doi: 10.3934/dcdsb.2020343

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

Received  April 2020 Revised  August 2020 Published  November 2020

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.

Citation: Adam Glick, Antonio Mastroberardino. Combined therapy for treating solid tumors with chemotherapy and angiogenic inhibitors. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020343
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. AltrockL. L. Liu and F. Michor, The mathematics of cancer: Integrating quantitative models, Nature Reviews Cancer, 15 (2015), 730-745.  doi: 10.1038/nrc4029.  Google Scholar

[4]

A. R. Anderson and V. Quaranta, Integrative mathematical oncology, Nature Reviews Cancer, 8 (2008), 227. Google Scholar

[5]

P. BeckettJ. EdwardsD. FennellR. HubbardI. 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.  Google Scholar

[6]

S. BenzekryG. ChapuisatJ. CiccoliniA. 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.  Google Scholar

[7]

G. Bergers and L. E. Benjamin, Angiogenesis: Tumorigenesis and the angiogenic switch, Nature Reviews Cancer, 3 (2003), 401-410.  doi: 10.1038/nrc1093.  Google Scholar

[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.  Google Scholar

[9]

R. M. BremnesT. DønnemS. Al-SaadK. Al-ShibliS. AndersenR. SireraC. CampsI. 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.  Google Scholar

[10]

F. M. Burnet, Immunological aspects of malignant disease, The Lancet, 289 (1967), 1171-1174.  doi: 10.1016/S0140-6736(67)92837-1.  Google Scholar

[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.  Google Scholar

[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. ChuN. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

L. G. de PillisA. 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.  Google Scholar

[18]

K. E. De VisserA. 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.  Google Scholar

[19]

G. P. DunnA. T. BruceH. IkedaL. J. Old and R. D. Schreiber, Cancer immunoediting: From immunosurveillance to tumor escape, Nature Immunology, 3 (2002), 991-998.  doi: 10.1038/ni1102-991.  Google Scholar

[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.  Google Scholar

[21]

A. d'OnofrioU. LedzewiczH. 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.  Google Scholar

[22]

A. ErgunK. 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.  Google Scholar

[23]

D. S. EttingerD. E. WoodW. AkerleyL. A. BazhenovaH. BorghaeiD. R. CamidgeR. T. CheneyL. R. ChirieacT. 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.  Google Scholar

[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.  Google Scholar

[26]

P. HahnfeldtD. PanigrahyJ. 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. HanT. PeyretM. MarchandA. QuartinoN. H. GosselinS. GirishD. 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.  Google Scholar

[28]

D. Hanahan and R. A. Weinberg, The hallmarks of cancer, Cell, 100 (2000), 57-70.  doi: 10.1016/S0092-8674(00)81683-9.  Google Scholar

[29]

N. G. Insights, Malignant pleural mesothelioma, Journal of the National Comprehensive Cancer Network, 14 (2016), 825-836.  doi: 10.6004/jnccn.2016.0087.  Google Scholar

[30]

R. KimM. 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.  Google Scholar

[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.  Google Scholar

[32]

V. A. KuznetsovI. A. MakalkinM. 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.  Google Scholar

[33]

L. LaplaneD. DulucN. LarmonierT. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

C. LetellierS. K. SasmalC. DraghiF. 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.  Google Scholar

[38]

L. A. Liotta and E. C. Kohn, The microenvironment of the tumour-host interface, Nature, 411 (2001), 375. Google Scholar

[39]

R. LiuB. D. FergusonY. ZhouK. NagaR. SalgiaP. 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.  Google Scholar

[40]

Á. G. LópezJ. 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.  Google Scholar

[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. MichelsonB. E. MillerA. 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.  Google Scholar

[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.  Google Scholar

[44]

B. MuzP. de la PuenteF. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[48]

S. T. R. PinhoF. S. BacelarR. 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.  Google Scholar

[49]

D. Ribatti, The concept of immune surveillance against tumors: The first theories, Oncotarget, 8 (2017), 7175. Google Scholar

[50]

K. RoeschD. Hasenclever and M. Scholz, Modelling lymphoma therapy and outcome, Bulletin of Mathematical Biology, 76 (2014), 401-430.  doi: 10.1007/s11538-013-9925-3.  Google Scholar

[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.  Google Scholar

[52]

H. SchättlerU. 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.  Google Scholar

[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.  Google Scholar

[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. TormoenM. 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.  Google Scholar

[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.  Google Scholar

[57]

L. VigerF. DenisM. 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.  Google Scholar

[58]

R. A. Weinberg and R. A. Weinberg, The Biology of Cancer, New York, 2013. doi: 10.1201/9780429258794.  Google Scholar

[59]

L. WuZ. YunT. TagawaK. 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.  Google Scholar

[60]

Y. XieY. ZhongT. GaoX. ZhangL. LiH. 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.  Google Scholar

[61]

S. YonucuD. YlmazC. PhippsM. 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.  Google Scholar

[62]

G. ZalcmanJ. MazieresJ. MargeryL. GreillierC. Audigier-ValetteD. Moro-SibilotO. MolinierR. CorreI. 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.  Google Scholar

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. AltrockL. L. Liu and F. Michor, The mathematics of cancer: Integrating quantitative models, Nature Reviews Cancer, 15 (2015), 730-745.  doi: 10.1038/nrc4029.  Google Scholar

[4]

A. R. Anderson and V. Quaranta, Integrative mathematical oncology, Nature Reviews Cancer, 8 (2008), 227. Google Scholar

[5]

P. BeckettJ. EdwardsD. FennellR. HubbardI. 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.  Google Scholar

[6]

S. BenzekryG. ChapuisatJ. CiccoliniA. 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.  Google Scholar

[7]

G. Bergers and L. E. Benjamin, Angiogenesis: Tumorigenesis and the angiogenic switch, Nature Reviews Cancer, 3 (2003), 401-410.  doi: 10.1038/nrc1093.  Google Scholar

[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.  Google Scholar

[9]

R. M. BremnesT. DønnemS. Al-SaadK. Al-ShibliS. AndersenR. SireraC. CampsI. 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.  Google Scholar

[10]

F. M. Burnet, Immunological aspects of malignant disease, The Lancet, 289 (1967), 1171-1174.  doi: 10.1016/S0140-6736(67)92837-1.  Google Scholar

[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.  Google Scholar

[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. ChuN. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

L. G. de PillisA. 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.  Google Scholar

[18]

K. E. De VisserA. 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.  Google Scholar

[19]

G. P. DunnA. T. BruceH. IkedaL. J. Old and R. D. Schreiber, Cancer immunoediting: From immunosurveillance to tumor escape, Nature Immunology, 3 (2002), 991-998.  doi: 10.1038/ni1102-991.  Google Scholar

[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.  Google Scholar

[21]

A. d'OnofrioU. LedzewiczH. 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.  Google Scholar

[22]

A. ErgunK. 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.  Google Scholar

[23]

D. S. EttingerD. E. WoodW. AkerleyL. A. BazhenovaH. BorghaeiD. R. CamidgeR. T. CheneyL. R. ChirieacT. 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.  Google Scholar

[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.  Google Scholar

[26]

P. HahnfeldtD. PanigrahyJ. 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. HanT. PeyretM. MarchandA. QuartinoN. H. GosselinS. GirishD. 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.  Google Scholar

[28]

D. Hanahan and R. A. Weinberg, The hallmarks of cancer, Cell, 100 (2000), 57-70.  doi: 10.1016/S0092-8674(00)81683-9.  Google Scholar

[29]

N. G. Insights, Malignant pleural mesothelioma, Journal of the National Comprehensive Cancer Network, 14 (2016), 825-836.  doi: 10.6004/jnccn.2016.0087.  Google Scholar

[30]

R. KimM. 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.  Google Scholar

[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.  Google Scholar

[32]

V. A. KuznetsovI. A. MakalkinM. 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.  Google Scholar

[33]

L. LaplaneD. DulucN. LarmonierT. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

C. LetellierS. K. SasmalC. DraghiF. 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.  Google Scholar

[38]

L. A. Liotta and E. C. Kohn, The microenvironment of the tumour-host interface, Nature, 411 (2001), 375. Google Scholar

[39]

R. LiuB. D. FergusonY. ZhouK. NagaR. SalgiaP. 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.  Google Scholar

[40]

Á. G. LópezJ. 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.  Google Scholar

[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. MichelsonB. E. MillerA. 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.  Google Scholar

[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.  Google Scholar

[44]

B. MuzP. de la PuenteF. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[48]

S. T. R. PinhoF. S. BacelarR. 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.  Google Scholar

[49]

D. Ribatti, The concept of immune surveillance against tumors: The first theories, Oncotarget, 8 (2017), 7175. Google Scholar

[50]

K. RoeschD. Hasenclever and M. Scholz, Modelling lymphoma therapy and outcome, Bulletin of Mathematical Biology, 76 (2014), 401-430.  doi: 10.1007/s11538-013-9925-3.  Google Scholar

[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.  Google Scholar

[52]

H. SchättlerU. 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.  Google Scholar

[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.  Google Scholar

[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. TormoenM. 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.  Google Scholar

[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.  Google Scholar

[57]

L. VigerF. DenisM. 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.  Google Scholar

[58]

R. A. Weinberg and R. A. Weinberg, The Biology of Cancer, New York, 2013. doi: 10.1201/9780429258794.  Google Scholar

[59]

L. WuZ. YunT. TagawaK. 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.  Google Scholar

[60]

Y. XieY. ZhongT. GaoX. ZhangL. LiH. 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.  Google Scholar

[61]

S. YonucuD. YlmazC. PhippsM. 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.  Google Scholar

[62]

G. ZalcmanJ. MazieresJ. MargeryL. GreillierC. Audigier-ValetteD. Moro-SibilotO. MolinierR. CorreI. 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.  Google Scholar

Figure 1.  Bifurcation diagram for tumor doubling time versus tumor cell count using default parameter values given in Table 1. A transcritical bifurcation occurs when $ \tau _t = 4.4069303 $ days
Figure 2.  Bifurcation diagram for immune cell killing rate of tumor cells versus tumor cell count using default parameter values given in Table 1 and $ \tau _t = 10 $ days. A transcritical bifurcation occurs when $ \alpha _{ti} = 4.852032\cdot 10^{-8} $ cell$ ^{-1} $ day$ ^{-1} $
Figure 3.  Numerical simulation using default parameter values given in Table 1 with initial conditions given in (40) and $ \tau _t = 10 $ days. The immune response is able to effectively eliminate the tumor cells so that the system approaches the tumor-free equilibrium solution $ \mathbb{E}_4 $ given in Tables 4
Figure 4.  Numerical simulation using parameter values given in Table 1 with $ \tau _t = 10 $ days, $ I(0) = 1.0\cdot 10^5 $ cells and all other initial conditions given in (40). The immune response is unable to eliminate the tumor so that the system approaches the equilibrium solution $ \mathbb{E}_7 $ given in Tables 4
Figure 5.  Numerical simulation using parameter values given in Table 1 with initial conditions given in (40) and $ \tau _t = 2 $ days. The immune response is unable to eliminate the tumor so that the system approaches the equilibrium solution $ \mathbb{E}_6 $ in Table 3, which corresponds to the equilibrium phase of immunoediting
Figure 6.  Numerical simulations using parameter values given in Table 1 and initial conditions given by Eqn. (40), where $ \tau _t = 2 $ days and the system is simulated for 250 days, showing a 'zoomed-in' view of Fig. 5
Figure 7.  Numerical simulations for the system undergoing 6 cycles of combined chemoangiogenesis therapy described in Sect. 4 using initial conditions given in Eqn. (40). We set $ \tau_{\epsilon} = 1.25 $ days, $ \tau_{\xi} = 19.6 $ days, and $ \tau _t = 2 $ days, and all other parameter values as listed in Tables 1$ - $2
Figure 8.  Numerical simulation for the system in which chemotherapy has no affect on the immune system using initial conditions given in (40). We set $ \epsilon _i = 0 $, $ \tau_{\epsilon} = 1.25 $ days, $ \tau_{\xi} = 19.6 $ days, $ \tau _t = 2 $ days, and all other parameter values as listed in Tables 1$ - $2
Figure 9.  Numerical simulation for the system in which chemotherapy has an adverse affect on the immune system at half the effectiveness as the tumor using initial conditions given in (40). We set $ \epsilon _i $ = $ -\frac{\epsilon _t}{2} $, $ \tau_{\epsilon} = 1.25 $ days, $ \tau_{\xi} = 19.6 $ days, $ \tau _t = 2 $ days, and all other parameter values as listed in Tables 1$ - $2. The tumor has a second peak early on that is not seen in Fig. 6$ - $Fig. 8, indicating that chemotherapy which harms the immune system makes the therapy more harmful to the patient than no therapy at all
Figure 10.  Numerical simulations of the system with therapy for (a) $ \tau _{\epsilon} = 0.417 $ days and $ \tau _{\xi} = 0.833 $ day, (b) $ \tau _{\epsilon} = 2.08 $ days and $ \tau _{\xi} = 0.833 $ days, (c) $ \tau _{\epsilon} = 0.417 $ days and $ \tau _{\xi} = 19.6 $ days, (d) $ \tau _{\epsilon} = 2.08 $ days and $ \tau _{\xi} = 19.6 $ days
Table 1.  Values of all relevant parameters without therapy
Parameter Description Value Units Source
$ r_h $ Host cell growth parameter 1.8$ \cdot 10^{-1} $ day$ ^{-1} $ [40]
$ r_t $ Tumor growth parameter $ \frac{\ln(2)}{\tau _t} $ day$ ^{-1} $ estimated
$ \tau _t $ Tumor doubling time 2 $ - $ 10 day estimated
$ r_e $ Endothelial cell growth parameter 2.15$ \cdot 10^{-1} $ day$ ^{-1} $ [60]
$ \rho_{et} $ Tumor (VEGF) recruitment of endothelial cells 9.22$ \cdot 10^{2} $ day$ ^{-1} $ [25]
$ \sigma_i $ Influx of immune cells 1.0$ \cdot10^{4} $ cell day$ ^{-1} $ [32]
$ \delta_i $ Immune cell natural death rate 7.0$ \cdot10^{-3} $ day$ ^{-1} $ [50]
$ b_h $ Inverse of host cell carrying capacity 1.0$ \cdot10^{-9} $ cell$ ^{-1} $ [40]
$ \gamma_{te} $ Tumor carrying capacity dependence parameter 0.8 no units estimated
$ K_{te} $ Tumor cell carrying capacity 1.0$ \cdot 10^{6} $ cells [25]
$ b_e $ Inverse of endothelial cell carrying capacity 1.0$ \cdot10^{-7} $ cell$ ^{-1} $ [25]
$ \alpha_{ht} $ Host cell killing rate by tumor cells 4.8$ \cdot 10^{-10} $ cell$ ^{-1} $ day$ ^{-1} $ [40]
$ \alpha_{ti} $ Immune cell response to tumor cell presence 1.101$ \cdot 10^{-7} $ cell$ ^{-1} $ day$ ^{-1} $ [40]
$ \alpha_{it} $ Linear immune cell inactivation rate by tumor cells 2.8$ \cdot 10^{-9} $ cell$ ^{-1} $ day$ ^{-1} $ [40]
$ \beta_{it} $ Quadratic immune cell inactivation rate by tumor cells 3.2$ \cdot 10^{-8} $ cell$ ^{-1} $ day$ ^{-1} $ [40]
Parameter Description Value Units Source
$ r_h $ Host cell growth parameter 1.8$ \cdot 10^{-1} $ day$ ^{-1} $ [40]
$ r_t $ Tumor growth parameter $ \frac{\ln(2)}{\tau _t} $ day$ ^{-1} $ estimated
$ \tau _t $ Tumor doubling time 2 $ - $ 10 day estimated
$ r_e $ Endothelial cell growth parameter 2.15$ \cdot 10^{-1} $ day$ ^{-1} $ [60]
$ \rho_{et} $ Tumor (VEGF) recruitment of endothelial cells 9.22$ \cdot 10^{2} $ day$ ^{-1} $ [25]
$ \sigma_i $ Influx of immune cells 1.0$ \cdot10^{4} $ cell day$ ^{-1} $ [32]
$ \delta_i $ Immune cell natural death rate 7.0$ \cdot10^{-3} $ day$ ^{-1} $ [50]
$ b_h $ Inverse of host cell carrying capacity 1.0$ \cdot10^{-9} $ cell$ ^{-1} $ [40]
$ \gamma_{te} $ Tumor carrying capacity dependence parameter 0.8 no units estimated
$ K_{te} $ Tumor cell carrying capacity 1.0$ \cdot 10^{6} $ cells [25]
$ b_e $ Inverse of endothelial cell carrying capacity 1.0$ \cdot10^{-7} $ cell$ ^{-1} $ [25]
$ \alpha_{ht} $ Host cell killing rate by tumor cells 4.8$ \cdot 10^{-10} $ cell$ ^{-1} $ day$ ^{-1} $ [40]
$ \alpha_{ti} $ Immune cell response to tumor cell presence 1.101$ \cdot 10^{-7} $ cell$ ^{-1} $ day$ ^{-1} $ [40]
$ \alpha_{it} $ Linear immune cell inactivation rate by tumor cells 2.8$ \cdot 10^{-9} $ cell$ ^{-1} $ day$ ^{-1} $ [40]
$ \beta_{it} $ Quadratic immune cell inactivation rate by tumor cells 3.2$ \cdot 10^{-8} $ cell$ ^{-1} $ day$ ^{-1} $ [40]
Table 2.  Values of parameters for combined therapy
Parameter Description Value Units Source
$ \xi_t $ AAT effect on tumor carrying capacity 8.9$ \cdot 10^{-1} $ no units estimated
$ \xi_e $ AAT effect on endothelial cells 9.088$ \cdot 10^{-1} $ no units estimated
$ \lambda_{\xi} $ Clearance rate of AAT agent $ \frac{ \ln(2) }{ \tau_{\xi}} $ day$ ^{-1} $ [15]
$ \tau_{\xi} $ Half life of AAT agent 0.833 $ - $ 19.6 day [61] [27]
$ \epsilon_h $ Chemotherapy effect on immune cells $ \frac{\epsilon_t}{2} $ day$ ^{-1} $ estimated
$ \epsilon_i $ Chemotherapy effect on immune cells 4.999$ \cdot 10^{-2} $ day$ ^{-1} $ estimated
$ \epsilon_t $ Chemotherapy effect on tumor cells 7.494$ \cdot 10^{-2} $ day$ ^{-1} $ estimated
$ \epsilon_e $ Chemotherapy effect on endothelial cells $ \frac{\epsilon_t}{2} $ day$ ^{-1} $ estimated
$ \lambda_{\epsilon} $ Clearance rate of chemotherapy agent $ \frac{ \ln(2) }{ \tau_{\epsilon}} $ day$ ^{-1} $ [15]
$ \tau_{\epsilon} $ Half life of chemotherapy agent 0.417 $ - $ 2.08 day [61]
Parameter Description Value Units Source
$ \xi_t $ AAT effect on tumor carrying capacity 8.9$ \cdot 10^{-1} $ no units estimated
$ \xi_e $ AAT effect on endothelial cells 9.088$ \cdot 10^{-1} $ no units estimated
$ \lambda_{\xi} $ Clearance rate of AAT agent $ \frac{ \ln(2) }{ \tau_{\xi}} $ day$ ^{-1} $ [15]
$ \tau_{\xi} $ Half life of AAT agent 0.833 $ - $ 19.6 day [61] [27]
$ \epsilon_h $ Chemotherapy effect on immune cells $ \frac{\epsilon_t}{2} $ day$ ^{-1} $ estimated
$ \epsilon_i $ Chemotherapy effect on immune cells 4.999$ \cdot 10^{-2} $ day$ ^{-1} $ estimated
$ \epsilon_t $ Chemotherapy effect on tumor cells 7.494$ \cdot 10^{-2} $ day$ ^{-1} $ estimated
$ \epsilon_e $ Chemotherapy effect on endothelial cells $ \frac{\epsilon_t}{2} $ day$ ^{-1} $ estimated
$ \lambda_{\epsilon} $ Clearance rate of chemotherapy agent $ \frac{ \ln(2) }{ \tau_{\epsilon}} $ day$ ^{-1} $ [15]
$ \tau_{\epsilon} $ Half life of chemotherapy agent 0.417 $ - $ 2.08 day [61]
Table 3.  Relevant equilibrium solutions without therapy using default parameter values given in Table 1 and $ \tau _t = 2 $ days
Equilibrium Solution Eigenvalues
$ \mathbb{E}_1 = (0, 1.429\cdot 10^6, 0, 0) $ $ (-0.007, 0.18, 0.19, 0.215) $
$ \mathbb{E}_2 = (1.000\cdot 10^9, 1.429\cdot 10^6, 0, 0) $ $ (-0.18, -0.007, 0.19, 0.215) $
$ \mathbb{E}_3 = (0, 1.429\cdot 10^6, 0, 1.000\cdot 10^7) $ $ (-0.215, -0.007, 0.18, 0.19) $
$ \mathbb{E}_4 = (1.000\cdot 10^9, 1.429\cdot 10^6, 0, 1.000\cdot 10^7) $ $ (-.215, -.18, -0.007, 0.19) $
$ \mathbb{E}_5 = (9.962\cdot 10^8, 3.126\cdot 10^6, 1.422\cdot 10^6, 2.520\cdot 10^{8}) $ $ (-10.62, -0.18, -0.0022 \pm 0.035i) $
$ \mathbb{E}_6 = (9.204\cdot 10^8, 3.045\cdot 10^6, 2.986\cdot 10^7, 1.137\cdot 10^{9}) $ $ (-48.67, -0.17, -0.16, 0.16) $
$ \mathbb{E}_7 = (0, 1.48\cdot 10^{-1}, 2.746\cdot 10^{10}, 3.432\cdot 10^{10}) $ $ (-67464.4, -1475.64, -13, -0.17) $
Equilibrium Solution Eigenvalues
$ \mathbb{E}_1 = (0, 1.429\cdot 10^6, 0, 0) $ $ (-0.007, 0.18, 0.19, 0.215) $
$ \mathbb{E}_2 = (1.000\cdot 10^9, 1.429\cdot 10^6, 0, 0) $ $ (-0.18, -0.007, 0.19, 0.215) $
$ \mathbb{E}_3 = (0, 1.429\cdot 10^6, 0, 1.000\cdot 10^7) $ $ (-0.215, -0.007, 0.18, 0.19) $
$ \mathbb{E}_4 = (1.000\cdot 10^9, 1.429\cdot 10^6, 0, 1.000\cdot 10^7) $ $ (-.215, -.18, -0.007, 0.19) $
$ \mathbb{E}_5 = (9.962\cdot 10^8, 3.126\cdot 10^6, 1.422\cdot 10^6, 2.520\cdot 10^{8}) $ $ (-10.62, -0.18, -0.0022 \pm 0.035i) $
$ \mathbb{E}_6 = (9.204\cdot 10^8, 3.045\cdot 10^6, 2.986\cdot 10^7, 1.137\cdot 10^{9}) $ $ (-48.67, -0.17, -0.16, 0.16) $
$ \mathbb{E}_7 = (0, 1.48\cdot 10^{-1}, 2.746\cdot 10^{10}, 3.432\cdot 10^{10}) $ $ (-67464.4, -1475.64, -13, -0.17) $
Table 4.  Relevant equilibrium solutions without therapy using default parameter values given in Table 1 and $ \tau _t = 10 $ days
Equilibrium Solution Eigenvalues
$ \mathbb{E}_1 = (0, 1.429\cdot 10^6, 0, 0) $ $ (-0.088, -0.007, 0.18, 0.215) $
$ \mathbb{E}_2 = (1.000\cdot 10^9, 1.429\cdot 10^6, 0, 0) $ $ (-0.18, -0.088, -0.007, 0.215) $
$ \mathbb{E}_3 = (0, 1.429\cdot 10^6, 0, 1.000\cdot 10^7) $ $ (-0.215, -0.088, -0.007, 0.18) $
$ \mathbb{E}_4 = (1.000\cdot 10^9, 1.429\cdot 10^6, 0, 1.000\cdot 10^7) $ $ (-.215, -.18, -0.088, -0.007) $
$ \mathbb{E}_6 = (9.085\cdot 10^8, 6.074\cdot 10^5, 3.433\cdot 10^7, 1.218\cdot 10^{9}) $ $ (-52.17, -0.16, -0.097, 0.079) $
$ \mathbb{E}_7 = (0, 1.48\cdot 10^{-1}, 2.746\cdot 10^{10}, 3.432\cdot 10^{10}) $ $ (-67464.3, -1475.5, -13, -0.035) $
Equilibrium Solution Eigenvalues
$ \mathbb{E}_1 = (0, 1.429\cdot 10^6, 0, 0) $ $ (-0.088, -0.007, 0.18, 0.215) $
$ \mathbb{E}_2 = (1.000\cdot 10^9, 1.429\cdot 10^6, 0, 0) $ $ (-0.18, -0.088, -0.007, 0.215) $
$ \mathbb{E}_3 = (0, 1.429\cdot 10^6, 0, 1.000\cdot 10^7) $ $ (-0.215, -0.088, -0.007, 0.18) $
$ \mathbb{E}_4 = (1.000\cdot 10^9, 1.429\cdot 10^6, 0, 1.000\cdot 10^7) $ $ (-.215, -.18, -0.088, -0.007) $
$ \mathbb{E}_6 = (9.085\cdot 10^8, 6.074\cdot 10^5, 3.433\cdot 10^7, 1.218\cdot 10^{9}) $ $ (-52.17, -0.16, -0.097, 0.079) $
$ \mathbb{E}_7 = (0, 1.48\cdot 10^{-1}, 2.746\cdot 10^{10}, 3.432\cdot 10^{10}) $ $ (-67464.3, -1475.5, -13, -0.035) $
[1]

Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251

[2]

Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457

[3]

Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341

[4]

Vieri Benci, Sunra Mosconi, Marco Squassina. Preface: Applications of mathematical analysis to problems in theoretical physics. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020446

[5]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118

[6]

Yining Cao, Chuck Jia, Roger Temam, Joseph Tribbia. Mathematical analysis of a cloud resolving model including the ice microphysics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 131-167. doi: 10.3934/dcds.2020219

2019 Impact Factor: 1.27

Article outline

Figures and Tables

[Back to Top]