American Institute of Mathematical Sciences

Advanced
2013, 10(3): 729-742. doi: 10.3934/mbe.2013.10.729

Hybrid discrete-continuous model of invasive bladder cancer

Eugene Kashdan 1, and  Svetlana Bunimovich-Mendrazitsky 2,

1. 

School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Ramat Aviv, 69978, Israel
2. 

Department of Computer Science and Mathematics, Ariel University Center of Samaria, Ariel, 40700, Israel

Received  June 2012 Revised  November 2012 Published  April 2013

Bladder cancer is the seventh most common cancer worldwide. Epidemiological studies and experiments implicated chemical penetration into urothelium (epithelial tissue surrounding bladder) in the etiology of bladder cancer. In this work we model invasive bladder cancer. This type of cancer starts in the urothelium and progresses towards surrounding muscles and tissues, causing metastatic disease. Our mathematical model of invasive BC consists of two coupled sub-models: (i) living cycle of the urothelial cells (normal and mutated) simulated using discrete technique of Cellular Automata and (ii) mechanism of tumor invasion described by the system of reaction-diffusion equations. Numerical simulations presented here are in good qualitative agreement with the experimental results and reproduce in vitro observations described in medical literature.
Keywords: cellular automata, Bladder cancer, reaction-diffusion equations., metaloproteinases.
Mathematics Subject Classification: Primary: 92C42; Secondary: 35K57, 37B1.
Citation: Eugene Kashdan, Svetlana Bunimovich-Mendrazitsky. Hybrid discrete-continuous model of invasive bladder cancer. Mathematical Biosciences & Engineering, 2013, 10 (3) : 729-742. doi: 10.3934/mbe.2013.10.729
References:
[1]

A. E. Anderson, A hybrid mathematical model of solid tumour invasion: The importance of cell adhesion, Mathematical Medicine and Biology, 22 (2005), 163-186. doi: 10.1093/imammb/dqi005.  Google Scholar

[2]

F. Andreu, V. Caselles and J. Mazan, Diffusion equations with finite speed of propagation, in "Functional Analysis and Evolution Equations" (eds. H. Amann et al), Birkhauser Basel, (2008), 17-34. doi: 10.1007/978-3-7643-7794-6_2.  Google Scholar

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[5]

C. E. Brinckerhoff and L. M.Matrisian, Matrix metalloproteinases: A tail of frog that became a prince, Nature Reviews, Molecular Cell Biology, 3 (2002) 207-214. Google Scholar

[6]

H. M Byrne, M. A. J. Chaplain, G. J. Pettet and D. L. S. McElwain, A mathematical model of trophoblast invasion, J. of Theor. Medicine, 1 (1999), 275-286. doi: 10.1080/10273669908833026.  Google Scholar

[7]

C. Chang and Z. Werb, The many faces of metalloproteases: cell growth, invasion, angiogenesis and metastasis, Trends in Cell Biology, 11 (2001), 37-43. Google Scholar

[8]

M. A. J. Chaplain and G. Lolas, Mathematical modeling of cancer invasion of tissue: Dynamic heterogeneity, Networks and Heterogeneous Media, 1 (2006), 399-439 doi: 10.3934/nhm.2006.1.399.  Google Scholar

[9]

L. M. Coussens, B. Fingleton and L. M. Matrisian, Matrix metalloproteinase inhibitors and cancer: Trials and tribulations, Science, 295 (2002), 2387-2392. doi: 10.1126/science.1067100.  Google Scholar

[10]

G. B. Ermentrout and L. Edelstein-Keshet, Cellular automata approaches to biological modelling, J. Theor. Biol., 160 (1993), 97-133. doi: 10.1006/jtbi.1993.1007.  Google Scholar

[11]

R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Research, 56 (1996), 5745-5753. Google Scholar

[12]

B. George, R. H. Datar and R. J. Cote, Molecular biology of bladder cancer: cell cycle alterations, in "Textbook of Bladder Cancer" (eds. S. P. Lerner, et al), Taylor & Francis, (2006), 107-122. Google Scholar

[13]

G. Giannelli, J. Falk-Marzillier, O. Schiraldi, W. G. Stetler-Stevenson and V. Quaranta, Induction of cell migration by matrix metalloprotease-2 cleavage of lamitiin-5, Science, 277 (1997), 225-228. Google Scholar

[14]

F. Graner and J. A. Glazier, Simulation of biological cell sorting using a two-dimensional extended Potts model, Phys. Rev. Lett., 69 (1992), 2013-2016. doi: 10.1103/PhysRevLett.69.2013.  Google Scholar

[15]

G. P. Hemstreet III and E. M, Messing, Early detection for bladder cancer, in "Textbook of Bladder Cancer" (eds. S. P. Lerner, et al), Taylor & Francis, (2006), 257-266. Google Scholar

[16]

A. Jemal, F. Bray, M. M. Center, J. Ferlay, E. Ward and D. Forman, Global cancer statistics, CA: A Cancer J. for Clinicians, 61 (2011), 69-90. doi: 10.3322/caac.20107.  Google Scholar

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T. Kakizoe, Development and progression of urothelial carcinoma, Cancer Science, 97 (1982), 821-828. doi: 10.1111/j.1349-7006.2006.00264.x.  Google Scholar

[18]

E. Kashdan and S. Bunimovich-Mendrazitsky, "Multi-Scale Model of Bladder Cancer Development," Discrete and Continuous Dynamical Systems, 2011, 803-812.  Google Scholar

[19]

M. Kirsch-Voldersa, M. Aardemab and A. Elhajoujic, Concepts of threshold in mutagenesis and carcinogenesis, Mutation Res., 464 (2000), 3-11. doi: 10.1016/S1383-5718(99)00161-8.  Google Scholar

[20]

C. J. Malemud, Matrix metalloproteinases (MMPs) in health and disease: an overview, Frontiers in Bioscience, 11 (2006), 1696-1701. doi: 10.2741/1915.  Google Scholar

[21]

B. P. Marchant, J. Norbury and J. A. Sherratt, Travelling wave solutions to a haptotaxis-dominated model of malignant invasion, Nonlinearity, 14 (2001), 1653-1671. doi: 10.1088/0951-7715/14/6/313.  Google Scholar

[22]

C. J. Marshall, L. M. Franks and A. W. Carbonell, Markers of neoplastic transformation in epithelial cell lines derived from human carcinomas, J. Natl. Cancer Inst., 58 (1977), 1743-1751. Google Scholar

[23]

L. L. Munn, C. Kunert and J. A. Tyrrell, Modeling tumor blood vessel dynamics, in "Mathematical Methods and Models in Biomedicine" (eds. U. Ledzewicz, et al), Springer, (2012), 113-142. doi: 10.1007/978-1-4614-4178-6_5.  Google Scholar

[24]

G. Murphy and J. Gavrilovic, The mathematical modelling of Proteolysis and cell migration: Creating a path?, Curr. Opin. Cell Biol., 11 (1999), 614-621. Google Scholar

[25]

K. Nabeshima, W. S. Lane and C. Biswas, Partial sequencing and characterisation of the tumour cell- derived collagenase stimulatory factor, Arch. Biochem. Biophys., 285 (1991), 90-96. Google Scholar

[26]

U. 0. Nseyo and D. L. Lamm, Immunotherapy of bladder cancer, Seminars in Surgical Oncology, 13 (1997), 342-349. Google Scholar

[27]

J. E. Nutt, G. C. Durkan, J. vK. Mellon and J. Lunce, Matrix metalloproteinases (MMPs) in bladder cancer: The induction of MMP9 by epidermal growth factor and its detection in urine, BJU International, 91 (2003), 99-104. doi: 10.1046/j.1464-410X.2003.04020.x.  Google Scholar

[28]

A. J. Perumpanani, J. A. Sherratt, J. Norbury and H. M. Byrne, A two parameter family of travelling waves with a singular barrier arising from the modelling of extracellular matrix mediated cellular invasion, Physica D, 126 (1999), 145-159. doi: 10.1016/S0167-2789(98)00272-3.  Google Scholar

[29]

V. Quaranta, K. A. Rejniak, P. Gerlee and A. R. A. Anderson, Invasion emerges from cancer cell adaptation to competitive microenvironments: Quantitative predictions from multiscale mathematical models, Seminars in Cancer Biology, 18 (2008), 338-348. doi: 10.1016/j.semcancer.2008.03.018.  Google Scholar

[30]

I. Ramis-Conde, M. A. J. Chaplain and A. R. A. Anderson, Mathematical modelling of cancer cell invasion of tissue, Mathematical and Computer Modelling, 47 (2008), 533-545. doi: 10.1016/j.mcm.2007.02.034.  Google Scholar

[31]

C. J. Sherr, Cancer cell cycles, Science, 274 (1996), 1672-1677. doi: 10.1126/science.274.5293.1672.  Google Scholar

[32]

W. G. Stetler- Stevenson, S. Aznavoorian and L. A. Liotta, Tumor cell interactions with the extracellular matrix during invasion and metastasis, Ann. Rev. Cell Biol., 9 (1993), 541-573. Google Scholar

[33]

J. Testa, Loss of metastatic phenotype by a human epidermoid carcinoma cell line hep-3 is accompanied by increased expression of tissue inhibitor of matrix metalloproteinase-2, Cancer Res., 52 (1992), 5597-5603. Google Scholar

[34]

Transitional epithelium of the urinary bladder, http://en.wikipedia.org/wiki/Urothelium., Image available for use under CCA license., ().  http://en.wikipedia.org/wiki/Urothelium.+" target="_new" title="Go to article in Google Scholar"> Google Scholar

[35]

S. Turner and J. A. Sheratt, Intercellular adhesion and cancer invasion: A discrete simulation using the extended potts model, J. Theor. Biol., 216 (2002), 85-100. doi: 10.1006/jtbi.2001.2522.  Google Scholar

[36]

K. Vasala, P. Paakko and T. Turpeenniemi-Hujanen, Matrix metalloproteinase-9 ( MMP-9) immunoreactive protein in urinary bladder cancer: A marker of favorable prognosis, Anticancer Research, 28 (2008), 1757-1762. Google Scholar

[37]

J. L. Vasquez, "Porous Medium Equation. Mathematical Theory," Oxford University Press, Oxford, 2007.  Google Scholar

[38]

S. M. Wnek, M. K. Medeirosa, K. E. Eblinb and A. J. Gandolfi, Persistence of DNA damage following exposure of human bladder cells to chronic monomethylarsonous acid, Tox. and Appl. Pharm., 241 (2009), 202-209. doi: 10.1016/j.taap.2009.08.016.  Google Scholar

show all references

References:
[1]

A. E. Anderson, A hybrid mathematical model of solid tumour invasion: The importance of cell adhesion, Mathematical Medicine and Biology, 22 (2005), 163-186. doi: 10.1093/imammb/dqi005.  Google Scholar

[2]

F. Andreu, V. Caselles and J. Mazan, Diffusion equations with finite speed of propagation, in "Functional Analysis and Evolution Equations" (eds. H. Amann et al), Birkhauser Basel, (2008), 17-34. doi: 10.1007/978-3-7643-7794-6_2.  Google Scholar

[3]

Atlas of Genetics and Cytogenetics in Oncology and Haematology, http://AtlasGeneticsOncology.org., Image available for use under CCA license., ().  http://AtlasGeneticsOncology.org.+" target="_new" title="Go to article in Google Scholar"> Google Scholar

[4]

A. H. Baker, D. R. Edwards and G. Murphy, Metalloproteinase inhibitors: Biological actions and therapeutic opportunities, J. of Cell Science, 115 (2002), 3719-3727. doi: 10.1242/jcs.00063.  Google Scholar

[5]

C. E. Brinckerhoff and L. M.Matrisian, Matrix metalloproteinases: A tail of frog that became a prince, Nature Reviews, Molecular Cell Biology, 3 (2002) 207-214. Google Scholar

[6]

H. M Byrne, M. A. J. Chaplain, G. J. Pettet and D. L. S. McElwain, A mathematical model of trophoblast invasion, J. of Theor. Medicine, 1 (1999), 275-286. doi: 10.1080/10273669908833026.  Google Scholar

[7]

C. Chang and Z. Werb, The many faces of metalloproteases: cell growth, invasion, angiogenesis and metastasis, Trends in Cell Biology, 11 (2001), 37-43. Google Scholar

[8]

M. A. J. Chaplain and G. Lolas, Mathematical modeling of cancer invasion of tissue: Dynamic heterogeneity, Networks and Heterogeneous Media, 1 (2006), 399-439 doi: 10.3934/nhm.2006.1.399.  Google Scholar

[9]

L. M. Coussens, B. Fingleton and L. M. Matrisian, Matrix metalloproteinase inhibitors and cancer: Trials and tribulations, Science, 295 (2002), 2387-2392. doi: 10.1126/science.1067100.  Google Scholar

[10]

G. B. Ermentrout and L. Edelstein-Keshet, Cellular automata approaches to biological modelling, J. Theor. Biol., 160 (1993), 97-133. doi: 10.1006/jtbi.1993.1007.  Google Scholar

[11]

R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Research, 56 (1996), 5745-5753. Google Scholar

[12]

B. George, R. H. Datar and R. J. Cote, Molecular biology of bladder cancer: cell cycle alterations, in "Textbook of Bladder Cancer" (eds. S. P. Lerner, et al), Taylor & Francis, (2006), 107-122. Google Scholar

[13]

G. Giannelli, J. Falk-Marzillier, O. Schiraldi, W. G. Stetler-Stevenson and V. Quaranta, Induction of cell migration by matrix metalloprotease-2 cleavage of lamitiin-5, Science, 277 (1997), 225-228. Google Scholar

[14]

F. Graner and J. A. Glazier, Simulation of biological cell sorting using a two-dimensional extended Potts model, Phys. Rev. Lett., 69 (1992), 2013-2016. doi: 10.1103/PhysRevLett.69.2013.  Google Scholar

[15]

G. P. Hemstreet III and E. M, Messing, Early detection for bladder cancer, in "Textbook of Bladder Cancer" (eds. S. P. Lerner, et al), Taylor & Francis, (2006), 257-266. Google Scholar

[16]

A. Jemal, F. Bray, M. M. Center, J. Ferlay, E. Ward and D. Forman, Global cancer statistics, CA: A Cancer J. for Clinicians, 61 (2011), 69-90. doi: 10.3322/caac.20107.  Google Scholar

[17]

T. Kakizoe, Development and progression of urothelial carcinoma, Cancer Science, 97 (1982), 821-828. doi: 10.1111/j.1349-7006.2006.00264.x.  Google Scholar

[18]

E. Kashdan and S. Bunimovich-Mendrazitsky, "Multi-Scale Model of Bladder Cancer Development," Discrete and Continuous Dynamical Systems, 2011, 803-812.  Google Scholar

[19]

M. Kirsch-Voldersa, M. Aardemab and A. Elhajoujic, Concepts of threshold in mutagenesis and carcinogenesis, Mutation Res., 464 (2000), 3-11. doi: 10.1016/S1383-5718(99)00161-8.  Google Scholar

[20]

C. J. Malemud, Matrix metalloproteinases (MMPs) in health and disease: an overview, Frontiers in Bioscience, 11 (2006), 1696-1701. doi: 10.2741/1915.  Google Scholar

[21]

B. P. Marchant, J. Norbury and J. A. Sherratt, Travelling wave solutions to a haptotaxis-dominated model of malignant invasion, Nonlinearity, 14 (2001), 1653-1671. doi: 10.1088/0951-7715/14/6/313.  Google Scholar

[22]

C. J. Marshall, L. M. Franks and A. W. Carbonell, Markers of neoplastic transformation in epithelial cell lines derived from human carcinomas, J. Natl. Cancer Inst., 58 (1977), 1743-1751. Google Scholar

[23]

L. L. Munn, C. Kunert and J. A. Tyrrell, Modeling tumor blood vessel dynamics, in "Mathematical Methods and Models in Biomedicine" (eds. U. Ledzewicz, et al), Springer, (2012), 113-142. doi: 10.1007/978-1-4614-4178-6_5.  Google Scholar

[24]

G. Murphy and J. Gavrilovic, The mathematical modelling of Proteolysis and cell migration: Creating a path?, Curr. Opin. Cell Biol., 11 (1999), 614-621. Google Scholar

[25]

K. Nabeshima, W. S. Lane and C. Biswas, Partial sequencing and characterisation of the tumour cell- derived collagenase stimulatory factor, Arch. Biochem. Biophys., 285 (1991), 90-96. Google Scholar

[26]

U. 0. Nseyo and D. L. Lamm, Immunotherapy of bladder cancer, Seminars in Surgical Oncology, 13 (1997), 342-349. Google Scholar

[27]

J. E. Nutt, G. C. Durkan, J. vK. Mellon and J. Lunce, Matrix metalloproteinases (MMPs) in bladder cancer: The induction of MMP9 by epidermal growth factor and its detection in urine, BJU International, 91 (2003), 99-104. doi: 10.1046/j.1464-410X.2003.04020.x.  Google Scholar

[28]

A. J. Perumpanani, J. A. Sherratt, J. Norbury and H. M. Byrne, A two parameter family of travelling waves with a singular barrier arising from the modelling of extracellular matrix mediated cellular invasion, Physica D, 126 (1999), 145-159. doi: 10.1016/S0167-2789(98)00272-3.  Google Scholar

[29]

V. Quaranta, K. A. Rejniak, P. Gerlee and A. R. A. Anderson, Invasion emerges from cancer cell adaptation to competitive microenvironments: Quantitative predictions from multiscale mathematical models, Seminars in Cancer Biology, 18 (2008), 338-348. doi: 10.1016/j.semcancer.2008.03.018.  Google Scholar

[30]

I. Ramis-Conde, M. A. J. Chaplain and A. R. A. Anderson, Mathematical modelling of cancer cell invasion of tissue, Mathematical and Computer Modelling, 47 (2008), 533-545. doi: 10.1016/j.mcm.2007.02.034.  Google Scholar

[31]

C. J. Sherr, Cancer cell cycles, Science, 274 (1996), 1672-1677. doi: 10.1126/science.274.5293.1672.  Google Scholar

[32]

W. G. Stetler- Stevenson, S. Aznavoorian and L. A. Liotta, Tumor cell interactions with the extracellular matrix during invasion and metastasis, Ann. Rev. Cell Biol., 9 (1993), 541-573. Google Scholar

[33]

J. Testa, Loss of metastatic phenotype by a human epidermoid carcinoma cell line hep-3 is accompanied by increased expression of tissue inhibitor of matrix metalloproteinase-2, Cancer Res., 52 (1992), 5597-5603. Google Scholar

[34]

Transitional epithelium of the urinary bladder, http://en.wikipedia.org/wiki/Urothelium., Image available for use under CCA license., ().  http://en.wikipedia.org/wiki/Urothelium.+" target="_new" title="Go to article in Google Scholar"> Google Scholar

[35]

S. Turner and J. A. Sheratt, Intercellular adhesion and cancer invasion: A discrete simulation using the extended potts model, J. Theor. Biol., 216 (2002), 85-100. doi: 10.1006/jtbi.2001.2522.  Google Scholar

[36]

K. Vasala, P. Paakko and T. Turpeenniemi-Hujanen, Matrix metalloproteinase-9 ( MMP-9) immunoreactive protein in urinary bladder cancer: A marker of favorable prognosis, Anticancer Research, 28 (2008), 1757-1762. Google Scholar

[37]

J. L. Vasquez, "Porous Medium Equation. Mathematical Theory," Oxford University Press, Oxford, 2007.  Google Scholar

[38]

S. M. Wnek, M. K. Medeirosa, K. E. Eblinb and A. J. Gandolfi, Persistence of DNA damage following exposure of human bladder cells to chronic monomethylarsonous acid, Tox. and Appl. Pharm., 241 (2009), 202-209. doi: 10.1016/j.taap.2009.08.016.  Google Scholar

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