A mechanical composite spheres analysis of engineered cartilage dynamics

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.

Cell – extracellular matrix interaction in glioma growth. In silico model

The study aims to investigate the role of viscoelastic interactions between cells and extracellular matrix (ECM) in avascular tumor growth. Computer simulations of glioma multicellular tumor spheroid (MTS) growth are being carried out for various conditions. The calculations are based on a continuous model, which simulates oxygen transport into MTS; transitions between three cell phenotypes, cell transport, conditioned by hydrostatic forces in cell–ECM composite system, cell motility and cell adhesion. Visco-elastic cell aggregation and elastic ECM scaffold represent two compressible constituents of the composite. Cell–ECM interactions form a Transition Layer on the spheroid surface, where mechanical characteristics of tumor undergo rapid transition. This layer facilitates tumor progression to a great extent. The study demonstrates strong effects of ECM stiffness, mechanical deformations of the matrix and cell–cell adhesion on tumor progression. The simulations show in particular that at certain, rather high degrees of matrix stiffness a formation of distant multicellular clusters takes place, while at further increase of ECM stiffness subtumors do not form. The model also illustrates to what extent mere mechanical properties of cell–ECM system may contribute into variations of glioma invasion scenarios.

Direct electrospinning of 3D auricle-shaped scaffolds for tissue engineering applications

Thirty-two poly(ε)caprolactone (PCL) scaffolds have been produced by electrospinning directly into an auricle-shaped mould and seeded with articular chondrocytes harvested from bovine ankle joints. After seeding, the auricle shaped constructs were cultured in vitro and analysed at days 1, 7, 14 and 21 for regional differences in total DNA, glycosaminoglycan (GAG) and collagen (COL) content as well as the expression of aggrecan (AGG), collagen type I and type II (COL1/2) and matrix metalloproteinase 3 and 13 (MMP3/13). Stress-relaxation indentation testing was performed to investigate regional mechanical properties of the electrospun constructs. Electrospinning into a conductive mould yielded stable 3D constructs both initially and for the whole in vitro culture period, with an equilibrium modulus in the MPa range. Rapid cell proliferation and COL accumulation was observed until week 3. Quantitative real time PCR analysis showed an initial increase in AGG, no change in COL2, a persistent increase in COL1, and only a slight decrease initially for MMP3. Electrospinning of fibrous scaffolds directly into an auricle-shape represents a promising option for auricular tissue engineering, as it can reduce the steps needed to achieve an implantable structure.

Cytoskeletal strains in modeled optohydrodynamically stressed healthy and diseased biological cells

Controlled external chemomechanical stimuli have been shown to influence cellular and tissue regeneration/degeneration, especially with regards to distinct disease sequelae or health maintenance. Recently, a unique three-dimensional stress state was mathematically derived to describe the experimental stresses applied to isolated living cells suspended in an optohydrodynamic trap (optical tweezers combined with microfluidics). These formulae were previously developed in two and three dimensions from the fundamental equations describing creeping flows past a suspended sphere. The objective of the current study is to determine the full-field cellular strain response due to the applied three-dimensional stress environment through a multiphysics computational simulation. In this investigation, the multiscale cytoskeletal structures are modeled as homogeneous, isotropic, and linearly elastic. The resulting computational biophysics can be directly compared with experimental strain measurements, other modeling interpretations of cellular mechanics including the liquid drop theory, and biokinetic models of biomolecule dynamics. The described multiphysics computational framework will facilitate more realistic cytoskeletal model interpretations, whose intracellular structures can be distinctly defined, including the cellular membrane substructures, nucleus, and organelles.

Ultrasonic wave propagation assessment of native cartilage explants and hydrogel scaffolds for tissue engineering

Non-destructive techniques characterising the mechanical properties of cells, tissues, and biomaterials provide baseline metrics for tissue engineering design. Ultrasonic wave propagation and attenuation has previously demonstrated the dynamics of extracellular matrix synthesis in chondrocyte-seeded hydrogel constructs. In this paper, we describe an ultrasonic method to analyse two of the construct elements used to engineer articular cartilage in real-time, native cartilage explants and an agarose biomaterial. Results indicated a similarity in wave propagation velocity ranges for both longitudinal (1500-1745 m/s) and transverse (350-950 m/s) waveforms. Future work will apply an acoustoelastic analysis to distinguish between the fluid and solid properties including the cell and matrix biokinetics as a validation of previous mathematical models.

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.

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.

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.

Mechanical stress analysis of microfluidic environments designed for isolated biological cell investigations

Advancements in technologies for assessing biomechanics at the cellular level have led to discoveries in mechanotransduction and the investigation of cell mechanics as a biomarker for disease. With the recent development of an integrated optical tweezer with micron resolution particle image velocimetry, the opportunity to apply controlled multiaxial stresses to suspended single cells is available (Neve, N., Lingwood, J. K., Zimmerman, J., Kohles, S. S., and Tretheway, D. C., 2008, "The muPIVOT: An Integrated Particle Image Velocimetry and Optical Tweezers Instrument for Microenvironment Investigations," Meas. Sci. Technol., 19(9), pp. 095403). A stress analysis was applied to experimental and theoretical flow velocity gradients of suspended cell-sized polystyrene microspheres demonstrating the relevant geometry of nonadhered spherical cells, as observed for osteoblasts, chondrocytes, and fibroblasts. Three flow conditions were assessed: a uniform flow field generated by moving the fluid sample with an automated translation stage, a gravity driven flow through a straight microchannel, and a gravity driven flow through a microchannel cross junction. The analysis showed that fluid-induced stresses on suspended cells (hydrodynamic shear, normal, and principal stresses in the range of 0.02-0.04 Pa) are generally at least an order of magnitude lower than adhered single cell studies for uniform and straight microchannel flows (0.5-1.0 Pa). In addition, hydrostatic pressures dominate (1-100 Pa) over hydrodynamic stresses. However, in a cross junction configuration, orders of magnitude larger hydrodynamic stresses are possible without the influence of physical contact and with minimal laser trapping power.

Two-Dimensional Modeling of Nanomechanical Strains in Healthy and Diseased Single-Cells During Microfluidic Stress Applications

Investigations in cellular and molecular engineering have explored the impact of nanotechnology and the potential for monitoring and control of human diseases. In a recent analysis, the dynamic fluid-induced stresses were characterized during microfluidic applications of an instrument with nanometer and picoNewton resolution as developed for single-cell biomechanics (Kohles, S. S., Nève, N., Zimmerman, J. D., and Tretheway, D. C., 2009, "Stress Analysis of Microfluidic Environments Designed for Isolated Biological Cell Investigations," ASME J. Biomech. Eng., 131(12), p. 121006). The results described the limited stress levels available in laminar, creeping-flow environments, as well as the qualitative cellular strain response to such stress applications. In this study, we present a two-dimensional computational model exploring the physical application of normal and shear stress profiles (with 0.1, 1.0, and 10.0 Pa peak amplitudes) potentially available within uniform and extensional flow states. The corresponding cellular strains and strain patterns were determined within cells modeled with healthy and diseased mechanical properties (5.0-0.1 kPa moduli, respectively). Strain energy density results integrated over the volume of the planar section indicated a strong mechanical sensitivity involving cells with disease-like properties. In addition, ex vivo microfluidic environments creating in vivo stress states would require freestream flow velocities of 2-7 mm/s. Knowledge of the nanomechanical stresses-strains necessary to illicit a biologic response in the cytoskeleton and cellular membrane will ultimately lead to refined mechanotransduction relationships.

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.

An inverse method for predicting tissue-level mechanics from cellular mechanical input

Extracellular matrix (ECM) provides a dynamic three-dimensional structure which translates mechanical stimuli to cells. This local mechanical stimulation may direct biological function including tissue development. Theories describing the role of mechanical regulators hypothesize the cellular response to variations in the external mechanical forces on the ECM. The exact ECM mechanical stimulation required to generate a specific pattern of localized cellular displacement is still unknown. The cell to tissue inverse problem offers an alternative approach to clarify this relationship. Developed for structural dynamics, the inverse dynamics problem translates measurements of local state variables (at the cell level) into an unknown or desired forcing function (at the tissue or ECM level). This paper describes the use of eigenvalues (resonant frequencies), eigenvectors (mode shapes), and dynamic programming to reduce the mathematical order of a simplified cell-tissue system and estimate the ECM mechanical stimulation required for a specified cellular mechanical environment. Finite element and inverse numerical analyses were performed on a simple two-dimensional model to ascertain the effects of weighting parameters and a reduction of analytical modes leading toward a solution. Simulation results indicate that the reduced number of mechanical modes (from 30 to 14 to 7) can adequately reproduce an unknown force time history on an ECM boundary. A representative comparison between cell to tissue (inverse) and tissue to cell (boundary value) modeling illustrates the multiscale applicability of the inverse model.

The μPIVOT: an integrated particle image velocimeter and optical tweezers instrument for microenvironment investigations

A novel instrument to manipulate and characterize the mechanical environment in and around microscale objects in a fluidic environment has been developed by integrating two laser-based techniques: micron-resolution particle image velocimetry (μPIV) and optical tweezers (OT). This instrument, the μPIVOT, enables a new realm of microscale studies, yet still maintains the individual capabilities of each optical technique. This was demonstrated with individual measurements of optical trap stiffness (∼70 pN μm(-1) for a 20 μm polystyrene sphere and a linear relationship between trap stiffness and laser power) and fluid velocities within 436 nm of a microchannel wall. The integrated device was validated by comparing computational flow predictions to the measured velocity profile around a trapped particle in either a uniform flow or an imposed, gravity-driven microchannel flow (R(2) = 0.988, RMS error = 13.04 μm s(-1)). Interaction between both techniques is shown to be negligible for 15 μm to 35 μm diameter trapped particles subjected to fluid velocities from 50 μm s(-1) to 500 μm s(-1) even at the highest laser power (1.45 W). The integrated techniques will provide a unique perspective toward understanding microscale phenomena including single-cell biomechanics, non-Newtonian fluid mechanics and single particle or particle-particle hydrodynamics.