Validation of the Boussinesq equation for use in traction field determination

Cell traction force plays an important role in many biological processes. Several traction force microscopy methods have been developed to determine cell traction forces based on the Boussinesq solution. This approach, however, is rooted in a half-space assumption. The purpose of this study was to determine the error induced in the half-space assumption using a finite element method (FEM). It demonstrates that displacement error between the FEM and the Boussinesq equation can be used to measure the accuracy of the Boussinesq equation, although singularity exists in the loading point. For one concentrated force, significant difference between the FEM and the Boussinesq equation occurs in the whole field; this difference decreases with an increase in the plate thickness. However, in the case of the balanced forces, the offset of the balanced forces decreases the errors in the middle area. Overall, this study demonstrates that increasing the thickness of the polyacrylamide gel is important for reducing the error of the Boussinesq equation when determining the displacement field of the gel under loads.     

Analytic study of electron flows with a virtual cathode

Abstrac The formation of a virtual cathode is studied with one-dimensional analytic self-consistent dynamic models describing the pulsed injection of an electron beam into equipotential regions: half-space or a planeparallel gap. __________ Translated from Fizika Plazmy, Vol. 26, No. 5, 2000, pp. 439–444. Original Russian Text Copyright ? 2000 by Dubinov.     

Finite-Difference Time-Domain Simulator for Half-Space Bioacoustic Problems

The finite-difference time-domain (FDTD) method provides a numerical technique for solving acoustic forward problems. The FDTD is particularly useful in geometries that encompass heterogeneities. Plane-wave modeling permits the simulation of many practical bioacoustic problems because the phase front in the focal zone of many acoustic transducers is nearly planar. The formulation presented in this paper provides a technique for solving half-space problems under local plane-wave illumination.     

Earthquakes site effects modeling by hybrid MS-BIEM: the case study of Sofia, Bulgaria

The study solves 2D elastodynamic model for seismic in-plane wave propagation in laterally inhomogeneous geological profiles situated in a vertically inhomogeneous half-space with a seismic source. For the aim, an efficient hybrid Modal Summation-Boundary Integral Equation Method (MSM-BIEM) is applied. The MSM is used as a tool for the simulation of wave propagation from the source position to the local multilayered laterally inhomogeneous geological profile, where the BIEM is applied. The BIEM is based on the frequency-dependent fundamental solution of the governing equation in elastodynamics and the hybrid method works in the frequency domain. The inverse FFT solution is applied to obtain time histories. The hybrid tool is applied to several models for the investigation of local site effects due to: (a) the impedance contrasts between soil layers, (b) surface topography and lateral inhomogeneity, (c) the seismic source properties and (d) the existence of water saturation in soils. The application of the modeling tool is a contribution to the seismic risk analysis of Sofia city.     

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.     

Electromagnetic pulse propagation in dispersive planar dielectrics

The responses of a plane-wave pulse train irradiating a lossy dispersive dielectric half-space are investigated. The incident pulse train is expressed as a Fourier series with summing done by the inverse fast Fourier transform. The Fourier series technique is adopted to avoid the many difficulties often encountered in finding the inverse Fourier transform when transform analyses are used. Calculations are made for propagation in pure water, and typical waveforms inside the dielectric half-space are presented. Higher harmonics are strongly attenuated, resulting in a single continuous sinusoidal waveform at the frequency of the fundamental depth in the material. The time-averaged specific absorption rate (SAR) for pulse-train propagation is shown to be the sum of the time-averaged SARs of the individual harmonic components of the pulse train. For the same average power, calculated SARs reveal that pulse trains generally penetrate deeper than carrier-frequency continuous waves but not deeper than continuous waves at frequencies approaching the fundamental of the pulse train. The effects of rise time on the propagating pulse train in the dielectrics are shown and explained. Since most practical pulsed systems are very limited in bandwidth, no pronounced differences between their response and continuous wave (CW) response would be expected. Typical results for pulse-train propagation in arrays of dispersive planar dielectric slabs are presented. Expressing the pulse train as a Fourier series provides a practical way of interpreting the dispersion characteristics from the spectral point of view.     

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.     

Transport calculations for light scattering in blood.

In vivo measurement of the oxygen saturation levels in blood may be obtained from relative amounts of backscattered monochromatic light at two different wavelengths, as measured with a fiber-optic catheter oximeter. Because of the short mean free path length of light in blood, the backscattering can be well approximated by a previously-derived, one-wavelength transport theory solution for the half-space searchlight problem. This solution, unlike simple diffusion approximations has the advantage that the boundary condition describing illumination of a localized area of blood by a monodirectional light beam can be rigorously satisfied. Sample calculations using the solution are compared with experimental values of the reflectance of blood.     

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.     

The effects of layer properties on shear disturbance propagation in skin

The effects of the stratum corneum and dermis on shear wave propagation along the skin surface was investigated using a mathematical model. The skin was modeled as two distinct viscoelastic layers, one representing the stratum corneum and the other representing the dermis. The layers were supported by a semi-infinite visco-elastic half-space representing the subcutaneous fat. Physical and mechanical properties of the materials in the model were determined from the literature and from our own experimental measurements. Although the stratum corneum is very thin (12-15 microns), results showed that it could have a strong effect on the wave propagation due to its high stiffness relative to the dermis. Results of the analysis are discussed with respect to an experimental procedure used to determine age-related changes in mechanical properties of skin.     

Role of the Ridley-Watkins-Hilsum mechanism in the stabilization of surface plasma waves

The stabilization of unstable harmonic oscillations of the surface charge density and electromagnetic field that are excited by a low-density electron beam propagating parallel to the plane boundary of a half-space occupied by gallium arsenide (GaAs) is studied in the hydrodynamic approximation. It is shown that, for gallium arsenide semiconductors (and, generally, for other so-called A^IIIB^V compound semiconductors), these oscillations are stabilized primarily by the Ridley-Watkins-Hilsum mechanism. The equilibrium field amplitudes and equilibrium surface charge densities of both the semiconductor and the beam are obtained for a slightly overcritical level, i.e., for electrons with energies slightly greater than the energy gap between the lower and higher valleys.     

A simple method to estimate the exponential material parameters of heart valve tissue based on analogy between uniaxial tension and micropipette aspiration

Micropipette aspiration (MA) has been widely used to measure the biomechanical properties of cells and biomaterials. To estimate material parameters from MA experimental data, analytical half-space models and inverse finite element (FE) analyses are typically used. The half-space model is easy to implement but cannot account for nonlinear material properties and complex geometrical boundary conditions that are inherent to MA. Inverse FE approaches can account for geometrical and material nonlinearities, but their implementation is resource-intensive and not widely available. Here, by making analogy between an analytical uniaxial tension model and a FE model of MA, we proposed an easily implementable and accurate method to estimate the material parameters of tissues tested by MA. We first adopted a strain invariant-based isotropic exponential constitutive model and implemented it in both the analytical uniaxial tension model and the FE model. The two models were fit to experimental data generated by MA of porcine aortic valve tissue (45 spots on four leaflets) to estimate material parameters. We found no significant differences between the effective moduli estimated by the two models ( $p > 0.39$ ), with the effective moduli estimated by the uniaxial tension model correlating significantly with those estimated by the FE model ( $p < 0.001; R^{2}= 0.96$ ) with a linear regression slope that was not different than unity ( $p = 0.38$ ). Thus, the analytical uniaxial tension model, which avoids solving resource-intensive numerical problems, is as accurate as the FE model in estimating the effective modulus of valve tissue tested by MA.     

Determining substrate displacement and cell traction fields--a new approach

This paper presents a new approach for the traction force microscopy (TFM) method which determines traction forces exerted by adherent cells on a thin, elastic polyacrylamide gel embedded with fluorescent microbeads. In this enhanced TFM method, a pattern recognition technique is first applied to match the pair of microbead embedded images before and after deformation, which subsequently provides the displacement field of the elastic substrate. Once the displacement field is obtained, the 3-D finite element method (FEM) is used to compute cell traction forces. The new TFM has been applied to determine traction forces of human tendon fibroblasts. Compared to existing TFM methods, the present method has the following advantages: (1) its displacement field obtained is associated with microbead movements; (2) it considers the finite thickness of the thin polyacrylamide gel and is therefore free from the infinite half-space approximation adopted by existing TFM methods; and (3) its computation procedure for determining cell traction forces is fast.     

Dissipation of Energy of an AC Electric Field in a Semi-Infinite Electron Plasma with Mirror Boundary Conditions

Absorption of the electromagnetic energy in a semi-infinite electron plasma is calculated for an arbitrary degree of the electron gas degeneracy. Absorption is determined by solving the boundary-value problem on the oscillations of electron plasma in a half-space with mirror boundary conditions for electrons. The Vlasov?Boltzmann kinetic equation with the Bhatnagar–Gross–Krook collision integral for the electron distribution function and Maxwell’s equation for the electric field are employed. The electron distribution function and the electric field inside plasma are searched for in the form of expansions in the eigenfunctions of the initial set of equations. The expansion coefficients are found for the case of mirror boundary conditions. The contribution of the plasma surface to absorption is analyzed. Cases with different degrees of electron gas degeneracy are considered. It is shown that absorption of the electromagnetic energy near the surface depends substantially on the ratio between the electric field frequency and the volumetric electron collision frequency.     

On atomic force microscopy and the constitutive behavior of living cells

Atomic force microscopy (AFM) is one of many new technologies available to study the mechanical properties and mechanobiological responses of living cells. Despite the widespread usage of this technology, there has been little attempt to develop new theoretical frameworks to interpret the associated data. Rather, most analyses rely on the classical Hertz solution for the indentation of an elastic half-space within the context of linearized elasticity. In contrast, we propose a fully nonlinear, constrained mixture model for adherent cells that allows one to account separately for the contributions of the three primary structural constituents of the cytoskeleton. Moreover, we extend a prior solution for a small indentation superimposed on a finite equibiaxial extension by incorporating in this mixture model for the special case of an initially random distribution of constituents (actin, intermediate filaments, and microtubules). We submit that this theoretical framework will allow an improved interpretation of indentation force-depth data from a sub-class of atomic force microscopy tests and will serve as an important analytical check for future finite element models. The latter will be necessary to exploit further the capabilities of both atomic force microscopy and nonlinear mixture theories for cell behavior.     

Surface deflection of primate fingertip under line load

A study of the biomechanics of the skin and the subcutaneous soft tissues is of fundamental importance in understanding the process of transduction at the mechanoreceptive nerve terminals responsible for the sense of touch. In the present investigation, the fingertips (distal phalanges) of three adult humans and four monkeys were indented in vivo using a line load delivered by a sharp wedge. The resulting skin surface deflection profile was photographed and used as a clue to infer the mechanical nature of the materials that make up the fingertip. It is shown that the modified Boussinesq solution used by Phillips and Johnson (1981), applicable when the fingertip is modeled as an elastic half-space in a state of plane strain, predicts a skin surface deflection profile that can only roughly approximate the empirically observed profiles. As an alternative, a simple model which views the fingertip as an elastic membrane filled with an incompressible fluid (like a 'waterbed') under plane strain conditions is proposed. It is shown that the predictions of this model, which takes into account the finite deformations that occur, agree very well with the photographed profiles in the region of interest (up to about 3 mm from the load).     

Rolling adhesion of leukocytes on soft substrates: Does substrate stiffness matter?

Cell rolling on vascular endothelium under hydrodynamic blood flow is critical for realization of many physiological and pathological processes, such as inflammatory response and tumor metastasis. The blood-borne cells are in direct contact with the inner layer of endothelium, formed by a highly compliant layer of endothelial cells. The effect of endothelial stiffness on the adhesion and motion of rolling cells is poorly understood. Inspired by recent in vitro studies, here we implemented a computational method to model the specific adhesion of a rolling cell onto a soft substrate, subjected to a creeping shear flow. The substrate is modeled as an elastic half-space, coated with P- and E-selectin receptors with specific affinity for the complementary ligands located on the moving cell. Of particular importance is to predict the effect of substrate stiffness on cell adhesion and its kinematics and kinetics of motion. Simulation results show that the effect of substrate compliance is minimal when coated with P-selectin. Conversely, the trajectory of rolling cells on E-selectin coated substrates is sensitive to the substrate compliance. This is attributed to the moderation of binding forces applied by the soft substrate which leads to a higher average translational velocity of cells.     

Torsional Tribological Behavior and Torsional Friction Model of Polytetrafluoroethylene against 1045 Steel

In this work, the plane-on-plane torsional fretting tribological behavior of polytetrafluoroethylene (PTFE) was studied. A model of a rigid, flat-ended punch acting on an elastic half-space was built according to the experimental conditions. The results indicate that the shape of T–θ curves was influenced by both the torsional angle and the normal load. The torsion friction torque and wear rate of PTFE exponentially decreased when the torsion angle rose. The torsional torque increased from 0.025 N·m under a normal load of 43 N to 0.082 N·m under a normal load of 123 N. With sequentially increasing normal load, the value of torque was maintained. With rising normal load, the wear mass loss of PTFE disks was increased and the wear rate was decreased. Good agreement was found with the calculated torque according to the model and the experimental torque except for that under a normal load of 163 N. The difference under a normal load of 163 N was caused by the coefficient of friction. Usually the coefficient of friction of a polymer decreases with increasing normal load, whereas a constant coefficient of friction was applied in the model.     

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.     

Flat medial-lateral conformity in total knee replacements does not minimize contact stresses.

The potential for wear in UHMWPE components for total knee replacements can be reduced by decreasing the stresses and strains arising from tibial-femoral contact. The conformity of the articular surfaces has a large effect on the resultant stresses, and components that achieve flat medial-lateral contact have been assumed to produce the lowest stresses due to their perfect conformity. We computed the stresses arising from curved and flat contact on a half-space using two-dimensional, plane strain elasticity solutions and finite element analyses to compare the performance of curved and flat indenters. These indenters were represented by a polynomial so the profiles could be continuously varied from curved to flat. Curved contact resulted in maximum stresses at the center of contact, while flat contact produced maximum stresses at the edge of contact. In addition, three contemporary tibial configurations (flat-on-flat, curved-on-flat, and curved-on-curved geometries) were analyzed using the finite element method with nonlinear material properties. The maximum contact stress, von Mises stress, and von Mises strain were lowest for the curved-on-curved model. The other configurations resulted in higher contact stresses, von Mises stresses, and von Mises strains. The perfect conformity arising from flat contact did not reduce the contact stresses in the UHMWPE component. The tensile stresses, however, were lowest for the flat-on-flat geometry compared with the other two configurations. Relating these distinct differences could prove useful in interpretation of data from simulator and retrieval studies.