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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 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.
Here we neglect osmotic pressure.
References
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)
D. Ambrosi, L. Preziosi, Cell adhesion mechanisms and stress relaxation in the mechanics of tumours. Biomech. Model. MechanoBiol. 8, 397–413 (2009)
A.R.A. Anderson, M.A.J. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis. Bull. Math. Biol. 60, 857–999 (1998)
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)
S. Astanin, A. Tosin, Mathematical model of tumour cord growth along the source of nutrient. Math. Model. Nat. Phenom. 2, 153–177 (2007)
I.V. Basov, V.V. Shelukhin, Generalized solutions to the equations of compressible Bingham flows. Z. Angew. Math. Mech. 79, 185–192 (1999)
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)
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)
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)
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)
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)
A. Bertuzzi, A. Fasano, A. Gandolfi, C. Sinisgalli, Interstitial pressure and extracellular fluid motion in tumour cords. Math. Biosci. Eng. 2, 445–460 (2005)
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)
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)
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)
A. Bertuzzi, C. Bruni, F. Papa, C. Sinisgalli, Optimal solution for a cancer radiotherapy problem. J. Math. Biol. 66, 311–349 (2013)
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)
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)
H.M. Byrne, L. Preziosi, Modelling solid tumour growth using the theory of mixtures. Math. Med. Biol. 20, 341–366 (2003)
M.A. Chaplain, S.R. McDougall, A.R. Anderson, Mathematical modeling of tumor-induced angiogenesis. Annu. Rev. Biomed. Eng. 8, 233–257 (2006)
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)
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)
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)
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)
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
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
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)
A. Fasano, M. Gabrielli, A. Gandolfi, The energy balance in stationary multicellular spheroids. Far East J. Math. Sci. 39, 105–128 (2010)
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)
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)
J. Folkman, M. Hochberg, Cell-regulation of growth in three dimensions. J. Exp. Med. 138, 745–753 (1973)
J. Folkman, E. Merler, C. Abernathy, G. Williams, Isolation of a tumor fraction responsible for angiogenesis. J. Exp. Med. 133, 275–288 (1971)
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)
A. Friedman, A hierarchy of cancer models and their mathematical challenges. Discrete Contin. Dyn. Syst. B 4, 147–159 (2004)
R.A. Gatenby, E.T. Gawlinski, A reaction-diffusion model for cancer invasion. Cancer Res. 56, 5745–5753 (1996)
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)
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)
P. Greenspan, Models for the growth of a solid tumour by diffusion. Stud. Appl. Math. 51, 317–340 (1972)
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)
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
T. Hillen, K.J. Painter, A user’s guide to pde models for chemotaxis. J. Math. Biol. 58, 183–217 (2009)
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)
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)
D.G. Hirst, J. Denekamp, Tumour cell proliferation in relation to the vasculature. Cell Tissue Kinet. 12, 31–42 (1979)
A. Iordan, A. Duperray, C. Verdier, A fractal approach to the rheology of concentrated cell suspensions. Phys. Rev. E 77, 011911 (2008)
R.K. Jain, Normalizing tumor vasculature with anti-angiogenic therapy: a new paradigm for combination therapy. Nat. Med. 7, 987–989 (2001)
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)
K.A. Landman, C.P. Please, Tumour dynamics and necrosis: surface tension and stability. IMA J. Math. Appl. Med. Biol. 18, 131–158 (2001)
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)
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)
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)
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)
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)
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)
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)
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)
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)
K.R. Rajagopal, L. Tao, Mechanics of Mixtures (World Scientific, Singapore, 1995)
D. Ribatti, A. Vacca, M. Presta, The discovery of angiogenic factors: a historical review. General Pharmacol. 35, 227–231 (2002)
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)
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)
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)
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)
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)
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)
Y. Yang, L. Xing, Optimization of radiotherapy dose-time fractionation with consideration of tumor specific biology. Med. Phys. 32, 3666–3677 (2005)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights 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
DOI: https://doi.org/10.1007/978-1-4939-0458-7_2
Published:
Publisher Name: Birkhäuser, New York, NY
Print ISBN: 978-1-4939-0457-0
Online ISBN: 978-1-4939-0458-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)