Stephen M. Klisch

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

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.

A Theory of Volumetric Growth for Compressible Elastic Biological Materials

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.

A Special Theory of Biphasic Mixtures and Experimental Results for Human Annulus Fibrosus Tested in Confined Compression

A finite deformation mixture theory is used to quantify the mechanical properties of the annulus fibrosus using experimental data obtained from a confined compression protocol. Certain constitutive assumptions are introduced to derive a special mixture of an elastic solid and an inviscid fluid, and the constraint of intrinsic incompressibility is introduced in a manner that is consistent with results obtained for the special theory. Thirty-two annulus fibrosus specimens oriented in axial (n = 16) and radial (n = 16) directions were obtained from the middle-lateral portion of intact intervertebral discs from human lumbar spines and tested in a stress-relaxation protocol. Material constants are determined by fitting the theory to experimental data representing the equilibrium stress versus stretch and the surface stress time history curves. No significant differences in material constants due to orientation existed, but significant differences existed due to the choice of theory used to fit the data. In comparison with earlier studies with healthy annular tissue, we report a lower aggregate modulus and a higher initial permeability constant. These differences are explained by the choice of reference configuration for the experimental studies.

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

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(PG) increased 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. Depth-related 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.

ARTICULAR CARTILAGE TENSILE INTEGRITY: MODULATION BY MATRIX DEPLETION IS MATURATION-DEPENDENT

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.

Volumetric Growth of Thermoelastic Materials and Mixtures

Abstract: 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.

Bioengineering Cartilage Growth, Maturation, and Form

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.

A Cartilage Growth Mixture Model With Collagen Remodeling: Validation Protocols

A cartilage growth mixture (CGM) model is proposed to address limitations of a model used in a previous study. New stress constitutive equations for the solid matrix are derived and collagen (COL) remodeling is incorporated into the CGM model by allowing the intrinsic COL material constants to evolve during growth. An analytical validation protocol based on experimental data from a recent in vitro growth study is developed. Available data included measurements of tissue volume, biochemical composition, and tensile modulus for bovine calf articular cartilage (AC) explants harvested at three depths and incubated for 13 days in 20% fetal borine serum (FBS) and 20% FBS+beta-aminopropionitrile. The proposed CGM model can match tissue biochemical content and volume exactly while predicting theoretical values of tensile moduli that do not significantly differ from experimental values. Also, theoretical values of a scalar COL remodeling factor are positively correlated with COL cross-link content, and mass growth functions are positively correlated with cell density. The results suggest that the CGM model may help us to guide in vitro growth protocols for AC tissue via the a priori prediction of geometric and biomechanical properties.

Mechanical Characterization of Native and Tissue-Engineered Cartilage

Cartilage functions as a low-friction, wear-resistant, load-bearing tissue. During a normal gait cycle, one cartilage surface rolls and slides against another, all the while being loaded and unloaded. The durability of the tissue also makes for an impressive material to study. However, when cartilage is damaged or diseased, the tissue has little capacity to repair itself. The goal of cell-based repair strategies to replace damaged or diseased tissue requires that the functional biomechanical properties of normal (developing or mature), diseased, and repair cartilage be restored. This chapter addresses some of the major methods used to assess the biomechanical properties of native and tissue-engineered cartilage. First, the traditional methods of testing by compression, tension, shear, and indentation are reviewed. Next, additional methods to evaluate interfacial mechanics and lubrication are described. Thus, a variety of mechanical tests may be used to assess functional properties for normal, diseased, and tissue-engineered cartilage.

Internally Constrained Mixtures of Elastic Continua

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.