We present a dynamic vascular tumor model combining a multiphase porous medium framework for avascular tumor growth in a consistent Arbitrary Lagrangian Eulerian formulation and a novel approach to incorporate angiogenesis. The multiphase model is based on Thermodynamically Constrained Averaging Theory and comprises the extracellular matrix as a porous solid phase and three fluid phases: (living and necrotic) tumor cells, host cells and the interstitial fluid. Angiogenesis is modeled by treating the neovasculature as a proper additional phase with volume fraction or blood vessel density. This allows us to define consistent inter-phase exchange terms between the neovasculature and the interstitial fluid. As a consequence, transcapillary leakage and lymphatic drainage can be modeled. By including these important processes we are able to reproduce the increased interstitial pressure in tumors which is a crucial factor in drug delivery and, thus, therapeutic outcome. Different coupling schemes to solve the resulting five-phase problem are realized and compared with respect to robustness and computational efficiency. We find that a fully monolithic approach is superior to both the standard partitioned and a hybrid monolithic-partitioned scheme for a wide range of parameters. The flexible implementation of the novel model makes further extensions (e.g., inclusion of additional phases and species) straightforward.

The aim of this work is to develop a novel computational approach to facilitate the modeling of angiogenesis during tumor growth. The preexisting vasculature is modeled as a 1D inclusion and embedded into the 3D tissue through a suitable coupling method, which allows for nonmatching meshes in 1D and 3D domain. The neovasculature, which is formed during angiogenesis, is represented in a homogenized way as a phase in our multiphase porous medium system. This splitting of models is motivated by the highly complex morphology, physiology, and flow patterns in the neovasculature, which are challenging and computationally expensive to resolve with a discrete, 1D angiogenesis and blood flow model. Moreover, it is questionable if a discrete representation generates any useful additional insight. By contrast, our model may be classified as a hybrid vascular multiphase tumor growth model in the sense that a discrete, 1D representation of the preexisting vasculature is coupled with a continuum model describing angiogenesis. It is based on an originally avascular model which has been derived via the thermodynamically constrained averaging theory. The new model enables us to study mass transport from the preexisting vasculature into the neovasculature and tumor tissue. We show by means of several illustrative examples that it is indeed capable of reproducing important aspects of vascular tumor growth phenomenologically.

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Summary Simulation approaches for fluid‐structure‐contact interaction, especially if requested to be consistent even down to the real contact scenarios, belong to the most challenging and still unsolved problems in computational mechanics. The main challenges are 2‐fold—one is to have a correct physical model for this scenario, and the other is to have a numerical method that is capable of working and being consistent down to a zero gap. Moreover, when analyzing such challenging setups of fluid‐structure interaction, which include contact of submersed solid components, it gets obvious that the influence of surface roughness effects is essential for a physical consistent modeling of such configurations. To capture this system behavior, we present a continuum mechanical model that is able to include the effects of the surface microstructure in a fluid‐structure‐contact interaction framework. An averaged representation for the mixture of fluid and solid on the rough surfaces, which is of major interest for the macroscopic response of such a system, is introduced therein. The inherent coupling of the macroscopic fluid flow and the flow inside the rough surfaces, the stress exchange of all contacting solid bodies involved, and the interaction between fluid and solid are included in the construction of the model. Although the physical model is not restricted to finite element–based methods, a numerical approach with its core based on the cut finite element method, enabling topological changes of the fluid domain to solve the presented model numerically, is introduced. Such a cut finite element method–based approach is able to deal with the numerical challenges mentioned above. Different test cases give a perspective toward the potential capabilities of the presented physical model and numerical approach.

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We present a computational model for the interaction of surface-and volume-bound scalar transport and reaction processes with a deformable porous medium. The application in mind is pericellular proteolysis, i.e. the dissolution of the solid phase of the extracellular matrix (ECM) as a response to the activation of certain chemical species at the cell membrane and in the vicinity of the cell. A poroelastic medium model represents the extra cellular scaffold and the interstitial fluid flow, while a surface-bound transport model accounts for the diffusion and reaction of membrane-bound chemical species. By further modelling the volumebound transport, we consider the advection, diffusion and reaction of sequestered chemical species within the extracellular scaffold. The chemo-mechanical coupling is established by introducing a continuum formulation for the interplay of reaction rates and the mechanical state of the ECM. It is based on known experimental insights and theoretical work on the thermodynamics of porous media and degradation kinetics of collagen fibres on the one hand and a damage-like effect of the fibre dissolution on the mechanical integrity of the ECM on the other hand. The resulting system of partial differential equations is solved via the finite-element method. To the best of our knowledge, it is the first computational model including contemporaneously the coupling between (i) advection-diffusion-reaction processes, (ii) interstitial flow and deformation of a porous medium, and (iii) the chemo-mechanical interaction

In this paper, we propose a finite element-based immersed method to treat the mechanical coupling between a deformable porous medium model (PM) and an immersed solid model (ISM). The PM is formulated as a homogenized, volume-coupled two-field model, comprising a nearly incompressible solid phase that interacts with an incompressible Darcy-Brinkman flow. The fluid phase is formulated with respect to the Lagrangian finite element mesh, following the solid phase deformation. The ISM is discretized with an independent Lagrangian mesh and may behave arbitrarily complex (it may, eg, be compressible, grow, and perform active deformations). We model two distinct types of interactions, namely, (1) the immersed fluid-structure interaction (FSI) between the ISM and the fluid phase in the PM and (2) the immersed structure-structure interaction (SSI) between the ISM and the solid phase in the PM. Within each time step, we solve both FSI and SSI, employing strongly coupled partitioned schemes. This novel finite element method establishes a main building block of an evolving computational framework for modeling and simulating complex biomechanical problems, with focus on key phenomena during cell migration. Cell movement is strongly influenced by mechanical interactions between the cell body and the surrounding tissue, ie, the extracellular matrix (ECM). In this context, the PM represents the ECM, ie, a fibrous scaffold of structural proteins interacting with interstitial flow, and the ISM represents the cell body. The FSI models the influence of fluid drag, and the SSI models the force transmission between cell and ECM at adhesions sites.

Summary In finite element analysis of volume coupled multiphysics, different meshes for the involved physical fields are often highly desirable in terms of solution accuracy and computational costs. We present a general methodology for volumetric coupling of different meshes within a monolithic solution scheme. A straightforward collocation approach is compared to a mortar‐based method for nodal information transfer. For the latter, dual shape functions based on the biorthogonality concept are used to build the projection matrices, thus further reducing the evaluation costs. We give a detailed explanation of the integration scheme and the construction of dual shape functions for general first‐order and second‐order Langrangian finite elements within the mortar method, as well as an analysis of the conservation properties of the projection operators. Moreover, possible incompatibilities due to different geometric approximations of curved boundaries are discussed. Numerical examples demonstrate the flexibility of the presented mortar approach for arbitrary finite element combinations in two and three dimensions and its applicability to different multiphysics coupling scenarios. Copyright © 2016 John Wiley & Sons, Ltd.