Anisotropic hydraulic permeability in compressed articular cartilage

The extent to which articular cartilage hydraulic permeability is anisotropic is largely unknown, despite its importance for understanding mechanisms of joint lubrication, load bearing, transport phenomena, and mechanotransduction. We developed and applied new techniques for the direct measurement of hydraulic permeability within statically compressed adult bovine cartilage explant disks, dissected such that disk axes were perpendicular to the articular surface. Applied pressure gradients were kept small to minimize flow-induced matrix compaction, and fluid outflows were measured by observation of a meniscus in a glass capillary under a microscope. Explant disk geometry under radially unconfined axial compression was measured by direct microscopic observation. Pressure, flow, and geometry data were input to a finite element model where hydraulic permeabilities in the disk axial and radial directions were determined. At less than 10% static compression, near free-swelling conditions, hydraulic permeability was nearly isotropic, with values corresponding to those of previous studies. With increasing static compression, hydraulic permeability decreased, but the radially directed permeability decreased more dramatically than the axially directed permeability such that strong anisotropy (a 10-fold difference between axial and radial directions) in the hydraulic permeability tensor was evident for static compression of 20-40%. Results correspond well with predictions of a previous microstructurally-based model for effects of tissue mechanical deformations on glycosaminoglycan architecture and cartilage hydraulic permeability. Findings inform understanding of structure-function relationships in cartilage matrix, and suggest several biomechanical roles for compression-induced anisotropic hydraulic permeability in articular cartilage.     

Glycosaminoglycan network geometry may contribute to anisotropic hydraulic permeability in cartilage under compression.

Resistance to fluid flow within cartilage extracellular matrix is provided primarily by a dense network of rod-like glycosaminoglycans (GAGs). If the geometrical organization of this network is random, the hydraulic permeability tensor of cartilage is expected to be isotropic. However, experimental data have suggested that hydraulic permeability may become anisotropic when the matrix is mechanically compressed, contributing to cartilage biomechanical functions such as lubrication. We hypothesized that this may be due to preferred GAG rod orientations and directionally-dependent reduction of inter-GAG spacings which reflect molecular responses to tissue deformations. To examine this hypothesis, we developed a model for effects of compression which allows the GAG rod network to deform consistently with tissue-scale deformations but while still respecting limitations imposed by molecular structure. This network deformation model was combined with a perturbation analysis of a classical analytical model for hydraulic permeability based on molecular structure. Finite element analyses were undertaken to ensure that this approach exhibited results similar to those emerging from more exact calculations. Model predictions for effects of uniaxial confined compression on the hydraulic permeability tensor were consistent with previous experimental results. Permeability decreased more rapidly in the direction perpendicular to compression than in the parallel direction, for matrix solid volume fractions associated with fluid transport in articular cartilage. GAG network deformations may therefore introduce anisotropy to the permeability (and other GAG-associated matrix properties) as physiological compression is applied, and play an important role in cartilage lubrication and other biomechanical functions.     

Anisotropic adaptive finite element method for modelling blood flow

In this study, we present an adaptive anisotropic finite element method (FEM) and demonstrate how computational efficiency can be increased when applying the method to the simulation of blood flow in the cardiovascular system. We use the SUPG formulation for the transient 3D incompressible Navier-Stokes equations which are discretised by linear finite elements for both the pressure and the velocity field. Given the pulsatile nature of the flow in blood vessels we have pursued adaptivity based on the average flow over a cardiac cycle. Error indicators are derived to define an anisotropic mesh metric field. Mesh modification algorithms are used to anisotropically adapt the mesh according to the desired size field. We demonstrate the efficiency of the method by first applying it to pulsatile flow in a straight cylindrical vessel and then to a porcine aorta with a stenosis bypassed by a graft. We demonstrate that the use of an anisotropic adaptive FEM can result in an order of magnitude reduction in computing time with no loss of accuracy compared to analyses obtained with uniform meshes.     

Finite element implementation of mechanochemical phenomena in neutral deformable porous media under finite deformation

Biological soft tissues and cells may be subjected to mechanical as well as chemical (osmotic) loading under their natural physiological environment or various experimental conditions. The interaction of mechanical and chemical effects may be very significant under some of these conditions, yet the highly nonlinear nature of the set of governing equations describing these mechanisms poses a challenge for the modeling of such phenomena. This study formulated and implemented a finite element algorithm for analyzing mechanochemical events in neutral deformable porous media under finite deformation. The algorithm employed the framework of mixture theory to model the porous permeable solid matrix and interstitial fluid, where the fluid consists of a mixture of solvent and solute. A special emphasis was placed on solute-solid matrix interactions, such as solute exclusion from a fraction of the matrix pore space (solubility) and frictional momentum exchange that produces solute hindrance and pumping under certain dynamic loading conditions. The finite element formulation implemented full coupling of mechanical and chemical effects, providing a framework where material properties and response functions may depend on solid matrix strain as well as solute concentration. The implementation was validated using selected canonical problems for which analytical or alternative numerical solutions exist. This finite element code includes a number of unique features that enhance the modeling of mechanochemical phenomena in biological tissues. The code is available in the public domain, open source finite element program FEBio (http:∕∕mrl.sci.utah.edu∕software).     

Elastic modulus and hydraulic permeability of MDCK monolayers

The critical role of cell mechanics in tissue health has led to the development of many in vitro methods that measure the elasticity of the cytoskeleton and whole cells, yet the connection between these local cell properties and bulk measurements of tissue mechanics remains unclear. To help bridge this gap, we have developed a monolayer indentation technique for measuring multi-cellular mechanics in vitro. Here, we measure the elasticity of cell monolayers and uncover the role of fluid permeability in these multi-cellular systems, finding that the resistance of fluid transport through cells controls their force–response at long times.     

Anisotropic adaptive finite element method for modelling blood flow

In this study, we present an adaptive anisotropic finite element method (FEM) and demonstrate how computational efficiency can be increased when applying the method to the simulation of blood flow in the cardiovascular system. We use the SUPG formulation for the transient 3D incompressible Navier–Stokes equations which are discretised by linear finite elements for both the pressure and the velocity field.

Given the pulsatile nature of the flow in blood vessels we have pursued adaptivity based on the average flow over a cardiac cycle. Error indicators are derived to define an anisotropic mesh metric field. Mesh modification algorithms are used to anisotropically adapt the mesh according to the desired size field. We demonstrate the efficiency of the method by first applying it to pulsatile flow in a straight cylindrical vessel and then to a porcine aorta with a stenosis bypassed by a graft. We demonstrate that the use of an anisotropic adaptive FEM can result in an order of magnitude reduction in computing time with no loss of accuracy compared to analyses obtained with uniform meshes.     

Finite element algorithm for frictionless contact of porous permeable media under finite deformation and sliding

This study formulates and implements a finite element contact algorithm for solid-fluid (biphasic) mixtures, accommodating both finite deformation and sliding. The finite element source code is made available to the general 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 addresses the need to automatically enforce free-draining conditions outside of the contact interface. The accuracy of the implementation is verified using contact problems, for which exact solutions are obtained by alternative analyses. Illustrations are also provided that demonstrate large deformations and sliding under configurations relevant to biomechanical applications such as articular contact. This study addresses an important computational need in the biomechanics of porous-permeable soft tissues. Placing the source code in the public domain provides a useful resource to the biomechanics community.     

Membrane hydraulic permeability changes during cooling of mammalian cells

In order to predict optimal cooling rates for cryopreservation of cells, the cell-specific membrane hydraulic permeability and corresponding activation energy for water transport need to be experimentally determined. These parameters should preferably be determined at subzero temperatures in the presence of ice. There is, however, a lack of methods to study membrane properties of cells in the presence of ice. We have used Fourier transform infrared spectroscopy to study freezing-induced membrane dehydration of mouse embryonic fibroblast (3T3) cells and derived the subzero membrane hydraulic permeability and the activation energy for water transport from these data. Coulter counter measurements were used to determine the suprazero membrane hydraulic permeability parameters from cellular volume changes of cells exposed to osmotic stress. The activation energy for water transport in the ice phase is about three fold greater compared to that at suprazero temperatures. The membrane hydraulic permeability at 0 °C that was extrapolated from suprazero measurements is about five fold greater compared to that extrapolated from subzero measurements. This difference is likely due to a freezing-induced dehydration of the bound water around the phospholipid head groups. Using Fourier transform infrared spectroscopy, two distinct water transport processes, that of free and membrane bound water, can be identified during freezing with distinct activation energies. Dimethylsulfoxide, a widely used cryoprotective agent, did not prevent freezing-induced membrane dehydration but decreased the activation energy for water transport.     

Gallbladder epithelial cell hydraulic water permeability and volume regulation

The hydraulic water permeability (Lp) of the cell membranes of Necturus gallbladder epithelial cells was estimated from the rate of change of cell volume after a change in the osmolality of the bathing solution. Cell volume was calculated from computer reconstruction of light microscopic images of epithelial cells obtained by the "optical slice" technique. The tissue was mounted in a miniature Ussing chamber designed to achieve optimal optical properties, rapid bath exchange, and negligible unstirred layer thickness. The control solution contained only 80% of the normal NaCl concentration, the remainder of the osmolality was made up by mannitol, a condition that did not significantly decrease the fluid absorption rate in gallbladder sac preparations. The osmotic gradient ranged from 11.5 to 41 mosmol and was achieved by the addition or removal of mannitol from the perfusion solutions. The Lp of the apical membrane of the cell was 1.0 X 10(-3) cm/s . osmol (Posm = 0.055 cm/s) and that of the basolateral membrane was 2.2 X 10(-3) cm/s . osmol (Posm = 0.12 cm/s). These values were sufficiently high so that normal fluid absorption by Necturus gallbladder could be accomplished by a 2.4-mosmol solute gradient across the apical membrane and a 1.1-mosmol gradient across the basolateral membrane. After the initial cell shrinkage or swelling resulting from the anisotonic mucosal or serosal medium, cell volume returned rapidly toward the control value despite the fact that one bathing solution remained anisotonic. This volume regulatory response was not influenced by serosal ouabain or reduction of bath NaCl concentration to 10 mM. Complete removal of mucosal perfusate NaCl abolished volume regulation after cell shrinkage. Estimates were also made of the reflection coefficient for NaCl and urea at the apical cell membrane and of the velocity of water flow across the cytoplasm.     

Prestressing in finite deformation abdominal aortic aneurysm simulation

In abdominal aortic aneurysm (AAA) simulation the patient-specific geometry of the object of interest is very often reconstructed from in vivo medical imaging such as CT scans. Such geometries represent a deformed configuration stressed by typical in vivo conditions. However, commonly, such structures are considered stress-free in simulation. In this contribution we sketch and compare two methods to introduce a physically meaningful stress/strain state to the obtained geometry for simulations in the finite strain regime and demonstrate the necessity of such prestressing techniques. One method is based on an inverse design analysis to calculate a stress-free reference configuration. The other method developed here is based on a modified updated Lagrangian formulation. Formulation of both methods is provided. Applicability and accurateness of both approaches are compared and evaluated utilizing fully three-dimensional patient-specific AAA structures in the finite strain regime.     

Effect of osmolality on the hydraulic permeability coefficient of red cells

The osmotic water permeability coefficient, L_p, for human and dog red cells has been measured as a function of medium osmolality, and found to depend on the osmolality of the bathing medium. In the case of human red cells L_p falls from 1.87 x 10^-11 cm³/dyne sec at 199 mOSM to 0.76 x 10^-11 cm³/dyne sec at 516 mOSM. A similar decrease was observed for dog red cells. Moreover, L_p was independent of the direction of water movement and the nature of the solute used to provide the osmotic pressure gradient; it depended only on the final osmolality of the medium. Furthermore, L_p was not affected by pH in the range of 6 to 8 nor by the presence of drugs such as valinomycin (1 x 10^-6 M) and tetrodotoxin (3.2 x 10^-6 M). The instantaneous nature of the response to changes in external osmolality suggests that the hydraulic conductivity of the membrane is controlled by a thin layer at the outer face of the membrane.     

Determination of the strain-dependent hydraulic permeability of the compressed bovine nucleus pulposus

The hydraulic permeability, k, of the nucleus pulposus (NP) is crucial, both in withstanding compressive stress and for convective transport of nutrients within the disc. Permeability has previously been determined using biphasic mathematical models, but has not been found by direct permeation experiments, which is the objective of this study. Bovine coccygeal nucleus samples (n=64), phi10mm and thickness 683+/-49microm (mean+/-S.D.) were compressed axially to one of lambda=1.0, 0.9, 0.8, 0.7, 0.6, 0.5, 0.4 or 0.3, where lambda is the stretch ratio. Ringer's solution was permeated through the sample, with an o-ring ensuring axial flow. During stress equilibrium, k was determined and fitted to four permeability-strain equations. Permeability decreased exponentially with compression, and was best described by Values of k were comparable to those arising from mathematical models, lending confidence to permeability being determined from such models.     

A power-law rheology-based finite element model for single cell deformation

Physical forces can elicit complex time- and space-dependent deformations in living cells. These deformations at the subcellular level are difficult to measure but can be estimated using computational approaches such as finite element (FE) simulation. Existing FE models predominantly treat cells as spring-dashpot viscoelastic materials, while broad experimental data are now lending support to the power-law rheology (PLR) model. Here, we developed a large deformation FE model that incorporated PLR and experimentally verified this model by performing micropipette aspiration on fibroblasts under various mechanical loadings. With a single set of rheological properties, this model recapitulated the diverse micropipette aspiration data obtained using three protocols and with a range of micropipette sizes. More intriguingly, our analysis revealed that decreased pipette size leads to increased pressure gradient, potentially explaining our previous counterintuitive finding that decreased pipette size leads to increased incidence of cell blebbing and injury. Taken together, our work leads to more accurate rheological interpretation of micropipette aspiration experiments than previous models and suggests pressure gradient as a potential determinant of cell injury.     

An experimental study of some indigenous drugs with special reference to hydraulic permeability

The effect of commonly used indigenous drugs for hepatic disorders i.e. Tinospora cordifolia, (Guduchi/Amrita), Andrographis paniculata (Kalmegha), Picrorhiza kurroa (Kutki), Phyllantnus niruri (Bhoomyamalaki) and Berberis aristata (Daruharidra) was tested on the hydraulic permeability of water in the presence of bile salt through a transport cell model. The data on hydraulic permeability were calculated as t (time). JV = Lp x AP, where Lp = hydraulic conductivity and AP is the pressure difference. It was observed that the value of controlled hydraulic permeability (0.49 x 10(-8) M3 S(-1) N(-1)) decreased in the presence of indigenous drugs and bile salt. The results suggest that these drugs might have the cell membrane stabilizing property which may lead to prevention of the toxic effect of bile salts in various hepatic disorders.     

Non-homogeneous analysis of three-dimensional transmural finite deformation in canine ventricular myocardium

A new method has been developed for analyzing transmural distributions of finite deformation in canine ventricular myocardium without the need to assume that the strain in a finite volume of the wall is homogeneous. The three-dimensional nodal geometric parameters of bilinear-cubic or bilinear-quadratic finite elements are fitted by least squares to the measured coordinates of 12-18 radiopaque markers implanted in the left ventricular free wall. For six dog hearts, root-mean-squared errors in the fitted in-plane coordinates ranged from 0.079-0.556 mm in the end-diastolic reference state and 0.142-0.622 mm at end-systole. The corresponding error ranges in the radial coordinate were 0.042-0.264 mm at end-diastole and 0.106-0.279 mm at end-systole. Smoothly continuous transmural profiles of wall strain computed as the element deformed during the cardiac cycle from end-diastole to end-systole showed good agreement with the discrete results of conventional homogeneous analysis. Using the kinematics of a thick-walled incompressible cylinder, overall absolute errors due to the non-homogeneity of myocardial deformation were found to be reduced in the new analysis by 30-35% for typical experimental parameters. Overall relative errors were also reduced (from 23 to 20%). Since measurement errors in the reconstructed marker coordinates were spatially smoothed by the fitting procedure, noise in the computed deformations was also substantially attenuated, and transmural gradients of three-dimensional strain components could be obtained with improved accuracy. Hence physiological factors affected by transmural stress and strain distributions, such as myocardial blood flow, ischemia and hypertrophy, may be better understood.     

Leukocyte deformability: finite element modeling of large viscoelastic deformation.

An axisymmetric deformation of a viscoelastic sphere bounded by a prestressed elastic thin shell in response to external pressure is studied by a finite element method. The research is motivated by the need for understanding the passive behavior of human leukocytes (white blood cells) and interpreting extensive experimental data in terms of the mechanical properties. The cell at rest is modeled as a sphere consisting of a cortical prestressed shell with incompressible Maxwell fluid interior. A large-strain deformation theory is developed based on the proposed model. General non-linear, large strain constitutive relations for the cortical shell are derived by neglecting the bending stiffness. A representation of the constitutive equations in the form of an integral of strain history for the incompressible Maxwell interior is used in the formulation of numerical scheme. A finite element program is developed, in which a sliding boundary condition is imposed on all contact surfaces. The mathematical model developed is applied to evaluate experimental data of pipette tests and observations of blood flow.     

A transversely isotropic viscoelastic constitutive equation for brainstem undergoing finite deformation

The objective of this study was to define the constitutive response of brainstem undergoing finite shear deformation. Brainstem was characterized as a transversely isotropic viscoelastic material and the material model was formulated for numerical implementation. Model parameters were fit to shear data obtained in porcine brainstem specimens undergoing finite shear deformation in three directions: parallel, perpendicular, and cross sectional to axonal fiber orientation and determined using a combined approach of finite element analysis (FEA) and a genetic algorithm (GA) optimizing method. The average initial shear modulus of brainstem matrix of 4-week old pigs was 12.7 Pa, and therefore the brainstem offers little resistance to large shear deformations in the parallel or perpendicular directions, due to the dominant contribution of the matrix in these directions. The fiber reinforcement stiffness was 121.2 Pa, indicating that brainstem is anisotropic and that axonal fibers have an important role in the cross-sectional direction. The first two leading relative shear relaxation moduli were 0.8973 and 0.0741, respectively, with corresponding characteristic times of 0.0047 and 1.4538 s, respectively, implying rapid relaxation of shear stresses. The developed material model and parameter estimation technique are likely to find broad applications in neural and orthopaedic tissues.