2013, 10(3): 565-578. doi: 10.3934/mbe.2013.10.565

Mathematical modeling of glioma therapy using oncolytic viruses

1. 

Laboratoire Interdisciplinaire des Environnements Continentaux, Université de Lorraine, CNRS UMR 7360, 8 rue du Général Delestraint, 57070 METZ, France

2. 

Laboratoire de Mathématiques Raphaël Salem, Université de Rouen, UMR 6085 CNRS, Avenue de l'Université, 76801 Saint Etienne du Rouvray, France

3. 

Department of Mathematics, Elmhurst College, 190 Prospect Avenue, Elmhurst, IL 60126, United States

Received  June 2012 Revised  February 2013 Published  April 2013

Diffuse infiltrative gliomas are adjudged to be the most common primary brain tumors in adults and they tend to blend in extensively in the brain micro-environment. This makes it difficult for medical practitioners to successfully plan effective treatments. In attempts to prolong the lengths of survival times for patients with malignant brain tumors, novel therapeutic alternatives such as gene therapy with oncolytic viruses are currently being explored. Based on such approaches and existing work, a spatio-temporal model that describes interaction between tumor cells and oncolytic viruses is developed. Conditions that lead to optimal therapy in minimizing cancer cell proliferation and otherwise are analytically demonstrated. Numerical simulations are conducted with the aim of showing the impact of virotherapy on proliferation or invasion of cancer cells and of estimating survival times.
Citation: 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
References:
[1]

E. C. Alvord Jr and C. M. Shaw, Neoplasms affecting the nervous system of the elderly,, in, (1991), 210. Google Scholar

[2]

D. D. Barker and A. J. Berk, Adenovirus proteins from both E1B reading frames are required for transformation of rodent cells by viral infection and DNA transfection,, Virology, 156 (1987), 107. doi: 10.1016/0042-6822(87)90441-7. Google Scholar

[3]

N. Bagheri, M. Shiina, D. A. Lauffenburger and W. M. Korn, A dynamical systems model for combinatorial cancer therapy enhances oncolytic adenovirus efficacy by MEK-Inhibition,, PLoS Comput. Biol., 7 (2011). doi: 10.1371/journal.pcbi.1001085. Google Scholar

[4]

F. G. Blankenberg, R. L. Teplitz, W. Ellis, M. S. Salamat, B. H. Min, L. Hall, D. B. Boothroyd, I. M. Johnstone and D. R. Enzmann, The influence of volumetric tumor doubling time, DNA ploidy, and histologic grade on the survival of patients with intracranial astrocytomas,, AJNR Am. J. Neuroradiol, 16 (1995), 1001. Google Scholar

[5]

P. C. Burger, E. R. Heinz, T. Shibata and P. Kleihues, Topographic anatomy and CT correlations in the untreated glioblastoma multiforme,, J. Neurosurg, 68 (1988), 698. doi: 10.3171/jns.1988.68.5.0698. Google Scholar

[6]

B. I. Camara and H. Mokrani, Analysis of wave solutions of an adhenovirus-tumor cell system,, Abstract and Applied Analysis, (2012), 1. doi: 10.1155/2012/590326. Google Scholar

[7]

G. Cherubini, T. Petouchoff, M. Grossi, S. Piersanti, E. Cundari and I. Saggio, E1B55K-deleted adenovirus (ONYX-015) overrides G1/S and G2/M checkpoints and causes mitotic catastrophe and endoreduplication in p53-proficient normal cells,, Cell Cycle, 5 (2006), 2244. Google Scholar

[8]

An. Claes, A. J. Idema and P. Wesseling, Diffuse glioma growth: A guerilla war,, Acta Neuropathol, 114 (2007), 443. doi: 10.1007/s00401-007-0293-7. Google Scholar

[9]

J. C. Concannon, S. Kramer S and R. Berry, The extent of intracranial gliomata at autopsy and its relation to techniques used in radiation therapy of brain tumors,, Am. J. Roentgenol. Radium Ther. Nucl. Med., 84 (1960), 99. Google Scholar

[10]

L. K. Csatary, G. Gosztonyi, J. Szeberenyi, Z. Fabian, V. Liszka, B. Bodey and C. M. Csatary, MTH-68/H oncolytic viral treatment in human high-grade gliomas,, J. Neurooncol, 67 (2004), 83. doi: 10.1023/B:NEON.0000021735.85511.05. Google Scholar

[11]

K. J. Excoffon, G. L. Traver and J. Zabner, The role of the extracellular domain in the biology of the coxsackievirus and adenovirus receptor,, Am. J. Respir. Cell Mol. Biol., 32 (2005), 498. doi: 10.1165/rcmb.2005-0031OC. Google Scholar

[12]

E. Fan, Extended tanh-function method and its applications to nonlinear equations,, Phys. Lett. A, 277 (2000), 212. doi: 10.1016/S0375-9601(00)00725-8. Google Scholar

[13]

A. Friedman and Y. Tao, Analysis of a model of a virus that replicates selectively in tumor cells,, J. Math. Biol., 47 (2003), 391. doi: 10.1007/s00285-003-0199-5. Google Scholar

[14]

X. Ge and M. Arcak, A new sufficient condition for additive D-stability and application to cyclic reaction-diffusion models,, American Control Conference, (2009), 2904. doi: 10.1109/ACC.2009.5160022. Google Scholar

[15]

H. L. Harpold, E. C. Alvord Jr. and K. R. Swanson, The evolution of mathematical modeling of glioma proliferation and invasion,, J. Neuropathol. Exp. Neurol., 66 (2007), 1. doi: 10.1097/nen.0b013e31802d9000. Google Scholar

[16]

D. Harrison, H. Sauthoff, S. Heitner, J. Jagirdar, W. N. Rom and J. G. Hay, Wild-type adenovirus decreases tumor xenograft growth, but despite viral persistence complete tumor responses are rarely achieved-deletion of the viral E1b-19-kD gene increases the viral oncolytic effect,, Hum. Gene. Ther., 12 (2001), 1323. doi: 10.1089/104303401750270977. Google Scholar

[17]

P. J. Kelly, C. Daumas-Duport, D. B. Kispert, B. A. Kall, B. W. Scheithaurer and J. J. Illig, Imaging-based sterotaxic serial biopsies in untreated intracranial glial neoplasms,, J. Neurosurg., 66 (1987), 865. doi: 10.3171/jns.1987.66.6.0865. Google Scholar

[18]

R. M. Lorence, A. L. Pecora, P. P. Major, S. J. Hotte, S. A. Laurie, M. S. Roberts, W. S. Groene and M. K. Bamat, Overview of phase I studies of intravenous administration of PV701, an oncolytic virus,, Curr. Opin. Mol. Ther., 5 (2003), 618. Google Scholar

[19]

D. Makower, A. Rozenblit, H. Kaufman, M. Edelman, M. E. Lane, J. Zwiebel, H. Haynes and S. Wadler, Phase II clinical trial of intralesional administration of the oncolytic adenovirus ONYX-015 in patients with hepatobiliary tumors with correlative p53 studies,, Clin. Cancer Res., 9 (2003), 693. Google Scholar

[20]

E. Mandonnet, J. Y. Delattre, M. L. Tanguy, K. R. Swanson, A. F. Carpentier, H. Duffau, P. Cornu, R. Van Effenterre, E. C. Alvord, Jr. and L. Capelle, Continuous growth of mean tumor diameter in a subset of grade II gliomas,, Ann. Neurol., 53 (2003), 524. Google Scholar

[21]

J. D. Murray, "Mathematical Biology II. Spatial Models and Biological Applications,", 3rd edition, (2003). Google Scholar

[22]

A. S. Novozhilov, F. S. Berezovskaya, E. V. Koonin and G. P. Karev, Mathematical modeling of tumor therapy with oncolytic viruses: Regimes with complete tumor elimination within the framework of deterministic models,, Biology Direct, 1 (2006), 1. Google Scholar

[23]

G. Paganelli, M. Bartolomei, C. Grana, M. Ferrari, P. Rocca and M. Chinol, Radioimmunotherapy of brain tumor,, Neurol. Res., 28 (2006), 518. Google Scholar

[24]

J. Pallud, E. Mandonnet, H. Duffau, M. Kujas, R. Guillevin, D. Galanaud, L. Taillandier and L. Capelle, Prognostic value of initial magnetic resonance imaging growth rates for World Health Organization grade II gliomas,, Ann. Neurol., 60 (2006), 380. Google Scholar

[25]

J. Peiffer, P. Kleihues and H. J. Scherer, Hans-Joachim Scherer (1906-1945), Pioneer in glioma research,, Brain Pathol., 9 (1999), 241. Google Scholar

[26]

R. Rockne, J. K. Rockhill, M. Mrugala M, A. M. Spence, I. Kalet, K. Hendrickson, A. Lai, T. Cloughesy, E. C. Alvord and K. R. Swanson, Predicting the efficacy of radiotherapy in individual glioblastoma patients in vivo: A mathematical modeling approach,, Phys. Med. Biol., 55 (2010), 3271. Google Scholar

[27]

D. C. Shrieve, E. Alexander III, P. Y. Wen, H. M. Kooy, P. M. Blackand and J. S. Loeffler, Comparison of sterotactic radiosurgery and brachytherapy in the treatment of recurrent glioblastoma multiforme,, Neurosurgery, 36 (1995), 275. Google Scholar

[28]

D. L. Silbergeld and M. R. Chicoine, Isolation and characterization of human malignant glioma cells from histologically normal brain,, J. Neurosurg., 86 (1997), 525. Google Scholar

[29]

K. R. Swanson, C. Bridge, J. D. Murray and E. C. Alvord Jr., Virtual and real brain tumors: Using mathematical modeling to quantify glioma growth and invasion,, J. Neurolog. Sci., 216 (2003), 1. Google Scholar

[30]

K. R. Swanson, R. C. Rostomily and E. C. Alvord Jr., A mathematical modeling tool for predicting the survival of individual patients following resection of glioblastoma: A proof of principle,, Br. J. Cancer, 98 (2008), 113. Google Scholar

[31]

T. Takayanagi and A. Ohuchi, A Mathematical analysis of the interactions between immunogenic cells and cytotoxic T Lymphocytes,, Microbiol. Immunol., 45 (2001), 709. Google Scholar

[32]

Y. Tao and Q. Guo, A mathematical model of combined therapies against cancer using viruses and inhibitors,, Science in China Series A: Mathematics, 51 (2008), 2315. doi: 10.1007/s11425-008-0070-7. Google Scholar

[33]

L. Wang and M. Y. Li, Diffusion-driven instability in reaction-diffusion systems,, J. Math. Analysis and Applications, 254 (2001), 138. doi: 10.1006/jmaa.2000.7220. Google Scholar

[34]

D. Wodarz and N. Komarova, "Computational Biology of Cancer: Lecture Notes and Mathematical Modeling,", World Scientific Publishing Company, (2005). Google Scholar

[35]

D. Wodarz, Viruses as antitumor weapons: Defining conditions for tumor remission,, Cancer Res., 61 (2001), 3501. Google Scholar

[36]

D. Wodarz and N. Komarova, Towards predictive computational models of oncolytic virus therapy: basis for experimental validation and model selection,, PLoS ONE, 4 (2009). doi: 10.1371/journal.pone.0004271. Google Scholar

[37]

J. T. Wu, H. M. Byrne, D. H. Kirn and L. M. Wein, Modeling and analysis of a virus that replicates selectively in tumor cells,, Bull. Math. Biol., 63 (2001), 731. Google Scholar

[38]

J. T. Wu, D. H. Kirn and L. M. Wein, Analysis of a three-way race between tumor growth, a replication- competent virus and an immune response,, Bull. Math. Biol., 66 (2004), 605. doi: 10.1016/j.bulm.2003.08.016. Google Scholar

[39]

R. Zurakowskia and D. Wodarz, Model-driven approaches for in vitro combination therapy using ONYX-015 replicating oncolytic adenovirus,, J. Theor. Biol., 245 (2007), 1. doi: 10.1016/j.jtbi.2006.09.029. Google Scholar

show all references

References:
[1]

E. C. Alvord Jr and C. M. Shaw, Neoplasms affecting the nervous system of the elderly,, in, (1991), 210. Google Scholar

[2]

D. D. Barker and A. J. Berk, Adenovirus proteins from both E1B reading frames are required for transformation of rodent cells by viral infection and DNA transfection,, Virology, 156 (1987), 107. doi: 10.1016/0042-6822(87)90441-7. Google Scholar

[3]

N. Bagheri, M. Shiina, D. A. Lauffenburger and W. M. Korn, A dynamical systems model for combinatorial cancer therapy enhances oncolytic adenovirus efficacy by MEK-Inhibition,, PLoS Comput. Biol., 7 (2011). doi: 10.1371/journal.pcbi.1001085. Google Scholar

[4]

F. G. Blankenberg, R. L. Teplitz, W. Ellis, M. S. Salamat, B. H. Min, L. Hall, D. B. Boothroyd, I. M. Johnstone and D. R. Enzmann, The influence of volumetric tumor doubling time, DNA ploidy, and histologic grade on the survival of patients with intracranial astrocytomas,, AJNR Am. J. Neuroradiol, 16 (1995), 1001. Google Scholar

[5]

P. C. Burger, E. R. Heinz, T. Shibata and P. Kleihues, Topographic anatomy and CT correlations in the untreated glioblastoma multiforme,, J. Neurosurg, 68 (1988), 698. doi: 10.3171/jns.1988.68.5.0698. Google Scholar

[6]

B. I. Camara and H. Mokrani, Analysis of wave solutions of an adhenovirus-tumor cell system,, Abstract and Applied Analysis, (2012), 1. doi: 10.1155/2012/590326. Google Scholar

[7]

G. Cherubini, T. Petouchoff, M. Grossi, S. Piersanti, E. Cundari and I. Saggio, E1B55K-deleted adenovirus (ONYX-015) overrides G1/S and G2/M checkpoints and causes mitotic catastrophe and endoreduplication in p53-proficient normal cells,, Cell Cycle, 5 (2006), 2244. Google Scholar

[8]

An. Claes, A. J. Idema and P. Wesseling, Diffuse glioma growth: A guerilla war,, Acta Neuropathol, 114 (2007), 443. doi: 10.1007/s00401-007-0293-7. Google Scholar

[9]

J. C. Concannon, S. Kramer S and R. Berry, The extent of intracranial gliomata at autopsy and its relation to techniques used in radiation therapy of brain tumors,, Am. J. Roentgenol. Radium Ther. Nucl. Med., 84 (1960), 99. Google Scholar

[10]

L. K. Csatary, G. Gosztonyi, J. Szeberenyi, Z. Fabian, V. Liszka, B. Bodey and C. M. Csatary, MTH-68/H oncolytic viral treatment in human high-grade gliomas,, J. Neurooncol, 67 (2004), 83. doi: 10.1023/B:NEON.0000021735.85511.05. Google Scholar

[11]

K. J. Excoffon, G. L. Traver and J. Zabner, The role of the extracellular domain in the biology of the coxsackievirus and adenovirus receptor,, Am. J. Respir. Cell Mol. Biol., 32 (2005), 498. doi: 10.1165/rcmb.2005-0031OC. Google Scholar

[12]

E. Fan, Extended tanh-function method and its applications to nonlinear equations,, Phys. Lett. A, 277 (2000), 212. doi: 10.1016/S0375-9601(00)00725-8. Google Scholar

[13]

A. Friedman and Y. Tao, Analysis of a model of a virus that replicates selectively in tumor cells,, J. Math. Biol., 47 (2003), 391. doi: 10.1007/s00285-003-0199-5. Google Scholar

[14]

X. Ge and M. Arcak, A new sufficient condition for additive D-stability and application to cyclic reaction-diffusion models,, American Control Conference, (2009), 2904. doi: 10.1109/ACC.2009.5160022. Google Scholar

[15]

H. L. Harpold, E. C. Alvord Jr. and K. R. Swanson, The evolution of mathematical modeling of glioma proliferation and invasion,, J. Neuropathol. Exp. Neurol., 66 (2007), 1. doi: 10.1097/nen.0b013e31802d9000. Google Scholar

[16]

D. Harrison, H. Sauthoff, S. Heitner, J. Jagirdar, W. N. Rom and J. G. Hay, Wild-type adenovirus decreases tumor xenograft growth, but despite viral persistence complete tumor responses are rarely achieved-deletion of the viral E1b-19-kD gene increases the viral oncolytic effect,, Hum. Gene. Ther., 12 (2001), 1323. doi: 10.1089/104303401750270977. Google Scholar

[17]

P. J. Kelly, C. Daumas-Duport, D. B. Kispert, B. A. Kall, B. W. Scheithaurer and J. J. Illig, Imaging-based sterotaxic serial biopsies in untreated intracranial glial neoplasms,, J. Neurosurg., 66 (1987), 865. doi: 10.3171/jns.1987.66.6.0865. Google Scholar

[18]

R. M. Lorence, A. L. Pecora, P. P. Major, S. J. Hotte, S. A. Laurie, M. S. Roberts, W. S. Groene and M. K. Bamat, Overview of phase I studies of intravenous administration of PV701, an oncolytic virus,, Curr. Opin. Mol. Ther., 5 (2003), 618. Google Scholar

[19]

D. Makower, A. Rozenblit, H. Kaufman, M. Edelman, M. E. Lane, J. Zwiebel, H. Haynes and S. Wadler, Phase II clinical trial of intralesional administration of the oncolytic adenovirus ONYX-015 in patients with hepatobiliary tumors with correlative p53 studies,, Clin. Cancer Res., 9 (2003), 693. Google Scholar

[20]

E. Mandonnet, J. Y. Delattre, M. L. Tanguy, K. R. Swanson, A. F. Carpentier, H. Duffau, P. Cornu, R. Van Effenterre, E. C. Alvord, Jr. and L. Capelle, Continuous growth of mean tumor diameter in a subset of grade II gliomas,, Ann. Neurol., 53 (2003), 524. Google Scholar

[21]

J. D. Murray, "Mathematical Biology II. Spatial Models and Biological Applications,", 3rd edition, (2003). Google Scholar

[22]

A. S. Novozhilov, F. S. Berezovskaya, E. V. Koonin and G. P. Karev, Mathematical modeling of tumor therapy with oncolytic viruses: Regimes with complete tumor elimination within the framework of deterministic models,, Biology Direct, 1 (2006), 1. Google Scholar

[23]

G. Paganelli, M. Bartolomei, C. Grana, M. Ferrari, P. Rocca and M. Chinol, Radioimmunotherapy of brain tumor,, Neurol. Res., 28 (2006), 518. Google Scholar

[24]

J. Pallud, E. Mandonnet, H. Duffau, M. Kujas, R. Guillevin, D. Galanaud, L. Taillandier and L. Capelle, Prognostic value of initial magnetic resonance imaging growth rates for World Health Organization grade II gliomas,, Ann. Neurol., 60 (2006), 380. Google Scholar

[25]

J. Peiffer, P. Kleihues and H. J. Scherer, Hans-Joachim Scherer (1906-1945), Pioneer in glioma research,, Brain Pathol., 9 (1999), 241. Google Scholar

[26]

R. Rockne, J. K. Rockhill, M. Mrugala M, A. M. Spence, I. Kalet, K. Hendrickson, A. Lai, T. Cloughesy, E. C. Alvord and K. R. Swanson, Predicting the efficacy of radiotherapy in individual glioblastoma patients in vivo: A mathematical modeling approach,, Phys. Med. Biol., 55 (2010), 3271. Google Scholar

[27]

D. C. Shrieve, E. Alexander III, P. Y. Wen, H. M. Kooy, P. M. Blackand and J. S. Loeffler, Comparison of sterotactic radiosurgery and brachytherapy in the treatment of recurrent glioblastoma multiforme,, Neurosurgery, 36 (1995), 275. Google Scholar

[28]

D. L. Silbergeld and M. R. Chicoine, Isolation and characterization of human malignant glioma cells from histologically normal brain,, J. Neurosurg., 86 (1997), 525. Google Scholar

[29]

K. R. Swanson, C. Bridge, J. D. Murray and E. C. Alvord Jr., Virtual and real brain tumors: Using mathematical modeling to quantify glioma growth and invasion,, J. Neurolog. Sci., 216 (2003), 1. Google Scholar

[30]

K. R. Swanson, R. C. Rostomily and E. C. Alvord Jr., A mathematical modeling tool for predicting the survival of individual patients following resection of glioblastoma: A proof of principle,, Br. J. Cancer, 98 (2008), 113. Google Scholar

[31]

T. Takayanagi and A. Ohuchi, A Mathematical analysis of the interactions between immunogenic cells and cytotoxic T Lymphocytes,, Microbiol. Immunol., 45 (2001), 709. Google Scholar

[32]

Y. Tao and Q. Guo, A mathematical model of combined therapies against cancer using viruses and inhibitors,, Science in China Series A: Mathematics, 51 (2008), 2315. doi: 10.1007/s11425-008-0070-7. Google Scholar

[33]

L. Wang and M. Y. Li, Diffusion-driven instability in reaction-diffusion systems,, J. Math. Analysis and Applications, 254 (2001), 138. doi: 10.1006/jmaa.2000.7220. Google Scholar

[34]

D. Wodarz and N. Komarova, "Computational Biology of Cancer: Lecture Notes and Mathematical Modeling,", World Scientific Publishing Company, (2005). Google Scholar

[35]

D. Wodarz, Viruses as antitumor weapons: Defining conditions for tumor remission,, Cancer Res., 61 (2001), 3501. Google Scholar

[36]

D. Wodarz and N. Komarova, Towards predictive computational models of oncolytic virus therapy: basis for experimental validation and model selection,, PLoS ONE, 4 (2009). doi: 10.1371/journal.pone.0004271. Google Scholar

[37]

J. T. Wu, H. M. Byrne, D. H. Kirn and L. M. Wein, Modeling and analysis of a virus that replicates selectively in tumor cells,, Bull. Math. Biol., 63 (2001), 731. Google Scholar

[38]

J. T. Wu, D. H. Kirn and L. M. Wein, Analysis of a three-way race between tumor growth, a replication- competent virus and an immune response,, Bull. Math. Biol., 66 (2004), 605. doi: 10.1016/j.bulm.2003.08.016. Google Scholar

[39]

R. Zurakowskia and D. Wodarz, Model-driven approaches for in vitro combination therapy using ONYX-015 replicating oncolytic adenovirus,, J. Theor. Biol., 245 (2007), 1. doi: 10.1016/j.jtbi.2006.09.029. Google Scholar

[1]

Reihaneh Mostolizadeh, Zahra Afsharnezhad, Anna Marciniak-Czochra. Mathematical model of Chimeric Anti-gene Receptor (CAR) T cell therapy with presence of cytokine. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 63-80. doi: 10.3934/naco.2018004

[2]

Alexander S. Bratus, Svetlana Yu. Kovalenko, Elena Fimmel. On viable therapy strategy for a mathematical spatial cancer model describing the dynamics of malignant and healthy cells. Mathematical Biosciences & Engineering, 2015, 12 (1) : 163-183. doi: 10.3934/mbe.2015.12.163

[3]

Zizi Wang, Zhiming Guo, Huaqin Peng. Dynamical behavior of a new oncolytic virotherapy model based on gene variation. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1079-1093. doi: 10.3934/dcdss.2017058

[4]

Dominik Wodarz. Computational modeling approaches to studying the dynamics of oncolytic viruses. Mathematical Biosciences & Engineering, 2013, 10 (3) : 939-957. doi: 10.3934/mbe.2013.10.939

[5]

Ben Sheller, Domenico D'Alessandro. Analysis of a cancer dormancy model and control of immuno-therapy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1037-1053. doi: 10.3934/mbe.2015.12.1037

[6]

Hsiu-Chuan Wei. Mathematical and numerical analysis of a mathematical model of mixed immunotherapy and chemotherapy of cancer. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1279-1295. doi: 10.3934/dcdsb.2016.21.1279

[7]

J. Ignacio Tello. On a mathematical model of tumor growth based on cancer stem cells. Mathematical Biosciences & Engineering, 2013, 10 (1) : 263-278. doi: 10.3934/mbe.2013.10.263

[8]

Marcello Delitala, Tommaso Lorenzi. Recognition and learning in a mathematical model for immune response against cancer. Discrete & Continuous Dynamical Systems - B, 2013, 18 (4) : 891-914. doi: 10.3934/dcdsb.2013.18.891

[9]

Shuo Wang, Heinz Schättler. Optimal control of a mathematical model for cancer chemotherapy under tumor heterogeneity. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1223-1240. doi: 10.3934/mbe.2016040

[10]

Joseph Malinzi, Rachid Ouifki, Amina Eladdadi, Delfim F. M. Torres, K. A. Jane White. Enhancement of chemotherapy using oncolytic virotherapy: Mathematical and optimal control analysis. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1435-1463. doi: 10.3934/mbe.2018066

[11]

Harsh Vardhan Jain, Avner Friedman. Modeling prostate cancer response to continuous versus intermittent androgen ablation therapy. Discrete & Continuous Dynamical Systems - B, 2013, 18 (4) : 945-967. doi: 10.3934/dcdsb.2013.18.945

[12]

Avner Friedman, Xiulan Lai. Antagonism and negative side-effects in combination therapy for cancer. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2237-2250. doi: 10.3934/dcdsb.2019093

[13]

Avner Friedman, Xiulan Lai. Free boundary problems associated with cancer treatment by combination therapy. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-18. doi: 10.3934/dcds.2019233

[14]

Avner Friedman. A hierarchy of cancer models and their mathematical challenges. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 147-159. doi: 10.3934/dcdsb.2004.4.147

[15]

Sebastián Ferrer, Francisco Crespo. Parametric quartic Hamiltonian model. A unified treatment of classic integrable systems. Journal of Geometric Mechanics, 2014, 6 (4) : 479-502. doi: 10.3934/jgm.2014.6.479

[16]

Ghendrih Philippe, Hauray Maxime, Anne Nouri. Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solution. Kinetic & Related Models, 2009, 2 (4) : 707-725. doi: 10.3934/krm.2009.2.707

[17]

Svetlana Bunimovich-Mendrazitsky, Yakov Goltser. Use of quasi-normal form to examine stability of tumor-free equilibrium in a mathematical model of bcg treatment of bladder cancer. Mathematical Biosciences & Engineering, 2011, 8 (2) : 529-547. doi: 10.3934/mbe.2011.8.529

[18]

Yangjin Kim, Avner Friedman, Eugene Kashdan, Urszula Ledzewicz, Chae-Ok Yun. Application of ecological and mathematical theory to cancer: New challenges. Mathematical Biosciences & Engineering, 2015, 12 (6) : i-iv. doi: 10.3934/mbe.2015.12.6i

[19]

M.A.J Chaplain, G. Lolas. Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity. Networks & Heterogeneous Media, 2006, 1 (3) : 399-439. doi: 10.3934/nhm.2006.1.399

[20]

Marzena Dolbniak, Malgorzata Kardynska, Jaroslaw Smieja. Sensitivity of combined chemo-and antiangiogenic therapy results in different models describing cancer growth. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 145-160. doi: 10.3934/dcdsb.2018009

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (2)

[Back to Top]