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Various Mathematical Models of Tumor Growth with Reference to Cancer Stem Cells: A Review

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Abstract

Using mathematical models to simulate biological systems has a long history. An increasing number of such models have been applied to various aspects of tumor growth, with the ultimate goal of controlling cancer. Nevertheless, very little has been done in the field of cancer stem cells. Herein, we have reviewed some mathematical models of tumor growth and their specific properties. Considering the importance of the role cancer stem cells play in the production, progression and recurrence of cancer, we have also examined a mathematical growth model describing the dynamics of tumor growth in the presence of cancer stem cells.

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References

  • Abbott RG, Forrest S, Pienta KJ (2006) Simulating the hallmarks of cancer. Artif Life 12(4):617–634

    Article  Google Scholar 

  • Adam JA (1988) A mathematical model of tumor growth by diffusion. Math Comput Model 11:455–456

    Article  MathSciNet  Google Scholar 

  • Adam JA, Maggelakis SA (1989) Mathematical models of tumor growth. iv. Effects of a necrotic core. Math Biosci 97(1):121–136

    Article  MATH  Google Scholar 

  • Alfonso VC, Flanagan DP (2018) Essentials of specific learning disability identification. Wiley, Hoboken

    Google Scholar 

  • Ambrosi D, Preziosi L (2002) On the closure of mass balance models for tumor growth. Math Models Methods Appl Sci 12(5):737–754

    Article  MathSciNet  MATH  Google Scholar 

  • Anderson AR, Quaranta V (2008) Integrative mathematical oncology. Nat Rev Cancer 8(3):227–234

    Article  Google Scholar 

  • Araujo RP, McElwain DLS (2004) A history of the study of solid tumour growth: the contribution of mathematical modelling. Bull Math Biol 66(5):1039–1091

    Article  MathSciNet  MATH  Google Scholar 

  • Arciero J, Jackson T, Kirschner D (2004) A mathematical model of tumor-immune evasion and siRNA treatment. Discrete Continuous Dyn Syst Ser B 4:39–58

    MathSciNet  MATH  Google Scholar 

  • Atuegwu NC, Arlinghaus LR, Li X, Chakravarthy a B, Abramson VG, Sanders ME, Yankeelov TE (2013) Parameterizing the logistic model of tumor growth by DW-MRI and DCE-MRI data to predict treatment response and changes in breast cancer cellularity during neoadjuvant chemotherapy. Transl Oncol 6(3):256–264

    Article  Google Scholar 

  • Badakhshan KP, Kamyad AV, Azemi A (2007) Using AVK method to solve nonlinear problems with uncertain parameters. Appl Math Comput 189:27–34

    MathSciNet  MATH  Google Scholar 

  • Bauer AL, Jackson TL, Jiang Y, Rohlf T (2010) Receptor cross-talk in angiogenesis: mapping environmental cues to cell phenotype using a stochastic, Boolean signaling network model. J Theor Biol 264(3):838–846

    Article  MATH  Google Scholar 

  • Bauer AL, Rohlf T (2012) Investigating the role of cross-talk between chemical and stromal factors in endothelial cell phenotype determination. Springer, Modeling tumor vasculature, pp 79–101

  • Bazmara H, Soltani M, Sefidgar M, Bazargan M, Naeenian MM, Rahmim A (2015) The vital role of blood flow-induced proliferation and migration in capillary network formation in a multiscale model of angiogenesis. PLoS One 10(6)

  • Bellomo N, Angelis ED, Preziosi L (2003) Review article: multiscale modeling and mathematical problems related to tumor evolution and medical therapy. J Theor Med 5(2):111–136

    Article  MATH  Google Scholar 

  • Berreta E, Capasso V, Morozova N (2012) Mathematical modeling of cancer stem cells population behavior. Math Model Nat Phenom 7:279–305

    Article  MathSciNet  MATH  Google Scholar 

  • Bertuzzi A, Fasano A, Gandolf A, Sinisgalli C (2007) Atp production and necrosis formation in a tumour spheroid model. Math Model Nat Phenom 2(3):30–46

    Article  MathSciNet  MATH  Google Scholar 

  • Bissell MJ, Radisky D (2001) Putting tumours in context. Nat Rev Cancer 1(1)

  • Breward C, Byrne H, Lewis C (2002) The role of cell–cell interactions in a two-phase model for avascular tumour growth. J Math Biol 45(2):125–152

    Article  MathSciNet  MATH  Google Scholar 

  • Breward C, Byrne H, Lewis C (2003) A multiphase model describing vascular tumour growth. Bull Math Biol 65(4):609–640

    Article  MATH  Google Scholar 

  • Borsi I, Fasano A, Primicerio M, Hillen T (2015) A non-local model for cancer stem cells and the tumor growth paradox. Math Med Biol

  • Burton A (1966) Rate of growth of solid tumors as a problem of diffusion. Growth 30:229–244

    Google Scholar 

  • Busini V, Arosio P, Masi M (2007) Mechanistic modelling of avascular tumor growth and pharmacokinetics influence part i. Chem Eng Sci 62(7):1877–1886

    Article  Google Scholar 

  • Byrne HM (1999) A weakly nonlinear analysis of a model of avascular solid tumour growth. J Math Biol 39(1):59–89

    Article  MathSciNet  MATH  Google Scholar 

  • Byrne HM (2010) Dissecting cancer through mathematics: from the cell to the animal model. Nat Rev Cancer 10(3):221

    Article  Google Scholar 

  • Byrne H, Preziosi L (2003) Modelling solid tumour growth using the theory of mixtures. Math Med Biol 20(4):341–366

    Article  MATH  Google Scholar 

  • Byrne HM, Drasdo D (2009) Individual-based and continuum models of growing cell populations: a comparison. J Math Biol 58(4):657–687

    Article  MathSciNet  MATH  Google Scholar 

  • Byrne HM, Alarcon T, Owen M, Webb S, Maini P (2006) Modelling aspects of cancer dynamics: a review. Philos Trans R Soc A Math Phys Eng Sci 364(1843):1563–1578

    Article  MathSciNet  Google Scholar 

  • Casciari J, Sotirchos SV, Sutherland RM (1992) Mathematical modelling of microenvironment and growth in emt6/ro multicellular tumour spheroids. Cell Prolif 25(1):1–22

    Article  Google Scholar 

  • Chakrabarty SP, Hanson FB (2005) Optimal control of drug delivery to brain tumors for a test of PDE driven models using the Galerkin finite element method. In: Proc. 44th IEEE Conf. Decis. Control 1(2)

  • Chaplain MAJ, Benson DL, Maini PK (1994) Nonlinear diffusion of a growth inhibitory factor in multicell spheroids. Math Biosci 121(1):1–13

    Article  MATH  Google Scholar 

  • Deakin A (1975) Model for the growth of a solid in vitro tumour. Growth 39:159–165

    Google Scholar 

  • Deisboeck TS, Stamatakos GS (2010) Multiscale cancer modeling. CRC Press (Taylor & Francis Group), Boca Raton

    Book  MATH  Google Scholar 

  • Deisboeck TS, Zhang L, Yoon J, Costa J (2009) In silico cancer modeling: Is it ready for prime time? Nat Clin Prac Oncol 6(1):34–42

    Article  Google Scholar 

  • Dittmar T, Zanker KS (2013) Role of cancer stem cells in cancer biology and therapy. CRC Press, Boca Raton, pp 257–282

    Google Scholar 

  • Drasdo D, Hoehme S, Block M (2007) On the role of physics in the growth and pattern formation of multi-cellular systems: What can we learn from individual-cell based models? J Stat Phys 128(1):287–345

    Article  MathSciNet  MATH  Google Scholar 

  • Durand R (1976) Cell cycle kinetics in an in vitro tumor model. Cell Prolif 9(5):403–412

    Article  Google Scholar 

  • Enderling H, Chaplain MAJ (2014) Mathematical modeling of tumor growth and treatment. Curr Pharm Des 20

  • Enderling H, Anderson ARA, Chaplain MAJ, Beheshti A, Hlatky L, Hahnfeldt P (2009) Paradoxical dependencies of tumor dormancy and progression on basic cell kinetics. Cancer Res 69(22):8814–8821

    Article  Google Scholar 

  • Figueredo GP, Siebers P, Aickelin U (2013) Investigating mathematical models of immune-interactions with early-stage cancer under an agent-based modelling perspective. BMC Bioinformatics 14

  • Folkman J, Hochberg M (1973) Self-regulation of growth in three dimensions. J Exp Med 138(4):745–753

    Article  Google Scholar 

  • Fornari C, Cordero F, Manini D, Balb G, Calogero RA (2011) Mathematical approach to predict the drug effects on cancer stem cell models. Electron Notes Theor Comput Sci 277:29–39

    Article  MATH  Google Scholar 

  • Franziska M (2008) Mathematical models of cancer stem cells. J Clin Oncol 26(17)

  • Glass L (1973) Instability and mitotic patterns in tissue growth, Transactions of the ASME Series G, J Dyn Syst Measure Control 95(3):324–336

    Article  Google Scholar 

  • Glazier JA, Graner F (1993) Simulation of the differential adhesion driven rearrangement of biological cells. Phys Rev 47(3)

  • Greenspan H (1972) Models for the growth of a solid tumor by diffusion. Stud Appl Math 52:317–340

    Article  MATH  Google Scholar 

  • Gruywagen CG, Woodward DE, Traqui P, Bartoo T, Murray DJ (1995) The modelling of diffusive tumors. J Biol Syst 3(4):937–945

    Article  Google Scholar 

  • Hermann B et al (2010) Three-dimensional multispecies nonlinear tumor growth-II: tumor invasion and angiogenesis. J Theor Biol 264:1254–1278

    Article  MathSciNet  MATH  Google Scholar 

  • Hillen T, Enderling H, Hahnfeldt P (2013) The tumor growth paradox and immune system-mediated selection for cancer stem cells. Bull Math Biol 75(1):161–184

    Article  MathSciNet  MATH  Google Scholar 

  • Hinow P et al (2009) A spatial model of tumor-host interaction: application of chemotherapy. Math Biosci Eng 6

  • Hogea C, Davatzikos C, Biros G (2008) An image-driven parameter estimation problem for a reaction-diffusion glioma growth model with mass effects. J Math Biol 56(6):793–825

    Article  MathSciNet  MATH  Google Scholar 

  • Johnston MD, Maini PK, Chapman SJ, Edwards CM, Bodme WF (2010) On the proportion of cancer stem cells in a tumor. J. Theor Biol 266(4):708–711

    Article  MATH  Google Scholar 

  • Kiran KL (2011) Modeling of tumor growth and optimization of therapeutic protocol design. PH.D. thesis

  • Kim Y, Stolarska MA (2007) A hybrid model for tumor spheroid growth in vitro i: theoretical development and early results. Math Models Methods Appl Sci 17:1173–1198

    Article  MathSciNet  MATH  Google Scholar 

  • Knopoff DA, Fernandea DR, Torres GA, Turner CV (2013) Adjoint method for a tumor growth PDE-Constrained optimization problem. Comput Math Appl

  • Kuznetsov V, Makalkin I, Taylor M, Perelson A (1994) Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis. Bull Math Biol 56(2):425–439

    Article  MATH  Google Scholar 

  • Liu Y, Sadowski SM, Weisbrod AB, Kebebew E, Summers RM, Yao J (2013) Tumor growth modeling based on dual phase Ct and Fdg-Pet. In: 2013 IEEE 10th international symposium on biomedical imaging, pp 394–397

  • Lowengrub JS, Frieboes HB, Jin F, Chuang Y-L, Li X, Macklin P, Wise S, Cristini V (2010) Nonlinear modelling of cancer: bridging the gap between cells and tumours. Nonlinearity 23(1):R1–R91

    Article  MathSciNet  MATH  Google Scholar 

  • Maddalena L (2014) Analysis of an integro- differential system modeling tumor. Appl Math Comput 245:152–157

    MathSciNet  MATH  Google Scholar 

  • Martins ML, Ferreira SC Jr, Vilela MJ (2007) Multiscale models for the growth of avascular tumors. Phys Life Rev 4(2):128–156

    Article  Google Scholar 

  • Marusic M, Bajzer E, Vuk-Pavlovic S, Freyer J (1994) Tumor growth in vivo and as multicellular spheroids compared by mathematical models. Bull Math Biol 56(4):617–631

    MATH  Google Scholar 

  • McElwain DLS, Ponzo PJ (1977) A model for the growth of a solid tumor with non-uniform oxygen consumption. Math Biosci 35(3–4):267–279

    Article  MATH  Google Scholar 

  • McElwain DLS, Morris LE (1978) Apoptosis as a volume loss mechanism in mathematical models of solid tumor growth. Math Biosci 39(1–2):147–157

    Article  Google Scholar 

  • Meghdadi N, Soltani M, Niroomand-Oscuii H, Ghalichi F (2016) Image based modeling of tumor growth. Australas Phys Eng Sci Med 39(3):601–613

    Article  Google Scholar 

  • Michor F (2008) Mathematical models of cancer stem cells. J Clin Oncol 26

  • Moreira J, Deutsch A (2002) Cellular automaton models of tumor development: a critical review. Adv Complex Syst 5(2):247–267

    Article  MathSciNet  MATH  Google Scholar 

  • Noori Skandaei MH, Erfanian HR, Kamyad AV, Mohammadi S (2012) Optimal control of bone marrow in cancer chemotherapy. Eur J Exp Biol 2:562–569

    Google Scholar 

  • Panetta JC, Chaplain M, Adam J (1998) The mathematical modelling of cancer: a review, mathematical models in medical and health sciences, innovations in applied mathematics. Vanderbilt University Press, pp 281–309

  • Perfahl H, Byrne HM, Chen T, Estrella V, Alarcón T, Lapin A, Gatenby RA, Gillies RJ, Lloyd MC, Maini PK (2011) Multiscale modelling of vascular tumour growth in 3D: the roles of domain size and boundary conditions. PLoS One 6(4)

  • Piantadosi S (1985) A model of growth with first-order birth and death rates. Comput Biomed Res 18(3):220–232

    Article  Google Scholar 

  • Preziosi L (2003) Cancer modelling and simulation. CRC Press (Taylor & Francis Group), Boca Raton

    Book  MATH  Google Scholar 

  • Quaranta V, Weaver A, Cummings PT, Anderson ARA (2005) Mathematical modeling of cancer: the future of prognosis and treatment. Clinica Chimica Acta 357(2):173–179

    Article  Google Scholar 

  • Quaranta V, Rejniak K, Gerlee P, Anderson ARA (2008) Invasion emerges from cancer cell adaptation to competitive microenvironments: quantitative predictions from multiscale mathematical models. Semin Cancer Biol 18(5):338–348

    Article  Google Scholar 

  • Rivaz A, Azizian M, Kamyad AV (under review) A numerical solution of a nonlinear mathematical model of tumor growth: an optimization approach

  • Rodrigues A, Minceva M (2005) Modelling and simulation in chemical engineering: tools for process innovation. Comput Chem Eng 29:1167–1183

    Article  Google Scholar 

  • Salavati H, Soltani M, Amanpour S (2018) The pivotal role of angiogenesis in a multi-scale modeling of tumor growth exhibiting the avascular and vascular phase. J Microvasc Res 119:105–116

    Article  Google Scholar 

  • Sanga S, Sinek JP, Frieboes HB, Ferrari M, Fruehauf JP, Cristini V (2006) Mathematical modeling of cancer progression and response to chemotherapy. Expert Rev Anticancer Ther 6(10):1361–1376

    Article  Google Scholar 

  • Schaller G, Meyer-Hermann M (2006) Continuum versus discrete model: a comparison for multicellular tumour spheroids. Philos Trans R Soc A Math Phys Eng Sci 364(1843):1443–1464

    Article  MathSciNet  Google Scholar 

  • Sefidgar M, Soltani M, Raahemifar K, Sadeghi M, Bazmara H, Bazargan M, Naeenian MM (2015) Numerical modeling of drug delivery in a dynamic solid tumor microvasculature. Microvasc Res 99:43–56

    Article  Google Scholar 

  • Sever R, Brugge JS (2015) Signal transduction in cancer. Cold Spring Harbor Perspectives in Medicine 5(4)

  • Sherratt JA, Chaplain MA (2001) A new mathematical model for avascular tumour growth. J Math Biol 43(4):291–312

    Article  MathSciNet  MATH  Google Scholar 

  • Shymko RM, Glass L (1976) Cellular and geometric control of tissue growth and mitotic instability. J Theor Biol 63(2):355–374

    Article  Google Scholar 

  • Soltani M, Chen P (2012) Effect of tumor shape and size on drug delivery to solid tumors. J Biol Eng 6:1–15

    Article  Google Scholar 

  • Soltani M, Chen P (2013) Numerical modeling of interstitial fluid flow coupled with blood flow through a remodeled solid tumor microvascular network. PLoS One 8(6)

  • Stolarska MA, Kim Y, Othmer HG (2009) Multi-scale models of cell and tissue dynamics’. Philos Trans R Soc A Math Phys Eng Sci 367:3525–3553

    Article  MathSciNet  MATH  Google Scholar 

  • Sutherland R, Durand R (1973) Hypoxic cells in an in vitro tumour model. Int J Radiat Biol 23:235–246

    Google Scholar 

  • Swanson KR, Alvord EC, Murray JD (2002) Virtual brain tumours (gliomas) enhance the reality of medical imaging and highlight inadequacies of current therapy. Br J Cancer 86(1):14–18

    Article  Google Scholar 

  • Swanson KR, Bridge C, Murray JD, Alvord EC (2003) Virtual and real brain tumors: using mathematical modeling to quantify glioma growth and invasion. J Neurol Sci 216(1):1–10

    Article  Google Scholar 

  • Tello JI (2013) On a mathematical model of tumor growth based on cancer stem cells. Math Biosci Eng 10:263–278

    Article  MathSciNet  MATH  Google Scholar 

  • Tiina R, Chapman SJ, Philip KM (2007) Mathematical models of avascular tumor growth. SIAM Rev 49(2):179–208

    Article  MathSciNet  MATH  Google Scholar 

  • Tindall MJ, Please C (2007) Modelling the cell cycle and cell movement in multicellular tumour spheroids. Bull Math Biol 69(4):1147–1165

    Article  MathSciNet  MATH  Google Scholar 

  • Tindall MJ, Please CP, Peddie MJ (2008) Modelling the formation of necrotic regions in avascular tumours. Math Biosci 211(1):34–55

    Article  MathSciNet  MATH  Google Scholar 

  • Tsygvintsev A, Marino S, Kirschner DE (2013) A mathematical model of gene therapy for the treatment of cancer. Math Methods Models Biomed

  • Turner C, Kohandel M (2010) Investigating the link between epithelial-mesenchymal transition and the cancer stem cell phenotype: a mathematical approch. J Theor Biol 265:329–335

    Article  MATH  Google Scholar 

  • Venkatasubramanian R, Henson MA, Forbes NS (2006) Incorporating energy metabolism into a growth model of multicellular tumor spheroids. J Theor Biol 242(2):440–453

    Article  MathSciNet  Google Scholar 

  • Ward JP, King JR (1997) Mathematical modelling of avascular-tumour growth. Math Med Biol 14(1):39–69

    Article  MATH  Google Scholar 

  • Weekes S, Barker B, Bober S, Cisneros K, Cline J, Thompson A, Hlatky L, Hahnfeldt P, Enderling H (2014) A multicompartment mathematical model of cancer stem cell-driven tumor growth dynamics. Bull Math Biol

  • Weinan E, Engquist B, Li X, Ren W, Vanden-Eijnden E (2007) Heterogeneous multiscale methods: a review. Commun Comput Phys 2(3):367–450

    MathSciNet  MATH  Google Scholar 

  • Zhu X, Zhou X, lewis MT, Xia L, Wong S (2011) Cancer stem cell, niche and egfr decide tumor development and treatment response: a bio-computational simulation study. J Theor Biol 269:138–149

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran

    Azim Rivaz

  2. Department of General Educations, Afzalipour School of Medicine, Kerman University of Medical Sciences, Kerman, Iran

    Mahdieh Azizian

  3. Department of Mechanical Engineering, K. N. Toosi University of Technology, Tehran, Iran

    Madjid Soltani

  4. Center for Biotechnology and Bioengineering (CBB), University of Waterloo, Waterloo, ON, Canada

    Madjid Soltani

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  1. Azim Rivaz
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  2. Mahdieh Azizian
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Correspondence to Mahdieh Azizian.

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Rivaz, A., Azizian, M. & Soltani, M. Various Mathematical Models of Tumor Growth with Reference to Cancer Stem Cells: A Review. Iran J Sci Technol Trans Sci 43, 687–700 (2019). https://doi.org/10.1007/s40995-019-00681-w

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