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

Abstract

In this article, we propose a mathematical model that describes hydrodynamics and deformation mechanics within a solid tumor which is embedded in or adjacent to a healthy (normal) tissue. The tumor and normal tissue regions are assumed to be deformable and the theory of mixtures is adapted to mass and momentum balance equations for fluid flow and tissue deformation mechanics in each region. The momentum balance equations are coupled via forces that interact between the phases (fluid and solid). Continuity of normal velocities, displacements, and normal stresses along with the Beaver–Joseph–Saffman condition are imposed at the interface between the tumor and tissue regions. The physiological transport parameters (such as hydraulic resistivity or permeability) are assumed to be heterogeneous and deformation dependent which makes the model nonlinear. We establish the existence of a weak solution using Galerkin and weak convergence methods. We show further that the solution is unique and depends continuously on the given data.

Keywords:

• In-vivo tumor
• biphasic mixture theory
• weak formulation
• Galerkin method
• weak convergence

2010 Mathematics Subject Classifications:

• 76Zxx
• 74F10
• 35Q74
• 35J66
• 35D30

Acknowledgments

First author acknowledges the Department of Science and Technology, Govt. of India, for financial support, grant No. DST/INSPIRE Fellowship/2015/IF150654, 30 Oct 2018 for PhD at Indian Institute of Technology (IIT) Kharagpur, India-721302. H. M. Byrne acknowledges the support under SGRIP International Faculty Outreach program which allowed her to visit Department of Mathematics, IIT Kharagpur, India.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Hereafter, throughout the manuscript, we use $K_o=K_o(\varsigma )$ and $K_{in}=K_{in}(\varsigma ),$ where $\varsigma =\{\begin{array}{ll}{\varsigma }^o,& \text{in}{\mathrm{\Omega }}_o\\ {\varsigma }^{in},& \text{in}{\mathrm{\Omega }}_{in}.\end{array}$

2 Please refer Appendix section for the definition of function spaces and preliminaries.

3 Similar approximate structure is used by Klanchar and Tarbell [35] in case of arterial tissue.

4 It is indeed expected to have such assumptions while dealing with uniqueness in case of nonlinear problems (e.g. please refer [10,42]).

5 This is the smallness condition on the fluid phase velocity in the tumor region.

Additional information

Funding

First author acknowledges the Department of Science and Technology, Govt. of India, for financial support, grant No. DST/INSPIRE Fellowship/2015/IF150654, 30 Oct 2018 for PhD at Indian Institute of Technology (IIT) Kharagpur, India-721302.