Title: Implementation and Validation of Finite Element Framework for Passive and Active Membrane Transport in Deformable Multiphasic Models of Biological Tissues and Cells
Description: The chondrocyte is the only cell type in articular cartilage, and its role is to maintain cartilage integrity by synthesizing and releasing macromolecules into the extracellular matrix (ECM) or breaking down its damaged constituents (Stockwell, 1991). The two major constituents of the ECM are type II collagen and aggrecans (aggregating proteoglycans). Proteoglycans have a high negative charge which attracts cations and increases the osmolarity, while also lowering the pH of the interstitial fluid. The fibrillar collagen matrix constrains ECM swelling that results from the Donnan osmotic pressure produced by proteoglycans (Wilkins et al., 2000). Activities of daily living produce fluctuating mechanical loads on the tissue which also alter the mechano-electro-chemical environment of chondrocytes embedded in the ECM. These conditions affect the physiology and function of chondrocytes directly (Wilkins et al., 2000; Guilak et al., 1995; Guilak et al., 1999). Relatively few studies of in situ chondrocyte mechanics have been reported in the biomechanics literature, in contrast to the more numerous experimental studies of the mechanobiological response of live cartilage explants to various culture and loading conditions.

Analyses of chondrocyte mechanics can shed significant insights in the interpretation of experimental mechanobiological responses. Predictions from carefully formulated biomechanics models may also generate hypotheses about the mechanisms that transduce signals to chondrocytes via mechanical, electrical and chemical pathways. Therefore, computational tools that can model the response of cells, embedded within a charged hydrated ECM, to various loading conditions may serve a valuable role in mechanobiological studies. Computational modeling has become a necessary tool to study biomechanics with complex geometries and mechanisms (De et al., 2010). Usually, theoretical and computational models of cell physiology and biophysics are formulated in 1D, deriving solutions by solving ordinary differential equations, such as cell volume regulation (Tosteson and Hoffman, 1960), pH regulation (Boron and De Weer, 1976), and Ca2+ regulation (Schuster et al., 2002). Cell modeling software, such as The Virtual Cell (vcell.org Moraru et al. (2008)), analyze stationary cell shapes and isolated cells.

To model the cell-ECM system while accounting for ECM deformation, the fibrillar nature of the ECM, interstitial fluid flow, solute transport, and electrical potential arising from Donnan or streaming effects, we adopt the multiphasic theory framework (Ateshian, 2007). This framework serves as the foundation of multiphasic analyses in the open source finite element software FEBio (Maas et al., 2012; Ateshian et al., 2013), which was developed specifically for biomechanics and biophysics, and offers a suitable environment to solve complex models of cell-ECM interactions in 3D. In the studies proposed here, we will extend the functionality of FEBio to further investigate the cell-ECM system. These extensions and studies are summarized in the following chapters: Chapter 1: This introductory chapter provides the general background and specific aims of this dissertation. Chapter 2: Cell-ECM interactions depend significantly on the ECM response to external loading conditions. For fibrillar soft tissues such as articular cartilage, it has been shown that modeling the ECM using a continuous fiber distribution produces much better agreement with experimental measurements of its response to loading.

However, evaluating the stress and elasticity tensors for such distributions is computationally very expensive in a finite element analysis. In this aim we develop a new numerical integration scheme to calculate these tensors more efficiently than standard techniques, only accounting for fibers that are in tension. Chapter 3: Cell-ECM interactions also depend significantly on accurate modeling of selective transport across the cell membrane. However, the thickness of this membrane is typically three orders of magnitude smaller than the cell size, which poses significant numerical challenges when modeling the membrane using the finite element method, such as element locking. To date, no existing finite element software offers a multiphasic membrane element. In this aim, we formulate and implement a new membrane element in FEBio, which can accommodate fluid and solute transport within the biphasic and multiphasic framework, to model passive and selective transport across the cell membrane.

Chapter 4: This aim extends Aim 2 to incorporate reactions across multiphasic membrane elements in FEBio, to model the conformational reactions of cell membrane transporters, such as carrier-mediated transporters and membrane pumps. This implementation is verified against standard models for the regulation of cell volume, pH, and Ca2+. Chapter 5: This final chapter provides a summary of the advances contributed in this dissertation, along with suggestions for future aims related to the topics covered here. With the completion of these aims, we have extended the modeling capabilities for cell physiology and mechanobiology to more complex multicellular systems embedded within their ECM, while subjected to a range of varying mechanical, electrical or chemical loading conditions.