Mansoor A. Haider

The Pericellular Matrix as a Transducer of Biomechanical and Biochemical Signals in Articular Cartilage

Abstract:  The pericellular matrix (PCM) is a narrow tissue region surrounding chondrocytes in articular cartilage, which together with the enclosed cell(s) has been termed the “chondron.” While the function of this region is not fully understood, it is hypothesized to have important biological and biomechanical functions. In this article, we review a number of studies that have investigated the structure, composition, mechanical properties, and biomechanical role of the chondrocyte PCM. This region has been shown to be rich in proteoglycans (e.g., aggrecan, hyaluronan, and decorin), collagen (types II, VI, and IX), and fibronectin, but is defined primarily by the presence of type VI collagen as compared to the extracellular matrix (ECM). Direct measures of PCM properties via micropipette aspiration of isolated chondrons have shown that the PCM has distinct mechanical properties as compared to the cell or ECM. A number of theoretical and experimental studies suggest that the PCM plays an important role in regulating the microenvironment of the chondrocyte. Parametric studies of cell–matrix interactions suggest that the presence of the PCM significantly affects the micromechanical environment of the chondrocyte in a zone‐dependent manner. These findings provide support for a potential biomechanical function of the chondrocyte PCM, and furthermore, suggest that changes in the PCM and ECM properties that occur with osteoarthritis may significantly alter the stress‐strain and fluid environments of the chondrocytes. An improved understanding of the structure and function of the PCM may provide new insights into the mechanisms that regulate chondrocyte physiology in health and disease.

The dynamic mechanical environment of the chondrocyte: A biphasic finite element model of cell-matrix interactions under cyclic compressive loading

Cyclic mechanical loading of articular cartilage results in a complex biomechanical environment at the scale of the chondrocytes that strongly affects cellular metabolic activity. Under dynamic loading conditions, the quantitative relationships between macroscopic loading characteristics and solid and fluid mechanical variables in the local cellular environment are not well understood. In this study, an axisymmetric multiscale model of linear biphasic cell-matrix interactions in articular cartilage was developed to investigate the cellular microenvironment in an explant subjected to cyclic confined compressive loading. The model was based on the displacement-velocity-pressure (u-v-p) mixed-penalty weighted residual formulation of linear biphasic theory that was implemented in the COMSOL MULTIPHYSICS software package. The microscale cartilage environment was represented as a three-zone biphasic region consisting of a spherical chondrocyte with encapsulating pericellular matrix (PCM) that was embedded in a cylindrical extracellular matrix (ECM) subjected to cyclic confined compressive loading boundary conditions. Biphasic material properties for the chondrocyte and the PCM were chosen based on previous in vitro micropipette aspiration studies of cells or chondrons isolated from normal or osteoarthritic cartilage. Simulations performed at four loading frequencies in the range 0.01-1.0 Hz supported the hypothesized dual role of the PCM as both a protective layer for the cell and a mechanical transducer of strain. Time varying biphasic variables at the cellular scale were strongly dependent on relative magnitudes of the loading period, and the characteristic gel diffusion times for the ECM, the PCM, and the chondrocyte. The multiscale simulations also indicated that axial strain was significantly amplified in the range 0.01-1.0 Hz, with a decrease in amplification factor and frequency insensitivity at the higher frequencies. Simulations of matrix degradation due to osteoarthritis indicated that strain amplification factors were more significantly altered when loss of matrix stiffness was exclusive to the PCM. The findings of this study demonstrate the complex dependence of dynamic mechanics in the local cellular environment of cartilage on macroscopic loading features and material properties of the ECM and the chondron.

An axisymmetric boundary integral model for assessing elastic cell properties in the micropipette aspiration contact problem

The micropipette aspiration technique has been used extensively in recent years to measure the mechanical properties of living cells. In the present study, a boundary integral formulation with quadratic elements is used to predict the elastic equilibrium response in the micropipette aspiration contact problem for a three-dimensional incompressible spherical continuum cell model (Youngs modulus E). In contrast to the halfspace model, the spherical cell model accounts for nonlinearities in the cell response which result from a consideration of geometric factors including the finite cell dimension (radius R), curvature of the cell boundary, evolution of the cell-micropipette contact region and curvature of the edges of the micropipette (inner radius a, edge curvature radius epsilon). The efficiency of the boundary element method facilitates the quantification of cell response as a function of the scaled pressure p/E, for the range of parameters a/R = 0.4-0.7, epsilon/a = 0.02-0.08, in terms of two measures that can be quantified using video microscopy. These are the aspiration length, which measures projection of the cell into the micropipette, and a characteristic strain, which measures stretching along the symmetry axis. For both measures of cell response, the resistance to aspiration is found to decrease with increasing values of the aspect ratio a/R and curvature parameter epsilon/a, and the nonlinearities in the cell response are most pronounced in the earlier portion of the aspiration test. The aspiration length is found to exhibit less sensitivity to the aspect ratio a/R than to the curvature parameter epsilon/a, whereas the characteristic strain, which provides a more realistic measure of overall cell stiffness, exhibits sensitivity to the aspect ratio a/R. The resistance to aspiration in the spherical cell model is initially less than that of the half space model but eventually exceeds the halfspace prediction and the deviation between the two models increases as the parameter epsilon/a decreases. Adjustment factors for the Youngs modulus E, as predicted by the halfspace model, are presented and the deviation from the spherical cell model is found to be as large as 35%, when measured locally on the response curve. In practice, the deviation will be less than the maximum figure but its precise value will depend on the number of data points available in the experiment and the specific curve-fitting procedure. The spherical cell model allows for efficient and more realistic simulations of the micropipette aspiration contact problem and quantifies two observable measures of cell response that, using video microscopy, can facilitate the determination of Youngs modulus for various cell populations while, simultaneously, providing a means of evaluating the validity of continuum cell models. Furthermore, this numerical model may be readily extended to account for more complex geometries, inhomogeneities in cellular properties, or more complex constitutive descriptions of the cell.

Analysis of Viscoelastic Wall Properties in Ovine Arteries

In this paper, we analyze how elastic and viscoelastic properties differ across seven locations along the large arteries in 11 sheep. We employ a two-parameter elastic model and a four-parameter Kelvin viscoelastic model to analyze experimental measurements of vessel diameter and blood pressure obtained in vitro at conditions mimicking in vivo dynamics. Elastic and viscoelastic wall properties were assessed via solutions to the associated inverse problem. We use sensitivity analysis to rank the model parameters from the most to the least sensitive, as well as to compute standard errors and confidence intervals. Results reveal that elastic properties in both models (including Youngs modulus and the viscoelastic relaxation parameters) vary across locations (smaller arteries are stiffer than larger arteries). We also show that for all locations, the inclusion of viscoelastic behavior is important to capture pressure-area dynamics.

Two-dimensional photonic crystal Fabry-Perot resonators with lossy dielectrics

Square and triangular lattice two-dimensional (2D) photonic crystals (PCs) composed of lossy dielectric rods in air were constructed with a microwave bandgap between 4-8 GHz. Fabry-Perot resonators of varying length were constructed from two of these PCs of adjustable thickness and reflectivity. The quality factor of cavity modes supported in the resonators was found to increase with increasing PC mirror thickness, but only to a point dictated by the lossiness of the dielectric rods. A 2-D periodic Greens function simulation was found to model the data accurately and quickly using physical parameters obtained in separate measurements. Simple rules are developed for designing optimal resonators in the presence of dielectric loss.

Linear and Nonlinear Viscoelastic Modeling of Aorta and Carotid Pressure–Area Dynamics Under In Vivo and Ex Vivo Conditions

A better understanding of the biomechanical properties of the arterial wall provides important insight into arterial vascular biology under normal (healthy) and pathological conditions. This insight has potential to improve tracking of disease progression and to aid in vascular graft design and implementation. In this study, we use linear and nonlinear viscoelastic models to predict biomechanical properties of the thoracic descending aorta and the carotid artery under ex vivo and in vivo conditions in ovine and human arteries. Models analyzed include a four-parameter (linear) Kelvin viscoelastic model and two five-parameter nonlinear viscoelastic models (an arctangent and a sigmoid model) that relate changes in arterial blood pressure to the vessel cross-sectional area (via estimation of vessel strain). These models were developed using the framework of Quasilinear Viscoelasticity (QLV) theory and were validated using measurements from the thoracic descending aorta and the carotid artery obtained from human and ovine arteries. In vivo measurements were obtained from 10 ovine aortas and 10 human carotid arteries. Ex vivo measurements (from both locations) were made in 11 male Merino sheep. Biomechanical properties were obtained through constrained estimation of model parameters. To further investigate the parameter estimates, we computed standard errors and confidence intervals and we used analysis of variance to compare results within and between groups. Overall, our results indicate that optimal model selection depends on the artery type. Results showed that for the thoracic descending aorta (under both experimental conditions), the best predictions were obtained with the nonlinear sigmoid model, while under healthy physiological pressure loading the carotid arteries nonlinear stiffening with increasing pressure is negligible, and consequently, the linear (Kelvin) viscoelastic model better describes the pressure–area dynamics in this vessel. Results comparing biomechanical properties show that the Kelvin and sigmoid models were able to predict the zero-pressure vessel radius; that under ex vivo conditions vessels are more rigid, and comparatively, that the carotid artery is stiffer than the thoracic descending aorta; and that the viscoelastic gain and relaxation parameters do not differ significantly between vessels or experimental conditions. In conclusion, our study demonstrates that the proposed models can predict pressure–area dynamics and that model parameters can be extracted for further interpretation of biomechanical properties.

Boundary integral calculations of two-dimensional electromagnetic scattering by photonic crystal Fabry-Perot structures

We compute the transmission properties of two-dimensional (2-D) electromagnetic transverse magnetic (TM) waves that are normally incident on a Fabry--Perot structure with mirrors consisting of photonic crystals. We use a boundary integral formulation with quadratic boundary elements and utilize the Ewald representation for Greens functions. We trace the frequencies of the Fabry--Perot cavity modes traversing the photonic bandgap as the cavity length increases and calculate corresponding Q-values.

Neural Network Analysis Identifies Scaffold Properties Necessary for In Vitro Chondrogenesis in Elastin-like Polypeptide Biopolymer Scaffolds

The successful design of biomaterial scaffolds for articular cartilage tissue engineering requires an understanding of the impact of combinations of material formulation parameters on diverse and competing functional outcomes of biomaterial performance. This study sought to explore the use of a type of unsupervised artificial network, a self-organizing map, to identify relationships between scaffold formulation parameters (crosslink density, molecular weight, and concentration) and 11 such outcomes (including mechanical properties, matrix accumulation, metabolite usage and production, and histological appearance) for scaffolds formed from crosslinked elastin-like polypeptide (ELP) hydrogels. The artificial neural network recognized patterns in functional outcomes and provided a set of relationships between ELP formulation parameters and measured outcomes. Mapping resulted in the best mean separation amongst neurons for mechanical properties and pointed to crosslink density as the strongest predictor of most outcomes, followed by ELP concentration. The map also grouped formulations together that simultaneously resulted in the highest values for matrix production, greatest changes in metabolite consumption or production, and highest histological scores, indicating that the network was able to recognize patterns amongst diverse measurement outcomes. These results demonstrated the utility of artificial neural network tools for recognizing relationships in systems with competing parameters, toward the goal of optimizing and accelerating the design of biomaterial scaffolds for articular cartilage tissue engineering.

A RADIAL BIPHASIC MODEL FOR LOCAL CELL-MATRIX MECHANICS IN ARTICULAR CARTILAGE ∗

Analytical and numerical solutions are presented for an interface problem that models deformation in the local cell-matrix unit (chondron) of articular cartilage. The cell and its protective pericellular matrix layer are modeled as isotropic biphasic continua deforming in small strain. A spherical geometry with purely radial deformation is assumed. Enforcement of the boundary and interface conditions results in an eigenvalue problem that is self-adjoint when the permeabilities of the cell and the layer are the same. In this case, a series solution of the interface problem is presented for a time-varying displacement prescribed at the boundary of the pericellular layer. The case of nonuniform permeability is considered via a numerical finite difference solution. The analytical and numerical solutions are used to conduct a parametric analysis of mechanical signal transmission due to an applied sinusoidal displacement. The dual role of the pericellular matrix as a mechanical signal transmitter and a protecti...

Boundary-Integral Calculations of Two-Dimensional Electromagnetic Scattering in Infinite Photonic Crystal Slabs: Channel Defects and Resonances

We compute the transmission of two-dimensional (2D) electromagnetic waves through a square lattice of lossless dielectric rods with a channel defect. The lattice is finite in the direction of propagation of the incident wave and periodic in a transverse direction. We revisit a boundary-integral formulation of 2D electromagnetic scattering [Venakides, Haider, and Papanicolaou, SIAM J. Appl. Math., 60 (2000), pp. 1686--1706] that is Fredholm of the first kind and develop a second-kind formulation. We refine the numerical implementation in the above paper by exploiting separability in the Greens function to evaluate the far-field influence more efficiently. The resulting cost savings in computing and solving the discretized linear system leads to an accelerated method. We use it to analyze E-polarized electromagnetic scattering of normally incident waves on a structure with a periodic channel defect. We find three categories of resonances: waveguide modes in the channel, high-amplitude fields in the crystal...