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December 2018, 15(6): 1435-1463. doi: 10.3934/mbe.2018066

Enhancement of chemotherapy using oncolytic virotherapy: Mathematical and optimal control analysis

1. 

Department of Mathematics and Applied Mathematics, University of Pretoria, Private Bag X 20, Hatfield, Pretoria 0028, South Africa

2. 

Department of Mathematics, The College of Saint Rose, Albany, New York, USA

3. 

Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

4. 

Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK

* Corresponding authors: Joseph Malinzi & Amina Eladdadi

Received  March 27, 2018 Accepted  July 10, 2018 Published  September 2018

Oncolytic virotherapy has been emerging as a promising novel cancer treatment which may be further combined with the existing therapeutic modalities to enhance their effects. To investigate how virotherapy could enhance chemotherapy, we propose an ODE based mathematical model describing the interactions between tumour cells, the immune response, and a treatment combination with chemotherapy and oncolytic viruses. Stability analysis of the model with constant chemotherapy treatment rates shows that without any form of treatment, a tumour would grow to its maximum size. It also demonstrates that chemotherapy alone is capable of clearing tumour cells provided that the drug efficacy is greater than the intrinsic tumour growth rate. Furthermore, virotherapy alone may not be able to clear tumour cells from body tissue but would rather enhance chemotherapy if viruses with high viral potency are used. To assess the combined effect of virotherapy and chemotherapy we use the forward sensitivity index to perform a sensitivity analysis, with respect to chemotherapy key parameters, of the virus basic reproductive number and the tumour endemic equilibrium. The results from this sensitivity analysis indicate the existence of a critical dose of chemotherapy above which no further significant reduction in the tumour population can be observed. Numerical simulations show that a successful combinational therapy of the chemotherapeutic drugs and viruses depends mostly on the virus burst size, infection rate, and the amount of drugs supplied. Optimal control analysis was performed, by means of the Pontryagin's maximum principle, to further refine predictions of the model with constant treatment rates by accounting for the treatment costs and sides effects. Results from this analysis suggest that the optimal drug and virus combination correspond to half their maximum tolerated doses. This is in agreement with the results from stability and sensitivity analyses.

Citation: 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
References:
[1]

M. Agarwal and A. S. Bhadauria, Mathematical modeling and analysis of tumor therapy with oncolytic virus, Journal of Applied Mathematics, 2 (2011), 131-140. doi: 10.4236/am.2011.21015.

[2]

T. AgrawalM. Saleem and S. Sahu, Optimal control of the dynamics of a tumor growth model with hollings' type-Ⅱ functional response, Computational and Applied Mathematics, 33 (2014), 591-606. doi: 10.1007/s40314-013-0083-x.

[3]

M. AlonsoC. Gomez-ManzanoH. JiangN. B. BekeleY. PiaoW. K. A. YungR. Alemany and J. Fueyo, Combination of the oncolytic adenovirus icovir-5 with chemotherapy provides enhanced anti-glioma effect in vivo, Journal of Cancer Gene Therapy, 14 (2007), 756-761. doi: 10.1038/sj.cgt.7701067.

[4]

Z. BajzerT. CarrK. JosicS. J. Russell and D. Dingli, Modeling of cancer virotherapy with recombinant measles viruses, Journal of Theoretical Biology, 252 (2008), 109-122. doi: 10.1016/j.jtbi.2008.01.016.

[5]

M. Bartkowski, S. Bridges, P. Came, H. Eggers, P. Fischer, H. Friedmann, M. Green, C. Gurgo, J. Hay, B. D. Korant et al., Chemotherapy of viral infections, vol. 61, Springer Science & Business Media, 2012.

[6]

S. BenzekryC. LamontA. BeheshtiA. Tracz and J. M. L. Ebos, Classical mathematical models for description and prediction of experimental tumor growth, PLoS Comput Biol, 10 (2014), e1003800. doi: 10.1371/journal.pcbi.1003800.

[7]

M. Bertau, E. Mosekilde and H. V. Westerhoff, Biosimulation in Drug Development, John Wiley & Sons, 2008. doi: 10.1002/9783527622672.

[8]

E. Binz and L. M. Ulrich, Chemovirotherapy: Combining chemotherapeutic treatment with oncolytic virotherapy, Oncolytic Virotherapy, 4 (2015), 39-48.

[9]

C. Bollard and H. HeslopS, T cells for viral infections after allogeneic hematopoietic stem cell transplant, Blood, 127 (2016), 3331-3340. doi: 10.1182/blood-2016-01-628982.

[10]

G. J. Bostol and S. Patil, Carboplatin in clinical stage Ⅰ seminoma: too much and too little at the same time, Journal of Clinical Oncology, 29 (2011), 949-952.

[11] T. D. Brock, The Emergence of Bacterial Genetics, Cold Spring Harbor Laboratory Press Cold Spring Harbor, New York, 1990.
[12]

R. W. Carlson and B. I. Sikic, Continuous infusion or bolus injection in cancer chemotherapy, Annals of Internal Medicine, 99 (1983), 823-833. doi: 10.7326/0003-4819-99-6-823.

[13]

J. CrivelliJ. FöldesP. Kim and J. Wares, A mathematical model for cell cycle-specific cancer virotherapy, Journal of Biological Dynamics, 6 (2012), 104-120. doi: 10.1080/17513758.2011.613486.

[14]

S. Dasari and P. Tchounwou, Cisplatin in cancer therapy: Molecular mechanisms of action, European Journal of Pharmacology, 740 (2014), 364-378. doi: 10.1016/j.ejphar.2014.07.025.

[15] R. J. de Boer, Modeling Population Dynamics: A Graphical Approach, Utrecht University, 2018.
[16]

L. de PillisK. R. FisterW. GuC. CollinsM. DaubD. GrossJ. Moore and B. Preskill, Mathematical model creation for cancer chemo-immunotherapy, Journal of Computational and Mathematical Methods in Medicine, 10 (2009), 165-184. doi: 10.1080/17486700802216301.

[17]

W. Fleming and R. Rishel, Deterministic and Stochastic Optimal Control, vol. 1, Springer-Verlag, Berlin-New York, 1975.

[18]

E. Frei III and G. P. Canellos, Dose: a critical factor in cancer chemotherapy, The American Journal of Medicine, 69 (1980), 585-594.

[19]

T. GajewskiH. Schreiber and Y. Fu, Innate and adaptive immune cells in the tumor microenvironment, Nature Immunology, 14 (2013), 1014-1022. doi: 10.1038/ni.2703.

[20]

K. Garber, China approves world's first oncolytic virus therapy for cancer treatment, Journal of the National Cancer Institute, 98 (2006), 298-300. doi: 10.1093/jnci/djj111.

[21]

V. GrohJ. WuC. Yee and T. Spies, Tumour-derived soluble MIC ligands impair expression of nkg2d and t-cell activation, Journal of Nature, 419 (2002), 734-738. doi: 10.1038/nature01112.

[22]

A. HowellsG. MarelliN. Lemoine and Y. Wang, Oncolytic viruses-interaction of virus and tumor cells in the battle to eliminate cancer, Frontiers in Oncology, 7 (2017), 195. doi: 10.3389/fonc.2017.00195.

[23]

E. Kelly and S. J. Russel, History of oncolytic viruses: Genesis to genetic engineering, Journal of Molecular Therapy, 15 (2007), 651-659. doi: 10.1038/sj.mt.6300108.

[24]

S. Khajanchi and S. Banerjee, Stability and bifurcation analysis of delay induced tumor immune interaction model, Applied Mathematics and Computation, 248 (2014), 652-671. doi: 10.1016/j.amc.2014.10.009.

[25]

D. Kirschner and J. Panetta, Modeling immunotherapy of the tumor-immune interaction, Journal of Mathematical Biology, 37 (1998), 235-252. doi: 10.1007/s002850050127.

[26]

A. KonstorumA. VellaA. Adler and R. Laubenbacher, Addressing current challenges in cancer immunotherapy with mathematical and computational modeling, The Royal Society Interface, (2017), 146902. doi: 10.1101/146902.

[27]

D. LeJ. Miller and V. Ganusov, Mathematical modeling provides kinetic details of the human immune response to vaccination, Frontiers in Cellular and Infection Microbiology, 7 (2015), 00177. doi: 10.3389/fcimb.2014.00177.

[28]

T. C. LiauE. Galanis and D. Kirn, Clinical trial results with oncolytic virotherapy: A century of promise, a decade of progress, Journal of Nature Clinical Practice Oncology, 4 (2007), 101-117. doi: 10.1038/ncponc0736.

[29]

W. Liu and H. I. Freedman, A mathematical model of vascular tumor treatment by chemotherapy, Journal of Mathematical and Computer Modelling, 42 (2005), 1089-1112. doi: 10.1016/j.mcm.2004.09.008.

[30]

J. MalinziA. Eladdadi and P. Sibanda, Modelling the spatiotemporal dynamics of chemovirotherapy cancer treatment, Journal of Biological Dynamics, 11 (2017), 244-274. doi: 10.1080/17513758.2017.1328079.

[31]

J. MalinziP. Sibanda and H. Mambili-Mamoboundou, Analysis of virotherapy in solid tumor invasion, Journal of Mathematical Biosciences, 263 (2015), 102-110. doi: 10.1016/j.mbs.2015.01.015.

[32]

S. NayarP. Dasgupta and C. Galustian, Extending the lifespan and efficacies of immune cells used in adoptive transfer for cancer immunotherapies-a review, Oncoimmunology, 4 (2015), e1002720. doi: 10.1080/2162402X.2014.1002720.

[33]

A. NguyenL. Ho and Y. Wan, Chemotherapy and oncolytic virotherapy: Advanced tactics in the war against cancer, Frontiers in Oncology, 4 (2014), 00145. doi: 10.3389/fonc.2014.00145.

[34]

A. S. NovozhilovF. S. BerezovskayaE. 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), 1-18.

[35]

R. T. D. OliverG. M. MeadG. J. RustinJ. S. GordonJ. K. JoffeN. AassR. ColemanP. P. R. Gabe and S. P. Stenning, Randomized trial of carboplatin versus radiotherapy for stage Ⅰ seminoma: mature results on relapse and contralateral testis cancer rates in MRC TE19/EORTC 30982 study (ISRCTN27163214), Journal of Clinical Oncology, 29 (2011), 957-962. doi: 10.1200/JCO.2009.26.4655.

[36]

P. K. OttolinoJ. S. DialloB. D. LichtyJ. C. Bell and J. A. McCart, Intelligent design: combination therapy with oncolytic viruses, Journal of Molecular Therapy, 18 (2010), 251-263. doi: 10.1038/mt.2009.283.

[37]

R. Ouifki and G. Witten, A model of HIV-1 infection with HAART therapy and intracellular delays, Discrete and Continous Dynamical Systems Series B, 8 (2007), 229-240. doi: 10.3934/dcdsb.2007.8.229.

[38]

S. T. R. PinhoH. I. Freedman and F. K. Nani, A chemotherapy model for the treatment of cancer with metastasis, Journal of Mathematical and Computer Modelling, 36 (2002), 773-803. doi: 10.1016/S0895-7177(02)00227-3.

[39]

S. T. R. PinhoD. S. Rodrigues and P. F. A. Mancera, A mathematical model of chemotherapy response to tumour growth, Canadian Applied Math Quarterly, 19 (2011), 369-384.

[40]

S. PinhoR. A. F. S. Bacelar and H. Freedman, A mathematical model for the effect of antiangiogenic 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.

[41] L. Pontryagin, Mathematical Theory of Optimal Processes, CRC Press, 1987. doi: 10.1201/9780203749319.
[42]

K. RelphH. PandhaG. SimpsonA. Melcher and K. Harrington, Cancer immunotherapy via combining oncolytic virotherapy with chemotherapy: recent advances, Oncolytic Virotherapy, 2016 (2016), 1-13.

[43]

S. J. RusselK. W. Pengl and J. C. Bell, Oncolytic virotherapy, Journal of Nature Biotechnology, 30 (2012), 658-670. doi: 10.1038/nbt.2287.

[44]

B. J. Schroers, Ordinary Differential Equations: A Practical Guide, Cambridge University Press, 2011. doi: 10.1017/CBO9781139057707.

[45]

J. S. SprattJ. S. Meyer and J. A. Spratt, Rates of growth of human solid neoplasms: Part i, Journal of Surgical Oncology, 60 (1995), 137-146. doi: 10.1002/jso.2930600216.

[46] H. Thieme, Mathematics in Population Biology, Princeton University Press, 2003.
[47]

J. P. Tian, The replicability of oncolytic virus: defining conditions in tumor virotherapy, Journal of Mathematical Biosciences and Engineering, 8 (2011), 841-860. doi: 10.3934/mbe.2011.8.841.

[48]

S. D. UndeviaA. G. Gomez and M. J. Ratain, Pharmacokinetic variability of anticancer agents, Nature Reviews Cancer, 5 (2005), 447-458. doi: 10.1038/nrc1629.

[49]

G. UngerechtsM. E. FrenzkeK. C. YaiwT. MiestP. B. Johnston and R. Cattaneo, Mantle cell lymphoma salvage regimen: Synergy between a reprogrammed oncolytic virus and two chemotherapeutics, Gene Therapy, 17 (2010), 1506-1516. doi: 10.1038/gt.2010.103.

[50]

J. R. Usher, Some mathematical models for cancer chemotherapy, Journal of Computers & Mathematics with Applications, 28 (1994), 73-80.

[51]

US Food and Drug Administration and others, FDA approves first-of-its-kind product for the treatment of melanoma. press release. october 27, 2015.

[52]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.

[53]

Y. WangJ. P. Tian and J. Wei, Lytic cycle: A defining process in oncolytic virotherapy, Journal of Applied Mathematical Modelling, 37 (2013), 5962-5978. doi: 10.1016/j.apm.2012.12.004.

[54]

D. Wodarz, Viruses as antitumor weapons defining conditions for tumor remission, Journal of Cancer Research, 61 (2001), 3501-3507.

[55]

M. YoshimoriH. OokuraY. ShimadaT. YoshidaN. OkazakiM. Yoshino and D. Saito, continuous infusion of anti-cancer drug with balloon infusors, Gan to Kagaku Ryoho. Cancer & Chemotherapy, 15 (1988), 3121-3125.

show all references

References:
[1]

M. Agarwal and A. S. Bhadauria, Mathematical modeling and analysis of tumor therapy with oncolytic virus, Journal of Applied Mathematics, 2 (2011), 131-140. doi: 10.4236/am.2011.21015.

[2]

T. AgrawalM. Saleem and S. Sahu, Optimal control of the dynamics of a tumor growth model with hollings' type-Ⅱ functional response, Computational and Applied Mathematics, 33 (2014), 591-606. doi: 10.1007/s40314-013-0083-x.

[3]

M. AlonsoC. Gomez-ManzanoH. JiangN. B. BekeleY. PiaoW. K. A. YungR. Alemany and J. Fueyo, Combination of the oncolytic adenovirus icovir-5 with chemotherapy provides enhanced anti-glioma effect in vivo, Journal of Cancer Gene Therapy, 14 (2007), 756-761. doi: 10.1038/sj.cgt.7701067.

[4]

Z. BajzerT. CarrK. JosicS. J. Russell and D. Dingli, Modeling of cancer virotherapy with recombinant measles viruses, Journal of Theoretical Biology, 252 (2008), 109-122. doi: 10.1016/j.jtbi.2008.01.016.

[5]

M. Bartkowski, S. Bridges, P. Came, H. Eggers, P. Fischer, H. Friedmann, M. Green, C. Gurgo, J. Hay, B. D. Korant et al., Chemotherapy of viral infections, vol. 61, Springer Science & Business Media, 2012.

[6]

S. BenzekryC. LamontA. BeheshtiA. Tracz and J. M. L. Ebos, Classical mathematical models for description and prediction of experimental tumor growth, PLoS Comput Biol, 10 (2014), e1003800. doi: 10.1371/journal.pcbi.1003800.

[7]

M. Bertau, E. Mosekilde and H. V. Westerhoff, Biosimulation in Drug Development, John Wiley & Sons, 2008. doi: 10.1002/9783527622672.

[8]

E. Binz and L. M. Ulrich, Chemovirotherapy: Combining chemotherapeutic treatment with oncolytic virotherapy, Oncolytic Virotherapy, 4 (2015), 39-48.

[9]

C. Bollard and H. HeslopS, T cells for viral infections after allogeneic hematopoietic stem cell transplant, Blood, 127 (2016), 3331-3340. doi: 10.1182/blood-2016-01-628982.

[10]

G. J. Bostol and S. Patil, Carboplatin in clinical stage Ⅰ seminoma: too much and too little at the same time, Journal of Clinical Oncology, 29 (2011), 949-952.

[11] T. D. Brock, The Emergence of Bacterial Genetics, Cold Spring Harbor Laboratory Press Cold Spring Harbor, New York, 1990.
[12]

R. W. Carlson and B. I. Sikic, Continuous infusion or bolus injection in cancer chemotherapy, Annals of Internal Medicine, 99 (1983), 823-833. doi: 10.7326/0003-4819-99-6-823.

[13]

J. CrivelliJ. FöldesP. Kim and J. Wares, A mathematical model for cell cycle-specific cancer virotherapy, Journal of Biological Dynamics, 6 (2012), 104-120. doi: 10.1080/17513758.2011.613486.

[14]

S. Dasari and P. Tchounwou, Cisplatin in cancer therapy: Molecular mechanisms of action, European Journal of Pharmacology, 740 (2014), 364-378. doi: 10.1016/j.ejphar.2014.07.025.

[15] R. J. de Boer, Modeling Population Dynamics: A Graphical Approach, Utrecht University, 2018.
[16]

L. de PillisK. R. FisterW. GuC. CollinsM. DaubD. GrossJ. Moore and B. Preskill, Mathematical model creation for cancer chemo-immunotherapy, Journal of Computational and Mathematical Methods in Medicine, 10 (2009), 165-184. doi: 10.1080/17486700802216301.

[17]

W. Fleming and R. Rishel, Deterministic and Stochastic Optimal Control, vol. 1, Springer-Verlag, Berlin-New York, 1975.

[18]

E. Frei III and G. P. Canellos, Dose: a critical factor in cancer chemotherapy, The American Journal of Medicine, 69 (1980), 585-594.

[19]

T. GajewskiH. Schreiber and Y. Fu, Innate and adaptive immune cells in the tumor microenvironment, Nature Immunology, 14 (2013), 1014-1022. doi: 10.1038/ni.2703.

[20]

K. Garber, China approves world's first oncolytic virus therapy for cancer treatment, Journal of the National Cancer Institute, 98 (2006), 298-300. doi: 10.1093/jnci/djj111.

[21]

V. GrohJ. WuC. Yee and T. Spies, Tumour-derived soluble MIC ligands impair expression of nkg2d and t-cell activation, Journal of Nature, 419 (2002), 734-738. doi: 10.1038/nature01112.

[22]

A. HowellsG. MarelliN. Lemoine and Y. Wang, Oncolytic viruses-interaction of virus and tumor cells in the battle to eliminate cancer, Frontiers in Oncology, 7 (2017), 195. doi: 10.3389/fonc.2017.00195.

[23]

E. Kelly and S. J. Russel, History of oncolytic viruses: Genesis to genetic engineering, Journal of Molecular Therapy, 15 (2007), 651-659. doi: 10.1038/sj.mt.6300108.

[24]

S. Khajanchi and S. Banerjee, Stability and bifurcation analysis of delay induced tumor immune interaction model, Applied Mathematics and Computation, 248 (2014), 652-671. doi: 10.1016/j.amc.2014.10.009.

[25]

D. Kirschner and J. Panetta, Modeling immunotherapy of the tumor-immune interaction, Journal of Mathematical Biology, 37 (1998), 235-252. doi: 10.1007/s002850050127.

[26]

A. KonstorumA. VellaA. Adler and R. Laubenbacher, Addressing current challenges in cancer immunotherapy with mathematical and computational modeling, The Royal Society Interface, (2017), 146902. doi: 10.1101/146902.

[27]

D. LeJ. Miller and V. Ganusov, Mathematical modeling provides kinetic details of the human immune response to vaccination, Frontiers in Cellular and Infection Microbiology, 7 (2015), 00177. doi: 10.3389/fcimb.2014.00177.

[28]

T. C. LiauE. Galanis and D. Kirn, Clinical trial results with oncolytic virotherapy: A century of promise, a decade of progress, Journal of Nature Clinical Practice Oncology, 4 (2007), 101-117. doi: 10.1038/ncponc0736.

[29]

W. Liu and H. I. Freedman, A mathematical model of vascular tumor treatment by chemotherapy, Journal of Mathematical and Computer Modelling, 42 (2005), 1089-1112. doi: 10.1016/j.mcm.2004.09.008.

[30]

J. MalinziA. Eladdadi and P. Sibanda, Modelling the spatiotemporal dynamics of chemovirotherapy cancer treatment, Journal of Biological Dynamics, 11 (2017), 244-274. doi: 10.1080/17513758.2017.1328079.

[31]

J. MalinziP. Sibanda and H. Mambili-Mamoboundou, Analysis of virotherapy in solid tumor invasion, Journal of Mathematical Biosciences, 263 (2015), 102-110. doi: 10.1016/j.mbs.2015.01.015.

[32]

S. NayarP. Dasgupta and C. Galustian, Extending the lifespan and efficacies of immune cells used in adoptive transfer for cancer immunotherapies-a review, Oncoimmunology, 4 (2015), e1002720. doi: 10.1080/2162402X.2014.1002720.

[33]

A. NguyenL. Ho and Y. Wan, Chemotherapy and oncolytic virotherapy: Advanced tactics in the war against cancer, Frontiers in Oncology, 4 (2014), 00145. doi: 10.3389/fonc.2014.00145.

[34]

A. S. NovozhilovF. S. BerezovskayaE. 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), 1-18.

[35]

R. T. D. OliverG. M. MeadG. J. RustinJ. S. GordonJ. K. JoffeN. AassR. ColemanP. P. R. Gabe and S. P. Stenning, Randomized trial of carboplatin versus radiotherapy for stage Ⅰ seminoma: mature results on relapse and contralateral testis cancer rates in MRC TE19/EORTC 30982 study (ISRCTN27163214), Journal of Clinical Oncology, 29 (2011), 957-962. doi: 10.1200/JCO.2009.26.4655.

[36]

P. K. OttolinoJ. S. DialloB. D. LichtyJ. C. Bell and J. A. McCart, Intelligent design: combination therapy with oncolytic viruses, Journal of Molecular Therapy, 18 (2010), 251-263. doi: 10.1038/mt.2009.283.

[37]

R. Ouifki and G. Witten, A model of HIV-1 infection with HAART therapy and intracellular delays, Discrete and Continous Dynamical Systems Series B, 8 (2007), 229-240. doi: 10.3934/dcdsb.2007.8.229.

[38]

S. T. R. PinhoH. I. Freedman and F. K. Nani, A chemotherapy model for the treatment of cancer with metastasis, Journal of Mathematical and Computer Modelling, 36 (2002), 773-803. doi: 10.1016/S0895-7177(02)00227-3.

[39]

S. T. R. PinhoD. S. Rodrigues and P. F. A. Mancera, A mathematical model of chemotherapy response to tumour growth, Canadian Applied Math Quarterly, 19 (2011), 369-384.

[40]

S. PinhoR. A. F. S. Bacelar and H. Freedman, A mathematical model for the effect of antiangiogenic 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.

[41] L. Pontryagin, Mathematical Theory of Optimal Processes, CRC Press, 1987. doi: 10.1201/9780203749319.
[42]

K. RelphH. PandhaG. SimpsonA. Melcher and K. Harrington, Cancer immunotherapy via combining oncolytic virotherapy with chemotherapy: recent advances, Oncolytic Virotherapy, 2016 (2016), 1-13.

[43]

S. J. RusselK. W. Pengl and J. C. Bell, Oncolytic virotherapy, Journal of Nature Biotechnology, 30 (2012), 658-670. doi: 10.1038/nbt.2287.

[44]

B. J. Schroers, Ordinary Differential Equations: A Practical Guide, Cambridge University Press, 2011. doi: 10.1017/CBO9781139057707.

[45]

J. S. SprattJ. S. Meyer and J. A. Spratt, Rates of growth of human solid neoplasms: Part i, Journal of Surgical Oncology, 60 (1995), 137-146. doi: 10.1002/jso.2930600216.

[46] H. Thieme, Mathematics in Population Biology, Princeton University Press, 2003.
[47]

J. P. Tian, The replicability of oncolytic virus: defining conditions in tumor virotherapy, Journal of Mathematical Biosciences and Engineering, 8 (2011), 841-860. doi: 10.3934/mbe.2011.8.841.

[48]

S. D. UndeviaA. G. Gomez and M. J. Ratain, Pharmacokinetic variability of anticancer agents, Nature Reviews Cancer, 5 (2005), 447-458. doi: 10.1038/nrc1629.

[49]

G. UngerechtsM. E. FrenzkeK. C. YaiwT. MiestP. B. Johnston and R. Cattaneo, Mantle cell lymphoma salvage regimen: Synergy between a reprogrammed oncolytic virus and two chemotherapeutics, Gene Therapy, 17 (2010), 1506-1516. doi: 10.1038/gt.2010.103.

[50]

J. R. Usher, Some mathematical models for cancer chemotherapy, Journal of Computers & Mathematics with Applications, 28 (1994), 73-80.

[51]

US Food and Drug Administration and others, FDA approves first-of-its-kind product for the treatment of melanoma. press release. october 27, 2015.

[52]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.

[53]

Y. WangJ. P. Tian and J. Wei, Lytic cycle: A defining process in oncolytic virotherapy, Journal of Applied Mathematical Modelling, 37 (2013), 5962-5978. doi: 10.1016/j.apm.2012.12.004.

[54]

D. Wodarz, Viruses as antitumor weapons defining conditions for tumor remission, Journal of Cancer Research, 61 (2001), 3501-3507.

[55]

M. YoshimoriH. OokuraY. ShimadaT. YoshidaN. OkazakiM. Yoshino and D. Saito, continuous infusion of anti-cancer drug with balloon infusors, Gan to Kagaku Ryoho. Cancer & Chemotherapy, 15 (1988), 3121-3125.

Figure 1.  Visual representation of the elasticity indices of $R_0$ with respect to model parameters. The bar graph shows that the virus burst size, infection and decay rates $b$, $\beta$ & $\gamma$ have the highest elasticity indices with virus decay being negatively correlated to $R_0$
Figure 2.  Plots of the elasticity indices, $e_q$, and $\Gamma^{U^*+ I^*}_q$ against the drug dosage $q$. Both Figures (a) and (b) depict that increasing the amount of drug infused reduces viral multiplication thus reducing the sensitivity indices. The figures further suggest that values of $q$ from 40 to 100 mg/l have minimal negative impact on viral replication
Figure 3.  (a) Plots of uninfected cell density with constant drug infusion and for different virus burst sizes. The plot shows that an increase in the virus burst size reduces the tumour density. (b)Plots of uninfected cell density with constant drug infusion and for different drug infusion rates. The plot shows that an increase in the drug infusion rate reduces the tumour density by a relatively small magnitude
Figure 4.  Plots of the virus and infected tumour densities for different virus burst size. The plots show that both densities increase with increasing virus burst size
Figure 5.  (a) Plots of the virus density for different virus infection rate. The plots show the virus density reduces with increasing virus infection rates. (b) Plots of infected tumour density for different infection rates. The figure shows that the infected tumour density increases as the infection rate increases
Figure 6.  Total tumour density with optimal control. The tumour density is reduced in a very short time period
Figure 7.  Optimal control variable variables $u_1$: The external supply of viruses and $u_2$: Drug dosage. The Figures depict 500 virions as the optimal number of viruses and the optimal drug dosage to be 50 m/g per
Table 1.  The model variables
Variable Description Units
$U(t)$ Uninfected tumour density cells per mm$^3$
$I(t)$ Virus infected tumour cell density cells per mm$^3$
$V(t)$ Free virus particles virions per mm$^3$
$E_v(t)$ Virus specific immune response cells per mm$^3$
$E_T(t)$ Tumour specific immune response cells per mm$^3$
$C(t)$ Drug concentration grams per millilitre (g/ml)
Variable Description Units
$U(t)$ Uninfected tumour density cells per mm$^3$
$I(t)$ Virus infected tumour cell density cells per mm$^3$
$V(t)$ Free virus particles virions per mm$^3$
$E_v(t)$ Virus specific immune response cells per mm$^3$
$E_T(t)$ Tumour specific immune response cells per mm$^3$
$C(t)$ Drug concentration grams per millilitre (g/ml)
Table 2.  The model parameters, their description and base values
Symbol Description Value & units Ref.
$K$ Tumour carrying capacity $10^6$ cells per mm$^3$ per day [4]
$\alpha$ Tumour growth rate $0.206$ cells per mm$^3$ per day [4]
$\beta$ Infection rate of tumour cells $0.001-0.1$ cells per mm$^3$ per day [4]
$\delta$ Infected tumour cells death $0.5115$ day$^{-1}$ [4]
$\gamma$ Rate of virus decay $0.01$ day$^{-1}$ [4]
$b$ Virus burst size $0-1000$ virions per mm$^3$ per day [11]
$\psi$ Rate drug decay $4.17$ millilitres per mm$^3$ per day [39]
$\delta_U$ Lysis rate of $U$ by the drug $50$ cells per mm$^3$ per day [39]
$\delta_I$ Lysis rate of $I$ by the drug $60$ cells per mm$^3$ per day [39]
$\phi$ $E_V$ production rate $0.7$ day$^{-1}$ [9]
$ \beta_T$ $E_T$ production rate $0.5$ cells per mm$^3$ per day [19,27]
$ \delta_v,\delta_T$ immune decay rates $0.01$ day$^{-1}$ [19,27]
$ K_u,K_c,\kappa$ Michaelis--Menten constants $10^5$ cells per mm$^3$ per day [25]
$ \nu_U$ Lysis rate of $U$ by $E_T$ $0.08$ cells per mm$^3$ per day est
$ \nu_I$ Lysis rate of $I$ by $E_T$ $0.1$ cells per mm$^3$ per day est
$ \tau$ Lysis rate of $I$ by $E_V$ $0.2$ cells per mm$^3$ per day est
Symbol Description Value & units Ref.
$K$ Tumour carrying capacity $10^6$ cells per mm$^3$ per day [4]
$\alpha$ Tumour growth rate $0.206$ cells per mm$^3$ per day [4]
$\beta$ Infection rate of tumour cells $0.001-0.1$ cells per mm$^3$ per day [4]
$\delta$ Infected tumour cells death $0.5115$ day$^{-1}$ [4]
$\gamma$ Rate of virus decay $0.01$ day$^{-1}$ [4]
$b$ Virus burst size $0-1000$ virions per mm$^3$ per day [11]
$\psi$ Rate drug decay $4.17$ millilitres per mm$^3$ per day [39]
$\delta_U$ Lysis rate of $U$ by the drug $50$ cells per mm$^3$ per day [39]
$\delta_I$ Lysis rate of $I$ by the drug $60$ cells per mm$^3$ per day [39]
$\phi$ $E_V$ production rate $0.7$ day$^{-1}$ [9]
$ \beta_T$ $E_T$ production rate $0.5$ cells per mm$^3$ per day [19,27]
$ \delta_v,\delta_T$ immune decay rates $0.01$ day$^{-1}$ [19,27]
$ K_u,K_c,\kappa$ Michaelis--Menten constants $10^5$ cells per mm$^3$ per day [25]
$ \nu_U$ Lysis rate of $U$ by $E_T$ $0.08$ cells per mm$^3$ per day est
$ \nu_I$ Lysis rate of $I$ by $E_T$ $0.1$ cells per mm$^3$ per day est
$ \tau$ Lysis rate of $I$ by $E_V$ $0.2$ cells per mm$^3$ per day est
Table 3.  Sensitivity and elasticity indices of $R_0$ with respect to model parameters
Parameter Sensitivity index Elasticity index
$b$ $S_b = 9.88 $ $e_b = 1$
$\beta$ $S_{\beta} = 988.4$ $e_{\beta}=1$
$\gamma$ $S_{\gamma} = - 9884$ $e_{\gamma}=-1$
$\delta$ $S_{\delta} = 2.2325$ $e_{\delta}= 0.011553$
$\alpha$ $S_{\alpha} = 2.4372$ $e_{\alpha} = 5.1 \times 10^{-12}$
$\delta_U$ $S_{\delta_U} = -1.004 \times 10^{-7}$ $e_{\delta_U} = -5.1 \times 10^{-12} $
$\delta_I$ $S_{\delta_I} = -90.32$ $e_{\delta_I} =- 0.011553$
$K_c$ $ S_{K_c}= 0.2048x$ $e_{K_c}= 0.0000414$
$K_u$ $S_{K_u}=-1.01\times 10^{-6}$ $ e_{K_u} = -2.05 \times 10^{-10}$
$\psi$ $S_{\psi} = 0.004$ $e_{\psi}= 0.000414$
$q$ $S_{q} = -0.008 $ $ e_{q} = - 0.000414$
Parameter Sensitivity index Elasticity index
$b$ $S_b = 9.88 $ $e_b = 1$
$\beta$ $S_{\beta} = 988.4$ $e_{\beta}=1$
$\gamma$ $S_{\gamma} = - 9884$ $e_{\gamma}=-1$
$\delta$ $S_{\delta} = 2.2325$ $e_{\delta}= 0.011553$
$\alpha$ $S_{\alpha} = 2.4372$ $e_{\alpha} = 5.1 \times 10^{-12}$
$\delta_U$ $S_{\delta_U} = -1.004 \times 10^{-7}$ $e_{\delta_U} = -5.1 \times 10^{-12} $
$\delta_I$ $S_{\delta_I} = -90.32$ $e_{\delta_I} =- 0.011553$
$K_c$ $ S_{K_c}= 0.2048x$ $e_{K_c}= 0.0000414$
$K_u$ $S_{K_u}=-1.01\times 10^{-6}$ $ e_{K_u} = -2.05 \times 10^{-10}$
$\psi$ $S_{\psi} = 0.004$ $e_{\psi}= 0.000414$
$q$ $S_{q} = -0.008 $ $ e_{q} = - 0.000414$
Table 4.  Selected sensitivity indices of the total tumour equilibria, $\Gamma ^{U+I} _p$, in response to drug, $q$ with the corresponding value of $R_0$
q (mg/l) 5 10 15 35 50 100
$\Gamma ^{U^*+I^*} _p$ $-8.3\times 10^{-5}$ $-2.5\times 10^{-5}$ $-1\times 10^{-5}$ $-6.1\times 10^{-5}$ $-9.6\times 10^{-6}$ $-2.4\times 10^{-6}$
$R_0$ $ 51.0476$ $51.0473$ $ 51.0470$ $51.0459$ $51.0450$ $ 51.0421$
q (mg/l) 5 10 15 35 50 100
$\Gamma ^{U^*+I^*} _p$ $-8.3\times 10^{-5}$ $-2.5\times 10^{-5}$ $-1\times 10^{-5}$ $-6.1\times 10^{-5}$ $-9.6\times 10^{-6}$ $-2.4\times 10^{-6}$
$R_0$ $ 51.0476$ $51.0473$ $ 51.0470$ $51.0459$ $51.0450$ $ 51.0421$
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