Modeling of Tumor Occurrence and Growth-III
УДК 519.87
DOI:
https://doi.org/10.14258/izvasu(2021)4-11Keywords:
mathematical modeling, tumor modeling, oncological disease, multiscale models, multiphase systemsAbstract
The last part of the article examines mathematical models of four types of oncological diseases: breast cancer (early stage), colorectal cancer (bowel cancer), glioma, and prostate cancer. Each of these models has its own individual characteristics and, accordingly, their approaches to modeling are different. The approach to modeling breast cancer involves complex interactions between tumor cells, fibroblasts, immunocytes, epithelial cells, extracellular matrix, vascular system, and cytokines. Colorectal cancer takes into account the multiscale approach, cell cycle, and gene mutations that were discussed in the previous sections. Glioma is one of the most aggressive brain tumors. Its model includes equations for glioma cell density, extracellular matrix concentration, matrix metalloproteinase concentration, and nutrient concentration. There is another model for glioma that considers an approach using oncolytic viruses. Prostate cancer takes into account the presence of testosterone and its effect on the further development of the disease.
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References
Антонцев С.Н., Папин А.А., Токарева М.А., Леонова Э.И., Гридюшко Е.А. Моделирование возникновения и роста опухолей-I // Известия Алт. гос. ун-та. 2020. № 4(114). DOI: 10.14258/izvasu(2020)4-11.
Антонцев С.Н., Папин А.А., Токарева М.А., Леонова Э.И., Гридюшко Е.А. Моделирование возникновения и роста опухолей-II // Извесття Алт. гос. ун-та. 2021. № 1(117). DOI: 10.14258/izvasu(2021)1-12.
Friedman A. Cancer as Multifaceted Disease // Math. Model. Nat. Phenom., 2012. 7. 1. DOI: 10.1051/mmnp/20127102.
Laia X., Stiffb A., Duggand M., Wesolowskie R., Carson W.E., Friedman A. Modeling combination therapy for breast cancer with BET and immune checkpoint inhibitors // PNAS. 2018. Vol. 115. № 21. DOI: 10.1073/pnas.1721559115.
Колпак Е.П., Французова И.С., Евтенова Е.О. Математические модели опухолей молочной железы // Молодой ученый. 2019. №22 (260). С. 17-35. URL: https://moluch.ru/archive/260/59754/.
Paterson C., Clevers H., Bozic I. Mathematical model of colorectal cancer initiation // PNAS. 2020. Vol. 117. № 34. DOI: 10.1073/pnas.2003771117.
Kirshtein A., Akbarinejad S., Hao W., Le T., Su S., Aronow R. A., Shahriyari L. Data Driven Mathematical Model of Colon Cancer Progression // Journal of Clinical Medicine. 2020. Vol. 9. № 12. DOI: 10.3390/jcm9123947.
Friedman A., Lai X. Free boundary problems associated with cancer treatment by combination therapy // Discrete & Continuous Dynamical Systems - A. 2019. 39. 12. DOI: 10.3934/dcds.2019233.
Ratajczyk E., Ledzewicz U., Leszczynski M., Friedman A. The role of TNF-a inhibitor in glioma virotherapy: A mathematical model // Mathematical Biosciences & Engineering. 2017. 14. 1. DOI: 10.3934/mbe.2017020.
Badziul D., Jakubczyk P., Chotorlishvili L., Toklikishvilie Z., Traciak J., J. Jakubowicz-Gil, Chmiel-Szajner S. Mathematical Prostate Cancer Evolution: Effect of Immunotherapy Based on Controlled Vaccination Strategy // Computational and Mathematical Methods in Medicine. 2020. Vol. 2020. DOI: 10.1155/2020/7970265.
Zazoua A., Wang W. Analysis of mathematical model of prostate cancer with androgen deprivation therapy // Communications in Nonlinear Science and Numerical Simulation. 2019. Vol. 66. DOI: 10.1016/j.cnsns.2018.06.004.
Franks S.J., King J.R. Interactions between a uniformly proliferating tumour and its surroundings: uniform material properties // Math. Med. Biol. 2003. 20. DOI: 10.1093/imammb/20.1.47.
Franks S.J.H., Byrne H.M., King J.P., Underwood J.C.E., Lewis C.E. Modeling the early growth of ductal carcinoma in situ of the breast // J. Math. Biol. 2003. 47. DOI: 10.1007/s00285-003-0214-x.
Franks S.J.H., Byrne H.M., King J.P., Underwood J.C.E., Lewis C.E. Mathematical modelling of comedo ductal carcinoma in situ of the breast // Math. Med. & Biol. 2003. 20. DOI: 10.1093/imammb/20.3.277.
Franks S.J.H., Byrne H.M., Underwood J.C.E., Lewis C.E. Biological inferences from a mathematical model of comedo ductal carcinoma in situ of the breast // J. Theor. Biol. 2005. 232. DOI: 10.1016/j.jtbi.2004.08.032.
Kim Y., Wallace J., Li F., Ostrowski M., Friedman A. Transformed epithelial cells and fibroblasts/myofibroblasts interaction in breast tumor: a mathematical model and experiments // J. Math. Biol. 2010. 61. DOI: 10.1007/s00285-009-0307-2.
Friedman A., Kim Y. Tumor cells proliferation and migration under the influence of their microenvironment // Math Biosci. & Engin. 2011. 8. DOI: 10.3934/mbe.2011.8.371.
Kim Y., Friedman A. Interaction of tumor with its microenvironment: a mathematical model // Bull. Math. Biol. 2010. 72. DOI: 10.1007/s11538-009-9481-z.
Aznavoorian S., Stracke M., Krutzsch H., Schiffmann E., Liotta L. Signal transduction for chemotaxis and haptotaxis by matrix molecules in tumor cells // J. Cell Biol. 1990. 110. 4. DOI: 10.1083/jcb.110.4.1427.
Perumpanani A., Byrne H. Extracellular matrix concentration exerts selection pressure on invasive cells // Eur. J. Cancer. 1999. 35. 8. DOI: 10.1016/s0959-8049(99)00125-2.
Ribba R., Colin T., Schnell S. A multiscale model of cancer, and its use in analyzing irradiation therapies // Theor. Biol. & Med. Mod. 2006. 3. № 7. DOI: 10.1186/1742-4682-3-7.
Ribba B., Sant O., Colin T., Bresch D., Grenien E., Boissel J.P. A multiscale model of avascular tumor growth to investigate agents // J. Theor. Biol. 2006. 243. DOI: 10.1016/j.jtbi.2006.07.013.
Friedman A., Hu B., Kao C-Y. Cell cycle control at the first restriction point and its effect on tissue growth // J. Math. Biol. 2010. 60. DOI: 10.1007/s00285-009-0290-7.
van Leeuwen I.M.M., Byrne H.M., Jensen O.E., King J.R. Crypt dynamics and colorectal cancer: advances in mathematical modeling // Cell Prolif. 2006. 39. DOI: 10.1111/j.1365-2184.2006.00378.x.
Harper P.R., Jones S.K. Mathematical models for the early detection and treatment of colorectal cancer // Health Care Management Science. 2005. 8. DOI: 10.1007/s10729-005-0393-7.
Komarova N.L., Lengauer C., Vogelstein B., Nowak M. Dynamics of genetic instability in sporadic and familial colorectal cancer // Cancer Biology & Therapy. 2002. 1. DOI: 10.4161/cbt.321.
Harpold H., Ec J., Swanson K. The evolution of mathematical modeling of glioma proliferation and invasion // J. Neuropathol. Exp. Neurol. 2007. 66. 1. DOI: 10.1097/nen.0b013e31802d9000.
Mandonnet E., Delattre J., Tanguy M., Swanson K., Carpentier A., Duffau H., Cornu P., Effenterre R., Ec J., Capelle L.J. Continuous growth of mean tumor diameter in a subset of grade ii gliomas // Ann. Neurol. 2003. 53. 4. DOI: 10.1002/ana.10528.
Swanson K., Ec J., Murray J. A quantitative model for differential motility of gliomas in grey and white matter // Cell Prolif. 2000. 33. 5. DOI: 10.1046/j.1365-2184.2000.00177.x.
Eikenberry S.E., Sankar T., Preul M.C., Kostelich E.J., Thalhauser C.J., Kuang Y. Virtual glioblastoma: growth, migration and treatment in a three-dimensional mathematical model // Cell Prolif. 2009. 42. DOI: 10.1111/j.1365-2184.2009.00613.x.
Kim Y., Lawler S., Nowicki M.O., Chiocca E.A., Friedman A. A mathematical model of brain tumor: pattern formation of glioma cells outside the tumor spheroid core // J. Theor. Biol. 2009. 260. DOI: 10.1016/j.jtbi.2009.06.025.
Armstrong N., Painter K., Sherratt J. A continuum approach to modeling cell-cell adhesion // J. Theor. Biol. 243. 1. DOI: 10.1016/j.jtbi.2006.05.030.
Sherratt J., Gourley S., Armstrong N., Painter K. Boundedness of solutions of a nonlocal reaction diffusion model for adhesion in cell aggregation and cancer invasion // Eur. J.Appl. Math. 2009. 20. DOI: 10.1017/S0956792508007742.
Fulci G., Breymann L., Gianni D., Kurozomi K., Rhee S., Yu J., Kaur B., Louis D., Weissleder R., Caligiuri M., Chiocca E.A. Cyclophosphamide enhances glioma virotherapy by inhibiting innate immune responses // PNAS. 2006. 103. DOI: 10.1073/pnas.0605496103.
Friedman A., Tao Y. Analysis of a model of virus that replicates selectively in tumor cells // J. Math. Biol. 2003. 47. DOI: 10.1007/s00285-003-0199-5.
Wu J.T., Byrne H.M., Kirn D.H., Wein L.M. Modeling and analysis of a virus that replicates selectively in tumor cells // Bull. Math. Biol. 2001. 63. DOI: 10.1006/bulm.2001.0245.
Wu J.T., Kirn D.H., Wein L.M. Analysis of a three-way race between tumor growth, a replication-competent virus and an immune response // Bull. Math. Biol. 2004. 66. DOI: 10.1016/j.bulm.2003.08.016.
Friedman A., Tian J.J., Fulci G., Chiocca E.A., Wang J. Glioma virotherapy: The effects of innate immune suppression and increased viral replication capacity // Cancer Research. 2006. 66. DOI: 10.1158/0008-5472.CAN-05-2661.
Eikenberry S.E., Nagy J.D., Kuang Y. The evolutionary impact of androgen levels on prostate cancer in a multi-scale mathematical model // Biol. Direct. 2010. 5. DOI: 10.1186/17456150-5-24.
Potter L.K., Zagar M.G., Barton H.A. Mathematical model for the androgenic regulation of the prostate in intact and castrated adult male rats // Am. J. Physiol. Endocrinol. Metab. 2006. 291. DOI: 10.1152/ajpendo.00545.2005.
Ideta A., Tanaka G., Takeuchi T., Aihara K. A Mathematical model of intermittent androgen suppression for prostate cancer // J. Nonlinear Sci. 2008. 18. DOI: 10.1007/s00332-008-9031-0.
Jackson T.L. A mathematical model of prostate tumor growth and androgen-independent relapse // Discrete Cont. Dyn-B. 2004. 4. DOI: 10.3934/dcdsb.2004.4.187.
Jackson T.L. A mathematical investigation of the multiple pathways to recurrent prostate cancer: comparison with experimental data // Neoplasia. 2004. 6. DOI: 10.1593/neo.04259.
Антонцев С.Н., Папин А.А., Токарева М.А., Леонова Э.И., Гридюшко Е.А. Моделирование возникновения и роста опухолей-II // Извесття Алт. гос. ун-та. 2021. № 1(117). DOI: 10.14258/izvasu(2021)1-12.
Friedman A. Cancer as Multifaceted Disease // Math. Model. Nat. Phenom., 2012. 7. 1. DOI: 10.1051/mmnp/20127102.
Laia X., Stiffb A., Duggand M., Wesolowskie R., Carson W.E., Friedman A. Modeling combination therapy for breast cancer with BET and immune checkpoint inhibitors // PNAS. 2018. Vol. 115. № 21. DOI: 10.1073/pnas.1721559115.
Колпак Е.П., Французова И.С., Евтенова Е.О. Математические модели опухолей молочной железы // Молодой ученый. 2019. №22 (260). С. 17-35. URL: https://moluch.ru/archive/260/59754/.
Paterson C., Clevers H., Bozic I. Mathematical model of colorectal cancer initiation // PNAS. 2020. Vol. 117. № 34. DOI: 10.1073/pnas.2003771117.
Kirshtein A., Akbarinejad S., Hao W., Le T., Su S., Aronow R. A., Shahriyari L. Data Driven Mathematical Model of Colon Cancer Progression // Journal of Clinical Medicine. 2020. Vol. 9. № 12. DOI: 10.3390/jcm9123947.
Friedman A., Lai X. Free boundary problems associated with cancer treatment by combination therapy // Discrete & Continuous Dynamical Systems - A. 2019. 39. 12. DOI: 10.3934/dcds.2019233.
Ratajczyk E., Ledzewicz U., Leszczynski M., Friedman A. The role of TNF-a inhibitor in glioma virotherapy: A mathematical model // Mathematical Biosciences & Engineering. 2017. 14. 1. DOI: 10.3934/mbe.2017020.
Badziul D., Jakubczyk P., Chotorlishvili L., Toklikishvilie Z., Traciak J., J. Jakubowicz-Gil, Chmiel-Szajner S. Mathematical Prostate Cancer Evolution: Effect of Immunotherapy Based on Controlled Vaccination Strategy // Computational and Mathematical Methods in Medicine. 2020. Vol. 2020. DOI: 10.1155/2020/7970265.
Zazoua A., Wang W. Analysis of mathematical model of prostate cancer with androgen deprivation therapy // Communications in Nonlinear Science and Numerical Simulation. 2019. Vol. 66. DOI: 10.1016/j.cnsns.2018.06.004.
Franks S.J., King J.R. Interactions between a uniformly proliferating tumour and its surroundings: uniform material properties // Math. Med. Biol. 2003. 20. DOI: 10.1093/imammb/20.1.47.
Franks S.J.H., Byrne H.M., King J.P., Underwood J.C.E., Lewis C.E. Modeling the early growth of ductal carcinoma in situ of the breast // J. Math. Biol. 2003. 47. DOI: 10.1007/s00285-003-0214-x.
Franks S.J.H., Byrne H.M., King J.P., Underwood J.C.E., Lewis C.E. Mathematical modelling of comedo ductal carcinoma in situ of the breast // Math. Med. & Biol. 2003. 20. DOI: 10.1093/imammb/20.3.277.
Franks S.J.H., Byrne H.M., Underwood J.C.E., Lewis C.E. Biological inferences from a mathematical model of comedo ductal carcinoma in situ of the breast // J. Theor. Biol. 2005. 232. DOI: 10.1016/j.jtbi.2004.08.032.
Kim Y., Wallace J., Li F., Ostrowski M., Friedman A. Transformed epithelial cells and fibroblasts/myofibroblasts interaction in breast tumor: a mathematical model and experiments // J. Math. Biol. 2010. 61. DOI: 10.1007/s00285-009-0307-2.
Friedman A., Kim Y. Tumor cells proliferation and migration under the influence of their microenvironment // Math Biosci. & Engin. 2011. 8. DOI: 10.3934/mbe.2011.8.371.
Kim Y., Friedman A. Interaction of tumor with its microenvironment: a mathematical model // Bull. Math. Biol. 2010. 72. DOI: 10.1007/s11538-009-9481-z.
Aznavoorian S., Stracke M., Krutzsch H., Schiffmann E., Liotta L. Signal transduction for chemotaxis and haptotaxis by matrix molecules in tumor cells // J. Cell Biol. 1990. 110. 4. DOI: 10.1083/jcb.110.4.1427.
Perumpanani A., Byrne H. Extracellular matrix concentration exerts selection pressure on invasive cells // Eur. J. Cancer. 1999. 35. 8. DOI: 10.1016/s0959-8049(99)00125-2.
Ribba R., Colin T., Schnell S. A multiscale model of cancer, and its use in analyzing irradiation therapies // Theor. Biol. & Med. Mod. 2006. 3. № 7. DOI: 10.1186/1742-4682-3-7.
Ribba B., Sant O., Colin T., Bresch D., Grenien E., Boissel J.P. A multiscale model of avascular tumor growth to investigate agents // J. Theor. Biol. 2006. 243. DOI: 10.1016/j.jtbi.2006.07.013.
Friedman A., Hu B., Kao C-Y. Cell cycle control at the first restriction point and its effect on tissue growth // J. Math. Biol. 2010. 60. DOI: 10.1007/s00285-009-0290-7.
van Leeuwen I.M.M., Byrne H.M., Jensen O.E., King J.R. Crypt dynamics and colorectal cancer: advances in mathematical modeling // Cell Prolif. 2006. 39. DOI: 10.1111/j.1365-2184.2006.00378.x.
Harper P.R., Jones S.K. Mathematical models for the early detection and treatment of colorectal cancer // Health Care Management Science. 2005. 8. DOI: 10.1007/s10729-005-0393-7.
Komarova N.L., Lengauer C., Vogelstein B., Nowak M. Dynamics of genetic instability in sporadic and familial colorectal cancer // Cancer Biology & Therapy. 2002. 1. DOI: 10.4161/cbt.321.
Harpold H., Ec J., Swanson K. The evolution of mathematical modeling of glioma proliferation and invasion // J. Neuropathol. Exp. Neurol. 2007. 66. 1. DOI: 10.1097/nen.0b013e31802d9000.
Mandonnet E., Delattre J., Tanguy M., Swanson K., Carpentier A., Duffau H., Cornu P., Effenterre R., Ec J., Capelle L.J. Continuous growth of mean tumor diameter in a subset of grade ii gliomas // Ann. Neurol. 2003. 53. 4. DOI: 10.1002/ana.10528.
Swanson K., Ec J., Murray J. A quantitative model for differential motility of gliomas in grey and white matter // Cell Prolif. 2000. 33. 5. DOI: 10.1046/j.1365-2184.2000.00177.x.
Eikenberry S.E., Sankar T., Preul M.C., Kostelich E.J., Thalhauser C.J., Kuang Y. Virtual glioblastoma: growth, migration and treatment in a three-dimensional mathematical model // Cell Prolif. 2009. 42. DOI: 10.1111/j.1365-2184.2009.00613.x.
Kim Y., Lawler S., Nowicki M.O., Chiocca E.A., Friedman A. A mathematical model of brain tumor: pattern formation of glioma cells outside the tumor spheroid core // J. Theor. Biol. 2009. 260. DOI: 10.1016/j.jtbi.2009.06.025.
Armstrong N., Painter K., Sherratt J. A continuum approach to modeling cell-cell adhesion // J. Theor. Biol. 243. 1. DOI: 10.1016/j.jtbi.2006.05.030.
Sherratt J., Gourley S., Armstrong N., Painter K. Boundedness of solutions of a nonlocal reaction diffusion model for adhesion in cell aggregation and cancer invasion // Eur. J.Appl. Math. 2009. 20. DOI: 10.1017/S0956792508007742.
Fulci G., Breymann L., Gianni D., Kurozomi K., Rhee S., Yu J., Kaur B., Louis D., Weissleder R., Caligiuri M., Chiocca E.A. Cyclophosphamide enhances glioma virotherapy by inhibiting innate immune responses // PNAS. 2006. 103. DOI: 10.1073/pnas.0605496103.
Friedman A., Tao Y. Analysis of a model of virus that replicates selectively in tumor cells // J. Math. Biol. 2003. 47. DOI: 10.1007/s00285-003-0199-5.
Wu J.T., Byrne H.M., Kirn D.H., Wein L.M. Modeling and analysis of a virus that replicates selectively in tumor cells // Bull. Math. Biol. 2001. 63. DOI: 10.1006/bulm.2001.0245.
Wu J.T., Kirn D.H., Wein L.M. Analysis of a three-way race between tumor growth, a replication-competent virus and an immune response // Bull. Math. Biol. 2004. 66. DOI: 10.1016/j.bulm.2003.08.016.
Friedman A., Tian J.J., Fulci G., Chiocca E.A., Wang J. Glioma virotherapy: The effects of innate immune suppression and increased viral replication capacity // Cancer Research. 2006. 66. DOI: 10.1158/0008-5472.CAN-05-2661.
Eikenberry S.E., Nagy J.D., Kuang Y. The evolutionary impact of androgen levels on prostate cancer in a multi-scale mathematical model // Biol. Direct. 2010. 5. DOI: 10.1186/17456150-5-24.
Potter L.K., Zagar M.G., Barton H.A. Mathematical model for the androgenic regulation of the prostate in intact and castrated adult male rats // Am. J. Physiol. Endocrinol. Metab. 2006. 291. DOI: 10.1152/ajpendo.00545.2005.
Ideta A., Tanaka G., Takeuchi T., Aihara K. A Mathematical model of intermittent androgen suppression for prostate cancer // J. Nonlinear Sci. 2008. 18. DOI: 10.1007/s00332-008-9031-0.
Jackson T.L. A mathematical model of prostate tumor growth and androgen-independent relapse // Discrete Cont. Dyn-B. 2004. 4. DOI: 10.3934/dcdsb.2004.4.187.
Jackson T.L. A mathematical investigation of the multiple pathways to recurrent prostate cancer: comparison with experimental data // Neoplasia. 2004. 6. DOI: 10.1593/neo.04259.
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2021-09-10
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Математика и механика
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Modeling of Tumor Occurrence and Growth-III: УДК 519.87. (2021). Izvestiya of Altai State University, 4(120), 71-80. https://doi.org/10.14258/izvasu(2021)4-11
