American Institute of Mathematical Sciences

Advanced
2011, 8(2): 239-252. doi: 10.3934/mbe.2011.8.239

Investigating the steady state of multicellular spheroids by revisiting the two-fluid model

Antonio Fasano 1, , Marco Gabrielli 2, and  Alberto Gandolfi 3,

1. 

Dipartimento di Matematica "U. Dini", Università di Firenze, Viale Morgagni 67/A, 50134 Firenze
2. 

Dipartimento di Matematica "U. Dini", Universita' di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy
3. 

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

Received  March 2010 Revised  August 2010 Published  April 2011

In this paper we examine the steady state of tumour spheroids considering a structure in which the central necrotic region contains an inner liquid core surrounded by dead cells that keep some mechanical integrity. This partition is a consequence of assuming that a finite delay is required for the degradation of dead cells into liquid. The phenomenological assumption of constant local volume fraction of cells is also made. The above structure is coupled with a simple mechanical model that views the cell component as a viscous fluid and the extracellular liquid as an inviscid fluid. By imposing the continuity of the normal stress throughout the whole spheroid, we show that a steady state can exist only if the forces on cells at the outer boundary (provided e.g. by a surface tension) are intense enough, and in such a case we can compute the stationary radius. By giving reasonable values to the parameters, the model predicts that the stationary radius decreases with the external oxygen concentration, as expected from experimental observations.
Keywords: Cancer modelling, flow of fluid mixtures, avascular tumours, free boundary problems..
Mathematics Subject Classification: Primary: 92C50; Secondary: 92C17, 76D27, 35R3.
Citation: Antonio Fasano, Marco Gabrielli, Alberto Gandolfi. Investigating the steady state of multicellular spheroids by revisiting the two-fluid model. Mathematical Biosciences & Engineering, 2011, 8 (2) : 239-252. doi: 10.3934/mbe.2011.8.239
References:
[1]

D. Ambrosi and L. Preziosi, Cell adhesion mechanisms and stress relaxation in the mechanics of tumours,, Biomech. Model. MechanoBiol., 8 (2009), 397.  doi: 10.1007/s10237-008-0145-y.  Google Scholar

[2]

S. Astanin and L. Preziosi, Multiphase models of tumour growth,, in, (2008), 223.  doi: 10.1007/978-0-8176-4713-1_9.  Google Scholar

[3]

S. Astanin and L. Preziosi, Mathematical modelling of the Warburg effect in tumour cords,, J. Theor. Biol., 258 (2009), 578.  doi: 10.1016/j.jtbi.2009.01.034.  Google Scholar

[4]

A. Bertuzzi, A. Fasano and 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. Analysis, 36 (2004), 882.  doi: 10.1137/S003614002406060.  Google Scholar

[5]

A. Bertuzzi, A. Fasano, A. Gandolfi and C. Sinisgalli, Necrotic core in EMT6/Ro tumor spheroids: Is it caused by an ATP deficit?,, J. Theor. Biol., 262 (2010), 142.  doi: 10.1016/j.jtbi.2009.09.024.  Google Scholar

[6]

A. Bertuzzi, C. Bruni, A. Fasano, A. Gandolfi, F. Papa and C. Sinisgalli, Response of tumor spheroids to radiation: Modeling and parameter identification,, Bull. Math. Biol., 72 (2010), 1069.  doi: 10.1007/s11538-009-9482-y.  Google Scholar

[7]

A. Bredel-Geissler, U. Karbach, S. Walenta, L. Vollrath and W. Mueller-Klieser, Proliferation-associated oxygen consumption and morphology of tumour cells in monolayer and spheroid culture,, J. Cell. Physiol., 153 (1992), 44.  doi: 10.1002/jcp.1041530108.  Google Scholar

[8]

H. M. Byrne and M. A. J. Chaplain, Growth of nonnecrotic tumors in the presence and absence of inhibitors,, Math. Biosci., 130 (1995), 151.  doi: 10.1016/0025-5564(94)00117-3.  Google Scholar

[9]

H. M. Byrne and M. A. J. Chaplain, Growth of necrotic tumors in the presence and absence of inhibitors,, Math. Biosci., 135 (1996), 187.  doi: 10.1016/0025-5564(96)00023-5.  Google Scholar

[10]

H. Byrne and L. Preziosi, Modeling solid tumor growth using the theory of mixtures,, Math. Med. Biol., 20 (2003), 341.  doi: 10.1093/imammb/20.4.341.  Google Scholar

[11]

H. M. Byrne, J. R. King, D. L. S. McElwain and L. Preziosi, A two-phase model of solid tumour growth,, Appl. Math. Lett., 16 (2003), 567.  doi: 10.1016/S0893-9659(03)00038-7.  Google Scholar

[12]

J. J. Casciari, S. V. Sotirchos and R. M. Sutherland, Variation in tumor cell growth rates and metabolism with oxygen concentration, glucose concentration, and extracellular pH,, J. Cell. Physiol., 151 (1992), 386.  doi: 10.1002/jcp.1041510220.  Google Scholar

[13]

V. Cristini, X. Li, J. S. Lowengrub and S. M. Wise, Nonlinear simulations of solid tumor growth using a mixture model: invasion and branching,, J. Math. Biol., 58 (2009), 723.  doi: 10.1007/s00285-008-0215-x.  Google Scholar

[14]

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

[15]

A. Friedman and F. Reitich, Symmetry-breaking bifurcations of analytic solutions to free boundary problems: An application to a model of tumor growth,, Trans. Amer. Math. Soc., 353 (2000), 1587.  doi: 10.1090/S0002-9947-00-02715-X.  Google Scholar

[16]

J. P. Freyer and R. M. Sutherland, A reduction in the in situ rates of oxygen and glucose consumption of cells in EMT6/Ro spheroids during growth,, J. Cell. Physiol., 124 (1985), 516.  doi: 10.1002/jcp.1041240323.  Google Scholar

[17]

J. P. Freyer and 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 (1986), 3504.   Google Scholar

[18]

G. Hamilton, Multicellular spheroids as an in vitro tumor model,, Cancer Lett., 131 (1998), 29.  doi: 10.1016/S0304-3835(98)00198-0.  Google Scholar

[19]

G. Helmlingen, P. A. Netti, H. C. Lichtembeld, R. J. Melder and R. K. Jain, Solid stress inhibits the growth of multicellular tumor spheroids,, Nature Biotech., 15 (1997), 778.  doi: 10.1038/nbt0897-778.  Google Scholar

[20]

A. Iordan, A. Duperray and C. Verdier, A fractal approach to the rheology of concentrated cell suspensions,, Phis. Rev. E, 77 (2008).  doi: 10.1103/PhysRevE.77.011911.  Google Scholar

[21]

P. A. Netti and R. K. Jain, Interstitial transport in solid tumours,, in, (2003), 51.  doi: 10.1201/9780203494899.ch3.  Google Scholar

[22]

K. A. Landman and C. P. Please, Tumour dynamics and necrosis: surface tension and stability,, IMA J. Math. Appl. Med. Biol., 18 (2001), 131.  doi: 10.1093/imammb/18.2.131.  Google Scholar

[23]

G. Lemon, J. R. King, H. M. Byrne, O. E. Jensen and K. M. Shakesheff, Mathematical modelling of engineered tissue growth using a multiphase porous flow mixture theory,, J. Math. Biol., 52 (2008), 571.  doi: 10.1007/s00285-005-0363-1.  Google Scholar

[24]

W. Mueller-Klieser, Method for the determination of oxygen consumption rates and diffusion coefficients in multicellular spheroids,, Biophysical Journal, 46 (1984), 343.  doi: 10.1016/S0006-3495(84)84030-8.  Google Scholar

[25]

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

[26]

K. R. Rajagopal and L. Tao, "Mechanics of Mixtures,'', World Scientific, (1995).   Google Scholar

[27]

K. Smallbone, R. A. Gatenby, R. J. Gillies, P. K. Maini and D. J. Gavaghan, Metabolic changes during carcinogenesis: Potential impact on invasiveness,, J. Theor. Biol., 244 (2007), 703.  doi: 10.1016/j.jtbi.2006.09.010.  Google Scholar

[28]

J. P. Ward and J. R. King, Mathematical modelling of avascular tumor growth I,, IMA J. Math. Appl. Med. Biol., 14 (1997), 36.  doi: 10.1093/imammb/14.1.39.  Google Scholar

[29]

J. P. Ward and J. R. King, Mathematical modelling of avascular tumor growth II. Modelling growth saturation,, IMA J. Math. Appl. Med. Biol., 16 (1999), 171.  doi: 10.1093/imammb/16.2.171.  Google Scholar

show all references

References:
[1]

D. Ambrosi and L. Preziosi, Cell adhesion mechanisms and stress relaxation in the mechanics of tumours,, Biomech. Model. MechanoBiol., 8 (2009), 397.  doi: 10.1007/s10237-008-0145-y.  Google Scholar

[2]

S. Astanin and L. Preziosi, Multiphase models of tumour growth,, in, (2008), 223.  doi: 10.1007/978-0-8176-4713-1_9.  Google Scholar

[3]

S. Astanin and L. Preziosi, Mathematical modelling of the Warburg effect in tumour cords,, J. Theor. Biol., 258 (2009), 578.  doi: 10.1016/j.jtbi.2009.01.034.  Google Scholar

[4]

A. Bertuzzi, A. Fasano and 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. Analysis, 36 (2004), 882.  doi: 10.1137/S003614002406060.  Google Scholar

[5]

A. Bertuzzi, A. Fasano, A. Gandolfi and C. Sinisgalli, Necrotic core in EMT6/Ro tumor spheroids: Is it caused by an ATP deficit?,, J. Theor. Biol., 262 (2010), 142.  doi: 10.1016/j.jtbi.2009.09.024.  Google Scholar

[6]

A. Bertuzzi, C. Bruni, A. Fasano, A. Gandolfi, F. Papa and C. Sinisgalli, Response of tumor spheroids to radiation: Modeling and parameter identification,, Bull. Math. Biol., 72 (2010), 1069.  doi: 10.1007/s11538-009-9482-y.  Google Scholar

[7]

A. Bredel-Geissler, U. Karbach, S. Walenta, L. Vollrath and W. Mueller-Klieser, Proliferation-associated oxygen consumption and morphology of tumour cells in monolayer and spheroid culture,, J. Cell. Physiol., 153 (1992), 44.  doi: 10.1002/jcp.1041530108.  Google Scholar

[8]

H. M. Byrne and M. A. J. Chaplain, Growth of nonnecrotic tumors in the presence and absence of inhibitors,, Math. Biosci., 130 (1995), 151.  doi: 10.1016/0025-5564(94)00117-3.  Google Scholar

[9]

H. M. Byrne and M. A. J. Chaplain, Growth of necrotic tumors in the presence and absence of inhibitors,, Math. Biosci., 135 (1996), 187.  doi: 10.1016/0025-5564(96)00023-5.  Google Scholar

[10]

H. Byrne and L. Preziosi, Modeling solid tumor growth using the theory of mixtures,, Math. Med. Biol., 20 (2003), 341.  doi: 10.1093/imammb/20.4.341.  Google Scholar

[11]

H. M. Byrne, J. R. King, D. L. S. McElwain and L. Preziosi, A two-phase model of solid tumour growth,, Appl. Math. Lett., 16 (2003), 567.  doi: 10.1016/S0893-9659(03)00038-7.  Google Scholar

[12]

J. J. Casciari, S. V. Sotirchos and R. M. Sutherland, Variation in tumor cell growth rates and metabolism with oxygen concentration, glucose concentration, and extracellular pH,, J. Cell. Physiol., 151 (1992), 386.  doi: 10.1002/jcp.1041510220.  Google Scholar

[13]

V. Cristini, X. Li, J. S. Lowengrub and S. M. Wise, Nonlinear simulations of solid tumor growth using a mixture model: invasion and branching,, J. Math. Biol., 58 (2009), 723.  doi: 10.1007/s00285-008-0215-x.  Google Scholar

[14]

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

[15]

A. Friedman and F. Reitich, Symmetry-breaking bifurcations of analytic solutions to free boundary problems: An application to a model of tumor growth,, Trans. Amer. Math. Soc., 353 (2000), 1587.  doi: 10.1090/S0002-9947-00-02715-X.  Google Scholar

[16]

J. P. Freyer and R. M. Sutherland, A reduction in the in situ rates of oxygen and glucose consumption of cells in EMT6/Ro spheroids during growth,, J. Cell. Physiol., 124 (1985), 516.  doi: 10.1002/jcp.1041240323.  Google Scholar

[17]

J. P. Freyer and 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 (1986), 3504.   Google Scholar

[18]

G. Hamilton, Multicellular spheroids as an in vitro tumor model,, Cancer Lett., 131 (1998), 29.  doi: 10.1016/S0304-3835(98)00198-0.  Google Scholar

[19]

G. Helmlingen, P. A. Netti, H. C. Lichtembeld, R. J. Melder and R. K. Jain, Solid stress inhibits the growth of multicellular tumor spheroids,, Nature Biotech., 15 (1997), 778.  doi: 10.1038/nbt0897-778.  Google Scholar

[20]

A. Iordan, A. Duperray and C. Verdier, A fractal approach to the rheology of concentrated cell suspensions,, Phis. Rev. E, 77 (2008).  doi: 10.1103/PhysRevE.77.011911.  Google Scholar

[21]

P. A. Netti and R. K. Jain, Interstitial transport in solid tumours,, in, (2003), 51.  doi: 10.1201/9780203494899.ch3.  Google Scholar

[22]

K. A. Landman and C. P. Please, Tumour dynamics and necrosis: surface tension and stability,, IMA J. Math. Appl. Med. Biol., 18 (2001), 131.  doi: 10.1093/imammb/18.2.131.  Google Scholar

[23]

G. Lemon, J. R. King, H. M. Byrne, O. E. Jensen and K. M. Shakesheff, Mathematical modelling of engineered tissue growth using a multiphase porous flow mixture theory,, J. Math. Biol., 52 (2008), 571.  doi: 10.1007/s00285-005-0363-1.  Google Scholar

[24]

W. Mueller-Klieser, Method for the determination of oxygen consumption rates and diffusion coefficients in multicellular spheroids,, Biophysical Journal, 46 (1984), 343.  doi: 10.1016/S0006-3495(84)84030-8.  Google Scholar

[25]

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

[26]

K. R. Rajagopal and L. Tao, "Mechanics of Mixtures,'', World Scientific, (1995).   Google Scholar

[27]

K. Smallbone, R. A. Gatenby, R. J. Gillies, P. K. Maini and D. J. Gavaghan, Metabolic changes during carcinogenesis: Potential impact on invasiveness,, J. Theor. Biol., 244 (2007), 703.  doi: 10.1016/j.jtbi.2006.09.010.  Google Scholar

[28]

J. P. Ward and J. R. King, Mathematical modelling of avascular tumor growth I,, IMA J. Math. Appl. Med. Biol., 14 (1997), 36.  doi: 10.1093/imammb/14.1.39.  Google Scholar

[29]

J. P. Ward and J. R. King, Mathematical modelling of avascular tumor growth II. Modelling growth saturation,, IMA J. Math. Appl. Med. Biol., 16 (1999), 171.  doi: 10.1093/imammb/16.2.171.  Google Scholar

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