Constitutive relation for red cell membrane. Correction.

The intention of this note is to correct a subtle and somewhat esoteric error that the author discovered in his previous publications on membrane elastic behavior. The consitutive relation between membrane force resultants and large, elastic deformations of a membrane surface involves a strain tensor, characterizing the finite deformations. The original strain tensor that appeared in the equations was the Lagrangian strain tensor; however, the proper strain representation (also Lagrangian in nature because it is "measured" relative to the undeformed material state) is transformed by rotations of coordinates in the deformed material state (whereas the Lagrangian strain tensor is transformed by rotations of coordinates in the undeformed state). The principal membrane tensions are unchanged by this correction; the material elastic constants remain the same; and therefore, the material behavior in shear and isotropic tension is the same. However, the tensor, constitutive relation can be properly applied to coordinate systems other than the principal axis system.     

Passive deformations and active motions of leukocytes

The purpose of this paper is to review the development of continuum mechanics models of single leukocytes in both passive deformations and active motions and to indicate some future directions. Models of passive deformations describe the overall rheological behavior of single leukocytes under externally applied forces and predict the average mechanical properties from experimental data. Various "apparent" viscoelastic coefficients are obtained depending on the models assumed and the types of test used. Models of spontaneous motions postulate active driving mechanisms which must be derived internally from the cell itself and probably have different bases for different kind of motions. For pseudopod protrusion on leukocytes, energy transduction from chemical potential to mechanical work associated with actin polymerization at the tip of the projection is assumed to supply the motive power. For pseudopod retraction, active contraction due to actin-myosin interaction is assumed to be the driving force. The feasibility of the hypotheses are tested via numerical examples and comparison of the theoretical results with experimental measurements.     

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

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

Epicardial suction: a new approach to mechanical testing of the passive ventricular wall

The lack of an appropriate three-dimensional constitutive relation for stress in passive ventricular myocardium currently limits the utility of existing mathematical models for experimental and clinical applications. Previous experiments used to estimate parameters in three-dimensional constitutive relations, such as biaxial testing of excised myocardial sheets or passive inflation of the isolated arrested heart, have not included significant transverse shear deformation or in-plane compression. Therefore, a new approach has been developed in which suction is applied locally to the ventricular epicardium to introduce a complex deformation in the region of interest, with transmural variations in the magnitude and sign of nearly all six strain components. The resulting deformation is measured throughout the region of interest using magnetic resonance tagging. A nonlinear, three-dimensional, finite element model is used to predict these measurements at several suction pressures. Parameters defining the material properties of this model are optimized by comparing the measured and predicted myocardial deformations. We used this technique to estimate material parameters of the intact passive canine left ventricular free wall using an exponential, transversely isotropic constitutive relation. We tested two possible models of the heart wall: first, that it was homogeneous myocardium, and second, that the myocardium was covered with a thin epicardium with different material properties. For both models, in agreement with previous studies, we found that myocardium was nonlinear and anisotropic with greater stiffness in the fiber direction. We obtained closer agreement to previously published strain data from passive filling when the ventricular wall was modeled as having a separate, isotropic epicardium. These results suggest that epicardium may play a significant role in passive ventricular mechanics.     

A microstructurally driven model for pulmonary artery tissue

A new constitutive model for elastic, proximal pulmonary artery tissue is presented here, called the total crimped fiber model. This model is based on the material and microstructural properties of the two main, passive, load-bearing components of the artery wall, elastin, and collagen. Elastin matrix proteins are modeled with an orthotropic neo-Hookean material. High stretch behavior is governed by an orthotropic crimped fiber material modeled as a planar sinusoidal linear elastic beam, which represents collagen fiber deformations. Collagen-dependent artery orthotropy is defined by a structure tensor representing the effective orientation distribution of collagen fiber bundles. Therefore, every parameter of the total crimped fiber model is correlated with either a physiologic structure or geometry or is a mechanically measured material property of the composite tissue. Further, by incorporating elastin orthotropy, this model better represents the mechanics of arterial tissue deformation. These advancements result in a microstructural total crimped fiber model of pulmonary artery tissue mechanics, which demonstrates good quality of fit and flexibility for modeling varied mechanical behaviors encountered in disease states.     

Hydrodynamic interaction of two unsteady model microorganisms

The study of pair-wise interactions between swimming microorganisms is fundamental to the understanding of the rheological and transport properties of semi-dilute suspensions. In this paper, the hydrodynamic interaction of two ciliated microorganisms is investigated numerically using a boundary-element method, and the microorganisms are modeled as spherical squirmers that swim by time-dependent surface deformations. The results show that the inclusion of the unsteady terms in the ciliary propulsion model has a large impact on the trajectories of the interacting cells, and causes a significant change in scattering angles with potential important consequences on the diffusion properties of semi-dilute suspensions. Furthermore, the analysis of the shear stress acting on the surface of the microorganisms revealed that the duration and the intensity of the near-field interaction are significantly modified by the presence of unsteadiness. This observation may account for the hydrodynamic nature of randomness in some biological reactions, and supersedes the distinction between intrinsic randomness and hydrodynamic interactions, adding a further element to the understanding and modeling of interacting microorganisms.     

Theoretical models in mechanics of the left ventricle

Different rheological concepts and theoretical studies have been recently presented using models of myocardial mechanics. Complex analysis of the mechanical behavior of the left ventricular wall have been developed in order to estimate the local stresses and deformations that occur during the heart cycle as well as the ventricular stroke volume and pressure. Theoretical models have taken into account non-linear and viscoelastic passive properties of the myocardium tissue, when subjected to large deformations, through given strain energy functions or stress-strain relations. Different prolate spheroid geometries have been considered for such thick shell cardiac structure. During the active state of the contraction, the rheological behavior of the fibers has been described using different muscle models and relationships between fiber tension and strain, and activation degree. A forthcoming approach for bridging the gap between the knowledge of the muscle fiber microrheological properties and the study of the mechanical behavior of the entire ventricle, consists in including anisotropic and inhomogeneous effects through fiber direction field.     

Cell nucleus as a microrheological probe to study the rheology of the cytoskeleton

Mechanical properties of the cell are important biomarkers for probing its architectural changes caused by cellular processes and/or pathologies. The development of microfluidic technologies has enabled measuring the cell’s mechanical properties at high throughput so that mechanical phenotyping can be applied to large samples in reasonable timescales. These studies typically measure the stiffness of the cell as the only mechanical biomarker and do not disentangle the rheological contributions of different structural components of the cell, including the cell cortex, the interior cytoplasm and its immersed cytoskeletal structures, and the nucleus. Recent advancements in high-speed fluorescent imaging have enabled probing the deformations of the cell cortex while also tracking different intracellular components in rates applicable to microfluidic platforms. We present a, to our knowledge, novel method to decouple the mechanics of the cell cortex and the cytoplasm by analyzing the correlation between the cortical deformations that are induced by external microfluidic flows and the nucleus displacements, induced by those cortical deformations, i.e., we use the nucleus as a high-throughput microrheological probe to study the rheology of the cytoplasm, independent of the cell cortex mechanics. To demonstrate the applicability of this method, we consider a proof-of-concept model consisting of a rigid spherical nucleus centered in a spherical cell. We obtain analytical expressions for the time-dependent nucleus velocity as a function of the cell deformations when the interior cytoplasm is modeled as a viscous, viscoelastic, porous, and poroelastic material and demonstrate how the nucleus velocity can be used to characterize the linear rheology of the cytoplasm over a wide range of forces and timescales/frequencies.     

An Experimental-Computational Approach to Quantify Blood Rheology in Sickle Cell Disease

In sickle cell disease, aberrant blood flow due to oxygen-dependent changes in red cell biomechanics is a key driver of pathology. Most studies to date have focused on the potential role of altered red cell deformability and blood rheology in precipitating vaso-occlusive crises. Numerous studies, however, have shown that sickle blood flow is affected even at high oxygen tensions, suggesting a potentially systemic role for altered blood flow in driving pathologies, including endothelial dysfunction, ischemia, and stroke. In this study, we applied a combined experimental-computation approach that leveraged an experimental platform that quantifies sickle blood velocity fields under a range of oxygen tensions and shear rates. We computationally fitted a continuum model to our experimental data to generate physics-based parameters that capture patient-specific rheological alterations. Our results suggest that sickle blood flow is altered systemically, from the arterial to the venous circulation. We also demonstrated the application of this approach as a tool to design patient-specific transfusion regimens. Finally, we demonstrated that patient-specific rheological parameters can be combined with patient-derived vascular models to identify patients who are at higher risk for cerebrovascular complications such as aneurysm and stroke. Overall, this study highlights that sickle blood flow is altered systemically, which can drive numerous pathologies, and this study demonstrates the potential utility of an experimentally parameterized continuum model as a predictive tool for patient-specific care.     

Combined finite-element and rigid-body analysis of human jaw joint dynamics

The jaw joint plays a crucial role in human mastication. It acts as a guidance for jaw movements and as a fulcrum for force generation. The joint is subjected to loading which causes tensions and deformations in its cartilaginous structures. These are assumed to be a major determinant for development, maintenance and also degeneration of the joint. To analyze the distribution of tensions and deformations in the cartilaginous structures of the jaw joint during jaw movement, a dynamical model of the human masticatory system has been constructed. Its movements are controlled by muscle activation. The articular cartilage layers and articular disc were included as finite-element (FE) models. As this combination of rigid-body and FE modeling had not been applied to musculoskeletal systems yet, its benefits and limitations were assessed by simulating both unloaded and loaded jaw movements. It was demonstrated that joint loads increase with muscle activation, irrespective of the external loads. With increasing joint load, the size of the stressed area of the articular surfaces was enlarged, whereas the peak stresses were much less affected. The results suggest that the articular disc enables distribution of local contact stresses over a much wider area of the very incongruent articular surfaces by transforming compressive principal stress into shear stress.     

FEA analysis of silicone MCP implant

This paper discusses finite element study of silicone rubber prosthesis for the metacarpophalangeal joint of the hand. Based on the experimental data, a material model which incorporates test data available for different stress states was chosen and calibrated. Finite element models for three commercially available silicone joint prosthesis were developed. All models incorporated the same material model and allowed for large deformations. These models were validated against the experimental data and analyzed under demanding loading conditions. Results such as highly non-linear material behavior, dependence on the loading history and large deformations near wrinkle formation in the hinge area of the joint clearly show the necessity and importance of using multi-stress- state non-linear material models and accounting for large deformations.     

Role of the membrane cortex in neutrophil deformation in small pipets.

The simplest model for a neutrophil in its "passive" state views the cell as consisting of a liquid-like cytoplasmic region surrounded by a membrane. The cell surface is in a state of isotropic contraction, which causes the cell to assume a spherical shape. This contraction is characterized by the cortical tension. The cortical tension shows a weak area dilation dependence, and it determines the elastic properties of the cell for small curvature deformations. At high curvature deformations in small pipets (with internal radii less than 1 micron), the measured critical suction pressure for cell flow into the pipet is larger than its estimate from the law of Laplace. A model is proposed where the region consisting of the cytoplasm membrane and the underlying cortex (having a finite thickness) is introduced at the cell surface. The mechanical properties of this region are characterized by the apparent cortical tension (defined as a free contraction energy per unit area) and the apparent bending modulus (introduced as a bending free energy per unit area) of its middle plane. The model predicts that for small curvature deformations (in pipets having radii larger than 1.2 microns) the role of the cortical thickness and the resistance for bending of the membrane-cortex complex is negligible. For high curvature deformations, they lead to elevated suction pressures above the values predicted from the law of Laplace. The existence of elevated suction pressures for pipets with radii from 1 micron down to 0.24 micron is found experimentally. The measured excess suction pressures cannot be explained only by the modified law of Laplace (for a cortex with finite thickness and negligible bending resistance), because it predicts unacceptable high cortical thicknesses (from 0.3 to 0.7 micron). It is concluded that the membrane-cortex complex has an apparent bending modulus from 1 x 10(-18) to 2 x 10(-18) J for a cortex with a thickness from 0.1 micron down to values much smaller than the radius of the smallest pipet (0.24 micron) used in this study.     

Load sharing between solid and fluid phases in articular cartilage: II--Comparison of experimental results and u-p finite element predictions.

Experimental measurements in conjunction with theoretical predictions were used to determine the extent of load supported by the fluid phase of cartilage at the articular surface. The u-p finite element model was used to simulate the loading of six separate porcine knee joints and to predict surface deformations of the cartilage layer on the lateral femoral condyle. Representative geometry for the condyle, contact pressures, and intrinsic material properties of the cartilage layer were supplied from experimental measures (see Part I). The u-p finite element predictions for surface deformations of the cartilage layer were obtained for several load partitioning states between the solid and fluid phases of cartilage at the articular surface. These were then compared to actual surface deformations obtained experimentally. It appeared from the comparison that approximately 75 percent of the applied load was borne by the fluid phase at the articular surface under this loading regime. This was qualitatively in agreement with the hypothesis that an applied load to articular joints is partitioned at the surface to the two phases according to the surface area ratios of the solid and fluid phases. It appeared that the solid phase was shielded from the total applied stress on the articular surface by the fluid and could be a reason for the excellent durability of the tissue under the demanding conditions in a diarthrodial joint.     

Modeling of the passive mechanical properties of the musculo-articular complex: Acute effects of cyclic and static stretching

The primary aim of this study was to implement a rheological model of the mechanical behavior of the passive musculo-articular complex (MAC). The second objective was to adapt this model to simulate changes in the passive MAC's mechanical properties induced by passive stretching protocols commonly performed in sport and rehabilitation programs. Nine healthy subjects performed passive ankle dorsi-flexion and plantar-flexion cycles at different velocities (from 0.035 to 2.09 rad s^?1) on an isokinetic dynamometer. This procedure enabled the articular angle to be controlled and the passive torque developed by the MAC in resistance to stretch to be measured. Our rheological model, dependent on nine parameters, was composed of two non-linear (exponential) springs for both plantar- and dorsi-flexion, a linear viscoelastic component and a solid friction component. The model was implemented with the Simulink software package, and the nine parameters were identified, for each subject, by minimizing the square-difference between experimental torque–angle relationships and modeled curves. This model is in good agreement with experiment, whatever the considered stretching velocity. Finally, the model was adapted to incorporate static stretching (4×2.5 min) and cyclic stretching (five loading/unloading cycles) protocols. Our results indicate that the model could be used to simulate the effects of stretching protocols by adjusting a single (different) parameter for each protocol.     

Emergent cell and tissue dynamics from subcellular modeling of active biomechanical processes

Cells and the tissues they form are not passive material bodies. Cells change their behavior in response to external biochemical and biomechanical cues. Behavioral changes, such as morphological deformation, proliferation and migration, are striking in many multicellular processes such as morphogenesis, wound healing and cancer progression. Cell-based modeling of these phenomena requires algorithms that can capture active cell behavior and their emergent tissue-level phenotypes. In this paper, we report on extensions of the subcellular element model to model active biomechanical subcellular processes. These processes lead to emergent cell and tissue level phenotypes at larger scales, including (i) adaptive shape deformations in cells responding to slow stretching, (ii) viscous flow of embryonic tissues, and (iii) streaming patterns of chemotactic cells in epithelial-like sheets. In each case, we connect our simulation results to recent experiments.     

Linear viscoelastic properties of the vertex model for epithelial tissues

Epithelial tissues act as barriers and, therefore, must repair themselves, respond to environmental changes and grow without compromising their integrity. Consequently, they exhibit complex viscoelastic rheological behavior where constituent cells actively tune their mechanical properties to change the overall response of the tissue, e.g., from solid-like to fluid-like. Mesoscopic mechanical properties of epithelia are commonly modeled with the vertex model. While previous studies have predominantly focused on the rheological properties of the vertex model at long time scales, we systematically studied the full dynamic range by applying small oscillatory shear and bulk deformations in both solid-like and fluid-like phases for regular hexagonal and disordered cell configurations. We found that the shear and bulk responses in the fluid and solid phases can be described by standard spring-dashpot viscoelastic models. Furthermore, the solid-fluid transition can be tuned by applying pre-deformation to the system. Our study provides insights into the mechanisms by which epithelia can regulate their rich rheological behavior.     

Modeling active muscle contraction in mitral valve leaflets during systole: a first approach

The present study addresses the effect of muscle activation contributions to mitral valve leaflet response during systole. State-of-art passive hyperelastic material modeling is employed in combination with a simple active stress part. Fiber families are assumed in the leaflets: one defined by the collagen and one defined by muscle activation. The active part is either assumed to be orthogonal to the collagen fibers or both orthogonal to and parallel with the collagen fibers (i.e. an orthotropic muscle fiber model). Based on data published in the literature and information herein on morphology, the size of the leaflet parts that contain muscle fibers is estimated. These parts have both active and passive materials, the remaining parts consist of passive material only. Several solid finite element analyses with different maximum activation levels are run. The simulation results are compared to corresponding echocardiography at peak systole for a porcine model. The physiologically correct flat shape of the closed valve is approached as the activation levels increase. The non-physiological bulging of the leaflet into the left atrium when using passive material models is reduced significantly. These results contribute to improved understanding of the physiology of the native mitral valve, and add evidence to the hypothesis that the mitral valve leaflets not are just passive elements moving as a result of hemodynamic pressure gradients in the left part of the heart.     

Isotropy and anisotropy of the arterial wall

The passive biomechanical response of intact cylindrical rat carotid arteries is studied in vitro and compared with the mechanical response of rubber tubes. Using true stress and natural strain in the definition of the incremental modulus of elasticity, the tissue wall properties are analyzed over wide ranges of simultaneous circumferential and longitudinal deformations. The type of loading chosen is 'physiological' i.e. symmetric: the cylindrical segments are subjected to internal pressure and axial prestretch without torsion or shear. Several aspects pertaining to the choice of parameters characterizing the material are discussed and the analysis pertaining to the deformational behavior of a hypothetical compliant tube with Hookean wall material is presented. The experimental results show that while rubber response can be adequately represented as linearly elastic and isotropic, the overall response of vascular tissue is highly non-linear and anisotropic. However, for states of deformation that occur in vivo, the elasticity of arteries is quite similar to that of rubber tubes and as such the arterial wall may be viewed as incrementally isotropic for the range of deformations that occur in vivo.     

Viscoelastic properties of living embryonic tissues: a quantitative study.

A number of properties of certain living embryonic tissues can be explained by considering them as liquids. Tissue fragments left in a shaker bath round up to form spherical aggregates, as do liquid drops. When cells comprising two distinct embryonic tissues are mixed, typically a nucleation-like process takes place, and one tissue sorts out from the other. The equilibrium configurations at the end of such sorting out phenomena have been interpreted in terms of tissue surface tensions arising from the adhesive interactions between individual cells. In the present study we go beyond these equilibrium properties and study the viscoelastic behavior of a number of living embryonic tissues. Using a specifically designed apparatus, spherical cell aggregates are mechanically compressed and their viscoelastic response is followed. A generalized Kelvin model of viscoelasticity accurately describes the measured relaxation curves for each of the four tissues studied. Quantitative results are obtained for the characteristic relaxation times and elastic and viscous parameters. Our analysis demonstrates that the cell aggregates studied here, when subjected to mechanical deformations, relax as elastic materials on short time scales and as viscous liquids on long time scales.     

The cytomechanics of axonal elongation and retraction

Neurites of PC12 and chick dorsal root ganglion neurons behave as viscoelastic solids in response to applied forces. This passive behavior can be modeled with three mechanical elements; a relatively stiff, undamped spring in series with a Voight element composed of a less stiff spring in parallel with a dashpot. In response to applied tensions greater than 100 microdynes, PC12 cells show lengthening behavior distinct from and in addition to the passive viscoelastic response. We interpret this as "towed growth" (Bray, D. 1984. Dev. Biol. 102:379-389) because the neurites can become twice as long without obvious thinning of the neurite and because in two cases neurite tensions fell below original rest tensions, a result that cannot be obtained with passive viscoelastic elements. The rate of towed growth showed a linear dependence of growth rate with applied tensions in 8 of 12 PC12 neurites exposed to applied tension greater than 100 microdynes. Both PC12 and chick sensory neurons showed evidence of retraction when neurite tensions were suddenly diminished. This response was measured as tension recovery after slackening in chick sensory neurites. In 62% of the cases, tension recovery exceeded and sometimes doubled the preexperimental steady-state tension. Our data indicate that this response is active tension generation by the neurite shaft. We conclude that neurite length is regulated by axial tension in both elongation and retraction. Our data suggest a three-way controller: above some tension set point, the neurite is stimulated to elongate. Below some different, lower tension threshold the neurite is stimulated to retract. Between these two tension thresholds, the neurite responds passively as a viscoelastic solid.