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.     

The bulk elastic modulus and the reversible properties of cell walls in developing Quercus leaves

We examined the relationship between the bulk elastic modulus (epsilon) of an individual leaf obtained by the pressure-volume (P-V) technique and the mechanical properties of cell walls in the leaf. The plants used were Quercus glauca and Q. serrata, an evergreen and a deciduous broad-leaved tree species, respectively. We compared epsilon and Young's modulus of leaf specimens determined by the stretch technique at various stages of their leaf development. The results showed that epsilon increased from approximately 5 to 20 MPa during leaf development, although other potential determinants of epsilon such as the apoplastic water content in the leaf and the diameter of a palisade tissue cells remained almost constant. epsilon in these two species was similar at every developmental stages, although the apparent mechanical strength of the leaf lamina and thickness of mesophyll cell walls were greater in Q. glauca. There were significant linear relationships between Young's modulus and epsilon (P < 0.01; R (2) = 0.78 and 0.84 in Q. glauca and Q. serrata, respectively) with small y-intercepts. From these results, we conclude that epsilon is closely related to the reversible properties of the cell walls. From the estimation of epsilon based on a physical model, we suggest that the effective thickness of cell walls responsible for epsilon is smaller than the observed wall thickness.     

An extended modeling of the micropipette aspiration experiment for the characterization of the Young's modulus and Poisson's ratio of adherent thin biological samples: numerical and experimental studies

The micropipette aspiration (MA) experiment remains a quite widely used micromanipulation technique for quantifying the elastic modulus of cells and, less frequently, of other biological samples. However, moduli estimations derived from MA experiments are only valid if the probed sample is non-adherent to the rigid substrate. This study extends this standard formulation by taking into account the influence of the sample adhesion. Using a finite element analysis of the sample aspiration into the micropipette, we derived a new expression of the aspirated length for linear elastic materials. Our results establish that (i) below a critical value, the thickness h of the probed sample must be considered to get an accurate value of its Young's modulus (ii) this critical value depends both on the Poisson's ratio and on the sample adhesivity. Additionally, we propose a novel method which allows the computation of the intrinsic Young's modulus of the adherent probed sample from its measured apparent elasticity modulus. Thanks to the set of computational graphs we derived from our theoretical analysis, we successfully validate this method by experiments performed on polyacrylamide gels. Interestingly, the original procedure we proposed allows a simultaneous quantification of the Young's modulus and of the Poisson's ratio of the adherent gel. Thus, our revisited analysis of MA experiments extends the application domain of this technique, while contributing to decrease the dispersion of elastic modulus values obtained by this method.     

Bending elastic modulus of red blood cell membrane derived from buckling instability in micropipet aspiration tests.

Observation of cell membrane buckling and cell folding in micropipette aspiration experiments was used to evaluate the bending rigidity of the red blood cell membrane. The suction pressure required to buckle the membrane surface initially was found to be about one-half to two-thirds of the pressure that caused the cell to fold and move up the pipet. A simple analytical model for buckling of a membrane disk supported at inner and outer radii correlates well with the observed buckling pressures vs. pipet radii. The buckling pressure is predicted to increase in inverse proportion to the cube of the pipet radius; also, the buckling pressure depends inversely on the radial distance to the toroidal rim of the cell, normalized by the pipet radius. As such, the pressure required to buckle the membrane with 1 X 10(-4) cm diam pipet would be about four times greater than with a 2 X 10(-4) cm pipet. This is the behavior observed experimentally. Based on analysis of the observed buckling data, the membrane bending or curvature elastic modulus is calculated to be 1.8 X 10(-12) dyn-cm.     

A mechanobiological investigation of platelets

Understanding mechanotransduction pathways leading to thrombosis will require progressive steps, including determination of the mechanical behavior of the platelet membrane in response to applied loads. The platelet membrane deformation capacity, as quantified by membrane progression into a borosilicate glass micropipette of defined internal diameter, was probed in murine platelets using a controlled range of negative pressure (0–7 cm H₂O). Based on our observations that the platelet portion outside the micropipette was mostly spherical and that the platelet volume did not change upon aspiration, a novel continuum mechanics-based model of the platelet micropipette aspiration experiment was created, and a new hyperelastic isotropic material model including membrane residual tension was proposed for the platelet membrane. Murine platelet membranes maintained an average linear deformation behavior: L _p/R _p = 146,100p _i × R _p + 19.923, where L _p is the platelet length aspirated in the micropipette (m), R _p is micropipette radius (m) and p _i is the aspiration pressure (Pa). The theoretical model was used to generate material constants for the murine platelet membrane that allowed for an accurate simulation of the micropipette aspiration experiments. From published results, another set of material constants was established for the human platelet membrane. Limited cases of platelet lysis upon aspiration were analyzed using the theoretical model to determine preliminary membrane tension strength values.     

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.     

The application of a homogeneous half-space model in the analysis of endothelial cell micropipette measurements

Experimental studies have shown that endothelial cells which have been exposed to shear stress maintain a flattened and elongated shape after detachment. Their mechanical properties, which are studied using the micropipette experiments, are influenced by the level as well as the duration of the shear stress. In the present paper, we analyze these mechanical properties with the aid of two mathematical models suggested by the micropipette technique and by the geometry peculiar to these cells in their detached post-exposure state. The two models differ in their treatment of the contact zone between the cell and the micropipette. The main results are expressions for an effective Young's modulus for the cells, which are used in conjunction with the micropipette data to determine an effective Young's modulus for bovine endothelial cells, and to discuss the dependence of this modulus upon exposure to shear stress.     

The human erythrocyte membrane skeleton may be an ionic gel. III. Micropipette aspiration of unswollen erythrocytes

We have carried out a theoretical analysis of micropipette aspiration of unswollen erythrocytes using the protein-gel-lipid-bilayer membrane model and taking into account that the modulus of area compression of the membrane skeleton may depend on the environmental conditions. Our analysis shows that the aspiration pressure needed to obtain a certain membrane projection length is strongly dependent on the ratio between the membrane skeleton modulus of area compression and the elastic shear modulus. Our analysis therefore predicts that micropipette aspiration of unswollen erythrocytes may be a sensitive method for detection of changes in this ratio. The analysis thus also shows that micropipette aspiration of unswollen erythrocytes can not be used to determine the membrane shear modulus unless something is known about the membrane skeleton modulus of area compression.     

Determination of bilayer membrane bending stiffness by tether formation from giant, thin-walled vesicles.

The curvature elastic modulus (bending stiffness) of stearoyloleoyl phosphatidylcholine (SOPC) bilayer membrane is determined from membrane tether formation experiments. R. E. Waugh and R. M. Hochmuth 1987. Biophys. J. 52:391-400) have shown that the radius of a bilayer cylinder (tether) is inversely related to the force supported along its axis. The coefficient that relates the axial force on the tether to the tether radius is the membrane bending stiffness. Thus, the bending stiffness can be calculated directly from measurements of the tether radius as a function of force. Giant (10-50-microns diam) thin-walled vesicles were aspirated into a micropipette and a tether was pulled out of the surface by gravitational forces on small glass beads that had adhered to the vesicle surface. Because the vesicle keeps constant surface area and volume, formation of the tether requires displacement of material from the projection of the vesicle in the pipette. Tethers can be made to grow longer or shorter or to maintain equilibrium by adjusting the aspiration pressure in the micropipette at constant tether force. The ratio of the change in the length of the tether to the change in the projection length is proportional to the ratio of the pipette radius to the tether radius. Thus, knowing the density and diameter of the glass beads and measuring the displacement of the projection as a function of tether length, independent determinations of the force on the tether and the tether radius were obtained. The bending stiffness for an SOPC bilayer obtained from these data is approximately 2.0 x 10(-12) dyn cm, for tether radii in the range of 20-100 nm.(ABSTRACT TRUNCATED AT 250 WORDS)     

Estimation of cell Young's modulus of adherent cells probed by optical and magnetic tweezers: influence of cell thickness and bead immersion

A precise characterization of cell elastic properties is crucial for understanding the mechanisms by which cells sense mechanical stimuli and how these factors alter cellular functions. Optical and magnetic tweezers are micromanipulation techniques which are widely used for quantifying the stiffness of adherent cells from their response to an external force applied on a bead partially embedded within the cell cortex. However, the relationships between imposed external force and resulting bead translation or rotation obtained from these experimental techniques only characterize the apparent cell stiffness. Indeed, the value of the estimated apparent cell stiffness integrates the effect of different geometrical parameters, the most important being the bead embedding angle 2gamma, bead radius R, and cell height h. In this paper, a three-dimensional finite element analysis was used to compute the cell mechanical response to applied force in tweezer experiments and to explicit the correcting functions which have to be used in order to infer the intrinsic cell Young's modulus from the apparent elasticity modulus. Our analysis, performed for an extensive set of values of gamma, h, and R, shows that the most relevant parameters for computing the correcting functions are the embedding half angle gamma and the ratio h(u)/2R, where h(u) is the under bead cell thickness. This paper provides original analytical expressions of these correcting functions as well as the critical values of the cell thickness below which corrections of the apparent modulus are necessary to get an accurate value of cell Young's modulus. Moreover, considering these results and taking benefit of previous results obtained on the estimation of cell Young's modulus of adherent cells probed by magnetic twisting cytometry (MTC) (Ohayon, J., and Tracqui, P., 2005, Ann. Biomed. Eng., 33, pp. 131-141), we were able to clarify and to solve the still unexplained discrepancies reported between estimations of elasticity modulus performed on the same cell type and probed with MTC and optical tweezers (OT). More generally, this study may strengthen the applicability of optical and magnetic tweezers techniques by insuring a more precise estimation of the intrinsic cell Young's modulus (CYM).     

Micropipette aspiration method for characterizing biological materials with surface energy

Many soft biological tissues possess a considerable surface stress, which plays a significant role in their biophysical functions, but most previous methods for characterizing their mechanical properties have neglected the effects of surface stress. In this work, we investigate the micropipette aspiration method to measure the mechanical properties of soft tissues and cells with surface effects. The neo-Hookean constitutive model is adopted to describe the hyperelasticity of the measured biological material, and the surface effect is taken into account by the finite element method. It is found that when the pipette radius or aspiration length is comparable to the elastocapillary length, surface energy may distinctly alter the aspiration response. Generally, both the aspiration length and the bulk normal stress decrease with increasing surface energy, and thus neglecting the surface energy would lead to an overestimation of elastic modulus. Through dimensional analysis and numerical simulations, we provide an explicit relation between the imposed pressure and the aspiration length. This method can be applied to determine the mechanical properties of soft biological tissues and organs, e.g., livers, tumors and embryos.     

Estimation of viscous dissipation inside an erythrocyte during aspirational entry into a micropipette.

Viscous dissipation inside the erythrocyte during its aspirational entry into a micropipette is analyzed. The motion of the intracellular fluid is approximated by a flow into the micropipette orifice from a half space (the portion of the erythrocyte outside the micropipette). The stream function and intracellular pressure (p) in the half space are obtained as a function of radial and axial positions near the orifice. Solution of the boundary value problem for a uniform stream entering a circular hole gives p = 2 eta HQ/pi R3p, where eta H is the intracellular viscosity, Q is the total discharge, and Rp is the pipette radius. The results indicate that the moving erythrocyte membrane helps to drive the intracellular fluid into the orifice. For normal erythrocytes, p is only approximately 0.5% of the total aspiration pressure (delta P). The contribution of p to delta P, however, may become significant when there is a large increase in eta H due to a markedly elevated intracellular hemoglobin concentration or an alteration of the physical state of hemoglobin.     

New Membrane Concept Applied to the Analysis of Fluid Shear- and Micropipette-Deformed Red Blood Cells

A two-dimensional elastomer material concept of the red cell membrane is applied to the analysis of fluid shear-deformed, point-attached red cells and micropipette aspiration of red cell disks. The elastic constant (corresponding to the “shear” modulus multiplied by the membrane thickness) is of the order 10^-2 dyn/cm for both cases. Additional experimental observations are in agreement with the membrane model, e.g. teardrop and “tether” formation of the sheared disks, pressure difference vs. aspirated length of the cell for micropipette experiments, etc     

Alterations in the mechanical properties of the human chondrocyte pericellular matrix with osteoarthritis

In articular cartilage, chondrocytes are surrounded by a pericellular matrix (PCM), which together with the chondrocyte have been termed the "chondron." While the precise function of the PCM is not know there has been considerable speculation that it plays a role in regulating the biomechanical environment of the chondrocyte. In this study, we measured the Young's modulus of the PCM from normal and osteoarthritic cartilage using the micropipette aspiration technique, coupled with a newly developed axisymmetric elastic layered half-space model of the experimental configuration. Viable, intact chondrons were extracted from human articular cartilage using a new microaspiration-based isolation technique. In normal cartilage, the Young's modulus of the PCM was similar in chondrons isolated from the surface zone (68.9 +/- 18.9 kPa) as compared to the middle and deep layers (62.0 +/- 30.5 kPa). However, the mean Young's modulus of the PCM (pooled for the two zones) was significantly decreased in osteoarthritic cartilage (66.5 +/- 23.3 kPa versus 41.3 +/- 21.1 kPa, p < 0.001). In combination with previous theoretical models of cell-matrix interactions in cartilage, these findings suggest that the PCM has an important influence on the stress-strain environment of the chondrocyte that potentially varies with depth from the cartilage surface. Furthermore, the significant loss of PCM stiffness that was observed in osteoarthritic cartilage may affect the magnitude and distribution of biomechanical signals perceived by the chondrocytes.     

Cell poking. Determination of the elastic area compressibility modulus of the erythrocyte membrane

Cell poking, a new method for measuring mechanical properties of single cells was used to determine the elastic area compressibility modulus of osmotically swollen human erythrocytes. With this method we determined the force required to indent cells attached to a glass coverslip (Petersen, N.O., W. B. McConnaughey , and E. L. Elson , 1982, Proc. Natl. Acad. Sci. USA, 79:5327. Forces on the order of one millidyne and indentations on the order of one micron were detected. An analysis of these data in terms of a simplified mechanical model yielded the elastic area compressibility modulus. This analysis used a variational approach to minimize the isothermal elastic potential energy density function given by E. A. Evans and R. Skalak (Mechanics and Thermodynamics of Biomembranes, 1980, CRC Press, Boca Raton , FL). Measurements on swollen erythrocytes gave a range of values, depending in part on the osmotic conditions, of 17.9 +/- 8.2 to 34.8 +/- 12.0 mdyn /micron for the elastic area compressibility modulus at 25 degrees C. Fractional area expansion greater than 2.6 +/- 0.8% produced rapid cell lysis. These values were not corrected for the reversible movement of water across the cell membrane in response to hydrostatic pressure gradients. Our results agree reasonably with those obtained by Evans et al. (Evans, E.A., R. Waugh , and L. Melnick , 1976, Biophys. J., 16:585-595.) using micropipette aspiration under similar conditions.     

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.     

Red cell membrane elasticity as determined by flow channel technique.

The elasticity of red cell membrane was determined in a rectangular flow channel under controlled shear flow. The relation between shear stress and cell extension ratio (lambda) has been analyzed with the use of Evans' two-dimensional model. The deformed cell shapes observed experimentally agreed well with the model with lambda up to 1.4. The best correlation was found at lambda = 1.2. The analysis suggests a nonlinear extensional membrane modulus in the low stress range encountered in the flow channel. In terms of an appropriate strain parameter, the elastic modulus is shown to rise toward the level encountered in micropipette aspiration experiments. The implications of the present findings in modeling of cell mechanics and in cell hemolysis are discussed.     

An extended relationship for the characterization of Young's modulus and Poisson's ratio of tunable polyacrylamide gels

Substrates with tunable mechanical properties are crucial for the study of cellular processes, and polyacrylamide gels (PAGs) are frequently used in this context. Several experimental techniques have been proposed to obtain the mechanical properties of PAGs. However, the range of the considered Poisson's ratio values remains quite large and no attempt has been made to propose an analytical relationship allowing the estimation of PAG Young's modulus when both bis-acrylamide and acrylamide concentrations are known. In order to complete the actual knowledge on the mechanical properties of PAGs, we took benefit of our original method based on the micropipette aspiration technique (Boudou et al., J. Biomech. 2006) for characterizing gels made with concentrations in the range 0.02% < or =[Bis]< or =0.20% and 3% < or =[Acry]< or =10%. We found that the PAGs Young's modulus varies nonlinearly with the acrylamide amount. Moreover, our study validates the quasi-incompressibility hypothesis usually made in studies using PAGs (mean Poisson's ratio of 0.480+/-0.012). More generally, and in agreement with data published by other groups, we propose an original nonlinear mathematical relationship allowing the computation of Young's modulus of PAG for any given acrylamide and bis-acrylamide amounts taken in the range of values we considered.     

Is the surface area of the red cell membrane skeleton locally conserved?

The incompressibility of the lipid bilayer keeps the total surface area of the red cell membrane constant. Local conservation of membrane surface area requires that each surface element of the membrane skeleton keeps its area when its aspect ratio is changed. A change in area would require a flow of lipids past the intrinsic proteins to which the skeleton is anchored. in fast red cell deformations, there is no time for such a flow. Consequently, the bilayer provides for local area conservation. In quasistatic deformations, the extent of local change in surface area is the smaller the larger the isotropic modulus of the skeleton in relation to the shear modulus. Estimates indicate: (a) the velocity of relative flow between lipid and intrinsic proteins is proportional to the gradient in normal tension within the skeleton and inversely proportional to the viscosity of the bilayer; (b) lateral diffusion of lipids is much slower than this flow; (c) membrane tanktreading at frequencies prevailing in vivo as well as the release of a membrane tongue from a micropipette are fast deformations; and (d) the slow phase in micropipette aspiration may be dominated by a local change in skeleton surface.