Elastic and viscoelastic properties of porcine subdermal fat using MRI and inverse FEA

There is a scarcity of investigation into the mechanical properties of subdermal fat. Recently, progress has been made in the determination of subdermal stress and strain distributions. This requires accurate constitutive modelling and consideration of the subdermal tissues. This paper reports the results of a study to estimate non-linear elastic and viscoelastic properties of porcine subdermal fat using a simple constitutive model. High-resolution magnetic resonance imaging (MRI) was used to acquire a time series of coincident images during a confined indentation experiment. Inverse finite element analysis was used to estimate the material parameters. The Neo Hookean model was used to represent the elastic behaviour (μ = 0.53 ± 0.31 kPa), while a single-element Prony series was used to model the viscoelastic response (α = 0.39 ± 0.03, τ = 700 ± 255 s).     

The prediction of stress-relaxation of ligaments and tendons using the quasi-linear viscoelastic model

Recent studies have questioned the ability of the quasi-linear viscoelastic (QLV) model to predict stresses and strains in response to loading conditions other than those used to fit the model. The objective of this study was to evaluate the ability of several models in the literature to predict the elastic stress response of ligament and tendon at strain levels higher than the levels used to fit the model. The constitutive models were then used to evaluate the ability of the QLV model to predict the overall stress response during stress relaxation. The models expressing stress as an exponential function of strain significantly overestimated stress when used at higher strain levels. The polynomial formulation of the Mooney–Rivlin model more accurately predicted the stress–strain behavior of ligament and tendon. The results demonstrate that the ability of the QLV model to accurately predict the stress-relaxation response is dependent in part on the accuracy of the function used to model the elastic response of the soft tissue.     

Growth-induced buckling of an epithelial layer

We use a proof-of-concept experiment and two mathematical models to explore growth-induced tissue buckling, as may occur in colorectal crypt formation. Our experiment reveals how growth of a cultured epithelial monolayer on a thin flexible substrate can cause out-of-plane substrate deflections. We describe this system theoretically using a ‘bilayer’ model in which a growing cell layer adheres to a thin compressible elastic beam. We compare this with the ‘supported-monolayer’ model due to Edwards and Chapman (Bull Math Biol 69:1927–1942, 2007) for an incompressible expanding beam (representing crypt epithelium), which incorporates viscoelastic tethering to underlying stroma. We show that the bilayer model can exhibit buckling via parametric growth (in which the system passes through a sequence of equilibrium states, parameterised by the total beam length); in this case, non-uniformities in cell growth and variations in cell–substrate adhesion are predicted to have minimal effect on the shape of resulting buckled states. The supported-monolayer model reveals how competition between lateral supports and stromal adhesion influences the wavelength of buckled states (in parametric growth), and how non-equilibrium relaxation of tethering forces influences post-buckled shapes. This model also predicts that non-uniformities in growth patterns have a much weaker influence on buckled shapes than non-uniformities in material properties. Together, the experiment and models support the concept of patterning by growth-induced buckling and suggest that targeted softening of a growing cell layer provides greater control in shaping tissues than non-uniform growth.     

Influence of boundary conditions on computed apparent elastic properties of cancellous bone

High-resolution finite element models of trabecular bone can be used to study trabecular structure–function relationships, elasticity, multiaxial strength, and tissue remodelling in more detail than experiments. Beside effects of the model size, scan/analysis resolution, segmentation process, etc., the type of the applied boundary conditions (BCs) have a strong influence on the predicted elastic properties. Appropriate BCs have to be applied on hexahedral digital finite element models in order to obtain effective elastic properties. Homogeneous displacement BCs as proposed by Van Rietbergen et al. (J Biomech 29(12):1653–1657, 1996) lead to “apparent” rather than to “effective” elastic properties. This study provides some answers concerning such differences by comparing various BC types (uniform displacement, mixed BCs, periodic BCs), different volume element definitions (original and mirrored models), and several bone volume fractions (BVTV ranging from 6.5 to 37.6%). First, the mixed BCs formulated by Hazanov (Arch Appl Mech 68(6):385–394, 1998) are theoretically extended to shear loading of a porous media. Second, six human bone samples are analyzed, their orthotropic Young’s moduli, shear moduli, and Poisson’s ratios computed and compared. It is found that the proposed mixed BCs give exactly the same effective elastic properties as periodic BCs if a periodic and orthotropic micro-structured material is used and thus denoted as “periodicity compatible” mixed uniform BCs (PMUBCs). As bone samples were shown to be nearly orthotropic for volume element side lengths ≥5 mm the proposed mixed BCs turn out to be the best choice because they give again essentially the same overall elastic properties as periodic BCs. For bone samples of smaller dimensions ( < 5 mm) with a strong anisotropy (beyond orthotropy) uniform displacement BCs remain applicable but they can significantly overestimate the effective stiffness. In Memoriam, Prof. Christian Huet.     

Physical properties of axenic maize root mucilage

Root mucilage was collected from 3–4 day-old axenically-grown maize seedlings (Zea mays L. cv. Freya). The water potential of the hydrated mucilage was measured by thermocouple psychrometry and the rheology at low deformation rates was studied using an oscillating cone and plate rheometer which provides information on both the elastic and viscous components of its behaviour. Water potential decreased as mucilage solute concentration increased, reaching a value of −60kPa at 1.2 mg mL⁻¹. At the lowest oscillation rate, the mucilage had a dynamic viscosity of 145 mPa s and behaved as a weak viscoelastic gel. After filtration to remove suspended root cap cells and other solid plant material, mucilage viscosity was reduced to 5–10 mPa s at low oscillation rates and the behaviour was that of a viscous liquid. The decrease in viscosity which occurs on filtration indicates that the root cap cells form an integral part of the gel system, either by interacting directly with each other or via the polysaccharide. Our observations provide further support for the idea that mucilage plays a major role in maintaining root-soil contact in the rhizosphere. This revised version was published online in June 2006 with corrections to the Cover Date.     

Thermal aggregation and ion-induced cold-gelation of bovine serum albumin

Protein cold-gelation has recently received particular attention for its relevance in bio and food technology. In this work, we report a study on bovine serum albumin cold-gelation induced by copper or zinc ions. Metal-induced cold-gelation of proteins requires two steps: during the first one, the heat treatment causes protein partial unfolding and aggregation; then, after cooling the solution to room temperature, gels are formed upon the addition of metal ions. The thermally induced behaviour has been mainly investigated through different techniques: Fourier transform infrared (FTIR) spectroscopy, circular dichroism, dynamic light scattering (DLS) and rheology. Data have shown that the aggregation process is mainly due to protein conformational changes—α-helices into β-aggregates—forming small aggregated structures with a mean diameter of about 20 nm a few minutes after heating. After metal ion addition, the viscoelastic properties of the gels have been investigated by rheological measurements. The behaviour of the elastic and viscous moduli as a function of time is discussed in terms of ion concentration and type. Our results show that: (1) the elastic behaviour depends on ion concentration and (2) at a given ion concentration, gels obtained in the presence of zinc exhibit an elastic value larger than that observed in the Cu²⁺ case. Data suggest that cold-gelation is the result of different mechanisms: the ion-mediated protein–protein interaction and the bridging effect due to the presence of divalent ions in solution.     

Energy model for contrast detection: spatiotemporal characteristics of threshold vision

A model for contrast detection of spatiotemporal stimuli is proposed which consists of a spatiotemporal linear filter, an energy device and a threshold device. Assuming the existence of independent intrinsic noise, the probability of stimulus detection was approximated by a Weibull function of the response energy. With this assumption, the stimulus energy is a constant at fixed detection probability. This energy model for contrast detection satisfactorily accounted for the elliptical threshold contours of line pairs at stimulus separations within the range 2–30 min and at stimulus onset asynchronies within the range 20–140 ms. The threshold contour at a large stimulus onset asynchrony (300 ms) was in the form of a rounded square. This finding was explained by assuming that the probability of seeing the line pair was determined by the joint probability that at least one stimulus had been detected. With the energy model, the temporal and spatial autocorrelation functions of the response to a flashed line were evaluated. The autocorrelation functions thus determined were used to predict the temporal contrast sensitivity function to a flickering line stimulus and the spatial contrast sensitivity function to flashed gratings, which were in agreement with the experimental data. The data obtained were fitted adequately by an impulse response approximated by a spatiotemporal Gabor-like function. Received: 08 December 1997 / Accepted in revised form: 26 January 1999     

Functional properties of fresh and cryopreserved carotid and femoral arteries,and of venous and synthetic grafts: comparison with arteries from normotensive and hypertensive patients

The ideal arterial graft must share identical functional properties with the host artery. Surgical reconstruction of the common carotid artery (CA) is performed in several clinical situations, using expanded polytetrafluoroethylene prosthesis (ePTFE) or saphenous vein (SV) grafts. At date there is interest in obtaining an arterial graft that improves the results of that nowadays available. The use of a fresh or cryopreserved/defrosted artery appears as an interesting alternative. However, if the fresh and cryopreserved/defrosted arteries allow an adequate viscoelastic and functional matching with the host arteries needs to be established. The aims were to compare the viscoelastic and functional performance of: (1) conduits used in CA reconstruction (SV and ePTFE) with those of the fresh and cryopreserved/defrosted CA and femoral arteries (FA), and (2) normotensive and hypertensive patients’ arteries with those of the arterial substitutes in vitro analyzed. Pressure, diameter and wall thickness of the CA were recorded in 15 normotensive and 15 hypertensive patients (in vivo studies), and in SV, fresh and cryopreserved/defrosted CA and FA (obtained from 15 donors), and ePTFE segments (in vitro studies). From stress–strain relationship we calculated elastic and viscous modulus, and the characteristic impedance. The local buffer and conduit functions were quantified as the viscous/elastic quotient and the inverse of the characteristic impedance. Fresh and cryopreserved/defrosted CA and FA were more alike, both in viscoelastic and functional levels, respect to normotensive and hypertensive patients’ arteries, than the ePTFE and SV grafts. CA and FA cryografts could be considered an important alternative for carotid reconstruction.     

Mechanical characteristics of synthetic polyelectrolyte gel as a physical model of the cytoskeleton

A physical model of the cytoskeleton based on synthetic polyelectrolyte hydrogel of polymethacrylic acid has been proposed. From the physicochemical point of view, the structures of polyelectrolyte gel and the cytoskeleton show a high degree of similarity. It has been shown that polyelectrolyte gel can shorten and produce mechanical stress in response to changes in the composition of the surrounding solution. The mechanical properties of the model gel have been evaluated: Young modulus (2–6 kPa), stress relaxation time (0.1–1 s), and apparent viscosity (0.3–3 kPa s). The viscoelastic properties of the gel depend on the degree of its swelling. It has been demonstrated that the mechanical properties of gels of polymethacrylic acid are close to those of biological objects.     

Nonlinear viscoelastic, thermodynamically consistent, models for biological soft tissue

The mechanical behavior of most biological soft tissue is nonlinear viscoelastic rather than elastic. Many of the models previously proposed for soft tissue involve ad hoc systems of springs and dashpots or require measurement of time-dependent constitutive coefficient functions. The model proposed here is a system of evolution differential equations, which are determined by the long-term behavior of the material as represented by an energy function of the type used for elasticity. The necessary empirical data is time independent and therefore easier to obtain. These evolution equations, which represent non-equilibrium, transient responses such as creep, stress relaxation, or variable loading, are derived from a maximum energy dissipation principle, which supplements the second law of thermodynamics. The evolution model can represent both creep and stress relaxation, depending on the choice of control variables, because of the assumption that a unique long-term manifold exists for both processes. It succeeds, with one set of material constants, in reproducing the loading-unloading hysteresis for soft tissue. The models are thermodynamically consistent so that, given data, they may be extended to the temperature-dependent behavior of biological tissue, such as the change in temperature during uniaxial loading. The Holzapfel et al. three-dimensional two-layer elastic model for healthy artery tissue is shown to generate evolution equations by this construction for biaxial loading of a flat specimen. A simplified version of the Shah-Humphrey model for the elastodynamical behavior of a saccular aneurysm is extended to viscoelastic behavior.     

Estimating absolute pollen productivity for some European Tertiary-relict taxa

Tertiary-relict plants are survivors from the pre-Quaternary periods. Today, most European Tertiary relicts are confined to small, isolated stands distributed in the Mediterranean and Black Sea regions. In the past, however, the fossil record indicates that these species were probably distributed over large parts of the European continent and may have been important constituents of the vegetation. Little is known about their pollen representation, which limits our ability to reconstruct this past vegetation with any accuracy. This paper draws on the results of pollen trapping experiments in Bulgaria and Georgia, where relict stands of Aesculus hippocastanum, Cercis siliquastrum, Fagus orientalis, Juglans regia and Pterocarya fraxinifolia are still in existence. We compared average pollen accumulation rates (PAR) to vegetation data from around the trapping locations to derive estimates of absolute pollen productivity using various pollen dispersal functions. Composite dispersal functions that model pollen components carried above the vegetation canopy and falling as rain provided better relationships between PAR and plant abundance than functions that consider only a single component or the ‘trunk-space’ component carried under the canopy. A composite dispersal function with a simple model for regional pollen and the best overall correlation statistics gave the following estimates of absolute pollen productivity (grains cm⁻² yr⁻¹ with 1 SE intervals): Carpinus betulus 19,000–28,700; Fagus orientalis 15,600–20,400; Juglans regia 27,200–36,200; Pterocarya fraxinifolia 182,000–192,600; Quercus spp. 21,700–24,800; Tilia begoniifolia 51,600–68,300; and T. tomentosa 14,700–18,200. These estimates were applied to fossil data from the Black Sea coast to reconstruct palaeovegetation using absolute and relative methods.     

Influence of a Superficial Tangential Zone Over Repairing Cartilage Defects: Implications for Tissue Engineering

The superficial tangential zone (STZ) plays a critical role in normal cartilage function but is not yet a focus for designing tissue-engineered constructs for cartilage repair. Without material properties of sufficient quality in this zone, transplanted constructs in vivo may have little chance of survival. This finite element study investigates the impact of the superficial tangential zone on the mechanical function of normal articular surfaces as well as those with transplanted constructs. The zone is modeled as a thin transversely isotropic material with strain dependent permeability. The analyses predict that a normal transversely isotropic STZ placed over a repair region reduces the axial compression (55–68%) of, and the rate of fluid loss (45–82%) from the articular surface. A reduction was also found in von Mises stress (26–57%), axial strain (22–56%), and radial strain (69–73%), and an increase in fluid pressure (19–45%) in repair tissue under the STZ. Incorporating a quality superficial tangential zone in tissue-engineered constructs may be a critical factor in achieving mechanical environments conducive for successful cartilage repairs.     

Characterization of human passive muscles for impact loads using genetic algorithm and inverse finite element methods

The objective of this study is to identify the dynamic material properties of human passive muscle tissues for the strain rates relevant to automobile crashes. A novel methodology involving genetic algorithm (GA) and finite element method is implemented to estimate the material parameters by inverse mapping the impact test data. Isolated unconfined impact tests for average strain rates ranging from 136 s⁻¹ to 262 s⁻¹ are performed on muscle tissues. Passive muscle tissues are modelled as isotropic, linear and viscoelastic material using three-element Zener model available in PAMCRASH^TM explicit finite element software. In the GA based identification process, fitness values are calculated by comparing the estimated finite element forces with the measured experimental forces. Linear viscoelastic material parameters (bulk modulus, short term shear modulus and long term shear modulus) are thus identified at strain rates 136 s⁻¹, 183 s⁻¹ and 262 s⁻¹ for modelling muscles. Extracted optimal parameters from this study are comparable with reported parameters in literature. Bulk modulus and short term shear modulus are found to be more influential in predicting the stress-strain response than long term shear modulus for the considered strain rates. Variations within the set of parameters identified at different strain rates indicate the need for new or improved material model, which is capable of capturing the strain rate dependency of passive muscle response with single set of material parameters for wide range of strain rates.     

Delayed feedback and multiple attractors in a host–parasitoid system

 Continuous-time, age structured, host–parasitoid models exhibit three types of cyclic dynamics: Lotka–Volterra-like consumer-resource cycles, discrete generation cycles, and “delayed feedback cycles” that occur if the gain to the parasitoid population (defined by the number of new female parasitoid offspring produced per host attacked) increases with the age of the host attacked. The delayed feedback comes about in the following way: an increase in the instantaneous density of searching female parasitoids increases the mortality rate on younger hosts, which reduces the density of future older and more productive hosts, and hence reduces the future per head recruitment rate of searching female parasitoids. Delayed feedback cycles have previously been found in studies that assume a step-function for the gain function. Here, we formulate a general host–parasitoid model with an arbitrary gain function, and show that stable, delayed feedback cycles are a general phenomenon, occurring with a wide range of gain functions, and strongest when the gain is an accelerating function of host age. We show by examples that locally stable, delayed feedback cycles commonly occur with parameter values that also yield a single, locally stable equilibrium, and hence their occurrence depends on initial conditions. A simplified model reveals that the mechanism responsible for the delayed feedback cycles in our host–parasitoid models is similar to that producing cycles and initial-condition-dependent dynamics in a single species model with age-dependent cannibalism. Received: 24 October 1997 / Revised version: 13 June 1998     

Cryopreservation procedure does not modify human carotid homografts mechanical properties: an isobaric and dynamic analysis

The viscoelastic and inertial properties of the arterial wall are responsible for the arterial functional role in the cardiovascular system. Cryopreservation is widely used to preserve blood vessels for vascular reconstruction but it is controversially suspected to affect the dynamic behaviour of these allografts. The aim of this work was to assess the cryopreservation's effects on human arteries mechanical properties. Common carotid artery (CCA) segments harvested from donors were divided into two groups: Fresh (n = 18), tested for 24–48 h after harvesting, and Cryopreserved (n = 18) for an average time of 30 days in gas-nitrogen phase, and finally defrosted. Each segment was tested in a circulation mock, and its pressure and diameter were registered at similar pump frequency, pulse and mean pressure levels, including those of normotensive and hipertensive conditions. A compliance transfer function (diameter/pressure) derived from a mathematical adaptive modelling was designed for the on line assessment of the arterial wall dynamics and its frequency response. Assessment of arterial wall dynamics was made by measuring its viscous (η), inertial (M) and elastic (E) properties, and creep and stress relaxation time constant (τ_C and τ_SR, respectively). The frequency response characterization allowed to evaluate the arterial wall filter or buffer function. Results showed that non-significant differences exist between wall dynamics and buffer function of fresh and cryopreserved segments of human CCA. In conclusion, our cryopreservation method maintains arterial wall functional properties, close to their fresh values.     

Valveless pumping in a fluid-filled closed elastic tube-system: one-dimensional theory with experimental validation

 An elastic rubber tube is connected with a stiffer rubber tube forming two halves of a torus and filled with water. Compressing one of the rubber tubes symmetrically and periodic at a point of asymmetry creates a remarkable unidirectional mean flow in the system. The size and the direction of the mean flow depend on the frequency of compression, the elasticity of the tubes, the compression ratio, and the type of compression with respect to time in a complicated manner. The system is modelled using a one-dimensional theory derived by averaging the Navier-Stokes equations ignoring higher order terms in a certain small quantity. The one-dimensional model is analysed partly analytically and partly numerically. A series of experiments on a physical realisation of the system are described. The theoretical findings and experimental results are compared; They show a remarkable agreement between the experiments and the predictions of the model. Frequencies at which the mean flow change direction are predicted numerically as well as analytically and the two results are compared. Received: 21 February 2002 / Revised version: 30 August 2002 / Published online: 17 January 2003 Key words or phrases: Flow – Elastic tubes – Valveless pumping – Navier-Stokes equations – Frequency dependent – One-dimensional model – Experimental validation     

An Inverse Power-Law Distribution of Molecular Bond Lifetimes Predicts Fractional Derivative Viscoelasticity in Biological Tissue

Viscoelastic characteristics of many materials falling under the category of soft glassy substances, including biological tissue, often exhibit a mechanical complex modulus Y(ω) well described by a fractional derivative model: Y(ω) = E(iω/ϕ)^k, where E = a generalized viscoelastic stiffness; i = (−1)^1/2; ω = angular frequency; ϕ = scaling factor; and k = an exponent valued between 0 and 1. The term “fractional derivative” refers to the value of k: when k = 0 the viscoelastic response is purely elastic, and when k = 1 the response is purely viscous. We provide an analytical derivation of the fractional derivative complex modulus based on the hypothesis that the viscoelastic response arises from many intermittent molecular crosslinks, whose lifetimes longer than a critical threshold lifetime, t_crit, are distributed with an inverse power law proportional to t^-(k+2). We demonstrate that E is proportional to the number and stiffness of crosslinks formed at any moment; the scaling factor ϕ is equivalent to reciprocal of t_crit; and the relative mean lifetime of the attached crosslinks is inversely proportional to the parameter k. To test whether electrostatic molecular bonds could be responsible for the fractional derivative viscoelasticity, we used chemically skinned human skeletal muscle as a one-dimensional model of a soft glassy substance. A reduction in ionic strength from 175 to 110 mEq resulted in a larger E with no change in k, consistent with a higher probability of interfilament molecular interactions. Thick to thin filament spacing was reduced by applying 4% w/v of the osmolyte Dextran T500, which also resulted in a larger E, indicating a greater probability of crosslink formation in proportion to proximity. A 10°C increase in temperature resulted in an increase in k, which corresponded to a decrease in cross-bridge attachment lifetime expected with higher temperatures. These theoretical and experimental results suggest that the fractional derivative viscoelasticity observed in some biological tissue arises as a mechanical consequence of electrostatic interactions, whose longest lifetimes are distributed with an inverse power law.     

Sensitivity analysis and parametric study of elastic properties of an unidirectional mineralized bone fibril-array using mean field methods

The key parameters determining the elastic properties of an unidirectional mineralized bone fibril-array decomposed in two further hierarchical levels are investigated using mean field methods. Modeling of the elastic properties of mineralized micro- and nanostructures requires accurate information about the underlying topology and the constituents’ material properties. These input data are still afflicted by great uncertainties and their influence on computed elastic constants of a bone fibril-array remains unclear. In this work, mean field methods are applied to model mineralized fibrils, the extra-fibrillar matrix and the resulting fibril-array. The isotropic or transverse isotropic elastic constants of these constituents are computed as a function of degree of mineralization, mineral distribution between fibrils and extra-fibrillar matrix, collagen stiffness and fibril volume fraction. The linear sensitivity of the elastic constants was assessed at a default set of the above parameters. The strain ratios between the constituents as well as the axial and transverse indentation moduli of the fibril-array were calculated for comparison with experiments. Results indicate that the degree of mineralization and the collagen stiffness dominate fibril-array elasticity. Interestingly, the stiffness of the extra-fibrillar matrix has a strong influence on transverse and shear moduli of the fibril-array. The axial strain of the intra-fibrillar mineral platelets is 30–90% of the applied fibril strain, depending on mineralization and collagen stiffness. The fibril-to-fibril-array strain ratio is essentially ~1. This study provides an improved insight in the parameters, which govern the fibril-array stiffness of mineralized tissues such as bone.     

On the number of balanced signed graphs

The classical enumeration theorem of Pólya (Acta Math.,68, 145–254, 1937) is applied to a modified version of Harary’s (Pacific J. Math.,8, 743–755, 1958) generating functions for counting bicolored graphs to derive a counting function for the number of balanced signed graphs. Methods for computing these counting polynomial functions are discussed.     

Electrically measuring viscoelastic parameters of adherent cell layers under controlled magnetic forces

Electrical cell-substrate impedance sensing (ECIS) was used to measure the time-dependence and frequency-dependence of impedance for current flowing underneath and between cells. Osteosarcoma cells with a topology similar to a short cylinder (coin-like) surmounted by a dome were used in this study. Application of a small step increase in net vertical stress to the cells (4 and 7 dyn/cm²), via magnetic beads bound to the dorsal (upper) surface, causes an increase in cell body height and an increase in cell-cell separation, as well as stretching of the cell-substrate adhesion bonds. This results in a fast drop in measured resistance (less than 2 s), followed by a slower change with a time constant of 60–150 s. This time constant is about 1.5 times longer at 22 °C than that at 37 °C; it also increases with applied stress. Our frequency scan data, as well as our data for the time course of resistance and capacitance, show that the fast change is associated with both the under-the-cells and between-the-cells resistance. The slower change in resistance mainly reflects the between-the-cells resistance. To obtain viscoelastic parameters from our data we use a simple viscoelastic model comprising viscous and elastic elements (i.e., a dashpot and two springs) for the cell body, and an elastic element (a spring) for the cell-substrate adhesion system. Our results show that the spring constants and the viscosity of the cell body components of this viscoelastic model decrease as the temperature increases, whereas the elastic modulus of cell-substrate adhesion increases with temperature. At 37 °C, for the cell body we obtain a value of about 10⁵ P for the viscous element of the viscoelastic model, and a spring constant expressed in units of an elastic modulus of about 10⁴ dyn/cm² for the spring in series with the viscous element, with another spring with a modulus of about 2×10³ dyn/cm² in parallel with these. In comparable units, we have a modulus for the cell-substrate adhesion system of about 3×10³ dyn/cm². Received: 23 March 1998 / Revised version: 23 June 1998 / Accepted: 1 July 1998