Computational Mechanics

, Volume 53, Issue 3, pp 465–484 | Cite as

Three phase flow dynamics in tumor growth

  • G. Sciumè
  • W. G. Gray
  • F. Hussain
  • M. Ferrari
  • P. Decuzzi
  • B. A. Schrefler
Original Paper


Existing tumor models generally consider only a single pressure for all the cell phases. Here, a three-fluid model originally proposed by the authors is further developed to allow for different pressures in the host cells (HC), the tumor cells (TC) and the interstitial fluid (IF) phases. Unlike traditional mixture theory models, this model developed within the thermodynamically constrained averaging theory contains all the necessary interfaces. Appropriate constitutive relationships for the pressure difference among the three fluid phases are introduced with respect to their relative wettability and fluid–fluid interfacial tensions, resulting in a more realistic modeling of cell adhesion and invasion. Five different tumor cases are studied by changing the interfacial tension between the three liquid phases, adhesion and dynamic viscosity. Since these parameters govern the relative velocities of the fluid phases and the adhesion of the phases to the extracellular matrix significant changes in tumor growth are observed. High interfacial tensions at the TC–IF and TC–HC interface support the lateral displacement of the healthy tissue in favor of a rapid growth of the malignant mass, with a relevant amount of HC which cannot be pushed out by TC and remain in place. On the other hand, lower TC–IF and TC–HC interfacial tensions tend to originate a more compact and dense tumor mass with a slower growth rate of the overall size. This novel computational model emphasizes the importance of characterizing the TC–HC interfacial properties to properly predict the temporal and spatial pattern evolution of tumor.


Multiphase model Interfacial tension Growth Necrosis Phase pressure TCAT 

List of symbols



Representative elementary volume


Thermodynamically constrained averaging theory

Roman letters

\(\mathbf{A}^{\alpha }\)

Fourth order tensor that accounts for the stress-rate of strain relationship


Coefficient of the pressure–saturations relationship

\(a_{\alpha }\)

Adhesion of the phase \(\alpha \)


Coefficient of the pressure–saturations relationship

\(\hbox {C}_{ij}\)

Nonlinear coefficient of the discretized capacity matrix

\(\mathbf{d}^{\overline{\overline{\alpha }}}\)

Rate of strain tensor


Diffusion coefficient for the species \(i\) in the phase \(l\)


Effective diffusion coefficient for the species \(i\) in the extracellular space


Tangent matrix of the solid skeleton


Total strain tensor


Elastic strain tensor


Visco-plastic strain tensor


Swelling strain tensor


Discretized source term associated with the primary variable \(v\)


Heaviside function


Non-linear coefficient of the discretized conduction matrix


Intrinsic permeability tensor

\(k_{rel}^\alpha \)

Relative permeability of phase \(\upalpha \)


Vector of shape functions related to the primary variable \(v\)

\(p^{\alpha }\)

Pressure in phase \(\upalpha \)

\(\mathbf{R}^{\alpha }\)

Resistance tensor

\(S^{\alpha }\)

Saturation degree of phase \(\upalpha \)


Effective stress tensor of solid phase s


Total stress tensor of solid phase s


Displacement vector of solid phase s


Solution vector

Greek letters

\(\bar{{\alpha }}\)

Biot’s coefficient

\(\gamma _{growth}^{t}\)

Growth coefficient

\(\gamma _{necrosis}^t\)

Necrosis coefficient

\(\gamma _{growth}^{\overline{nl}}\)

Nutrient consumption coefficient related to growth

\(\gamma _{0}^{\overline{nl}}\)

Nutrient consumption coefficient not related to growth

\(\theta ^{\overline{\overline{\alpha }}}\)

Macroscale temperature of phase \(\upalpha \)

\(\delta \)

Exponent in the effective diffusion function for oxygen

\(\varepsilon \)


\(\varepsilon ^{\alpha }\)

Volume fraction of phase \(\upalpha \)

\(\mu ^{\alpha }\)

Dynamic viscosity of phase \(\upalpha \)

\(\rho ^{\alpha }\)

Density of phase \(\upalpha \)

\(\sigma _{\alpha \beta }\)

Interfacial tension between phases \(\alpha \) and \(\beta \)

\(\varsigma ^{\overline{\alpha }}\)

Chemical potential

\(\psi ^{\overline{\alpha }}\)

Gravitational potential

\(\chi ^{\alpha }\)

Solid surface fraction in contact with phase \(\upalpha \)

\(\omega ^{N\overline{t}}\)

Mass fraction of necrotic cells in the tumor cells phase

\(\omega ^{\overline{nl}}\)

Nutrient mass fraction in the phase \(l\)

\(\omega _{crit}^{\overline{nl}}\)

Critical nutrient mass fraction for growth

\(\omega _{env}^{\overline{nl}}\)

Reference nutrient mass fraction in the environment

TCAT symbols

\(\mathop {M}\limits ^{\kappa \rightarrow \alpha }\)

Inter-phase mass transfer from \(k\) to \(\alpha \) phase

\(\varepsilon ^{\alpha }r^{i\alpha }\)

Reaction term i.e. intra-phase mass transformation

\(\mathop {\mathbf{T}}\limits ^{\kappa \rightarrow \alpha }\)

Inter-phase momentum transfer from \(k\) to \(\alpha \) phase

Subscripts and superscripts


Critical value for growth


Host cell phase (healthy cells of the host issue)


Interstitial fluid






Tumor cell phase

\(\upalpha \)

Phase indicator with \(\alpha = h,\, l,\, s\), or \(t\)



GS and BS acknowledge partial support from the Strategic Research Project “Algorithms and Architectures for Computational Science and Engineering”—AACSE (STPD08JA32—2008) of the University of Padova (Italy) and the partial support of Università Italo Francese within the Vinci Program. WGG acknowledges partial support from the U.S. National Science Foundation Grant ATM-0941235 and the U.S. Department of Energy Grant DE-SC0002163. PD and MF acknowledge partial support from the NIH/NCI grants U54CA143837 and U54CA151668. MF acknowledges the Ernest Cockrell Jr. Distinguished Endowed Chair.


  1. 1.
    Amack JD, Manning ML (2012) Knowing the boundaries: extending the differential adhesion hypothesis in embryonic cell sorting. Science 338(6104):212–215CrossRefGoogle Scholar
  2. 2.
    Ambrosi D, Preziosi L, Vitale G (2012) The interplay between stress and growth in solid tumors. Mech Res Commun 42:87–91CrossRefGoogle Scholar
  3. 3.
    Bidan CM, Kommareddy KP, Rumpler M, Kollmannsberger P, Bréchet YJM, Fratzl P, Dunlop JWC (2012) How linear tension converts to curvature: geometric control of bone tissue growth. PLoS ONE 7(5):e36336. doi: 10.1371/journal.pone.0036336
  4. 4.
    Brooks RH, Corey AT (1964) Hydraulic properties of porous media. Hydrol Pap 3, Colorado State University, Fort CollinsGoogle Scholar
  5. 5.
    Brooks RH, Corey AT (1966) Properties of porous media affecting fluid flow. J Irrigation Drainage Div Am Soc Civ Eng 92(IR2):61–88Google Scholar
  6. 6.
    Cahn JW, Hilliard JE (1958) Free energy of a nonuniform system: I. Interfacial free energy. J Chem Phys 28:256–267CrossRefGoogle Scholar
  7. 7.
    Corey AT, Rathjens CH, Henderson JH, Wyllie MRJ (1956) Three-phase relative permeability. Trans AIME 207:349–351Google Scholar
  8. 8.
    Deisboeck TS, Wang Z, Macklin P, Cristini V (2011) Multiscale cancer modeling. Annu Rev Biomed Eng 13:127–155CrossRefGoogle Scholar
  9. 9.
    Dunlop JW, Gamsjäger E, Bidan C, Kommareddy KP, Kollmansberger P, Rumpler M, Fischer FD, Fratzl P (2011) The modeling of tissue growth in confined geometries, effect of surface tension. In: Proceedings of CMM-2011 (Warsaw) computer methods in mechanicsGoogle Scholar
  10. 10.
    Gonzalez-Rodriguez D, Guevorkian K, Douezan S, Françoise Brochard-Wyart F (2012) Soft matter models of developing tissues and tumors. Science 338:910–917CrossRefGoogle Scholar
  11. 11.
    Gray WG, Miller CT (2005) Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 1. Motivation and overview. Adv Water Resour 28:161–180CrossRefGoogle Scholar
  12. 12.
    Gray WG, Miller CT (2009) Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 5. Single-fluid-phase transport. Adv Water Resour 32:681–711CrossRefGoogle Scholar
  13. 13.
    Gray WG, Schrefler BA (2007) Analysis of the solid stress tensor in multiphase porous media. Int J Num Anal Methods Geomech 31:541–581CrossRefMATHGoogle Scholar
  14. 14.
    Hawkins-Daarud A, van der Zee KG, Oden JT (2012) Numerical simulation of a thermodynamically consistent four-species tumor growth model. Int J Numer Methods Biomed Eng 28:3–24CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Hughes TJR, Franca LP, Mallet (1987) A new finite element formulation for computational fluid dynamics: VI. Convergence analysis of the generalized SUPG formulation for linear time-dependent multidimensional advective-diffusive systems. Comput Methods Appl Mech Eng 63:97–112CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Jackson AS, Miller CT, Gray WG (2009) Thermodynamically constrained averaging theory approach for modelling flow and transport phenomena in porous medium systems: 6. Two-fluid-phase flow. Adv Water Resour 32:779–795CrossRefGoogle Scholar
  17. 17.
    Johnson GC, Bamman DJ (1984) A discussion of stress rates in finite deformation problems. Int J Solids Struct 8:725–737CrossRefGoogle Scholar
  18. 18.
    Leverett MC, Lewis WB, True ME (1942) Dimensional-model studies of oil-field behavior. Pet. Technol. Tech. Paper 1413, January: 175–193Google Scholar
  19. 19.
    Lewis RW, Schrefler BA (1998) The finite element method in the static and dynamic deformation and consolidation of porous media. Wiley, ChichesterMATHGoogle Scholar
  20. 20.
    Lowengrub JS, Frieboes HB, Jin F, Chuang Y-L, Li X, Macklin P, Wise SM, Cristini V (2010) Nonlinear modeling of cancer: bridging the gap between cells and tumors. Nonlinearity 23(1):R1–R9. doi: 10.1088/0951-7715/23/1/R01 CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Molina JR, Hayashi Y, Stephens C, Georgescu M-M (2010) Invasive glioblastoma cells acquire stemness and increased Akt activation. Neoplasia 12(6):453–463Google Scholar
  22. 22.
    Murthy V, Valliappan S, Khalili-Naghadeh N (1989) Time step constraints in finite element analysis of the Poisson type equation. Comput Struct 31:269–273CrossRefGoogle Scholar
  23. 23.
    Parker JC, Lenhard RJ (1987) A model for hysteretic constitutive relations governing multiphase flow. 1. Saturation-pressure relations. Water Resour Res 23:2187–2196CrossRefGoogle Scholar
  24. 24.
    Parker JC, Lenhard RJ (1990) Determining three-phase permeability saturation-pressure relations from two-phase measurements. J Petroleum Sci Eng 4:57–65CrossRefGoogle Scholar
  25. 25.
    Preisig M, Prévost JH (2011) Stabilization procedures in coupled poromechanics problems: a critical assessment. Int J Numer Anal Methods Geomech 35(11):1207–1225CrossRefGoogle Scholar
  26. 26.
    Preziosi L, Vitale G (2011) A multiphase model of tumour and tissue growth including cell adhesion and plastic re-organisation. Math Models Methods Appl Sci 21(9):1901–1932CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Rank E, Katz C, Werner H (1983) On the importance of the discrete maximum principle in transient analysis using finite element methods. Int J Num Meth Eng 19:1771–1782CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Roose T, Chapman SJ, Maini PK (2007) Mathematical models of avascular tumor growth. SIAM Rev 49(2):179–208CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Salomoni V, Schrefler BA (2005) A CBS-type stabilizing algorithm for the consolidation of saturated porous media. Int J Numer Methods Eng 63:502–527CrossRefMATHGoogle Scholar
  30. 30.
    Schrefler BA, Zhan XY, Simoni L (1995) A coupled model for water flow, airflow and heat flow in deformable porous media. Int J Heat Fluid Flow 5:531–547CrossRefMATHGoogle Scholar
  31. 31.
    Sciumè G, Gray WG, Ferrari M, Decuzzi P, Schrefler BA (2013) On computational modeling in tumor growth. Arch Comput Methods Eng. doi: 10.1007/s11831-013-9090-8
  32. 32.
    Sciumè G, Shelton SE, Gray WG, Miller CT, Hussain F, Ferrari M, Decuzzi P, Schrefler BA (2013) A multiphase model for three dimensional tumor growth. New J Phys 15: 15005. doi: 10.1088/1367-2630/15/1/015005
  33. 33.
    Sciumè G, Shelton SE, Gray WG, Miller CT, Hussain F, Ferrari M, Decuzzi P, Schrefler BA (2012) Tumor growth modeling from the perspective of multiphase porous media mechanics. Mol Cell Biomech 9(3):193–212 Google Scholar
  34. 34.
    Turska E, Wisniewski K, Schrefler BA (1994) Error propagation of staggered solution procedures for transient problems. Comput Methods Appl Mech Eng 144:177–188Google Scholar
  35. 35.
    van Genuchten MT (1980) A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci Soc Am J 44:892–898CrossRefGoogle Scholar
  36. 36.
    Wise SM, Lowengrub JS, Frieboes HB, Cristini V (2008) Three-dimensional multispecies nonlinear tumor growth model and numerical method. J Theor Biol 253:524–543CrossRefMathSciNetGoogle Scholar
  37. 37.
    Zavarise G, Wrigger P, Schrefler BA (1995) On augmented Lagrangian algorithms for thermomechanical contact problems with friction. Int J Num Methods Eng 38:2929–2949CrossRefMATHGoogle Scholar
  38. 38.
    Zienkiewicz OC, Taylor RL (2000) The finite element method. Solid mechanics, vol 2. Butterworth Heinemann, OxfordGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • G. Sciumè
    • 1
  • W. G. Gray
    • 2
  • F. Hussain
    • 3
    • 4
  • M. Ferrari
    • 4
    • 5
  • P. Decuzzi
    • 4
    • 6
  • B. A. Schrefler
    • 1
    • 4
  1. 1.Department of Civil, Environmental and Architectural EngineeringUniversity of PaduaPaduaItaly
  2. 2.Department of Environmental Sciences and EngineeringUniversity of North Carolina at Chapel HillChapel HillUSA
  3. 3.Department of Mechanical EngineeringTexas Tech UniversityLubbockUSA
  4. 4.Department of NanomedicineThe Methodist Hospital Research InstituteHoustonUSA
  5. 5.Department of MedicineWeill Cornell Medical College of Cornell UniversityNew YorkUSA
  6. 6.Department of Translational ImagingThe Methodist Hospital Research InstituteHoustonUSA

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