SpringerLink
Log in

Mathematical Modelling of Cancer Invasion: A Review

  • Conference paper
  • First Online:
Methods of Mathematical Oncology (MMDS 2020)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 370))

Abstract

A defining feature of cancer is the capability to spread locally into the surrounding tissue, with cancer cells spreading beyond any normal boundaries. Cancer invasion is a complex phenomenon involving many inter-connected processes at different spatial and temporal scales. A key component of invasion is the ability of cancer cells to alter and degrade the extracellular matrix through the secretion of matrix-degrading enzymes. Combined with excessive cell proliferation and cell migration (individual and collective), this facilitates the spread of cancer cells into the local tissue. Along with tumour-induced angiogenesis, invasion is a critical component of metastatic spread, ultimately leading to the formation of secondary tumours in other parts of the host body. In this paper we present an overview of the various mathematical models and different modelling techniques and approaches that have been developed over the past 25 years or so and which focus on various aspects of the invasive process.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  • Anderson, A.R.A., Chaplain, M.A.J., Newman, E.L., Steele, R.J.C., Thompson, A.M.: Mathematical modelling of tumour invasion and metastasis. Comput. Math. Methods Med. 2(2), 129–154 (2000)

    MATH  Google Scholar 

  • Armstrong, N.J., Painter, K.J., Sherratt, J.A.: A continuum approach to modelling cell-cell adhesion. J. Theor. Biol. 243(1), 98–113 (2006)

    Article  MathSciNet  Google Scholar 

  • Brady, R., Enderling, H.: Mathematical models of cancer: when to predict novel therapies, and when not to. Bull. Math. Biol. 81, 3722–3731 (2019)

    Article  MathSciNet  Google Scholar 

  • Byrne, H.M., Chaplain, M.A.J., Pettet, G.J., McElwain, D.L.S.: An analysis of a mathematical model of trophoblast invasion. Appl. Math. Lett. 14, 1005–1010 (2000)

    Article  Google Scholar 

  • Chaplain, M.A.J., Lolas, G.: Mathematical modelling of cancer cell invasion of tissue: the role of the urokinase plasminogen activation system. Math. Models Methods Appl. Sci. 15(11), 1685–1734 (2005)

    Article  MathSciNet  Google Scholar 

  • Cruywagen, G.C., Woodward, D.E., Tracqui, P., Bartoo, G.T., Murray, J.D., Alvord Jr., E.C.: The modelling of difusive tumors. J. Biol. Syst. 3, 937–945 (1995)

    Google Scholar 

  • Domschke, P., Trucu, D., Gerisch, A., Chaplain, M.A.J.: Mathematical modelling of cancer invasion: implications of cell adhesion variability for tumour infiltrative growth patterns. J. Theor. Biol. 361, 41–60 (2014)

    Article  MathSciNet  Google Scholar 

  • Folkman, J.: Tumor angiogenesis: therapeutic implications. New Engl. J. Med. 285, 1182–1186 (1971)

    Article  Google Scholar 

  • Franssen, L.C., Chaplain, M.A.J.: A mathematical multi-organ model for bidirectional epithelial-mesenchymal transitions in the metastatic spread of cancer. IMA J. Appl. Math. 85, 724–761 (2020)

    Article  MathSciNet  Google Scholar 

  • Franssen, L.C., Lorenzi, T., Burgess, A.E.F., Chaplain, M.A.J.: A mathematical framework for modelling the metastatic spread of cancer. Bull. Math. Biol. 81, 1965–2010 (2019)

    Article  MathSciNet  Google Scholar 

  • Franssen, L.C., Sfakianakis, N., Chaplain, M.A.J.: A novel 3D atomistic-continuum cancer invasion model: in silico simulations of an in vitro organotypic invasion assay. J. Theor. Biol. (2021, in press)

    Google Scholar 

  • Friedl, P., Wolf, K.: Tumour-cell invasion and migration: diversity and escape mechanisms. Nat. Rev. Cancer 5, 362–374 (2003)

    Article  Google Scholar 

  • Gatenby, R.A.: Models of tumor-host interaction as competing populations: implications for tumor biology and treatment. J. Theor. Biol. 176(4), 447–455 (1995)

    Article  MathSciNet  Google Scholar 

  • Gatenby, R.A., Gawlinski, E.T.: A reaction-diffusion model of cancer invasion. Cancer Res. 56(24), 5745–5753 (1996)

    Google Scholar 

  • Gatenby, R.A., Gawlinski, E.T., Gmitro, A.F., Kaylor, B., Gillies, R.J.: Acid-mediated tumor invasion: a multidisciplinary study. Cancer Res. 66(10), 5216–5223 (2006)

    Article  Google Scholar 

  • Gerisch, A., Chaplain, M.A.J.: Mathematical modelling of cancer cell invasion of tissue: local and non-local models and the effect of adhesion. J. Theor. Biol. 250(4), 684–704 (2008)

    Article  MathSciNet  Google Scholar 

  • Hanahan, D., Weinberg, R.A.: The Hallmarks of Cancer. Cell 100, 57–70 (2000)

    Article  Google Scholar 

  • Hanahan, D., Weinberg, R.A.: The hallmarks of cancer: the next generation. Cell 144, 646–671 (2011)

    Article  Google Scholar 

  • Marchant, B.P., Nornbury, J., Perumpanani, A.J.: Traveling shock waves arising in a model of malignant invasion. SIAM J. Appl. Math. 60(2), 463–476 (2000)

    Article  MathSciNet  Google Scholar 

  • Marchant, B.P., Nornbury, J., Sherratt, J.A.: Travelling wave solutions to a haptotaxis-dominated model of malignant invasion. Nonlinearity 14, 1653–1671 (2001)

    Article  MathSciNet  Google Scholar 

  • Marchant, B.P., Norbury, J., Byrne, H.M.: Biphasic behaviour in malignant invasion. Math. Med. Biol. 23, 173–196 (2006)

    Article  Google Scholar 

  • Nyström, M., Thomas, G.J., Stone, I.C., Mackenzie, M., Hart, I.R., Marshall, J.F.: Development of a quantitative method to analyse tumour cell invasion in organotypic culture. J. Pathol. 205, 468–475 (2005)

    Article  Google Scholar 

  • Peng, L., Trucu, D., Lin, P., Thompson, A., Chaplain, M.A.J.: A multiscale mathematical model of tumour invasive growth. Bull. Math. Biol. 79(3), 389–429 (2017)

    Article  MathSciNet  Google Scholar 

  • Perumpanani, A.J., Sherratt, J.A., Norbury, J., Byrne, H.M.: Biological inferences from a mathematical model for malignant invasion. Invas Metast 16, 209–221 (1996)

    Google Scholar 

  • Perumpanani, A.J., et al.: Extarcellular matrix-mediated chemotaxis can impede cell migration. Proc. R. Soc. London B 265, 2347–2352 (1998)

    Article  Google Scholar 

  • Perumpanani, A.J., Sherratt, J.A., Norbury, J., Byrne, H.M.: A two parameter family of travelling waves with a singular barrier arising from the modelling of extracellular matrix mediated cellular invasion. Phys. D 126, 145–159 (1999)

    Article  Google Scholar 

  • Recamier, J.C.: Recherches sur le traitement du cancer sur la compression méthodique simple ou combinée et sur l’histoire générale de la meme maladie. Tome Second. Paris, Chez Gabon, Libraire-Editeur (1829)

    Google Scholar 

  • Sfakianakis, N., Madzvamuse, A., Chaplain, M.A.J.: A hybrid multiscale model for cancer invasion of the extracellular matrix. Multisc. Model. Simul. 18(2), 824–50 (2020)

    Article  MathSciNet  Google Scholar 

  • Shuttleworth, R., Trucu, D.: Multiscale dynamics of a heterotypic cancer cell population within a fibrous extracellular matrix. J. Theor. Biol. 486, 1–22 (2019a)

    Google Scholar 

  • Shuttleworth, R., Trucu, D.: Multiscale modelling of fibres dynamics and cell adhesion within moving boundary cancer invasion. Bull. Math. Biol. 81, 2176–2219 (2019b)

    Google Scholar 

  • Shuttleworth, R., Trucu, D.: Cell-scale degradation of peritumoural extracellular matrix fibre network and its role within tissue-scale cancer invasion. Bull. Math. Biol. 82, 1–47 (2019c)

    Google Scholar 

  • Smallbone, K., Gavaghan, D.J., Gatenby, R.A., Maini, P.K.: The role of acidity in solid tumour growth and invasion. J. Theor. Biol. 235, 476–484 (2005)

    Article  MathSciNet  Google Scholar 

  • Smallbone, K., Gatenby, R.A., Gillies, R.J., Maini, P.K., Gavaghan, D.J.: Metabolic changes during carcinogenesis: potential impact on invasiveness. J. Theor. Biol. 244, 703–713 (2007)

    Article  MathSciNet  Google Scholar 

  • Smallbone, K., Gatenby, R.A., Maini, P.K.: Mathematical modelling of tumour acidity. J. Theor. Biol. 255, 106–112 (2008)

    Article  MathSciNet  Google Scholar 

  • Swanson, K.R.: Quantifying glioma cell growth and invasion in vitro. Math. Comput. Model. 47, 638–648 (2008)

    Article  MathSciNet  Google Scholar 

  • Swanson, K.R., Alvord Jr., E.C., Murray, J.D.: A quantitative model for differential motility of gliomas in grey and white matter. Cell Prolif. 33, 317–329 (2000)

    Google Scholar 

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

    Google Scholar 

  • Talmadge, J.E., Fidler, I.J.: Aacr centennial series: the biology of cancer metastasis: historical perspective. Cancer Res. 70, 5649–5669 (2010)

    Article  Google Scholar 

  • Tracqui, P., Cruywagen, G.C., Woodward, D.E., Bartoo, G.T., Murray, J.D., Alvord Jr., E.C.: A mathematical model of glioma growth: the effect of chemotherapy on spatio-temporal growth. Cell Prolif. 28, 17–31 (1995)

    Google Scholar 

  • Trucu, D., Lin, P., Chaplain, M.A.J., Wang, Y.: A multiscale moving boundary model arising in cancer invasion. Multisc. Model. Simul. 11(1), 309–335 (2013)

    Article  MathSciNet  Google Scholar 

  • Turner, S., Sherratt, J.A.: Intercellular adhesion and cancer invasion: a discrete simulation using the extended Potts model. J. Theor. Biol. 216(1), 85–100 (2002)

    Article  MathSciNet  Google Scholar 

  • Valster, A., Tran, N.L., Nakada, M., Berens, M.E., Chan, A.Y., Symons, M.: Cell migration and invasion assays. Methods 37, 208–215 (2005)

    Article  Google Scholar 

  • Woodward, D.E., Cook, J., Tracqui, P., Cruywagen, G.C., Murray, J.D., Alvord Jr., E.C.: A mathematical model of glioma growth: the effect of extent of surgical resection. Cell Prolif. 29, 269–288 (1996)

    Google Scholar 

Download references

Acknowledgements

MAJC gratefully acknowledges the support of EPSRC Grant No. EP/S030875/1 (EPSRC SofTMech\(^{\wedge }\)MP Centre-to-Centre Award).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, University of St Andrews, St Andrews, KY16 9SS, Scotland

    Nikolaos Sfakianakis & Mark A. J. Chaplain

Authors
  1. Nikolaos Sfakianakis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mark A. J. Chaplain
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mark A. J. Chaplain .

Editor information

Editors and Affiliations

  1. Centre for Mathematical Modeling and Data Science, Osaka University, Toyonaka, Japan

    Takashi Suzuki

  2. Research Team MONC, Institut de Mathématiques de Bordeaux (CNRS UMR 5251), Talence, France

    Clair Poignard

  3. School of Mathematics and Statistics, MI, University of St Andrews, St Andrews, UK

    Mark Chaplain

  4. Quantitative Systems Biology Center, Vanderbilt University School of Medicine, Nashville, TN, USA

    Vito Quaranta

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Singapore Pte Ltd.

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Sfakianakis, N., Chaplain, M.A.J. (2021). Mathematical Modelling of Cancer Invasion: A Review. In: Suzuki, T., Poignard, C., Chaplain, M., Quaranta, V. (eds) Methods of Mathematical Oncology. MMDS 2020. Springer Proceedings in Mathematics & Statistics, vol 370. Springer, Singapore. https://doi.org/10.1007/978-981-16-4866-3_10

Download citation

Publish with us

Policies and ethics

Search

Navigation