Skip to main content
SpringerLink
Log in
Book cover

Mathematical Oncology 2013 pp 27–61Cite as

Birkhäuser

Conservation Laws in Cancer Modeling

  • Chapter
  • First Online:

  • 1133 Accesses

Abstract

We review mathematical models of tumor growth based on conservation laws in the full system of cells and interstitial liquid. First we deal with tumor cords evolving in axisymmetric geometry, where cells motion is simply passive and compatible with the saturation condition. The model is characterized by the presence of free boundaries with constraints driving the free boundary conditions, which in our opinion are particularly important, especially in the presence of treatments. Then a tumor spheroid is considered in the framework of the so-called two-fluid scheme. In a multicellular spheroid, on the appearance of a fully degraded necrotic core, the analysis of mechanical stresses becomes necessary to determine the motion via momentum balance, requiring the specification of the constitutive law for the “cell fluid.” We have chosen a Bingham-type law that presents considerable difficulties because of the presence of a yield stress, particularly with reference to the determination of an asymptotic configuration. Finally, we report some recent PDE-based models addressing complex processes in multicomponent tumors, more oriented to clinical practice.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    Some authors adopt the extreme view point that proliferation takes place only at the tumor surface because of contact inhibition (e.g., [17]) and then migrate, driven by the surface curvature. Here we stick to the experimental observation that in the tumors we are talking about proliferation occurs in the tumor mass, whenever enough oxygen is available.

  2. 2.

    Here we neglect osmotic pressure.

References

  1. T. Alarcón, H.M. Byrne, P.K. Maini, A cellular automaton model for tumour growth in inhomogeneous environment. J. Theor. Biol. 225, 257–274 (2003)

    Article  Google Scholar 

  2. D. Ambrosi, L. Preziosi, Cell adhesion mechanisms and stress relaxation in the mechanics of tumours. Biomech. Model. MechanoBiol. 8, 397–413 (2009)

    Article  Google Scholar 

  3. A.R.A. Anderson, M.A.J. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis. Bull. Math. Biol. 60, 857–999 (1998)

    Article  MATH  Google Scholar 

  4. R.P. Araujo, D.L.S. McElwain, A history of the study of solid tumour growth: the contribution of mathematical modelling. Bull. Math. Biol. 66, 1039–1091 (2004)

    Article  MathSciNet  Google Scholar 

  5. S. Astanin, A. Tosin, Mathematical model of tumour cord growth along the source of nutrient. Math. Model. Nat. Phenom. 2, 153–177 (2007)

    Article  MathSciNet  Google Scholar 

  6. I.V. Basov, V.V. Shelukhin, Generalized solutions to the equations of compressible Bingham flows. Z. Angew. Math. Mech. 79, 185–192 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. N. Bellomo, N.K. Li, P.K. Maini, On the foundations of cancer modelling: selected topics, speculations, and perspectives. Math. Mod. Meth. Appl. Sci. 18, 593–646 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. A. Bertuzzi, A. d’Onofrio, A. Fasano, A. Gandolfi, Regression and regrowth of tumour cords following single-dose anticancer treatment. Bull. Math. Biol. 65, 903–931 (2003)

    Google Scholar 

  9. A. Bertuzzi, A. Fasano, A. Gandolfi, A free boundary problem with unilateral constraints describing the evolution of a tumour cord under the influence of cell killing agents. SIAM J. Math. Anal. 36, 882–915 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Bertuzzi, A. Fasano, A. Gandolfi, A mathematical model for tumor cords incorporating the flow of interstitial fluid. Math. Mod. Meth. Appl. Sci. 15, 1735–1777 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Bertuzzi, A. Fasano, L. Filidoro, A. Gandolfi, C. Sinisgalli, Dynamics of tumour cords following changes in oxygen availability: a model including a delayed exit from quiescence. Math. Comput. Model. 41, 1119–1135 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Bertuzzi, A. Fasano, A. Gandolfi, C. Sinisgalli, Interstitial pressure and extracellular fluid motion in tumour cords. Math. Biosci. Eng. 2, 445–460 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Bertuzzi, A. Fasano, A. Gandolfi, C. Sinisgalli, Cell resensitization after delivery of a cycle-specific anticancer drug and effect of dose splitting: learning from tumour cords. J. Theor. Biol. 244, 388–399 (2007)

    Article  MathSciNet  Google Scholar 

  14. A. Bertuzzi, A. Fasano, A. Gandolfi, C. Sinisgalli, Reoxygenation and split-dose response to radiation in a tumour model with Krogh-type vascular geometry. Bull. Math. Biol. 70, 992–1012 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Bertuzzi, A. Fasano, A. Gandolfi, C. Sinisgalli, Modelling the evolution of a tumoural multicellular spheroid as a two-fluid Bingham-like system. Math. Mod. Meth. Appl. Sci. 23, 2561–2602 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. A. Bertuzzi, C. Bruni, F. Papa, C. Sinisgalli, Optimal solution for a cancer radiotherapy problem. J. Math. Biol. 66, 311–349 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. A. Brú, S. Albertos, J.L. Subiza, J. López García-Asenjo, I. Brú, The universal dynamics of tumor growth. Biophys. J. 85, 2948–2961 (2003)

    Article  Google Scholar 

  18. H.M. Byrne, J.R. King, D.L.S. McElwain, L. Preziosi, A two-phase model of solid tumour growth. Appl. Math. Lett. 16, 567–573 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  20. M.A. Chaplain, S.R. McDougall, A.R. Anderson, Mathematical modeling of tumor-induced angiogenesis. Annu. Rev. Biomed. Eng. 8, 233–257 (2006)

    Article  Google Scholar 

  21. D. Chen, J.M. Roda, C.B. Marsh, T.D. Eubank, A. Friedman, Hypoxia inducible factors-mediated inhibition of cancer by GM-CSF: a mathematical model. Bull. Math. Biol. 74, 2752–2777 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. A. d’Onofrio, A. Gandolfi, Chemotherapy of vascularised tumours: role of vessel density and the effect of vascular “pruning”. J. Theor. Biol. 264, 253–265 (2010)

    Google Scholar 

  23. T. Eubank, R.D. Roberts, M. Khan, J. Curry, G.J. Nuovo, P. Kuppusamyl, C. Marsh, Granulocyte macrophage Colony-Stimulating factor inhibits breast cancer growth and metastasis by invoking an anti-angiogenic program in tumor-educated macrophages. Cancer Res. 69, 2133–2140 (2009)

    Article  Google Scholar 

  24. A. Fasano, Glucose metabolism in multicellular spheroids, ATP production and effects of acidity, in New Challenges for Cancer Systems Biomedicine, ed. by A. d’Onofrio, Z. Agur, P. Cerrai, A. Gandolfi (Springer, to appear)

    Google Scholar 

  25. A. Fasano, A. Gandolfi, The steady state of multicellular tumour spheroids: a modelling challenge, in Mathematical Methods and Models in Biomedicine, ed. by U. Ledzewicz, H. Schaettler, A. Friedman, E. Kashdan (Springer, New York, 2012), pp. 161–179

    Google Scholar 

  26. A. Fasano, A. Bertuzzi, A. Gandolfi, Mathematical modelling of tumour growth and treatment. In: Complex Systems in Biomedicine, ed. by A. Quarteroni, L. Formaggia, A. Veneziani (Springer, Italia, Milano, 2006), pp. 71–108

    Chapter  Google Scholar 

  27. A. Fasano, M.A. Herrero, M. Rocha Rodrigo, Slow and fast invasion waves in a model of acid-mediated tumour growth. Math. Biosci. 220, 45–56 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. A. Fasano, M. Gabrielli, A. Gandolfi, The energy balance in stationary multicellular spheroids. Far East J. Math. Sci. 39, 105–128 (2010)

    MathSciNet  MATH  Google Scholar 

  29. A. Fasano, M. Gabrielli, A. Gandolfi, Investigating the steady state of multicellular spheroids by revisiting the two-fluid model. Math. Biosci. Eng. 8, 239–252 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. A. Fasano, M. Gabrielli, A. Gandolfi, Erratum to: investigating the steady state of multicellular spheroids by revisiting the two-fluid model. Math. Biosci. Eng. 9, 697 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. J. Folkman, M. Hochberg, Cell-regulation of growth in three dimensions. J. Exp. Med. 138, 745–753 (1973)

    Article  Google Scholar 

  32. J. Folkman, E. Merler, C. Abernathy, G. Williams, Isolation of a tumor fraction responsible for angiogenesis. J. Exp. Med. 133, 275–288 (1971)

    Article  Google Scholar 

  33. J.P. Freyer, R.M. Sutherland, Regulation of growth saturation and development of necrosis in EMT6/Ro multicellular spheroids by the glucose and oxygen supply. Cancer Res. 46, 3504–3512 (1986)

    Google Scholar 

  34. A. Friedman, A hierarchy of cancer models and their mathematical challenges. Discrete Contin. Dyn. Syst. B 4, 147–159 (2004)

    Article  MATH  Google Scholar 

  35. R.A. Gatenby, E.T. Gawlinski, A reaction-diffusion model for cancer invasion. Cancer Res. 56, 5745–5753 (1996)

    Google Scholar 

  36. J.B. Gillen, E.A. Gaffney, N.K. Martin, P.K. Maini, A general reaction-diffusion model of acidity in cancer invasion. J. Math. Biol. (2013)

    Google Scholar 

  37. S. Goel, D.G. Duda, L. Xu, L.L. Munn, Y. Boucher, D. Fukumura, R.K. Jain, Normalization of the vasculature for treatment of cancer and other diseases. Physiol. Rev. 91, 1071–1121 (2011)

    Article  Google Scholar 

  38. P. Greenspan, Models for the growth of a solid tumour by diffusion. Stud. Appl. Math. 51, 317–340 (1972)

    MATH  Google Scholar 

  39. P. Hahnfeldt, D. Panigrahy, J. Folkman, L. Hlatky, Tumor development under angiogenic signaling: a dynamical theory of tumor growth, treatment response, and postvascular dormancy. Cancer Res. 59, 4770–4775 (1999)

    Google Scholar 

  40. M.A. Herrero, Reaction-diffusion systems: a mathematical biology approach, in Cancer Modelling and Simulation, ed. by L. Preziosi (Chapman and Hall, Boca Raton, 2003), pp. 367–420

    Google Scholar 

  41. T. Hillen, K.J. Painter, A user’s guide to pde models for chemotaxis. J. Math. Biol. 58, 183–217 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  43. P. Hinow, P. Gerlee, L.J. McCawley, V. Quaranta, M. Ciobanu, J.M. Graham, B.P. Ayati, J. Claridge, K.R. Swanson, M. Loveless, A.R.A. Anderson, A spatial model of tumor-host interaction: application of chemotherapy. Math. Biosci. Eng. 6, 521–546 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  44. D.G. Hirst, J. Denekamp, Tumour cell proliferation in relation to the vasculature. Cell Tissue Kinet. 12, 31–42 (1979)

    Google Scholar 

  45. A. Iordan, A. Duperray, C. Verdier, A fractal approach to the rheology of concentrated cell suspensions. Phys. Rev. E 77, 011911 (2008)

    Article  Google Scholar 

  46. R.K. Jain, Normalizing tumor vasculature with anti-angiogenic therapy: a new paradigm for combination therapy. Nat. Med. 7, 987–989 (2001)

    Article  Google Scholar 

  47. H.V. Jain, A. Friedman, Modeling prostate cancer response to continuous versus intermittent androgen ablation therapy. Discrete Contin. Dyn. Syst. B 18, 945–967 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  48. K.A. Landman, C.P. Please, Tumour dynamics and necrosis: surface tension and stability. IMA J. Math. Appl. Med. Biol. 18, 131–158 (2001)

    Article  MATH  Google Scholar 

  49. U. Ledzewicz, M. Naghnaeian, H. Schättler, Optimal response to chemotherapy for a mathematical model of tumor-immune dynamics. J. Math. Biol. 64, 557–77 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  50. H.A. Levine, B.D. Sleeman, M. Nilsen-Hamilton, Mathematical modeling of the onset of capillary formation initiating angiogenesis. J. Math. Biol. 42, 195–238 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  51. J.S. Lowengrub, H.B. Frieboes, F. Jin, Y.L. Chuang, X. Li, P. Macklin, S.M. Wise, V. Cristini, Nonlinear modelling of cancer: bridging the gap between cells and tumours. Nonlinearity 23, 1–91 (2010)

    Article  MathSciNet  Google Scholar 

  52. J.V. Moore, H.A. Hopkins, W.B. Looney, Dynamic histology of a rat hepatoma and the response to 5-fluorouracil. Cell Tissue Kinet. 13, 53–63 (1980)

    Google Scholar 

  53. J.V. Moore, P.S. Hasleton, C.H. Buckley, Tumour cords in 52 human bronchial and cervical squamous cell carcinomas: inferences for their cellular kinetics and radiobiology. Br. J. Cancer 51, 407–413 (1985)

    Article  Google Scholar 

  54. M. Neeman, K.A. Jarrett, L.O. Sillerud, J.P. Freyer, Self-diffusion of water in multicellular spheroids measured by magnetic resonance microimaging. Cancer Res. 51, 4072–4079 (1991)

    Google Scholar 

  55. P. Panorchan, M.S. Thompson, K.J. Davis, Y. Tseng, K. Konstantopoulos, D. Wirtz, Single-molecule analysis of cadherin-mediated cell–cell adhesion. J. Cell Sci. 119, 66–74 (2006)

    Article  Google Scholar 

  56. V.M. Perez-Garcia, G.F. Calvo, J. Belmonte-Beitia, D. Diego, L. Perez-Romasanta, Bright solitary waves in malignant gliomas. Phys. Rev. E 84, 1–6 (2011)

    Article  Google Scholar 

  57. L. Preziosi, G. Vitale, A multiphase model of tumor and tissue growth including cell adhesion and plastic reorganization. Math. Mod. Meth. Appl. Sci. 21, 1901–1932 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  58. K.R. Rajagopal, L. Tao, Mechanics of Mixtures (World Scientific, Singapore, 1995)

    MATH  Google Scholar 

  59. D. Ribatti, A. Vacca, M. Presta, The discovery of angiogenic factors: a historical review. General Pharmacol. 35, 227–231 (2002)

    Article  Google Scholar 

  60. J.M. Roda, L.A. Summer, R. Evans, G.S. Philips, C.B. Marsh, T.D. Eubank, Hypoxia-inducible factor-2α regulates GM-CSF-derived soluble vascular endothelial growth factor receptor 1 production from macrophages and inhibits tumor growth and angiogenesis. J. Immunol. 187, 1970–1976 (2011)

    Article  Google Scholar 

  61. J.M. Roda, Y. Wang, L. Sumner, G. Phillips, T.D. Eubank, C. Marsh, Stabilization of HIF-2α induces SVEGFR-1 production from Tumor-associated macrophages and enhances the Anti-tumor effects of GM-CSF in murine melanoma model. J. Immunol. 189, 3168–3177 (2012)

    Article  Google Scholar 

  62. A. Stephanou, S.R. McDougall, A.R.A. Anderson, M.A.J. Chaplain, Mathematical modelling of flow in 2d and 3d vascular networks: applications to anti-angiogenic and chemotherapeutic drug strategies. Math. Comput. Model. 41, 1137–1156 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  64. K.R. Swanson, R. Rockne, J. Claridge, M.A. Chaplain, E.C. Alvord Jr., A.R.A. Anderson, Quantifying the role of angiogenesis in malignant progression of gliomas: In silico modeling integrates imaging and histology. Cancer Res. 71, 7366–7375 (2011)

    Article  Google Scholar 

  65. I.F. Tannock, The relation between cell proliferation and the vascular system in a transplanted mouse mammary tumour. Br. J. Cancer 22, 258–273 (1968)

    Article  Google Scholar 

  66. Y. Yang, L. Xing, Optimization of radiotherapy dose-time fractionation with consideration of tumor specific biology. Med. Phys. 32, 3666–3677 (2005)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica “U. Dini”, Universita’ di Firenze, Viale Morgagni 67/A, 50134, Firenze, Italy

    Antonio Fasano

  2. Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti” - CNR, Viale Manzoni 30, 00185, Roma, Italy

    Antonio Fasano, Alessandro Bertuzzi & Carmela Sinisgalli

Authors
  1. Antonio Fasano
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alessandro Bertuzzi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Carmela Sinisgalli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Antonio Fasano .

Editor information

Editors and Affiliations

  1. International Prevention Research Institute, Lyon, France

    Alberto d'Onofrio

  2. Istituto di Analisi dei Sistemi ed Informatica "Antonio Ruberti"—CNR, Rome, Italy

    Alberto Gandolfi

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media New York

About this chapter

Cite this chapter

Fasano, A., Bertuzzi, A., Sinisgalli, C. (2014). Conservation Laws in Cancer Modeling. In: d'Onofrio, A., Gandolfi, A. (eds) Mathematical Oncology 2013. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-0458-7_2

Download citation

Publish with us

Policies and ethics

Search

Navigation