
-
Previous Article
Long time localization of modified surface quasi-geostrophic equations
- DCDS-B Home
- This Issue
-
Next Article
Weak pullback attractors for stochastic Ginzburg-Landau equations in Bochner spaces
Combined therapy for treating solid tumors with chemotherapy and angiogenic inhibitors
1. | Department of Nuclear Engineering, University of California at Berkeley, Berkeley, CA 94270, USA |
2. | Department of Mathematics and Statistics, Loyola University Chicago, Chicago, IL 60660, USA |
Anti-angiogenesis therapy has been an emerging cancer treatment which may be further combined with chemotherapy to enhance overall survival of cancer patients. In this paper, we investigate a system of nonlinear ordinary differential equations describing a microenvironment consisting of host cells, tumor cells, immune cells and endothelial cells while incorporating treatment combination with chemotherapy and anti-angiogenesis therapy. We perform a dynamical systems analysis demonstrating that our model is able to capture the three phases of cancer immunoediting: elimination, equilibrium, and escape. In addition, we present transcritical bifurcations for relevant parameter values that correspond to the progression from the elimination phase to the equilibrium phase. A range of medically useful tumor doubling times were simulated to determine how combined therapy affects the tumor microenvironment over the course of a 250 day treatment. This analysis found two additional bifurcation parameters that move the system of equations from the equilibrium phase to the elimination phase. We determine that the most important aspect of an effective therapy is the activation of the anti-tumor immune response.
References:
[1] |
Cancer statistics, URL https://www.cancer.org/research/cancer-facts-statistics/, Accessed: 2020-03-24 Google Scholar |
[2] |
Scipy.integrate.ode, https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.ode.html, Accessed: 2019-02-19 Google Scholar |
[3] |
P. M. Altrock, L. L. Liu and F. Michor,
The mathematics of cancer: Integrating quantitative models, Nature Reviews Cancer, 15 (2015), 730-745.
doi: 10.1038/nrc4029. |
[4] |
A. R. Anderson and V. Quaranta, Integrative mathematical oncology, Nature Reviews Cancer, 8 (2008), 227. Google Scholar |
[5] |
P. Beckett, J. Edwards, D. Fennell, R. Hubbard, I. Woolhouse and M. Peake,
Demographics, management and survival of patients with malignant pleural mesothelioma in the national lung cancer audit in england and wales, Lung Cancer, 88 (2015), 344-348.
doi: 10.1016/j.lungcan.2015.03.005. |
[6] |
S. Benzekry, G. Chapuisat, J. Ciccolini, A. Erlinger and F. Hubert,
A new mathematical model for optimizing the combination between antiangiogenic and cytotoxic drugs in oncology, Comptes Rendus Mathematique, 350 (2012), 23-28.
doi: 10.1016/j.crma.2011.11.019. |
[7] |
G. Bergers and L. E. Benjamin,
Angiogenesis: Tumorigenesis and the angiogenic switch, Nature Reviews Cancer, 3 (2003), 401-410.
doi: 10.1038/nrc1093. |
[8] |
M. J. Bissell and W. C. Hines,
Why don't we get more cancer? A proposed role of the microenvironment in restraining cancer progression, Nature Medicine, 17 (2011), 320-329.
doi: 10.1038/nm.2328. |
[9] |
R. M. Bremnes, T. Dønnem, S. Al-Saad, K. Al-Shibli, S. Andersen, R. Sirera, C. Camps, I. Marinez and L.-T. Busund,
The role of tumor stroma in cancer progression and prognosis: Emphasis on carcinoma-associated fibroblasts and non-small cell lung cancer, Journal of Thoracic Oncology, 6 (2011), 209-217.
doi: 10.1097/JTO.0b013e3181f8a1bd. |
[10] |
F. M. Burnet,
Immunological aspects of malignant disease, The Lancet, 289 (1967), 1171-1174.
doi: 10.1016/S0140-6736(67)92837-1. |
[11] |
H. M. Byrne, Dissecting cancer through mathematics: From the cell to the animal model, Nature Reviews Cancer, 10 (2010), 221.
doi: 10.1038/nri2758. |
[12] |
L. Cancer, Mesothelioma lung protocols, Journal of the National Comprehensive Cancer Network, 1 (2015), 01-14. Google Scholar |
[13] |
M. Carbone, Y. I. Baris, P. Bertino, B. Brass, S. Comertpay, A. U. Dogan, G. Gaudino, S. Jube, S. Kanodia, C. R. Partridge, et al, Erionite exposure in north dakota and turkish villages with mesothelioma, Proceedings of the National Academy of Sciences, 108 (2011), 618–13,623. Google Scholar |
[14] |
G. J. Chu, N. van Zandwijk and J. E. J. Rasko,
The immune microenvironment in mesothelioma: Mechanisms of resistance to immunotherapy, Frontiers in Oncology, 69 (2019), 1-12.
doi: 10.3389/fonc.2019.01366. |
[15] |
L. de Pillis, W. Gu and A. E. Radunskaya, Mixed immunotherapy and chemotherapy of tumors: modeling, applications and biological interpretations, Journal of Theoretical Biology, 238 (2006), 841–862. URL http://www.sciencedirect.com/science/article/pii/S0022519305002936
doi: 10.1016/j.jtbi.2005.06.037. |
[16] |
L. G. De Pillis and A. Radunskaya,
A mathematical tumor model with immune resistance and drug therapy: an optimal control approach, Computational and Mathematical Methods in Medicine, 3 (2001), 79-100.
doi: 10.1080/10273660108833067. |
[17] |
L. G. de Pillis, A. E. Radunskaya and C. L. Wiseman,
A validated mathematical model of cell-mediated immune response to tumor growth, Cancer Research, 65 (2005), 7950-7958.
doi: 10.1158/0008-5472.CAN-05-0564. |
[18] |
K. E. De Visser, A. Eichten and L. M. Coussens,
Paradoxical roles of the immune system during cancer development, Nature Reviews Cancer, 6 (2006), 24-37.
doi: 10.1038/nrc1782. |
[19] |
G. P. Dunn, A. T. Bruce, H. Ikeda, L. J. Old and R. D. Schreiber,
Cancer immunoediting: From immunosurveillance to tumor escape, Nature Immunology, 3 (2002), 991-998.
doi: 10.1038/ni1102-991. |
[20] |
A. d'Onofrio and A. Gandolfi,
Tumour eradication by antiangiogenic therapy: Analysis and extensions of the model by hahnfeldt et al.(1999), Mathematical Biosciences, 191 (2004), 159-184.
doi: 10.1016/j.mbs.2004.06.003. |
[21] |
A. d'Onofrio, U. Ledzewicz, H. Maurer and H. Schättler,
On optimal delivery of combination therapy for tumors, Mathematical Biosciences, 222 (2009), 13-26.
doi: 10.1016/j.mbs.2009.08.004. |
[22] |
A. Ergun, K. Camphausen and L. M. Wein,
Optimal scheduling of radiotherapy and angiogenic inhibitors, Bulletin of Mathematical Biology, 65 (2003), 407-424.
doi: 10.1016/S0092-8240(03)00006-5. |
[23] |
D. S. Ettinger, D. E. Wood, W. Akerley, L. A. Bazhenova, H. Borghaei, D. R. Camidge, R. T. Cheney, L. R. Chirieac, T. A. D'amico and T. Dilling, Nccn guidelines insights: malignant pleural mesothelioma, version 3.2016, Journal of the National Comprehensive Cancer Network, 14 (2016), 825-836. Google Scholar |
[24] |
R. A. Gatenby,
Application of competition theory to tumour growth: Implications for tumour biology and treatment, European Journal of Cancer, 32 (1996), 722-726.
doi: 10.1016/0959-8049(95)00658-3. |
[25] |
A. E. Glick and A. Mastroberardino, An optimal control approach for the treatment of solid tumors with angiogenesis inhibitors, Mathematics, 5 (2017), 49.
doi: 10.3390/math5040049. |
[26] |
P. Hahnfeldt, D. Panigrahy, J. Folkman and L. Hlatky, Tumor development under angiogenic signaling: A dynamical theory of tumor growth, treatment response, and postvascular dormancy, Cancer Research, 59 (1999), 4770-4775. Google Scholar |
[27] |
K. Han, T. Peyret, M. Marchand, A. Quartino, N. H. Gosselin, S. Girish, D. E. Allison and J. Jin,
Population pharmacokinetics of bevacizumab in cancer patients with external validation, Cancer Chemotherapy and Pharmacology, 78 (2016), 341-351.
doi: 10.1007/s00280-016-3079-6. |
[28] |
D. Hanahan and R. A. Weinberg,
The hallmarks of cancer, Cell, 100 (2000), 57-70.
doi: 10.1016/S0092-8674(00)81683-9. |
[29] |
N. G. Insights,
Malignant pleural mesothelioma, Journal of the National Comprehensive Cancer Network, 14 (2016), 825-836.
doi: 10.6004/jnccn.2016.0087. |
[30] |
R. Kim, M. Emi and K. Tanabe,
Cancer immunoediting from immune surveillance to immune escape, Immunology, 121 (2007), 1-14.
doi: 10.1111/j.1365-2567.2007.02587.x. |
[31] |
D. Kirschner and J. C. Panetta,
Modeling immunotherapy of the tumor–immune interaction, Journal of Mathematical Biology, 37 (1998), 235-252.
doi: 10.1007/s002850050127. |
[32] |
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.
doi: 10.1016/S0092-8240(05)80260-5. |
[33] |
L. Laplane, D. Duluc, N. Larmonier, T. Pradeu and A. Bikfalvi,
The multiple layers of the tumor environment, Trends in Cancer, 4 (2018), 802-809.
doi: 10.1016/j.trecan.2018.10.002. |
[34] |
U. Ledzewicz, H. Maurer and H. Schaettler, Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy, Math. Biosci. Eng., 8 (2011), 307–323.
doi: 10.3934/mbe.2011.8.307. |
[35] |
U. Ledzewicz and H. Schättler,
Optimal bang-bang controls for a two-compartment model in cancer chemotherapy, Journal of Optimization Theory and Applications, 114 (2002), 609-637.
doi: 10.1023/A:1016027113579. |
[36] |
U. Ledzewicz and H. Schättler,
Antiangiogenic therapy in cancer treatment as an optimal control problem, SIAM Journal on Control and Optimization, 46 (2007), 1052-1079.
doi: 10.1137/060665294. |
[37] |
C. Letellier, S. K. Sasmal, C. Draghi, F. Denis and D. Ghosh,
A chemotherapy combined with an anti-angiogenic drug applied to a cancer model including angiogenesis, Chaos, Solitons & Fractals, 99 (2017), 297-311.
doi: 10.1016/j.chaos.2017.04.013. |
[38] |
L. A. Liotta and E. C. Kohn, The microenvironment of the tumour-host interface, Nature, 411 (2001), 375. Google Scholar |
[39] |
R. Liu, B. D. Ferguson, Y. Zhou, K. Naga, R. Salgia, P. S. Gill and V. Krasnoperov,
Ephb4 as a therapeutic target in mesothelioma, BMC Cancer, 13 (2013), 01-07.
doi: 10.1186/1471-2407-13-269. |
[40] |
Á. G. López, J. M. Seoane and M. A. F. Sanjuán,
A validated mathematical model of tumor growth including tumor–host interaction, cell-mediated immune response and chemotherapy, Bulletin of Mathematical Biology, 76 (2014), 2884-2906.
doi: 10.1007/s11538-014-0037-5. |
[41] |
J. McGeachie, Blue histology - vascular system more about... endothelial cells, http://www.lab.anhb.uwa.edu.au/mb140/moreabout/endothel.htm Google Scholar |
[42] |
S. Michelson, B. E. Miller, A. S. Glicksman and J. T. Leith,
Tumor micro-ecology and competitive interactions, Journal of Theoretical Biology, 128 (1987), 233-246.
doi: 10.1016/S0022-5193(87)80171-6. |
[43] |
L. Mutti,
Scientific advances and new frontiers in mesothelioma therapeutics, Journal of Thoracic Oncology, 13 (2018), 1269-1283.
doi: 10.1016/j.jtho.2018.06.011. |
[44] |
B. Muz, P. de la Puente, F. Azab and A. K. Azab,
The role of hypoxia in cancer progression, angiogenesis, metastasis, and resistance to therapy, Hypoxia, 3 (2015), 83-92.
doi: 10.2147/HP.S93413. |
[45] |
K. R. Fister and J. C. Panetta,
Optimal control applied to competing chemotherapeutic cell-kill strategies, SIAM Journal on Applied Mathematics, 63 (2003), 1954-1971.
doi: 10.1137/S0036139902413489. |
[46] |
C. R. Parish,
Cancer immunotherapy: The past, the present and the future, Immunology and Cell Biology, 81 (2003), 106-113.
doi: 10.1046/j.0818-9641.2003.01151.x. |
[47] |
L. Petzold,
Automatic selection of methods for solving stiff and nonstiff systems of ordinary differential equations, SIAM Journal on Scientific and Statistical Computing, 4 (1983), 137-148.
doi: 10.1137/0904010. |
[48] |
S. T. R. Pinho, F. S. Bacelar, R. F. S. Andrade and H. I. Freedman,
A mathematical model for the effect of anti-angiogenic therapy in the treatment of cancer tumours by chemotherapy, Nonlinear Analysis: Real World Applications, 14 (2013), 815-828.
doi: 10.1016/j.nonrwa.2012.07.034. |
[49] |
D. Ribatti, The concept of immune surveillance against tumors: The first theories, Oncotarget, 8 (2017), 7175. Google Scholar |
[50] |
K. Roesch, D. Hasenclever and M. Scholz,
Modelling lymphoma therapy and outcome, Bulletin of Mathematical Biology, 76 (2014), 401-430.
doi: 10.1007/s11538-013-9925-3. |
[51] |
H. Schättler and U. Ledzewicz, Optimal Control for Mathematical Models of Cancer Therapies, vol. 42., Springer, New York, 2015
doi: 10.1007/978-1-4939-2972-6. |
[52] |
H. Schättler, U. Ledzewicz and B. Amini,
Dynamical properties of a minimally parameterized mathematical model for metronomic chemotherapy, Journal of Mathematical Biology, 72 (2016), 1255-1280.
doi: 10.1007/s00285-015-0907-y. |
[53] |
H. E. Skipper,
On mathematical modeling of critical variables in cancer treatment (goals: Better understanding of the past and better planning in the future), Bulletin of Mathematical Biology, 48 (1986), 253-278.
doi: 10.1016/S0092-8240(86)90027-3. |
[54] |
N. V. Stepanova, Course of the immune reaction during the development of a malignant tumor, Biophysics, 24 (1980), 917-923. Google Scholar |
[55] |
G. W. Tormoen, M. R. Crittenden and M. J. Gough,
Role of the immunosuppressive microenvironment in immunotherapy, Advances in Radiation Oncology, 3 (2018), 520-526.
doi: 10.1016/j.adro.2018.08.018. |
[56] |
N. S. Vasudev and A. R. Reynolds,
Anti-angiogenic therapy for cancer: Current progress, unresolved questions and future directions, Angiogenesis, 17 (2014), 471-494.
doi: 10.1007/s10456-014-9420-y. |
[57] |
L. Viger, F. Denis, M. Rosalie and C. Letellier,
A cancer model for the angiogenic switch, Journal of Theoretical Biology, 360 (2014), 21-33.
doi: 10.1016/j.jtbi.2014.06.020. |
[58] |
R. A. Weinberg and R. A. Weinberg, The Biology of Cancer, New York, 2013.
doi: 10.1201/9780429258794. |
[59] |
L. Wu, Z. Yun, T. Tagawa, K. Rey-McIntyre and M. de Perrot,
Ctla-4 blockade expands infiltrating t cells and inhibits cancer cell repopulation during the intervals of chemotherapy in murine mesothelioma, Molecular Cancer Therapeutics, 11 (2012), 1809-1819.
doi: 10.1158/1535-7163.MCT-11-1014. |
[60] |
Y. Xie, Y. Zhong, T. Gao, X. Zhang, L. Li, H. Ruan and D. Li,
Human lymphatic endothelial cells contribute to epithelial ovarian carcinoma metastasis by promoting lymphangiogenesis and tumour cell invasion, Experimental and Therapeutic Medicine, 11 (2016), 1587-1594.
doi: 10.3892/etm.2016.3134. |
[61] |
S. Yonucu, D. Ylmaz, C. Phipps, M. B. Unlu and M. Kohandel,
Quantifying the effects of antiangiogenic and chemotherapy drug combinations on drug delivery and treatment efficacy, PLOS Computational Biology, 13 (2017), 1-17.
doi: 10.1371/journal.pcbi.1005724. |
[62] |
G. Zalcman, J. Mazieres, J. Margery, L. Greillier, C. Audigier-Valette, D. Moro-Sibilot, O. Molinier, R. Corre, I. Monnet and V. Gounant,
Bevacizumab for newly diagnosed pleural mesothelioma in the mesothelioma avastin cisplatin pemetrexed study (maps): A randomised, controlled, open-label, phase 3 trial, The Lancet, 387 (2016), 1405-1414.
doi: 10.1016/S0140-6736(15)01238-6. |
show all references
References:
[1] |
Cancer statistics, URL https://www.cancer.org/research/cancer-facts-statistics/, Accessed: 2020-03-24 Google Scholar |
[2] |
Scipy.integrate.ode, https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.ode.html, Accessed: 2019-02-19 Google Scholar |
[3] |
P. M. Altrock, L. L. Liu and F. Michor,
The mathematics of cancer: Integrating quantitative models, Nature Reviews Cancer, 15 (2015), 730-745.
doi: 10.1038/nrc4029. |
[4] |
A. R. Anderson and V. Quaranta, Integrative mathematical oncology, Nature Reviews Cancer, 8 (2008), 227. Google Scholar |
[5] |
P. Beckett, J. Edwards, D. Fennell, R. Hubbard, I. Woolhouse and M. Peake,
Demographics, management and survival of patients with malignant pleural mesothelioma in the national lung cancer audit in england and wales, Lung Cancer, 88 (2015), 344-348.
doi: 10.1016/j.lungcan.2015.03.005. |
[6] |
S. Benzekry, G. Chapuisat, J. Ciccolini, A. Erlinger and F. Hubert,
A new mathematical model for optimizing the combination between antiangiogenic and cytotoxic drugs in oncology, Comptes Rendus Mathematique, 350 (2012), 23-28.
doi: 10.1016/j.crma.2011.11.019. |
[7] |
G. Bergers and L. E. Benjamin,
Angiogenesis: Tumorigenesis and the angiogenic switch, Nature Reviews Cancer, 3 (2003), 401-410.
doi: 10.1038/nrc1093. |
[8] |
M. J. Bissell and W. C. Hines,
Why don't we get more cancer? A proposed role of the microenvironment in restraining cancer progression, Nature Medicine, 17 (2011), 320-329.
doi: 10.1038/nm.2328. |
[9] |
R. M. Bremnes, T. Dønnem, S. Al-Saad, K. Al-Shibli, S. Andersen, R. Sirera, C. Camps, I. Marinez and L.-T. Busund,
The role of tumor stroma in cancer progression and prognosis: Emphasis on carcinoma-associated fibroblasts and non-small cell lung cancer, Journal of Thoracic Oncology, 6 (2011), 209-217.
doi: 10.1097/JTO.0b013e3181f8a1bd. |
[10] |
F. M. Burnet,
Immunological aspects of malignant disease, The Lancet, 289 (1967), 1171-1174.
doi: 10.1016/S0140-6736(67)92837-1. |
[11] |
H. M. Byrne, Dissecting cancer through mathematics: From the cell to the animal model, Nature Reviews Cancer, 10 (2010), 221.
doi: 10.1038/nri2758. |
[12] |
L. Cancer, Mesothelioma lung protocols, Journal of the National Comprehensive Cancer Network, 1 (2015), 01-14. Google Scholar |
[13] |
M. Carbone, Y. I. Baris, P. Bertino, B. Brass, S. Comertpay, A. U. Dogan, G. Gaudino, S. Jube, S. Kanodia, C. R. Partridge, et al, Erionite exposure in north dakota and turkish villages with mesothelioma, Proceedings of the National Academy of Sciences, 108 (2011), 618–13,623. Google Scholar |
[14] |
G. J. Chu, N. van Zandwijk and J. E. J. Rasko,
The immune microenvironment in mesothelioma: Mechanisms of resistance to immunotherapy, Frontiers in Oncology, 69 (2019), 1-12.
doi: 10.3389/fonc.2019.01366. |
[15] |
L. de Pillis, W. Gu and A. E. Radunskaya, Mixed immunotherapy and chemotherapy of tumors: modeling, applications and biological interpretations, Journal of Theoretical Biology, 238 (2006), 841–862. URL http://www.sciencedirect.com/science/article/pii/S0022519305002936
doi: 10.1016/j.jtbi.2005.06.037. |
[16] |
L. G. De Pillis and A. Radunskaya,
A mathematical tumor model with immune resistance and drug therapy: an optimal control approach, Computational and Mathematical Methods in Medicine, 3 (2001), 79-100.
doi: 10.1080/10273660108833067. |
[17] |
L. G. de Pillis, A. E. Radunskaya and C. L. Wiseman,
A validated mathematical model of cell-mediated immune response to tumor growth, Cancer Research, 65 (2005), 7950-7958.
doi: 10.1158/0008-5472.CAN-05-0564. |
[18] |
K. E. De Visser, A. Eichten and L. M. Coussens,
Paradoxical roles of the immune system during cancer development, Nature Reviews Cancer, 6 (2006), 24-37.
doi: 10.1038/nrc1782. |
[19] |
G. P. Dunn, A. T. Bruce, H. Ikeda, L. J. Old and R. D. Schreiber,
Cancer immunoediting: From immunosurveillance to tumor escape, Nature Immunology, 3 (2002), 991-998.
doi: 10.1038/ni1102-991. |
[20] |
A. d'Onofrio and A. Gandolfi,
Tumour eradication by antiangiogenic therapy: Analysis and extensions of the model by hahnfeldt et al.(1999), Mathematical Biosciences, 191 (2004), 159-184.
doi: 10.1016/j.mbs.2004.06.003. |
[21] |
A. d'Onofrio, U. Ledzewicz, H. Maurer and H. Schättler,
On optimal delivery of combination therapy for tumors, Mathematical Biosciences, 222 (2009), 13-26.
doi: 10.1016/j.mbs.2009.08.004. |
[22] |
A. Ergun, K. Camphausen and L. M. Wein,
Optimal scheduling of radiotherapy and angiogenic inhibitors, Bulletin of Mathematical Biology, 65 (2003), 407-424.
doi: 10.1016/S0092-8240(03)00006-5. |
[23] |
D. S. Ettinger, D. E. Wood, W. Akerley, L. A. Bazhenova, H. Borghaei, D. R. Camidge, R. T. Cheney, L. R. Chirieac, T. A. D'amico and T. Dilling, Nccn guidelines insights: malignant pleural mesothelioma, version 3.2016, Journal of the National Comprehensive Cancer Network, 14 (2016), 825-836. Google Scholar |
[24] |
R. A. Gatenby,
Application of competition theory to tumour growth: Implications for tumour biology and treatment, European Journal of Cancer, 32 (1996), 722-726.
doi: 10.1016/0959-8049(95)00658-3. |
[25] |
A. E. Glick and A. Mastroberardino, An optimal control approach for the treatment of solid tumors with angiogenesis inhibitors, Mathematics, 5 (2017), 49.
doi: 10.3390/math5040049. |
[26] |
P. Hahnfeldt, D. Panigrahy, J. Folkman and L. Hlatky, Tumor development under angiogenic signaling: A dynamical theory of tumor growth, treatment response, and postvascular dormancy, Cancer Research, 59 (1999), 4770-4775. Google Scholar |
[27] |
K. Han, T. Peyret, M. Marchand, A. Quartino, N. H. Gosselin, S. Girish, D. E. Allison and J. Jin,
Population pharmacokinetics of bevacizumab in cancer patients with external validation, Cancer Chemotherapy and Pharmacology, 78 (2016), 341-351.
doi: 10.1007/s00280-016-3079-6. |
[28] |
D. Hanahan and R. A. Weinberg,
The hallmarks of cancer, Cell, 100 (2000), 57-70.
doi: 10.1016/S0092-8674(00)81683-9. |
[29] |
N. G. Insights,
Malignant pleural mesothelioma, Journal of the National Comprehensive Cancer Network, 14 (2016), 825-836.
doi: 10.6004/jnccn.2016.0087. |
[30] |
R. Kim, M. Emi and K. Tanabe,
Cancer immunoediting from immune surveillance to immune escape, Immunology, 121 (2007), 1-14.
doi: 10.1111/j.1365-2567.2007.02587.x. |
[31] |
D. Kirschner and J. C. Panetta,
Modeling immunotherapy of the tumor–immune interaction, Journal of Mathematical Biology, 37 (1998), 235-252.
doi: 10.1007/s002850050127. |
[32] |
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.
doi: 10.1016/S0092-8240(05)80260-5. |
[33] |
L. Laplane, D. Duluc, N. Larmonier, T. Pradeu and A. Bikfalvi,
The multiple layers of the tumor environment, Trends in Cancer, 4 (2018), 802-809.
doi: 10.1016/j.trecan.2018.10.002. |
[34] |
U. Ledzewicz, H. Maurer and H. Schaettler, Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy, Math. Biosci. Eng., 8 (2011), 307–323.
doi: 10.3934/mbe.2011.8.307. |
[35] |
U. Ledzewicz and H. Schättler,
Optimal bang-bang controls for a two-compartment model in cancer chemotherapy, Journal of Optimization Theory and Applications, 114 (2002), 609-637.
doi: 10.1023/A:1016027113579. |
[36] |
U. Ledzewicz and H. Schättler,
Antiangiogenic therapy in cancer treatment as an optimal control problem, SIAM Journal on Control and Optimization, 46 (2007), 1052-1079.
doi: 10.1137/060665294. |
[37] |
C. Letellier, S. K. Sasmal, C. Draghi, F. Denis and D. Ghosh,
A chemotherapy combined with an anti-angiogenic drug applied to a cancer model including angiogenesis, Chaos, Solitons & Fractals, 99 (2017), 297-311.
doi: 10.1016/j.chaos.2017.04.013. |
[38] |
L. A. Liotta and E. C. Kohn, The microenvironment of the tumour-host interface, Nature, 411 (2001), 375. Google Scholar |
[39] |
R. Liu, B. D. Ferguson, Y. Zhou, K. Naga, R. Salgia, P. S. Gill and V. Krasnoperov,
Ephb4 as a therapeutic target in mesothelioma, BMC Cancer, 13 (2013), 01-07.
doi: 10.1186/1471-2407-13-269. |
[40] |
Á. G. López, J. M. Seoane and M. A. F. Sanjuán,
A validated mathematical model of tumor growth including tumor–host interaction, cell-mediated immune response and chemotherapy, Bulletin of Mathematical Biology, 76 (2014), 2884-2906.
doi: 10.1007/s11538-014-0037-5. |
[41] |
J. McGeachie, Blue histology - vascular system more about... endothelial cells, http://www.lab.anhb.uwa.edu.au/mb140/moreabout/endothel.htm Google Scholar |
[42] |
S. Michelson, B. E. Miller, A. S. Glicksman and J. T. Leith,
Tumor micro-ecology and competitive interactions, Journal of Theoretical Biology, 128 (1987), 233-246.
doi: 10.1016/S0022-5193(87)80171-6. |
[43] |
L. Mutti,
Scientific advances and new frontiers in mesothelioma therapeutics, Journal of Thoracic Oncology, 13 (2018), 1269-1283.
doi: 10.1016/j.jtho.2018.06.011. |
[44] |
B. Muz, P. de la Puente, F. Azab and A. K. Azab,
The role of hypoxia in cancer progression, angiogenesis, metastasis, and resistance to therapy, Hypoxia, 3 (2015), 83-92.
doi: 10.2147/HP.S93413. |
[45] |
K. R. Fister and J. C. Panetta,
Optimal control applied to competing chemotherapeutic cell-kill strategies, SIAM Journal on Applied Mathematics, 63 (2003), 1954-1971.
doi: 10.1137/S0036139902413489. |
[46] |
C. R. Parish,
Cancer immunotherapy: The past, the present and the future, Immunology and Cell Biology, 81 (2003), 106-113.
doi: 10.1046/j.0818-9641.2003.01151.x. |
[47] |
L. Petzold,
Automatic selection of methods for solving stiff and nonstiff systems of ordinary differential equations, SIAM Journal on Scientific and Statistical Computing, 4 (1983), 137-148.
doi: 10.1137/0904010. |
[48] |
S. T. R. Pinho, F. S. Bacelar, R. F. S. Andrade and H. I. Freedman,
A mathematical model for the effect of anti-angiogenic therapy in the treatment of cancer tumours by chemotherapy, Nonlinear Analysis: Real World Applications, 14 (2013), 815-828.
doi: 10.1016/j.nonrwa.2012.07.034. |
[49] |
D. Ribatti, The concept of immune surveillance against tumors: The first theories, Oncotarget, 8 (2017), 7175. Google Scholar |
[50] |
K. Roesch, D. Hasenclever and M. Scholz,
Modelling lymphoma therapy and outcome, Bulletin of Mathematical Biology, 76 (2014), 401-430.
doi: 10.1007/s11538-013-9925-3. |
[51] |
H. Schättler and U. Ledzewicz, Optimal Control for Mathematical Models of Cancer Therapies, vol. 42., Springer, New York, 2015
doi: 10.1007/978-1-4939-2972-6. |
[52] |
H. Schättler, U. Ledzewicz and B. Amini,
Dynamical properties of a minimally parameterized mathematical model for metronomic chemotherapy, Journal of Mathematical Biology, 72 (2016), 1255-1280.
doi: 10.1007/s00285-015-0907-y. |
[53] |
H. E. Skipper,
On mathematical modeling of critical variables in cancer treatment (goals: Better understanding of the past and better planning in the future), Bulletin of Mathematical Biology, 48 (1986), 253-278.
doi: 10.1016/S0092-8240(86)90027-3. |
[54] |
N. V. Stepanova, Course of the immune reaction during the development of a malignant tumor, Biophysics, 24 (1980), 917-923. Google Scholar |
[55] |
G. W. Tormoen, M. R. Crittenden and M. J. Gough,
Role of the immunosuppressive microenvironment in immunotherapy, Advances in Radiation Oncology, 3 (2018), 520-526.
doi: 10.1016/j.adro.2018.08.018. |
[56] |
N. S. Vasudev and A. R. Reynolds,
Anti-angiogenic therapy for cancer: Current progress, unresolved questions and future directions, Angiogenesis, 17 (2014), 471-494.
doi: 10.1007/s10456-014-9420-y. |
[57] |
L. Viger, F. Denis, M. Rosalie and C. Letellier,
A cancer model for the angiogenic switch, Journal of Theoretical Biology, 360 (2014), 21-33.
doi: 10.1016/j.jtbi.2014.06.020. |
[58] |
R. A. Weinberg and R. A. Weinberg, The Biology of Cancer, New York, 2013.
doi: 10.1201/9780429258794. |
[59] |
L. Wu, Z. Yun, T. Tagawa, K. Rey-McIntyre and M. de Perrot,
Ctla-4 blockade expands infiltrating t cells and inhibits cancer cell repopulation during the intervals of chemotherapy in murine mesothelioma, Molecular Cancer Therapeutics, 11 (2012), 1809-1819.
doi: 10.1158/1535-7163.MCT-11-1014. |
[60] |
Y. Xie, Y. Zhong, T. Gao, X. Zhang, L. Li, H. Ruan and D. Li,
Human lymphatic endothelial cells contribute to epithelial ovarian carcinoma metastasis by promoting lymphangiogenesis and tumour cell invasion, Experimental and Therapeutic Medicine, 11 (2016), 1587-1594.
doi: 10.3892/etm.2016.3134. |
[61] |
S. Yonucu, D. Ylmaz, C. Phipps, M. B. Unlu and M. Kohandel,
Quantifying the effects of antiangiogenic and chemotherapy drug combinations on drug delivery and treatment efficacy, PLOS Computational Biology, 13 (2017), 1-17.
doi: 10.1371/journal.pcbi.1005724. |
[62] |
G. Zalcman, J. Mazieres, J. Margery, L. Greillier, C. Audigier-Valette, D. Moro-Sibilot, O. Molinier, R. Corre, I. Monnet and V. Gounant,
Bevacizumab for newly diagnosed pleural mesothelioma in the mesothelioma avastin cisplatin pemetrexed study (maps): A randomised, controlled, open-label, phase 3 trial, The Lancet, 387 (2016), 1405-1414.
doi: 10.1016/S0140-6736(15)01238-6. |







Parameter | Description | Value | Units | Source |
Host cell growth parameter | 1.8 |
day |
[40] | |
Tumor growth parameter | day |
estimated | ||
Tumor doubling time | 2 |
day | estimated | |
Endothelial cell growth parameter | 2.15 |
day |
[60] | |
Tumor (VEGF) recruitment of endothelial cells | 9.22 |
day |
[25] | |
Influx of immune cells | 1.0 |
cell day |
[32] | |
Immune cell natural death rate | 7.0 |
day |
[50] | |
Inverse of host cell carrying capacity | 1.0 |
cell |
[40] | |
Tumor carrying capacity dependence parameter | 0.8 | no units | estimated | |
Tumor cell carrying capacity | 1.0 |
cells | [25] | |
Inverse of endothelial cell carrying capacity | 1.0 |
cell |
[25] | |
Host cell killing rate by tumor cells | 4.8 |
cell |
[40] | |
Immune cell response to tumor cell presence | 1.101 |
cell |
[40] | |
Linear immune cell inactivation rate by tumor cells | 2.8 |
cell |
[40] | |
Quadratic immune cell inactivation rate by tumor cells | 3.2 |
cell |
[40] |
Parameter | Description | Value | Units | Source |
Host cell growth parameter | 1.8 |
day |
[40] | |
Tumor growth parameter | day |
estimated | ||
Tumor doubling time | 2 |
day | estimated | |
Endothelial cell growth parameter | 2.15 |
day |
[60] | |
Tumor (VEGF) recruitment of endothelial cells | 9.22 |
day |
[25] | |
Influx of immune cells | 1.0 |
cell day |
[32] | |
Immune cell natural death rate | 7.0 |
day |
[50] | |
Inverse of host cell carrying capacity | 1.0 |
cell |
[40] | |
Tumor carrying capacity dependence parameter | 0.8 | no units | estimated | |
Tumor cell carrying capacity | 1.0 |
cells | [25] | |
Inverse of endothelial cell carrying capacity | 1.0 |
cell |
[25] | |
Host cell killing rate by tumor cells | 4.8 |
cell |
[40] | |
Immune cell response to tumor cell presence | 1.101 |
cell |
[40] | |
Linear immune cell inactivation rate by tumor cells | 2.8 |
cell |
[40] | |
Quadratic immune cell inactivation rate by tumor cells | 3.2 |
cell |
[40] |
Parameter | Description | Value | Units | Source |
AAT effect on tumor carrying capacity | 8.9 |
no units | estimated | |
AAT effect on endothelial cells | 9.088 |
no units | estimated | |
Clearance rate of AAT agent | day |
[15] | ||
Half life of AAT agent | 0.833 |
day | [61] [27] | |
Chemotherapy effect on immune cells | day |
estimated | ||
Chemotherapy effect on immune cells | 4.999 |
day |
estimated | |
Chemotherapy effect on tumor cells | 7.494 |
day |
estimated | |
Chemotherapy effect on endothelial cells | day |
estimated | ||
Clearance rate of chemotherapy agent | day |
[15] | ||
Half life of chemotherapy agent | 0.417 |
day | [61] |
Parameter | Description | Value | Units | Source |
AAT effect on tumor carrying capacity | 8.9 |
no units | estimated | |
AAT effect on endothelial cells | 9.088 |
no units | estimated | |
Clearance rate of AAT agent | day |
[15] | ||
Half life of AAT agent | 0.833 |
day | [61] [27] | |
Chemotherapy effect on immune cells | day |
estimated | ||
Chemotherapy effect on immune cells | 4.999 |
day |
estimated | |
Chemotherapy effect on tumor cells | 7.494 |
day |
estimated | |
Chemotherapy effect on endothelial cells | day |
estimated | ||
Clearance rate of chemotherapy agent | day |
[15] | ||
Half life of chemotherapy agent | 0.417 |
day | [61] |
Equilibrium Solution | Eigenvalues |
Equilibrium Solution | Eigenvalues |
Equilibrium Solution | Eigenvalues |
Equilibrium Solution | Eigenvalues |
[1] |
Didier Bresch, Thierry Colin, Emmanuel Grenier, Benjamin Ribba, Olivier Saut. A viscoelastic model for avascular tumor growth. Conference Publications, 2009, 2009 (Special) : 101-108. doi: 10.3934/proc.2009.2009.101 |
[2] |
Baba Issa Camara, Houda Mokrani, Evans K. Afenya. Mathematical modeling of glioma therapy using oncolytic viruses. Mathematical Biosciences & Engineering, 2013, 10 (3) : 565-578. doi: 10.3934/mbe.2013.10.565 |
[3] |
Vieri Benci, Sunra Mosconi, Marco Squassina. Preface: Applications of mathematical analysis to problems in theoretical physics. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : i-i. doi: 10.3934/dcdss.2020446 |
[4] |
Luigi Barletti, Giovanni Nastasi, Claudia Negulescu, Vittorio Romano. Mathematical modelling of charge transport in graphene heterojunctions. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021010 |
[5] |
A. K. Misra, Anupama Sharma, Jia Li. A mathematical model for control of vector borne diseases through media campaigns. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1909-1927. doi: 10.3934/dcdsb.2013.18.1909 |
[6] |
Ritu Agarwal, Kritika, Sunil Dutt Purohit, Devendra Kumar. Mathematical modelling of cytosolic calcium concentration distribution using non-local fractional operator. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021017 |
2019 Impact Factor: 1.27
Tools
Metrics
Other articles
by authors
[Back to Top]