Modeling the dynamic composition of engineered cartilage

Mechanics of 3D Cell-Hydrogel Interactions: Experiments, Models, and Mechanisms

Hydrogels are highly water-swollen molecular networks that are ideal platforms to create tissue mimetics owing to their vast and tunable properties. As such, hydrogels are promising cell-delivery vehicles for applications in tissue engineering and have also emerged as an important base for ex vivo models to study healthy and pathophysiological events in a carefully controlled three-dimensional environment. Cells are readily encapsulated in hydrogels resulting in a plethora of biochemical and mechanical communication mechanisms, which recapitulates the natural cell and extracellular matrix interaction in tissues. These interactions are complex, with multiple events that are invariably coupled and spanning multiple length and time scales. To study and identify the underlying mechanisms involved, an integrated experimental and computational approach is ideally needed. This review discusses the state of our knowledge on cell-hydrogel interactions, with a focus on mechanics and transport, and in this context, highlights recent advancements in experiments, mathematical and computational modeling. The review begins with a background on the thermodynamics and physics fundamentals that govern hydrogel mechanics and transport. The review focuses on two main classes of hydrogels, described as semiflexible polymer networks that represent physically cross-linked fibrous hydrogels and flexible polymer networks representing the chemically cross-linked synthetic and natural hydrogels. In this review, we highlight five main cell-hydrogel interactions that involve key cellular functions related to communication, mechanosensing, migration, growth, and tissue deposition and elaboration. For each of these cellular functions, recent experiments and the most up to date modeling strategies are discussed and then followed by a summary of how to tune hydrogel properties to achieve a desired functional cellular outcome. We conclude with a summary linking these advancements and make the case for the need to integrate experiments and modeling to advance our fundamental understanding of cell-matrix interactions that will ultimately help identify new therapeutic approaches and enable successful tissue engineering.

Application of flexural and membrane stress analysis to distinguish tensile and compressive moduli of biologic materials

Three-point bending is often used during the mechanical determination of tissue material properties. When taken to failure, the test samples often experience high deformations. The objective of this study was to present beam and plate theories as analytical tools for determining tensile and compressive elastic moduli during the transition from flexure to membrane stress states. Samples of cartilage, a highly flexible connective tissue having differing tensile and compressive moduli, were tested. Three-point bending tests were conducted on auricular (ear) and costal (rib) cartilage harvested from pigs. The influence of span length variation and Poisson's ratio assumptions were statistically assessed. Tensile elastic moduli of the ear (3.886 MPa) and rib (6.131 MPa) were derived from high-deformation bending tests. The functional assessment described here can be applied as a design input approach for tissue reconstruction and tissue engineering, considering both hard and soft tissue applications.

A Hybrid Model of Cartilage Regeneration Capturing the Interactions Between Cellular Dynamics and Porosity

To accelerate the development of strategies for cartilage tissue engineering, models are necessary to investigate the interactions between cellular dynamics and the local microenvironment. We use a discrete framework to capture the individual behavior of cells, modeling experiments where cells are seeded in a porous scaffold or hydrogel and over the time course of a month, the scaffold slowly degrades while cells divide and synthesize extracellular matrix constituents. The movement of cells and the ability to proliferate is a function of the local porosity, defined as the volume fraction of fluid in the surrounding region. A phenomenological approach is used to capture a continuous profile for the degrading scaffold and accumulating matrix, which will then change the local porosity throughout the construct. We parameterize the model by first matching total cell counts in the construct to chondrocytes seeded in a polyglycolic acid scaffold (Freed et al. in Biotechnol Bioeng 43:597-604, 1994). We investigate the influence of initial scaffold porosity on the total cell count and spatial profiles of cell and ECM in the construct. Cell counts were higher at day 30 in scaffolds of lower initial porosity, and similar cell counts were obtained using different models of scaffold degradation and matrix accumulation (either uniform or cell-specific). Using this modeling framework, we study the interplay between a phenomenological representation of scaffold architecture and porosity as well as the potential continuous application of growth factors. We determine parameter regimes where large cellular aggregates occur, which can hinder matrix accumulation and cellular proliferation. The developed modeling framework can easily be extended and can be used to identify optimal scaffolds and culture conditions that lead to a desired distribution of extracellular matrix and cell counts throughout the construct.

Comparison of patient-specific computational models vs. clinical follow-up, for adjacent segment disc degeneration and bone remodelling after spinal fusion

Spinal fusion is a standard surgical treatment for patients suffering from low back pain attributed to disc degeneration. However, results are somewhat variable and unpredictable. With fusion the kinematic behaviour of the spine is altered. Fusion and/or stabilizing implants carrying considerable load and prevent rotation of the fused segments. Associated with these changes, a risk for accelerated disc degeneration at the adjacent levels to fusion has been demonstrated. However, there is yet no method to predict the effect of fusion surgery on the adjacent tissue levels, i.e. bone and disc. The aim of this study was to develop a coupled and patient-specific mechanoregulated model to predict disc generation and changes in bone density after spinal fusion and to validate the results relative to patient follow-up data. To do so, a multiscale disc mechanoregulation adaptation framework was developed and coupled with a previously developed bone remodelling algorithm. This made it possible to determine extra cellular matrix changes in the intervertebral disc and bone density changes simultaneously based on changes in loading due to fusion surgery. It was shown that for 10 cases the predicted change in bone density and degeneration grade conforms reasonable well to clinical follow-up data. This approach helps us to understand the effect of surgical intervention on the adjacent tissue remodelling. Thereby, providing the first insight for a spine surgeon as to which patient could potentially be treated successfully by spinal fusion and in which patient has a high risk for adjacent tissue changes.

Heterogeneity is key to hydrogel-based cartilage tissue regeneration

Degradable hydrogels have been developed to provide initial mechanical support to encapsulated cells while facilitating the growth of neo-tissues. When cells are encapsulated within degradable hydrogels, the process of neo-tissue growth is complicated by the coupled phenomena of transport of large extracellular matrix macromolecules and the rate of hydrogel degradation. If hydrogel degradation is too slow, neo-tissue growth is hindered, whereas if it is too fast, complete loss of mechanical integrity can occur. Therefore, there is a need for effective modelling techniques to predict hydrogel designs based on the growth parameters of the neo-tissue. In this article, hydrolytically degradable hydrogels are investigated due to their promise in tissue engineering. A key output of the model focuses on the ability of the construct to maintain overall structural integrity as the construct transitions from a pure hydrogel to engineered neo-tissue. We show that heterogeneity in cross-link density and cell distribution is the key to this successful transition and ultimately to achieve tissue growth. Specifically, we find that optimally large regions of weak cross-linking around cells in the hydrogel and well-connected and dense cell clusters create the optimum conditions needed for neo-tissue growth while maintaining structural integrity. Experimental observations using cartilage cells encapsulated in a hydrolytically degradable hydrogel are compared with model predictions to show the potential of the proposed model.

Customized biomaterials to augment chondrocyte gene therapy

A persistent challenge in enhancing gene therapy is the transient availability of the target gene product. This is particularly true in tissue engineering applications. The transient exposure of cells to the product could be insufficient to promote tissue regeneration. Here we report the development of a new material engineered to have a high affinity for a therapeutic gene product. We focus on insulin-like growth factor-I (IGF-I) for its highly anabolic effects on many tissues such as spinal cord, heart, brain and cartilage. One of the ways that tissues store IGF-I is through a group of insulin like growth factor binding proteins (IGFBPs), such as IGFBP-5. We grafted the IGF-I binding peptide sequence from IGFBP-5 onto alginate in order to retain the endogenous IGF-I produced by transfected chondrocytes. This novel material bound IGF-I and released the growth factor for at least 30days in culture. We found that this binding enhanced the biosynthesis of transfected cells up to 19-fold. These data demonstrate the coordinated engineering of cell behavior and material chemistry to greatly enhance extracellular matrix synthesis and tissue assembly, and can serve as a template for the enhanced performance of other therapeutic proteins.

STATEMENT OF SIGNIFICANCE

The present manuscript focuses on the enhancement of chondrocyte gene therapy through the modification of scaffold materials to enhance the retention of targeted gene products. This study combined tissue engineering and gene therapy, where customized biomaterials augmented the action of IGF-I by enhancing the retention of protein produced by transfection of the IGF-I gene. This approach enabled tuning of binding of IGF-I to alginate, which increased GAG and HYPRO production by transfected chondrocytes. To our knowledge, peptide-based modification of materials to augment growth factor-targeted gene therapy has not been reported previously.

A mathematical model of tissue-engineered cartilage development under cyclic compressive loading

In this work a coupled model of solute transport and uptake, cell proliferation, extracellular matrix synthesis and remodeling of mechanical properties accounting for the impact of mechanical loading is presented as an advancement of a previously validated coupled model for free-swelling tissue-engineered cartilage cultures. Tissue-engineering constructs were modeled as biphasic with a linear elastic solid, and relevant intrinsic mechanical stimuli in the constructs were determined by numerical simulation for use as inputs of the coupled model. The mechanical dependent formulations were derived from a calibration and parametrization dataset and validated by comparison of normalized ratios of cell counts, total glycosaminoglycans and collagen after 24-h continuous cyclic unconfined compression from another dataset. The model successfully fit the calibration dataset and predicted the results from the validation dataset with good agreement, with average relative errors up to 3.1 and 4.3 %, respectively. Temporal and spatial patterns determined for other model outputs were consistent with reported studies. The results suggest that the model describes the interaction between the simultaneous factors involved in in vitro tissue-engineered cartilage culture under dynamic loading. This approach could also be attractive for optimization of culture protocols, namely through the application to longer culture times and other types of mechanical stimuli.

Tuning tissue growth with scaffold degradation in enzyme-sensitive hydrogels: a mathematical model

Despite tremendous advances in the field of tissue engineering, a number of obstacles remain that hinder its successful translation to the clinic. One challenge that relates to the use of cells encapsulated in a hydrogel is identifying a hydrogel design that can provide an appropriate environment for cells to successfully synthesize and deposit new matrix molecules while providing a mechanical support that can resist physiological loads at the early stage of implementation. A solution to this problem has been to balance tissue growth and hydrogel degradation. However, identifying this balance is difficult due to the complexity of coupling diffusion, deposition, and degradation mechanisms. Very little is known about the complex behavior of these mechanisms, emphasizing the need for a rigorous mathematical approach that can assist and guide experimental advances. To address this issue, this paper discusses a model for interstitial growth based on mixture theory, that can capture the coupling between cell-mediated hydrogel degradation (i.e., hydrogels containing enzyme-sensitive crosslinks) and the transport of extracellular matrix (ECM) molecules released by encapsulated cells within a hydrogel. Taking cartilage tissue engineering as an example, the model investigates the role of enzymatic degradation on ECM diffusion and its impact on two important outcomes: the extent of ECM transport (and deposition) and the evolution of the hydrogel's mechanical integrity. Numerical results based on finite element analysis show that if properly tuned, enzymatic degradation yields the appearance of a highly localized degradation front propagating away from the cell, which can be immediately followed by a front of growing neotissue. We show that this situation is key to maintaining mechanical properties (e.g., stiffness) while allowing for deposition of new ECM molecules. Overall, our study suggests a hydrogel design that could enable successful tissue engineering (e.g., of cartilage, bone, etc.) where mechanical integrity is important.

Nanostructured injectable cell microcarriers for tissue regeneration

Biodegradable polymer microspheres have emerged as cell carriers for the regeneration and repair of irregularly shaped tissue defects due to their injectability, controllable biodegradability and capacity for drug incorporation and release. Notably, recent advances in nanotechnology allowed the manipulation of the physical and chemical properties of the microspheres at the nanoscale, creating nanostructured microspheres mimicking the composition and/or structure of natural extracellular matrix. These nanostructured microspheres, including nanocomposite microspheres and nanofibrous microspheres, have been employed as cell carriers for tissue regeneration. They enhance cell attachment and proliferation, promote positive cell-carrier interactions and facilitate stem cell differentiation for target tissue regeneration. This review highlights the recent advances in nanostructured microspheres that are employed as injectable, biomimetic and cell-instructive cell carriers.

Extracellular Matrix Deposition in Engineered Micromass Cartilage Pellet Cultures: Measurements and Modelling

This article explores possible mechanisms governing extracellular matrix deposition in engineered cartilaginous cell pellets. A theoretical investigation is carried out alongside an experimental study measuring proteoglycan and collagen volume fractions within murine chondrogenic (ATDC-5) cell pellets. The simple mathematical model, which adopts a nutrient-dependent proteoglycan production rate, successfully reproduces the periphery-dominated proteoglycan deposition, characteristic of the growth pattern observed experimentally within pellets after 21 days of culture. The results suggest that this inhomogeneous proteoglycan production is due to nutrient deficiencies at the pellet centre. Our model analysis further indicates that a spatially uniform distribution of proteoglycan matrix could be maintained by initiating the culture process with a smaller-sized pellet. Finally, possible extensions are put forward with an aim to improve the model predictions for the early behaviour, where different mechanisms appear to dominate the matrix production within the pellets.

Influence of the scaffold geometry on the spatial and temporal evolution of the mechanical properties of tissue-engineered cartilage: insights from a mathematical model

The production of tissue-engineered cartilage in vitro with inhomogeneous mechanical properties is a problem yet to be solved. Different geometries have been studied to overcome this caveat; however, the reported measurements are limited to average values of some properties and qualitative measures of spatial distributions. We will apply a coupled model to extend knowledge about the introduction of a macrochannel in a scaffold by calculating spatiotemporal patterns for several interest variables related to the remodeling of the mechanical properties. Model parameters were estimated based on experimental data on the temporal patterns of glycosaminoglycans, collagen and compressive Young's modulus for channel-free constructs. The model reproduced the experimental data trends in both geometries, with experimental-numerical correlations between 0.84 and 0.97. The channel had a higher impact on the reduction in spatial heterogeneities and delay of saturation of core properties than in the improvement of average properties. Despite the possible improvement of cell densities for longer periods than 56 days, it is estimated that it will not cause further significant improvements of the mechanical properties. The degrees of spatial heterogeneity of the Young's modulus and permeability in the channeled geometry are 23 and 27 % of the channel-free values. While the average Young's modulus values are in the range of native cartilage, the permeabilities are one to three degrees of magnitude higher than the native cartilage, suggesting that limiting factors such as scaffold porosity and initial permeability are more relevant than scaffold geometry to effectively decrease the tissue permeability.

A novel scaffold geometry for chondral applications: theoretical model and in vivo validation

A theoretical model of the 3D scaffold internal architecture has been implemented with the aim to predict the effects of some geometrical parameters on total porosity, Young modulus, buckling resistance and permeability of the graft. This model has been adopted to produce porous poly-caprolacton based grafts for chondral tissue engineering applications, best tuning mechanical and functional features of the scaffolds. Material prototypes were produced with an internal geometry with parallel oriented cylindrical pores of 200 μm of radius (r) and an interpore distance/pores radius (d/r) ratio of 1. The scaffolds have been then extensively characterized; progenitor cells were then used to test their capability to support cartilaginous matrix deposition in an ectopic model. Scaffold prototypes fulfill both the chemical-physical requirements, in terms of Young's modulus and permeability, and the functional needs, such as surface area per volume and total porosity, for an enhanced cellular colonization and matrix deposition. Moreover, the grafts showed interesting chondrogenic potential in vivo, besides offering adequate mechanical performances in vitro, thus becoming a promising candidate for chondral tissues repair. Finally, a very good agreement was found between the prediction of the theoretical model and the experimental data. Many assumption of this theoretical model, hereby applied to cartilage, may be transposed to other tissue engineering applications, such as bone substitutes.

A phase-field model for articular cartilage regeneration in degradable scaffolds

Degradable scaffolds represent a promising solution for tissue engineering of damaged or degenerated articular cartilage which due to its avascular nature, is characterized by a low self-repair capacity. To estimate the articular cartilage regeneration process employing degradable scaffolds, we propose a mathematical model as the extension of Olson and Haider's work (Int. J. Pure Appl. Math. 53:333-353, 2009). The simulated tissue engineering procedure consists in (i) the explant of a cylindrical sample, (ii) the removal of the inner core region, and (iii) the filling of the inner region with hydrogels, degradable scaffolds enriched with nutrients, such as oxygen and glucose. The phase-field model simulates the cartilage regeneration process at the scaffold-cartilage interface. It embeds reaction-diffusion equations, which are used to model the nutrient and regenerated extracellular matrix. The equations are solved using an unconditionally stable hybrid numerical scheme. Cartilage repair processes with full-thickness defects, which are controlled by properties of hydrogel materials and cartilage explant culture based on biological interest are observed. The implemented mathematical model shows the capability to simulate cartilage repairing processes, which can be virtually controlled evaluating hydrogel and cartilage material properties including nutrient supply and defected magnitude. In particular, the adopted methodology is able to explain the regeneration time of cartilage within hydrogel environments. With the numerical scheme, the numerical simulations are demonstrated for the potential improvement of hydrogel structures.

Novel technique for online characterization of cartilaginous tissue properties

The goal of tissue engineering is to use substitutes to repair and restore organ function. Bioreactors are an indispensable tool for monitoring and controlling the unique environment for engineered constructs to grow. However, in order to determine the biochemical properties of engineered constructs, samples need to be destroyed. In this study, we developed a novel technique to nondestructively online-characterize the water content and fixed charge density of cartilaginous tissues. A new technique was developed to determine the tissue mechano-electrochemical properties nondestructively. Bovine knee articular cartilage and lumbar annulus fibrosus were used in this study to demonstrate that this technique could be used on different types of tissue. The results show that our newly developed method is capable of precisely predicting the water volume fraction (less than 3% disparity) and fixed charge density (less than 16.7% disparity) within cartilaginous tissues. This novel technique will help to design a new generation of bioreactors which are able to actively determine the essential properties of the engineered constructs, as well as regulate the local environment to achieve the optimal conditions for cultivating constructs.

A cell-matrix model of anabolic and catabolic dynamics during cartilage biomolecule regulation

Physiologic regulation of extracellular matrix (ECM) in articular cartilage tissue is controlled by cellular and molecular mechanisms which are not fully understood. It has been observed that the synthesis of the ECM structural molecules, glycosaminoglycan and collagen are promoted by growth factors such as IGF-1 and TGF-β. Concomitant ECM degradation is promoted by a variety of cytokines such as IL-1. The clinical need for reparative therapies of articular cartilage is linked with its poor intrinsic healing capacity. The following modelling approach was applied to engineered cartilage as a platform for exploring cartilage biology and to introduce a predictive tool as a bioinformatic support system supporting regenerative therapies. Systems biology was adapted through a mathematical framework producing a computational intelligence paradigm to explore a controlled phasic regulatory influence of the inhibition and production of ECM biomolecules. Model outcomes describe a steady synthesis of ECM as a dependence on a cyclic influence of the catabolic action of proteases and anabolic action of growth factors. This relationship is shown quantitatively in a governing harmonic equation representing the simplified biological mechanisms of biomolecule homeostasis.

Modeling of bioreactor hydrodynamic environment and its effects on tissue growth

The design of optimal bioreactor systems for tissue engineering applications requires a sophisticated understanding of the complexities of the bioreactor environment and the role that it plays in the formation of engineered tissues. To this end, a tissue growth model is developed to characterize the tissue growth and extracellular matrix synthesis by chondrocytes seeded and cultivated on polyglycolic acid scaffolds in a wavy-walled bioreactor for a period of 4 weeks. This model consists of four components: (1) a computational fluid dynamics (CFD) model to characterize the complex hydrodynamic environment in the bioreactor, (2) a kinetic growth model to characterize the cell growth and extracellular matrix production dynamics, (3) an artificial neural network (ANN) that empirically correlates hydrodynamic parameters with kinetic constants, and (4) a second ANN that correlates the biochemical composition of constructs with their material properties. In tandem, these components enable the prediction of the dynamics of tissue growth, as well as the final compositional and mechanical properties of engineered cartilage. The growth model methodology developed in this study serves as a tool to predict optimal bioprocessing conditions required to achieve desired tissue properties.

Three-Dimensional Culture of Cells and Matrix Biomolecules for Engineered Tissue Development and Biokinetics Model Validation

There has been considerable progress in cellular and molecular engineering due to recent advances in multiscale technology. Such technologies allow controlled manipulation of physiochemical interactions among cells in tissue culture. In particular, a novel chemomechanical bioreactor has recently been designed for the study of bone and cartilage tissue development, with particular focus on extracellular matrix formation. The bioreactor is equally significant as a tool for validation of mathematical models that explore biokinetic regulatory thresholds (Saha, A. K., and Kohles, S. S., 2010, "A Distinct Catabolic to Anabolic Threshold Due to Single-Cell Nanomechanical Stimulation in a Cartilage Biokinetics Model," J. Nanotechnol. Eng. Med., 1(3), p. 031005; 2010, "Periodic Nanomechanical Stimulation in a Biokinetics Model Identifying Anabolic and Catabolic Pathways Associated With Cartilage Matrix Homeostasis," J. Nanotechnol. Eng. Med., 1(4), p. 041001). In the current study, three-dimensional culture protocols are described for maintaining the cellular and biomolecular constituents within defined parameters. Preliminary validation of the bioreactor's form and function, expected bioassays of the resulting matrix components, and application to biokinetic models are described. This approach provides a framework for future detailed explorations combining multiscale experimental and mathematical analyses, at nanoscale sensitivity, to describe cell and biomolecule dynamics in different environmental regimes.

Biokinetic Mechanisms Linked With Musculoskeletal Health Disparities: Stochastic Models Applying Tikhonov's Theorem to Biomolecule Homeostasis

Multiscale technology and advanced mathematical models have been developed to control and characterize physicochemical interactions, respectively, enhancing cellular and molecular engineering progress. Ongoing tissue engineering development studies have provided experimental input for biokinetic models examining the influence of static or dynamic mechanical stimuli (Saha, A. K., and Kohles, S. S., 2010, "A Distinct Catabolic to Anabolic Threshold Due to Single-Cell Nanomechanical Stimulation in a Cartilage Biokinetics Model," J. Nanotechnol. Eng. Med., 1(3) p. 031005; 2010, "Periodic Nanomechanical Stimulation in a Biokinetics Model Identifying Anabolic and Catabolic Pathways Associated With Cartilage Matrix Homeostasis," J. Nanotechnol. Eng. Med., 1(4), p. 041001). In the current study, molecular regulatory thresholds associated with specific disease disparities are further examined through applications of stochastic mechanical stimuli. The results indicate that chondrocyte bioregulation initiates the catabolic pathway as a secondary response to control anabolic processes. In addition, high magnitude loading produced as a result of stochastic input creates a destabilized balance in homeostasis. This latter modeled result may be reflective of an injurious state or disease progression. These mathematical constructs provide a framework for single-cell mechanotransduction and may characterize transitions between healthy and disease states.

The role of mass balance equations in growth mechanics illustrated in surface and volume dissolutions

Growth mechanics problems require the solution of mass balance equations that include supply terms and account for mass exchanges among constituents of a mixture. Though growth may often be accompanied by a variety of concomitant phenomena that increase modeling complexity, such as solid matrix deformation, evolving traction-free configurations, cell division, and active cell contraction, it is important to distinguish these accompanying phenomena from the fundamental growth process that consists of deposition or removal of mass from the solid matrix. Therefore, the objective of this study is to present a canonical problem of growth, namely, dissolution of a rigid solid matrix in a solvent. This problem illustrates a case of negative growth (loss of mass) of the solid in a mixture framework that includes three species, a solid, a solvent, and a solute, where the solute is the product of the solid dissolution. By analyzing both volumetric and surface dissolutions, the two fundamental modes of growth are investigated within the unified framework of mixture theory.

A phenomenological mixture model for biosynthesis and linking of cartilage extracellular matrix in scaffolds seeded with chondrocytes

A phenomenological mixture model is presented for interactions between biosynthesis of extracellular matrix (ECM) constituents and ECM linking in a scaffold seeded with chondrocytes. A system of three ordinary differential equations for average apparent densities of unlinked ECM, linked ECM and scaffold is developed along with associated initial conditions for scaffold material properties. Equations for unlinked ECM synthesis and ECM linking include an inhibitory mechanism where associated rates decrease as unlinked ECM concentration in the interstitial fluid increases. Linking rates are proposed to depend on average porosity in the evolving tissue construct. The resulting initial value problem contains nine independent parameters that account for scaffold biomaterial properties and interacting mechanisms in the engineered system. Effects of parameter variations on model variables are analyzed relative to a baseline case with emphasis on the evolution of solid phase apparent density, which is often correlated with the compressive elastic modulus of the tissue construct. The new model provides an additional quantitative framework for assessing and optimizing the design of engineered cell-scaffold systems and guiding strategies for articular cartilage tissue engineering.

Periodic Nanomechanical Stimulation in a Biokinetics Model Identifying Anabolic and Catabolic Pathways Associated With Cartilage Matrix Homeostasis

Enhancing the available nanotechnology to describe physicochemical interactions during biokinetic regulation will strongly support cellular and molecular engineering efforts. In a recent mathematical model developed to extend the applicability of a statically loaded, single-cell biomechanical analysis, a biokinetic regulatory threshold was presented (Saha and Kohles, 2010, "A Distinct Catabolic to Anabolic Threshold Due to Single-Cell Static Nanomechanical Stimulation in a Cartilage Biokinetics Model," J. Nanotechnol. Eng. Med., 1(3), p. 031005). Results described multiscale mechanobiology in terms of catabolic to anabolic pathways. In the present study, we expand the mathematical model to continue exploring the nanoscale biomolecular response within a controlled microenvironment. Here, we introduce a dynamic mechanical stimulus for regulating cartilage molecule synthesis. Model iterations indicate the identification of a biomathematical mechanism balancing the harmony between catabolic and anabolic states. Relative load limits were defined to distinguish between "healthy" and "injurious" biomolecule accumulations. The presented mathematical framework provides a specific algorithm from which to explore biokinetic regulation.

Scaffold percolative efficiency: in vitro evaluation of the structural criterion for electrospun mats

Fibrous scaffolds of engineered structures can be chosen as promising porous environments when an approved criterion validates their applicability for a specific medical purpose. For such biomaterials, this paper sought to investigate various structural characteristics in order to determine whether they are appropriate descriptors. A number of poly(3-hydroxybutyrate) scaffolds were electrospun; each of which possessed a distinguished architecture when their material and processing conditions were altered. Subsequent culture of mouse fibroblast cells (L929) was carried out to evaluate the cells viability on each scaffold after their attachment for 24 h and proliferation for 48 and 72 h. The scaffolds' porosity, pores number, pores size and distribution were quantified and none could establish a relationship with the viability results. Virtual reconstruction of the mats introduced an authentic criterion, "Scaffold Percolative Efficiency" (SPE), with which the above descriptors were addressed collectively. It was hypothesized to be able to quantify the efficacy of fibrous scaffolds by considering the integration of porosity and interconnectivity of the pores. There was a correlation of 80% as a good agreement between the SPE values and the spectrophotometer absorbance of viable cells; a viability of more than 350% in comparison to that of the controls.

A Brief Review of the Modelling of the Time Dependent Mechanical Properties of Tissue Engineering Scaffolds

The functionality of tissue scaffolds in vivo plays a critical role in the treatment process. Due to the time dependent nature of the mechanical properties of the constituent phases of the scaffold, a wide range of mechanical property histories may be observed during the treatment process, possibly influencing outcomes. The critical nature of the mechanical properties in load bearing applications indicates a need for the simultaneous modelling of both scaffold degradation and tissue regeneration with time, and the resulting effective properties of the tissue engineering construct. To this end, a review of the literature is conducted to identify the various existing approaches to modelling scaffold degradation, tissue behavior, and the dependency of the two processes on one another.

A Distinct Catabolic to Anabolic Threshold Due to Single-Cell Static Nanomechanical Stimulation in a Cartilage Biokinetics Model

Understanding physicochemical interactions during biokinetic regulation will be critical for the creation of relevant nanotechnology supporting cellular and molecular engineering. The impact of nanoscale influences in medicine and biology can be explored in detail through mathematical models as an in silico testbed. In a recent single-cell biomechanical analysis, the cytoskeletal strain response due to fluid-induced stresses was characterized (Wilson, Z. D., and Kohles, S. S., 2010, "Two-Dimensional Modeling of Nanomechanical Strains in Healthy and Diseased Single-Cells During Microfluidic Stress Applications," J. Nanotech. Eng. Med., 1(2), p. 021005). Results described a microfluidic environment having controlled nanometer and piconewton resolution for explorations of multiscale mechanobiology. In the present study, we constructed a mathematical model exploring the nanoscale biomolecular response to that controlled microenvironment. We introduce mechanical stimuli and scaling factor terms as specific input values for regulating a cartilage molecule synthesis. Iterative model results for this initial multiscale static load application have identified a transition threshold load level from which the mechanical input causes a shift from a catabolic state to an anabolic state. Modeled molecule homeostatic levels appear to be dependent upon the mechanical stimulus as reflected experimentally. This work provides a specific mathematical framework from which to explore biokinetic regulation. Further incorporation of nanomechanical stresses and strains into biokinetic models will ultimately lead to refined mechanotransduction relationships at the cellular and molecular levels.

In vitro ovine articular chondrocyte proliferation: experiments and modelling

This study focuses on analysis of in vitro cultures of chondrocytes from ovine articular cartilage. Isolated cells were seeded in Petri dishes, then expanded to confluence and phenotypically characterized by flow cytometry. The sigmoidal temporal profile of total counts was obtained by classic haemocytometry and corresponding cell size distributions were measured electronically using a Coulter Counter. A mathematical model recently proposed (1) was adopted for quantitative interpretation of these experimental data. The model is based on a 1-D (that is, mass-structured), single-staged population balance approach capable of taking into account contact inhibition at confluence. The model's parameters were determined by fitting measured total cell counts and size distributions. Model reliability was verified by predicting cell proliferation counts and corresponding size distributions at culture times longer than those used when tuning the model's parameters. It was found that adoption of cell mass as the intrinsic characteristic of a growing chondrocyte population enables sigmoidal temporal profiles of total counts in the Petri dish, as well as cell size distributions at 'balanced growth', to be adequately predicted.

A validated model of GAG deposition, cell distribution, and growth of tissue engineered cartilage cultured in a rotating bioreactor

In this work a new phenomenological model of growth of cartilage tissue cultured in a rotating bioreactor is developed. It represents an advancement of a previously derived model of deposition of glycosaminoglycan (GAG) in engineered cartilage by (i) introduction of physiological mechanisms of proteoglycan accumulation in the extracellular matrix (ECM) as well as by correlating (ii) local cell densities and (iii) tissue growth to the ECM composition. In particular, previously established predictions and correlations of local oxygen concentrations and GAG synthesis rates are extended to distinguish cell secreted proteoglycan monomers free to diffuse in cell surroundings and outside from the engineered construct, from large aggrecan molecules, which are constrained within the ECM and practically immovable. The model includes kinetics of aggregation, that is, transformation of mobile GAG species into immobile aggregates as well as maintenance of the normal ECM composition after the physiological GAG concentration is reached by incorporation of a product inhibition term. The model also includes mechanisms of the temporal evolution of cell density distributions and tissue growth under in vitro conditions. After a short initial proliferation phase the total cell number in the construct remains constant, but the local cell distribution is leveled out by GAG accumulation and repulsion due to negative molecular charges. Furthermore, strong repulsive forces result in expansion of the local tissue elements observed macroscopically as tissue growth (i.e., construct enlargement). The model is validated by comparison with experimental data of (i) GAG distribution and leakage, (ii) spatial-temporal distributions of cells, and (iii) tissue growth reported in previous works. Validation of the model predictive capability--against a selection of measured data that were not used to construct the model--suggests that the model successfully describes the interplay of several simultaneous processes carried out during in vitro cartilage tissue regeneration and indicates that this approach could also be attractive for application in other tissue engineering systems.

Enabling tools for engineering collagenous tissues integrating bioreactors, intravital imaging, and biomechanical modeling

Many investigators have engineered diverse connective tissues having good mechanical properties, yet few tools enable a global understanding of the associated formation of collagen fibers, the primary determinant of connective tissue stiffness. Toward this end, we developed a biomechanical model for collagenous tissues grown on polymer scaffolds that accounts for the kinetics of polymer degradation as well as the synthesis and degradation of multiple families of collagen fibers in response to cyclic strains imparted in a bioreactor. The model predicted well both overall thickness and stress-stretch relationships for tubular engineered vessels cultured for 8 weeks, and suggested that a steady state had not yet been reached. To facilitate future refinements of the model, we also developed bioreactors that enable intravital nonlinear optical microscopic imaging. Using these tools, we found that collagen fiber alignment was driven strongly by nondegraded polymer fibers at early times during culture, with subsequent mechano-stimulated dispersal of fiber orientations as polymer fibers degraded. In summary, mathematical models of growth and remodeling of engineered tissues cultured on polymeric scaffolds can predict evolving tissue morphology and mechanics after long periods of culture, and related empirical observations promise to further our understanding of collagen matrix development in vitro.

Simulating the growth of articular cartilage explants in a permeation bioreactor to aid in experimental protocol design

Recently a cartilage growth finite element model (CGFEM) was developed to solve nonhomogeneous and time-dependent growth boundary-value problems (Davol et al., 2008, "A Nonlinear Finite Element Model of Cartilage Growth," Biomech. Model. Mechanobiol., 7, pp. 295-307). The CGFEM allows distinct stress constitutive equations and growth laws for the major components of the solid matrix, collagens and proteoglycans. The objective of the current work was to simulate in vitro growth of articular cartilage explants in a steady-state permeation bioreactor in order to obtain results that aid experimental design. The steady-state permeation protocol induces different types of mechanical stimuli. When the specimen is initially homogeneous, it directly induces homogeneous permeation velocities and indirectly induces nonhomogeneous solid matrix shear stresses; consequently, the steady-state permeation protocol is a good candidate for exploring two competing hypotheses for the growth laws. The analysis protocols were implemented through the alternating interaction of the two CGFEM components: poroelastic finite element analysis (FEA) using ABAQUS and a finite element growth routine using MATLAB. The CGFEM simulated 12 days of growth for immature bovine articular cartilage explants subjected to two competing hypotheses for the growth laws: one that is triggered by permeation velocity and the other by maximum shear stress. The results provide predictions for geometric, biomechanical, and biochemical parameters of grown tissue specimens that may be experimentally measured and, consequently, suggest key biomechanical measures to analyze as pilot experiments are performed. The combined approach of CGFEM analysis and pilot experiments may lead to the refinement of actual experimental protocols and a better understanding of in vitro growth of articular cartilage.

EDC/NHS-crosslinked type II collagen-chondroitin sulfate scaffold: characterization and in vitro evaluation

Three-dimensional biodegradable porous type II collagen scaffolds are interesting materials for cartilage tissue engineering. This study reports the preparation of porous type II collagen-chondroitin sulfate (CS) scaffold using variable concentrations of 1-ethyl-3(3-dimethyl aminopropyl) carbodiimide (EDC) and N-hydroxysuccinimide (NHS). The physico-chemical properties and ultrastructural morphology of the collagen scaffolds were determined. Then, isolated chondrocytes were cultured in porous type II collagen scaffolds either in the presence and/or absence of covalently attached CS up to 14 days. Cell proliferation, the total amount of proteoglycans and type II collagen retained in the scaffold and chondrocytes morphology were evaluated. The results suggest that EDC-crosslinking improves the mechanical stability of collagen-CS scaffolds with increasing EDC concentration. Cell proliferation and the total amount of proteoglycans and type II collagen retained in the scaffolds were higher in type II collagen-CS scaffolds. Histological analysis showed the formation of a denser cartilaginous layer at the scaffold periphery. Scanning electron microscopy (SEM) revealed chondrocytes distributed the porous surface of both scaffolds maintained their spherical morphology. The results of the present study also indicate that type II collagen-CS scaffolds have potential for use in tissue engineering.

Theoretical analysis of engineered cartilage oxygenation: influence of construct thickness and media flow rate

A novel parallel-plate bioreactor has been shown to modulate the mechanical and biochemical properties of engineered cartilage by the application of fluid-induced shear stress. Flow or perfusion bioreactors may improve tissue development via enhanced transport of nutrients or gases as well as the application of mechanical stimuli, or a combination of these factors. The goal of this study was to complement observed experimental responses to flow by simulating oxygen transport within cartilage constructs of different thicknesses (250 microm or 1 mm). Using numerical computation of convection-diffusion equations, the evaluation of the tissue oxygenation is performed. Four culture conditions are defined based on tissue thickness and flow rates ranging from 0 to approximately 25 mL min(-1). Under these experimental conditions results show a mean oxygen concentration within the tissue varying from 0.01 to 0.19 mol m(-3) as a function of the tissue thickness and the magnitude of the applied shear stress. More generally, the influence of shear stress varying (via flow rate modification) from 10(-3) to 10 dynes cm(-2) on the tissue oxygenation is studied. The influence on the results of important physical parameters such as the maximal oxygen consumption rate of cells is discussed. Lastly, the importance of oxygen concentration in the lower chamber and its relevance to tissue oxygenation are highlighted by the model results.

A mechanical composite spheres analysis of engineered cartilage dynamics

In the preparation of bioengineered reparative strategies for damaged or diseased tissues, the processes of biomaterial degradation and neotissue synthesis combine to affect the developing mechanical state of multiphase, composite engineered tissues. Here, cell-polymer constructs for engineered cartilage have been fabricated by seeding chondrocytes within three-dimensional scaffolds of biodegradable polymers. During culture, synthetic scaffolds degraded passively as the cells assembled an extracellular matrix (ECM) composed primarily of glycosaminoglycan and collagen. Biochemical and biomechanical assessment of the composite (cells, ECM, and polymer scaffold) were modeled at a unit-cell level to mathematically solve stress-strain relationships and thus construct elastic properties (n=4 samples per seven time points). This approach employed a composite spheres, micromechanical analysis to determine bulk moduli of: (1) the cellular-ECM inclusion within the supporting scaffold structure; and (2) the cellular inclusion within its ECM. Results indicate a dependence of constituent volume fractions with culture time (p<0.05). Overall mean bulk moduli were variably influenced by culture, as noted for the cell-ECM inclusion (K(c-m)=29.7 kPa, p=0.1439), the cellular inclusion (K(c)=5.5 kPa, p=0.0067), and its surrounding ECM (K(m)=373.9 kPa, p=0.0748), as well as the overall engineered construct (K=165.0 kPa, p=0.6899). This analytical technique provides a framework to describe the time-dependent contribution of cells, accumulating ECM, and a degrading scaffold affecting bioengineered construct mechanical properties.

A novel simulation model for stem cells differentiation

A novel mathematical model to simulate mesenchymal stem cells differentiation into specialized cells is proposed. The model is based upon material balances for extracellular matrix compounds, growth factors and nutrients coupled with a mass-structured population balance describing cell growth, proliferation and differentiation. The proposed model is written in a general form and it may be used to simulate a generic cell differentiation pathway occurring in vivo or during in vitro cultivation when specific growth factors are used. Literature experimental data concerning the differentiation of mesenchymal stem cells into chondrocytes in terms of total DNA and glycosaminoglycan content are successfully compared with model results, thus demonstrating the validity of the proposed model as well as its predictive capability. A further test of the model capability is performed for the case of in vivo fracture healing during which mesenchymal stem cells differentiate into chondrocytes and osteoblasts. Considerations about the extension of the proposed model to different pathologies beside fracture healing are reported. Finally, sensitivity analysis of model parameters is also performed in order to clarify what mechanisms most strongly influence differentiation and the distribution of cell types.

Enabling tools for tissue engineering

Tissue engineering is a multidisciplinary field that combines engineering, physical sciences, biology, and medicine to restore or replace tissues and organs functions. In this review, enabling tools for tissue engineering are discussed in the context of four key areas or pillars: prediction, production, performance, and preservation. Prediction refers to the computational modeling where the ability to simulate cellular behavior in complex three-dimensional environments will be essential for design of tissues. Production refer imaging modalities that allow high resolution, non-invasive monitoring of the development and incorporation of tissue engineered constructs. Lastly, preservation includes biochemical tools that permit cryopreservation, vitrification, and freeze-drying of cells and tissues. Recent progress and future perspectives for development in each of these key areas are presented.

A Novel Simulation Model for Engineered Cartilage Growth in Static Systems

A novel mathematical model to simulate the growth of engineered cartilage in static systems is proposed. This model is based on material balances for the involved species (glycosaminoglycan and collagen, both pertaining to extracellular matrix), as well as mass-structured population balance for simulating cell growth and its proliferation within the scaffold. This model may simulate tissue growth on static culture taking place in Petri dishes, static flasks, and well plates for different types of scaffolds (i.e., poly(glycolic acid) [PGA], PGA/poly(l-lactic acid), and collagen sponge). This work aimed to demonstrate that the model approach proposed in previous works, regarding engineered cartilage growth on PGA scaffolds performed in rotating bioreactors, may also be applied to different scaffolds and system configurations. In particular, the balance equation for simulating collagen production is introduced, as well as the use of spatial averaging over the spatial region to compare experimental data with the model. Experimental data from the literature in terms of cells, glycosaminoglycans, and collagen content have been successfully compared with model results, thus demonstrating the validity of the proposed model, as well as its predictive capability.

Modulation of depth-dependent properties in tissue-engineered cartilage with a semi-permeable membrane and perfusion: a continuum model of matrix metabolism and transport

The functional properties of cartilaginous tissues are determined predominantly by the content, distribution, and organization of proteoglycan and collagen in the extracellular matrix. Extracellular matrix accumulates in tissue-engineered cartilage constructs by metabolism and transport of matrix molecules, processes that are modulated by physical and chemical factors. Constructs incubated under free-swelling conditions with freely permeable or highly permeable membranes exhibit symmetric surface regions of soft tissue. The variation in tissue properties with depth from the surfaces suggests the hypothesis that the transport processes mediated by the boundary conditions govern the distribution of proteoglycan in such constructs. A continuum model (DiMicco and Sah in Transport Porus Med 50:57-73, 2003) was extended to test the effects of membrane permeability and perfusion on proteoglycan accumulation in tissue- engineered cartilage. The concentrations of soluble, bound, and degraded proteoglycan were analyzed as functions of time, space, and non-dimensional parameters for several experimental configurations. The results of the model suggest that the boundary condition at the membrane surface and the rate of perfusion, described by non-dimensional parameters, are important determinants of the pattern of proteoglycan accumulation. With perfusion, the proteoglycan profile is skewed, and decreases or increases in magnitude depending on the level of flow-based stimulation. Utilization of a semi-permeable membrane with or without unidirectional flow may lead to tissues with depth-increasing proteoglycan content, resembling native articular cartilage.

Computational Modeling to Predict the Temporal Regulation of Chondrocyte Metabolism in Response to Various Dynamic Compression Regimens

Based on previously published experimental work, computational models were developed to simulate the effect of different dynamic compression regimens on the activity of chondrocytes seeded in agarose constructs. In particular, the balance between proliferation and matrix synthesis can be adjusted by applying different intervals of continuous or intermittent mechanical compression. A phenomenological compartment based-modeling approach was used as first model. A more mechanistic cell cycle model was used as the second model. The compartment-based modeling approach was found to be useful in representing a balance between proliferation and proteoglycan synthesis, when the effect of a certain stimulation protocol is known. In order to predict the response to different intervals of mechanical stimulation, however, a more mechanistic cell cycle-based approach is required. The cell cycle model supports an important role of the onset of loading. In addition, an inhibitory effect of further loading is required, which is more likely to be related to cell cycle progression velocity than to a decreased probability of commitment to the cell cycle. The mechanisms behind this inhibitory effect and the computational implementation, however, require further investigation.

Biodendrimer-based hydrogel scaffolds for cartilage tissue repair

Photo-crosslinkable dendritic macromolecules are attractive materials for the preparation of cartilage tissue engineering scaffolds that may be optimized for in situ formation of hydrated, mechanically stable, and well-integrated hydrogel scaffolds supporting chondrocytes and chondrogenesis. We designed and synthesized a novel hydrogel scaffold for cartilage repair, based on a multivalent and water-soluble tri-block copolymer consisting of a poly(ethylene glycol) core and methacrylated poly(glycerol succinic acid) dendrimer terminal blocks. The terminal methacrylates allow mild and biocompatible photo-crosslinking with a visible light, facilitating in vivo filling of irregularly shaped defects with the dendrimer-based scaffold. The multivalent dendrimer constituents allow high crosslink densities that inhibit swelling after crosslinking while simultaneously introducing biodegradation sites. The mechanical properties and water content of the hydrogel can easily be tuned by changing the biodendrimer concentration. In vitro chondrocyte encapsulation studies demonstrate significant synthesis of neocartilaginous material, containing proteoglycans and type II collagen.

Epitope tagging for tracking elastin-like polypeptides

Elastin-like polypeptides (ELPs) are a class of biocompatible, non-immunogenic and crosslinkable biomaterials that offer promise for use as an injectable scaffold for cartilage repair. In this study, an oligohistidine (His(6)) epitope tag was incorporated at the N-terminus of an ELP using recombinant DNA techniques to permit tracking without compromising on material biocompatibility. His(6)-tagged ELPs were successfully detected by Western blot analysis and quantified by ELISAs following digestion with trypsin. The mass of His(6) tagged ELP fragments freed from a crosslinked ELP hydrogel after digestion with trypsin correlated highly with hydrogel weight loss, providing evidence of the tag's capability to enable tracking of enzymatic degradation of the ELP hydrogel. The His(6) tag also facilitated recognition of crosslinked ELPs from background staining of articular cartilage. These results suggest that the His(6) epitope tag has the potential to track ELP scaffold loss independently of newly formed tissue mass for evaluating matrix remodeling in vivo.

The local matrix distribution and the functional development of tissue engineered cartilage, a finite element study

Assessment of the functionality of tissue engineered cartilage constructs is hampered by the lack of correlation between global measurements of extra cellular matrix constituents and the global mechanical properties. Based on patterns of matrix deposition around individual cells, it has been hypothesized previously, that mechanical functionality arises when contact occurs between zones of matrix associated with individual cells. The objective of this study is to determine whether the local distribution of newly synthesized extracellular matrix components contributes to the evolution of the mechanical properties of tissue engineered cartilage constructs. A computational homogenization approach was adopted, based on the concept of a periodic representative volume element. Local transport and immobilization of newly synthesized matrix components were described. Mechanical properties were taken dependent on the local matrix concentration and subsequently the global aggregate modulus and hydraulic permeability were derived. The transport parameters were varied to assess the effect of the evolving matrix distribution during culture. The results indicate that the overall stiffness and permeability are to a large extent insensitive to differences in local matrix distribution. This emphasizes the need for caution in the visual interpretation of tissue functionality from histology and underlines the importance of complementary measurements of the matrix's intrinsic molecular organization.

Dynamic matrix composition in engineered cartilage with stochastic supplementation of growth factors

Dynamic extracellular matrix (ECM) synthesis is explored in a hypothesized engineered cartilage construct. Growth (alpha) and decay (beta) rate parameters are developed from a previous engineered cartilage model. The presented mathematical model was constructed from the parameterized experimental data using a deterministic and stochastic examination of ECM synthesis based on a negative feedback control mechanism. A growth factor supplementation is incorporated in a probabilistic mathematical approach. The growth factor component modified an initial deterministic model through a Gaussian white noise fluctuation. As the primary constituents of ECM, the mathematical tool is intended to characterize the probable steady state distribution of glycosaminoglycan (GAG) and collagen molecules as well as mean mass accumulation at homeostasis. Computer simulation of the models is applied to reported data from four similar chondrocyte-polymer construct culture systems. The range in rate ratios reflect the differing nature of GAG and collagen synthesis (alphaGAG/betaGAG = 4.2 to 148.6; alphacollagen/betacollagen = 8.1 to 2590.4). This technique reduced the influencing synthesis factors to a few key descriptive parameters. Additional anabolic and catabolic factors may further be built into the models.

Poly(lactide-co-glycolide) microspheres as a moldable scaffold for cartilage tissue engineering

This study demonstrates the use of biodegradable poly(lactide-co-glycolide) (PLG) microspheres as a moldable scaffold for cartilage tissue engineering. Chondrocytes were delivered to a cylindrical mold with or without PLG microspheres and cultured in vitro for up to 8 weeks. Cartilagenous tissue formed using chondrocytes and microspheres maintained thickness, shape, and chondrocyte collagen type II phenotype, as indicated by type II collagen staining. The presence of microspheres further enhanced total tissue mass and the amount of glycosaminoglycan that accumulated. Evaluation of microsphere composition demonstrated effects of polymer molecular weight, end group chemistry, and buffer inclusion on tissue-engineered cartilage growth. Higher molecular weight PLG resulted in a larger mass of cartilage-like tissue formed and a higher content of proteoglycans. Cartilage-like tissue formed using microspheres made from low molecular weight and free carboxylic acid end groups did not display increases in tissue mass, yet a modest increased proteoglycan accumulation was detected. Microspheres comprised of PLG with methyl ester end groups yielded a steady increase in tissue mass, with no real increase in matrix accumulation. The microencapsulation of Mg(OH)(2) had negative effects on tissue mass and matrix accumulation. The data herein reflect the potential utility of a moldable PLG-chondrocyte system for tissue-engineering applications.

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

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.

Prediction of growth factor effects on engineered cartilage composition using deterministic and stochastic modeling

In the design of engineered tissues, guided balance of biomaterial degeneration with tissue synthesis offers refined control of construct development. The objective of this study was to develop a mathematical model that describes the steady state metabolism of extracellular matrix molecules (ECM: glycosaminoglycan and collagen) in an engineered cartilage construct taking into account localized environmental changes that may arise because of the application of growth factors. The variable effects of growth factors were incorporated in the form of random noise rather than the difference in rates of synthesis and catabolism. Thus, the frequency of ECM accumulation for each matrix molecule in the steady state under the random influence of growth factor was produced relative to the matrix carrying capacity. Published synthesis-rate time constants and steady state ECM conditions from chondrocyte-polymer scaffold composites provided both input and validation for the model. Although the presence of growth factors in the presented system dynamics were considered randomized, the results described a positive feedback or promotional ECM synthesis at low levels of growth factors. While a negative feedback or inhibition of ECM synthesis was characterized at higher levels of growth factors. This transition phenomenon is based on a comparison with the results of a steady state condition in the form of a deterministic model and supports previous reports of guided accumulation in musculoskeletal, connective, and neuronal tissues.

Effect of environmental fluctuations on the dynamic composition of engineered cartilage: A deterministic model in stochastic environment

Dynamics of extracellular matrix (ECM) deposition and scaffold degradation in cell-polymer constructs have been studied in a random fluctuating environment created due to the applications of growth factors into the in vitro generation of cartilaginous constructs. Existing models of cell-polymer constructs for the design of engineered cartilage have been discussed and then a new deterministic scheme in random environment proposed taking into account the effects of growth factors as the environmental variability in the form of Gaussian white noise. Steady-state probability distribution of each individual component of the ECM in its homeostasis is found explicitly. The computer-simulated results of the model have been discussed and then compared with the data from a variety of scaffold systems and culture conditions.

Dynamics of the cell and its extracellular matrix-a simple mathematical approach

The extracellular matrix (ECM) is produced by the cells and secreted into the surrounding medium, and consists of a complex mixture of structural and functional proteins. It has been recently observed that the ECM can influence the behavior of cell growth in vitro quite remarkably. A simple mathematical model has been constructed based on negative feedback control mechanisms to represent the dynamics of ECM deposition and cellular differentiation. The model analysis shows a strong relationship between the numerical solution and the experimental observations in cell-polymer constructs for the design of engineered cartilage. The current paper may be a useful guide for those who want to explore the studies on cell-matrix interactions.