Interaction between a fluid and a cellular biological medium in vessels and porous media

Fluid flow through porous media and networks of interconnected vessels that contain deposits of biological origin attached to the surface of the pores (or vessels) is of key importance in several natural phenomena and technological processes. Some notable examples include the flow of aqueous solutions through porous media during the formation of microbial biofilms, the flow of blood in a circulatory system afflicted by thrombi and atheromatous plaques, and the flow of nutrient solutions through porous scaffolds during the in vitro construction of artificial tissues from stem cells. In all the aforementioned processes, the attached cellular biological media alter the geometrical and topological properties of the porous structure and, thus, affect the local flow field as well as the averaged characteristics of flow at the system level (e.g. flow rate, pressure drop). In turn, at the pore level, fluid flow affects the morphology and structure of the cellular biological media as well as the transport of dissolved and particulate matter past and through them. Of particular importance are the forces, which are exerted by the flowing fluid on the attached biomaterials and can cause rupture and even detachment of fragments or of entire formations depending on their strength and cohesion. The analysis of the interactions between the flowing fluid and the attached cellular biological media in the physical systems under consideration is challenging. First and foremost, microbial biofilms, tissues, thrombi, and other cellular biological media are multiphase complex systems, which consist of biological cells along with their extracellular matrix (ECM), and exhibit a dynamically evolving and highly organized hierarchical structure. In this regard, an accurate description of momentum transport within such biomaterials is a difficult task by itself. Furthermore, in the context of a network of pores (or vessels), a high degree of complexity stems also from: (a) the three-dimensional structure of the network of pores (or vessels), (b) the flow regime, and (c) the variability in size, shape, position and intrinsic material properties of the attached biomaterials. In the laboratory setting, significant reduction in the degree of complexity is usually achieved with the development and use of simplified experimental systems representing a single pore or vessel. Such systems include microfluidic devices and flow chambers with prescribed pore geometry, well defined flow regime (shear driven, pressure driven or forced flow), and the potential of visualization and quantification of the morphology of the biomaterial. In parallel, the development of rigorous mathematical models is required for thorough investigation, better understanding, and quantification of fluid-biomaterial interactions in the context of both simplified and more complex real systems. The scope of this dissertation is the theoretical and computational modeling of fluidbiomaterial interactions at two levels. At the biomaterial level, the objective is to develop a theoretical model for the description of momentum transfer within a poroelastic biomaterial, taking into account the interaction between the extracellular fluid and the solid skeleton that consists of cells and ECM (Chapters 1 & 2). At the vessel level, the objective is to develop a computational model of the interaction between a poroelastic biomaterial, which is attached to the vessel wall, and a Newtonian free-fluid that flows through the vessel (Chapters 3 & 4). The first chapter presents an overview of modeling momentum transfer in cellular biological media with respect to two fundamental issues: (i) the formulation of balance equations and constitutive relations at the macroscopic scale, and (ii) the proper connection of the macroscopic constitutive parameters with geometrical and physicochemical properties of the system at the molecular- and cellular- scales. With regard to the governing equations, significant work has been done with single scale approaches (e.g. monophasic models, mixture theory), whereas upscaling methods (e.g. homogenization, spatial averaging) or multiscale equation-free approaches have received much less attention. The underlying concepts, strengths and limitations of each approach, as well as examples of their implementation in the field of biomaterials are presented. In the second chapter, a theoretical model is developed for the description of momentum transfer in a biphasic fluid-solid system at a scale of observation with characteristic length much larger than those of the constituent phases. A continuum based formulation of momentum transport in a fluid-solid system at the finer spatial scale is used as starting point, and then the method of local spatial averaging with a weight function is implemented in order to establish the partial differential equations that describe the dynamics of fluid flow and matrix deformation at the coarser scale. The hypotheses and conditions which delimit the domain of validity of the upscaled equations are clearly stated. In the special case of a homogeneous medium and under certain other conditions, the derived equations become similar to those which are postulated in the theory of interacting continua and Biot's theory of poroelasticity. In addition, closure problems are developed for the consistent calculation of the parameters which appear in the upscaled constitutive relations for the stress tensors and the fluid structure interaction forces. As a validation check, the developed closure problem for the static hydraulic permeability tensor is solved numerically in the context of a periodic unit cell model. Excellent agreement is observed with results from analytical and numerical solutions published in the literature for regular and random arrays of spheres. The third chapter is concerned with the development of a benchmark problem for fluidbiomaterial interactions in a vessel. To this end, an analytical solution is developed for the problem of plane Couette-Poiseuille flow past a poroelastic layer. A novel feature of the studied problem is that the compressibility of the poroelastic layer is taken into account and, thus, the solid stress problem becomes two-dimensional, albeit the flow problem remains unidirectional. The lateral stresses and displacements are caused by the action of pressure. Existing analytical solutions for Poiseuille flow past a rigid porous layer and an incompressible poroelastic layer are found to be special cases of the solution presented here. The usefulness of the established analytical solution is twofold. First and foremost, the established analytical solution might serve as a decent benchmark problem for the verification of numerical codes. In this regard, the problem under consideration is also solved with a finite element method and the numerical calculations are compared to the exact analytical results. Second, the analytical solution can be used to gain insight into how several system parameters affect the process. To this end, a systematic parametric study is conducted to examine the effects of material properties and characteristics of the flow regime on the fields of fluid velocity and solid stress as well as on averaged quantities, such as the flow rate through the poroelastic layer and the interaction forces exerted on material interfaces. The fourth and final chapter presents results from computer simulations of plane Poiseuille flow past and through a semi-elliptical poroelastic biomaterial, which is attached to the surface of a straight vessel. Fluid flow in the clear fluid region is described by the NavierStokes equations, and momentum transfer within the biomaterial is described by the upscaled biphasic equations established in the second chapter. The governing equations are solved numerically with a finite element method using both structured and unstructured meshes (for reasons of comparison). The effect of the Reynolds and Darcy number that characterize the flow past and through the biomaterial, respectively, is investigated for obstacles with different configuration with respect to flow (semicircle, oblate semi-ellipse, prolate semi-ellipse). The distribution of the von Mises stress within the biomaterial is determined, and also, the drag and lift forces exerted by the fluid on the biomaterial are calculated. Polynomial regression analysis has shown that the drag and lift forces can be described very satisfactorily with a finite sum of power law functions of Reynolds. This work is expected to be useful in the analysis of fracture initiation within and detachment from biomaterials like biofilms, thrombi and other which are found attached to the walls of porous media and networks of interconnected vessels