), 1-18. [10] W. Hao and A Friedman, Serum uPAR as biomarker in breast cancer recurrence: A mathematical model, PLoS ONE, 11 (2016), e0153508. [11] W. Hao, H. M. Komar, P. A. Hart, D. L. Conwell, G. B. Lesinski and A. Friedman, A mathematical model of chronic pancreatitis, Proc. Natl. Acad. Sci. USA, 114 (2017), 5011-5016.  doi: 10.1073/pnas.1620264114. [12] J. M. T. Janco, P. Lamichhane, L. Karyampudi and K. L. Knutson, Tumor-infiltrating dendritic cells in cancer pathogenesis, J. Immunol., 194 (2015), 2985-2991. [13] Y. Kim, S. Lawler, M. O. Nowicki, E. A. Chiocca and A. Friedman, A mathematical model for pattern formation of glioma cells outside the tumor spheroid core, J. Theor. Biol., 260 (2009), 359-371.  doi: 10.1016/j.jtbi.2009.06.025. [14] Y. Kim, J. Wallace, F. Li, M. Ostrowski and A. Friedman, Transformed epithelial cells and fibroblasts/myofibroblasts interaction in breast tumor: a mathematical model and experiments, J. Theor. Biol., 61 (2010), 401-421.  doi: 10.1007/s00285-009-0307-2. [15] X. Lai and A. Friedman, Combination therapy of cancer with BRAF inhibitor and immune checkpoint inhibitor: A mathematical model, BMC System Biology, 11 (2017), 1-18. [16] X. Lai, A. Stiff, M. Duggan, R. Wesolowski, W. E. Carson III and A. Friedman, Modeling combination therapy for breast cancer with BET and immune checkpoint inhibitors, Proc. Natl. Acad. Sci. USA, 115 (2018), 5534-5539. [17] K. L. Liao, X. F. Bai and A. Friedman, Mathematical modeling of interleukin-27 induction of anti-tumor T cells response, PLoS ONE, 9 (2014), e91844. [18] Y. Ma, G. V. Shurin1, Z. Peiyuan and M. R. Shurin, Dendritic Cells in the Cancer Microenvironment., J. Cancer, 4 (2013), 36-44. [19] M. L. Maitland, C. Hudoba, K. L. Snider and M. J. Ratain, Analysis of the yield of phase Ⅱ combination therapy trials in medical oncology, Clinical Cancer Research, (2010), 1078-0432. [20] J. Markowitz, R. Wesolowski, T. Papenfuss, T. R. Brooks and W. E. Carson, Myeloid-derived suppressor cells in breast cancer., Breast Cancer Res. Treat., 140 (2013), 13-21. [21] R. L. Mautea, S. R. Gordona, A. T. Mayere, M. N. McCrackena, A. Natarajane and N. G. Ring, et al., Engineering high-affinity PD-1 variants for optimized immunotherapy and immuno-PET imaging., Proc. Natl. Acad. Sci. USA, 112 (2015), E6506-E6514. [22] P. Muller, K. Martin, S. Theurich, M. V. Bergwelt-Baildon and A. Zippelius, Cancer chemotherapy agents target intratumoral dendritic cells to potentiate antitumor immunity, Oncoimmunology, 3 (2014). doi: 10.4161/21624011.2014.954460. [23] M. R. Muppidi and S. George, Immune Checkpoint Inhibitors in Renal Cell Carcinoma, Journal of Targeted Therapies in Cancer 2015, 4 (2015), 47-52. [24] E. Obeid, R. Nanda, Y.-X. Fu and O. I. Olopade, The role of tumor-associated macrophages in breast cancer progression (review), Int. J. Oncol., 43 (2013), 5-12. [25] J. Palucka and J. Banchereau, Cancer immunotherapy via dendritic cells, Nat. Rev. Cancer, 12 (2012), 265-277. [26] C. Y. Perrot, D. Javelaud and A. Mauviel, Insights into the Transforming Growth Factor-beta Signaling Pathway in Cutaneous Melanoma, Ann. Dermatol., 25 (2013), 135-144. [27] R. Saenz, D. Futalan, L. Leutenez, F. Eekhout, J. F. Fecteau and S. Sundelius, et al., TLR4- dependent activation of dendritic cells by an HMGB1-derived peptide adjuvant. J. Transl. Med., 12 (2014), 1-11. [28] M. R. Sharma, W. M. Stadler and M. J. Ratain, Randomized phase Ⅱ trials: a long-term investment with promising returns, Journal of the National Cancer Institute, 103 (2011), 1093-1100. [29] 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, J. Hematol. Oncol., 6 (2013). [30] Y. B. Shui, X. Wang, J. S. Hu, S. P. Wang, C. M. Garcia and et al., Vascular endothelial growth factor expression and signaling in the lens. Invest. Ophthalmol. Vis. Sci., 44 (2003), 3911-3919. [31] G. P. Sims, D. C. Rowe, S. T. Rietdijk, R. Herbst and A. J. Coyle, HMGB1 and RAGE in inflammation and cancer, Annu. Rev. Immunol., 28 (2010), 367-388. [32] S. Singh, N. Mehta, J. Lilan, M. B. Budhthoki, F. Chao and L. Yong, Initiative action of tumor-associated macrophage during tumor metastasis, Biochim Open, 4 (2017), 8-18.  doi: 10.1016/j.biopen.2016.11.002. [33] V. Umansky, C. Blattner, C. Gebhardt and J. Utikal, The role of myeloid-derived suppressor cells (MDSC) in cancer progression, Vaccines, 4 (2016), 1-16.  doi: 10.3390/vaccines4040036. [34] E. Vacchelli, Y. Ma, E. E. Baracco, A. Sistigu, D. P. Enot and F. Pietrocola, et al., Chemotherapy-induced antitumor immunity requires formyl peptide receptor 1, Science, 350 (2015), 972-978. [35] T. L. Whiteside, The role of regulatory T cells in cancer immunology, Immunotargets Ther., 4 (2015), 159-171. [36] M. E. Young, Estimation of diffusion coefficients of proteins, Biotechnology and Bioengineering, XXII (1980), 947-955. [37] L. Zandarashvili, D. Sahu, K. Lee, Y. S. Lee, P. Singh and K. Rajarathnam, et al., Real-time kinetics of high-mobility group box 1 (HMGB1) oxidation in extracellular fluids studied by in situ protein NMR spectroscopy, J. Biol. Chem., 288 (2013), 11621-11627. [38] J. Zhang, M. B. Patel, R. Griffiths, A. Mao, Y. soo Song and N. S. Karlovich, et al., Tumor necrosis factor-alpha produced in the kidney contributes to angiotensin Ⅱ-dependent hypertension, Hypertension, 64 (2014), 1275-1281.

show all references

##### References:
 [1] G. Cantelli, E. Crosas-Molist1, M. Georgouli and V. Sanz-Moreno, TGFB-induced transcription in cancer, Seminars in Cancer Biology, 42 (2017), 60-69.  doi: 10.1016/j.semcancer.2016.08.009. [2] T. Chanmee, P. Ontong, K. Konno and N. Itano, Tumor-associated macrophages as major players in the tumor microenvironment, Cancers, 6 (2014), 1670-1690. [3] X. Chen and J. J. Oppenheim, Contrasting effects of TNF and anti-TNF on the activation of effector T cells and regulatory T cells in autoimmunity, FEBS Lett., 585 (2011), 3611-3618. [4] D. Chen, A. A. Bobko, A. C. Gross, R. Evans, C. B. Marsh and V. V. Khramtsov, et al., Involvement of tumor macrophage HIFs in chemotherapy effectiveness: mathematical modeling of oxygen, pH, and glutathione., PLoS ONE, 9 (2014), e107511. [5] X. Cheng, V. Veverka, A. Radhakrishnan, L. C. Waters, F. W. Muskett and S. H. Morgan, 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. [6] T. Condamine and D. I. Gabrilovich, Molecular mechanisms regulating myeloid-derived suppressor cell differentiation and function, Trends Immunol., 32 (2011), 19-25. [7] A. Friedman and X. Lai, Combination therapy for cancer with oncolytic virus and checkpoint inhibitor: A mathematical model, PLoS ONE, 2 (2018), e0192449. [8] W. Hao and A. Friedman, The role of exosomes in pancreatic cancer microenvironment, Bull. Math. Biol., 80 (2018), 1111-1133.  doi: 10.1007/s11538-017-0254-9. [9] W. Hao and A. Friedman, Mathematical model on Alzheimer's disease, BMC Syst. Biol., 10 (2016), 1-18. [10] W. Hao and A Friedman, Serum uPAR as biomarker in breast cancer recurrence: A mathematical model, PLoS ONE, 11 (2016), e0153508. [11] W. Hao, H. M. Komar, P. A. Hart, D. L. Conwell, G. B. Lesinski and A. Friedman, A mathematical model of chronic pancreatitis, Proc. Natl. Acad. Sci. USA, 114 (2017), 5011-5016.  doi: 10.1073/pnas.1620264114. [12] J. M. T. Janco, P. Lamichhane, L. Karyampudi and K. L. Knutson, Tumor-infiltrating dendritic cells in cancer pathogenesis, J. Immunol., 194 (2015), 2985-2991. [13] Y. Kim, S. Lawler, M. O. Nowicki, E. A. Chiocca and A. Friedman, A mathematical model for pattern formation of glioma cells outside the tumor spheroid core, J. Theor. Biol., 260 (2009), 359-371.  doi: 10.1016/j.jtbi.2009.06.025. [14] Y. Kim, J. Wallace, F. Li, M. Ostrowski and A. Friedman, Transformed epithelial cells and fibroblasts/myofibroblasts interaction in breast tumor: a mathematical model and experiments, J. Theor. Biol., 61 (2010), 401-421.  doi: 10.1007/s00285-009-0307-2. [15] X. Lai and A. Friedman, Combination therapy of cancer with BRAF inhibitor and immune checkpoint inhibitor: A mathematical model, BMC System Biology, 11 (2017), 1-18. [16] X. Lai, A. Stiff, M. Duggan, R. Wesolowski, W. E. Carson III and A. Friedman, Modeling combination therapy for breast cancer with BET and immune checkpoint inhibitors, Proc. Natl. Acad. Sci. USA, 115 (2018), 5534-5539. [17] K. L. Liao, X. F. Bai and A. Friedman, Mathematical modeling of interleukin-27 induction of anti-tumor T cells response, PLoS ONE, 9 (2014), e91844. [18] Y. Ma, G. V. Shurin1, Z. Peiyuan and M. R. Shurin, Dendritic Cells in the Cancer Microenvironment., J. Cancer, 4 (2013), 36-44. [19] M. L. Maitland, C. Hudoba, K. L. Snider and M. J. Ratain, Analysis of the yield of phase Ⅱ combination therapy trials in medical oncology, Clinical Cancer Research, (2010), 1078-0432. [20] J. Markowitz, R. Wesolowski, T. Papenfuss, T. R. Brooks and W. E. Carson, Myeloid-derived suppressor cells in breast cancer., Breast Cancer Res. Treat., 140 (2013), 13-21. [21] R. L. Mautea, S. R. Gordona, A. T. Mayere, M. N. McCrackena, A. Natarajane and N. G. Ring, et al., Engineering high-affinity PD-1 variants for optimized immunotherapy and immuno-PET imaging., Proc. Natl. Acad. Sci. USA, 112 (2015), E6506-E6514. [22] P. Muller, K. Martin, S. Theurich, M. V. Bergwelt-Baildon and A. Zippelius, Cancer chemotherapy agents target intratumoral dendritic cells to potentiate antitumor immunity, Oncoimmunology, 3 (2014). doi: 10.4161/21624011.2014.954460. [23] M. R. Muppidi and S. George, Immune Checkpoint Inhibitors in Renal Cell Carcinoma, Journal of Targeted Therapies in Cancer 2015, 4 (2015), 47-52. [24] E. Obeid, R. Nanda, Y.-X. Fu and O. I. Olopade, The role of tumor-associated macrophages in breast cancer progression (review), Int. J. Oncol., 43 (2013), 5-12. [25] J. Palucka and J. Banchereau, Cancer immunotherapy via dendritic cells, Nat. Rev. Cancer, 12 (2012), 265-277. [26] C. Y. Perrot, D. Javelaud and A. Mauviel, Insights into the Transforming Growth Factor-beta Signaling Pathway in Cutaneous Melanoma, Ann. Dermatol., 25 (2013), 135-144. [27] R. Saenz, D. Futalan, L. Leutenez, F. Eekhout, J. F. Fecteau and S. Sundelius, et al., TLR4- dependent activation of dendritic cells by an HMGB1-derived peptide adjuvant. J. Transl. Med., 12 (2014), 1-11. [28] M. R. Sharma, W. M. Stadler and M. J. Ratain, Randomized phase Ⅱ trials: a long-term investment with promising returns, Journal of the National Cancer Institute, 103 (2011), 1093-1100. [29] 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, J. Hematol. Oncol., 6 (2013). [30] Y. B. Shui, X. Wang, J. S. Hu, S. P. Wang, C. M. Garcia and et al., Vascular endothelial growth factor expression and signaling in the lens. Invest. Ophthalmol. Vis. Sci., 44 (2003), 3911-3919. [31] G. P. Sims, D. C. Rowe, S. T. Rietdijk, R. Herbst and A. J. Coyle, HMGB1 and RAGE in inflammation and cancer, Annu. Rev. Immunol., 28 (2010), 367-388. [32] S. Singh, N. Mehta, J. Lilan, M. B. Budhthoki, F. Chao and L. Yong, Initiative action of tumor-associated macrophage during tumor metastasis, Biochim Open, 4 (2017), 8-18.  doi: 10.1016/j.biopen.2016.11.002. [33] V. Umansky, C. Blattner, C. Gebhardt and J. Utikal, The role of myeloid-derived suppressor cells (MDSC) in cancer progression, Vaccines, 4 (2016), 1-16.  doi: 10.3390/vaccines4040036. [34] E. Vacchelli, Y. Ma, E. E. Baracco, A. Sistigu, D. P. Enot and F. Pietrocola, et al., Chemotherapy-induced antitumor immunity requires formyl peptide receptor 1, Science, 350 (2015), 972-978. [35] T. L. Whiteside, The role of regulatory T cells in cancer immunology, Immunotargets Ther., 4 (2015), 159-171. [36] M. E. Young, Estimation of diffusion coefficients of proteins, Biotechnology and Bioengineering, XXII (1980), 947-955. [37] L. Zandarashvili, D. Sahu, K. Lee, Y. S. Lee, P. Singh and K. Rajarathnam, et al., Real-time kinetics of high-mobility group box 1 (HMGB1) oxidation in extracellular fluids studied by in situ protein NMR spectroscopy, J. Biol. Chem., 288 (2013), 11621-11627. [38] J. Zhang, M. B. Patel, R. Griffiths, A. Mao, Y. soo Song and N. S. Karlovich, et al., Tumor necrosis factor-alpha produced in the kidney contributes to angiotensin Ⅱ-dependent hypertension, Hypertension, 64 (2014), 1275-1281.
Interaction of immune cells with cancer cells. Sharp arrows indicate proliferation/activation, blocked arrow indicates killing/blocking, inverted sharp arrow indicates recruitment/chemoattraction. $C$: cancer cells, $T$: effector T cells
Average densities/concentrations, in ${\rm g}/{\rm cm}^3$, of all the variables of the model in control case (no drugs). All parameter values are the same as in Tables 2 and 3, for a mouse model
Growth of tumor volume under treatment with $\gamma_X$ or $\gamma_A$, or combination ($\gamma_X,\gamma_A$). The chemotherapy or/and anti-PD-1 treatments. (a) $\gamma_X = 5\times 10^{-13}$ ${\rm g}/{\rm cm}^3\cdot {\rm day}$, $\gamma_A = 2\times 10^{-11}$ ${\rm g}/{\rm cm}^3\cdot {\rm day}$; (b) $\gamma_X = 3\times 10^{-13}$ ${\rm g}/{\rm cm}^3\cdot {\rm day}$, $\gamma_A = 5\times 10^{-11}$ ${\rm g}/{\rm cm}^3\cdot {\rm day}$. All other parameter values are the same as in Tables 2 and 3, for a mouse model
Efficacy of combination therapy at day 30 for different pair of $(\gamma_X, \gamma_A)$. Here (a) $\gamma_X = 0 - 2\times 10^{-12}$ ${\rm g}/{\rm cm}^3$ and $\gamma_A = 0 - 1.8\times 10^{-10}$ ${\rm g}/{\rm cm}^3$. (b) $\gamma_X = 0.9\times 10^{-12} - 2\times 10^{-12}$ ${\rm g}/{\rm cm}^3$ and $\gamma_A = 1\times 10^{-10} - 1.8\times 10^{-10}$ ${\rm g}/{\rm cm}^3$. All other parameter values are the same as in Tables 2 and 3
Average T cell density at day 30 for different pair of $(\gamma_X, \gamma_A)$. Here $\gamma_X = 0 - 2\times 10^{-12}$ ${\rm g}/{\rm cm}^3$ and $\gamma_A = 0 - 1.8\times 10^{-10}$ ${\rm g}/{\rm cm}^3$. All other parameter values are the same as in Tables 2 and 3
Average concentration of TNF-$\alpha$ at day 30 for different pair of $(\gamma_X, \gamma_A)$. Here $\gamma_X = 0 - 2\times 10^{-12}$ ${\rm g}/{\rm cm}^3$ and $\gamma_A = 0 - 1.8\times 10^{-10}$ ${\rm g}/{\rm cm}^3$. All other parameter values are the same as in Tables 2 and 3
List of variables (in units of g/${\rm cm}^3$)
 Notation Description Notation Description $D$ density of DCs $I_{12}$ IL-12 concentration $T$ density of effector T cells $M_P$ MCP-1 (CCL2) concentration $M_1$ density of proinflammatory macrophages M1 $I_{10}$ IL-10 concentration $M_2$ density of anti-proinflammatory macrophages M2 $T_\beta$ TGF- $\beta$ : concentration $C$ density of cancer cells $P$ PD-1 concentration $A$ concentration of anti-PD-1 $L$ PD-L1 concentration $X$ concentration of a chemotherapy agent $Q$ PD-1-PD-L1 concentration
 Notation Description Notation Description $D$ density of DCs $I_{12}$ IL-12 concentration $T$ density of effector T cells $M_P$ MCP-1 (CCL2) concentration $M_1$ density of proinflammatory macrophages M1 $I_{10}$ IL-10 concentration $M_2$ density of anti-proinflammatory macrophages M2 $T_\beta$ TGF- $\beta$ : concentration $C$ density of cancer cells $P$ PD-1 concentration $A$ concentration of anti-PD-1 $L$ PD-L1 concentration $X$ concentration of a chemotherapy agent $Q$ PD-1-PD-L1 concentration
Summary of parameter values
 Notation Description Value used References $\delta_D$ diffusion coefficient of DCs $8.64 \times 10^{-7}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [8] $\delta_T$ diffusion coefficient of T cells $8.64\times 10^{-7}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [8] $\delta_M$ diffusion coefficient of macrophages $8.64\times 10^{-7}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [8] $\delta_C$ diffusion coefficient of tumor cells $8.64\times 10^{-7}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [8] $\delta_{I_{12}}$ diffusion coefficient of IL-12 $6.05\times 10^{-2}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [16] $\delta_{T_{\beta}}$ diffusion coefficient of TGF-$\beta$ $8.52\times 10^{-2}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [16] $\delta_{I_{10}}$ diffusion coefficient of IL-10 $9.11\times 10^{-2}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [16] $\delta_{T_\alpha}$ diffusion coefficient of TNF-$\alpha$ $8.46\times 10^{-2}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [16] $\delta_{M_P}$ diffusion coefficient of MCP-1 $1.12\times 10^{-1}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [16] $\delta_A$ diffusion coefficient of anti-PD-1 $4.73\times 10^{-2}$ ${\rm cm}^2$ ${\rm day}^{-1}$ estimated $\delta_X$ diffusion coefficient of $X$ $0.27$ ${\rm cm}^2$ ${\rm day}^{-1}$ estimated $\sigma_0$ flux rate of $T$ cells at the boundary 1 ${\rm cm}^{-1}$ [8] $\chi_M$ chemoattraction coefficient of MCP-1 $10$ ${\rm cm}^5/{\rm g}\cdot {\rm day}$ [13,14] $\lambda_{DC}$ activation rate of DCs by tumor cells $10$ ${\rm g}/{\rm cm}^3\cdot{\rm day}$ [16] $\lambda_{TI_{12}}$ activation rate of T cells by IL-12 $16.2$ ${\rm day}^{-1}$ estimated $\lambda_{M_1}$ activation rate of M1 macrophages $1.35$ ${\rm day}^{-1}$ [16] $\lambda_{M_2}$ activation rate of M2 macrophages $1.01$ ${\rm day}^{-1}$ [16] $\beta_{M_1}$ phenotype change rate of M1 to M2 macrophages $0.3$ ${\rm day}^{-1}$ estimated $\beta_{M_2}$ phenotype change rate of M2 to M1 macrophages $4.68\times 10^{-3}$ ${\rm day}^{-1}$ estimated $\lambda_{C}$ growth rate of cancer cells $1.92$ ${\rm day}^{-1}$ estimated $\lambda_{I_{12}D}$ production rate of IL-12 by DCs $1.38\times 10^{-6}$ ${\rm day}^{-1}$ [16] $\lambda_{I_{12}M_1}$ production rate of IL-12 by M1 macrophages $5.52\times 10^{-6}$ ${\rm day}^{-1}$ [16] $\lambda_{T_\beta C}$ production rate of TGF-$\beta$ by cancer cells $2.79\times 10^{-10}$ ${\rm day}^{-1}$ estimated $\lambda_{T_\beta M_2}$ production rate of TGF-$\beta$ by M2 macrophages $6.97 \times 10^{-9}$ ${\rm day}^{-1}$ estimated $\lambda_{I_{10} C}$ production rate of IL-10 by cancer cells $2.07\times 10^{-8}$ ${\rm day}^{-1}$ [16] $\lambda_{I_{10} M_2}$ production rate of IL-10 by M2 macrophages $1.65\times 10^{-9}$ ${\rm day}^{-1}$ [16] $\lambda_{T_\alpha M_1}$ production rate of TNF $\alpha$ by M1 macrophages $1.36\times 10^{-5}$ ${\rm day}^{-1}$ [16] $\lambda_{T_\alpha T}$ production rate of TNF $\alpha$ by Th1 cells $9.06\times 10^{-8}$ ${\rm day}^{-1}$ estimated $\lambda_{M_PM_2}$ production rate of MCP-1 by M2 macrophages $1.2\times 10^{-8}$ ${\rm day}^{-1}$ [16] $\lambda_{M_PC}$ production rate of MCP-1 by cancer cells $8.24\times 10^{-7}$ ${\rm day}^{-1}$ [16] $d_{D}$ death rate of DCs 0.1 ${\rm day}^{-1}$ [8] $d_{T}$ death rate of T cells $0.18$ ${\rm day}^{-1}$ [8] $d_{M_1}$ death rate of M1 macrophage $0.015$ ${\rm day}^{-1}$ [9] $d_{M_2}$ death rate of M2 macrophage $0.015$ ${\rm day}^{-1}$ [9] $d_{C}$ death rate of tumor cells $0.17$ ${\rm day}^{-1}$ [8] $d_{I_{12}}$ degradation rate of IL-12 $1.38$ ${\rm day}^{-1}$ [8] $d_{T_\beta}$ degradation rate of TGF-$\beta$ $499.066$ ${\rm day}^{-1}$ [16] $d_{I_{10}}$ degradation rate of IL-10 $8.3178$ ${\rm day}^{-1}$ [16] $d_{T_\alpha}$ degradation rate of TGF-$\alpha$ $55.01$ ${\rm day}^{-1}$ [16] $d_{M_P}$ degradation rate of MCP-1 $55.01$ ${\rm day}^{-1}$ [10] $d_{A}$ degradation rate of anti-PD-1 $0.047$ ${\rm day}^{-1}$ [16] $d_X$ degradation rate of docetexel $1.11$ ${\rm day}^{-1}$ estimated
 Notation Description Value used References $\delta_D$ diffusion coefficient of DCs $8.64 \times 10^{-7}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [8] $\delta_T$ diffusion coefficient of T cells $8.64\times 10^{-7}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [8] $\delta_M$ diffusion coefficient of macrophages $8.64\times 10^{-7}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [8] $\delta_C$ diffusion coefficient of tumor cells $8.64\times 10^{-7}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [8] $\delta_{I_{12}}$ diffusion coefficient of IL-12 $6.05\times 10^{-2}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [16] $\delta_{T_{\beta}}$ diffusion coefficient of TGF-$\beta$ $8.52\times 10^{-2}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [16] $\delta_{I_{10}}$ diffusion coefficient of IL-10 $9.11\times 10^{-2}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [16] $\delta_{T_\alpha}$ diffusion coefficient of TNF-$\alpha$ $8.46\times 10^{-2}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [16] $\delta_{M_P}$ diffusion coefficient of MCP-1 $1.12\times 10^{-1}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [16] $\delta_A$ diffusion coefficient of anti-PD-1 $4.73\times 10^{-2}$ ${\rm cm}^2$ ${\rm day}^{-1}$ estimated $\delta_X$ diffusion coefficient of $X$ $0.27$ ${\rm cm}^2$ ${\rm day}^{-1}$ estimated $\sigma_0$ flux rate of $T$ cells at the boundary 1 ${\rm cm}^{-1}$ [8] $\chi_M$ chemoattraction coefficient of MCP-1 $10$ ${\rm cm}^5/{\rm g}\cdot {\rm day}$ [13,14] $\lambda_{DC}$ activation rate of DCs by tumor cells $10$ ${\rm g}/{\rm cm}^3\cdot{\rm day}$ [16] $\lambda_{TI_{12}}$ activation rate of T cells by IL-12 $16.2$ ${\rm day}^{-1}$ estimated $\lambda_{M_1}$ activation rate of M1 macrophages $1.35$ ${\rm day}^{-1}$ [16] $\lambda_{M_2}$ activation rate of M2 macrophages $1.01$ ${\rm day}^{-1}$ [16] $\beta_{M_1}$ phenotype change rate of M1 to M2 macrophages $0.3$ ${\rm day}^{-1}$ estimated $\beta_{M_2}$ phenotype change rate of M2 to M1 macrophages $4.68\times 10^{-3}$ ${\rm day}^{-1}$ estimated $\lambda_{C}$ growth rate of cancer cells $1.92$ ${\rm day}^{-1}$ estimated $\lambda_{I_{12}D}$ production rate of IL-12 by DCs $1.38\times 10^{-6}$ ${\rm day}^{-1}$ [16] $\lambda_{I_{12}M_1}$ production rate of IL-12 by M1 macrophages $5.52\times 10^{-6}$ ${\rm day}^{-1}$ [16] $\lambda_{T_\beta C}$ production rate of TGF-$\beta$ by cancer cells $2.79\times 10^{-10}$ ${\rm day}^{-1}$ estimated $\lambda_{T_\beta M_2}$ production rate of TGF-$\beta$ by M2 macrophages $6.97 \times 10^{-9}$ ${\rm day}^{-1}$ estimated $\lambda_{I_{10} C}$ production rate of IL-10 by cancer cells $2.07\times 10^{-8}$ ${\rm day}^{-1}$ [16] $\lambda_{I_{10} M_2}$ production rate of IL-10 by M2 macrophages $1.65\times 10^{-9}$ ${\rm day}^{-1}$ [16] $\lambda_{T_\alpha M_1}$ production rate of TNF $\alpha$ by M1 macrophages $1.36\times 10^{-5}$ ${\rm day}^{-1}$ [16] $\lambda_{T_\alpha T}$ production rate of TNF $\alpha$ by Th1 cells $9.06\times 10^{-8}$ ${\rm day}^{-1}$ estimated $\lambda_{M_PM_2}$ production rate of MCP-1 by M2 macrophages $1.2\times 10^{-8}$ ${\rm day}^{-1}$ [16] $\lambda_{M_PC}$ production rate of MCP-1 by cancer cells $8.24\times 10^{-7}$ ${\rm day}^{-1}$ [16] $d_{D}$ death rate of DCs 0.1 ${\rm day}^{-1}$ [8] $d_{T}$ death rate of T cells $0.18$ ${\rm day}^{-1}$ [8] $d_{M_1}$ death rate of M1 macrophage $0.015$ ${\rm day}^{-1}$ [9] $d_{M_2}$ death rate of M2 macrophage $0.015$ ${\rm day}^{-1}$ [9] $d_{C}$ death rate of tumor cells $0.17$ ${\rm day}^{-1}$ [8] $d_{I_{12}}$ degradation rate of IL-12 $1.38$ ${\rm day}^{-1}$ [8] $d_{T_\beta}$ degradation rate of TGF-$\beta$ $499.066$ ${\rm day}^{-1}$ [16] $d_{I_{10}}$ degradation rate of IL-10 $8.3178$ ${\rm day}^{-1}$ [16] $d_{T_\alpha}$ degradation rate of TGF-$\alpha$ $55.01$ ${\rm day}^{-1}$ [16] $d_{M_P}$ degradation rate of MCP-1 $55.01$ ${\rm day}^{-1}$ [10] $d_{A}$ degradation rate of anti-PD-1 $0.047$ ${\rm day}^{-1}$ [16] $d_X$ degradation rate of docetexel $1.11$ ${\rm day}^{-1}$ estimated
Summary of parameter values
 Notation Description Value used References $K_D$ half-saturation of dendritic cells $4\times 10^{-4}$ g/${\rm cm}^3$ [16] $K_{T}$ half-saturation of T cells $3\times 10^{-3}$ g/${\rm cm}^3$ [16] $K_{M_1}$ half-saturation of M1 macrophages $10^{-4}$ ${\rm g}/{\rm cm}^3$ [16] $K_{M_2}$ half-saturation of M2 macrophages $3.2\times 10^{-3}$ ${\rm g}/{\rm cm}^3$ [16] $K_{C}$ half-saturation of tumor cells $0.4$ g/${\rm cm}^3$ [8] $K_{I_{12}}$ half-saturation of IL-12 $8\times 10^{-10}$ g/${\rm cm}^3$ [16] $K_{T_\beta}$ half-saturation of TGF-$\beta$ $2.68\times 10^{-13}$ ${\rm g}/{\rm cm}^3$ [16] $K_{I_{10}}$ half-saturation of IL-10 $8.75\times 10^{-11}$ g/${\rm cm}^3$ [16] $K_{T_\alpha}$ half-saturation of TNF-$\alpha$ $3\times 10^{-11}$ g/${\rm cm}^3$ [11] $K_{M_P}$ half-saturation of MCP-1 $2\times 10^{-7}$ ${\rm g}/{\rm cm}^3$ [10] $K_{X}$ half-saturation of $X$ $8.02\times 10^{-11}$ ${\rm g}/{\rm cm}^3$ estimated $K_{TI_{10}}$ inhibition of function of T cells by IL-10 $4.375\times 10^{-11}$ g/${\rm cm}^3$ [16] $K_{TQ}$ inhibition of function of T cells by PD-1-PD-L1 $4.86\times 10^{-20}$ ${\rm g}^2/{\rm cm}^6$ estimated $K_{CX}$ inhibition of proliferation of cancer cells by docetexel $8.02\times 10^{-10}$ ${\rm g}/{\rm cm}^3$ estimated $D_0$ density of inactive DCs $2\times 10^{-5}$ g/${\rm cm}^3$ [8] $T_{0}$ density of naive T cells in tumor $6\times 10^{-4}$ g/${\rm cm}^3$ estimated $M_{10}$ density of monocytes $1.2\times 10^{-4}$ ${\rm g}/{\rm cm}^3$ [16] $M_{20}$ density of monocytes $3.84\times 10^{-3}$ ${\rm g}/{\rm cm}^3$ [16] $C_M$ carrying capacity of cancer cells $0.8$ g/${\rm cm}^3$ [8] $\hat T$ density of T cells from lymph node $6\times 10^{-3}$ g/${\rm cm}^3$ [16] $\eta$ killing rate of tumor cells by T cells $210$ ${\rm cm}^3/{\rm g}\cdot {\rm day}$ [16] $\mu_{XC}$ absorbtion rate of $X$ by cancer cells estimated $\rho_P$ expression of PD-1 in T cells $2.49\times 10^{-7}$ [15] $\rho_L$ expression of PD-L1 in T cells $3.25\times 10^{-7}$ [15]
 Notation Description Value used References $K_D$ half-saturation of dendritic cells $4\times 10^{-4}$ g/${\rm cm}^3$ [16] $K_{T}$ half-saturation of T cells $3\times 10^{-3}$ g/${\rm cm}^3$ [16] $K_{M_1}$ half-saturation of M1 macrophages $10^{-4}$ ${\rm g}/{\rm cm}^3$ [16] $K_{M_2}$ half-saturation of M2 macrophages $3.2\times 10^{-3}$ ${\rm g}/{\rm cm}^3$ [16] $K_{C}$ half-saturation of tumor cells $0.4$ g/${\rm cm}^3$ [8] $K_{I_{12}}$ half-saturation of IL-12 $8\times 10^{-10}$ g/${\rm cm}^3$ [16] $K_{T_\beta}$ half-saturation of TGF-$\beta$ $2.68\times 10^{-13}$ ${\rm g}/{\rm cm}^3$ [16] $K_{I_{10}}$ half-saturation of IL-10 $8.75\times 10^{-11}$ g/${\rm cm}^3$ [16] $K_{T_\alpha}$ half-saturation of TNF-$\alpha$ $3\times 10^{-11}$ g/${\rm cm}^3$ [11] $K_{M_P}$ half-saturation of MCP-1 $2\times 10^{-7}$ ${\rm g}/{\rm cm}^3$ [10] $K_{X}$ half-saturation of $X$ $8.02\times 10^{-11}$ ${\rm g}/{\rm cm}^3$ estimated $K_{TI_{10}}$ inhibition of function of T cells by IL-10 $4.375\times 10^{-11}$ g/${\rm cm}^3$ [16] $K_{TQ}$ inhibition of function of T cells by PD-1-PD-L1 $4.86\times 10^{-20}$ ${\rm g}^2/{\rm cm}^6$ estimated $K_{CX}$ inhibition of proliferation of cancer cells by docetexel $8.02\times 10^{-10}$ ${\rm g}/{\rm cm}^3$ estimated $D_0$ density of inactive DCs $2\times 10^{-5}$ g/${\rm cm}^3$ [8] $T_{0}$ density of naive T cells in tumor $6\times 10^{-4}$ g/${\rm cm}^3$ estimated $M_{10}$ density of monocytes $1.2\times 10^{-4}$ ${\rm g}/{\rm cm}^3$ [16] $M_{20}$ density of monocytes $3.84\times 10^{-3}$ ${\rm g}/{\rm cm}^3$ [16] $C_M$ carrying capacity of cancer cells $0.8$ g/${\rm cm}^3$ [8] $\hat T$ density of T cells from lymph node $6\times 10^{-3}$ g/${\rm cm}^3$ [16] $\eta$ killing rate of tumor cells by T cells $210$ ${\rm cm}^3/{\rm g}\cdot {\rm day}$ [16] $\mu_{XC}$ absorbtion rate of $X$ by cancer cells estimated $\rho_P$ expression of PD-1 in T cells $2.49\times 10^{-7}$ [15] $\rho_L$ expression of PD-L1 in T cells $3.25\times 10^{-7}$ [15]
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