Nanobiomechanics of living cells: a review
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paper_title: Comparative study on the differential mechanical properties of human liver cancer and normal cells

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Abstract Although cancerous cells and normal cells are known to have different elasticity values, there have been inconsistent reports in terms of the actual and relative values for these two cell types depending on the experimental conditions. This paper investigated the mechanical characterization of normal hepatocytes (THLE-2) and hepatocellular carcinoma cells (HepG2) using atomic force microscopy indentation experiments and the Hertz–Sneddon model, and the results were confirmed by an independent de-adhesion assay. To improve the reliability of the data, we considered the effects of tip geometry and indentation depth on the measured elasticity of the cells. This study demonstrated that THLE-2 cells had a higher elastic modulus compared with the HepG2 cells and that this difference was more significant when a conical tip was used. The inhibitor study indicated that this difference in the mechanical properties of THLE-2 and HepG2 cells was mainly attributed to differential arrangements in the cytoskele...
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paper_title: Confocal microscopy indentation system for studying in situ chondrocyte mechanics

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Abstract Chondrocytes synthesize extracellular matrix molecules, thus they are essential for the development, adaptation and maintenance of articular cartilage. Furthermore, it is well accepted that the biosynthetic activity of chondrocytes is influenced by the mechanical environment. Therefore, their response to mechanical stimuli has been studied extensively. Much of the knowledge in this area of research has been derived from testing of isolated cells, cartilage explants, and fixed cartilage specimens: systems that differ in important aspects from chondrocytes embedded in articular cartilage and observed during loading conditions. In this study, current model systems have been improved by working with the intact cartilage in real time. An indentation system was designed on a confocal microscope that allows for simultaneous loading and observation of chondrocytes in their native environment. Cell mechanics were then measured under precisely controlled loading conditions. The indentation system is based on a light transmissible cylindrical glass indentor of 0.17 mm thickness and 1.64 mm diameter that is aligned along the focal axis of the microscope and allows for real time observation of live cells in their native environment. The system can be used to study cell deformation and biological responses, such as calcium sparks, while applying prescribed loads on the cartilage surface. It can also provide novel information on the relationship between cell loading and cartilage adaptive/degenerative processes in the intact tissue.
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paper_title: Transient adhesion mediated by ligand–receptor interaction on surfaces of variable nanotopography

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Surface microtopography and nanotopography have been shown to influence cell adhesion and function, including proliferation and differentiation, leading both to fundamental questions and practical applications in the field of biomaterials and nanomedicine. However, the mechanisms of how cells sense topography remain obscure. In this study, we measured directly the effect of nanotopography on the kinetics of association and dissociation of ligand–receptor bonds, which are critically involved in the first steps of cell adhesion. We designed models of biological functionalised surfaces with controlled roughness varying from 2 to 400 nm of root mean square, and controlled ligand density. Tests of transient adhesion of receptor–coated microspheres on these surfaces were performed, using a laminar flow chamber assay. We probed Intercellular Adhesion Molecule ICAM–1–anti–ICAM–1 bond adhesion kinetics in the single molecule limit on smooth and rough substrates. Frequency of adhesion did not exhibit any noticeable dependence on roughness parameter, except at high bead velocity. Detachment rate was also independent of roughness. Finally, leucocyte transient adhesion tests were performed on similar substrates, using variable activating incubating media. Here also, no strong effect of roughness was observed in these conditions. Results are rationalised in terms of the role of local geometry on the access of ligands to receptors.
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paper_title: Cellular Tensegrity and Mechanochemical Transduction

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To explain how biological tissues form and function, we must first understand how different types of regulatory signals, both chemical and mechanical, integrate inside the cell. A clue to the mechanism of signal integration comes from recognition that the action of a force on any mass, regardless of scale, will result in a change in three dimensional structure. This is critical because recent studies reveal that many of the molecules that mediate signal transduction and stimulus-response coupling are physically bound to insoluble structural scaffoldings within the cytoskeleton and nucleus (Ingber 1993a). In this type of “solid-state” regulatory system, mechanically-induced structural arrangements could provide a mechanism for regulating cellular biochemistry and hence, efficiently integrating structure and function. However, this is a difficult question to address using conventional molecular biological approaches because this problem is not based on changes in chemical composition or local binding interactions. Rather, it is a question of architecture. As a result of this challenge, a new scientific discipline of “Molecular Cell Engineering” is beginning to emerge which combines elements of molecular cell biology, bioengineering, architecture, and biomechanics.
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paper_title: MECHANICS OF THE HUMAN RED BLOOD CELL DEFORMED BY OPTICAL TWEEZERS

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The mechanical deformation characteristics of living cells are known to influence strongly their chemical and biological functions and the onset, progression and consequences of a number of human diseases. The mechanics of the human red blood cell (erythrocyte) subjected to large deformation by optical tweezers forms the subject of this paper. Video photography of the cell deformed in a phosphate buffered saline solution at room temperature during the imposition of controlled stretching forces, in the tens to several hundreds picoNewton range, is used to assess experimentally the deformation characteristics. The mechanical responses of the cell during loading and upon release of the optical force are then analysed to extract the elastic properties of the cell membrane by recourse to several different constitutive formulations of the elastic and viscoelastic behavior within the framework of a fully three-dimensional finite element analysis. A parametric study of various geometric, loading and structural factors is also undertaken in order to develop quantitative models for the mechanics of deformation by means of optical tweezers. The outcome of the experimental and computational analyses is then compared with the information available on the mechanical response of the red blood cell from other independent experimental techniques. Potential applications of the optical tweezers method described in this paper to the study of mechanical deformation of living cells under different stress states and in response to the progression of some diseases are also highlighted.
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paper_title: Nanotopographical control of human osteoprogenitor differentiation

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Current load-bearing orthopaedic implants are produced in ‘bio-inert’ materials such as titanium alloys. When inserted into the reamed bone during hip or knee replacement surgery the implants interact with mesenchymal populations including the bone marrow. Bio-inert materials are shielded from the body by differentiation of the cells along the fibroblastic lineage producing scar tissue and inferior healing. This is exacerbated by implant micromotion, which can lead to capsule formation. Thus, next-generation implant materials will have to elicit influence over osteoprogenitor differentiation and mesenchymal populations in order to recruit osteoblastic cells and produce direct bone apposition onto the implant. A powerful method of delivering cues to cells is via topography. Micro-scale topography has been shown to affect cell adhesion, migration, cytoskeleton, proliferation and differentiation of a large range of cell types (thus far all cell types tested have been shown to be responsive to topographical cues). More recent research with nanotopography has also shown a broad range of cell response, with fibroblastic cells sensing down to 10 nm in height. Initial studies with human mesenchymal populations and osteoprogenitor populations have again shown strong cell responses to nanofeatures with increased levels of osteocalcin and osteopontin production from the cells on certain topographies. This is indicative of increased osteoblastic activity on the nanotextured materials. Looking at preliminary data, it is tempting to speculate that progenitor cells are, in fact, more responsive to topography than more mature cell types and that they are actively seeking cues from their environment. This review will investigate the range of nanotopographies available to researchers and our present understanding of mechanisms of progenitor cell response. Finally, it will make some speculations of the future of nanomaterials and progenitor cells in tissue engineering.
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paper_title: Signal transduction pathways involved in mechanical regulation of HB-GAM expression in osteoblastic cells.

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Protein kinase C (PKC), protein kinase A (PKA), prostaglandin synthesis, and various mitogen-activated protein kinases (MAPKs) have been reported to be activated in bone cells by mechanical loading. We studied the involvement of these signal transduction pathways in the downregulation of HB-GAM expression in osteoblastic cells after cyclic stretching. Specific antagonists and agonists of these signal transduction pathways were added to cells before loading and to non-loaded control cells. Quantitative RT-PCR was used to evaluate gene expression. The data demonstrated that the extracellular signal-regulated kinase (ERK) 1/2 pathway, PKC, PKA, p38, and c-Jun N-terminal kinase MAPK participated in the mechanical downregulation of HB-GAM expression, whereas prostaglandin synthesis did not seem to be involved.
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paper_title: Cellular mechanotransduction: putting all the pieces together again

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Analysis of cellular mechanotransduction, the mechanism by which cells convert mechanical signals into biochemical responses, has focused on identification of critical mechanosensitive molecules and cellular components. Stretch-activated ion channels, caveolae, integrins, cadherins, growth factor receptors, myosin motors, cytoskeletal filaments, nuclei, extracellular matrix, and numerous other structures and signaling molecules have all been shown to contribute to the mechanotransduction response. However, little is known about how these different molecules function within the structural context of living cells, tissues, and organs to produce the orchestrated cellular behaviors required for mechanosensation, embryogenesis, and physiological control. Recent work from a wide range of fields reveals that organ, tissue, and cell anatomy are as important for mechanotransduction as individual mechanosensitive proteins and that our bodies use structural hierarchies (systems within systems) composed of interconne...
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paper_title: Mechanical models for living cells—a review☆

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Abstract As physical entities, living cells possess structural and physical properties that enable them to withstand the physiological environment as well as mechanical stimuli occurring within and outside the body. Any deviation from these properties will not only undermine the physical integrity of the cells, but also their biological functions. As such, a quantitative study in single cell mechanics needs to be conducted. In this review, we will examine some mechanical models that have been developed to characterize mechanical responses of living cells when subjected to both transient and dynamic loads. The mechanical models include the cortical shell–liquid core (or liquid drop) models which are widely applied to suspended cells; the solid model which is generally used for adherent cells; the power-law structural damping model which is more suited for studying the dynamic behavior of adherent cells; and finally, the biphasic model which has been widely used to study musculoskeletal cell mechanics. Based upon these models, future attempts can be made to develop even more detailed and accurate mechanical models of living cells once these three factors are adequately addressed: structural heterogeneity, appropriate constitutive relations for each of the distinct subcellular regions and components, and active forces acting within the cell. More realistic mechanical models of living cells can further contribute towards the study of mechanotransduction in cells.
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paper_title: Nanosize and vitality : TiO2 nanotube diameter directs cell fate

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We generated, on titanium surfaces, self-assembled layers of vertically oriented TiO2 nanotubes with defined diameters between 15 and 100 nm and show that adhesion, spreading, growth, and differentiation of mesenchymal stem cells are critically dependent on the tube diameter. A spacing less than 30 nm with a maximum at 15 nm provided an effective length scale for accelerated integrin clustering/focal contact formation and strongly enhances cellular activities compared to smooth TiO2 surfaces. Cell adhesion and spreading were severely impaired on nanotube layers with a tube diameter larger than 50 nm, resulting in dramatically reduced cellular activity and a high extent of programmed cell death. Thus, on a TiO2 nanotube surface, a lateral spacing geometry with openings of 30−50 nm represents a critical borderline for cell fate.
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paper_title: Manipulation and isolation of single cells and nuclei.

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The heterogeneous behavior of cells within a cell population makes measurements at the multicellular level insensitive to changes in single cells. Single-cell and single-nucleus analyses are therefore important to address this deficiency which will aid in the understanding of fundamental biology at both the cellular and subcellular levels. Recent technological advancements have enabled the development of new methodologies capable of handling these new challenges. This review highlights various techniques used in single-cell and single-nucleus manipulation and isolation. In particular, the applications related to microfluidics, electrical, optical, and physical methods will be discussed. Ultimately, it is hoped that these techniques will enable fundamental tests to be conducted on single cells and nuclei. One important potential outcome is that this will contribute not only towards detection and isolation of diseased cells but also more accurate diagnosis and prognosis of human diseases.
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paper_title: Unconfined creep compression of chondrocytes

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Abstract The study of single cell mechanics offers a valuable tool for understanding cellular milieus. Specific knowledge of chondrocyte biomechanics could lead to elucidation of disease etiologies and the biomechanical factors most critical to stimulating regenerative processes in articular cartilage. Recent studies in our laboratory have suggested that it may be acceptable to approximate the shape of a single chondrocyte as a disc. This geometry is easily utilized for generating models of unconfined compression. In this study, three continuum mechanics models of increasing complexity were formulated and used to fit unconfined compression creep data. Creep curves were obtained from middle/deep zone chondrocytes ( n =15) and separately fit using the three continuum models. The linear elastic solid model yielded a Young's modulus of 2.55±0.85 kPa. The viscoelastic model (adapted from the Kelvin model) generated an instantaneous modulus of 2.47±0.85 kPa, a relaxed modulus of 1.48±0.35 kPa, and an apparent viscosity of 1.92±1.80 kPa-s. Finally, a linear biphasic model produced an aggregate modulus of 2.58±0.87 kPa, a permeability of 2.57×10 −12 ±3.09 m 4 /N-s, and a Poisson's ratio of 0.069±0.021. The results of this study demonstrate that similar values for the cell modulus can be obtained from three models of increasing complexity. The elastic model provides an easy method for determining the cell modulus, however, the viscoelastic and biphasic models generate additional material properties that are important for characterizing the transient response of compressed chondrocytes.
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paper_title: Viscoelastic properties of chondrocytes from normal and osteoarthritic human cartilage

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The deformation behavior and mechanical properties of articular chondrocytes are believed to play an important role in their response to mechanical loading of the extracellular matrix. This study utilized the micropipette aspiration test to measure the viscoelastic properties of chondrocytes isolated from macroscopically normal or end-stage osteoarthritic cartilage. A three-parameter standard linear solid was used to model the viscoelastic behavior of the cells. Significant differences were found between the mechanical properties of chondrocytes isolated from normal and osteoarthritic cartilage. Specifically, osteoarthritic chondrocytes exhibited a significantly higher equilibrium modulus (0.33 +/- 0.23 compared with 0.24 +/- 0.11 kPa), instantaneous modulus (0.63 +/- 0.51 compared with 0.41 +/- 0.17 kPa), and apparent viscosity (5.8 +/- 6.5 compared with 3.0 +/- 1.8 kPa-s) compared with chondrocytes isolated from macroscopically normal, nonosteoarthritic cartilage. The elastic moduli and relaxation time constant determined experimentally in this study were used to estimate the apparent biphasic properties of the chondrocyte on the basis of the equation for the gel relaxation time of a biphasic material. The differences in viscoelastic properties may reflect alterations in the structure and composition of the chondrocyte cytoskeleton that have previously been associated with osteoarthritic cartilage. Coupled with earlier theoretical models of cell-matrix interactions in articular cartilage, the increased elastic and viscous properties suggest that the mechanical environment of the chondrocyte may be altered in osteoarthritic cartilage.
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paper_title: Mechanobiology of the Intervertebral Disc and Relevance to Disc Degeneration

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Mechanical loading of the intervertebral disc may contribute to disc degeneration by initiating degeneration or by regulating cell-mediated remodeling events that occur in response to the mechanical stimuli of daily activity. This article is a review of the current knowledge of the role of mechanical stimuli in regulating intervertebral disc cellular responses to loading and the cellular changes that occur with degeneration. ::: ::: Intervertebral disc cells exhibit diverse biologic responses to mechanical stimuli, depending on the loading type, magnitude, duration, and anatomic zone of cell origin. The innermost cells respond to low-to-moderate magnitudes of static compression, osmotic pressure, or hydrostatic pressure with increases in anabolic cell responses. Higher magnitudes of loading may give rise to catabolic responses marked by elevated protease gene or protein expression or activity. The key regulators of these mechanobiologic responses for intervertebral disc cells will be the micromechanical stimuli experienced at the cellular level, which are predicted to differ from that measured for the extracellular matrix. Large hydrostatic pressures, but little volume change, are predicted to occur for cells of the nucleus pulposus during compression, while the highly oriented cells of the anulus fibrosus may experience deformations in tension or compression during matrix deformations. In general, the pattern of biologic response to applied loads suggests that the cells of the nucleus pulposus and inner portion of the anulus fibrosus experience comparable micromechanical stimuli in situ and may respond more similarly than cells of the outer portion of the anulus fibrosus. Changes in these features with degeneration are critically understudied, particularly degeneration-associated changes in cell-level mechanical stimuli and the associated mechanobiology. ::: ::: Little is known of the mechanisms that regulate cellular responses to intervertebral mechanobiology, nor is much known with regard to the precise mechanical stimuli experienced by cells during loading. Mechanical factors appear to regulate responses of the intervertebral disc cells through mechanisms involving intracellular Ca2+ transients and cytoskeletal remodeling that may regulate downstream effects such as gene expression and posttranslational biosynthesis. Future studies should address the broader biologic responses to mechanical stimuli in intervertebral disc mechanobiology, the involved signaling mechanisms, and the apparently important interactions among mechanical factors, genetic factors, cytokines, and inflammatory mediators that may be critical in the regulation of intervertebral disc degeneration.
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paper_title: The Deformation of an Erythrocyte Under the Radiation Pressure by Optical Stretch

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, p. 149, 1993) for liposomes. The case under study is the erythrocytestretched by a pair of laser beams in opposite directions within buffer solutions. Thestudy aims to elucidate the effect of radiation pressure from the optical laser because upto now little is known about its inﬂuence on the cell deformation. Following an earlierstudy by Guck et al. (Phys. Rev. Lett.,
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paper_title: Viscoelasticity of the human red blood cell

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We report here the first measurements of the complex modulus of the isolated red blood cell (RBC). Because the RBC is often larger than capillary diameter, important determinants of microcirculatory function are RBC deformability and its changes with pathologies, such as sickle cell disease and malaria. A functionalized ferrimagnetic microbead was attached to the membrane of healthy RBC and then subjected to an oscillatory magnetic field. The resulting torque caused cell deformation. From the oscillatory forcing and resulting bead motions, which were tracked optically, we computed elastic and frictional moduli, g' and g", respectively, from 0.1 to 100 Hz. The g' was nearly frequency independent and dominated the response at all but the highest frequencies measured. Over three frequency decades, g" increased as a power law with an exponent of 0.64, a result not predicted by any simple model. These data suggest that RBC relaxation times that have been reported previously, and any models that rest upon them, are artifactual; the artifact, we suggest, arises from forcing to an exponential fit data of limited temporal duration. A linear range of response was observed, but, as forcing amplitude increased, nonlinearities became clearly apparent. A finite element model suggests that membrane bending was localized to the vicinity of the bead and dominated membrane shear. While the mechanisms accounting for these RBC dynamics remain unclear, methods described here establish new avenues for the exploration of connections among the mechanical, chemical, and biological characteristics of the RBC in health and disease.
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paper_title: Biomechanics approaches to studying human diseases

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Nanobiomechanics has recently been identified as an emerging field that can potentially make significant contributions in the study of human diseases. Research into biomechanics at the cellular and molecular levels of some human diseases has not only led to a better elucidation of the mechanisms behind disease progression, because diseased cells differ physically from healthy ones, but has also provided important knowledge in the fight against these diseases. This article highlights some of the cell and molecular biomechanics research carried out on human diseases such as malaria, sickle cell anemia and cancer and aims to provide further important insights into the pathophysiology of such diseases. It is hoped that this can lead to new methods of early detection, diagnosis and treatment.
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paper_title: Geometry as a Factor for Tissue Growth: Towards Shape Optimization of Tissue Engineering Scaffolds

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Scaffolds for tissue engineering are usually designed to support cell viability with large adhesion surfaces and high permeability to nutrients and oxygen. Recent experiments support the idea that, in addition to surface roughness, elasticity and chemistry, the macroscopic geometry of the substrate also contributes to control the kinetics of tissue deposition. In this study, a previously proposed model for the behavior of osteoblasts on curved surfaces is used to predict the growth of bone matrix tissue in pores of different shapes. These predictions are compared to in vitro experiments with MC3T3-E1 pre-osteoblast cells cultivated in two-millimeter thick hydroxyapatite plates containing prismatic pores with square- or cross-shaped sections. The amount and shape of the tissue formed in the pores measured by phase contrast microscopy confirms the predictions of the model. In cross-shaped pores, the initial overall tissue deposition is twice as fast as in square-shaped pores. These results suggest that the optimization of pore shapes may improve the speed of ingrowth of bone tissue into porous scaffolds.
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paper_title: The enhancement of chondrogenic differentiation of human mesenchymal stem cells by enzymatically regulated RGD functionalities

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A thiol-acrylate photopolymerization was used to incorporate enzymatically cleavable peptide sequences into PEG hydrogels to induce chondrogenic differentiation of encapsulated human mesenchymal stem cells (hMSCs). An adhesive sequence, RGD, was designed with an MMP-13 specific cleavable linker. RGD promotes survival of hMSCs encapsulated in PEG gels and has shown to induce early stages of chondrogenesis, while its persistence can limit complete differentiation. Therefore, an MMP-13 cleavage site was incorporated into the peptide sequence to release RGD mimicking the native differentiation timeline. Active MMP-13 production of encapsulated hMSCs was seen to increase from days 9–14 and only in chondrogenic differentiating cultures. Seeded hMSCs attached to the material prior to enzymatic cleavage, but a significant population of the cells detach after cleavage and release of RGD. Finally, hMSCs encapsulated in RGD-releasing gels produce 10 times as much glycosaminoglycan as cells with uncleavable RGD functionalities, by day 21 of culture. Furthermore, 75% of the cells stain positive for collagen type II deposition where RGD is cleavable, as compared to 19% for cultures where RGD persists. Collectively, this data provides evidence that temporal regulation of integrin-binding peptides is important in the design of niches in differentiating hMSCs to chondrocytes.
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paper_title: Cell mechanics, structure, and function are regulated by the stiffness of the three-dimensional microenvironment.

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This study adopts a combined computational and experimental approach to determine the mechanical, structural, and metabolic properties of isolated chondrocytes cultured within three-dimensional hydrogels. A series of linear elastic and hyperelastic finite-element models demonstrated that chondrocytes cultured for 24 h in gels for which the relaxation modulus is <5 kPa exhibit a cellular Young's modulus of ∼5 kPa. This is notably greater than that reported for isolated chondrocytes in suspension. The increase in cell modulus occurs over a 24-h period and is associated with an increase in the organization of the cortical actin cytoskeleton, which is known to regulate cell mechanics. However, there was a reduction in chromatin condensation, suggesting that changes in the nucleus mechanics may not be involved. Comparison of cells in 1% and 3% agarose showed that cells in the stiffer gels rapidly develop a higher Young's modulus of ∼20 kPa, sixfold greater than that observed in the softer gels. This was associated with higher levels of actin organization and chromatin condensation, but only after 24 h in culture. Further studies revealed that cells in stiffer gels synthesize less extracellular matrix over a 28-day culture period. Hence, this study demonstrates that the properties of the three-dimensional microenvironment regulate the mechanical, structural, and metabolic properties of living cells.
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paper_title: Cytoindentation for obtaining cell biomechanical properties.

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A novel biomechanical testing methodology was developed to obtain the intrinsic material properties of an individual cell attached to a rigid substrate. With use of a newly designed cell-indentation apparatus (cytoindenter), displacement-controlled indentation tests were conducted on the surface of individual MG63 cells and the corresponding surface reaction force of each cell was measured. The cells were modeled with a linear elasticity solution of half-space indentation and the linear biphasic theory on the assumption that the viscoelastic behavior of each cell was due to the interaction between the solid cytoskeletal matrix and the cytoplasmic fluid. To obtain the intrinsic material properties (aggregate modulus, Poisson's ratio, and permeability), the data for experimental surface reaction force and deformation were curve-fitted with use of solutions predicted with a linear biphasic finite element code in conjunction with optimization routines. The MG63 osteoblast-like cells had a compressive aggregate modulus of 2.05+/-0.89 kPa, which is two to three orders of magnitude smaller than that of articular cartilage, six to seven orders smaller than that of compact bone, and quite similar to that of leukocytes. The permeability was 1.18+/-0.65 (x10(-10)) m4/N-s, which is four to six orders of magnitude larger than that of cartilage. The Poisson's ratio was 0.37+/-0.03. The intrinsic material properties of the individual cell in this study can be useful in precisely quantifying mechanical stimuli acting on cells. This information is also needed for theories attempting to establish mechanotransductional relationships.
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paper_title: AFM indentation study of breast cancer cells

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Mechanical properties of individual living cells are known to be closely related to the health and function of the human body. Here, atomic force microscopy (AFM) indentation using a micro-sized spherical probe was carried out to characterize the elasticity of benign (MCF-10A) and cancerous (MCF-7) human breast epithelial cells. AFM imaging and confocal fluorescence imaging were also used to investigate their corresponding sub-membrane cytoskeletal structures. Malignant (MCF-7) breast cells were found to have an apparent Young's modulus significantly lower (1.4-1.8 times) than that of their non-malignant (MCF-10A) counterparts at physiological temperature (37 degrees C), and their apparent Young's modulus increase with loading rate. Both confocal and AFM images showed a significant difference in the organization of their sub-membrane actin structures which directly contribute to their difference in cell elasticity. This change may have facilitated easy migration and invasion of malignant cells during metastasis.
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paper_title: Determination of the Poisson's ratio of the cell: recovery properties of chondrocytes after release from complete micropipette aspiration.

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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.
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paper_title: Micropipette aspiration of living cells.

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The mechanical behavior of living cells is studied with micropipette suction in which the surface of a cell is aspirated into a small glass tube while tracking the leading edge of its surface. Such edges can be tracked in a light microscope to an accuracy of +/-25 nm and suction pressures as small as 0.1-0.2 pN/microm2 can be imposed on the cell. Both soft cells, such as neutrophils and red cells, and more rigid cells, such as chondrocytes and endothelial cells, are studied with this technique. Interpretation of the measurements with basic continuum models leads to values for a cell's elastic and viscous properties. In particular, neutrophils are found to behave as a liquid drop with a cortical (surface) tension of about 30 pN/microm and a viscosity on the order of 100 Pa s. On the other hand, chondrocytes and endothelial cells behave as solids with an elastic modulus of the order of 500 pN/microm2 (0.5 kPa).
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paper_title: Measuring the Mechanical Properties of Living Cells Using Atomic Force Microscopy

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Mechanical properties of cells and extracellular matrix (ECM) play important roles in many biological processes including stem cell differentiation, tumor formation, and wound healing. Changes in stiffness of cells and ECM are often signs of changes in cell physiology or diseases in tissues. Hence, cell stiffness is an index to evaluate the status of cell cultures. Among the multitude of methods applied to measure the stiffness of cells and tissues, micro-indentation using an Atomic Force Microscope (AFM) provides a way to reliably measure the stiffness of living cells. This method has been widely applied to characterize the micro-scale stiffness for a variety of materials ranging from metal surfaces to soft biological tissues and cells. The basic principle of this method is to indent a cell with an AFM tip of selected geometry and measure the applied force from the bending of the AFM cantilever. Fitting the force-indentation curve to the Hertz model for the corresponding tip geometry can give quantitative measurements of material stiffness. This paper demonstrates the procedure to characterize the stiffness of living cells using AFM. Key steps including the process of AFM calibration, force-curve acquisition, and data analysis using a MATLAB routine are demonstrated. Limitations of this method are also discussed.
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paper_title: Connections between single-cell biomechanics and human disease states: gastrointestinal cancer and malaria.

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We investigate connections between single-cell mechanical properties and subcellular structural reorganization from biochemical factors in the context of two distinctly different human diseases: gastrointestinal tumor and malaria. Although the cell lineages and the biochemical links to pathogenesis are vastly different in these two cases, we compare and contrast chemomechanical pathways whereby intracellular structural rearrangements lead to global changes in mechanical deformability of the cell. This single-cell biomechanical response, in turn, seems to mediate cell mobility and thereby facilitates disease progression in situations where the elastic modulus increases or decreases due to membrane or cytoskeleton reorganization. We first present new experiments on elastic response and energy dissipation under repeated tensile loading of epithelial pancreatic cancer cells in force- or displacement-control. Energy dissipation from repeated stretching significantly increases and the cell's elastic modulus decreases after treatment of Panc-1 pancreatic cancer cells with sphingosylphosphorylcholine (SPC), a bioactive lipid that influences cancer metastasis. When the cell is treated instead with lysophosphatidic acid, which facilitates actin stress fiber formation, neither energy dissipation nor modulus is noticeably affected. Integrating recent studies with our new observations, we ascribe these trends to possible SPC-induced reorganization primarily of keratin network to perinuclear region of cell; the intermediate filament fraction of the cytoskeleton thus appears to dominate deformability of the epithelial cell. Possible consequences of these results to cell mobility and cancer metastasis are postulated. We then turn attention to progressive changes in mechanical properties of the human red blood cell (RBC) infected with the malaria parasite Plasmodium falciparum. We present, for the first time, continuous force-displacement curves obtained from in-vitro deformation of RBC with optical tweezers for different intracellular developmental stages of parasite. The shear modulus of RBC is found to increase up to 10-fold during parasite development, which is a noticeably greater effect than that from prior estimates. By integrating our new experimental results with published literature on deformability of Plasmodium-harbouring RBC, we examine the biochemical conditions mediating increases or decreases in modulus, and their implications for disease progression. Some general perspectives on connections among structure, single-cell mechanical properties and biological responses associated with pathogenic processes are also provided in the context of the two diseases considered in this work.
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paper_title: Quantitative analysis of the viscoelastic properties of thin regions of fibroblasts using atomic force microscopy.

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Viscoelasticity of the leading edge, i.e., the lamellipodium, of a cell is the key property for a deeper understanding of the active extension of a cell's leading edge. The fact that the lamellipodium of a cell is very thin (<1000 nm) imparts special challenges for accurate measurements of its viscoelastic behavior. It requires addressing strong substrate effects and comparatively high stresses (>1 kPa) on thin samples. We present the method for an atomic force microscopy-based microrheology that allows us to fully quantify the viscoelastic constants (elastic storage modulus, viscous loss modulus, and the Poisson ratio) of thin areas of a cell (<1000 nm) as well as those of thick areas. We account for substrate effects by applying two different models-a model for well-adhered regions (Chen model) and a model for nonadhered regions (Tu model). This method also provides detailed information about the adhered regions of a cell. The very thin regions relatively near the edge of NIH 3T3 fibroblasts can be identified by the Chen model as strongly adherent with an elastic strength of approximately 1.6 +/- 0.2 kPa and with an experimentally determined Poisson ratio of approximately 0.4 to 0.5. Further from the edge of these cells, the adherence decreases, and the Tu model is effective in evaluating its elastic strength ( approximately 0.6 +/- 0.1 kPa). Thus, our AFM-based microrheology allows us to correlate two key parameters of cell motility by relating elastic strength and the Poisson ratio to the adhesive state of a cell. This frequency-dependent measurement allows for the decomposition of the elastic modulus into loss and storage modulus. Applying this decomposition and Tu's and Chen's finite depth models allow us to obtain viscoelastic signatures in a frequency range from 50 to 300 Hz, showing a rubber plateau-like behavior.
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paper_title: Nanomechanical characterization of tissue engineered bone grown on titanium alloy in vitro

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Intensive work has been performed on the characterization of the mechanical properties of mineralised tissues formed in vivo. However, the mechanical properties of bone-like tissue formed in vitro have rarely been characterised. Most research has either focused on compact cortical bone or cancellous bone, whilst leaving woven bone unaddressed. In this study, bone-like mineralised matrix was produced by osteoblasts cultured in vitro on the surface of titanium alloys. The volume of this tissue-engineered bone is so small that the conventional tensile tests or bending tests are implausible. Therefore, nanoindentation techniques which allow the characterization of the test material from the nanoscale to the microscale were adopted. These reveal the apparent elastic modulus and hardness of the calcospherulite crystals (a representative element for woven bone) are 2.35 ± 0.73 and 0.41 ± 0.15 GPa, respectively. The nanoscale viscoelasticity of such woven bone was further assessed by dynamic indentation analysis.
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paper_title: A compact torsional reference device for easy, accurate and traceable AFM piconewton calibration.

paper_content:

The invention of the atomic force microscope led directly to the possibility of carrying out nanomechanical tests with forces below the nanonewton and the ability to test nanomaterials and single molecules. As a result there is a pressing need for accurate and traceable force calibration of AFM measurements that is not satisfactorily met by existing calibration methods. Here we present a force reference device that makes it possible to calibrate the normal stiffness of typical AFM microcantilevers down to 90?pN?nm?1 with very high accuracy and repeatability and describe how it can be calibrated traceably to the International System of Units via the ampere and the metre, avoiding in that way the difficulties associated with traceability to the SI kilogram. We estimate the total uncertainty associated with cantilever calibration including traceability to be better than 3.5%, thus still offering room for future improvement.
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paper_title: Introduction to Polymers

paper_content:

CONCEPTS, NOMENCLATURE AND SYNTHESIS OF POLYMERS Concepts and Nomenclature The Origins of Polymer Science and the Polymer Industry Basic Definitions and Nomenclature Molar Mass and Degree of Polymerization Principles of Polymerization Introduction Classification of Polymerization Reactions Monomer Functionality and Polymer Skeletal Structure Functional Group Reactivity and Molecular Size: The Principle of Equal Reactivity Step Polymerization Introduction Linear Step Polymerization Non-Linear Step Polymerization Radical Polymerization Introduction to Radical Polymerization The Chemistry of Conventional Free-Radical Polymerization Kinetics of Conventional Free-Radical Polymerization Free-Radical Polymerization Processes Reversible-Deactivation (`Living') Radical Polymerizations Non-Linear Radical Polymerizations Ionic Polymerization Introduction to Ionic Polymerization Cationic Polymerization Anionic Polymerization Group-Transfer Polymerization Stereochemistry and Coordination Polymerization Introduction to Stereochemistry of Polymerization Tacticity of Polymers Geometric Isomerism in Polymers Prepared from Conjugated Dienes Ziegler-Natta Coordination Polymerization Metallocene Coordination Polymerization Ring-Opening Polymerization Introduction to Ring-Opening Polymerization Cationic Ring-Opening Polymerization Anionic Ring-Opening Polymerization Free-Radical Ring-Opening Polymerization Ring-Opening Metathesis Polymerization Specialized Methods of Polymer Synthesis Introduction Solid-State Topochemical Polymerization Polymerization by Oxidative Coupling Precursor Routes to Intractable Polymers Supramolecular Polymerization (Polyassociation) Copolymerization Introduction Step Copolymerization Chain Copolymerization Block Copolymer Synthesis Graft Copolymer Synthesis CHARACTERIZATION OF POLYMERS Theoretical Description of Polymers in Solution Introduction Thermodynamics of Polymer Solutions Chain Dimensions Frictional Properties of Polymer Molecules in Dilute Solution Number-Average Molar Mass Introduction to Measurements of Number-Average Molar Mass Membrane Osmometry Vapour Pressure Osmometry Ebulliometry and Cryoscopy End-Group Analysis Effects of Low Molar Mass Impurities upon Mn Scattering Methods Introduction Static Light Scattering Dynamic Light Scattering Small-Angle X-Ray and Neutron Scattering Frictional Properties of Polymers in Solution Introduction Dilute Solution Viscometry Ultracentrifugation Molar Mass Distribution Introduction Fractionation Gel Permeation Chromatography Field-Flow Fractionation Mass Spectroscopy Chemical Composition and Molecular Microstructure Introduction Principles of Spectroscopy Ultraviolet and Visible Light Absorption Spectroscopy Infrared Spectroscopy Raman Spectroscopy Nuclear Magnetic Resonance Spectroscopy Mass Spectroscopy PHASE STRUCTURE AND MORPHOLOGY OF BULK POLYMERS The Amorphous State Introduction The Glass Transition Factors Controlling the Tg Macromolecular Dynamics The Crystalline State Introduction Determination of Crystal Structure Polymer Single Crystals Semi-Crystalline Polymers Liquid Crystalline Polymers Defects in Crystalline Polymers Crystallization Melting Multicomponent Polymer Systems Introduction Polymer Blends Block Copolymers PROPERTIES OF BULK POLYMERS Elastic Deformation Introduction Elastic Deformation Elastic Deformation of Polymers Viscoelasticity Introduction Viscoelastic Mechanical Models Boltzmann Superposition Principle Dynamic Mechanical Testing Frequency Dependence of Viscoelastic Behaviour Transitions and Polymer Structure Time-Temperature Superposition Effect of Entanglements Non-Linear Viscoelasticity Elastomers Introduction Thermodynamics of Elastomer Deformation Statistical Theory of Elastomer Deformation Stress-Strain Behaviour of Elastomers Factors Affecting Mechanical Behaviour Yield and Crazing Introduction Phenomenology of Yield Yield Criteria Deformation Mechanisms Crazing Fracture and Toughening Introduction Fundamentals of Fracture Mechanics of Fracture Fracture Phenomena Toughened Polymers Polymer Composites Introduction to Composite Materials Matrix Materials Types of Reinforcement Composite Composition Particulate Reinforcement Fibre Reinforcement Nanocomposites Electrical Properties Introduction to Electrical Properties Dielectric Properties Conduction in Polymers Polymer Electronics Answers to Problems Index Problems and Further Reading appear at the end of each chapter.
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paper_title: Cytoindentation for obtaining cell biomechanical properties.

paper_content:

A novel biomechanical testing methodology was developed to obtain the intrinsic material properties of an individual cell attached to a rigid substrate. With use of a newly designed cell-indentation apparatus (cytoindenter), displacement-controlled indentation tests were conducted on the surface of individual MG63 cells and the corresponding surface reaction force of each cell was measured. The cells were modeled with a linear elasticity solution of half-space indentation and the linear biphasic theory on the assumption that the viscoelastic behavior of each cell was due to the interaction between the solid cytoskeletal matrix and the cytoplasmic fluid. To obtain the intrinsic material properties (aggregate modulus, Poisson's ratio, and permeability), the data for experimental surface reaction force and deformation were curve-fitted with use of solutions predicted with a linear biphasic finite element code in conjunction with optimization routines. The MG63 osteoblast-like cells had a compressive aggregate modulus of 2.05+/-0.89 kPa, which is two to three orders of magnitude smaller than that of articular cartilage, six to seven orders smaller than that of compact bone, and quite similar to that of leukocytes. The permeability was 1.18+/-0.65 (x10(-10)) m4/N-s, which is four to six orders of magnitude larger than that of cartilage. The Poisson's ratio was 0.37+/-0.03. The intrinsic material properties of the individual cell in this study can be useful in precisely quantifying mechanical stimuli acting on cells. This information is also needed for theories attempting to establish mechanotransductional relationships.
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paper_title: Plastic indentation in metals with cones

paper_content:

Abstract Q uasi-static indentations with diamond cones have been made in specimens of copper and mild steel work-hardened by varying amounts. The relationship between the hardness so found, the cone angle, and the yield stress is given. The results agree well with the indentation theory of L ockett (1968a) based on classical slip line analysis. This shows that for a work-hardened material the hardness increases with increasing cone angle for cone angles above 105°. Lockett's theory is degenerate below 105° and the experiments here also show a change in deformation mode in this region. Although this agreement with theory appears satisfactory, the deformation process appears to be substantially different from that implied in Lockett's analysis. For large angle cones the deformation resembles a ‘radial’ compression similar to that described by S amuels and M ulhearn (1957): for small cone angles the deformation resembles more clearly the classical slip-line form given by Lockett, though this is precisely the region where the theory becomes degenerate. The effective strain produced by each cone during the indentation process itself is determined for each cone and it is shown that it is possible to construct the stress-strain curve of a given material from a series of cone indentations.
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paper_title: Plastic indentation in metals with cones

paper_content:

Abstract Q uasi-static indentations with diamond cones have been made in specimens of copper and mild steel work-hardened by varying amounts. The relationship between the hardness so found, the cone angle, and the yield stress is given. The results agree well with the indentation theory of L ockett (1968a) based on classical slip line analysis. This shows that for a work-hardened material the hardness increases with increasing cone angle for cone angles above 105°. Lockett's theory is degenerate below 105° and the experiments here also show a change in deformation mode in this region. Although this agreement with theory appears satisfactory, the deformation process appears to be substantially different from that implied in Lockett's analysis. For large angle cones the deformation resembles a ‘radial’ compression similar to that described by S amuels and M ulhearn (1957): for small cone angles the deformation resembles more clearly the classical slip-line form given by Lockett, though this is precisely the region where the theory becomes degenerate. The effective strain produced by each cone during the indentation process itself is determined for each cone and it is shown that it is possible to construct the stress-strain curve of a given material from a series of cone indentations.
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paper_title: On the factors affecting the critical indenter penetration for measurement of coating hardness

paper_content:

Abstract The nanoindentation test is the only viable approach to assess the properties of very thin coatings (
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paper_title: On the relationship between plastic zone radius and maximum depth during nanoindentation

paper_content:

The relationship between plastic zone size and plastic depth during indentation has been studied by a number of workers and an expression relating the plastic zone radius to the residual indentation depth (assumed to be the plastic depth) was developed by Lawn et al. [B.R. Lawn, A.G. Evans and D.B. Marshall, J. Am. Ceram. Soc. 63 (1980)198.] based on microindentation testing. In this study, the relationship between the plastic zone radius and residual indentation depth was examined using finite element analysis for conical indentation in elastic-perfectly-plastic bulk materials. The simulations show that the Lawn method overestimates the plastic zone size for different materials with a wide range of Young's modulus over hardness ratio and for indenters with different geometries and it does not consider tip rounding effects. Therefore, an analytical expression is outlined which agrees well with finite element data. For practical application, the relationship between the radius of the plastic deformation zone and the maximum penetration depth is developed here which has been used to modify the energy-based model developed at Newcastle University to predict the hardness and Young's modulus of coated systems. It is found that the new relationship is easy to apply and predictions of the hardness and Young's modulus of coated glass show good agreement with experimental results.
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paper_title: Cell deformation behavior in mechanically loaded rabbit articular cartilage 4 weeks after anterior cruciate ligament transection

paper_content:

OBJECTIVE ::: Chondrocyte stresses and strains in articular cartilage are known to modulate tissue mechanobiology. Cell deformation behavior in cartilage under mechanical loading is not known at the earliest stages of osteoarthritis. Thus, the aim of this study was to investigate the effect of mechanical loading on volume and morphology of chondrocytes in the superficial tissue of osteoarthritic cartilage obtained from anterior cruciate ligament transected (ACLT) rabbit knee joints, 4 weeks after intervention. ::: ::: ::: METHODS ::: A unique custom-made microscopy indentation system with dual-photon microscope was used to apply controlled 2 MPa force-relaxation loading on patellar cartilage surfaces. Volume and morphology of chondrocytes were analyzed before and after loading. Also global and local tissue strains were calculated. Collagen content, collagen orientation and proteoglycan content were quantified with Fourier transform infrared microspectroscopy, polarized light microscopy and digital densitometry, respectively. ::: ::: ::: RESULTS ::: Following the mechanical loading, the volume of chondrocytes in the superficial tissue increased significantly in ACLT cartilage by 24% (95% confidence interval (CI) 17.2-31.5, P < 0.001), while it reduced significantly in contralateral group tissue by -5.3% (95% CI -8.1 to -2.5, P = 0.003). Collagen content in ACLT and contralateral cartilage were similar. PG content was reduced and collagen orientation angle was increased in the superficial tissue of ACLT cartilage compared to the contralateral cartilage. ::: ::: ::: CONCLUSIONS ::: We found the novel result that chondrocyte deformation behavior in the superficial tissue of rabbit articular cartilage is altered already at 4 weeks after ACLT, likely because of changes in collagen fibril orientation and a reduction in PG content.
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paper_title: Confocal microscopy indentation system for studying in situ chondrocyte mechanics

paper_content:

Abstract Chondrocytes synthesize extracellular matrix molecules, thus they are essential for the development, adaptation and maintenance of articular cartilage. Furthermore, it is well accepted that the biosynthetic activity of chondrocytes is influenced by the mechanical environment. Therefore, their response to mechanical stimuli has been studied extensively. Much of the knowledge in this area of research has been derived from testing of isolated cells, cartilage explants, and fixed cartilage specimens: systems that differ in important aspects from chondrocytes embedded in articular cartilage and observed during loading conditions. In this study, current model systems have been improved by working with the intact cartilage in real time. An indentation system was designed on a confocal microscope that allows for simultaneous loading and observation of chondrocytes in their native environment. Cell mechanics were then measured under precisely controlled loading conditions. The indentation system is based on a light transmissible cylindrical glass indentor of 0.17 mm thickness and 1.64 mm diameter that is aligned along the focal axis of the microscope and allows for real time observation of live cells in their native environment. The system can be used to study cell deformation and biological responses, such as calcium sparks, while applying prescribed loads on the cartilage surface. It can also provide novel information on the relationship between cell loading and cartilage adaptive/degenerative processes in the intact tissue.
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paper_title: Biomechanics of single zonal chondrocytes

paper_content:

Articular cartilage has a distinct zonal architecture, and previous work has shown that chondrocytes from different zones exhibit variations in gene expression and biosynthesis. In this study, the material properties of single chondrocytes from the superficial and middle/deep zones of bovine distal metatarsal articular cartilage were determined using unconfined compression and digital videocapture. To determine the viscoelastic properties of zonal chondrocytes, unconfined creep compression experiments were performed and the resulting creep curves of individual cells were fit using a standard linear viscoelastic solid model. In the model, a fixed value of the Poisson's ratio was used, determined optically from direct compression of middle/deep chondrocytes. The two approaches used in this study yielded the following average material properties of single chondrocytes: Poisson's ratio of 0.26±0.08, instantaneous modulus of 1.06±0.82 kPa, relaxed modulus of 0.78±0.58 kPa, and apparent viscosity of 4.08±7.20 kPa s. Superficial zone chondrocytes were found to be significantly stiffer than middle/deep zone chondrocytes. Attachment time did not affect the stiffness of the cells. The zonal variation in viscoelastic properties may result from the distinct mechanical environments experienced by the cells in vivo. Identifying intrinsic differences in the biomechanics of superficial and middle/deep zone chondrocytes is an important component in understanding how biomechanics influence articular cartilage health and disease.
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paper_title: Mechanical characterization of differentiated human embryonic stem cells

paper_content:

Human embryonic stem cells (hESCs) possess an immense potential in a variety of regenerative applications. A firm understanding of hESC mechanics, on the single cell level, may provide great insight into the role of biophysical forces in the maintenance of cellular phenotype and elucidate mechanical cues promoting differentiation along various mesenchymal lineages. Moreover, cellular biomechanics can provide an additional tool for characterizing stem cells as they follow certain differentiation lineages, and thus may aid in identifying differentiated hESCs, which are most suitable for tissue engineering. This study examined the viscoelastic properties of single undifferentiated hESCs, chondrogenically differentiated hESC subpopulations, mesenchymal stem cells (MSCs), and articular chondrocytes (ACs). hESC chondrogenesis was induced using either transforming growth factor-β1 (TGF-β1) or knock out serum replacer as differentiation agents, and the resulting cell populations were separated based on density. All cell groups were mechanically tested using unconfined creep cytocompression. Analyses of subpopulations from all differentiation regimens resulted in a spectrum of mechanical and morphological properties spanning the range of hESCs to MSCs to ACs. Density separation was further successful in isolating cellular subpopulations with distinct mechanical properties. The instantaneous and relaxed moduli of subpopulations from TGF-β1 differentiation regimen were statistically greater than those of undifferentiated hESCs. In addition, two subpopulations from the TGF-β1 group were identified, which were not statistically different from native articular chondrocytes in their instantaneous and relaxed moduli, as well as their apparent viscosity. Identification of a differentiated hESC subpopulation with similar mechanical properties as native chondrocytes may provide an excellent cell source for tissue engineering applications. These cells will need to withstand any mechanical stimulation regimen employed to augment the mechanical and biochemical characteristics of the neotissue. Density separation was effective at purifying distinct populations of cells. A differentiated hESC subpopulation was identified with both similar mechanical and morphological characteristics as ACs. Future research may utilize this cell source in cartilage regeneration efforts.
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paper_title: TENSEGRITY: THE ARCHITECTURAL BASIS OF CELLULAR MECHANOTRANSDUCTION

paper_content:

Physical forces of gravity, hemodynamic stresses, and movement play a critical role in tissue development. Yet, little is known about how cells convert these mechanical signals into a chemical response. This review attempts to place the potential molecular mediators of mechanotransduction (e.g. stretch-sensitive ion channels, signaling molecules, cytoskeleton, integrins) within the context of the structural complexity of living cells. The model presented relies on recent experimental findings, which suggests that cells use tensegrity architecture for their organization. Tensegrity predicts that cells are hard-wired to respond immediately to mechanical stresses transmitted over cell surface receptors that physically couple the cytoskeleton to extracellular matrix (e.g. integrins) or to other cells (cadherins, selectins, CAMs). Many signal transducing molecules that are activated by cell binding to growth factors and extracellular matrix associate with cytoskeletal scaffolds within focal adhesion complexes. Mechanical signals, therefore, may be integrated with other environmental signals and transduced into a biochemical response through force-dependent changes in scaffold geometry or molecular mechanics. Tensegrity also provides a mechanism to focus mechanical energy on molecular transducers and to orchestrate and tune the cellular response.
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paper_title: Cell shape, cytoskeletal mechanics, and cell cycle control in angiogenesis

paper_content:

Capillary endothelial cells can be switched between growth and differentiation by altering cell-extracellular matrix interactions and thereby, modulating cell shape. Studies were carried out to determine when cell shape exerts its growth-regulatory influence during cell cycle progression and to explore the role of cytoskeletal structure and mechanics in this control mechanism. When G0-synchronized cells were cultured in basic fibroblast growth factor (FGF)-containing defined medium on dishes coated with increasing densities of fibronectin or a synthelic integrin ligand (RGD-containing peptide), cell spreading, nuclear extention, and DNA synthesis all increased in parallel. To determine the minimum time cells must be adherent and spread on extracellular matrix (ECM) to gain entry into S phase, cells were removed with trypsin or induced to retract using cytochalasin D at different times after plating. Both approaches revealed that cells must remain extended for approximately 12–15 h and hence, most of G1, in order to enter S phase. After this restriction point was passed, normally ‘anchorage-dependent’ endothelial cells turned on DNA synthesis even when round and in suspension. The importance of actin-containing microfilaments in shape-dependent growth control was confirmed by culturing cells in the presence of cytochalasin D (25–1000 ng ml−1): dose-dependent inhibition of cell spreading, nuclear extension, and DNA synthesis resulted. In contrast, induction of microtubule disassembly using nocodazole had little effect on cell or nuclear spreading and only partially inhibited DNA synthesis. Interestingly, combination of nocodazole with a suboptimal dose of cytochalasin D (100 ng ml−1) resulted in potent inhibition of both spreading and growth, suggesting that microtubules are redundant structural elements which can provide critical load-bearing functions when microfilaments are partially compromised. Similar synergism between nocodazole and cytochalasin D was observed when cytoskeletal stiffness was measured directly in living cells using magnetic twisting cytometry. These results emphasize the importance of matrix-dependent changes in cell and nuclear shape as well as higher order structural interactions between different cytoskeletal filament systems for control of capillary cell growth during angiogenesis.
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paper_title: The mechanical environment of the chondrocyte: a biphasic finite element model of cell-matrix interactions in articular cartilage.

paper_content:

Abstract Mechanical compression of the cartilage extracellular matrix has a significant effect on the metabolic activity of the chondrocytes. However, the relationship between the stress–strain and fluid-flow fields at the macroscopic “tissue” level and those at the microscopic “cellular” level are not fully understood. Based on the existing experimental data on the deformation behavior and biomechanical properties of articular cartilage and chondrocytes, a multi-scale biphasic finite element model was developed of the chondrocyte as a spheroidal inclusion embedded within the extracellular matrix of a cartilage explant. The mechanical environment at the cellular level was found to be time-varying and inhomogeneous, and the large difference (∼3 orders of magnitude) in the elastic properties of the chondrocyte and those of the extracellular matrix results in stress concentrations at the cell–matrix border and a nearly two-fold increase in strain and dilatation (volume change) at the cellular level, as compared to the macroscopic level. The presence of a narrow “pericellular matrix” with different properties than that of the chondrocyte or extracellular matrix significantly altered the principal stress and strain magnitudes within the chondrocyte, suggesting a functional biomechanical role for the pericellular matrix. These findings suggest that even under simple compressive loading conditions, chondrocytes are subjected to a complex local mechanical environment consisting of tension, compression, shear, and fluid pressure. Knowledge of the local stress and strain fields in the extracellular matrix is an important step in the interpretation of studies of mechanical signal transduction in cartilage explant culture models.
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paper_title: Determination of the Poisson's ratio of the cell: recovery properties of chondrocytes after release from complete micropipette aspiration.

paper_content:

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.
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paper_title: Finite element modelling of nanoindentation based methods for mechanical properties of cells.

paper_content:

The viscoelastic properties of the living cells are for quantifying the biomechanical effects of drug treatment, diseases and aging. Nanoindentation techniques have proven effective to characterize the viscoelastic properties of living cells. However, most studies utilized the Hertz contact model and assumed the Heaviside step loading, which does not represent real tests. Therefore, new mathematical models have been developed to determine the viscoelastic properties of the cells for nanoindentation tests. Finite element method was used to determine the empirical correction parameter in the mathematical model to account for large deformation, in which case the combined effect of finite lateral and vertical dimensions of the cell is essential. The viscoelastic integral operator was used to account for the realistic deformation rate. The predictive model captures the mechanical responses of the cells observed from previous experimental study. This work has demonstrated that the new model consistently predicts viscoelastic properties for both ramping and stress relaxation periods, which cannot be achieved by the commonly used models. Utilization of this new model can enrich the experimental cell mechanics in interpretation of nanoindentation of cells.
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paper_title: Plastic indentation in metals with cones

paper_content:

Abstract Q uasi-static indentations with diamond cones have been made in specimens of copper and mild steel work-hardened by varying amounts. The relationship between the hardness so found, the cone angle, and the yield stress is given. The results agree well with the indentation theory of L ockett (1968a) based on classical slip line analysis. This shows that for a work-hardened material the hardness increases with increasing cone angle for cone angles above 105°. Lockett's theory is degenerate below 105° and the experiments here also show a change in deformation mode in this region. Although this agreement with theory appears satisfactory, the deformation process appears to be substantially different from that implied in Lockett's analysis. For large angle cones the deformation resembles a ‘radial’ compression similar to that described by S amuels and M ulhearn (1957): for small cone angles the deformation resembles more clearly the classical slip-line form given by Lockett, though this is precisely the region where the theory becomes degenerate. The effective strain produced by each cone during the indentation process itself is determined for each cone and it is shown that it is possible to construct the stress-strain curve of a given material from a series of cone indentations.
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paper_title: How can cells sense the elasticity of a substrate? An analysis using a cell tensegrity model

paper_content:

A eukaryotic cell attaches and spreads on substrates, whether it is the extracellular matrix naturally produced by the cell itself, or artificial materials, such as tissue-engineered scaffolds. Attachment and spreading require the cell to apply forces in the nN range to the substrate via adhesion sites, and these forces are balanced by the elastic response of the substrate. This mechanical interaction is one determinant of cell morphology and, ultimately, cell phenotype. In this paper we use a finite element model of a cell, with a tensegrity structure to model the cytoskeleton of actin filaments and microtubules, to explore the way cells sense the stiffness of the substrate and thereby adapt to it. To support the computational results, an analytical 1D model is developed for comparison. We find that (i) the tensegrity hypothesis of the cytoskeleton is sufficient to explain the matrix-elasticity sensing, (ii) cell sensitivity is not constant but has a bell-shaped distribution over the physiological matrix-elasticity range, and (iii) the position of the sensitivity peak over the matrix-elasticity range depends on the cytoskeletal structure and in particular on the F-actin organisation. Our model suggests that F-actin reorganisation observed in mesenchymal stem cells (MSCs) in response to change of matrix elasticity is a structural-remodelling process that shifts the sensitivity peak towards the new value of matrix elasticity. This finding discloses a potential regulatory role of scaffold stiffness for cell differentiation.
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paper_title: A three-dimensional finite element model of an adherent eukaryotic cell.

paper_content:

Mechanical stimulation is known to cause alterations in the behaviour of cells adhering to a substrate. The mechanisms by which forces are transduced into biological responses within the cell remain largely unknown. Since cellular deformation is likely involved, further understanding of the biomechanical origins of alterations in cellular response can be aided by the use of computational models in describing cellular structural behaviour and in determining cellular deformation due to imposed loads of various magnitudes. In this paper, a finite element modelling approach that can describe the biomechanical behaviour of adherent eukaryotic cells is presented. It fuses two previous modelling approaches by incorporating, in an idealised geometry, all cellular components considered structurally significant, i.e. prestressed cytoskeleton, cytoplasm, nucleus and membrane components. The aim is to determine if we can use this model to describe the non-linear structural behaviour of an adherent cell and to determine the contribution of the various cellular components to cellular stability. Results obtained by applying forces (in the picoNewton range) to the model membrane nodes suggest a key role for the cytoskeleton in determining cellular stiffness. The model captures non-linear structural behaviours such as strain hardening and prestress effects (in the region of receptor sites), and variable compliance along the cell surface. The role of the cytoskeleton in stiffening a cell during the process of cell spreading is investigated by applying forces to five increasingly spread cell geometries. Parameter studies reveal that material properties of the cytoplasm (elasticity and compressibility) also have a large influence on cellular stiffness. The computational model of a single cell developed here is proposed as one that is sufficiently complex to capture the non-linear behaviours of the cell response to forces whilst not being so complex that the parameters cannot be specified. The model could be very useful in computing cellular structural behaviour in response to various in vitro mechanical stimuli (e.g. fluid flow, substrate strain), or for use in algorithms that attempt to simulate mechanobiological processes.
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paper_title: Cellular tensegrity: exploring how mechanical changes in the cytoskeleton regulate cell growth, migration, and tissue pattern during morphogenesis.

paper_content:

Publisher Summary This chapter focuses on the role of the intracellular cytoskeleton (CSK) in cell shape determination and tissue morphogenesis. The role of mechanical changes in the CSK during embryological development is reviewed. The chapter focuses on the mechanism by which mechanical forces are transmitted across the cell surface and through the CSK, as well as how they regulate cell shape. An analysis of the biomechanical basis of cell shape control addresses two central questions: (1) how do changes in mechanical forces alter CSK organization, and (2) how do changes in CSK structure regulate cell growth and function. The results from recent studies showing that the CSK can respond directly to mechanical stress are also reviewed. The particular type of mechanical response that living cells exhibit is consistent with a theory of CSK architecture that is based on tensional integrity and is known as “tensegrity”. Inherent to the tensegrity model is an efficient mechanism for integrating changes in structure and function at the tissue, cell, nuclear, and molecular levels. The chapter explores the possibility that CSK tensegrity may also provide a mechanical basis for cell locomotion as well as a structural mechanism for coupling mechanical and chemical signaling pathways inside the cell.
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paper_title: Cell mechanics, structure, and function are regulated by the stiffness of the three-dimensional microenvironment.

paper_content:

This study adopts a combined computational and experimental approach to determine the mechanical, structural, and metabolic properties of isolated chondrocytes cultured within three-dimensional hydrogels. A series of linear elastic and hyperelastic finite-element models demonstrated that chondrocytes cultured for 24 h in gels for which the relaxation modulus is <5 kPa exhibit a cellular Young's modulus of ∼5 kPa. This is notably greater than that reported for isolated chondrocytes in suspension. The increase in cell modulus occurs over a 24-h period and is associated with an increase in the organization of the cortical actin cytoskeleton, which is known to regulate cell mechanics. However, there was a reduction in chromatin condensation, suggesting that changes in the nucleus mechanics may not be involved. Comparison of cells in 1% and 3% agarose showed that cells in the stiffer gels rapidly develop a higher Young's modulus of ∼20 kPa, sixfold greater than that observed in the softer gels. This was associated with higher levels of actin organization and chromatin condensation, but only after 24 h in culture. Further studies revealed that cells in stiffer gels synthesize less extracellular matrix over a 28-day culture period. Hence, this study demonstrates that the properties of the three-dimensional microenvironment regulate the mechanical, structural, and metabolic properties of living cells.
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paper_title: AFM indentation study of breast cancer cells

paper_content:

Mechanical properties of individual living cells are known to be closely related to the health and function of the human body. Here, atomic force microscopy (AFM) indentation using a micro-sized spherical probe was carried out to characterize the elasticity of benign (MCF-10A) and cancerous (MCF-7) human breast epithelial cells. AFM imaging and confocal fluorescence imaging were also used to investigate their corresponding sub-membrane cytoskeletal structures. Malignant (MCF-7) breast cells were found to have an apparent Young's modulus significantly lower (1.4-1.8 times) than that of their non-malignant (MCF-10A) counterparts at physiological temperature (37 degrees C), and their apparent Young's modulus increase with loading rate. Both confocal and AFM images showed a significant difference in the organization of their sub-membrane actin structures which directly contribute to their difference in cell elasticity. This change may have facilitated easy migration and invasion of malignant cells during metastasis.
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paper_title: Biomechanical properties of single chondrocytes and chondrons determined by micromanipulation and finite-element modelling

paper_content:

A chondrocyte and its surrounding pericellular matrix (PCM) are defined as a chondron. Single chondrocytes and chondrons isolated from bovine articular cartilage were compressed by micromanipulation between two parallel surfaces in order to investigate their biomechanical properties and to discover the mechanical significance of the PCM. The force imposed on the cells was measured directly during compression to various deformations and then holding. When the nominal strain at the end of compression was 50 per cent, force relaxation showed that the cells were viscoelastic, but this viscoelasticity was generally insignificant when the nominal strain was 30 per cent or lower. The viscoelastic behaviour might be due to the mechanical response of the cell cytoskeleton and/or nucleus at higher deformations. A finite-element analysis was applied to simulate the experimental force-displacement/time data and to obtain mechanical property parameters of the chondrocytes and chondrons. Because of the large strains in the cells, a nonlinear elastic model was used for simulations of compression to 30 per cent nominal strain and a nonlinear viscoelastic model for 50 per cent. The elastic model yielded a Young’s modulus of 14+ 1 kPa (mean+ s.e.) for chondrocytes and 19+ 2 kPa for chondrons, respectively. The viscoelastic model generated an instantaneous elastic modulus of 21+ 3a nd 27+ 4 kPa, a long-term modulus of 9.3+ 0.8 and 12+ 1 kPa and an apparent viscosity of 2.8+ 0.5 and 3.4+ 0.6 kPa s for chondrocytes and chondrons, respectively. It was concluded that chondrons were generally stiffer and showed less viscoelastic behaviour than chondrocytes, and that the PCM significantly influenced the mechanical properties of the cells.
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paper_title: Large elastic deformations of isotropic materials VII. Experiments on the deformation of rubber

paper_content:

It is shown in this part how the theory of large elastic deformations of incompressible isotropic materials, developed in previous parts, can be used to interpret the load-deformation curves obtained for certain simple types of deformation of vulcanized rubber test-pieces in terms of a single stored-energy function. The types of experiment described are: (i) the pure homogeneous deformation of a thin sheet of rubber in which the deformation is varied in such a manner that one of the invariants of the strain, I 1 or I 2 , is maintained constant; (ii) pure shear of a thin sheet of rubber (i.e. pure homogeneous deformation in which one of the extension ratios in the plane of the sheet is maintained at unity, while the other is varied); (iii) simultaneous simple extension and pure shear of a thin sheet (i.e. pure homogeneous deformation in which one of the extension ratios in the plane of the sheet is maintained constant at a value less than unity, while the other is varied); (iv) simple extension of a strip of rubber; (v) simple compression (i.e. simple extension in which the extension ratio is less than unity); (vi) simple torsion of a right-circular cylinder; (vii) superposed axial extension and torsion of a right-circular cylindrical rod. It is shown that the load-deformation curves in all these cases can be interpreted on the basis of the theory in terms of a stored-energy function W which is such that δ W /δ I 1 is independent of I 1 and I 2 and the ratio (δ W /δ I 2 ) (δ W /δ I 1 ) is independent of I 1 and falls, as I 2 increases, from about 0*25 at I 2 = 3.
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paper_title: A microstructural approach to cytoskeletal mechanics based on tensegrity.

paper_content:

Mechanical properties of living cells are commonly described in terms of the laws of continuum mechanics. The purpose of this report is to consider the implications of an alternative approach that emphasizes the discrete nature of stress bearing elements in the cell and is based on the known structural properties of the cytoskeleton. We have noted previously that tensegrity architecture seems to capture essential qualitative features of cytoskeletal shape distortion in adherent cells (Ingber, 1993a; Wang et al., 1993). Here we extend those qualitative notions into a formal microstructural analysis. On the basis of that analysis we attempt to identify unifying principles that might underlie the shape stability of the cytoskeleton. For simplicity, we focus on a tensegrity structure containing six rigid struts interconnected by 24 linearly elastic cables. Cables carry initial tension ("prestress") counterbalanced by compression of struts. Two cases of interconnectedness between cables and struts are considered: one where they are connected by pin-joints, and the other where the cables run through frictionless loops at the junctions. At the molecular level, the pinned structure may represent the case in which different cytoskeletal filaments are cross-linked whereas the looped structure represents the case where they are free to slip past one another. The system is then subjected to uniaxial stretching. Using the principal of virtual work, stretching force vs. extension and structural stiffness vs. stretching force relationships are calculated for different prestresses. The stiffness is found to increase with increasing prestress and, at a given prestress, to increase approximately linearly with increasing stretching force. This behavior is consistent with observations in living endothelial cells exposed to shear stresses (Wang & Ingber, 1994). At a given prestress, the pinned structure is found to be stiffer than the looped one, a result consistent with data on mechanical behavior of isolated, cross-linked and uncross-linked actin networks (Wachsstock et al., 1993). On the basis of our analysis we concluded that architecture and the prestress of the cytoskeleton might be key features that underlie a cell's ability to regulate its shape.
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paper_title: In situ mechanical properties of the chondrocyte cytoplasm and nucleus

paper_content:

Abstract The way in which the nucleus experiences mechanical forces has important implications for understanding mechanotransduction. Knowledge of nuclear material properties and, specifically, their relationship to the properties of the bulk cell can help determine if the nucleus directly experiences mechanical loads, or if it is a signal transduction mechanism secondary to cell membrane deformation that leads to altered gene expression. Prior work measuring nuclear material properties using micropipette aspiration suggests that the nucleus is substantially stiffer than the bulk cell [Guilak, F., Tedrow, J.R., Burgkart, R., 2000. Viscoelastic properties of the cell nucleus. Biochem. Biophys. Res. Commun. 269, 781–786], whereas recent work with unconfined compression of single chondrocytes showed a nearly one-to-one correlation between cellular and nuclear strains [Leipzig, N.D., Athanasiou, K.A., 2008. Static compression of single chondrocytes catabolically modifies single-cell gene expression. Biophys. J. 94, 2412–2422]. In this study, a linearly elastic finite element model of the cell with a nuclear inclusion was used to simulate the unconfined compression data. Cytoplasmic and nuclear stiffnesses were varied from 1 to 7 kPa for several combinations of cytoplasmic and nuclear Poisson's ratios. It was found that the experimental data were best fit when the ratio of cytoplasmic to nuclear stiffness was 1.4, and both cytoplasm and nucleus were modeled as incompressible. The cytoplasmic to nuclear stiffness ratio is significantly lower than prior reports for isolated nuclei. These results suggest that the nucleus may behave mechanically different in situ than when isolated.
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paper_title: Determining the Instantaneous Modulus of Viscoelastic Solids Using Instrumented Indentation Measurements

paper_content:

Instrumented indentation is often used in the study of small-scale mechanical behavior of "soft" matters that exhibit viscoelastic behavior. A number of techniques have recently been proposed to obtain the viscoelastic properties from indentation load-dis
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paper_title: Cell mechanics, structure, and function are regulated by the stiffness of the three-dimensional microenvironment.

paper_content:

This study adopts a combined computational and experimental approach to determine the mechanical, structural, and metabolic properties of isolated chondrocytes cultured within three-dimensional hydrogels. A series of linear elastic and hyperelastic finite-element models demonstrated that chondrocytes cultured for 24 h in gels for which the relaxation modulus is <5 kPa exhibit a cellular Young's modulus of ∼5 kPa. This is notably greater than that reported for isolated chondrocytes in suspension. The increase in cell modulus occurs over a 24-h period and is associated with an increase in the organization of the cortical actin cytoskeleton, which is known to regulate cell mechanics. However, there was a reduction in chromatin condensation, suggesting that changes in the nucleus mechanics may not be involved. Comparison of cells in 1% and 3% agarose showed that cells in the stiffer gels rapidly develop a higher Young's modulus of ∼20 kPa, sixfold greater than that observed in the softer gels. This was associated with higher levels of actin organization and chromatin condensation, but only after 24 h in culture. Further studies revealed that cells in stiffer gels synthesize less extracellular matrix over a 28-day culture period. Hence, this study demonstrates that the properties of the three-dimensional microenvironment regulate the mechanical, structural, and metabolic properties of living cells.
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paper_title: Separating poroviscoelastic deformation mechanisms in hydrogels

paper_content:

Hydrogels have applications in drug delivery, mechanical actuation, and regenerative medicine. When hydrogels are deformed, load-relaxation arising from fluid flow—poroelasticity—and from rearrangement of the polymer network—viscoelasticity—is observed. The physical mechanisms are different in that poroelastic relaxation varies with experimental length-scale while viscoelastic does not. Here, we show that poroviscoelastic load-relaxation is the product of the two individual responses. The difference in length-scale dependence of the two mechanisms can be exploited to uniquely determine poroviscoelastic properties from simultaneous analysis of multi-scale indentation experiments, providing insight into hydrogel physical behavior.
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paper_title: Nanoindentation of biological and biomimetic materials

paper_content:

Nanoindentation techniques have recently been adapted for the study of biological materials. This feature will consider the experimental adaptations required for such studies. Following a brief review of the structure and constitutive behavior of biological materials, we examine the experimental aspects in detail, including working with hydrated samples, time-dependent mechanical behavior and extremely compliant materials. The analysis of experimental data, consistent with the constitutive response of the material, will then be treated. Examples of nanoindentation data collected using commercially-available instruments are shown, including nanoindentation creep curves of biological materials and relaxation responses of biomimetic hydrogels. Finally, we conclude by examining the current state and future needs of the biological nanoindentation community.
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paper_title: Unconfined creep compression of chondrocytes

paper_content:

Abstract The study of single cell mechanics offers a valuable tool for understanding cellular milieus. Specific knowledge of chondrocyte biomechanics could lead to elucidation of disease etiologies and the biomechanical factors most critical to stimulating regenerative processes in articular cartilage. Recent studies in our laboratory have suggested that it may be acceptable to approximate the shape of a single chondrocyte as a disc. This geometry is easily utilized for generating models of unconfined compression. In this study, three continuum mechanics models of increasing complexity were formulated and used to fit unconfined compression creep data. Creep curves were obtained from middle/deep zone chondrocytes ( n =15) and separately fit using the three continuum models. The linear elastic solid model yielded a Young's modulus of 2.55±0.85 kPa. The viscoelastic model (adapted from the Kelvin model) generated an instantaneous modulus of 2.47±0.85 kPa, a relaxed modulus of 1.48±0.35 kPa, and an apparent viscosity of 1.92±1.80 kPa-s. Finally, a linear biphasic model produced an aggregate modulus of 2.58±0.87 kPa, a permeability of 2.57×10 −12 ±3.09 m 4 /N-s, and a Poisson's ratio of 0.069±0.021. The results of this study demonstrate that similar values for the cell modulus can be obtained from three models of increasing complexity. The elastic model provides an easy method for determining the cell modulus, however, the viscoelastic and biphasic models generate additional material properties that are important for characterizing the transient response of compressed chondrocytes.
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paper_title: Using indentation to characterize the poroelasticity of gels

paper_content:

When an indenter is pressed into a gel to a fixed depth, the solvent in the gel migrates, and the force on the indenter relaxes. Within the theory of poroelasticity, the force relaxation curves for indenters of several types are obtained in a simple form, enabling indentation to be used with ease as a method for determining the elastic constants and permeability of the gel. The method is demonstrated with a conical indenter on an alginate hydrogel.
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paper_title: General Theory of Three‐Dimensional Consolidation

paper_content:

The settlement of soils under load is caused by a phenomenon called consolidation, whose mechanism is known to be in many cases identical with the process of squeezing water out of an elasticporous medium. The mathematical physical consequences of this viewpoint are established in the present paper. The number of physical constants necessary to determine the properties of the soil is derived along with the general equations for the prediction of settlements and stresses in three‐dimensional problems. Simple applications are treated as examples. The operational calculus is shown to be a powerful method of solution of consolidation problems.
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paper_title: Incompressible porous media models by use of the theory of mixtures

paper_content:

Abstract This work illustrates the use of the thermodynamics of mixtures to formulate incompressible porous media models. An incompressible porous material is a mixture where the solid and the fluid constituents are each incompressible. Among the results are formulas which show how the chemical potentials for the constituents determine the stress tensor for the mixture and the pore pressure for each pore fluid. The general results are specialized to the forms necessary to produce models used in the applications. For example, the classical models used to study the flow of immiscible fluids in deformable and rigid solids are given.
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paper_title: A Mixture Theory for Charged-Hydrated Soft Tissues Containing Multi-electrolytes: Passive Transport and Swelling Behaviors

paper_content:

A new mixture theory was developed to model the mechano-electrochemical behaviors of charged-hydrated soft tissues containing multi-electrolytes. The mixture is composed of n + 2 constituents (1 charged solid phase, 1 noncharged solvent phase, and n ion species ). Results from this theory show that three types offorce are involved in the transport of ions and solvent through such materials : (1) a mechanochemical force (including hydraulic and osmotic pressures); (2) an electrochemical force; and (3) an electrical force. Our results also show that three types of material coefficients are required to characterize the transport rates of these ions and solvent: (1) a hydraulic permeability ; (2) mechano-electrochemical coupling coefficients; and (3) an ionic conductance matrix. Specifically, we derived the fundamental governing relationships between these forces and material coefficients to describe such mechano-electrochemical transduction effects as streaming potential, streaming current, diffusion (membrane) potential, electro-osmosis, and anomalous (negative) osmosis. As an example, we showed that the well-known formula for the resting cell membrane potential (Hodgkin and Huxley, 1952a, b) could be derived using our new n + 2 mixture model (a generalized triphasic theory). In general, the n + 2 mixture theory is consistent with and subsumes all previous theories pertaining to specific aspects of charged-hydrated tissues. In addition, our results provided the stress, strain, and fluid velocity fields within a tissue of finite thickness during a one-dimensional steady diffusion process. Numerical results were provided for the exchange of Na + and Ca ++ through the tissue. These numerical results support our hypothesis that tissue fixed charge density (c F ) plays a significant role in modulating kinetics of ions and solvent transport through charged-hydrated soft tissues.
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paper_title: The biphasic poroviscoelastic behavior of articular cartilage: role of the surface zone in governing the compressive behavior.

paper_content:

Surface fibrillation of articular cartilage is an early sign of degenerative changes in the development of osteoarthritis. To assess the influence of the surface zone on the viscoelastic properties of cartilage under compressive loading, we prepared osteochondral plugs from skeletally mature steers, with and without the surface zone of articular cartilage, for study in the confined compression creep experiment. The relative contributions of two viscoelastic mechanisms, i.e. a flow-independent mechanism [Hayes and Bodine, J. Biomechanics 11, 407-419 (1978)], and a flow-dependent mechanism [Mow et al. J. biomech. Engng 102, 73-84 (1980)], to the compressive creep response of these two types of specimens were determined using the biphasic poroviscoelastic theory proposed by Mak. [J. Biomechanics 20, 703-714 (1986)]. From the experimental results and the biphasic poroviscoelastic theory, we found that frictional drag associated with interstitial fluid flow and fluid pressurization are the dominant mechanisms of load support in the intact specimens, i.e. the flow-dependent mechanisms alone were sufficient to describe normal articular cartilage compressive creep behavior. For specimens with the surface removed, we found an increased creep rate which was derived from an increased tissue permeability, as well as significant changes in the flow-independent parameters of the viscoelastic solid matrix. permeability, as well as significant changes in the flow-independent parameters of the viscoelastic solid matrix. From these tissue properties and the biphasic poroviscoelastic theory, we determined that the flow-dependent mechanisms of load support, i.e. frictional drag and fluid pressurization, were greatly diminished in cartilage without the articular surface. Calculations based upon these material parameters show that for specimens with the surface zone removed, the cartilage solid matrix became more highly loaded during the early stages of creep. This suggests that an important function of the articular surface is to provide for a low fluid permeability, and thereby serve to restrict fluid exudation and increase interstitial fluid pressurization. Thus, it is likely that with increasing severity of damage to the articular surface, load support in cartilage under compression shifts from the flow-dependent modes of fluid drag and pressurization to increased solid matrix stress. This suggests that it is important to maintain the integrity of the articular surface in preserving normal compressive behavior of the tissue and normal load carriage in the joint.
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paper_title: Scaling the Microrheology of Living Cells

paper_content:

We report a scaling law that governs both the elastic and frictional properties of a wide variety of living cell types, over a wide range of time scales and under a variety of biological interventions. This scaling identifies these cells as soft glassy materials existing close to a glass transition, and implies that cytoskeletal proteins may regulate cell mechanical properties mainly by modulating the effective noise temperature of the matrix. The practical implications are that the effective noise temperature is an easily quantified measure of the ability of the cytoskeleton to deform, flow, and reorganize.
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paper_title: The relation between load and penetration in the axisymmetric boussinesq problem for a punch of arbitrary profile

paper_content:

Abstract A solution of the axisymmetric Boussinesq problem is derived from which are deduced simple formulae for the depth of penetration of the tip of a punch of arbitrary profile and for the total load which must be applied to the punch to achieve this penetration. Simple expressions are also derived for the distribution of pressure under the punch and for the shape of the deformed surface. The results are illustrated by the evaluation of the expressions for several simple punch shapes.
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paper_title: A method for interpreting the data from depth-sensing indentation instruments

paper_content:

Depth-sensing indentation instruments provide a means for studying the elastic and plastic properties of thin films. A method for obtaining hardness and Young's modulus from the data obtained from these types of instruments is described. Elastic displacements are determined from the data obtained during unloading of the indentation. Young's modulus can be calculated from these measurements. In addition, the elastic contribution to the total displacement can be removed in order to calculate hardness. Determination of the exact shape of the indenter at the tip is critical to the measurement of both hardness and elastic modulus for indentation depths less than a micron. Hardness is shown to depend on strain rate, especially when the hardness values are calculated from the data along the loading curves.
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paper_title: Non-equilibration of hydrostatic pressure in blebbing cells

paper_content:

Current models for protrusive motility in animal cells focus on cytoskeleton-based mechanisms, where localized protrusion is driven by local regulation of actin biochemistry. In plants and fungi, protrusion is driven primarily by hydrostatic pressure. For hydrostatic pressure to drive localized protrusion in animal cells, it would have to be locally regulated, but current models treating cytoplasm as an incompressible viscoelastic continuum or viscous liquid require that hydrostatic pressure equilibrates essentially instantaneously over the whole cell. Here, we use cell blebs as reporters of local pressure in the cytoplasm. When we locally perfuse blebbing cells with cortex-relaxing drugs to dissipate pressure on one side, blebbing continues on the untreated side, implying non-equilibration of pressure on scales of approximately 10 microm and 10 s. We can account for localization of pressure by considering the cytoplasm as a contractile, elastic network infiltrated by cytosol. Motion of the fluid relative to the network generates spatially heterogeneous transients in the pressure field, and can be described in the framework of poroelasticity.
---
paper_title: Using indentation to characterize the poroelasticity of gels

paper_content:

When an indenter is pressed into a gel to a fixed depth, the solvent in the gel migrates, and the force on the indenter relaxes. Within the theory of poroelasticity, the force relaxation curves for indenters of several types are obtained in a simple form, enabling indentation to be used with ease as a method for determining the elastic constants and permeability of the gel. The method is demonstrated with a conical indenter on an alginate hydrogel.
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paper_title: Estimating the sensitivity of mechanosensitive ion channels to membrane strain and tension

paper_content:

Bone adapts to its environment by a process in which osteoblasts and osteocytes sense applied mechanical strain. One possible pathway for the detection of strain involves mechanosensitive channels and we sought to determine their sensitivity to membrane strain and tension. We used a combination of experimental and computational modeling techniques to gain new insights into cell mechanics and the regulation of mechanosensitive channels. Using patch-clamp electrophysiology combined with video microscopy, we recorded simultaneously the evolution of membrane extensions into the micropipette, applied pressure, and membrane currents. Nonselective mechanosensitive cation channels with a conductance of 15 pS were observed. Bleb aspiration into the micropipette was simulated using finite element models incorporating the cytoplasm, the actin cortex, the plasma membrane, cellular stiffening in response to strain, and adhesion between the membrane and the micropipette. Using this model, we examine the relative importance of the different cellular components in resisting suction into the pipette and estimate the membrane strains and tensions needed to open mechanosensitive channels. Radial membrane strains of 800% and tensions of 5 10−4 N.m−1 were needed to open 50% of mechanosensitive channels. We discuss the relevance of these results in the understanding of cellular reactions to mechanical strain and bone physiology.
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paper_title: Flat‐punch indentation of viscoelastic material

paper_content:

The indentation of standard viscoelastic solids, that is, the three-element viscoelastic material, by an axisymmetric, flat-ended indenter has been investigated theoretically. Under the boundary conditions of flat-punch indentation of a viscoelastic half-space, the solutions of the equations of viscoelastic deformation are derived for the standard viscoelastic material. Their generality resides in their inclusion of compressible as well as incompressible solids. They cover the two transient situations: flat-punch creep test and load-relaxation test. In experimental tests of their applicability, nanoindentation and microindentation probes under creep and relaxation conditions yielded a modulus from 0.1 to 1.1 GPa and viscosity from 1 to 37 Gpa · s for a crosslinked glassy polyurethane coatings. For bulk polystyrene, the values vary from 1 to 2 GPa and from 20 to 40 Gpa · s, respectively. The analysis here provides a fundamental basis for probing elastic and viscous properties of coatings with nanoindentation or microindentation tests. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 10–22, 2000
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paper_title: Quantitative analysis of the viscoelastic properties of thin regions of fibroblasts using atomic force microscopy.

paper_content:

Viscoelasticity of the leading edge, i.e., the lamellipodium, of a cell is the key property for a deeper understanding of the active extension of a cell's leading edge. The fact that the lamellipodium of a cell is very thin (<1000 nm) imparts special challenges for accurate measurements of its viscoelastic behavior. It requires addressing strong substrate effects and comparatively high stresses (>1 kPa) on thin samples. We present the method for an atomic force microscopy-based microrheology that allows us to fully quantify the viscoelastic constants (elastic storage modulus, viscous loss modulus, and the Poisson ratio) of thin areas of a cell (<1000 nm) as well as those of thick areas. We account for substrate effects by applying two different models-a model for well-adhered regions (Chen model) and a model for nonadhered regions (Tu model). This method also provides detailed information about the adhered regions of a cell. The very thin regions relatively near the edge of NIH 3T3 fibroblasts can be identified by the Chen model as strongly adherent with an elastic strength of approximately 1.6 +/- 0.2 kPa and with an experimentally determined Poisson ratio of approximately 0.4 to 0.5. Further from the edge of these cells, the adherence decreases, and the Tu model is effective in evaluating its elastic strength ( approximately 0.6 +/- 0.1 kPa). Thus, our AFM-based microrheology allows us to correlate two key parameters of cell motility by relating elastic strength and the Poisson ratio to the adhesive state of a cell. This frequency-dependent measurement allows for the decomposition of the elastic modulus into loss and storage modulus. Applying this decomposition and Tu's and Chen's finite depth models allow us to obtain viscoelastic signatures in a frequency range from 50 to 300 Hz, showing a rubber plateau-like behavior.
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paper_title: Non-equilibration of hydrostatic pressure in blebbing cells

paper_content:

Current models for protrusive motility in animal cells focus on cytoskeleton-based mechanisms, where localized protrusion is driven by local regulation of actin biochemistry. In plants and fungi, protrusion is driven primarily by hydrostatic pressure. For hydrostatic pressure to drive localized protrusion in animal cells, it would have to be locally regulated, but current models treating cytoplasm as an incompressible viscoelastic continuum or viscous liquid require that hydrostatic pressure equilibrates essentially instantaneously over the whole cell. Here, we use cell blebs as reporters of local pressure in the cytoplasm. When we locally perfuse blebbing cells with cortex-relaxing drugs to dissipate pressure on one side, blebbing continues on the untreated side, implying non-equilibration of pressure on scales of approximately 10 microm and 10 s. We can account for localization of pressure by considering the cytoplasm as a contractile, elastic network infiltrated by cytosol. Motion of the fluid relative to the network generates spatially heterogeneous transients in the pressure field, and can be described in the framework of poroelasticity.
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paper_title: Slow Stress Propagation in Adherent Cells

paper_content:

Mechanical cues influence a wide range of cellular behaviors including motility, differentiation, and tumorigenesis. Although previous studies elucidated the role of specific players such as ion channels and focal adhesions as local mechanosensors, the investigation of how mechanical perturbations propagate across the cell is necessary to understand the spatial coordination of cellular processes. Here we quantify the magnitude and timing of intracellular stress propagation, using atomic force microscopy and particle tracking by defocused fluorescence microscopy. The apical cell surface is locally perturbed by atomic force microscopy cantilever indentation, and distal displacements are measured in three dimensions by tracking integrin-bound fluorescent particles. We observe an immediate response and slower equilibration, occurring over times that increase with distance from perturbation. This distance-dependent equilibration occurs over several seconds and can be eliminated by disruption of the actin cytoskeleton. Our experimental results are not explained by traditional viscoelastic models of cell mechanics, but they are consistent with predictions from poroelastic models that include both cytoskeletal deformation and flow of the cytoplasm. Our combined atomic force microscopy-particle tracking measurements provide direct evidence of slow, distance-dependent dissipative stress propagation in response to external mechanical cues and offer new insights into mechanical models and physiological behaviors of adherent cells.
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paper_title: The cytoplasm of living cells behaves as a poroelastic material

paper_content:

The cytoplasm is the largest part of the cell by volume and hence its rheology sets the rate at which cellular shape changes can occur. Recent experimental evidence suggests that cytoplasmic rheology can be described by a poroelastic model, in which the cytoplasm is treated as a biphasic material consisting of a porous elastic solid meshwork (cytoskeleton, organelles, macromolecules) bathed in an interstitial fluid (cytosol). In this picture, the rate of cellular deformation is limited by the rate at which intracellular water can redistribute within the cytoplasm. However, direct supporting evidence for the model is lacking. Here we directly validate the poroelastic model to explain cellular rheology at short timescales using microindentation tests in conjunction with mechanical, chemical and genetic treatments. Our results show that water redistribution through the solid phase of the cytoplasm (cytoskeleton and macromolecular crowders) plays a fundamental role in setting cellular rheology at short timescales.
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paper_title: Poro-Viscoelastic Behavior of Gelatin Hydrogels Under Compression-Implications for Bioelasticity Imaging

paper_content:

Ultrasonic elasticity imaging enables visualization of soft tissue deformation for medical diagnosis. Our aim is to understand the role of flow-dependent and flow-independent viscoelastic mechanisms in the response of biphasic polymeric media, including biological tissues and hydrogels, to low-frequency forces. Combining the results of confined and unconfined compression experiments on gelatin hydrogels with finite element analysis (FEA) simulations of the experiments, we explore the role of polymer structure, loading, and boundary conditions in generating contrast for viscoelastic features. Feature estimation is based on comparisons between the biphasic poro-elastic and biphasic poro-viscoelastic (BPVE) material models, where the latter adds the viscoelastic response of the solid polymer matrix. The approach is to develop a consistent FEA material model (BPVE) from confined compression-stress relaxation measurements to extract the strain dependent hydraulic permeability variation and cone-plate rheometer measurements to obtain the flow-independent viscoelastic constants for the solid-matrix phase. The model is then applied to simulate the unconfined compression experiment to explore the mechanics of hydropolymers under conditions of quasi-static elasticity imaging. The spatiotemporal distributions of fluid and solid-matrix behavior within the hydrogel are studied to propose explanations for strain patterns that arise during the elasticity imaging of heterogeneous media.
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paper_title: A Mixture Theory for Charged-Hydrated Soft Tissues Containing Multi-electrolytes: Passive Transport and Swelling Behaviors

paper_content:

A new mixture theory was developed to model the mechano-electrochemical behaviors of charged-hydrated soft tissues containing multi-electrolytes. The mixture is composed of n + 2 constituents (1 charged solid phase, 1 noncharged solvent phase, and n ion species ). Results from this theory show that three types offorce are involved in the transport of ions and solvent through such materials : (1) a mechanochemical force (including hydraulic and osmotic pressures); (2) an electrochemical force; and (3) an electrical force. Our results also show that three types of material coefficients are required to characterize the transport rates of these ions and solvent: (1) a hydraulic permeability ; (2) mechano-electrochemical coupling coefficients; and (3) an ionic conductance matrix. Specifically, we derived the fundamental governing relationships between these forces and material coefficients to describe such mechano-electrochemical transduction effects as streaming potential, streaming current, diffusion (membrane) potential, electro-osmosis, and anomalous (negative) osmosis. As an example, we showed that the well-known formula for the resting cell membrane potential (Hodgkin and Huxley, 1952a, b) could be derived using our new n + 2 mixture model (a generalized triphasic theory). In general, the n + 2 mixture theory is consistent with and subsumes all previous theories pertaining to specific aspects of charged-hydrated tissues. In addition, our results provided the stress, strain, and fluid velocity fields within a tissue of finite thickness during a one-dimensional steady diffusion process. Numerical results were provided for the exchange of Na + and Ca ++ through the tissue. These numerical results support our hypothesis that tissue fixed charge density (c F ) plays a significant role in modulating kinetics of ions and solvent transport through charged-hydrated soft tissues.
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paper_title: Poro-viscoelastic properties of anisotropic cylindrical composite materials

paper_content:

A new poro-viscoelastic mechanical model, which is able to predict strain creep, stress relaxation and instant elasticity for anisotropic cylindrical composite materials based on the characteristics of dentinal biomaterials of cylindrical microstructures, is described. The model enables evaluation of the poro-viscoelastic properties of dentinal biomaterials with no cracks and with fatigue-cracks. The predicted poro-viscoelastic properties obtained from the present anisotropic constitutive model agree well with the available experimental data for both wet and dry situations. The present model can be applied to anisotropic cylindrical composite materials other than biomaterials.
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paper_title: Viscoelastic properties of transformed cells: Role in tumor cell progression and metastasis formation

paper_content:

The micropipette aspiration technique was used to investigate the deformation properties of a panel of nontransformed and transformed rat fibroblasts derived from the same normal cell line. In this method, a step negative pressure is applied to the cell via a micropipette and the aspiration distance into the pipette as a function of time is determined using video techniques. A standard solid viscoelastic model was then used to analyze the viscoelastic properties of the cell. From these results, it is concluded that a direct correlation exists between an increase in deformability and progression of the transformed phenotype from a nontumorigenic cell line into a tumorigenic, metastatic cell line.
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paper_title: Large Deformation Finite Element Analysis of Micropipette Aspiration to Determine the Mechanical Properties of the Chondrocyte

paper_content:

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.
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paper_title: Poro-viscoelastic constitutive modeling of unconfined creep of hydrogels using finite element analysis with integrated optimization method

paper_content:

Abstract Hydrogels are cross-linked polymer networks swollen with water and are being considered as potential replacements for deceased load bearing tissues such as cartilage. Hydrogels show nonlinear time dependent behavior, and are a challenge to model. A three-element poro-viscoelastic constitutive model was developed based on the structure and nature of the hydrogel. To identify the material parameters, an inverse finite element (FE) technique was used that combines experimental results with FE modeling and an optimization method. Unconfined compression creep tests were conducted on poly(vinyl alcohol) (PVA) and poly(ethylene-co-vinyl alcohol)–poly(vinyl pyrrolidone) (EVAL–PVP) hydrogels manufactured by injection molding. Results from the creep experiments showed that for PVA hydrogels, an increase in polymer concentration correlates with a decrease in the equilibrium water content (EWC) and the creep strain. In EVAL–PVP hydrogels, an increase in the hydrophobic segments (EVAL) correlates with a decrease in the EWC as well as the creep strain. An inverse FE method was used to identify the viscoelastic material parameters of the hydrogels in combination with creep testing using the poro-viscoelastic three-element constitutive model. The elastic modulus estimated from the inverse FE technique showed good agreement with the modulus estimated directly from the test data.
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paper_title: The biphasic poroviscoelastic behavior of articular cartilage: role of the surface zone in governing the compressive behavior.

paper_content:

Surface fibrillation of articular cartilage is an early sign of degenerative changes in the development of osteoarthritis. To assess the influence of the surface zone on the viscoelastic properties of cartilage under compressive loading, we prepared osteochondral plugs from skeletally mature steers, with and without the surface zone of articular cartilage, for study in the confined compression creep experiment. The relative contributions of two viscoelastic mechanisms, i.e. a flow-independent mechanism [Hayes and Bodine, J. Biomechanics 11, 407-419 (1978)], and a flow-dependent mechanism [Mow et al. J. biomech. Engng 102, 73-84 (1980)], to the compressive creep response of these two types of specimens were determined using the biphasic poroviscoelastic theory proposed by Mak. [J. Biomechanics 20, 703-714 (1986)]. From the experimental results and the biphasic poroviscoelastic theory, we found that frictional drag associated with interstitial fluid flow and fluid pressurization are the dominant mechanisms of load support in the intact specimens, i.e. the flow-dependent mechanisms alone were sufficient to describe normal articular cartilage compressive creep behavior. For specimens with the surface removed, we found an increased creep rate which was derived from an increased tissue permeability, as well as significant changes in the flow-independent parameters of the viscoelastic solid matrix. permeability, as well as significant changes in the flow-independent parameters of the viscoelastic solid matrix. From these tissue properties and the biphasic poroviscoelastic theory, we determined that the flow-dependent mechanisms of load support, i.e. frictional drag and fluid pressurization, were greatly diminished in cartilage without the articular surface. Calculations based upon these material parameters show that for specimens with the surface zone removed, the cartilage solid matrix became more highly loaded during the early stages of creep. This suggests that an important function of the articular surface is to provide for a low fluid permeability, and thereby serve to restrict fluid exudation and increase interstitial fluid pressurization. Thus, it is likely that with increasing severity of damage to the articular surface, load support in cartilage under compression shifts from the flow-dependent modes of fluid drag and pressurization to increased solid matrix stress. This suggests that it is important to maintain the integrity of the articular surface in preserving normal compressive behavior of the tissue and normal load carriage in the joint.
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paper_title: Determination of the Poisson's ratio of the cell: recovery properties of chondrocytes after release from complete micropipette aspiration.

paper_content:

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.
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paper_title: Microrheology of Human Lung Epithelial Cells Measured by Atomic Force Microscopy

paper_content:

Lung epithelial cells are subjected to large cyclic forces from breathing. However, their response to dynamic stresses is poorly defined. We measured the complex shear modulus (G*(ω)) of human alveolar (A549) and bronchial (BEAS-2B) epithelial cells over three frequency decades (0.1–100 Hz) and at different loading forces (0.1–0.9 nN) with atomic force microscopy. G*(ω) was computed by correcting force-indentation oscillatory data for the tip-cell contact geometry and for the hydrodynamic viscous drag. Both cell types displayed similar viscoelastic properties. The storage modulus G′(ω) increased with frequency following a power law with exponent ∼0.2. The loss modulus G″(ω) was ∼2/3 lower and increased similarly to G′(ω) up to ∼10 Hz, but exhibited a steeper rise at higher frequencies. The cells showed a weak force dependence of G′(ω) and G″(ω). G*(ω) conformed to the power-law model with a structural damping coefficient of ∼0.3, indicating a coupling of elastic and dissipative processes within the cell. Power-law behavior implies a continuum distribution of stress relaxation time constants. This complex dynamics is consistent with the rheology of soft glassy materials close to a glass transition, thereby suggesting that structural disorder and metastability may be fundamental features of cell architecture.
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paper_title: Creep indentation of single cells.

paper_content:

An apparatus for creep indentation of individual adherent cells was designed, developed, and experimentally validated. The creep cytoindentation apparatus (CCA) can perform stress-controlled experiments and measure the corresponding deformation of single anchorage-dependent cells. The apparatus can resolve forces on the order of 1 nN and cellular deformations on the order of 0.1 micron. Experiments were conducted on bovine articular chondrocytes using loads on the order of 10 nN. The experimentally observed viscoelastic behavior of these cells was modeled using the punch problem and standard linear solid. The punch problem yielded a Young's modulus of 1.11 +/- 0.48 kPa. The standard linear solid model yielded an instantaneous elastic modulus of 8.00 +/- 4.41 kPa, a relaxed modulus of 1.09 +/- 0.54 kPa, an apparent viscosity of 1.50 +/- 0.92 kPa-s, and a time constant of 1.32 +/- 0.65 s. To our knowledge, this is the first time that stress-controlled indentation testing has been applied at the single cell level. This methodology represents a new tool in understanding the mechanical nature of anchorage-dependent cells and mechanotransductional pathways.
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paper_title: Dimensional and mechanical dynamics of active and stable edges in motile fibroblasts investigated by using atomic force microscopy

paper_content:

The atomic force microscope (AFM) was employed to investigate the extension and retraction dynamics of protruding and stable edges of motile 3T3 fibroblasts in culture. Such dynamics closely paralleled the results of earlier studies employing video microscopy that indicated that the AFM force-mapping technique does not appreciably perturb these dynamics. Force scans permitted height determinations of active and stable edges. Whereas the profiles of active edges are flat with average heights of 0.4–0.8 μm, stable edges smoothly ascend to 2–3 μm within about 6 μm of the edge. In the region of the leading edge, the height fluctuates up to 50% (SD) of the mean value, much more than the stable edge; this fluctuation presumably reflects differences in underlying cytoskeletal activity. In addition, force mapping yields an estimate of the local Young’s modulus or modulus of elasticity (E, the cortical stiffness). This stiffness will be related to “cortical tension,” can be accurately calculated for the stable edges, and is ≈12 kPa in this case. The thinness of the leading edge precludes accurate estimation of the E values, but within 4 μm of the margin it is considerably smaller than that for stable edges, which have an upper limit of 3–5 kPa. Although blebbing cannot absolutely be ruled out as a mechanism of extension, the data are consistent with an actin polymerization and/or myosin motor mechanism in which the average material properties of the extending margin would be nearly constant to the edge. Because the leading edge is softer than the stable edge, these data also are consistent with the notion that extension preferentially occurs in regions of lower cortical tension.
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paper_title: Unconfined creep compression of chondrocytes

paper_content:

Abstract The study of single cell mechanics offers a valuable tool for understanding cellular milieus. Specific knowledge of chondrocyte biomechanics could lead to elucidation of disease etiologies and the biomechanical factors most critical to stimulating regenerative processes in articular cartilage. Recent studies in our laboratory have suggested that it may be acceptable to approximate the shape of a single chondrocyte as a disc. This geometry is easily utilized for generating models of unconfined compression. In this study, three continuum mechanics models of increasing complexity were formulated and used to fit unconfined compression creep data. Creep curves were obtained from middle/deep zone chondrocytes ( n =15) and separately fit using the three continuum models. The linear elastic solid model yielded a Young's modulus of 2.55±0.85 kPa. The viscoelastic model (adapted from the Kelvin model) generated an instantaneous modulus of 2.47±0.85 kPa, a relaxed modulus of 1.48±0.35 kPa, and an apparent viscosity of 1.92±1.80 kPa-s. Finally, a linear biphasic model produced an aggregate modulus of 2.58±0.87 kPa, a permeability of 2.57×10 −12 ±3.09 m 4 /N-s, and a Poisson's ratio of 0.069±0.021. The results of this study demonstrate that similar values for the cell modulus can be obtained from three models of increasing complexity. The elastic model provides an easy method for determining the cell modulus, however, the viscoelastic and biphasic models generate additional material properties that are important for characterizing the transient response of compressed chondrocytes.
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paper_title: Quantitative analysis of the viscoelastic properties of thin regions of fibroblasts using atomic force microscopy.

paper_content:

Viscoelasticity of the leading edge, i.e., the lamellipodium, of a cell is the key property for a deeper understanding of the active extension of a cell's leading edge. The fact that the lamellipodium of a cell is very thin (<1000 nm) imparts special challenges for accurate measurements of its viscoelastic behavior. It requires addressing strong substrate effects and comparatively high stresses (>1 kPa) on thin samples. We present the method for an atomic force microscopy-based microrheology that allows us to fully quantify the viscoelastic constants (elastic storage modulus, viscous loss modulus, and the Poisson ratio) of thin areas of a cell (<1000 nm) as well as those of thick areas. We account for substrate effects by applying two different models-a model for well-adhered regions (Chen model) and a model for nonadhered regions (Tu model). This method also provides detailed information about the adhered regions of a cell. The very thin regions relatively near the edge of NIH 3T3 fibroblasts can be identified by the Chen model as strongly adherent with an elastic strength of approximately 1.6 +/- 0.2 kPa and with an experimentally determined Poisson ratio of approximately 0.4 to 0.5. Further from the edge of these cells, the adherence decreases, and the Tu model is effective in evaluating its elastic strength ( approximately 0.6 +/- 0.1 kPa). Thus, our AFM-based microrheology allows us to correlate two key parameters of cell motility by relating elastic strength and the Poisson ratio to the adhesive state of a cell. This frequency-dependent measurement allows for the decomposition of the elastic modulus into loss and storage modulus. Applying this decomposition and Tu's and Chen's finite depth models allow us to obtain viscoelastic signatures in a frequency range from 50 to 300 Hz, showing a rubber plateau-like behavior.
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paper_title: Impact and contact stress analysis in multilayer media

paper_content:

Abstract The contact problem in a multilayer medium is analyzed based upon classical elasticity theory. The mixed boundary value problem is reformulated into a general approximation technique suitable for calculation on a digital computer. The Boussinesq problem demonstrates that this approximate method is much more accurate than an “exact” analysis recently published. A second example, which is most commonly applied in engineering practice, involves the parabolic punch. Numerical results are presented for both examples, to illustrate the physically significant effects of soft and hard layers and different layer thicknesses. The analysis is then extended to quasi-static impact. Results of experiments dropping a steel ball on nylon and rubber layers over a granite base are given and good agreement is found between the analytical and experimental results.
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paper_title: A thin-layer model for viscoelastic, stress-relaxation testing of cells using atomic force microscopy: do cell properties reflect metastatic potential?

paper_content:

Atomic force microscopy has rapidly become a valuable tool for quantifying the biophysical properties of single cells. The interpretation of atomic force microscopy-based indentation tests, however, is highly dependent on the use of an appropriate theoretical model of the testing configuration. In this study, a novel, thin-layer viscoelastic model for stress relaxation was developed to quantify the mechanical properties of chondrosarcoma cells in different configurations to examine the hypothesis that viscoelastic properties reflect the metastatic potential and invasiveness of the cell using three well-characterized human chondrosarcoma cell lines (JJ012, FS090, 105KC) that show increasing chondrocytic differentiation and decreasing malignancy, respectively. Single-cell stress relaxation tests were conducted at 2 h and 2 days after plating to determine cell mechanical properties in either spherical or spread morphologies and analyzed using the new theoretical model. At both time points, JJ012 cells had the lowest moduli of the cell lines examined, whereas FS090 typically had the highest. At 2 days, all cells showed an increase in stiffness and a decrease in apparent viscosity compared to the 2-h time point. Fluorescent labeling showed that the F-actin structure in spread cells was significantly different between FS090 cells and JJ012/105KC cells. Taken together with results of previous studies, these findings indicate that cell transformation and tumorigenicity are associated with a decrease in cell modulus and apparent viscosity, suggesting that cell mechanical properties may provide insight into the metastatic potential and invasiveness of a cell.
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paper_title: Spherical indentation testing of poroelastic relaxations in thin hydrogel layers

paper_content:

In this work, we present the Poroelastic Relaxation Indentation (PRI) testing approach for quantifying the mechanical and transport properties of thin layers of poly(ethylene glycol) hydrogels with thicknesses on the order of 200 μm. Specifically, PRI characterizes poroelastic relaxation in hydrogels by indenting the material at fixed depth and measuring the contact area-dependent load relaxation process as a function of time. With the aid of a linear poroelastic theory developed for thin or geometrically confined swollen polymer networks, we demonstrate that PRI can quantify the water diffusion coefficient, shear modulus and average pore size of the hydrogel layer. This approach provides a simple methodology to quantify the material properties of thin swollen polymer networks relevant to transport phenomena.
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paper_title: Microrheology of Human Lung Epithelial Cells Measured by Atomic Force Microscopy

paper_content:

Lung epithelial cells are subjected to large cyclic forces from breathing. However, their response to dynamic stresses is poorly defined. We measured the complex shear modulus (G*(ω)) of human alveolar (A549) and bronchial (BEAS-2B) epithelial cells over three frequency decades (0.1–100 Hz) and at different loading forces (0.1–0.9 nN) with atomic force microscopy. G*(ω) was computed by correcting force-indentation oscillatory data for the tip-cell contact geometry and for the hydrodynamic viscous drag. Both cell types displayed similar viscoelastic properties. The storage modulus G′(ω) increased with frequency following a power law with exponent ∼0.2. The loss modulus G″(ω) was ∼2/3 lower and increased similarly to G′(ω) up to ∼10 Hz, but exhibited a steeper rise at higher frequencies. The cells showed a weak force dependence of G′(ω) and G″(ω). G*(ω) conformed to the power-law model with a structural damping coefficient of ∼0.3, indicating a coupling of elastic and dissipative processes within the cell. Power-law behavior implies a continuum distribution of stress relaxation time constants. This complex dynamics is consistent with the rheology of soft glassy materials close to a glass transition, thereby suggesting that structural disorder and metastability may be fundamental features of cell architecture.
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paper_title: Viscoelastic properties of human mesenchymally-derived stem cells and primary osteoblasts, chondrocytes, and adipocytes

paper_content:

The mechanical properties of single cells play important roles in regulating cell-matrix interactions, potentially influencing the process of mechanotransduction. Recent studies also suggest that cellular mechanical properties may provide novel biological markers, or “biomarkers,” of cell phenotype, reflecting specific changes that occur with disease, differentiation, or cellular transformation. Of particular interest in recent years has been the identification of such biomarkers that can be used to determine specific phenotypic characteristics of stem cells that separate them from primary, differentiated cells. The goal of this study was to determine the elastic and viscoelastic properties of three primary cell types of mesenchymal lineage (chondrocytes, osteoblasts, and adipocytes) and to test the hypothesis that primary differentiated cells exhibit distinct mechanical properties compared to adult stem cells (adipose-derived or bone marrow-derived mesenchymal stem cells). In an adherent, spread configuration, chondrocytes, osteoblasts, and adipocytes all exhibited significantly different mechanical properties, with osteoblasts being stiffer than chondrocytes and both being stiffer than adipocytes. Adipose-derived and mesenchymal stem cells exhibited similar properties to each other, but were mechanically distinct from primary cells, particularly when comparing a ratio of elastic to relaxed moduli. These findings will help more accurately model the cellular mechanical environment in mesenchymal tissues, which could assist in describing injury thresholds and disease progression or even determining the influence of mechanical loading for tissue engineering efforts. Furthermore, the identification of mechanical properties distinct to stem cells could result in more successful sorting procedures to enrich multipotent progenitor cell populations.
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paper_title: Finite element modelling of nanoindentation based methods for mechanical properties of cells.

paper_content:

The viscoelastic properties of the living cells are for quantifying the biomechanical effects of drug treatment, diseases and aging. Nanoindentation techniques have proven effective to characterize the viscoelastic properties of living cells. However, most studies utilized the Hertz contact model and assumed the Heaviside step loading, which does not represent real tests. Therefore, new mathematical models have been developed to determine the viscoelastic properties of the cells for nanoindentation tests. Finite element method was used to determine the empirical correction parameter in the mathematical model to account for large deformation, in which case the combined effect of finite lateral and vertical dimensions of the cell is essential. The viscoelastic integral operator was used to account for the realistic deformation rate. The predictive model captures the mechanical responses of the cells observed from previous experimental study. This work has demonstrated that the new model consistently predicts viscoelastic properties for both ramping and stress relaxation periods, which cannot be achieved by the commonly used models. Utilization of this new model can enrich the experimental cell mechanics in interpretation of nanoindentation of cells.
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paper_title: Nanomechanical properties of individual chondrocytes and their developing growth factor-stimulated pericellular matrix

paper_content:

Abstract The nanomechanical properties of individual cartilage cells (chondrocytes) and their aggrecan and collagen-rich pericellular matrix (PCM) were measured via atomic force microscope nanoindentation using probe tips of two length scales (nanosized and micron-sized). The properties of cells freshly isolated from cartilage tissue (devoid of PCM) were compared to cells that were cultured for selected times (up to 28 days) in 3-D alginate gels which enabled PCM assembly and accumulation. Cells were immobilized and kept viable in pyramidal wells microfabricated into an array on silicon chips. Hertzian contact mechanics and finite element analyses were employed to estimate apparent moduli from the force versus depth curves. The effects of culture conditions on the resulting PCM properties were studied by comparing 10% fetal bovine serum to medium containing a combination of insulin growth factor-1 (IGF-1)+osteogenic protein-1 (OP-1). While both systems showed increases in stiffness with time in culture between days 7 and 28, the IGF-1+OP-1 combination resulted in a higher stiffness for the cell-PCM composite by day 28 and a higher apparent modulus of the PCM which is compared to the FBS cultured cells. These studies give insight into the temporal evolution of the nanomechanical properties of the pericellar matrix relevant to the biomechanics and mechanobiology of tissue-engineered constructs for cartilage repair.
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paper_title: Immobilizing live bacteria for AFM imaging of cellular processes.

paper_content:

Coccoid cells of the bacterial species Staphylococcus aureus have been mechanically trapped in lithographically patterned substrates and imaged under growth media using atomic force microscopy (AFM) in order to follow cellular processes. The cells are not perturbed as there is no chemical linkage to the surface. Confinement effects are minimized compared to trapping the cells in porous membranes or soft gels. S. aureus cells have been imaged undergoing cell division whilst trapped in the patterned substrates. Entrapment in lithographically patterned substrates provides a novel way for anchoring bacterial cells so that the AFM tip will not push the cells off during imaging, whilst allowing the bacteria to continue with cellular processes.
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paper_title: Understanding the nanoindentation mechanisms of a microsphere for biomedical applications

paper_content:

Nanoindentation techniques have proven effective to characterize nanomaterials and soft biomaterials. Using microfabricated wells to hold microspheres will enable automated indentation of microspheres. However, the existing contact mechanics based models such as the Hertz model and other modified models (e.g. thin layer models) only deal with indenting the specimen placed on a flat surface (i.e. the bottom surface is constrained vertically) without lateral constraint. Therefore, new mathematical models have been developed in this study to investigate the nanoindentation responses for a microsphere sitting in a well. Finite element simulation was employed to determine the empirical correction parameter in the mathematical model to account for the constraint imposed by the well. Utilization of this new model can also enrich the experimental contact mechanics.
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paper_title: Cell mechanics, structure, and function are regulated by the stiffness of the three-dimensional microenvironment.

paper_content:

This study adopts a combined computational and experimental approach to determine the mechanical, structural, and metabolic properties of isolated chondrocytes cultured within three-dimensional hydrogels. A series of linear elastic and hyperelastic finite-element models demonstrated that chondrocytes cultured for 24 h in gels for which the relaxation modulus is <5 kPa exhibit a cellular Young's modulus of ∼5 kPa. This is notably greater than that reported for isolated chondrocytes in suspension. The increase in cell modulus occurs over a 24-h period and is associated with an increase in the organization of the cortical actin cytoskeleton, which is known to regulate cell mechanics. However, there was a reduction in chromatin condensation, suggesting that changes in the nucleus mechanics may not be involved. Comparison of cells in 1% and 3% agarose showed that cells in the stiffer gels rapidly develop a higher Young's modulus of ∼20 kPa, sixfold greater than that observed in the softer gels. This was associated with higher levels of actin organization and chromatin condensation, but only after 24 h in culture. Further studies revealed that cells in stiffer gels synthesize less extracellular matrix over a 28-day culture period. Hence, this study demonstrates that the properties of the three-dimensional microenvironment regulate the mechanical, structural, and metabolic properties of living cells.
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paper_title: Indentation and adhesive probing of a cell membrane with AFM: theoretical model and experiments

paper_content:

In probing adhesion and cell mechanics by atomic force microscopy (AFM), the mechanical properties of the membrane have an important if neglected role. Here we theoretically model the contact of an AFM tip with a cell membrane, where direct motivation and data are derived from a prototypical ligand-receptor adhesion experiment. An AFM tip is functionalized with a prototypical ligand, SIRPα, and then used to probe its native receptor on red cells, CD47. The interactions prove specific and typical in force, and also show in detachment, a sawtooth-shaped disruption process that can extend over hundreds of nm. The theoretical model here that accounts for both membrane indentation as well as membrane extension in tip retraction incorporates membrane tension and elasticity as well as AFM tip geometry and stochastic disruption. Importantly, indentation depth proves initially proportional to membrane tension and does not follow the standard Hertz model. Computations of detachment confirm nonperiodic disruption with membrane extensions of hundreds of nm set by membrane tension. Membrane mechanical properties thus clearly influence AFM probing of cells, including single molecule adhesion experiments.
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paper_title: On the contact and adhesion of rough surfaces

paper_content:

The elastic contact of rough surfaces has been studied by Greenwood and Williamson [Proc. R. Soc. London, Ser. A 295, 300 (1966)] for Hertzian contacts and by Fuller and Tabor [Proc. R. Soc. London, Ser. A 345, 327 (1975)] for Johnson-Kendall-Roberts contacts. The theory for Deryagin-Muller-Toporov contacts is proposed here and compared with previous results. An extra load due to adhesion forces acting around the contacts appears, which can account for the friction under negative load and the increase in the apparent friction coefficient on clean surfaces.
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paper_title: Nanomechanical properties of individual chondrocytes and their developing growth factor-stimulated pericellular matrix

paper_content:

Abstract The nanomechanical properties of individual cartilage cells (chondrocytes) and their aggrecan and collagen-rich pericellular matrix (PCM) were measured via atomic force microscope nanoindentation using probe tips of two length scales (nanosized and micron-sized). The properties of cells freshly isolated from cartilage tissue (devoid of PCM) were compared to cells that were cultured for selected times (up to 28 days) in 3-D alginate gels which enabled PCM assembly and accumulation. Cells were immobilized and kept viable in pyramidal wells microfabricated into an array on silicon chips. Hertzian contact mechanics and finite element analyses were employed to estimate apparent moduli from the force versus depth curves. The effects of culture conditions on the resulting PCM properties were studied by comparing 10% fetal bovine serum to medium containing a combination of insulin growth factor-1 (IGF-1)+osteogenic protein-1 (OP-1). While both systems showed increases in stiffness with time in culture between days 7 and 28, the IGF-1+OP-1 combination resulted in a higher stiffness for the cell-PCM composite by day 28 and a higher apparent modulus of the PCM which is compared to the FBS cultured cells. These studies give insight into the temporal evolution of the nanomechanical properties of the pericellar matrix relevant to the biomechanics and mechanobiology of tissue-engineered constructs for cartilage repair.
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paper_title: Contribution of the nucleus to the mechanical properties of endothelial cells.

paper_content:

The cell nucleus plays a central role in the response of the endothelium to mechanical forces, possibly by deforming during cellular adaptation. The goal of this work was to precisely quantify the mechanical properties of the nucleus. Individual endothelial cells were subjected to compression between glass microplates. This technique allows measurement of the uniaxial force applied to the cell and the resulting deformation. Measurements were made on round and spread cells to rule out the influence of cell morphology on the nucleus mechanical properties. Tests were also carried out with nuclei isolated from cell cultures by a chemical treatment. The non-linear force-deformation curves indicate that round cells deform at lower forces than spread cells and nuclei. Finite-element models were also built with geometries adapted to actual morphometric measurements of round cells, spread cells and isolated nuclei. The nucleus and the cytoplasm were modeled as separate homogeneous hyperelastic materials. The models simulate the compression and yield the force-deformation curve for a given set of elastic moduli. These parameters are varied to obtain a best fit between the theoretical and experimental data. The elastic modulus of the cytoplasm is found to be on the order of 500N/m(2) for spread and round cells. The elastic modulus of the endothelial nucleus is on the order of 5000N/m(2) for nuclei in the cell and on the order of 8000N/m(2) for isolated nuclei. These results represent an unambiguous measurement of the nucleus mechanical properties and will be important in understanding how cells perceive mechanical forces and respond to them.
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paper_title: Influence of cell spreading and contractility on stiffness measurements using AFM

paper_content:

Atomic Force Microscopy (AFM) is widely used for measuring mechanical properties of cells, and to understand how cells respond to their mechanical environments. A standard method for obtaining cell stiffness from experimental force–indentation curves is based on the simplified Hertz theory developed for studying the indentation of a semi-infinite elastic body by a spherical punch, assumptions that do not hold for biological cells. The modified Hertz theory developed by Dimitriadis et al., which takes the finite sample height into account, is widely used by experimentalists for greater accuracy. However, neither of these two models account for the finite lateral spread of the cells and cellular contractility. In this paper, we address the influence of cell geometry, cell pre-stress, and nuclear properties on cell stiffness measurements by modeling indentation of a cell of prescribed geometry with a spherical AFM probe using the finite element method. Using parametric studies, we develop scaling relationships between the effective stiffness probed by AFM and the bulk cell stiffness, taking cell and tip geometry into account. Taken together, our results demonstrate the need to take cell geometry into account while estimating the cell stiffness and provide simple expressions for doing so.
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paper_title: In situ mechanical properties of the chondrocyte cytoplasm and nucleus

paper_content:

Abstract The way in which the nucleus experiences mechanical forces has important implications for understanding mechanotransduction. Knowledge of nuclear material properties and, specifically, their relationship to the properties of the bulk cell can help determine if the nucleus directly experiences mechanical loads, or if it is a signal transduction mechanism secondary to cell membrane deformation that leads to altered gene expression. Prior work measuring nuclear material properties using micropipette aspiration suggests that the nucleus is substantially stiffer than the bulk cell [Guilak, F., Tedrow, J.R., Burgkart, R., 2000. Viscoelastic properties of the cell nucleus. Biochem. Biophys. Res. Commun. 269, 781–786], whereas recent work with unconfined compression of single chondrocytes showed a nearly one-to-one correlation between cellular and nuclear strains [Leipzig, N.D., Athanasiou, K.A., 2008. Static compression of single chondrocytes catabolically modifies single-cell gene expression. Biophys. J. 94, 2412–2422]. In this study, a linearly elastic finite element model of the cell with a nuclear inclusion was used to simulate the unconfined compression data. Cytoplasmic and nuclear stiffnesses were varied from 1 to 7 kPa for several combinations of cytoplasmic and nuclear Poisson's ratios. It was found that the experimental data were best fit when the ratio of cytoplasmic to nuclear stiffness was 1.4, and both cytoplasm and nucleus were modeled as incompressible. The cytoplasmic to nuclear stiffness ratio is significantly lower than prior reports for isolated nuclei. These results suggest that the nucleus may behave mechanically different in situ than when isolated.
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paper_title: Mechanical regulation of nuclear structure and function.

paper_content:

Mechanical loading induces both nuclear distortion and alterations in gene expression in a variety of cell types. Mechanotransduction is the process by which extracellular mechanical forces can activate a number of well-studied cytoplasmic signaling cascades. Inevitably, such signals are transduced to the nucleus and induce transcription factor-mediated changes in gene expression. However, gene expression also can be regulated through alterations in nuclear architecture, providing direct control of genome function. One putative transduction mechanism for this phenomenon involves alterations in nuclear architecture that result from the mechanical perturbation of the cell. This perturbation is associated with direct mechanical strain or osmotic stress, which is transferred to the nucleus. This review describes the current state of knowledge relating the nuclear architecture and the transfer of mechanical forces to the nucleus mediated by the cytoskeleton, the nucleoskeleton, and the LINC (linker of the nucleoskeleton and cytoskeleton) complex. Moreover, remodeling of the nucleus induces alterations in nuclear stiffness, which may be associated with cell differentiation. These phenomena are discussed in relation to the potential influence of nuclear architecture-mediated mechanoregulation of transcription and cell fate.
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paper_title: Cell mechanics, structure, and function are regulated by the stiffness of the three-dimensional microenvironment.

paper_content:

This study adopts a combined computational and experimental approach to determine the mechanical, structural, and metabolic properties of isolated chondrocytes cultured within three-dimensional hydrogels. A series of linear elastic and hyperelastic finite-element models demonstrated that chondrocytes cultured for 24 h in gels for which the relaxation modulus is <5 kPa exhibit a cellular Young's modulus of ∼5 kPa. This is notably greater than that reported for isolated chondrocytes in suspension. The increase in cell modulus occurs over a 24-h period and is associated with an increase in the organization of the cortical actin cytoskeleton, which is known to regulate cell mechanics. However, there was a reduction in chromatin condensation, suggesting that changes in the nucleus mechanics may not be involved. Comparison of cells in 1% and 3% agarose showed that cells in the stiffer gels rapidly develop a higher Young's modulus of ∼20 kPa, sixfold greater than that observed in the softer gels. This was associated with higher levels of actin organization and chromatin condensation, but only after 24 h in culture. Further studies revealed that cells in stiffer gels synthesize less extracellular matrix over a 28-day culture period. Hence, this study demonstrates that the properties of the three-dimensional microenvironment regulate the mechanical, structural, and metabolic properties of living cells.
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paper_title: A Triphasic Theory for the Swelling and Deformation Behaviors of Articular Cartilage

paper_content:

Swelling of articular cartilage depends on its fixed charge density and distribution, the stiffness of its collagen-proteoglycan matrix, and the ion concentrations in the interstitium. A theory for a tertiary mixture has been developed, including the two fluid-solid phases (biphasic), and an ion phase, representing cation and anion of a single salt, to describe the deformation and stress fields for cartilage under chemical and/or mechanical loads. This triphasic theory combines the physico-chemical theory for ionic and polyionic (proteoglycan) solutions with the biphasic theory for cartilage. The present model assumes the fixed charge groups to remain unchanged, and that the counter-ions are the cations of a single salt of the bathing solution. The momentum equation for the neutral salt and for the intersitial water are expressed in terms of their chemical potentials whose gradients are the driving forces for their movements. These chemical potentials depend on fluid pressure p, salt concentration c, solid matrix dilatation e and fixed charge density cF . For a uni-uni valent salt such as NaCl, they are given by μi = μo i + (RT/Mi )ln[γ± 2 c (c + c F )] and μW = μo w + [p − RTφ(2c + cF ) + Bw e]/ρT w , where R, T, Mi , γ± , φ, ρT w and Bw are universal gas constant, absolute temperature, molecular weight, mean activity coefficient of salt, osmotic coefficient, true density of water, and a coupling material coefficient, respectively. For infinitesimal strains and material isotropy, the stress-strain relationship for the total mixture stress is σ = − pI − Tc I + λs (trE)I + 2μs E, where E is the strain tensor and (λs ,μs ) are the Lame constants of the elastic solid matrix. The chemical-expansion stress (− Tc ) derives from the charge-to-charge repulsive forces within the solid matrix. This theory can be applied to both equilibrium and non-equilibrium problems. For equilibrium free swelling problems, the theory yields the well known Donnan equilibrium ion distribution and osmotic pressure equations, along with an analytical expression for the “pre-stress” in the solid matrix. For the confined-compression swelling problem, it predicts that the applied compressive stress is shared by three load support mechanisms: 1) the Donnan osmotic pressure; 2) the chemical-expansion stress; and 3) the solid matrix elastic stress. Numerical calculations have been made, based on a set of equilibrium free-swelling and confined-compression data, to assess the relative contribution of each mechanism to load support. Our results show that all three mechanisms are important in determining the overall compressive stiffness of cartilage.
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paper_title: Biomechanical properties of single chondrocytes and chondrons determined by micromanipulation and finite-element modelling

paper_content:

A chondrocyte and its surrounding pericellular matrix (PCM) are defined as a chondron. Single chondrocytes and chondrons isolated from bovine articular cartilage were compressed by micromanipulation between two parallel surfaces in order to investigate their biomechanical properties and to discover the mechanical significance of the PCM. The force imposed on the cells was measured directly during compression to various deformations and then holding. When the nominal strain at the end of compression was 50 per cent, force relaxation showed that the cells were viscoelastic, but this viscoelasticity was generally insignificant when the nominal strain was 30 per cent or lower. The viscoelastic behaviour might be due to the mechanical response of the cell cytoskeleton and/or nucleus at higher deformations. A finite-element analysis was applied to simulate the experimental force-displacement/time data and to obtain mechanical property parameters of the chondrocytes and chondrons. Because of the large strains in the cells, a nonlinear elastic model was used for simulations of compression to 30 per cent nominal strain and a nonlinear viscoelastic model for 50 per cent. The elastic model yielded a Young’s modulus of 14+ 1 kPa (mean+ s.e.) for chondrocytes and 19+ 2 kPa for chondrons, respectively. The viscoelastic model generated an instantaneous elastic modulus of 21+ 3a nd 27+ 4 kPa, a long-term modulus of 9.3+ 0.8 and 12+ 1 kPa and an apparent viscosity of 2.8+ 0.5 and 3.4+ 0.6 kPa s for chondrocytes and chondrons, respectively. It was concluded that chondrons were generally stiffer and showed less viscoelastic behaviour than chondrocytes, and that the PCM significantly influenced the mechanical properties of the cells.
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paper_title: On the factors affecting the critical indenter penetration for measurement of coating hardness

paper_content:

Abstract The nanoindentation test is the only viable approach to assess the properties of very thin coatings (
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paper_title: A Mixture Theory for Charged-Hydrated Soft Tissues Containing Multi-electrolytes: Passive Transport and Swelling Behaviors

paper_content:

A new mixture theory was developed to model the mechano-electrochemical behaviors of charged-hydrated soft tissues containing multi-electrolytes. The mixture is composed of n + 2 constituents (1 charged solid phase, 1 noncharged solvent phase, and n ion species ). Results from this theory show that three types offorce are involved in the transport of ions and solvent through such materials : (1) a mechanochemical force (including hydraulic and osmotic pressures); (2) an electrochemical force; and (3) an electrical force. Our results also show that three types of material coefficients are required to characterize the transport rates of these ions and solvent: (1) a hydraulic permeability ; (2) mechano-electrochemical coupling coefficients; and (3) an ionic conductance matrix. Specifically, we derived the fundamental governing relationships between these forces and material coefficients to describe such mechano-electrochemical transduction effects as streaming potential, streaming current, diffusion (membrane) potential, electro-osmosis, and anomalous (negative) osmosis. As an example, we showed that the well-known formula for the resting cell membrane potential (Hodgkin and Huxley, 1952a, b) could be derived using our new n + 2 mixture model (a generalized triphasic theory). In general, the n + 2 mixture theory is consistent with and subsumes all previous theories pertaining to specific aspects of charged-hydrated tissues. In addition, our results provided the stress, strain, and fluid velocity fields within a tissue of finite thickness during a one-dimensional steady diffusion process. Numerical results were provided for the exchange of Na + and Ca ++ through the tissue. These numerical results support our hypothesis that tissue fixed charge density (c F ) plays a significant role in modulating kinetics of ions and solvent transport through charged-hydrated soft tissues.
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paper_title: Finite element modelling of nanoindentation based methods for mechanical properties of cells.

paper_content:

The viscoelastic properties of the living cells are for quantifying the biomechanical effects of drug treatment, diseases and aging. Nanoindentation techniques have proven effective to characterize the viscoelastic properties of living cells. However, most studies utilized the Hertz contact model and assumed the Heaviside step loading, which does not represent real tests. Therefore, new mathematical models have been developed to determine the viscoelastic properties of the cells for nanoindentation tests. Finite element method was used to determine the empirical correction parameter in the mathematical model to account for large deformation, in which case the combined effect of finite lateral and vertical dimensions of the cell is essential. The viscoelastic integral operator was used to account for the realistic deformation rate. The predictive model captures the mechanical responses of the cells observed from previous experimental study. This work has demonstrated that the new model consistently predicts viscoelastic properties for both ramping and stress relaxation periods, which cannot be achieved by the commonly used models. Utilization of this new model can enrich the experimental cell mechanics in interpretation of nanoindentation of cells.
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paper_title: Silicone rubber substrata: a new wrinkle in the study of cell locomotion

paper_content:

When tissue cells are cultured on very thin sheets of cross-linked silicone fluid, the traction forces the cells exert are made visible as elastic distortion and wrinkling of this substratum. Around explants this pattern of wrinkling closely resembles the "center effects" long observed in plasma clots and traditionally attributed to dehydration shrinkage.
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Title: Nanobiomechanics of living cells: a review

Section 1: Introduction
Description 1: Provide an overview of the significance of mechanical properties of living cells, various testing methods, and the importance of understanding nanoindentation in cell interactions.

Section 2: Experimental aspects
Description 2: Discuss the details of experimental setups used in nanoindentation, including nanoindenter apparatus and atomic force microscope (AFM).

Section 3: Choice of appropriate atomic force microscope tips
Description 3: Detail the different AFM tip geometries and their respective advantages and disadvantages in nanoindentation.

Section 4: Flat punch
Description 4: Explain the specific use and characteristics of flat punch indenters for probing cell mechanics.

Section 5: Spherical tip
Description 5: Discuss the properties and applications of spherical tips in nanoindentation of cells.

Section 6: Pyramid tip
Description 6: Describe the use of pyramid tips for probing the fine features of cytoskeleton and associated drawbacks.

Section 7: Conical tip
Description 7: Provide details on conical tips, their use, and how they compare to pyramid tips.

Section 8: Extended atomic force microscope testing rigs
Description 8: Introduce the concept of cytocompression and its differentiation from cytoindentation.

Section 9: Mechanical modelling
Description 9: Review different mechanical models used to estimate cell mechanics, including structure-based and continuum models.

Section 10: Percolation models
Description 10: Explain how percolation models describe cell structure and mechanics.

Section 11: Summary of structure-based models
Description 11: Summarize the tensegrity and percolation models, emphasizing their complimentary nature.

Section 12: Continuum models
Description 12: Discuss the quantifiable mechanical properties of cells using continuum models and their relevance.

Section 13: Elastic model
Description 13: Examine the use of elastic models for determining the mechanical properties of cells under equilibrium conditions.

Section 14: Poroelastic model
Description 14: Describe the poroelastic model and its application in explaining the mechanical behavior of cells.

Section 15: Spring-dashpot viscoelastic models
Description 15: Discuss viscoelasticity models using spring-dashpot elements and their application in cell mechanics.

Section 16: Power-law rheology
Description 16: Introduce power-law rheology for describing the mechanics of certain cell types.

Section 17: Nanoindentation models
Description 17: Review nanoindentation models suitable for different AFM tips, including elastic, poroelastic, and viscoelastic models.

Section 18: Cell mechanics determined by different models
Description 18: Compare the mechanical properties of cells as determined by various models.

Section 19: Cell mechanics determined by different indenters
Description 19: Discuss the impact of indenter tip geometry on the measured mechanical properties of cells.

Section 20: Cell morphology on cell mechanics
Description 20: Evaluate how cell morphology affects cell mechanics and the importance of substrate influence.

Section 21: Cell -tip adhesion
Description 21: Address the issue of tip-cell adhesion during nanoindentation and the models considering adhesion forces.

Section 22: Inverse finite-element analysis
Description 22: Discuss the use of three-dimensional finite-element analysis to determine cell mechanical properties.

Section 23: Summary of strategy of selecting tip geometry and mechanical models
Description 23: Provide guidelines for selecting appropriate tip geometries and mechanical models based on cell morphology and experimental conditions.

Section 24: Conclusion and perspectives
Description 24: Summarize the importance of nanobiomechanics in understanding cell processes, and provide an outlook on future directions in mechanical modelling and experimental protocols.