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.

在本文中，我们提出了一个数学模型，该模型描述了嵌入或邻近健康（正常）组织的实体瘤内的流体动力学和变形力学。假设肿瘤和正常组织区域是可变形的，混合理论适用于每个区域的流体流动和组织变形力学的质量和动量平衡方程。动量平衡方程通过相（流体和固体）之间相互作用的力耦合。正常速度、位移和正常应力的连续性以及 Beaver-Joseph-Saffman 条件被施加在肿瘤和组织区域之间的界面处。生理输运参数（例如水力电阻率或渗透率）被假定为非均质且依赖于变形，这使得模型非线性。我们使用 Galerkin 和弱收敛方法建立了弱解的存在性。我们进一步表明，该解决方案是独一无二的，并且持续依赖于给定的数据。