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

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

Application of the micropipette technique to the measurement of cultured porcine aortic endothelial cell viscoelastic properties

The viscoelastic deformation of porcine aortic endothelial cells grown under static culture conditions was measured using the micropipette technique. Experiments were conducted both for control cells (mechanically or trypsin detached from the substrate) and for cells in which cytoskeletal elements were disrupted by cytochalasin B or colchicine. The time course of the aspirated length into the pipette was measured after applying a stepwise increase in aspiration pressure. To analyze the data, a standard linear viscoelastic half-space model of the endothelial cell was used. The aspirated length was expressed as an exponential function of time. The actin microfilaments were found to be the major cytoskeletal component determining the viscoelastic response of endothelial cells grown in static culture.     

Passive deformation analysis of human leukocytes

The following analysis presents an experimental and theoretical study of the passive viscoelastic behavior of human leukocytes. Individual neutrophils in EDTA were observed both during their partial aspiration into a small micropipette and after expulsion from a large micropipette where the cell had been totally aspirated and deformed into a sausage shape. To analyze the data, a passive model of leukocyte rheology has been developed consisting of a cortical shell containing a Maxwell fluid which describes the average properties of the cell cytoplasm. The cortical shell represents a crosslinked actin layer near the surface of the cell and is assumed to be under pre-stressed tension. This model can reproduce the results of experiments using micropipette for both short-time small deformation and slow recovery data after large deformation. In addition, a finite element scheme has been established for the same model which shows close agreement with the analytical solution.     

An axisymmetric boundary integral model for incompressible linear viscoelasticity: application to the micropipette aspiration contact problem

The micropipette aspiration test has been used extensively in recent years as a means of quantifying cellular mechanics and molecular interactions at the microscopic scale. However, previous studies have generally modeled the cell as an infinite half-space in order to develop an analytical solution for a viscoelastic solid cell. In this study, an axisymmetric boundary integral formulation of the governing equations of incompressible linear viscoelasticity is presented and used to simulate the micropipette aspiration contact problem. The cell is idealized as a homogeneous and isotropic continuum with constitutive equation given by three-parameter (E, tau 1, tau 2) standard linear viscoelasticity. The formulation is used to develop a computational model via a "correspondence principle" in which the solution is written as the sum of a homogeneous (elastic) part and a nonhomogeneous part, which depends only on past values of the solution. Via a time-marching scheme, the solution of the viscoelastic problem is obtained by employing an elastic boundary element method with modified boundary conditions. The accuracy and convergence of the time-marching scheme are verified using an analytical solution. An incremental reformulation of the scheme is presented to facilitate the simulation of micropipette aspiration, a nonlinear contact problem. In contrast to the halfspace model (Sato et al., 1990), this computational model accounts for nonlinearities in the cell response that result from a consideration of geometric factors including the finite cell dimension (radius R), curvature of the cell boundary, evolution of the cell-micropipette contact region, and curvature of the edges of the micropipette (inner radius a, edge curvature radius epsilon). Using 60 quadratic boundary elements, a micropipette aspiration creep test with ramp time t* = 0.1 s and ramp pressure p*/E = 0.8 is simulated for the cases a/R = 0.3, 0.4, 0.5 using mean parameter values for primary chondrocytes. Comparisons to the half-space model indicate that the computational model predicts an aspiration length that is less stiff during the initial ramp response (t = 0-1 s) but more stiff at equilibrium (t = 200 s). Overall, the ramp and equilibrium predictions of aspiration length by the computational model are fairly insensitive to aspect ratio a/R but can differ from the half-space model by up to 20 percent. This computational approach may be readily extended to account for more complex geometries or inhomogeneities in cellular properties.     

Deformation properties of articular chondrocytes: a critique of three separate techniques

This paper presents a series of techniques, which examine the deformation characteristics of bovine articular chondrocytes. The direct contact approach employs well established methodology, involving AFM and micropipette aspiration, to yield structural properties of local regions of isolated chondrocytes. The former technique yields a non-linear response with increased structural stiffness in a central location on a projected image of the chondrocyte. A simple viscoelastic model can be used with data from the micropipette aspiration technique to yield a mean value of Young's modulus, which is similar to that recently reported (Jones et al., 1999). An indirect approach is also described, involving the response of chondrocytes seeded within compressed agarose constructs. For 1% agarose constructs, the resulting cell strain, yields a gross cell modulus of 2.7 kPa. The study highlights the difficulties in establishing unique mechanical parameters, which reflect the deformation behaviour of articular chondrocytes.     

Viscoelastic behavior of erythrocyte membrane.

A nonlinear viscoelastic relation is developed to describe the viscoelastic properties of erythrocyte membrane. This constitutive equation is used in the analysis of the time-dependent aspiration of an erythrocyte membrane into a micropipette. Equations governing this motion are reduced to a nonlinear integral equation of the Volterra type. A numerical procedure based on a finite difference scheme is used to solve the integral equation and to match the experimental data. The data, aspiration length vs. time, is used to determine the relaxation function at each time step. The inverse problem of obtaining the time dependence of the aspiration length from a given relaxation function is also solved. Analytical results obtained are applied to the experimental data of Chien et al. 1978. Biophys. J. 24:463-487. A relaxation function similar to that of a four-parameter solid with a shear-thinning viscous term is proposed.     

Passive mechanical properties of human leukocytes.

Micropipette experiments are used to determine the rheological properties of human leukocytes. Individual cells in EDTA are subjected to a known aspiration pressure via a micropipette, and their surface deformation from the undeformed spherical shape is recorded on a television monitor. The cells are mathematically modeled as homogeneous spheres, and a standard solid viscoelastic model is found to describe accurately the deformation of the cell for small strains. These experimental and theoretical studies provide the basis for further investigations of leukocyte rheology in health and disease.     

A power-law rheology-based finite element model for single cell deformation

Physical forces can elicit complex time- and space-dependent deformations in living cells. These deformations at the subcellular level are difficult to measure but can be estimated using computational approaches such as finite element (FE) simulation. Existing FE models predominantly treat cells as spring-dashpot viscoelastic materials, while broad experimental data are now lending support to the power-law rheology (PLR) model. Here, we developed a large deformation FE model that incorporated PLR and experimentally verified this model by performing micropipette aspiration on fibroblasts under various mechanical loadings. With a single set of rheological properties, this model recapitulated the diverse micropipette aspiration data obtained using three protocols and with a range of micropipette sizes. More intriguingly, our analysis revealed that decreased pipette size leads to increased pressure gradient, potentially explaining our previous counterintuitive finding that decreased pipette size leads to increased incidence of cell blebbing and injury. Taken together, our work leads to more accurate rheological interpretation of micropipette aspiration experiments than previous models and suggests pressure gradient as a potential determinant of cell injury.     

Alterations in the Young's modulus and volumetric properties of chondrocytes isolated from normal and osteoarthritic human cartilage

The mechanical environment of the chondrocyte is an important factor that influences the maintenance of the articular cartilage extracellular matrix. Previous studies have utilized theoretical models of chondrocytes within articular cartilage to predict the stress-strain and fluid flow environments around the cell, but little is currently known regarding the cellular properties which are required for implementation of these models. The objectives of this study were to characterize the mechanical behavior of primary human chondrocytes and to determine the Young's modulus of chondrocytes from non-osteoarthritic ('normal') and osteoarthritic cartilage. A second goal was to quantify changes in the volume of isolated chondrocytes in response to mechanical deformation. The micropipette aspiration technique was used to measure the deformation of a single chondrocyte into a glass micropipette in response to a prescribed pressure. The results of this study indicate that the human chondrocyte behaves as a viscoelastic solid. No differences were found between the Young's moduli of normal (0.65+/-0.63 kPa, n = 44) and osteoarthritic chondrocytes (0.67+/-0.86 kPa, n = 69, p = 0.93). A significant difference in cell volume was observed immediately and 600 s after complete aspiration of the cell into the pipette (p < 0.001), and the magnitude of this volume change between normal (11+/-11%, n = 40) and osteoarthritic (20+/-11%, n = 41) chondroctyes was significantly different at both time points (p < 0.002). This finding suggests that chondrocytes from osteoarthritic cartilage may have altered volume regulation capabilities in response to mechanical deformation. The mechanical and volumetric properties determined in this study will be of use in analytical and finite element models of chondrocyte-matrix interactions in order to better predict the mechanical environment of the cell in vivo.     

Viscoelastic behaviour of human mesenchymal stem cells

Background  

In this study, we have investigated the viscoelastic behaviour of individual human adult bone marrow-derived mesenchymal stem cells (hMSCs) and the role of F-actin filaments in maintaining these properties, using micropipette aspiration technique together with a standard linear viscoelastic solid model.     

Mechanics and deformation of the nucleus in micropipette aspiration experiment

Robust biomechanical models are essential for the study of nuclear mechanics and deformation and can help shed light on the underlying mechanisms of stress transition in nuclear elements. Here, we develop a computational model for an isolated nucleus undergoing micropipette aspiration. Our model includes distinct components representing the nucleoplasm and nuclear envelope. The nuclear envelope itself comprises three layers: inner and outer nuclear membranes and one thicker layer representing the nuclear lamina. The nucleoplasm is modeled as a viscoelastic Maxwell material with a single time constant, while a modified Maxwell model, equivalent to a spring and a dashpot in series and both in parallel with a spring, is adopted for the inner and outer nuclear membranes. The nuclear envelope layer is taken as a linear elastic material. The proposed computational model, validated using experimental observations of Guilak et al. [2000. Viscoelastic properties of the cell nucleus. Biochemical and Biophysical Research Communications 269, 781-786] and Deguchi et al. [2005, Flow-induced hardening of endothelial nucleus as an intracellular stress-bearing organelle. Journal of Biomechanics 38, 1751-1759], is employed to study nuclear mechanics and deformation in micropipette aspiration and to shed light on the contribution of individual nuclear components on the response. The results indicate that the overall response of an isolated nucleus in micropipette aspiration is highly sensitive to the apparent stiffness of the nuclear lamina. This observation suggests that micropipette aspiration is an effective technique for examining the influence of various kinds of alteration in the nuclear lamina, such as mutations in the gene encoding lamin A, and also structural remodeling due to mechanical perturbation.     

Leukocyte relaxation properties.

Study of the mechanical properties of leukocytes is useful to understand their passage through narrow capillaries and interaction with other cells. Leukocytes are known to be viscoelastic and their properties have been established by micropipette aspiration techniques. Here, the recovery of leukocytes to their normal spherical form is studied after prolonged deformation in a pipette which is large enough to permit complete entry of the leukocyte. The recovery history is characterized by the time history of the major diameter (d1) and minor diameter (d2). When the cell is removed from the pipette, it shows initially a small rapid recoil followed by a slower asymptotic recovery to the spherical shape. In the presence of cell activation and formation of pseudopods, the time history for recovery is prolonged compared with passive cell recovery. If a protopod pre-existed during the holding period, the recovery only begins when the protopod starts to retract.     

Comparison of analytical and inverse finite element approaches to estimate cell viscoelastic properties by micropipette aspiration

The viscoelastic properties of cells are important in predicting cell deformation under mechanical loading and may reflect cell phenotype or pathological transition. Previous studies have demonstrated that viscoelastic parameters estimated by finite element (FE) analyses of micropipette aspiration (MA) data differ from those estimated by the analytical half-space model. However, it is unclear whether these differences are statistically significant, as previous studies have been based on average cell properties or parametric analyses that do not reflect the inherent experimental and biological variability of real experimental data. To determine whether cell material parameters estimated by the half-space model are significantly different from those predicted by the FE method, we implemented an inverse FE method to estimate the viscoelastic parameters of a population of primary porcine aortic valve interstitial cells tested by MA. We found that inherent differences between the analytical and inverse FE estimation methods resulted in statistically significant differences in individual cell properties. However, in cases with small pipette to cell radius ratios and short loading periods, model-dependent differences were masked by experimental and cell-to-cell variability. Analytical models that account for finite cell-size and loading rate further relaxed the experimental conditions for which accurate cell material parameter estimates could be obtained. These data provide practical guidelines for analysis of MA data that account for the wide range of conditions encountered in typical experiments.     

Characterizing Cell Adhesion by Using Micropipette Aspiration

We have developed a technique to directly quantify cell-substrate adhesion force using micropipette aspiration. The micropipette is positioned perpendicular to the surface of an adherent cell and a constant-rate aspiration pressure is applied. Since the micropipette diameter and the aspiration pressure are our control parameters, we have direct knowledge of the aspiration force, whereas the cell behavior is monitored either in brightfield or interference reflection microscopy. This setup thus allows us to explore a range of geometric parameters, such as projected cell area, adhesion area, or pipette size, as well as dynamical parameters such as the loading rate. We find that cell detachment is a well-defined event occurring at a critical aspiration pressure, and that the detachment force scales with the cell adhesion area (for a given micropipette diameter and loading rate), which defines a critical stress. Taking into account the cell adhesion area, intrinsic parameters of the adhesion bonds, and the loading rate, a minimal model provides an expression for the critical stress that helps rationalize our experimental results.     

Power-law rheology analysis of cells undergoing micropipette aspiration

Accurate quantification of the mechanical properties of living cells requires the combined use of experimental techniques and theoretical models. In this paper, we investigate the viscoelastic response of suspended NIH 3T3 fibroblasts undergoing micropipette aspiration using power-law rheology model. As an important first step, we examine the pipette size effect on cell deformation and find that pipettes larger than ~7 μm are more suitable for bulk rheological measurements than smaller ones and the cell can be treated as effectively continuum. When the large pipettes are used to apply a constant pressure to a cell, the creep deformation is better fitted with the power-law rheology model than with the liquid drop or spring-dashpot models; magnetic twisting cytometry measurement on the rounded cell confirms the power-law behavior. This finding is further extended to suspended cells treated with drugs targeting their cytoskeleton. As such, our results suggest that the application of relatively large pipettes can provide more effective assessment of the bulk material properties as well as support application of power-law rheology to cells in suspension.     

Effect of colchicine on viscoelastic properties of neutrophils

The effect of colchicine (15-60 micrograms/ml) on the viscoelastic properties of human neutrophils was studied by the micropipette technique. The small deformation of the neutrophil in response to a step aspiration pressure was analyzed by using a three-element model in which an elastic element, K1, is in parallel with a Maxwell element composed of another elastic element, K2, in series with a viscous element, mu. Colchicine treatment of neutrophils caused decreases in K2 and mu without affecting K1. The results indicate that the integrity of the microtubules plays a significant role in providing the viscoelastic resistance (as represented by the Maxwell element in the model) of neutrophils to deforming stress.     

Mechanical properties of cancer cytoskeleton depend on actin filaments to microtubules content: Investigating different grades of colon cancer cell lines

Biomechanical properties of cancer cells have been proposed as promising biomarkers to investigate cancer progression. Cytoskeletal reorganization alters these characteristics in different grades of cancer cells. In the present study based on the micropipette aspiration method, whole body evaluation for two different colon cancer cells was performed to determine viscoelastic parameters of the cells. A finite element model was developed for verification of experiments and predicting some behaviors of cells. Western blot analysis and fluorescence intensity for actin microfilaments and microtubules were performed to measure cell content of the proteins. It was illustrated that the proportion of microtubules and actin microfilaments is different in grade I and grade IV colon cancer cells in a manner that microtubules attain an effectual role in progressive reorganization of cytoskeleton in transition from nonaggressive to malignant phenotypes in cancer cells. Furthermore, it was concluded that larger instantaneous Young's modulus value for high grade cells is related to the existence of extensively build-up actin networks at the cell cortex. Based on the cell mechanics results, a simple parameter is suggested for sorting different grades of colon cancer cells.     

Micropipette aspiration on the outer hair cell lateral wall

The mechanical properties of the lateral wall of the guinea pig cochlear outer hair cell were studied using the micropipette aspiration technique. A fire-polished micropipette with an inner diameter of approximately 4 microm was brought into contact with the lateral wall and negative pressure was applied. The resulting deformation of the lateral wall was recorded on videotape and subjected to morphometric analysis. The relation between the length of the aspirated portion of the cell and aspiration pressure is characterized by the stiffness parameter, K(s) = 1.07 +/- 0.24 (SD) dyn/cm (n = 14). Values of K(s) do not correlate with the original cell length, which ranges from 29 to 74 microm. Theoretical analysis based on elastic shell theory applied to the experimental data yields an estimate of the effective elastic shear modulus, mu = 15.4 +/- 3.3 dyn/cm. These data were obtained at subcritical aspiration pressures, typically less than 10 cm H2O. After reaching a critical (vesiculation) pressure, the cytoplasmic membrane appeared to separate from the underlying structures, a vesicle with a length of 10-20 microm was formed, and the cytoplasmic membrane resealed. This vesiculation process was repeated until a cell-specific limit was reached and no more vesicles were formed. Over 20 vesicles were formed from the longest cells in the experiment.     

Alteration of viscoelastic properties is associated with a change in cytoskeleton components of ageing chondrocytes from rabbit knee articular cartilage

The cytoskeleton network is believed to play an important role in the biomechanical properties of the chondrocyte. Ours and other laboratories have demonstrated that chondrocytes exhibit a viscoelastic solid creep behavior in vitro and that viscoelastic properties decrease in osteoarthritic chondrocytes. In this study, we aimed to understand whether the alteration of viscoelastic properties is associated with changes in cytoskeleton components of ageing chondrocytes from rabbit knee articular cartilage. Three age groups were used for this study: young (2-months-old, N=23), adult (8-months-old, N=23), and old (31-months-old, N=23) rabbit groups. Cartilage structure and proteoglycan and type II collagen content were determined by H&E and Toluidine Blue staining, and type II collagen antibody. The detailed structure of the chondrocytes in all groups was visualized using transmission electron microscopy (TEM). Chondrocytes were isolated from full-thickness knee cartilage of rabbits from all groups and their viscoelastic properties were quantified within 2 hours of isolation using a micropipette aspiration technique combined with a standard linear viscoelastic solid model. The components and network of the cytoskeleton within the cells were analyzed by laser scanning confocal microscopy (LSCM) with immunofluorescence staining as well as real time PCR and western blotting. With ageing, articular cartilage contained less chondrocytes and less proteoglycans and type II collagen. TEM observations showed that the cell membranes were not clearly defined, organelles were fewer and the nuclei were deformed or shrunk in the old cells compared with the young and adult cells. In suspension, chondrocytes from all three age groups showed significant viscoelastic creep behavior, but the deformation rate and amplitude of old chondrocytes were increased under the same negative pressure when compared to young and adult chondrocytes. Viscoelastic properties of the old cells, including equilibrium modulus (E infinity), instantaneous modulus (E0) and apparent viscosity (mu) were significantly lower than that those of the young and adult ones (P < 0.001). No significant differences were detected between young and adult chondrocytes (P > 0.05). Moreover, we found that the cytoskeletal networks of old cells were sparser, and that the contents of the various components of the intracellular networks were reduced in old cells, compared with adult and young cells. Aged chondrocytes had a different response to mechanical stimulation when compared to young and adult chondrocytes due to alteration of their viscoelastic properties, which was in turn associated with changes in cell structure and cytoskeleton composition.