The Effect of Tumor Angiogenesis Agents on Tumor Growth Dynamics: A Mathematical Model
,
Abstract
Purpose: Tumor-induced angiogenesis is a dangerous state of the tumor growth process in which solid tumors have a blood supply. Modeling has been a very important tool in studying tumor growth and angiogenesis. In this paper, we developed a cancer model by introducing tumor angiogenesis agents to better highlight the role of these chemical substances in tumor-induced vascularization. Our model can reconstruct the transition from a pre-angiogenic to a post-angiogenic state.
Materials and Methods: The proposed model comprises five variables: host cells (normal cells), immune cells, tumor cells, endothelial cells, and tumor angiogenesis agents. Chaotic behavior in the production of different populations of cells during vascular growth may confer survival advantages to tumors. Our model has a chaotic regime, which is an indication of tumor-induced angiogenesis dynamics. The fixed points are analyzed biologically, and stability analysis is performed via their eigenvalues. We analyzed the model dynamics via observability and bifurcation analysis.
Results: The numerical simulations illustrate biological and clinical findings about vascular tumors. The results show that the proposed model with the existence of tumor angiogenesis agents could capture both avascular and vascular stages of tumor growth. There is no effect of tumor cell killing rate via immune cells on the system dynamics. However, the increase of inhibitory factors of tumor angiogenesis agents leads to the termination of chaos.
Conclusion: Our results show the ineffectiveness of targeted treatments on the immune system, which has been confirmed by many negative treatment methods in immunotherapies. Tumor-secreted inhibitor factors are essential to regulating the angiogenesis process. However, increasing inhibitor factors via anti-angiogenic drugs would be a more effective therapeutic approach to eradicate metastasis.
1- Robert M Sutherland, "Cell and environment interactions in tumor microregions: the multicell spheroid model." Science, Vol. 240 (No. 4849), pp. 177-84, (1988).
2- Judah Folkman, "Angiogenesis in cancer, vascular, rheumatoid and other disease." Nature medicine, Vol. 1 (No. 1), pp. 27-30, (1995).
3- Judah Folkman, "Tumor angiogenesis: therapeutic implications." N Engl j Med, Vol. 285 (No. 21), pp. 1182-86, (1971).
4- Anil Bagri, Hosein Kouros-Mehr, Kevin G Leong, and Greg D Plowman, "Use of anti-VEGF adjuvant therapy in cancer: challenges and rationale." Trends in molecular medicine, Vol. 16 (No. 3), pp. 122-32, (2010).
5- Alexander RA Anderson and Mark AJ Chaplain, "Continuous and discrete mathematical models of tumor-induced angiogenesis." Bulletin of mathematical biology, Vol. 60 (No. 5), pp. 857-99, (1998).
6- Joseph J Casciari, Stratis V Sotirchos, and Robert M Sutherland, "Mathematical modelling of microenvironment and growth in EMT6/Ro multicellular tumour spheroids." Cell proliferation, Vol. 25 (No. 1), pp. 1-22, (1992).
7- John A Adam and SA Maggelakis, "Mathematical models of tumor growth. IV. Effects of a necrotic core." Mathematical biosciences, Vol. 97 (No. 1), pp. 121-36, (1989).
8- Alan C Burton, "Rate of growth of solid tumours as a problem of diffusion." Growth, Vol. 30 (No. 2), pp. 157-76, (1966).
9- HM Byrne and Mark AJ Chaplain, "Free boundary value problems associated with the growth and development of multicellular spheroids." European Journal of Applied Mathematics, Vol. 8 (No. 6), pp. 639-58, (1997).
10- Jonathan A Sherratt, "Traveling wave solutions of a mathematical model for tumor encapsulation." SIAM Journal on Applied Mathematics, Vol. 60 (No. 2), pp. 392-407, (2000).
11- BP Marchant, J Norbury, and JA Sherratt, "Travelling wave solutions to a haptotaxis-dominated model of malignant invasion." Nonlinearity, Vol. 14 (No. 6), p. 1653, (2001).
12- Sabine Dormann and Andreas Deutsch, "Modeling of self-organized avascular tumor growth with a hybrid cellular automaton." In silico biology, Vol. 2 (No. 3), pp. 393-406, (2002).
13- W Düchting and Th Vogelsaenger, "Recent progress in modelling and simulation of three-dimensional tumor growth and treatment." Biosystems, Vol. 18 (No. 1), pp. 79-91, (1985).
14- Emma L Stott, Nick F Britton, James A Glazier, and Mark Zajac, "Stochastic simulation of benign avascular tumour growth using the Potts model." Mathematical and Computer Modelling, Vol. 30 (No. 5-6), pp. 183-98, (1999).
15- Renfrey Burnard Potts, "Some generalized order-disorder transformations." in Mathematical proceedings of the cambridge philosophical society, (1952), Vol. 48 (No. 1): Cambridge University Press, pp. 106-09.
16- Adriaan Daniël Fokker, "Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld." Annalen der Physik, Vol. 348 (No. 5), pp. 810-20, (1914).
17- VM Planck, "Über einen Satz der statistischen Dynamik und seine Erweiterung in der Quantentheorie." Sitzungberichte der, (1917).
18- Lisette G De Pillis and Ami Radunskaya, "The dynamics of an optimally controlled tumor model: A case study." Mathematical and Computer Modelling, Vol. 37 (No. 11), pp. 1221-44, (2003).
19- Vladimir A Kuznetsov, Iliya A Makalkin, Mark A Taylor, and Alan S Perelson, "Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis." Bulletin of mathematical biology, Vol. 56 (No. 2), pp. 295-321, (1994).
20- Mehmet Itik and Stephen P Banks, "Chaos in a three-dimensional cancer model." International Journal of Bifurcation and Chaos, Vol. 20 (No. 01), pp. 71-79, (2010).
21- Jorge Duarte, Cristina Januario, Carla Rodrigues, and Josep Sardanyes, "Topological complexity and predictability in the dynamics of a tumor growth model with Shilnikov's chaos." International Journal of Bifurcation and Chaos, Vol. 23 (No. 07), p. 1350124, (2013).
22- Christophe Letellier, Fabrice Denis, and Luis A Aguirre, "What can be learned from a chaotic cancer model?" Journal of theoretical biology, Vol. 322pp. 7-16, (2013).
23- Álvaro G López, Juan Sabuco, Jesús M Seoane, Jorge Duarte, Cristina Januário, and Miguel AF Sanjuán, "Avoiding healthy cells extinction in a cancer model." Journal of theoretical biology, Vol. 349pp. 74-81, (2014).
24- Judah Folkman, "Clinical applications of research on angiogenesis." New England Journal of Medicine, Vol. 333 (No. 26), pp. 1757-63, (1995).
25- M Baum, MAJ Chaplain, ARA Anderson, M Douek, and JS Vaidya, "Does breast cancer exist in a state of chaos?" European journal of cancer, Vol. 35 (No. 6), pp. 886-91, (1999).
26- ARA Anderson, MAJ Chaplain, C Garcia-Reimbert, and CA Vargas, "A gradient-driven mathematical model of antiangiogenesis." Mathematical and Computer Modelling, Vol. 32 (No. 10), pp. 1141-52, (2000).
27- Michael S O'Reilly et al., "Angiostatin: a novel angiogenesis inhibitor that mediates the suppression of metastases by a Lewis lung carcinoma." cell, Vol. 79 (No. 2), pp. 315-28, (1994).
28- SA Maggelakis, "The effects of tumor angiogenesis factor (TAF) and tumor inhibitor factors (TIFs) on tumor vascularization: A mathematical model." Mathematical and Computer Modelling, Vol. 23 (No. 6), pp. 121-33, (1996).
29- Sajad Shafiekhani, Hojat Dehghanbanadaki, Azam Sadat Fatemi, Sara Rahbar, Jamshid Hadjati, and Amir Homayoun Jafari, "Prediction of anti-CD25 and 5-FU treatments efficacy for pancreatic cancer using a mathematical model." BMC cancer, Vol. 21pp. 1-21, (2021).
30- Yongwoon Jung, Pavel Kraikivski, Sajad Shafiekhani, Scott S Terhune, and Ranjan K Dash, "Crosstalk between Plk1, p53, cell cycle, and G2/M DNA damage checkpoint regulation in cancer: computational modeling and analysis." npj Systems Biology and Applications, Vol. 7 (No. 1), p. 46, (2021).
31- Sajad Shafiekhani, Nematollah Gheibi, and Azam Janati Esfahani, "Combination of anti-PD-L1 and radiotherapy in hepatocellular carcinoma: a mathematical model with uncertain parameters." Simulation, Vol. 99 (No. 4), pp. 313-25, (2023).
32- Sadjad Shafiekhani, Sara Rahbar, Fahimeh Akbarian, Jamshid Hadjati, Armin Allahverdy, and Amir Homayoun Jafari, "Providing a Stochastic Petri Net Model for Interactions of the Immune System and B16-F10 Tumor Cells in order to Investigate the Effect of Myeloid-Derived Suppressor Cells (MDSC) on Behavioral States of Tumor." Frontiers in Biomedical Technologies, Vol. 4 (No. 1-2), pp. 31-37, (2017).
33- Sajad Shafiekhani, Sara Rahbar, Fahimeh Akbarian, and Amir Homayoun Jafari, "Fuzzy stochastic petri net with uncertain kinetic parameters for modeling tumor-immune system." in 2018 25th National and 3rd International Iranian Conference on Biomedical Engineering (ICBME), (2018): IEEE, pp. 1-5.
34- Louise Viger, Fabrice Denis, Martin Rosalie, and Christophe Letellier, "A cancer model for the angiogenic switch." Journal of theoretical biology, Vol. 360pp. 21-33, (2014).
35- Christophe Letellier, Sourav Kumar Sasmal, Clément Draghi, Fabrice Denis, and Dibakar Ghosh, "A chemotherapy combined with an anti-angiogenic drug applied to a cancer model including angiogenesis." Chaos, Solitons & Fractals, Vol. 99pp. 297-311, (2017).
36- Konstantin E Starkov, "A cancer model for the angiogenic switch and immunotherapy: Tumor eradication in analysis of ultimate dynamics." International Journal of Bifurcation and Chaos, Vol. 30 (No. 10), p. 2050150, (2020).
37- Parthasakha Das, Samhita Das, Pritha Das, Fathalla A Rihan, Muhammet Uzuntarla, and Dibakar Ghosh, "Optimal control strategy for cancer remission using combinatorial therapy: a mathematical model-based approach." Chaos, Solitons & Fractals, Vol. 145p. 110789, (2021).
38- A Mohseni, M Pooyan, S Raiesdana, and M. B. Menhaj, "A cancer model including tumor angiogenesis agents." Power System Technology, Journal Article Vol. 482, pp. 973-94, (2024).
39- Femke Hillen and Arjan W Griffioen, "Tumour vascularization: sprouting angiogenesis and beyond." Cancer and Metastasis Reviews, Vol. 26 (No. 3), pp. 489-502, (2007).
40- Ilaria Brazzoli, Elena De Angelis, and Pierre‐Emmanuel Jabin, "A mathematical model of immune competition related to cancer dynamics." Mathematical Methods in the applied sciences, Vol. 33 (No. 6), pp. 733-50, (2010).
41- Mitchell J Feigenbaum, "Quantitative universality for a class of nonlinear transformations." Journal of statistical physics, Vol. 19 (No. 1), pp. 25-52, (1978).
42- Aniruddha Choudhury, Szilvia Mosolits, Parviz Kokhaei, Lotta Hansson, Marzia Palma, and Håkan Mellstedt, "Clinical results of vaccine therapy for cancer: learning from history for improving the future." Advances in cancer research, Vol. 95pp. 147-202, (2006).
43- Ming Chi and Arkadiusz Z Dudek, "Vaccine therapy for metastatic melanoma: systematic review and meta-analysis of clinical trials." Melanoma research, Vol. 21 (No. 3), pp. 165-74, (2011).
44- Markus R Owen, Tomás Alarcón, Philip K Maini, and Helen M Byrne, "Angiogenesis and vascular remodelling in normal and cancerous tissues." Journal of mathematical biology, Vol. 58pp. 689-721, (2009).
2- Judah Folkman, "Angiogenesis in cancer, vascular, rheumatoid and other disease." Nature medicine, Vol. 1 (No. 1), pp. 27-30, (1995).
3- Judah Folkman, "Tumor angiogenesis: therapeutic implications." N Engl j Med, Vol. 285 (No. 21), pp. 1182-86, (1971).
4- Anil Bagri, Hosein Kouros-Mehr, Kevin G Leong, and Greg D Plowman, "Use of anti-VEGF adjuvant therapy in cancer: challenges and rationale." Trends in molecular medicine, Vol. 16 (No. 3), pp. 122-32, (2010).
5- Alexander RA Anderson and Mark AJ Chaplain, "Continuous and discrete mathematical models of tumor-induced angiogenesis." Bulletin of mathematical biology, Vol. 60 (No. 5), pp. 857-99, (1998).
6- Joseph J Casciari, Stratis V Sotirchos, and Robert M Sutherland, "Mathematical modelling of microenvironment and growth in EMT6/Ro multicellular tumour spheroids." Cell proliferation, Vol. 25 (No. 1), pp. 1-22, (1992).
7- John A Adam and SA Maggelakis, "Mathematical models of tumor growth. IV. Effects of a necrotic core." Mathematical biosciences, Vol. 97 (No. 1), pp. 121-36, (1989).
8- Alan C Burton, "Rate of growth of solid tumours as a problem of diffusion." Growth, Vol. 30 (No. 2), pp. 157-76, (1966).
9- HM Byrne and Mark AJ Chaplain, "Free boundary value problems associated with the growth and development of multicellular spheroids." European Journal of Applied Mathematics, Vol. 8 (No. 6), pp. 639-58, (1997).
10- Jonathan A Sherratt, "Traveling wave solutions of a mathematical model for tumor encapsulation." SIAM Journal on Applied Mathematics, Vol. 60 (No. 2), pp. 392-407, (2000).
11- BP Marchant, J Norbury, and JA Sherratt, "Travelling wave solutions to a haptotaxis-dominated model of malignant invasion." Nonlinearity, Vol. 14 (No. 6), p. 1653, (2001).
12- Sabine Dormann and Andreas Deutsch, "Modeling of self-organized avascular tumor growth with a hybrid cellular automaton." In silico biology, Vol. 2 (No. 3), pp. 393-406, (2002).
13- W Düchting and Th Vogelsaenger, "Recent progress in modelling and simulation of three-dimensional tumor growth and treatment." Biosystems, Vol. 18 (No. 1), pp. 79-91, (1985).
14- Emma L Stott, Nick F Britton, James A Glazier, and Mark Zajac, "Stochastic simulation of benign avascular tumour growth using the Potts model." Mathematical and Computer Modelling, Vol. 30 (No. 5-6), pp. 183-98, (1999).
15- Renfrey Burnard Potts, "Some generalized order-disorder transformations." in Mathematical proceedings of the cambridge philosophical society, (1952), Vol. 48 (No. 1): Cambridge University Press, pp. 106-09.
16- Adriaan Daniël Fokker, "Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld." Annalen der Physik, Vol. 348 (No. 5), pp. 810-20, (1914).
17- VM Planck, "Über einen Satz der statistischen Dynamik und seine Erweiterung in der Quantentheorie." Sitzungberichte der, (1917).
18- Lisette G De Pillis and Ami Radunskaya, "The dynamics of an optimally controlled tumor model: A case study." Mathematical and Computer Modelling, Vol. 37 (No. 11), pp. 1221-44, (2003).
19- Vladimir A Kuznetsov, Iliya A Makalkin, Mark A Taylor, and Alan S Perelson, "Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis." Bulletin of mathematical biology, Vol. 56 (No. 2), pp. 295-321, (1994).
20- Mehmet Itik and Stephen P Banks, "Chaos in a three-dimensional cancer model." International Journal of Bifurcation and Chaos, Vol. 20 (No. 01), pp. 71-79, (2010).
21- Jorge Duarte, Cristina Januario, Carla Rodrigues, and Josep Sardanyes, "Topological complexity and predictability in the dynamics of a tumor growth model with Shilnikov's chaos." International Journal of Bifurcation and Chaos, Vol. 23 (No. 07), p. 1350124, (2013).
22- Christophe Letellier, Fabrice Denis, and Luis A Aguirre, "What can be learned from a chaotic cancer model?" Journal of theoretical biology, Vol. 322pp. 7-16, (2013).
23- Álvaro G López, Juan Sabuco, Jesús M Seoane, Jorge Duarte, Cristina Januário, and Miguel AF Sanjuán, "Avoiding healthy cells extinction in a cancer model." Journal of theoretical biology, Vol. 349pp. 74-81, (2014).
24- Judah Folkman, "Clinical applications of research on angiogenesis." New England Journal of Medicine, Vol. 333 (No. 26), pp. 1757-63, (1995).
25- M Baum, MAJ Chaplain, ARA Anderson, M Douek, and JS Vaidya, "Does breast cancer exist in a state of chaos?" European journal of cancer, Vol. 35 (No. 6), pp. 886-91, (1999).
26- ARA Anderson, MAJ Chaplain, C Garcia-Reimbert, and CA Vargas, "A gradient-driven mathematical model of antiangiogenesis." Mathematical and Computer Modelling, Vol. 32 (No. 10), pp. 1141-52, (2000).
27- Michael S O'Reilly et al., "Angiostatin: a novel angiogenesis inhibitor that mediates the suppression of metastases by a Lewis lung carcinoma." cell, Vol. 79 (No. 2), pp. 315-28, (1994).
28- SA Maggelakis, "The effects of tumor angiogenesis factor (TAF) and tumor inhibitor factors (TIFs) on tumor vascularization: A mathematical model." Mathematical and Computer Modelling, Vol. 23 (No. 6), pp. 121-33, (1996).
29- Sajad Shafiekhani, Hojat Dehghanbanadaki, Azam Sadat Fatemi, Sara Rahbar, Jamshid Hadjati, and Amir Homayoun Jafari, "Prediction of anti-CD25 and 5-FU treatments efficacy for pancreatic cancer using a mathematical model." BMC cancer, Vol. 21pp. 1-21, (2021).
30- Yongwoon Jung, Pavel Kraikivski, Sajad Shafiekhani, Scott S Terhune, and Ranjan K Dash, "Crosstalk between Plk1, p53, cell cycle, and G2/M DNA damage checkpoint regulation in cancer: computational modeling and analysis." npj Systems Biology and Applications, Vol. 7 (No. 1), p. 46, (2021).
31- Sajad Shafiekhani, Nematollah Gheibi, and Azam Janati Esfahani, "Combination of anti-PD-L1 and radiotherapy in hepatocellular carcinoma: a mathematical model with uncertain parameters." Simulation, Vol. 99 (No. 4), pp. 313-25, (2023).
32- Sadjad Shafiekhani, Sara Rahbar, Fahimeh Akbarian, Jamshid Hadjati, Armin Allahverdy, and Amir Homayoun Jafari, "Providing a Stochastic Petri Net Model for Interactions of the Immune System and B16-F10 Tumor Cells in order to Investigate the Effect of Myeloid-Derived Suppressor Cells (MDSC) on Behavioral States of Tumor." Frontiers in Biomedical Technologies, Vol. 4 (No. 1-2), pp. 31-37, (2017).
33- Sajad Shafiekhani, Sara Rahbar, Fahimeh Akbarian, and Amir Homayoun Jafari, "Fuzzy stochastic petri net with uncertain kinetic parameters for modeling tumor-immune system." in 2018 25th National and 3rd International Iranian Conference on Biomedical Engineering (ICBME), (2018): IEEE, pp. 1-5.
34- Louise Viger, Fabrice Denis, Martin Rosalie, and Christophe Letellier, "A cancer model for the angiogenic switch." Journal of theoretical biology, Vol. 360pp. 21-33, (2014).
35- Christophe Letellier, Sourav Kumar Sasmal, Clément Draghi, Fabrice Denis, and Dibakar Ghosh, "A chemotherapy combined with an anti-angiogenic drug applied to a cancer model including angiogenesis." Chaos, Solitons & Fractals, Vol. 99pp. 297-311, (2017).
36- Konstantin E Starkov, "A cancer model for the angiogenic switch and immunotherapy: Tumor eradication in analysis of ultimate dynamics." International Journal of Bifurcation and Chaos, Vol. 30 (No. 10), p. 2050150, (2020).
37- Parthasakha Das, Samhita Das, Pritha Das, Fathalla A Rihan, Muhammet Uzuntarla, and Dibakar Ghosh, "Optimal control strategy for cancer remission using combinatorial therapy: a mathematical model-based approach." Chaos, Solitons & Fractals, Vol. 145p. 110789, (2021).
38- A Mohseni, M Pooyan, S Raiesdana, and M. B. Menhaj, "A cancer model including tumor angiogenesis agents." Power System Technology, Journal Article Vol. 482, pp. 973-94, (2024).
39- Femke Hillen and Arjan W Griffioen, "Tumour vascularization: sprouting angiogenesis and beyond." Cancer and Metastasis Reviews, Vol. 26 (No. 3), pp. 489-502, (2007).
40- Ilaria Brazzoli, Elena De Angelis, and Pierre‐Emmanuel Jabin, "A mathematical model of immune competition related to cancer dynamics." Mathematical Methods in the applied sciences, Vol. 33 (No. 6), pp. 733-50, (2010).
41- Mitchell J Feigenbaum, "Quantitative universality for a class of nonlinear transformations." Journal of statistical physics, Vol. 19 (No. 1), pp. 25-52, (1978).
42- Aniruddha Choudhury, Szilvia Mosolits, Parviz Kokhaei, Lotta Hansson, Marzia Palma, and Håkan Mellstedt, "Clinical results of vaccine therapy for cancer: learning from history for improving the future." Advances in cancer research, Vol. 95pp. 147-202, (2006).
43- Ming Chi and Arkadiusz Z Dudek, "Vaccine therapy for metastatic melanoma: systematic review and meta-analysis of clinical trials." Melanoma research, Vol. 21 (No. 3), pp. 165-74, (2011).
44- Markus R Owen, Tomás Alarcón, Philip K Maini, and Helen M Byrne, "Angiogenesis and vascular remodelling in normal and cancerous tissues." Journal of mathematical biology, Vol. 58pp. 689-721, (2009).
| Files | ||
| Issue | Articles in Press | |
| Section | Original Article(s) | |
| Keywords | ||
| Mathematical model Chaos Vascular tumor Tumor angiogenesis agents Theoretical biology Computational biology. | ||
| Rights and permissions | |
|
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. |
How to Cite
1.
Mohseni A, Pooyan M, Raiesdana S, Menhaj MB. The Effect of Tumor Angiogenesis Agents on Tumor Growth Dynamics: A Mathematical Model. Frontiers Biomed Technol. 2024;.
