Publication Cover
Applicable Analysis
An International Journal
Volume 101, 2022 - Issue 15
385
Views
1
CrossRef citations to date
0
Altmetric
Research Article

Existence and uniqueness results on biphasic mixture model for an in-vivo tumor

, ORCID Icon &
Pages 5442-5468 | Received 15 May 2020, Accepted 12 Feb 2021, Published online: 03 Mar 2021

References

  • Roose T, Jonathan Chapman S, Maini PK. Mathematical models of avascular tumor growth. SIAM Rev. 2007;49(2):179–208.
  • Preziosi L. Cancer modelling and simulation. Chapman and Hall/CRC Press, London 2003.
  • Netti PA, Baxter LT, Boucher Y, et al. Macro-and microscopic fluid transport in living tissues: application to solid tumors. AIChE J. 1997;43(3):818–834.
  • Baxter LT, Jain RK. Transport of fluid and macromolecules in tumors. I. Role of interstitial pressure and convection. Microvasc Res. 1989;37(1):77–104.
  • Araujo RP, Sean McElwain DL. A history of the study of solid tumour growth: the contribution of mathematical modelling. Bull Math Biol. 2004;66(5):1039–1091.
  • Greenspan HP. On the growth and stability of cell cultures and solid tumors. J Theor Biol. 1976;56(1):229–242.
  • Ambrosi D, Preziosi L. On the closure of mass balance models for tumor growth. Math Models Methods Appl Sci. 2002;12(05):737–754.
  • Byrne H, Preziosi L. Modelling solid tumour growth using the theory of mixtures. Math Med Biol. 2003;20(4):341–366.
  • Preziosi L, Tosin A. Multiphase modelling of tumour growth and extracellular matrix interaction: mathematical tools and applications. J Math Biol. 2009;58(4):625–656.
  • Girault V, Rivière B. DG approximation of coupled Navier–Stokes and darcy equations by Beaver–Joseph–Saffman interface condition. SIAM J Numer Anal. 2009;47(3):2052–2089.
  • Çeşmelioğlu A, Riviere B. Analysis of time-dependent Navier–Stokes flow coupled with darcy flow. J Numer Math. 2008;16(4):249–280.
  • Cesmelioglu A, Girault V, Riviere B. Time-dependent coupling of Navier–Stokes and darcy flows. ESAIM: Math Model Numer Anal. 2013;47(2):539–554.
  • Kohr M, Raja Sekhar GP. Existence and uniqueness result for two-dimensional porous media flows with porous inclusions based on Brinkman equation. Eng Anal Bound Elem. 2007;31(7):604–613.
  • Kohr M, Raja Sekhar GP, Ului EM, et al. Two-dimensional Stokes–Brinkman cell model a boundary integral formulation. Appl Anal. 2012;91(2):251–275.
  • Muha B, Čanić S. Existence of a solution to a fluid-multi-layered-structure interaction problem. J Differ Equ. 2014;256(2):658–706.
  • Muha B, Čanić S. Existence of a weak solution to a fluid–elastic structure interaction problem with the Navier slip boundary condition. J Differ Equ. 2016;260(12):8550–8589.
  • Badia S, Quaini A, Quarteroni A. Coupling biot and Navier–Stokes equations for modelling fluid–poroelastic media interaction. J Comput Phys. 2009;228(21):7986–8014.
  • Bukač M, Yotov I, Zakerzadeh R, et al. Partitioning strategies for the interaction of a fluid with a poroelastic material based on a Nitsche's coupling approach. Comput Methods Appl Mech Eng. 2015;292:138–170.
  • Cesmelioglu A. Analysis of the coupled Navier–Stokes/biot problem. J Math Anal Appl. 2017;456(2):970–991.
  • Showalter RE. Diffusion in poro-elastic media. J Math Anal Appl. 2000;251(1):310–340.
  • Alam M, Dey B, Raja Sekhar GP. Mathematical analysis of hydrodynamics and tissue deformation inside an isolated solid tumor. Theor Appl Mech. 2018;45(2):253–278.
  • Alam M, Dey B, Raja Sekhar GP. Mathematical modeling and analysis of hydroelastodynamics inside a solid tumor containing deformable tissue. ZAMM-J Appl Math Mech/Z Angew Math Mech. 2019;99(5):e201800223.
  • Barry SI, Aldis GK. Comparison of models for flow induced deformation of soft biological tissue. J Biomech. 1990;23(7):647–654.
  • Holmes MH, Mow VC. The nonlinear characteristics of soft gels and hydrated connective tissues in ultrafiltration. J Biomech. 1990;23(11):1145–1156.
  • Ateshian GA, Weiss JA. Anisotropic hydraulic permeability under finite deformation. J Biomech Eng. 2010;132(11):111004.
  • Federico S, Herzog W. On the anisotropy and inhomogeneity of permeability in articular cartilage. Biomech Model Mechanobiol. 2008;7(5):367–378.
  • Stefano SDi, Ramírez-Torres A, Penta R, et al. Self-influenced growth through evolving material inhomogeneities. Int J Non Linear Mech. 2018;106:174–187.
  • Giverso C, Preziosi L. Influence of the mechanical properties of the necrotic core on the growth and remodelling of tumour spheroids. Int J Non Linear Mech. 2019;108:20–32.
  • Dey B, Raja Sekhar GP. Hydrodynamics and convection enhanced macromolecular fluid transport in soft biological tissues: application to solid tumor. J Theor Biol. 2016;395:62–86.
  • Dey B, Raja Sekhar GP, Kanti Mukhopadhyay S. In vivo mimicking model for solid tumor towards hydromechanics of tissue deformation and creation of necrosis. J Biol Phys. 2018; 44(3),361–400.
  • Damiano ER, Duling BR, Ley K, et al. Axisymmetric pressure-driven flow of rigid pellets through a cylindrical tube lined with a deformable porous wall layer. J Fluid Mech. 1996;314:163–189.
  • Dey B, Raja Sekhar GP. A theoretical study on the elastic deformation of cellular phase and creation of necrosis due to the convection reaction process inside a spherical tumor. Int J Biomathematics. 2016;9(06):1650095.
  • Rajagopal KR. On a hierarchy of approximate models for flows of incompressible fluids through porous solids. Math Models Methods Appl Sci. 2007;17(02):215–252.
  • Michael Lai W, Mow VC. Drag-induced compression of articular cartilage during a permeation experiment. Biorheology. 1980;17(1–2):111–123.
  • Klanchar M, Tarbell JM. Modeling water flow through arterial tissue. Bull Math Biol. 1987;49(6):651–669.
  • Parker KH, Mehta RV, Caro CG. Steady flow in porous, elastically deformable materials. J Appl Mech. 1987;54(4):794–800.
  • Ehrhardt M. An introduction to fluid-porous interface coupling. Prog Comput Phys (PiCP), Coupled Fluid Flow Energy, Biol Environ Res. 2010;2:3–12.
  • Grosan T, Postelnicu A, Pop I. Brinkman flow of a viscous fluid through a spherical porous medium embedded in another porous medium. Transp Porous Media. 2010;81(1):89.
  • Hou JS, Holmes MH, Lai WM, et al. Boundary conditions at the cartilage-synovial fluid interface for joint lubrication and theoretical verifications. J Biomech Eng. 1989;111(1):78–87.
  • Young Y-N, Mori Y, Miksis MJ. Slightly deformable darcy drop in linear flows. Phys Rev Fluids. 2019;4(6):063601.
  • Temam R. Navier–Stokes equations: theory and numerical analysis, volume 343. AMS Chelsea Publishing, Providence, RI 2001.
  • Cao Y, Chen S, Meir AJ. Steady flow in a deformable porous medium. Math Methods Appl Sci. 2014;37(7):1029–1041.
  • Martin RH. Nonlinear operators and differential equations in Banach spaces. Wiley-Interscience, New York 1976.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.