An integrated finite-element approach to mechanics, transport and biosynthesis in tissue engineering

Intervertebral Disk Nutrients and Transport Mechanisms in Relation to Disk Degeneration: A Narrative Literature Review

Objective

The purpose of this paper was to review the literature regarding the mechanisms leading to degeneration in intervertebral disks and to discuss contributing mechanical and biological factors.

Methods

The inclusion criteria for the literature review were research studies conducted in the last 3 decades with free full-text available in English. Review articles and articles pertaining to temporomandibular joints and joints of the body other than the intervertebral disk were excluded. The following databases were searched: PubMed, EBSCOhost, and Google Scholar through September 9, 2016.

Results

A total of 57 articles were used in this review. Intervertebral disk cells require glucose for sustainability and oxygen to synthesize matrix components. Nutrients enter the disk via 2 vascular supply routes: capillary beds of end plates and the peripheral annulus fibrosus. Solute size, shape and charge, compression, and metabolic demand all influence the efficiency of nutrient transport, and alterations of any of these factors may have effects on nutrient transport and, potentially, disk degeneration.

Conclusions

Progressive nutrient transport disruptions may actively contribute in advancing the phases of degenerative disk disease. Such disruptions include dysfunctional loading and spinal position, lack of motion, high frequency loading, disk injury, aging, smoking, an acidic environment, and a lack of nutrient bioavailability.

Relevant biological processes for tissue development with stem cells and their mechanistic modeling: A review

A potential alternative for tissue transplants is tissue engineering, in which the interaction of cells and biomaterials can be optimized. Tissue development in vitro depends on the complex interaction of several biological processes such as extracellular matrix synthesis, vascularization and cell proliferation, adhesion, migration, death, and differentiation. The complexity of an individual phenomenon or of the combination of these processes can be studied with phenomenological modeling techniques. This work reviews the main biological phenomena in tissue development and their mathematical modeling, focusing on mesenchymal stem cell growth in three-dimensional scaffolds.

A biochemo-mechano coupled, computational model combining membrane transport and pericellular proteolysis in tissue mechanics

We present a computational model for the interaction of surface- and volume-bound scalar transport and reaction processes with a deformable porous medium. The application in mind is pericellular proteolysis, i.e. the dissolution of the solid phase of the extracellular matrix (ECM) as a response to the activation of certain chemical species at the cell membrane and in the vicinity of the cell. A poroelastic medium model represents the extra cellular scaffold and the interstitial fluid flow, while a surface-bound transport model accounts for the diffusion and reaction of membrane-bound chemical species. By further modelling the volume-bound transport, we consider the advection, diffusion and reaction of sequestered chemical species within the extracellular scaffold. The chemo-mechanical coupling is established by introducing a continuum formulation for the interplay of reaction rates and the mechanical state of the ECM. It is based on known experimental insights and theoretical work on the thermodynamics of porous media and degradation kinetics of collagen fibres on the one hand and a damage-like effect of the fibre dissolution on the mechanical integrity of the ECM on the other hand. The resulting system of partial differential equations is solved via the finite-element method. To the best of our knowledge, it is the first computational model including contemporaneously the coupling between (i) advection-diffusion-reaction processes, (ii) interstitial flow and deformation of a porous medium, and (iii) the chemo-mechanical interaction impelled by the dissolution of the ECM. Our numerical examples show good agreement with experimental data. Furthermore, we outline the capability of the methodology to extend existing numerical approaches towards a more comprehensive model for cellular biochemo-mechanics.

A poroelastic mixture model of mechanobiological processes in biomass growth: theory and application to tissue engineering

In this article we propose a novel mathematical description of biomass growth that combines poroelastic theory of mixtures and cellular population models. The formulation, potentially applicable to general mechanobiological processes, is here used to study the engineered cultivation in bioreactors of articular chondrocytes, a process of Regenerative Medicine characterized by a complex interaction among spatial scales (from nanometers to centimeters), temporal scales (from seconds to weeks) and biophysical phenomena (fluid-controlled nutrient transport, delivery and consumption; mechanical deformation of a multiphase porous medium). The principal contribution of this research is the inclusion of the concept of cellular "force isotropy" as one of the main factors influencing cellular activity. In this description, the induced cytoskeletal tensional states trigger signalling transduction cascades regulating functional cell behavior. This mechanims is modeled by a parameter which estimates the influence of local force isotropy by the norm of the deviatoric part of the total stress tensor. According to the value of the estimator, isotropic mechanical conditions are assumed to be the promoting factor of extracellular matrix production whereas anisotropic conditions are assumed to promote cell proliferation. The resulting mathematical formulation is a coupled system of nonlinear partial differential equations comprising: conservation laws for mass and linear momentum of the growing biomass; advection-diffusion-reaction laws for nutrient (oxygen) transport, delivery and consumption; and kinetic laws for cellular population dynamics. To develop a reliable computational tool for the simulation of the engineered tissue growth process the nonlinear differential problem is numerically solved by: (1) temporal semidiscretization; (2) linearization via a fixed-point map; and (3) finite element spatial approximation. The biophysical accuracy of the mechanobiological model is assessed in the analysis of a simplified 1D geometrical setting. Simulation results show that: (1) isotropic/anisotropic conditions are strongly influenced by both maximum cell specific growth rate and mechanical boundary conditions enforced at the interface between the biomass construct and the interstitial fluid; (2) experimentally measured features of cultivated articular chondrocytes, such as the early proliferation phase and the delayed extracellular matrix production, are well described by the computed spatial and temporal evolutions of cellular populations.

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.

A Finite Element Model for Mixed Porohyperelasticity with Transport, Swelling, and Growth

The purpose of this manuscript is to establish a unified theory of porohyperelasticity with transport and growth and to demonstrate the capability of this theory using a finite element model developed in MATLAB. We combine the theories of volumetric growth and mixed porohyperelasticity with transport and swelling (MPHETS) to derive a new method that models growth of biological soft tissues. The conservation equations and constitutive equations are developed for both solid-only growth and solid/fluid growth. An axisymmetric finite element framework is introduced for the new theory of growing MPHETS (GMPHETS). To illustrate the capabilities of this model, several example finite element test problems are considered using model geometry and material parameters based on experimental data from a porcine coronary artery. Multiple growth laws are considered, including time-driven, concentration-driven, and stress-driven growth. Time-driven growth is compared against an exact analytical solution to validate the model. For concentration-dependent growth, changing the diffusivity (representing a change in drug) fundamentally changes growth behavior. We further demonstrate that for stress-dependent, solid-only growth of an artery, growth of an MPHETS model results in a more uniform hoop stress than growth in a hyperelastic model for the same amount of growth time using the same growth law. This may have implications in the context of developing residual stresses in soft tissues under intraluminal pressure. To our knowledge, this manuscript provides the first full description of an MPHETS model with growth. The developed computational framework can be used in concert with novel in-vitro and in-vivo experimental approaches to identify the governing growth laws for various soft tissues.

A triphasic constrained mixture model of engineered tissue formation under in vitro dynamic mechanical conditioning

While it has become axiomatic that mechanical signals promote in vitro engineered tissue formation, the underlying mechanisms remain largely unknown. Moreover, efforts to date to determine parameters for optimal extracellular matrix (ECM) development have been largely empirical. In the present work, we propose a two-pronged approach involving novel theoretical developments coupled with key experimental data to develop better mechanistic understanding of growth and development of dense connective tissue under mechanical stimuli. To describe cellular proliferation and ECM synthesis that occur at rates of days to weeks, we employ mixture theory to model the construct constituents as a nutrient-cell-ECM triphasic system, their transport, and their biochemical reactions. Dynamic conditioning protocols with frequencies around 1 Hz are described with multi-scale methods to couple the dissimilar time scales. Enhancement of nutrient transport due to pore fluid advection is upscaled into the growth model, and the spatially dependent ECM distribution describes the evolving poroelastic characteristics of the scaffold-engineered tissue construct. Simulation results compared favorably to the existing experimental data, and most importantly, distinguish between static and dynamic conditioning regimes. The theoretical framework for mechanically conditioned tissue engineering (TE) permits not only the formulation of novel and better-informed mechanistic hypothesis describing the phenomena underlying TE growth and development, but also the exploration/optimization of conditioning protocols in a rational manner.

Evaluation of the growth environment of a hydrostatic force bioreactor for preconditioning of tissue-engineered constructs

Bioreactors have been widely acknowledged as valuable tools to provide a growth environment for engineering tissues and to investigate the effect of physical forces on cells and cell-scaffold constructs. However, evaluation of the bioreactor environment during culture is critical to defining outcomes. In this study, the performance of a hydrostatic force bioreactor was examined by experimental measurements of changes in dissolved oxygen (O2), carbon dioxide (CO2), and pH after mechanical stimulation and the determination of physical forces (pressure and stress) in the bioreactor through mathematical modeling and numerical simulation. To determine the effect of hydrostatic pressure on bone formation, chick femur skeletal cell-seeded hydrogels were subjected to cyclic hydrostatic pressure at 0-270 kPa and 1 Hz for 1 h daily (5 days per week) over a period of 14 days. At the start of mechanical stimulation, dissolved O2 and CO2 in the medium increased and the pH of the medium decreased, but remained within human physiological ranges. Changes in physiological parameters (O2, CO2, and pH) were reversible when medium samples were placed in a standard cell culture incubator. In addition, computational modeling showed that the distribution and magnitude of physical forces depends on the shape and position of the cell-hydrogel constructs in the tissue culture format. Finally, hydrostatic pressure was seen to enhance mineralization of chick femur skeletal cell-seeded hydrogels.

A mixture approach to investigate interstitial growth in engineering scaffolds

Controlling biological growth within a cell-laden polymeric scaffold is a critical challenge in the tissue engineering community. Indeed, construct growth must often be balanced with scaffold degradation and is often coupled to varying degrees of deformation that originate from swelling, external forces and the effects of confinement. These factors have been shown to affect growth in many ways, but to date, our understanding is mostly qualitative. While cell sensing, molecular transport and scaffold/tissue interactions are believed to be important players, it will be critical to quantify, predict and control these effects in order to eventually optimize tissue growth in the laboratory. The aim of this paper was thus to provide a theoretical framework to better understand how the scaffold-mediated mechanisms of transport, deposition (and possibly degradation) and elasticity affect the overall growth of a tissue subjected to finite deformations. We propose a formulation in which the macroscopic evolutions in tissue size, density as well as the appearance of residual stresses can be directly related to changes in internal composition by considering three fundamental principles: mechanical equilibrium, chemical equilibrium and molecular incompressibility. The resulting model allows us to pay particular attention to features that are critical to the interaction between growth and deformation: osmotic pressure and swelling, the strain mismatch between old and newly deposited material as well as the mechano-sensitive cell-mediated production. We show that all of these phenomena may indeed strongly affect the overall growth of a construct under finite deformations.

Computational modeling of chemical reactions and interstitial growth and remodeling involving charged solutes and solid-bound molecules

Mechanobiological processes are rooted in mechanics and chemistry, and such processes may be modeled in a framework that couples their governing equations starting from fundamental principles. In many biological applications, the reactants and products of chemical reactions may be electrically charged, and these charge effects may produce driving forces and constraints that significantly influence outcomes. In this study, a novel formulation and computational implementation are presented for modeling chemical reactions in biological tissues that involve charged solutes and solid-bound molecules within a deformable porous hydrated solid matrix, coupling mechanics with chemistry while accounting for electric charges. The deposition or removal of solid-bound molecules contributes to the growth and remodeling of the solid matrix; in particular, volumetric growth may be driven by Donnan osmotic swelling, resulting from charged molecular species fixed to the solid matrix. This formulation incorporates the state of strain as a state variable in the production rate of chemical reactions, explicitly tying chemistry with mechanics for the purpose of modeling mechanobiology. To achieve these objectives, this treatment identifies the specific theoretical and computational challenges faced in modeling complex systems of interacting neutral and charged constituents while accommodating any number of simultaneous reactions where reactants and products may be modeled explicitly or implicitly. Several finite element verification problems are shown to agree with closed-form analytical solutions. An illustrative tissue engineering analysis demonstrates tissue growth and swelling resulting from the deposition of chondroitin sulfate, a charged solid-bound molecular species. This implementation is released in the open-source program FEBio ( www.febio.org ). The availability of this framework may be particularly beneficial to optimizing tissue engineering culture systems by examining the influence of nutrient availability on the evolution of inhomogeneous tissue composition and mechanical properties, the evolution of construct dimensions with growth, the influence of solute and solid matrix electric charge on the transport of cytokines, the influence of binding kinetics on transport, the influence of loading on binding kinetics, and the differential growth response to dynamically loaded versus free-swelling culture conditions.

The effect of oxygen tension on human articular chondrocyte matrix synthesis: integration of experimental and computational approaches

Significant oxygen gradients occur within tissue engineered cartilaginous constructs. Although oxygen tension is an important limiting parameter in the development of new cartilage matrix, its precise role in matrix formation by chondrocytes remains controversial, primarily due to discrepancies in the experimental setup applied in different studies. In this study, the specific effects of oxygen tension on the synthesis of cartilaginous matrix by human articular chondrocytes were studied using a combined experimental-computational approach in a “scaffold-free” 3D pellet culture model. Key parameters including cellular oxygen uptake rate were determined experimentally and used in conjunction with a mathematical model to estimate oxygen tension profiles in 21-day cartilaginous pellets. A threshold oxygen tension (pO₂ ≈ 8% atmospheric pressure) for human articular chondrocytes was estimated from these inferred oxygen profiles and histological analysis of pellet sections. Human articular chondrocytes that experienced oxygen tension below this threshold demonstrated enhanced proteoglycan deposition. Conversely, oxygen tension higher than the threshold favored collagen synthesis. This study has demonstrated a close relationship between oxygen tension and matrix synthesis by human articular chondrocytes in a “scaffold-free” 3D pellet culture model, providing valuable insight into the understanding and optimization of cartilage bioengineering approaches. Biotechnol. Bioeng. 2014;111: 1876–1885.

Multiphasic finite element framework for modeling hydrated mixtures with multiple neutral and charged solutes

Computational tools are often needed to model the complex behavior of biological tissues and cells when they are represented as mixtures of multiple neutral or charged constituents. This study presents the formulation of a finite element modeling framework for describing multiphasic materials in the open-source finite element software febio.1 Multiphasic materials may consist of a charged porous solid matrix, a solvent, and any number of neutral or charged solutes. This formulation proposes novel approaches for addressing several challenges posed by the finite element analysis of such complex materials: The exclusion of solutes from a fraction of the pore space due to steric volume and short-range electrostatic effects is modeled by a solubility factor, whose dependence on solid matrix deformation and solute concentrations may be described by user-defined constitutive relations. These solute exclusion mechanisms combine with long-range electrostatic interactions into a partition coefficient for each solute whose value is dependent upon the evaluation of the electric potential from the electroneutrality condition. It is shown that this electroneutrality condition reduces to a polynomial equation with only one valid root for the electric potential, regardless of the number and valence of charged solutes in the mixture. The equation of charge conservation is enforced as a constraint within the equation of mass balance for each solute, producing a natural boundary condition for solute fluxes that facilitates the prescription of electric current density on a boundary. It is also shown that electrical grounding is necessary to produce numerical stability in analyses where all the boundaries of a multiphasic material are impermeable to ions. Several verification problems are presented that demonstrate the ability of the code to reproduce known or newly derived solutions: (1) the Kedem-Katchalsky model for osmotic loading of a cell; (2) Donnan osmotic swelling of a charged hydrated tissue; and (3) current flow in an electrolyte. Furthermore, the code is used to generate novel theoretical predictions of known experimental findings in biological tissues: (1) current-generated stress in articular cartilage and (2) the influence of salt cation charge number on the cartilage creep response. This generalized finite element framework for multiphasic materials makes it possible to model the mechanoelectrochemical behavior of biological tissues and cells and sets the stage for the future analysis of reactive mixtures to account for growth and remodeling.

Synthesis rates and binding kinetics of matrix products in engineered cartilage constructs using chondrocyte-seeded agarose gels

Large-sized cartilage constructs suffer from inhomogeneous extracellular matrix deposition due to insufficient nutrient availability. Computational models of nutrient consumption and tissue growth can be utilized as an efficient alternative to experimental trials to optimize the culture of large constructs; models require system-specific growth and consumption parameters. To inform models of the [bovine chondrocyte]-[agarose gel] system, total synthesis rate (matrix accumulation rate+matrix release rate) and matrix retention fractions of glycosaminoglycans (GAG), collagen, and cartilage oligomeric matrix protein (COMP) were measured either in the presence (continuous or transient) or absence of TGF-β3 supplementation. TGF-β3's influences on pyridinoline content and mechanical properties were also measured. Reversible binding kinetic parameters were characterized using computational models. Based on our recent nutrient supplementation work, we measured glucose consumption and critical glucose concentration for tissue growth to computationally simulate the culture of a human patella-sized tissue construct, reproducing the experiment of Hung et al. (2003). Transient TGF-β3 produced the highest GAG synthesis rate, highest GAG retention ratio, and the highest binding affinity; collagen synthesis was elevated in TGF-β3 supplementation groups over control, with the highest binding affinity observed in the transient supplementation group; both COMP synthesis and retention were lower than those for GAG and collagen. These results informed the modeling of GAG deposition within a large patella construct; this computational example was similar to the previous experimental results without further adjustments to modeling parameters. These results suggest that these nutrient consumption and matrix synthesis models are an attractive alternative for optimizing the culture of large-sized constructs.

Subject-specific analysis of joint contact mechanics: application to the study of osteoarthritis and surgical planning

Advances in computational mechanics, constitutive modeling, and techniques for subject-specific modeling have opened the door to patient-specific simulation of the relationships between joint mechanics and osteoarthritis (OA), as well as patient-specific preoperative planning. This article reviews the application of computational biomechanics to the simulation of joint contact mechanics as relevant to the study of OA. This review begins with background regarding OA and the mechanical causes of OA in the context of simulations of joint mechanics. The broad range of technical considerations in creating validated subject-specific whole joint models is discussed. The types of computational models available for the study of joint mechanics are reviewed. The types of constitutive models that are available for articular cartilage are reviewed, with special attention to choosing an appropriate constitutive model for the application at hand. Issues related to model generation are discussed, including acquisition of model geometry from volumetric image data and specific considerations for acquisition of computed tomography and magnetic resonance imaging data. Approaches to model validation are reviewed. The areas of parametric analysis, factorial design, and probabilistic analysis are reviewed in the context of simulations of joint contact mechanics. Following the review of technical considerations, the article details insights that have been obtained from computational models of joint mechanics for normal joints; patient populations; the study of specific aspects of joint mechanics relevant to OA, such as congruency and instability; and preoperative planning. Finally, future directions for research and application are summarized.

Multiscale mechanics of articular cartilage: potentials and challenges of coupling musculoskeletal, joint, and microscale computational models

Articular cartilage experiences significant mechanical loads during daily activities. Healthy cartilage provides the capacity for load bearing and regulates the mechanobiological processes for tissue development, maintenance, and repair. Experimental studies at multiple scales have provided a fundamental understanding of macroscopic mechanical function, evaluation of the micromechanical environment of chondrocytes, and the foundations for mechanobiological response. In addition, computational models of cartilage have offered a concise description of experimental data at many spatial levels under healthy and diseased conditions, and have served to generate hypotheses for the mechanical and biological function. Further, modeling and simulation provides a platform for predictive risk assessment, management of dysfunction, as well as a means to relate multiple spatial scales. Simulation-based investigation of cartilage comes with many challenges including both the computational burden and often insufficient availability of data for model development and validation. This review outlines recent modeling and simulation approaches to understand cartilage function from a mechanical systems perspective, and illustrates pathways to associate mechanics with biological function. Computational representations at single scales are provided from the body down to the microstructure, along with attempts to explore multiscale mechanisms of load sharing that dictate the mechanical environment of the cartilage and chondrocytes.

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

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

Solute transport across a contact interface in deformable porous media

A finite element formulation of neutral solute transport across a contact interface between deformable porous media is implemented and validated against analytical solutions. By reducing the integral statements of external virtual work on the two contacting surfaces into a single contact integral, the algorithm automatically enforces continuity of solute molar flux across the contact interface, whereas continuity of the effective solute concentration (a measure of the solute mechano-chemical potential) is achieved using a penalty method. This novel formulation facilitates the analysis of problems in biomechanics where the transport of metabolites across contact interfaces of deformable tissues may be of interest. This contact algorithm is the first to address solute transport across deformable interfaces, and is made available in the public domain, open-source finite element code FEBio (http://www.febio.org).

Effect of intervertebral disc degeneration on disc cell viability: a numerical investigation

Degeneration of the intervertebral disc may be initiated and supported by impairment of the nutrition processes of the disc cells. The effects of degenerative changes on cell nutrition are, however, only partially understood. In this work, a finite volume model was used to investigate the effect of endplate calcification, water loss, reduction of disc height and cyclic mechanical loading on the sustainability of the disc cell population. Oxygen, lactate and glucose diffusion, production and consumption were modelled with non-linear coupled partial differential equations. Oxygen and glucose consumption and lactate production were expressed as a function of local oxygen concentration, pH and cell density. The cell viability criteria were based on local glucose concentration and pH. Considering a disc with normal water content, cell death was initiated in the centre of the nucleus for oxygen, glucose, and lactate diffusivities in the cartilaginous endplate below 20% of the physiological values. The initial cell population could not be sustained even in the non-calcified endplates when a reduction of diffusion inside the disc due to water loss was modelled. Alterations in the disc shape such as height loss, which shortens the transport route between the nutrient sources and the cells, and cyclic mechanical loads, could enhance cell nutrition processes.

Triphasic mixture model of cell-mediated enzymatic degradation of hydrogels

One critical component of engineering living tissue equivalents is the design scaffolds (often made of hydrogels) whose degradation kinetics can match that of matrix production by cells. However, cell-mediated enzymatic degradation of a hydrogel is a highly complex and nonlinear process that is challenging to comprehend based solely on experimental observations. To address this issue, this study presents a triphasic mixture model of the enzyme-hydrogel system, which consists of a solid polymer network, water and enzyme. On the basis mixture theory, the rubber elasticity theory and the Michaelis-Menton kinetics for degradation, the model naturally incorporates a strong coupling between gel mechanical properties, the kinetics of degradation and the transport of enzyme through the gel. The model is then used to investigate the particular problem of a single spherical enzyme-producing cell, embedded in a spherical hydrogel domain, for which the governing equations can be cast within the cento-symmetric assumptions. The governing equations are subsequently solved using an implicit nonlinear finite element procedure to obtain the evolution of enzyme concentration and gel degradation through time and space. The model shows that two regimes of degradation behaviour exist, whereby degradation is dominated either by diffusion or dominated by reaction kinetics. Depending on the enzyme properties and the initial hydrogel design, the temporal and spatial changes in gel cross-linking are dramatically impacted, a feature that is likely to strongly affect new tissue development.

Coupled porohyperelastic mass transport (PHEXPT) finite element models for soft tissues using ABAQUS

Finite element models (FEMs) including characteristic large deformations in highly nonlinear materials (hyperelasticity and coupled diffusive/convective transport of neutral mobile species) will allow quantitative study of in vivo tissues. Such FEMs will provide basic understanding of normal and pathological tissue responses and lead to optimization of local drug delivery strategies. We present a coupled porohyperelastic mass transport (PHEXPT) finite element approach developed using a commercially available ABAQUS finite element software. The PHEXPT transient simulations are based on sequential solution of the porohyperelastic (PHE) and mass transport (XPT) problems where an Eulerian PHE FEM is coupled to a Lagrangian XPT FEM using a custom-written FORTRAN program. The PHEXPT theoretical background is derived in the context of porous media transport theory and extended to ABAQUS finite element formulations. The essential assumptions needed in order to use ABAQUS are clearly identified in the derivation. Representative benchmark finite element simulations are provided along with analytical solutions (when appropriate). These simulations demonstrate the differences in transient and steady state responses including finite deformations, total stress, fluid pressure, relative fluid, and mobile species flux. A detailed description of important model considerations (e.g., material property functions and jump discontinuities at material interfaces) is also presented in the context of finite deformations. The ABAQUS-based PHEXPT approach enables the use of the available ABAQUS capabilities (interactive FEM mesh generation, finite element libraries, nonlinear material laws, pre- and postprocessing, etc.). PHEXPT FEMs can be used to simulate the transport of a relatively large neutral species (negligible osmotic fluid flux) in highly deformable hydrated soft tissues and tissue-engineered materials.

Substrates for cardiovascular tissue engineering

Cardiovascular tissue engineering aims to find solutions for the suboptimal regeneration of heart valves, arteries and myocardium by creating 'living' tissue replacements outside (in vitro) or inside (in situ) the human body. A combination of cells, biomaterials and environmental cues of tissue development is employed to obtain tissues with targeted structure and functional properties that can survive and develop within the harsh hemodynamic environment of the cardiovascular system. This paper reviews the up-to-date status of cardiovascular tissue engineering with special emphasis on the development and use of biomaterial substrates. Key requirements and properties of these substrates, as well as methods and readout parameters to test their efficacy in the human body, are described in detail and discussed in the light of current trends toward designing biologically inspired microenviroments for in situ tissue engineering purposes.

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.

Validation of a fluid-structure interaction model of solute transport in pores of cyclically deformed tissue scaffolds

Convection induced by repetitive compression of porous tissue scaffolds enhances solute transport inside the scaffold. Our previous experiments have shown that pore size, shape, and orientation with respect to strain direction greatly influence loading-induced solute transport. The objective of this study was to develop a computational model of deformation-induced solute transport in porous tissue scaffolds, which included the pore geometry of the scaffold. This geometry consisted of a cubic scaffold with single channel in the middle of the scaffold, immersed in a fluid reservoir. Cylindrical pores with circular or elliptic cross section, and spheroid pores were modeled. The scaffold was cyclically compressed from one side, causing fluid motion and dispersion of solute inside the scaffold pore. Scaffold deformation was solved using the finite element method, and fluid flow and solute transport were solved using the finite volume method. The distortion of the scaffold-fluid interface was transferred as a boundary condition to the fluid flow solver. Both convection and diffusion were included in the computations. The solute transport rates in the different scaffold pore geometries agreed well with our previous experimental results obtained with X-ray microimaging. This model will be used to explore transport properties of a spectrum of novel scaffold designs.

Pericellular Matrix Mechanics in the Anulus Fibrosus Predicted by a Three-Dimensional Finite Element Model and In Situ Morphology

Anulus fibrosus (AF) cells have been demonstrated to exhibit dramatic differences in morphology and biologic responses to different types of mechanical stimuli. AF cells may reside as single cell, paired or multiple cells in a contiguous pericellular matrix (PCM), whose structure and properties are expected to have a significant influence on the mechanical stimuli that these cells may experience during physiologic loading of the spine, as well as in tissue degeneration and regeneration. In this study, a computational model was developed to predict the micromechanical stimuli, such as stress and strain, fluid pressure and flow, of cells and their surrounding PCM in the AF tissue using three-dimensional (3D) finite element models based on in situ morphology. 3D solid geometries of cell-PCM regions were registered from serial confocal images obtained from mature rat AF tissues by custom codes. Distinct cell-matrix units were modeled with a custom 3D biphasic finite element code (COMSOL Multiphysics), and simulated to experience uni-axial tensile strain along the local collagen fiber direction. AF cells were predicted to experience higher volumetric strain with a strain amplification ratio (relative to that in the extracellular matrix) of ~ 3.1 - 3.8 at equilibrium, as compared to the PCM domains (1.3 - 1.9). The strain concentrations were generally found at the cell/PCM interface and stress concentration at the PCM/ECM interface. Increased numbers of cells within a contiguous PCM was associated with an apparent increase of strain levels and decreased rate of fluid pressurization in the cell, with magnitudes dependent on the cell size, shape and relative position inside the PCM. These studies provide spatio-temporal information on micromechanics of AF cells in understanding the mechanotransduction in the intervertebral disc.

Cyclic deformation-induced solute transport in tissue scaffolds with computer designed, interconnected, pore networks: experiments and simulations

Nutrient supply and waste removal in porous tissue engineering scaffolds decrease from the periphery to the center, leading to limited depth of ingrowth of new tissue into the scaffold. However, as many tissues experience cyclic physiological strains, this may provide a mechanism to enhance solute transport in vivo before vascularization of the scaffold. The hypothesis of this study was that pore cross-sectional geometry and interconnectivity are of major importance for the effectiveness of cyclic deformation-induced solute transport. Transparent elastic polyurethane scaffolds, with computer-programmed design of pore networks in the form of interconnected channels, were fabricated using a 3D printing and injection molding technique. The scaffold pores were loaded with a colored tracer for optical contrast, cyclically compressed with deformations of 10 and 15% of the original undeformed height at 1.0 Hz. Digital imaging was used to quantify the spatial distribution of the tracer concentration within the pores. Numerical simulations of a fluid-structure interaction model of deformation-induced solute transport were compared to the experimental data. The results of experiments and modeling agreed well and showed that pore interconnectivity heavily influences deformation-induced solute transport. Pore cross-sectional geometry appears to be of less relative importance in interconnected pore networks. Validated computer models of solute transport can be used to design optimal scaffold pore geometries that will enhance the convective transport of nutrients inside the scaffold and the removal of waste, thus improving the cell survivability deep inside the scaffold.

Mathematical modelling of fibre-enhanced perfusion inside a tissue-engineering bioreactor

We develop a simple mathematical model for forced flow of culture medium through a porous scaffold in a tissue-engineering bioreactor. Porous-walled hollow fibres penetrate the scaffold and act as additional sources of culture medium. The model, based on Darcy's law, is used to examine the nutrient and shear-stress distributions throughout the scaffold. We consider several configurations of fibres and inlet and outlet pipes. Compared with a numerical solution of the full Navier-Stokes equations within the complex scaffold geometry, the modelling approach is cheap, and does not require knowledge of the detailed microstructure of the particular scaffold being used. The potential of this approach is demonstrated through quantification of the effect the additional flow from the fibres has on the nutrient and shear-stress distribution.

The micromechanics of fluid-solid interactions during growth in porous soft biological tissue

In this paper, we address some modelling issues related to biological growth. Our treatment is based on a formulation for growth that was proposed within the context of mixture theory (J Mech Phys Solids 52:1595-1625, 2004). We aim to make this treatment more appropriate for the physics of porous soft tissues, paying particular attention to the nature of fluid transport, and mechanics of fluid and solid phases. The interactions between transport and mechanics have significant implications for growth and swelling. We also reformulate the governing differential equations for reaction-transport of solutes to represent the incompressibility constraint on the fluid phase of the tissue. This revision enables a straightforward implementation of numerical stabilisation for the advection-dominated limit of these equations. A finite element implementation with operator splitting is used to solve the coupled, non-linear partial differential equations that arise from the theory. We carry out a numerical and analytic study of the convergence of the operator splitting scheme subject to strain- and stress-homogenisation of the mechanics of fluid-solid interactions. A few computations are presented to demonstrate aspects of the physical mechanisms, and the numerical performance of the formulation.

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.

Biphasic finite element model of solute transport for direct infusion into nervous tissue

Infusion-based techniques are promising drug delivery methods for treating diseases of the nervous system. Direct infusion into tissue parenchyma circumvents the blood-brain barrier, localizes delivery, and facilitates transport of macromolecular agents. Computational models that predict interstitial flow and solute transport may aid in protocol design and optimization. We have developed a biphasic finite element (FE) model that accounts for local, flow-induced tissue swelling around an infusion cavity. It solves for interstitial fluid flow, tissue deformation, and solute transport in surrounding isotropic gray matter. FE solutions for pressure-controlled infusion were validated by comparing with analytical solutions. The influence of deformation-dependent hydraulic permeability was considered. A transient, nonlinear relationship between infusion pressure and infusion rate was determined. The sensitivity of convection-dominated solute transport (i.e., albumin) over a range of nervous tissue properties was also simulated. Solute transport was found to be sensitive to pressure-induced swelling effects mainly in regions adjacent to the infusion cavity (r/a 0 <or= 5 where a 0 is the outer cannula radius) for short times infusion simulated (3 min). Overall, the biphasic approach predicted enhanced macromolecular transport for small volume infusions (e.g., 2 microL over 1 h). Solute transport was enhanced by decreasing Young's modulus and increasing hydraulic permeability of the tissue.

Interstitial flow and its effects in soft tissues

Interstitial flow plays important roles in the morphogenesis, function, and pathogenesis of tissues. To investigate these roles and exploit them for tissue engineering or to overcome barriers to drug delivery, a comprehensive consideration of the interstitial space and how it controls and affects such processes is critical. Here we attempt to review the many physical and mathematical correlations that describe fluid and mass transport in the tissue interstitium; the factors that control and affect them; and the importance of interstitial transport on cell biology, tissue morphogenesis, and tissue engineering. Finally, we end with some discussion of interstitial transport issues in drug delivery, cell mechanobiology, and cell homing toward draining lymphatics.

Mathematical modelling of engineered tissue growth using a multiphase porous flow mixture theory

This paper outlines the framework of a porous flow mixture theory for the mathematical modelling of in vitro tissue growth, and gives an application of this theory to an aspect of tissue engineering. The problem is formulated as a set of partial differential equations governing the space and time dependence of the amounts of each component of the tissue (phase), together with the physical stresses in each component. The theory requires constitutive relations to specify the material properties of each phase, and also requires relations to specify the stresses developed due to mechanical interactions, both within each phase and between different phases. An application of the theory is given to the study of the mobility and aggregation of a population of cells seeded into an artificial polymeric scaffold. Stability analysis techniques show that the interplay of the forces between the tissue constituents results in two different regimes: either the cells form aggregates or disperse through the scaffold.

Dynamic finite element modeling of poroviscoelastic soft tissue

Clinical evidences relative to biomechanical factors have demonstrated their important contribution to the behaviour of soft tissues. Finite element (FE) analysis is used to study the mechanical behaviour of soft tissue because it can provide numerical solutions to problems that are intractable to analytic solutions. This study focuses on the development of a FE model of a poroelastic biological tissue, which incorporates the viscoelastic material behaviour, finite deformation and inertial effect. The FE formulation is based on the weak form derived from the governing equation, and Newmark-beta method as well as Newton's method is incorporated into the implicit non-linear solutions. One-dimensional analytical solutions were used to verify the theoretical formulation and the numerical implementation of the proposed model. This study was further extended to analyze two-dimensional biomechanical models and the results clearly demonstrate the importance of including finite deformation, viscoelasticity and inertial effects.

Computational study of culture conditions and nutrient supply in cartilage tissue engineering

Different culture conditions for cartilage tissue engineering were evaluated with respect to the supply of oxygen and glucose and the accumulation of lactate. A computational approach was adopted in which the culture configurations were modeled as a batch process and transport was considered within constructs seeded at high cell concentrations and of clinically relevant dimensions. To assess the extent to which mass transfer can be influenced theoretically, extreme cases were distinguished in which the culture medium surrounding the construct was assumed either completely static or well mixed and fully oxygenated. It can be concluded that severe oxygen depletion and lactate accumulation can occur within constructs for cartilage tissue engineering. However, the results also indicate that transport restrictions are not insurmountable, providing that the medium is well homogenized and oxygenated and the construct's surfaces are sufficiently exposed to the medium. The large variation in uptake rates of chondrocytes indicates that for any specific application the quantification of cellular utilization rates, depending on the cell source and culture conditions, is an essential starting point for optimizing culture protocols.

Determination of the Poisson's ratio of the cell: recovery properties of chondrocytes after release from complete micropipette aspiration

Chondrocytes in articular cartilage are regularly subjected to compression and recovery due to dynamic loading of the joint. Previous studies have investigated the elastic and viscoelastic properties of chondrocytes using micropipette aspiration techniques, but in order to calculate cell properties, these studies have generally assumed that cells are incompressible with a Poisson's ratio of 0.5. The goal of this study was to measure the Poisson's ratio and recovery properties of the chondrocyte by combining theoretical modeling with experimental measures of complete cellular aspiration and release from a micropipette. Chondrocytes isolated from non-osteoarthritic and osteoarthritic cartilage were fully aspirated into a micropipette and allowed to reach mechanical equilibrium. Cells were then extruded from the micropipette and cell volume and morphology were measured throughout the experiment. This experimental procedure was simulated with finite element analysis, modeling the chondrocyte as either a compressible two-mode viscoelastic solid, or as a biphasic viscoelastic material. By fitting the experimental data to the theoretically predicted cell response, the Poisson's ratio and the viscoelastic recovery properties of the cell were determined. The Poisson's ratio of chondrocytes was found to be 0.38 for non-osteoarthritic cartilage and 0.36 for osteoarthritic chondrocytes (no significant difference). Osteoarthritic chondrocytes showed an increased recovery time following full aspiration. In contrast to previous assumptions, these findings suggest that chondrocytes are compressible, consistent with previous studies showing cell volume changes with compression of the extracellular matrix.

Can the growth factors PTHrP, Ihh and VEGF, together regulate the development of a long bone?

Endochondral ossification is the process of differentiation of cartilaginous into osseous tissue. Parathyroid hormone related protein (PTHrP), Indian hedgehog (Ihh) and vascular endothelial growth factor (VEGF), which are synthesized in different zones of the growth plate, were found to have crucial roles in regulating endochondral ossification. The aim of this study was to evaluate whether the three growth factors PTHrP, Ihh and VEGF, together, could regulate longitudinal growth in a normal human, fetal femur. For this purpose, a one-dimensional finite element (FE) model, incorporating growth factor signaling, was developed of the human, distal, femoral growth plate. It included growth factor synthesis in the relevant zones, their transport and degradation and their effects. Simulations ran from initial hypertrophy in the center of the bone until secondary ossification starts at approximately 3.5 months postnatal. For clarity, we emphasize that no mechanical stresses were considered. The FE model showed a stable growth plate in which the bone growth rate was constant and the number of cells per zone oscillated around an equilibrium. Simulations incorporating increased and decreased PTHrP and Ihh synthesis rates resulted, respectively, in more and less cells per zone and in increased and decreased bone growth rates. The FE model correctly reflected the development of a growth plate and the rate of bone growth in the femur. Simulations incorporating increased and decreased PTHrP and Ihh synthesis rates reflected growth plate pathologies and growth plates in PTHrP-/- and Ihh-/- mice. The three growth factors, PTHrP, Ihh and VEGF, could potentially together regulate tissue differentiation.

Bioreactor design for tissue engineering

Bioreactor systems play an important role in tissue engineering, as they enable reproducible and controlled changes in specific environmental factors. They can provide technical means to perform controlled studies aimed at understanding specific biological, chemical or physical effects. Furthermore, bioreactors allow for a safe and reproducible production of tissue constructs. For later clinical applications, the bioreactor system should be an advantageous method in terms of low contamination risk, ease of handling and scalability. To date the goals and expectations of bioreactor development have been fulfilled only to some extent, as bioreactor design in tissue engineering is very complex and still at an early stage of development. In this review we summarize important aspects for bioreactor design and provide an overview on existing concepts. The generation of three dimensional cartilage-carrier constructs is described to demonstrate how the properties of engineered tissues can be improved significantly by combining biological and engineering knowledge. In the future, a very intimate collaboration between engineers and biologists will lead to an increased fundamental understanding of complex issues that can have an impact on tissue formation in bioreactors.

Nutrient Utilization by Bovine Articular Chondrocytes: A Combined Experimental and Theoretical Approach

A combined experimental-numerical approach was adopted to characterize glucose and oxygen uptake and lactate production by bovine articular chondrocytes in a model system. For a wide range of cell concentrations, cells in agarose were supplemented with either low or high glucose medium. During an initial culture phase of 48h, oxygen was monitored noninvasively using a biosensor system. Glucose and lactate were determined by medium sampling. In order to quantify glucose and oxygen uptake, a finite element approach was adopted to describe diffusion and uptake in the experimental model. Numerical predictions of lactate, based on simple relations for cell metabolism, were found to agree well for low glucose, but not for high glucose medium. Oxygen did not play a role in either case. Given the close association between chondrocyte energy metabolism and matrix synthesis, a quantifiable prediction of utilization can present a valuable contribution in the optimization of tissue engineering conditions.

Large Deformation Finite Element Analysis of Micropipette Aspiration to Determine the Mechanical Properties of the Chondrocyte

Chondrocytes, the cells in articular cartilage, exhibit solid-like viscoelastic behavior in response to mechanical stress. In modeling the creep response of these cells during micropipette aspiration, previous studies have attributed the viscoelastic behavior of chondrocytes to either intrinsic viscoelasticity of the cytoplasm or to biphasic effects arising from fluid-solid interactions within the cell. However, the mechanisms responsible for the viscoelastic behavior of chondrocytes are not fully understood and may involve one or both of these phenomena. In this study, the micropipette aspiration experiment was modeled using a large strain finite element simulation that incorporated contact boundary conditions. The cell was modeled using finite strain incompressible and compressible elastic models, a two-mode compressible viscoelastic model, or a biphasic elastic or viscoelastic model. Comparison of the model to the experimentally measured response of chondrocytes to a step increase in aspiration pressure showed that a two-mode compressible viscoelastic formulation accurately captured the creep response of chondrocytes during micropipette aspiration. Similarly, a biphasic two-mode viscoelastic analysis could predict all aspects of the cell's creep response to a step aspiration. In contrast, a biphasic elastic formulation was not capable of predicting the complete creep response, suggesting that the creep response of the chondrocytes under micropipette aspiration is predominantly due to intrinsic viscoelastic phenomena and is not due to the biphasic behavior.