Research

Reactive Mixtures

Mixture theory, which can combine continuum theories for the motion and deformation of solids and fluids with general principles of chemistry, is well suited for modeling the complex responses of biological tissues, including tissue growth and remodeling, tissue engineering, mechanobiology of cells and a variety of other active processes. A comprehensive presentation of the equations of reactive mixtures of charged solid and fluid constituents has been lacking in the biomechanics literature. Therefore, a theoretical component of our studies focused on providing the conservation laws and entropy inequality, as well as interface jump conditions, for reactive mixtures consisting of a constrained solid mixture and multiple fluid constituents.

In this framework, the constituents were assumed to be intrinsically incompressible and could carry an electrical charge. The interface jump condition on the mass flux of individual constituents was shown to define a surface growth equation that predicts deposition or removal of material from the solid matrix, complementing the description of volume growth described by the conservation of mass. The formulation postulated that the reference configuration of the solid matrix was time-invariant. State variables were defined that could account for solid matrix growth and remodeling. Constitutive constraints were also provided on the stresses and momentum supplies of the various constituents, as well as the interface jump conditions for the electrochemical potential of the fluids. Simplifications appropriate for biological tissues were proposed, which help reduce the governing equations into a more practical format. It was shown that explicit mechanisms of growth-induced residual stresses can be predicted in this framework due to alterations in the fixed-charge density of solid matrix-bound molecules.

Growth by Cell Division

In a follow-up study, a framework was formulated within the theory of mixtures for continuum modeling of biological tissue growth that explicitly addressed cell division, using a homogenized representation of cells and their extracellular matrix (ECM). The model relied on the description of the cell as containing a solution of water and osmolytes, and having a porous solid matrix. The division of a cell into two nearly identical daughter cells was modeled as the doubling of the cell solid matrix and osmolyte content, producing an increase in water uptake via osmotic effects. This framework was also generalized to account for the growth of ECM-bound molecular species that impart a fixed charge density (FCD) to the tissue, such as proteoglycans. This FCD similarly induced osmotic effects, resulting in extracellular water uptake and osmotic pressurization of the ECM interstitial fluid, with concomitant swelling of its solid matrix.

Interstitial Fluid Pressurization

Over the last two decades, considerable progress has been reported in the field of cartilage mechanics that impacts our understanding of the role of interstitial fluid pressurization on cartilage lubrication. In our laboratory, theoretical and experimental studies have demonstrated that the interstitial fluid of cartilage pressurizes considerably under loading, potentially supporting most of the applied load under various transient or steady-state conditions. The fraction of the total load supported by fluid pressurization has been called the interstitial fluid load support, and our experiments have shown that this load support can be predicted from mixture models of cartilage very accurately. A significant component of the doctoral dissertations of Drs. Michael A. Soltz and Seonghun Park have focused on this topic.

Correlation with Frictional Response

Our experimental studies have demonstrated for the first time that the friction coefficient of cartilage correlates negatively with this variable, achieving remarkably low values when the fluid load support is greatest. A theoretical framework that embodies this relationship has been validated against experiments, predicting and explaining various outcomes, and demonstrating that a low friction coefficient can be maintained for prolonged loading durations under normal physiological function. These studies were performed as part of Dr. Ramaswamy Krishnan's dissertation, and continued by Mr. Matteo Caligaris.

Boundary Lubrication by Chondroitin Sulfate

Furthermore, we have shown that interstitial fluid load support and the frictional response of cartilage may be compromised significantly by degradative changes that accompany osteoarthritis. We discovered that chondroitin sulfate, a natural constituent of cartilage, also contributes to reducing friction in cartilage via a boundary lubrication mechanism. The role of this molecule was deduced from a two-pronged approach: Enzymatic degradation of the chondroitin sulfate of cartilage was shown to increase the equilibrium friction coefficient, and supplementation of the tissue bath with chondroitin sulfate was shown to decrease it. These studies were the main topic of Dr. Ines M. Basalo's dissertation.

Modeling the Cell Response to Osmotic Loading Using Mixture Theory

The fundamental physical mechanisms of solute and water transport across the cell membrane have long been studied in the field of cell membrane biophysics, and there exist a number of formalisms aiming to characterize transport through membrane channels and/or lipid bilayers. These formalisms include a one-parameter (solute permeability) model, a classic two-parameter (water and solute permeability) model and a commonly used three-parameter model (water and solute permeability and a solute-solvent interaction term) developed by Kedem and Katchalsky. The parameters of interest (permeabilities) can be extracted from the formulation when the cell volume change is measured in the experiment, assuming that the cell volume change is due purely to the volume of the water (and solute) that enters or exudes from the cell. These formulations have been derived from the general theory of irreversible thermodynamics.

In more recent decades, the field of rational mechanics has addressed problems in the mixture of fluids (solvent and solutes) as well as the mixture of fluids and solids (deformable porous media). In our studies we have applied the theory of mixtures to the analysis of the passive response of cells to osmotic loading. We have generalized the formulation to incorporate partition coefficients for the solutes in the cytoplasm relative to the external solution, and accounts for cell membrane tension.

Biomimetic Analysis of Alginate Beads in Dextran Solutions

To help assess whether this more elaborate model of the cell was justified, we performed a study to investigate the response of spherical gels to osmotic loading, both from experiments and theory. In the experimental component of the study alginate was used as the model gel, and it was osmotically loaded with dextran solutions of various concentrations and molecular weight, to verify the predictions from the theoretical analysis. Results showed that the mixture framework could accurately predict the transient and equilibrium response of alginate gels to osmotic loading with dextran solutions. It was found that the partition coefficient of dextran in alginate regulates the equilibrium volume response and can explain partial volume recovery based on passive transport mechanisms. This study was the first to demonstrate that gels may exhibit a partial volume recovery when loaded osmotically. The validation of this framework facilitated our subsequent investigations of the role of the protoplasm in the response of cells to osmotic loading.

Partitioning of Osmolytes by the Cytoplasm

Due to the dense organization of organelles, cytoskeletal elements, and protein complexes that make up the intracellular environment, we hypothesized that membrane-permeant solutes may be excluded from a fraction of the interstitial space of the cytoplasm via steric restrictions, electrostatic interactions and other long-range intermolecular forces. We performed experiments to investigate the hypothesis that the intracellular partitioning of membrane-permeant solutes manifests itself as a partial volume recovery in response to hyperosmotic loading, based on our prior theoretical and biomimetic experimental studies. Osmotic loading experiments were performed on immature bovine chondrocytes using culture conditions where regulatory volume responses were shown to be insignificant. Osmotic loading with membrane-permeant glycerol (92 Da) and urea (60 Da) were observed to produce partial volume recoveries consistent with the proposed hypothesis, whereas loading with 1,2-propanediol (76 Da) produced complete volume recovery. Combining these experimental results with the previous theoretical framework produced a measure for the intracellular partition coefficient of each of these solutes. The finding that intracellular partitioning of membrane-permeant solutes manifests itself as a partial volume recovery under osmotic loading offered a simple method for characterizing the partition coefficient. These measurements suggested that significant partitioning may occur even for small membrane-permeant osmolytes. Furthermore, a positive correlation was observed suggesting that a solute’s cytoplasmic partition coefficient increases with increasing hydrophobicity. The biomimetic studies of alginate in dextran and osmotic loading of chondrocytes represent a significant component of the dissertation of Michael Albro.

Water Transport Through Chondrocytes

Because of the avascular nature of adult cartilage nutrients and waste products are transported to and from the chondrocytes by diffusion and convection through the extracellular matrix. The convective interstitial fluid flow within and around chondrocytes is poorly understood. From a theoretical study, we demonstrated that the incorporation of a semi-permeable membrane when modeling the chondrocyte leads to the following findings: Under mechanical loading of an isolated chondrocyte the intracellular fluid pressure is on the order of tens of Pascals and the transmembrane fluid outflow, on the order of picometers per second, takes several days to subside; consequently the chondrocyte behaves practically as an incompressible solid whenever the loading duration is on the order of minutes or hours. When embedded in its extracellular matrix, the chondrocyte response is substantially different. Mechanical loading of the tissue leads to a fluid pressure difference between intracellular and extracellular compartments on the order of tens of kilopascals and the transmembrane outflow, on the order of a nanometer per second, subsides in about one hour. The volume of the chondrocyte decreases concomitantly with that of the extracellular matrix. The interstitial fluid flow in the extracellular matrix is directed around the cell, with peak values on the order of tens of nanometers per second. The viscous fluid shear stress acting on the cell surface is orders of magnitude smaller than the solid matrix shear stresses resulting from the extracellular matrix deformation. These results provided new insight toward our understanding of water transport in chondrocytes.

Our finite element modeling studies are conducted in collaboration with the Musculoskeletal Research Laboratories of Prof. Jeffrey Weiss at the University of Utah. The finite element code FEBio developed in collaboration with Prof. Weiss can be downloaded from febio.org.

Short-Time Biphasic and Incompressible Elastic Material Responses

Porous-permeable tissues have often been modeled using porous media theories such as the biphasic theory. Finite element contact analyses of porous deformable media under large deformations are not generally available in existing commercial codes. Therefore we performed a study to examine the equivalence of the short-time biphasic and incompressible elastic responses for arbitrary deformations and constitutive relations from first principles. This equivalence was illustrated in problems of unconfined compression of a disk, and of articular contact under finite deformation, using two different constitutive relations for the solid matrix of cartilage, one of which accounts for the large disparity observed between the tensile and compressive moduli in this tissue. Demonstrating this equivalence under general conditions provided a rationale for using available finite element codes for incompressible elastic materials as a practical substitute for biphasic analyses, so long as only the short-time biphasic response is sought.

Finite Element Algorithm for Frictionless Contact of Porous Permeable Media Under Finite Deformation and Sliding

Since existing commercial codes do not provide a consistent implementation of finite element contact for porous deformable media, we subsequently formulated and implemented a finite element contact algorithm for solid-fluid (biphasic) mixtures, accommodating both finite deformation and sliding. The finite element source code is currently available to the public. The algorithm uses a penalty method regularized with an augmented Lagrangian method to enforce the continuity of contact traction and normal component of fluid flux across the contact interface. The formulation addressed the need to automatically enforce free-draining conditions outside of the contact interface. The accuracy of the implementation was verified using contact problems for which exact solutions were obtained by alternative analyses. Illustrations were also provided that demonstrate large deformations and sliding under configurations relevant to biomechanical applications such as articular contact. This study addressed an important computational need in the biomechanics of porous-permeable soft tissues. Placing the source code in the public domain will provide a useful resource to the biomechanics community.

Dynamic Loading of Gels and Tissues

In many biological tissues, it has been shown that dynamic loading can promote biosynthetic activity and prevent tissue degradation. For example, in articular cartilage, cyclical loading in the range of physiologically relevant frequencies promotes glycosaminoglycan and protein synthesis, whereas static loading inhibits this process. This type of response has also been observed in cartilage tissue engineering studies, where dynamic compression has been shown to promote better mechanical properties in comparison to unloaded controls. It is generally believed that dynamic loading may increase cellular activity via two pathways: A mechano-transduction pathway whereby cells respond to the loading environment directly, through integrin-mediated responses or molecular conformational changes transduced to the nucleus; and an enhanced solute transport pathway whereby loading increases the supply of nutrients, growth factors, cytokines, morphogens, etc., leading to greater matrix synthesis.

In an effort to elucidate the role of solute transport in dynamically loaded porous tissues, our laboratory previously extended the framework of mixture theory to specifically accommodate the interaction of solutes with the solid matrix of a tissue, generalizing the classical framework of Fick's diffusion where solutes only interact with the solvent. In this framework, two diffusion coefficients emerged naturally from the formulation, one that describes solute diffusion in free solution, involving the frictional drag between solute and solvent, and another that describes solute diffusion in the porous tissue, also involving the frictional drag between solute and solid matrix. Thus, this theoretical framework could naturally describe the well-attested observation that solutes diffuse at different rates inside a porous medium than in free solution.

An intriguing theoretical prediction emerged from this framework, which had not been previously anticipated: Accordingly, under sustained dynamic loading, this theory predicted that the solute concentration inside the tissue would rise to values far exceeding those achieved under passive diffusion (no loading). In effect, the theory suggested that dynamic loading could pump the solute from the external bathing solution into the tissue, against its concentration gradient, thereby describing an active transport process. Since this outcome had never been previously reported from experiments, we subsequently performed a series of studies where agarose hydrogels of various porosities were exposed to dextran solutions of various molecular weights, and confirmed these earlier theoretical predictions unequivocally. For certain combinations of gel concentrations and dextran molecular weights, solute uptake was enhanced up to sixteen-fold relative to passive diffusion, and results suggested that higher values could be achieved with longer loading durations. Thus, a fundamental mechanism of active transport in deformable porous tissues was discovered through these studies, initially motivated from theory. Importantly, the theory would not have predicted these enhancements in solute uptake if the diffusive drag between solute and solid matrix had been neglected; indeed, the pumping mechanism described by this theory is contingent on the solid matrix exchanging momentum with the solute.

The initial theoretical study of enhanced solute uptake was a component of the dissertation of Dr. Robert Mauck. Experimental studies that confirmed the predictions of this framework were initiated in the dissertation of Dr. Nadeen Chahine and have continued as part of the dissertation of Michael Albro.

Solute Partitioning in Cells

Another interesting theoretical development introduced in our studies was the formulation of a constitutive relation for the chemical potential of solutes that explicitly accommodates the well-known phenomenon of solute partitioning in a porous medium. Though well recognized experimentally, it appears that the concept of partitioning had not been previously incorporated into theoretical frameworks of transport in porous media. In a subsequent theoretical study, we adapted this framework to the analysis of osmotic loading of cells and discovered that we could extend the classical model formulated by Kedem and Katchalsky: In the K-K model, which models the cell as a fluid-filled membrane, when a cell is loaded osmotically with a membrane-permeant solute, its volume decreases initially but eventually recovers to its initial value as the internal concentration of the membrane-permeating solute eventually rises to the external value. In our extended framework, the cell is modeled as a gel-filled membrane, where the membrane-permeant solute may be partially partitioned out of the porous gel (representing the cell protoplasm). A consequence of this partitioning is that the cell does not recover its initial volume at steady state, since a smaller cell volume is required to match the internal and external solute concentrations.

Thus, our theoretical framework predicted that partial volume recovery following osmotic loading with a membrane-permeant solute could be explained by passive mechanisms (solute partitioning). Prior experimental studies either overlooked this partial volume recovery or attributed it to active transport processes known to regulate cell volume. To verify that our theoretical framework described a feasible mechanism, we conducted osmotic loading experiments on alginate beads, using dextran solutions of various concentrations and molecular weights and confirmed that partial volume recovery could indeed occur even in the absence of a membrane and active membrane transporters, in a manner fully consistent with the theory. We have since applied this framework to analyze the volumetric response of chondrocytes to various osmolytes with similar success.

The experimental studies on the partial volume recovery of alginate gels and chondrocytes in response to osmotic loading are a significant component of the dissertation of Michael Albro.

The Role of Theoretical Analyses

A critical concept emerging from these studies is that suitably formulated theoretical frameworks for solute transport in porous deformable media may motivate novel hypotheses relevant to the study of biological tissues and cells, and guide the design of experiments to test these hypotheses. When combining solute transport with porous deformable media, we find that theoretical frameworks are still under development, since basic concepts long recognized from experiments (such as hindered solute diffusion, and solute partitioning in a porous medium) can be incorporated in a variety of ways into theory, sometimes yielding new and unexpected predictions.