Stephen M. Klisch

Contribution of Proteoglycan Osmotic Swelling Pressure to the Compressive Properties of Articular Cartilage

EunHee Han et al. — Tue, 24 Jan 2012 14:12:52 PST

The negatively charged proteoglycans (PG) provide compressive resistance to articular cartilage by means of their fixed charge density (FCD) and high osmotic pressure (π_PG), and the collagen network (CN) provides the restraining forces to counterbalance π_PG. Our objectives in this work were to: 1), account for collagen intrafibrillar water when transforming biochemical measurements into a FCD-π_PG relationship; 2), compute π_PG and CN contributions to the compressive behavior of full-thickness cartilage during bovine growth (fetal, calf, and adult) and human adult aging (young and old); and 3), predict the effect of depth from the articular surface on π_PG in human aging. Extrafibrillar FCD (FCD_EF) and π_PGincreased with bovine growth due to an increase in CN concentration, whereas PG concentration was steady. This maturation-related increase was amplified by compression. With normal human aging, FCD_EF and π_PG decreased. The π_PG-values were close to equilibrium stress (σ_EQ) in all bovine and young human cartilage, but were only approximately half of σ_EQ in old human cartilage. Depthrelated variations in the strain, FCD_EF,π_PG, and CN stress profiles in human cartilage suggested a functional deterioration of the superficial layer with aging. These results suggest the utility of the FCD-π_PG relationship for elucidating the contribution of matrix macromolecules to the biomechanical properties of cartilage.

Modeling the collagen fibril network of biological tissues as a nonlinearly elastic material using a continuous volume fraction distribution function

Reza Shirazi et al. — Wed, 21 Dec 2011 14:40:16 PST

Despite distinct mechanical functions, biological soft tissues have a common microstructure in which a ground matrix is reinforced by a collagen fibril network. The microstructural properties of the collagen network contribute to continuum mechanical tissue properties that are strongly anisotropic with tensile-compressive asymmetry. In this study, a novel approach based on a continuous distribution of collagen fibril volume fractions is developed to model fibril reinforced soft tissues as nonlinearly elastic and anisotropic material. Compared with other approaches that use a normalized number of fibrils for the definition of the distribution function, this representation is based on a distribution parameter (i.e. volume fraction) that is commonly measured experimentally while also incorporating pre-stress of the collagen fibril network in a tissue natural configuration. After motivating the form of the collagen strain energy function, examples are provided for two volume fraction distribution functions. Consequently, collagen second-Piola Kirchhoff stress and elasticity tensors are derived, first in general form and then specifically for a model that may be used for immature bovine articular cartilage. It is shown that the proposed strain energy is a convex function of the deformation gradient tensor and, thus, is suitable for the formation of a polyconvex tissue strain energy function.

Differential Regulation of Immature Articular Cartilage Compressive Moduli and Poisson's Ratios by in Vitro Stimulation with IGF-1 and TGF-β1

Gregory M. Williams et al. — Wed, 21 Dec 2011 14:40:12 PST

Mechanisms of articular cartilage growth and maturation have been elucidated by studying composition-function dynamics during in vivo development and in vitro culture with stimuli such as insulin-like growth factor-1 (IGF-1) and transforming growth factor-beta 1 (TGF-β1). This study tested the hypothesis that IGF-1 and TGF-β1 regulate immature cartilage compressive moduli and Poisson’s ratios in a manner consistent with known effects on tensile properties. Bovine calf articular cartilage from superficial-articular (S) and middle-growth (M) regions were analyzed fresh or following culture in medium with IGF-1 or TGF-β1. Mechanical properties in confined (CC) and unconfined (UCC) compression, cartilage matrix composition, and explant size were assessed. Culture with IGF-1 resulted in softening in CC and UCC, increased Poisson’s ratios, substantially increased tissue volume, and accumulation of glycosaminoglycan (GAG) and collagen (COL). Culture with TGF-β1 promoted maturational changes in the S layer, including stiffening in CC and UCC and increased concentrations of GAG, COL, and pyridinoline crosslinks (PYR), but little growth. Culture of M layer explants with TGF-β1 was nearly homeostatic. Across treatment groups, compressive moduli in CC and UCC were positively related to GAG, COL, and PYR concentrations, while Poisson’s ratios were negatively related to concentrations of these matrix components. Thus, IGF-1 and TGF-β1 differentially regulate the compressive mechanical properties and size of immature articular cartilage in vitro. Prescribing tissue growth, maturation, or homeostasis by controlling the in vitro biochemical environment with such growth factors may have applications in cartilage repair and tissue engineering.

Differential Regulation of Immature Articular Cartilage Compressive Moduli and Poisson’s Ratios by in vitro Stimulation With IGF-1 and TGF-β1

Gregory M. Williams et al. — Thu, 07 Oct 2010 14:31:41 PDT

A Nonlinear Constituent Based Viscoelastic Model for Articular Cartilage and Analysis of Tissue Remodeling Due to Altered Glycosaminoglycan-Collagen Interactions

Gregory C. Thomas et al. — Fri, 16 Oct 2009 11:11:24 PDT

A constituent based nonlinear viscoelastic (VE) model was modified from a previous study (Vena, et al., 2006, “A Constituent-Based Model for the Nonlinear Viscoelastic Behavior of Ligaments,” J. Biomech. Eng., 128, pp. 449–457) to incorporate a glycosaminoglycan (GAG)-collagen (COL) stress balance using compressible elastic stress constitutive equations specific to articular cartilage (AC). For uniaxial loading of a mixture of quasilinear VE constituents, time constant and relaxation ratio equations are derived to highlight how a mixture of constituents with distinct quasilinear VE properties is one mechanism that produces a nonlinear VE tissue. Uniaxial tension experiments were performed with newborn bovine AC specimens before and after ~55% and ~85% GAG depletion treatment with guanidine. Experimental tissue VE parameters were calculated directly from stress relaxation data, while intrinsic COL VE parameters were calculated by curve fitting the data with the nonlinear VE model with intrinsic GAG viscoelasticity neglected. Select tissue and intrinsic COL VE parameters were significantly different from control and experimental groups and correlated with GAG content, suggesting that GAG-COL interactions exist to modulate tissue and COL mechanical properties. Comparison of the results from this and other studies that subjected more mature AC tissue to GAG depletion treatment suggests that the GAGs interact with the COL network in a manner that may be beneficial for rapid volumetric expansion during developmental growth while protecting cells from excessive matrix strains. Furthermore, the underlying GAG-COL interactions appear to diminish as the tissue matures, indicating a distinctive remodeling response during developmental growth.

A Theory of Volumetric Growth for Compressible Elastic Biological Materials

Stephen M. Klisch et al. — Wed, 01 Apr 2009 10:23:12 PDT

A general theory of volumetric growth for compressible elastic materials is presented. The authors derive a complete set of governing equations in the present configuration for an elastic material undergoing a continuous growth process. In particular, they obtain two constitutive restrictions from a work-energy principle. First, the authors show that a growing elastic material behaves as a Green-elastic material. Second, they obtain an expression that relates the stress power due to growth to the rate of energy change due to growth. Then, the governing equations for a small increment of growth are derived from the more general theory. The equations for the incremental growth boundary-value problem provide an intuitive description of the quantities that describe growth and are used to implement the theory. The main features of the theory are illustrated with specific examples employing two strain energy functions that have been used to model biological materials.

Internally Constrained Mixtures of Elastic Continua

Stephen M. Klisch — Wed, 01 Apr 2009 10:23:11 PDT

A treatment of internally constrained mixtures of elastic continua at a common temperature is developed. Internal constraints involving the deformation gradient tensors and the common mixture temperature are represented by a constraint manifold, and an internally constrained mixture of elastic continua is associated with each unique equivalence class of unconstrained mixtures. The example of intrinsic incompressibility of each constituent first proposed by Mills is discussed.

Bioengineering Cartilage Growth, Maturation, and Form

Gregory M. Williams et al. — Wed, 11 Mar 2009 15:21:16 PDT

Cartilage of articular joints grows and matures to achieve characteristic sizes, forms, and functional properties. Through these processes, the tissue not only serves as a template for bone growth but also yields mature articular cartilage providing joints with a low-friction, wear-resistant bearing material. The study of cartilage growth and maturation is a focus of both cartilage biologists and bioengineers with one goal of trying to create biologic tissue substitutes for the repair of damaged joints. Experimental approaches both in vivo and in vitro are being used to better understand the mechanisms and regulation of growth and maturation processes. This knowledge may facilitate the controlled manipulation of cartilage size, shape, and maturity to meet the criteria needed for successful clinical applications. Mathematical models are also useful tools for quantitatively describing the dynamically changing composition, structure and function of cartilage during growth and maturation and may aid the development of tissue engineering solutions. Recent advances in methods of cartilage formation and culture which control the size, shape, and maturity of these tissues are numerous and provide contrast to the physiologic development of cartilage.

Regulation of immature cartilage growth by IGF-I, TGF- β 1, BMP-7, and PDGF-AB: role of metabolic balance between fixed charge and collagen network

Anna Asanbaeva et al. — Wed, 11 Mar 2009 15:14:04 PDT

Cartilage growth may involve alterations in the balance between the swelling tendency of proteoglycans and the restraining function of the collagen network. Growth factors, including IGF-I, TGF-β1, BMP-7, and PDGF-AB, regulate chondrocyte metabolism and, consequently, may regulate cartilage growth. Immature bovine articular cartilage explants from the superficial and middle zones were incubated for 13 days in basal medium or medium supplemented with serum, IGF-I, TGF-β1, BMP-7, or PDGF-AB. Variations in tissue size, accumulation of proteoglycan and collagen, and tensile properties were assessed. The inclusion of serum, IGF-I, or BMP-7 resulted in expansive tissue growth, stimulation of proteoglycan deposition but not of collagen, and a diminution of tensile integrity. The regulation of cartilage metabolism by TGF-β1 resulted in tissue homeostasis, with maintenance of size, composition, and function. Incubation in basal medium or with PDGF-AB resulted in small volumetric and compositional changes, but a marked decrease in tensile integrity. These results demonstrate that the phenotype of cartilage growth, and the associated balance between proteoglycan content and integrity of the collagen network, is regulated differentially by certain growth factors.

Cartilage growth and remodeling: modulation of balance between proteoglycan and collagen network in vitro with β-aminopropionitrile

Anna Asanbaeva et al. — Wed, 11 Mar 2009 15:14:03 PDT

Objective

To examine the effect of β-aminopropionitrile (BAPN), an inhibitor of lysyl oxidase, on growth and remodeling of immature articular cartilage in vitro.

Design

Immature bovine articular cartilage explants from the superficial and middle layers were cultured for 13 days in serum-containing medium with or without BAPN. Variations in tissue size, accumulation of proteoglycan and collagen (COL), and tensile mechanical properties were assessed.

Results

The inclusion of serum resulted in expansive tissue growth, stimulation of proteoglycan and COL deposition, and a diminution of tensile integrity. Supplementation of medium with BAPN accentuated this phenotype in terms of a further increase in tissue size in explants from the superficial layer and further diminution of tensile integrity, without affecting the contents of proteoglycan and COL in explants from both the superficial and middle layers.

Conclusion

COL crosslinking is a major factor in modulating the phenotype of cartilage growth and the associated balance between proteoglycan content and integrity of the COL network.

A Mixture of Elastic Materials with Different Constituent Temperatures and Internal Constraints

Stephen M. Klisch — Wed, 11 Mar 2009 15:14:02 PDT

A novel treatment of the 2nd Law of Thermodynamics and the development of general thermomechanical constraints are introduced for a mixture of two elastic materials in which the constituents may have different temperatures. First, a homothermal quasi-static process at a common mixture temperature is introduced. Part I of the 2nd Law of Thermodynamics is invoked to assert that the Clausius integrals are path-independent, which leads to a prescription, or an identification, of the partial entropy functions. Then, two assumptions are introduced that establish the values of the partial entropy functions for general processes, including those for which the constituent temperatures are not equal. Constitutive restrictions are derived for path-independent processes from the mixture energy equation, and further constitutive restrictions are derived for general processes upon invoking the Clausius–Duhem inequality as a statement of Part II of the 2nd Law of Thermodynamics. The complete set of constitutive restrictions are then shown to equal those derived by other authors, a result which supports the adopted assumptions concerning the partial entropy functions for general processes. Then, an internal constraint involving the deformation gradient tensors and the constituent temperatures is represented by a constraint manifold, and an internally constrained mixture of elastic materials is associated with each unique equivalence class of unconstrained mixtures. The examples of a mixture constrained to have a common temperature and a mixture constrained by temperature-dependent intrinsic compressibility are discussed.

Application of a fiber-reinforced continuum theory to multiple deformations of the annulus fibrosus

Stephen M. Klisch et al. — Wed, 11 Mar 2009 15:14:02 PDT

Accurate tissue stress predictions for the annulus fibrosus are essential for understanding the factors that cause or contribute to disc degeneration and mechanical failure. Current computational models used to predict in vivo disc stresses utilize material laws for annular tissue that are not rigorously validated against experimental data. Consequently, predictions of disc stress resulting from physical activities may be inaccurate and therefore unreliable as a basis for defining mechanical–biologic injury criteria. To address this need we present a model for the annulus as an isotropic ground substance reinforced with two families of collagen fibers, and an approach for determining the material constants by simultaneous consideration of multiple experimental data sets. Two strain energy functions for the annulus are proposed and used in the theory to derive the constitutive equations relating the stress to pure stretch deformations. These equations are applied to four distinct experimental protocols and the material constants are determined from a simultaneous, nonlinear regression analysis. Good agreement between theory and experiment is achieved when the invariants are included within multiple, separate exponentials in the strain energy function.

A Bimodular Theory for Finite Deformations: Comparison of orthotropic second-order and exponential stress constitutive equations for articular cartilage

Stephen M. Klisch — Wed, 11 Mar 2009 15:02:58 PDT

Cartilaginous tissues, such as articular cartilage and the annulus fibrosus, exhibit orthotropic behavior with highly asymmetric tensile–compressive responses. Due to this complex behavior, it is difficult to develop accurate stress constitutive equations that are valid for finite deformations. Therefore, we have developed a bimodular theory for finite deformations of elastic materials that allows the mechanical properties of the tissue to differ in tension and compression. In this paper, we derive an orthotropic stress constitutive equation that is second-order in terms of the Biot strain tensor as an alternative to traditional exponential type equations. Several reduced forms of the bimodular second-order equation, with six to nine parameters, and a bimodular exponential equation, with seven parameters, were fit to an experimental dataset that captures the highly asymmetric and orthotropic mechanical response of cartilage. The results suggest that the bimodular second-order models may be appealing for some applications with cartilaginous tissues.

A cartilage growth mixture model for infinitesimal strains: solutions of boundary-value problems related to in vitro growth experiments

Stephen M. Klisch et al. — Wed, 11 Mar 2009 15:02:58 PDT

A cartilage growth mixture (CGM) model is linearized for infinitesimal elastic and growth strains. Parametric studies for equilibrium and nonequilibrium boundary-value problems representing the in vitro growth of cylindrical cartilage constructs are solved. The results show that the CGM model is capable of describing the main biomechanical features of cartilage growth. The solutions to the equilibrium problems reveal that tissue composition, constituent pre-stresses, and geometry depend on collagen remodeling activity, growth symmetry, and differential growth. Also, nonhomogeneous growth leads to nonhomogeneous tissue composition and constituent pre-stresses. The solution to the nonequilibrium problem reveals that the tissue is nearly in equilibrium at all time points. The results suggest that the CGM model may be used in the design of tissue engineered cartilage constructs for the repair of cartilage defects; for example, to predict how dynamic mechanical loading affects the development of nonuniform properties during in vitro growth. Furthermore, the results lay the foundation for future analyses with nonlinear models that are needed to develop realistic models of cartilage growth.

A Nonlinear Finite Element Model of Cartilage Growth

Andrew Davol et al. — Wed, 11 Mar 2009 14:58:38 PDT

The long range objective of this work is to develop a cartilage growth finite element model (CGFEM), based on the theories of growing mixtures that has the capability to depict the evolution of the anisotropic and inhomogeneous mechanical properties, residual stresses, and nonhomogeneities that are attained by native adult cartilage. The CGFEM developed here simulates isotropic in vitro growth of cartilage with and without mechanical stimulation. To accomplish this analysis a commercial finite element code (ABAQUS) is combined with an external program (MATLAB) to solve an incremental equilibrium boundary value problem representing one increment of growth. This procedure is repeated for as many increments as needed to simulate the desired growth protocol. A case study is presented utilizing a growth law dependent on the magnitude of the diffusive fluid velocity to simulate an in vitro dynamic confined compression loading protocol run for 2 weeks. The results include changes in tissue size and shape, nonhomogeneities that develop in the tissue, as well as the variation that occurs in the tissue constitutive behavior from growth.

Volumetric Growth of Thermoelastic Materials and Mixtures

Stephen M. Klisch et al. — Wed, 11 Mar 2009 14:58:38 PDT

The proteoglycan and collagen constituents of cartilage serve distinct mechanical roles. Changes to the mechanical loading conditions during cartilage growth lead to changes in the concentrations of these molecules and, consequently, the mechanical properties. The main aim of this paper is to present a theory that can describe the mechanical aspects of cartilage growth. The model for cartilage growth is based on a general thermomechanical theory for a mixture of an arbitrary number of growing elastic constituents and an inviscid fluid. Our development of a growth mixture theory is accomplished in two steps. First, the thermodynamics of growing elastic materials are considered. The resulting theory of growing thermoelastic materials is extended to continuum mixture theory. Using this general growth mixture theory, we then propose a cartilage growth model that includes two special types of internal constraints that are relevant to cartilage.

Articular Cartilage Mechanical and Biochemical Property Relations Before and After In Vitro Growth

Timothy Ficklin et al. — Wed, 11 Mar 2009 14:58:37 PDT

The aim of this study was to design in vitro growth protocols that can comprehensively quantify articular cartilage structure–function relations via measurement of mechanical and biochemical properties. Newborn bovine patellofemoral groove articular cartilage explants were tested sequentially in confined compression (CC), unconfined compression (UCC), and torsional shear before (D0, i.e. day zero) and after (D14, i.e. day 14) unstimulated in vitro growth. The contents of collagen (COL), collagen-specific pyridinoline (PYR) crosslinks, glycosaminoglycan, and DNA significantly decreased during in vitro growth; consequently, a wide range of biochemical properties existed for investigating structure–function relations when pooling the D0 and D14 groups. All D0 mechanical properties were independent of compression strain while only Poisson's ratios were dependent on direction (i.e. anisotropic). Select D0 and D14 group mechanical properties were correlated with biochemical measures; including (but not limited to) results that CC/UCC moduli and UCC Poisson's ratios were correlated with COL and PYR. COL network weakening during in vitro growth due to reduced COL and PYR was accompanied by reduced CC/UCC moduli and increased UCC Poisson's ratios.

A Bimodular Polyconvex Anisotropic Strain Energy Function for Articular Cartilage

Stephen M. Klisch — Thu, 05 Mar 2009 16:02:45 PST

A strain energy function for finite deformations is developed that has the capability to describe the nonlinear, anisotropic, and asymmetric mechanical response that is typical of articular cartilage. In particular, the bimodular feature is employed by including strain energy terms that are only mechanically active when the corresponding fiber directions are in tension. Furthermore, the strain energy function is a polyconvex function of the deformation gradient tensor so that it meets material stability criteria. A novel feature of the model is the use of bimodular and polyconvex “strong interaction terms” for the strain invariants of orthotropic materials. Several regression analyses are performed using a hypothetical experimental dataset that captures the anisotropic and asymmetric behavior of articular cartilage. The results suggest that the main advantage of a model employing the strong interaction terms is to provide the capability for modeling anisotropic and asymmetric Poisson's ratios, as well as axial stress–axial strain responses, in tension and compression for finite deformations.

Articular Cartilage Tensile Integrity: Modulation by matrix depletion is maturation-dependent

Anna Asanbaeva et al. — Thu, 05 Mar 2009 16:02:44 PST

Articular cartilage function depends on the molecular composition and structure of its extracellular matrix (ECM). The collagen network (CN) provides cartilage with tensile integrity, but must also remodel during growth. Such remodeling may depend on matrix molecules interacting with the CN to modulate the tensile behavior of cartilage. The objective of this study was to determine the effects of increasingly selective matrix depletion on tensile properties of immature and mature articular cartilage, and thereby establish a framework for identifying molecules involved in CN remodeling. Depletion of immature cartilage with guanidine, chondroitinase ABC, chondroitinase AC, and Streptomyces hyaluronidase markedly increased tensile integrity, while the integrity of mature cartilage remained unaltered after depletion with guanidine. The enhanced tensile integrity after matrix depletion suggests that certain ECM components of immature matrix serve to inhibit CN interactions and may act as modulators of physiological alterations of cartilage geometry and tensile properties during growth/maturation.

A Growth Mixture Theory for Cartilage With Application to Growth-Related Experiments on Cartilage Explants

Stephen M. Klisch et al. — Thu, 05 Mar 2009 16:02:44 PST

In this paper, we present a growth mixture model for cartilage. The main features of this model are illustrated in a simple equilibrium boundary-value problem that is chosen to illustrate how a mechanical theory of cartilage growth may be applied to growth-related experiments on cartilage explants. The cartilage growth mixture model describes the independent growth of the proteoglycan and collagen constituents due to volumetric mass deposition, which leads to the remodeling of the composition and the mechanical properties of the solid matrix. The model developed here also describes how the material constants of the collagen constituent depend on a scalar parameter that may change over time (e.g., crosslink density); this leads to a remodeling of the structural and mechanical properties of the collagen constituent. The equilibrium boundary-value problem that describes the changes observed in cartilage explants harvested at different stages of a growth or a degenerative process is formulated. This boundary-value problem is solved using existing experimental data for developing bovine cartilage explants harvested at three developmental stages. The solution of the boundary-value problem in conjunction with existing experimental data suggest the types of experimental studies that need to be conducted in the future to determine model parameters and to further refine the model.