Rheological behavior of mammalian cells.

Rheological properties of living cells determine how cells interact with their mechanical microenvironment and influence their physiological functions. Numerous experimental studies have show that mechanical contractile stress borne by the cytoskeleton and weak power-law viscoelasticity are governing principles of cell rheology, and that the controlling physics is at the level of integrative cytoskeletal lattice properties. Based on these observations, two concepts have emerged as leading models of cytoskeletal mechanics. One is the tensegrity model, which explains the role of the contractile stress in cytoskeletal mechanics, and the other is the soft glass rheology model, which explains the weak power-law viscoelasticity of cells. While these two models are conceptually disparate, the phenomena that they describe are often closely associated in living cells for reasons that are largely unknown. In this review, we discuss current understanding of cell rheology by emphasizing the underlying biophysical mechanism and critically evaluating the existing rheological models.

Solitary fibrous tumors of the pleura.

Two cases of female patients with solitary fibrous tumor of the pleura are described. Chest radiography and computed tomography showed the presence of a giant mass with inhomogeneous density in the hemithorax in both cases. Transthoracic needle biopsy of the masses showed benign fibrous tissue. Both patients underwent thoracotomy. The tumors arose from the visceral pleura, were peduncular, appeared well-circumscribed with exterior surface smooth, firm and multinodular. The size of tumors were 16x11x6 cm and 15x11x4.5 cm, weighed 650 g and 573 g and both were successfully resected. The pathological diagnosis was benign solitary fibrous tumor of the pleura. Seven years after the operations the clinical condition and chest radiography are normal in both women.

Models of cytoskeletal mechanics of adherent cells.

Adherent cells sense their mechanical environment, which, in turn, regulates their functions. During the past decade, a growing body of evidence has indicated that a deformable, solid-state intracellular structure known as the cytoskeleton (CSK) plays a major role in transmitting and distributing mechanical stresses within the cell as well as in their conversion into a chemical response. Therefore in order to understand mechanical regulation and control of cellular functions, one needs to understand mechanisms that determine how the CSK changes its shape and mechanics in response to stress. In this survey, we examined commonly used structurally based models of the CSK. In particular, we focused on two classes of these models: open-cell foam networks and stress-supported structures. We identified the underlying mechanisms that determine deformability of those models and compare model predictions with data previously obtained from mechanical tests on cultured living adherent cells at steady state. We concluded that stress-supported structures appear more suitable for describing cell deformability because this class of structures can explain the central role that the cytoskeletal contractile prestress plays in cellular mechanics.

Mechanical behavior in living cells consistent with the tensegrity model.

Alternative models of cell mechanics depict the living cell as a simple mechanical continuum, porous filament gel, tensed cortical membrane, or tensegrity network that maintains a stabilizing prestress through incorporation of discrete structural elements that bear compression. Real-time microscopic analysis of cells containing GFP-labeled microtubules and associated mitochondria revealed that living cells behave like discrete structures composed of an interconnected network of actin microfilaments and microtubules when mechanical stresses are applied to cell surface integrin receptors. Quantitation of cell tractional forces and cellular prestress by using traction force microscopy confirmed that microtubules bear compression and are responsible for a significant portion of the cytoskeletal prestress that determines cell shape stability under conditions in which myosin light chain phosphorylation and intracellular calcium remained unchanged. Quantitative measurements of both static and dynamic mechanical behaviors in cells also were consistent with specific a priori predictions of the tensegrity model. These findings suggest that tensegrity represents a unified model of cell mechanics that may help to explain how mechanical behaviors emerge through collective interactions among different cytoskeletal filaments and extracellular adhesions in living cells.

Invited review: engineering approaches to cytoskeletal mechanics.

An outstanding problem in cell biology is how cells sense mechanical forces and how those forces affect cellular functions. Various biophysical and biochemical mechanisms have been invoked to answer this question. A growing body of evidence indicates that the deformable cytoskeleton (CSK), an intracellular network of interconnected filamentous biopolymers, provides a physical basis for transducing mechanical signals into biochemical signals. Therefore, to understand how mechanical forces regulate cellular functions, it is important to know how cells respond to changes in the CSK force balance and to identify the underlying mechanisms that control transmission of mechanical forces throughout the CSK and bring it to equilibrium. Recent developments of new experimental techniques for measuring cell mechanical properties and novel theoretical models of cellular mechanics make it now possible to identify and quantitate the contributions of various CSK structures to the overall balance of mechanical forces in the cell. This review focuses on engineering approaches that have been used in the past two decades in studies of the mechanics of the CSK.

A microstructural model of elastostatic properties of articular cartilage in confined compression.

A microstructural model of cartilage was developed to investigate the relative contribution of tissue matrix components to its elastostatic properties. Cartilage was depicted as a tensed collagen lattice pressurized by the Donnan osmotic swelling pressure of proteoglycans. As a first step in modeling the collagen lattice, two-dimensional networks of tensed, elastic, interconnected cables were studied as conceptual models. The models were subjected to the boundary conditions of confined compression and stress-strain curves and elastic moduli were obtained as a function of a two-dimensional equivalent of swelling pressure. Model predictions were compared to equilibrium confined compression moduli of calf cartilage obtained at different bath concentrations ranging from 0.01 to 0.50 M NaCl. It was found that a triangular cable network provided the most consistent correspondence to the experimental data. The model showed that the cartilage collagen network remained tensed under large confined compression strains and could therefore support shear stress. The model also predicted that the elastic moduli increased with increasing swelling pressure in a manner qualitatively similar to experimental observations. Although the model did not preclude potential contributions of other tissue components and mechanisms, the consistency of model predictions with experimental observations suggests that the cartilage collagen network, prestressed by proteoglycan swelling pressure, plays an important role in supporting compression.

Contribution of intermediate filaments to cell stiffness, stiffening, and growth.

It has been shown previously that intermediate filament (IF) gels in vitro exhibit stiffening at high-applied stress, and it was suggested that this stiffening property of IFs might be important for maintaining cell integrity at large deformations (Janmey PA, Evtenever V, Traub P, and Schliwa M, J Cell Biol 113: 155-160, 1991). In this study, the contribution of IFs to cell mechanical behavior was investigated by measuring cell stiffness in response to applied stress in adherent wild-type and vimentin-deficient fibroblasts using magnetic twisting cytometry. It was found that vimentin-deficient cells were less stiff and exhibited less stiffening than wild-type cells, except at the lowest applied stress (10 dyn/cm(2)) where the difference in the stiffness was not significant. Similar results were obtained from measurements on wild-type fibroblasts and endothelial cells after vimentin IFs were disrupted by acrylamide. If, however, cells were plated over an extended period of time (16 h), they exhibited a significantly greater stiffness before than after acrylamide, even at the lowest applied stress. A possible reason could be that the initially slack IFs became fully extended due to a high degree of cell spreading and thus contributed to the transmission of mechanical stress across the cell. Taken together, these findings were consistent with the notion that IFs play important roles in the mechanical properties of the cell during large deformation. The experimental data also showed that depleting or disrupting IFs reduced, but did not entirely abolish, cell stiffening. This residual stiffening might be attributed to the effect of geometrical realignment of cytoskeletal filaments in the direction of applied load. It was also found that vimentin-deficient cells exhibited a slower rate of proliferation and DNA synthesis than wild-type cells. This could be a direct consequence of the absence of the intracellular IFs that may be necessary for efficient mediation of mechanical signals within the cell. Taken together, results of this study suggest that IFs play important roles in the mechanical properties of cells and in cell growth.

A quantitative model of cellular elasticity based on tensegrity.

A tensegrity structure composed of six struts interconnected with 24 elastic cables is used as a quantitative model of the steady-state elastic response of cells, with the struts and cables representing microtubules and actin filaments, respectively. The model is stretched uniaxially and the Young's modulus (E0) is obtained from the initial slope of the stress versus strain curve of an equivalent continuum. It is found that E0 is directly proportional to the pre-existing tension in the cables (or compression in the struts) and inversely proportional to the cable (or strut) length square. This relationship is used to predict the upper and lower bounds of E0 of cells, assuming that the cable tension equals the yield force of actin (approximately 400 pN) for the upper bound, and that the strut compression equals the critical buckling force of microtubules for the lower bound. The cable (or strut) length is determined from the assumption that model dimensions match the diameter of probes used in standard mechanical tests on cells. Predicted values are compared to reported data for the Young's modulus of various cells. If the probe diameter is greater than or equal to 3 microns, these data are closer to the lower bound than to the upper bound. This, in turn, suggests that microtubules of the CSK carry initial compression that exceeds their critical buckling force (order of 10(0)-10(1) pN), but is much smaller than the yield force of actin. If the probe diameter is less than or equal to 2 microns, experimental data fall outside the region defined by the upper and lower bounds.

The role of prestress and architecture of the cytoskeleton and deformability of cytoskeletal filaments in mechanics of adherent cells: a quantitative analysis.

Mechanical properties of adherent cells were investigated using methods of engineering mechanics. The cytoskeleton (CSK) was modeled as a filamentous network and key mechanisms and corresponding molecular structures which determine cell elastic behavior were identified. Three models of the CSK were considered: open-cell foam networks, prestressed cable nets, and a tensegrity model as a special case of the latter. For each model, the modulus of elasticity (i.e. an index of resistance to small deformation) was given as a function of mechanical and geometrical properties of CSK filaments whose values were determined from the data in the literature. Quantitative predictions for the elastic modulus were compared with data obtained previously from mechanical tests on adherent cells. The open-cell foam model yielded the elastic modulus (10(3)-10(4)Pa) which was consistent with measurements which apply a large compressive stress to the cell. This suggests that bending of CSK filaments is the key mechanism for resisting large compression. The prestressed cable net and tensegrity model yielded much lower elastic moduli (10(1)-10(2)Pa) which were consistent with values determined from equilibrium measurements at low applied stress. This suggests that CSK prestress and architecture are the primary determinants of the cell elastic response. The tensegrity model revealed the possibility that buckling of microtubules of the CSK also contributed to cell elasticity.

Confined and unconfined stress relaxation of cartilage: appropriateness of a transversely isotropic analysis.

Previous studies have shown that stress relaxation behavior of calf ulnar growth plate and chondroepiphysis cartilage can be described by a linear transverse isotropic biphasic model. The model provides a good fit to the observed unconfined compression transients when the out-of-plane Poisson's ratio is set to zero. This assumption is based on the observation that the equilibrium stress in the axial direction (deltaz) is the same in confined and unconfined compression, which implies that the radial stress deltar = 0 in confined compression. In our study, we further investigated the ability of the transversely isotropic model to describe confined and unconfined stress relaxation behavior of calf cartilage. A series of confined and unconfined stress relaxation tests were performed on calf articular cartilage (4.5 mm diameter, approximately 3.3 mm height) in a displacement-controlled compression apparatus capable of measuring delta(z) and delta(r). In equilibrium, delta(r) > 0 and delta(z) in confined compression was greater than in unconfined compression. Transient data at each strain were fitted by the linear transversely isotropic biphasic model and the material parameters were estimated. Although the model could provide good fits to the unconfined transients, the estimated parameters overpredicted the measured delta(r). Conversely, if the model was constrained to match equilibrium delta(r), the fits were poor. These findings suggest that the linear transversely isotropic biphasic model could not simultaneously describe the observed stress relaxation and equilibrium behavior of calf cartilage.

Mathematical modeling of the first inflation of degassed lungs.

The pressure-volume (P-V) relationship of degassed lungs during the first inflation is different from that in consecutive inflations. We developed a mathematical model of the P-V curve of the first inflation by assuming that (1) central airways are open leading to many subtrees of n generations that are initially closed; (2) an airway opens when inflation pressure reaches the opening threshold pressure of that segment; and (3) the opening threshold pressures do not depend on airway generation. In this model, airway opening occurs in cascades or avalanches. To test the model which contains only two parameters, n and a pressure, P(low), at which at least one subtree completely opens, we measured the first inflation P-V curves of 15 excised and degassed rabbit lungs. By fitting these data, we found that n=17+/-5, P(low)=23+/-4 cmH2O, and that there is a wide distribution of threshold pressures for airways with diameters <2 mm. Analysis of the P-V curve in a lung which was lavaged with a liquid of constant surface tension and in which airways are presumably open demonstrated that the distribution of threshold pressures is narrow, and hence no avalanches occur during inflation. We conclude that in normal lungs the first inflation is dominated by avalanche behavior of airway opening providing information on the global distribution of threshold pressures and the average site of airway closure.

Is cytoskeletal tension a major determinant of cell deformability in adherent endothelial cells?

We tested the hypothesis that mechanical tension in the cytoskeleton (CSK) is a major determinant of cell deformability. To confirm that tension was present in adherent endothelial cells, we either cut or detached them from their basal surface by a microneedle. After cutting or detachment, the cells rapidly retracted. This retraction was prevented, however, if the CSK actin lattice was disrupted by cytochalasin D (Cyto D). These results confirmed that there was preexisting CSK tension in these cells and that the actin lattice was a primary stress-bearing component of the CSK. Second, to determine the extent to which that preexisting CSK tension could alter cell deformability, we developed a stretchable cell culture membrane system to impose a rapid mechanical distension (and presumably a rapid increase in CSK tension) on adherent endothelial cells. Altered cell deformability was quantitated as the shear stiffness measured by magnetic twisting cytometry. When membrane strain increased 2.5 or 5%, the cell stiffness increased 15 and 30%, respectively. Disruption of actin lattice with Cyto D abolished this stretch-induced increase in stiffness, demonstrating that the increased stiffness depended on the integrity of the actin CSK. Permeabilizing the cells with saponin and washing away ATP and Ca2+ did not inhibit the stretch-induced stiffening of the cell. These results suggest that the stretch-induced stiffening was primarily due to the direct mechanical changes in the forces distending the CSK but not to ATP- or Ca(2+)-dependent processes. Taken together, these results suggest preexisting CSK tension is a major determinant of cell deformability in adherent endothelial cells.

A tensegrity model of the cytoskeleton in spread and round cells.

Measurements on adherent cells have shown that spreading affects their mechanics. Highly spread cells are stiffer than less spread cells. The stiffness increases approximately linearly with increasing applied stress and more so in highly spread cells than in less spread cells. In this study, a six-strut tensegrity model of the cytoskeleton is used to analyze the effect of spreading on cellular mechanics. Two configurations are considered: a "round" configuration where a spherically shaped model is anchored to a flat rigid surface at three joints, and a "spread" configuration, where three additional joints of the model are attached to the surface. In both configurations a pulling force is applied at a free joint, distal from the anchoring surface, and the corresponding deformation is determined from equations of equilibrium. The model stiffness is obtained as the ratio of applied force to deformation. It is found that the stiffness changes with spreading consistently with the observations in cells. These findings suggest the possibility that the spreading-induced changes of the mechanical properties of the cell are the result of the concomitant changes in force distribution and microstructural geometry of the cytoskeleton.

A microstructural approach to cytoskeletal mechanics based on tensegrity.

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

Dynamic behavior of lung parenchyma in shear.

Dynamic shear properties of excised rabbit lungs were studied by measuring creep deformation after application of a step indentation force to the pleural surfaces by a rigid cylindrical punch. The punch diameter was 9.5 mm, and punch forces were 2,4, and 6 g. Measurements were made at lung volumes of 40, 60, and 90% of the total lung capacity before and after lavage with 3-dimethyl siloxane, which provided a constant surface tension of 16 dyn/cm at the alveolar surfaces. A power-law model was fitted to creep data and then transformed into the frequency (f) domain by using Laplace transforms. The optimum model parameters were used to calculate shear elastance (E mu), shear resistance (R mu), and shear hysteresivity (2 pi fR mu/E mu) between 0.01 and 2.0 Hz. It was found that E mu slightly increased and R mu decreased nearly hyperbolically with increasing f, both decreased with increasing indentation force, and both increased with increasing mean lung volume. Shear hysteresivity decreased sharply from 0.01 to 0.25 Hz and then assumed a nearly steady value that was an order of magnitude lower than the value reported previously for uniformly oscillated lungs. Changes in E mu and R mu after lavage were correlated with changes in transpulmonary pressure and not with changes in surface film properties. These results suggest that in the breathing range of frequencies 1) the energy loss of lung parenchyma is a much smaller fraction of the stored elastic energy in shear than in uniformly oscillated lungs and 2) transpulmonary pressure, not dynamic properties of surface film, is the primary determinant of lung dynamic properties in shear.

Effect of surface forces on oscillatory behavior of lungs.

The effect of alveolar surface tension on lung dynamic behavior was investigated by measuring total lung and tissue impedances in excised rabbit lungs at breathing frequencies of 0.2-0.8 Hz and tidal volumes of 10, 20, and 30 ml before and after lavage with 3-dimethyl siloxane, which provided a constant surface tension of 16 dyn/cm. The lungs were oscillated around the mean deflation pressures of 5 (control) and 8 cmH2O (lavaged), i.e., lung volume of 60% of total lung capacity. The total lung impedance was calculated from measurements of pressure and airflow at the trachea, and tissue impedance was measured by the alveolar capsule technique. The airway contribution was obtained as the difference between total lung and tissue impedances. In the lavaged lungs, dynamic elastance (Edyn) decreased and tissue resistance (Rti) increased relative to the control values over the entire frequency range. Airway resistance increased at the higher flow rates only. The decrease in Edyn could be attributed to the absence of surface film elastance in the lavaged lungs. The increase in airway resistance could be attributed to accentuated flow dependence due to changes in airway geometry and residual lavage liquid. However, the most intriguing result was the increase in Rti in the lavaged lungs. It could be attributed to altered mechanics at the alveolar duct level after lavage. It is concluded that dissipative properties of lung tissue are major determinants of Rti, whereas elastic properties of both tissue and surface film are important determinants of Edyn.

Dynamic moduli of rabbit lung tissue and pigeon ligamentum propatagiale undergoing uniaxial cyclic loading.

In fibrous connective tissue networks, mechanical loads may be transferred from one fiber to the next by friction between slipping fibers (J. Appl. Physiol. 74: 665-681, 1993). Here we tested that hypothesis; it predicts that elastance of fibrous networks increases with increasing frequency, decreases with increasing strain amplitude (delta epsilon), and decreases with tissue swelling by solvent. Similarly, it predicts that hysteresivity (eta) decreases with increasing frequency, increases with increasing delta epsilon, decreases with tissue swelling, and, importantly, exceeds that of isolated fibrous constituents of the matrix. Elastance and eta of two structurally dissimilar connective tissues were measured, the rabbit lung parenchymal strip (a loose collagenous tissue) and the pigeon ligamentum propatagiale (an elastin-rich tissue). Experiments covered the frequency range 0.03125-3.125 Hz. Elastance of lung parenchyma was substantially lower than that of propatagial ligament, increased linearly with the logarithm of frequency, and decreased with delta epsilon; that of ligamentum propatagiale was insensitive to both frequency and delta epsilon. eta of lung parenchyma decreased moderately with increasing frequency and assumed values of approximately 0.1, but eta of ligamentum propatagiale was frequency and delta epsilon invariant and assumed values an order of magnitude smaller. These tissues also showed disparate mechanical responses when exposed to hypertonic bath solutions. Although there were some quantitative differences between predictions and experimental observations, the dynamic behavior of lung parenchyma was generally consistent with that of a network in which load is transferred from one fiber to the next by the agency of friction acting at slipping interface surfaces.

Alternative model of respiratory tissue viscoplasticity.

Respiratory tissue impedance exhibits both tidal volume and frequency dependences in the ranges of normal breathing. Hildebrandt argued that these indicate tissue viscoplasticity and offered a model in support of his argument consisting of viscoelastic and plastoelastic compartments, both mechanically in parallel (J. Appl. Physiol. 28: 365-372, 1970). Although the model appears to be qualitatively consistent with oscillatory behavior of a wide variety of respiratory tissues, it yields only moderately good quantitative correspondences despite a relatively large number of parameters, eight. One reason may be the model topology, which implies that rate-dependent and amplitude-dependent processes are decoupled. This is contrary to observed behavior. In this study we offer a model in which viscoelastic and plastoelastic compartments are mechanically coupled through a serial arrangement. The total number of parameters in the model is four. Using a least squares technique, we fitted this model to impedance data of chest wall, healthy lungs, and edematous lungs, all measured in vivo. We found that the model could account for the major, as well as the more subtle, features of the chest wall data with fewer parameters and fewer ad hoc assumptions than Hildebrandt's model. Although it lacks anatomic specifics, the model suggests that the observed chest wall behavior may stem from the actin-myosin cross-bridge kinetics. It also seems applicable to lung tissue, although the requirements for the plastoelastic compartment are less certain. In the case of edematous lungs, the applicability of the model is difficult to establish.

Toward a kinetic theory of connective tissue micromechanics.

The aim of this study is to develop unifying concepts at the microstructural level to account for macroscopic connective tissue dynamics. We establish the hypothesis that rate-dependent and rate-independent dissipative stresses arise in the interaction among fibers in the connective tissue matrix. A quantitative theoretical analysis is specified in terms of geometry and material properties of connective tissue fibers and surrounding constituents. The analysis leads to the notion of slip and diffusion boundary layers, which become unifying concepts in understanding mechanisms that underlie connective tissue elasticity and energy dissipation during various types of loading. The complex three-dimensional fiber network is simplified to the interaction of two ideally elastic fibers that dissipate energy on slipping interface surfaces. The effects of such interactions are assumed to be expressed in the aggregate matrix. Special solutions of the field equations are obtained analytically, whereas the general solution of the model field equations is obtained numerically. The solutions lead to predictions of tissue behavior that are qualitatively, if not quantitatively, consistent with reports of a variety of dynamic moduli, their dependencies on the rate and amplitude of load application, and some features associated with preconditioning.