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A Progressive Rupture Model of Soft Tissue Stress Relaxation
Abstract
A striking feature of stress relaxation in biological soft tissue is that it frequently follows a power law in time with an exponent that is independent of strain even when the elastic properties of the tissue are highly nonlinear. This kind of behavior is an example of quasi-linear viscoelasticity, and is usually modeled in a purely empirical fashion. The goal of the present study was to account for quasi-linear viscoelasticity in mechanistic terms based on our previously developed hypothesis that it arises as a result of isolated micro-yield events occurring in sequence throughout the tissue, each event passing the stress it was sustaining on to other regions of the tissue until they themselves yield. We modeled stress relaxation computationally in a collection of stress-bearing elements. Each element experiences a stochastic sequence of either increases in elastic equilibrium length or decreases in stiffness according to the stress imposed upon it. This successfully predicts quasi-linear viscoelastic behavior, and in addition predicts power-law stress relaxation that proceeds at the same slow rate as observed in real biological soft tissue.
... Saez et al [30] developed an anisotropic multi-scale damage model for characterizing progressive rupture of fibrous soft tissues and tested it using finite element modeling on a blood vessel post angioplasty. Bates et al [31] studied progressive rupture in a model of soft tissue stress relaxation. Here, isolated micro-yield events occurring in sequence throughout the tissue, its interaction with other regions, and material anisotropy, were simulated, and their effect on the overall tissue yielding was characterized. ...
The development of foot ulcers is a common consequence of severe diabetes. Due to vascular disorders and impeded healing caused by the disease, most foot ulcers have been reported to be affected by body weight and progress with time. Also, abnormal distribution of plantar pressures has been observed to cause the formation of additional ulcers, which may collectively lead to traumatic amputations. While a study of such pathophysiology is not possible through experiments, a few computational modelling works have investigated diabetic foot ulcers. To date, ulcers with a few sizes and locations have been studied, and their effect on the plantar stresses has been quantified. In this work, we have attempted to study the effect of all possible ulcer locations on the generated plantar peak stresses and peak stress locations where additional ulcers may form. Also, the effect of ulcer location on the possible ulcer growth was investigated. A full-scale foot model was developed and a total of 52 ulcer locations were simulated separately, with standing and walking loads. The generated stresses were normalised with the foot size and statistically analysed to develop novel formulations for predicting peak plantar stresses and their locations for any known ulcer location. The results from this study are anticipated to provide important guidelines to doctors and medical practitioners for predicting foot ulcer progression in diabetic patients with existing ulcers and allow the administration of timely preventive interventions.
... It is possible, however, when fibers come into contact, the resulting frictional forces might create some form of more transient connection. Such a mechanism of temporary bond formation and subsequent breakage could be a mechanism for viscoelastic behavior, as shown by Bates and Ma 25 . 26 . ...
Fibrotic diseases are characterized by progressive and often irreversible scarring of connective tissue in various organs, leading to substantial changes in tissue mechanics largely as a result of alterations in collagen structure. This is particularly important in the lung because its bulk modulus is so critical to the volume changes that take place during breathing. Nevertheless, it remains unclear how fibrotic abnormalities in the mechanical properties of pulmonary connective tissue can be linked to the stiffening of its individual collagen fibers. To address this question, we developed a network model of randomly oriented collagen and elastin fibers to represent pulmonary alveolar wall tissue. We show that the stress–strain behavior of this model arises via the interactions of collagen and elastin fiber networks and is critically dependent on the relative fiber stiffnesses of the individual collagen and elastin fibers themselves. We also show that the progression from linear to nonlinear stress–strain behavior of the model is associated with the percolation of stress across the collagen fiber network, but that the location of the percolation threshold is influenced by the waviness of collagen fibers.
... As one microstructure yields, it passes the stress it was sustaining to another region until it also yields, with this process occurring slowly for very long periods of time. 5 This appears to be in addition to the quick, sharper decay. While this quicker mode of relaxation may solely be caused by individual collagen fibres relaxing, the passage of fluid through the channels may also contribute. ...
The knee meniscus is a highly porous structure which exhibits a grading architecture through the depth of the tissue. The superficial layers on both femoral and tibial sides are constituted by a fine mesh of randomly distributed collagen fibers while the internal layer is constituted by a network of collagen channels of a mean size of 22.14 μ m aligned at a 30 ∘ inclination with respect to the vertical. Horizontal dog-bone samples extracted from different depths of the tissue were mechanically tested in uniaxial tension to examine the variation of elastic and viscoelastic properties across the meniscus. The tests show that a random alignment of the collagen fibers in the superficial layers leads to stiffer mechanical responses ( E = 105 and 189 MPa) in comparison to the internal regions ( E = 34 MPa). All regions exhibit two modes of relaxation at a constant strain ( τ 1 = 6.4 to 7.7 s, τ 2 = 49.9 to 59.7 s).
... Furthermore, nonlinearities in relaxation behavior have previously been captured by models in which stress is relieved through sequential rupturing of Maxwell bodies, where each micro-yield event passes the stress on to other regions of the tissue. 63,64 The results from these models give an exponential decay with a long-tail power law. However, as chemical bonds have a finite yield strength, in reality, curves may flatten, behaving as stretched exponentials. ...
Characterizing the time-dependent mechanical properties of cells is not only necessary to determine how they deform but also to understand how external forces trigger biochemical-signaling cascades to govern their behavior. At present, mechanical properties are largely assessed by applying local shear or compressive forces on single cells grown in isolation on non-physiological 2D surfaces. In comparison, we developed the microfabricated vacuum actuated stretcher to measure tensile loading of 3D multicellular "microtissue" cultures. Using this approach, we here assessed the time-dependent stress relaxation and recovery responses of microtissues and quantified the spatial viscoelastic deformation following step length changes. Unlike previous results, stress relaxation and recovery in microtissues measured over a range of step amplitudes and pharmacological treatments followed an augmented stretched exponential behavior describing a broad distribution of inter-related timescales. Furthermore, despite the variety of experimental conditions, all responses led to a single linear relationship between the residual elastic stress and the degree of stress relaxation, suggesting that these mechanical properties are coupled through interactions between structural elements and the association of cells with their matrix. Finally, although stress relaxation could be quantitatively and spatially linked to recovery, they differed greatly in their dynamics; while stress recovery acted as a linear process, relaxation time constants changed with an inverse power law with the step size. This assessment of microtissues offers insights into how the collective behavior of cells in a 3D collagen matrix generates the dynamic mechanical properties of tissues, which is necessary to understand how cells deform and sense mechanical forces in vivo.
... Several researchers previously measured the tissue strains responsible for stressing and rupturing the lung's fiber network [26][27][28]. However, simulation studies of strains on lung tissue due to aging have not been extensively investigated. ...
Elderly patients with obstructive lung diseases often receive mechanical ventilation to support their breathing and restore respiratory function. However, mechanical ventilation is known to increase the severity of ventilator-induced lung injury (VILI) in the elderly. Therefore, it is important to investigate the effects of aging to better understand the lung tissue mechanics to estimate the severity of ventilator-induced lung injuries. Two age-related geometric models involving human bronchioles from generation G10 to G23 and alveolar sacs were developed. The first is for a 50-year-old (normal) and second is for an 80-year old (aged) model. Lung tissue mechanics of normal and aged models were investigated under mechanical ventilation through computational simulations. Results obtained indicated that lung tissue strains during inhalation (t = 0.2 s) decreased by about 40% in the alveolar sac (G23) and 27% in the bronchiole (G20), respectively, for the 80-year-old as compared to the 50-year-old. The respiratory mechanics parameters (work of breathing per unit volume and maximum tissue strain) over G20 and G23 for the 80-year-old decreased by about 64% (three-fold) and 80% (four-fold), respectively, during the mechanical ventilation breathing cycle. However, there was a significant increase (by about threefold) in lung compliance for the 80-year-old in comparison to the 50-year-old. These findings from the computational simulations demonstrated that lung mechanical characteristics are significantly compromised in aging tissues, and these effects were quantified in this study.
Regional viscoelastic properties of thoracic tissues in acute respiratory distress syndrome (ARDS) and their change with position and positive end-expiratory pressure (PEEP) are unknown. In an experimental porcine ARDS, dorsal and ventral lung (R 2 ,L and E 2 ,L) and chest wall (R 2 ,cw and E 2 ,cw) viscoelastic resistive(R) and elastic(E) parameters were measured at 20, 15, 10 and 5 cmH 2 O PEEP in supine and prone position. E 2 and R 2 were obtained by fitting the decay of pressure after end-inspiratory occlusion to the equation: Pviscmax(t)=R 2 e(-t/τ 2 ), where t is length of occlusion and τ 2 time constant. E 2 was = R 2 /τ 2 . R 2 ,cw and E 2 ,cw were measured from esophageal, dorsal and ventral pleural pressures. Global R 2 ,L and E 2 ,L were obtained from the global trans-pulmonary pressure (airway pressure-esophageal pressure), and regional R 2 ,L and E 2 ,L from the dorsal and ventral airway pressure-pleural pressure difference. Lung ventilation was measured by electrical impedance tomography (EIT). Global R 2 ,cw and E 2 ,cw did not change with PEEP or position. Global R 2 ,L (median(Q1-Q3)) was 37.1(11.0-65.1), 5.1(4.3-5.5), 12.1(8.4-19.5), and 41.0 (26.6-53.5) cmH 2 O/L/s in supine, and 15.3 (9.1-41.9), 7.9 (5.7-11.0), 8.0 (5.1-12.1) and 12.9 (6.4-19.4) cmH 2 O/L in prone from 20 to 5 cmH 2 O PEEP (P=0.06 for PEEP and P=0.06 for position). Dorsal R 2 ,L significantly and positively correlated with amount of collapse measured with EIT. Global and regional lung and chest wall viscoelastic parameters can be described by a simple rheological model. Regional E 2 and R 2 were uninfluenced by PEEP and position except for PEEP on dorsal E 2 ,L and position on dorsal E 2 ,cw.
Understanding lung and airway behavior presents a number of challenges, both experimental and theoretical, but the potential rewards are great in terms of both potential treatments for disease and interesting biophysical phenomena. This presents an opportunity for modeling to contribute to greater understanding, and here, we focus on modeling efforts that work toward understanding the behavior of airways in vivo, with an emphasis on asthma. We look particularly at those models that address not just isolated airways but many of the important ways in which airways are coupled both with each other and with other structures. This includes both interesting phenomena involving the airways and the layer of airway smooth muscle that surrounds them, and also the emergence of spatial ventilation patterns via dynamic airway interaction. WIREs Syst Biol Med 2016, 8:459–467. doi: 10.1002/wsbm.1349
This article is categorized under:
• Analytical and Computational Methods > Dynamical Methods
• Models of Systems Properties and Processes > Mechanistic Models
• Models of Systems Properties and Processes > Organ, Tissue, and Physiological Models
We show that dynamical systems with spatial degrees of freedom naturally evolve into a self-organized critical point. Flicker noise, or 1/f noise, can be identified with the dynamics of the critical state. This picture also yields insight into the origin of fractal objects.
One of the hallmarks of biopolymer gels is their nonlinear viscoelastic response to stress, making the measurement of the mechanics of these gels very challenging. Various rheological protocols have been proposed for this; however, a thorough understanding of the techniques and their range of applicability as well as a careful comparison between these methods are still lacking. Using both strain ramp and differential prestress protocols, we investigate the nonlinear response of a variety of systems ranging from extracellular fibrin gels to intracellular F-actin solutions and F-actin cross-linked with permanent and physiological transient linkers. We find that the prestress and strain ramp results agree well for permanently cross-linked networks over two decades of strain rates, while the protocols only agree at high strain rates for more transient networks. Surprisingly, the nonlinear response measured with the prestress protocol is insensitive to creep; although a large applied steady stress can lead to significant flow, this has no significant effect on either the linear or nonlinear response of the system. A simple model is presented to provide insight into these observations.
The purpose of this study was to characterize the viscoelastic behavior of diabetic and non-diabetic plantar soft tissue at six ulcer-prone/load-bearing locations beneath the foot to determine any changes that may play a role in diabetic ulcer formation and subsequent amputation in this predisposed population. Four older diabetic and four control fresh frozen cadaveric feet were each dissected to isolate plantar tissue specimens from the hallux, first, third, and fifth metatarsals, lateral midfoot, and calcaneus. Stress relaxation experiments were used to quantify the viscoelastic tissue properties by fitting the data to the quasi-linear viscoelastic (QLV) theory using two methods, a traditional frequency-insensitive approach and an indirect frequency-sensitive approach, and by measuring several additional parameters from the raw data including the rate and amount of overall relaxation. The stress relaxation response of both diabetic and non-diabetic specimens was unexpectedly similar and accordingly few of the QLV parameters for either fit approach and none of raw data parameters differed. Likewise, no differences were found between plantar locations. The accuracy of both fit methods was comparable, however, neither approach predicted the ramp behavior. Further, fit coefficients varied considerably from one method to the other, making it hard to discern meaningful trends. Future testing using alternate loading modes and intact feet may provide more insight into the role that time-dependent properties play in diabetic foot ulceration.
To further our understanding of the mechanisms underlying chest wall mechanics, we investigated the dynamic response of the isolated passive rat diaphragm strip. Stress adaptation of the tissue was measured from 0.05 to 60 s after subjecting the strips to strain steps of normalized strain amplitudes from 0.005 to 0.04. The tissue resistance (R), elastance (E), and hysteresivity (eta) were measured in the same range of amplitudes by sinusoidally straining the strip at frequencies from 0.03125 to 10 Hz. The stress (T) depended exponentially on the strain (epsilon) and relaxed and recovered linearly with the logarithm of time. E increased linearly with the logarithm of frequency and decreased with increasing amplitude. R fell hyperbolically with frequency and showed an amplitude dependence similar to that of E. To interpret the strong nonlinear behavior, we extended the viscoelastic model of Hildebrandt (J. Appl. Physiol. 28: 365-372, 1970) to include an exponential stress-strain relationship. Accordingly, the step response was described by T - Tr = Tr(e alpha delta epsilon - 1)(1 - gamma log t), where delta epsilon is the strain amplitude, Tr is the initial operating stress, alpha is a measure of the stress-strain nonlinearity, and gamma is the rate of stress adaptation. The oscillatory response of the model was computed by applying Fung's quasi-linear viscoelastic theory. This quasi-linear viscoelastic model fitted the step and oscillatory data fairly well but only if alpha depended negatively on delta epsilon, as might be expected in a plastic material.
Tracheal pressure, central airflow, and alveolar capsule pressures in cardiac lobes were measured in open-chest dogs during 0.1- to 20-Hz pseudorandom forced oscillations applied at the airway opening. In the interval 0.1-4.15 Hz, the input impedance data were fitted by four-parameter models including frequency-independent airway resistance and inertance and tissue parts featuring a marked negative frequency dependence of resistance and a slight elevation of elastance with frequency. The models gave good fits both in the control state and during histamine infusion. At the same time, the regional transfer impedances (alveolar pressure-to-central airflow ratios) showed intralobar and interlobar variabilities of similar degrees, which increased with frequency and were exaggerated during histamine infusion. Results of simulation studies based on a lung model consisting of a central airway and a number of peripheral units with airway and tissue parameters that were given independent wide distributions were in agreement with the experimental findings and showed that even an extremely inhomogeneous lung structure can produce virtually homogeneous mechanical behavior at the input.
When lungs are emptied during exhalation, peripheral airways close up. For people with lung disease, they may not reopen for a significant portion of inhalation, impairing gas exchange. A knowledge of the mechanisms that govern reinflation of collapsed regions of lungs is therefore central to the development of ventilation strategies for combating respiratory problems. Here we report measurements of the terminal airway resistance, Rt, during the opening of isolated dog lungs. When inflated by a constant flow, Rt decreases in discrete jumps. We find that the probability distribution of the sizes of the jumps and of the time intervals between them exhibit power-law behaviour over two decades. We develop a model of the inflation process in which 'avalanches' of airway openings are seen--with power-law distributions of both the size of avalanches and the time intervals between them--which agree quantitatively with those seen experimentally, and are reminiscent of the power-law behaviour observed for self-organized critical systems. Thus power-law distributions, arising from avalanches associated with threshold phenomena propagating down a branching tree structure, appear to govern the recruitment of terminal airspaces.
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.
Systems as diverse as genetic networks or the world wide web are best
described as networks with complex topology. A common property of many large
networks is that the vertex connectivities follow a scale-free power-law
distribution. This feature is found to be a consequence of the two generic
mechanisms that networks expand continuously by the addition of new vertices,
and new vertices attach preferentially to already well connected sites. A model
based on these two ingredients reproduces the observed stationary scale-free
distributions, indicating that the development of large networks is governed by
robust self-organizing phenomena that go beyond the particulars of the
individual systems.
The objective of this study was to produce linear and nonlinear viscoelastic models of eight major ligaments in the human ankle/foot complex for use in computer models of the lower extremity. The ligaments included in this study were the anterior talofibular (ATaF), anterior tibiofibular (ATiF), anterior tibiotalar (ATT), calcaneofibular (CF), posterior talofibular (PTaF), posterior tibiofibular (PTiF), posterior tibiotalar (PTT), and tibiocalcaneal (TiC) ligaments. Step relaxation and ramp tests were performed. Back-extrapolation was used to correct for vibration effects and the error introduced by the finite rise time in step relaxation tests. Ligament behavior was found to be nonlinear viscoelastic, but could be adequately modeled up to 15 percent strain using Fung's quasilinear viscoelastic (QLV) model. Failure properties and the effects of preconditioning were also examined.
Spatially or chemically isolated functional modules composed of several cellular components and carrying discrete functions are considered fundamental building blocks of cellular organization, but their presence in highly integrated biochemical networks lacks quantitative support. Here, we show that the metabolic networks of 43 distinct organisms are organized into many small, highly connected topologic modules that combine in a hierarchical manner into larger, less cohesive units, with their number and degree of clustering following a power law. Within Escherichia coli, the uncovered hierarchical modularity closely overlaps with known metabolic functions. The identified network architecture may be generic to system-level cellular organization.
Since its introduction by Fung (1972), the law of quasilinear viscoelasticity (QLV) was applied for several tissues including, among others, tendons (Haut and Little, 1972), ligaments (Geiger et al, 1983; Woo et al, 1981), articular cartilage under tension (Woo et al, 1980), heart valves (Rousseau et al, 1983), papillary muscle (Pinto and Patitucci, 1980), and smooth muscle (Price et al, 1977). Fung based the QLY law on the observation that, in several tissues, the time-varying response to a step change in uniaxial stretch \(
(from{\kern 1pt} \lambda = 1.0{\kern 1pt} to{\kern 1pt} \lambda > {\kern 1pt} 1.0)
\), can be expressed by: (xxx) where G(t) is the reduced relaxation function and T
(e)
(sI) is the elastic (immediate) response.
With mathematical and computational models furthering our understanding of lung mechanics, function and disease, this book provides an all-inclusive introduction to the topic from a quantitative standpoint. Focusing on inverse modeling, the reader is guided through the theory in a logical progression, from the simplest models up to state-of-the-art models that are both dynamic and nonlinear. Key tools used in biomedical engineering research, such as regression theory, linear and nonlinear systems theory, and the Fourier transform, are explained. Derivations of important physical principles, such as the Poiseuille equation and the wave speed equation, from first principles are also provided. Example applications to experimental data throughout illustrate physiological relevance, whilst problem sets at the end of each chapter provide practice and test reader comprehension. This book is ideal for biomedical engineering and biophysics graduate students and researchers wishing to understand this emerging field.
A generic mechanism for the ubiquitous phenomenon of 1/f noise is
reviewed. This mechanism arises in random processes expressible as a
product of several random variables. Under mild conditions this product
form leads to the log-normal distribution which we show
straightforwardly generates 1/f noise. Thus, 1/f noise is tied directly
to a probability limit distribution. A second mechanism involving
scaling is introduced to provide a natural crossover from log-normal to
inverse power-law behavior and generates 1/fα noise
instead of pure 1/f noise. Examples of these distributions and the
transitions between them are drawn from such diverse areas as economics,
scientific productivity, bronchial structure and cardiac activity.
The separation of electrophoresing DNA molecules of varying lengths, actuated by their size-dependent collision with a stationary obstacle or array of obstacles, has recently gained prominence. To gain insight into how a chain initially unravels subsequent to a polymer-obstacle collision, we investigate the stretching dynamics of a tethered polymer chain initially at equilibrium, following the imposition of a uniform flow of solvent. The solution for the Rouse model of the polymer chain is obtained via an analysis into normal modes. We examine the consequences of finite chain extensibility by performing Brownian dynamics simulations of the wormlike chain model, which describes DNA elasticity. Detailed results are presented for the propagation of tension with time in Rouse and wormlike chains. Our results suggest the diffusive propagation of tension in Rouse chains, whereas a convective mechanism of tension propagation in wormlike chains under conditions of strong flow is demonstrated.
Very large amplitude pseudorandom uniaxial perturbations containing frequencies between 0.125 and 12.5 Hz were applied to five dog lung tissue strips. Three different nonlinear block-structured models in nonparametric form were fit to the data. These models consisted of (1) a static nonlinear block followed by a dynamic linear block (Hammerstein model); (2) the same blocks in reverse order (Wiener model); and (3) the blocks in parallel (parallel model). Both the Hammerstein and Wiener models performed well for a given input perturbation, each accounting for greater than 99% of the measured stress signal variance. However, the Wiener and parallel model parameters showed some dependence on the strain amplitude and the mean stress. In contrast, a single Hammerstein model accounted for the data at all strain amplitudes and operating stresses. A Hammerstein model featuring a fifth-order polynomial static nonlinearity and a linear impulse response function of 1 s duration accounted for the most output variance (99.84% 0.13%, mean standard deviations for perturbations of 50% strain at 1.5 kPa stress). The static nonlinear behavior of the Hammerstein model also matched the quasistatic stress–strain behavior obtained at the same strain amplitude and operating stress. These results show that the static nonlinear behavior of the dog lung tissue strip is separable from its linear dynamic behavior. 1998 Biomedical Engineering Society.
PAC98: 8745Bp, 8710+e
A comparative study of the Schottky barrier height variation on sulfide‐treated GaAs(100) surfaces with low work function metal contacts was made using current‐voltage and capacitance‐voltage measurements. Five different wet chemical sulfide treatments were found to cause little variation in the Sm (0.72 eV) and Mg (0.59 eV) Schottky barrier heights, but caused significant variation in the Al (0.58–0.75 eV) barrier heights when compared to the untreated control diodes. A low temperature (160 °C) anneal was found to cause variation in all of these, uniformly raising the barrier heights of the Sm (+0.07 eV) and Al (+0.04 eV) contacts, and degrading the Mg contacts. These results demonstrate the critical importance of both the reaction specifics and the stability of the interface on the Schottky barrier height.
In this work we introduce mechanical networks which highlight the relation between viscoelastic and structural properties of chemical systems at the sol-gel transition. Cross-linking polymers at the gel point show in general a power law behavior of the complex modulus, ie., G*(w) = (iw)textasciicircuma (0 textless a textless 1), which is related to the (constitutive) gel equation. We present a mechanical ladder model whose stress-strain relation obeys the gel equation with a =⅟2 and which consists of an infinite number of springs and dashpots. Furthermore, we investigate terminated ladder arrangements which mimic pre- and postgel behavior. To elucidate the complex dependence of a on structural properties which one observes for systems near to the gel point, we analyze mechanical fractal networks.
Although the active properties of airway smooth muscle (ASM) have garnered much modeling attention, the passive mechanical properties are not as well studied. In particular, there are important dynamic effects observed in passive ASM, particularly strain-induced fluidization, which have been observed both experimentally and in models; however, to date these models have left an incomplete picture of the biophysical, mechanistic basis for these behaviors. The well-known Huxley cross-bridge model has for many years successfully described many of the active behaviors of smooth muscle using sliding filament theory; here, we propose to extend this theory to passive biological soft tissue, particularly ASM, using as a basis the attachment and detachment of cross-linker proteins at a continuum of cross-linker binding sites. The resulting mathematical model exhibits strain-induced fluidization, as well as several types of force recovery, at the same time suggesting a new mechanistic basis for the behavior. The model is validated by comparison to new data from experimental preparations of rat tracheal airway smooth muscle. Furthermore, experiments in noncontractile tissue show qualitatively similar behavior, suggesting support for the protein-filament theory as a biomechanical basis for the behavior.
Neurofilaments are found in abundance in the cytoskeleton of neurons, where they act as an intracellular framework protecting the neuron from external stresses. To elucidate the nature of the mechanical properties that provide this protection, we measure the linear and nonlinear viscoelastic properties of networks of neurofilaments. These networks are soft solids that exhibit dramatic strain stiffening above critical strains of 30-70%. Surprisingly, divalent ions such as Mg(2+), Ca(2+), and Zn(2+) act as effective cross-linkers for neurofilament networks, controlling their solidlike elastic response. This behavior is comparable to that of actin-binding proteins in reconstituted filamentous actin. We show that the elasticity of neurofilament networks is entropic in origin and is consistent with a model for cross-linked semiflexible networks, which we use to quantify the cross-linking by divalent ions.
We report a scaling law that governs both the elastic and frictional properties of a wide variety of living cell types, over a wide range of time scales and under a variety of biological interventions. This scaling identifies these cells as soft glassy materials existing close to a glass transition, and implies that cytoskeletal proteins may regulate cell mechanical properties mainly by modulating the effective noise temperature of the matrix. The practical implications are that the effective noise temperature is an easily quantified measure of the ability of the cytoskeleton to deform, flow, and reorganize.
Airway smooth muscle (ASM) is cyclically stretched during breathing, even in the active state, yet the factors determining its dynamic force-length behavior remain incompletely understood. We developed a model of the activated ASM strip and compared its behavior to that observed in strips of rat trachealis muscle stimulated with methacholine. The model consists of a nonlinear viscoelastic element (Kelvin body) in series with a force generator obeying the Hill force-velocity relationship. Isometric force in the model is proportional to the number of bound crossbridges, the attachment of which follows first-order kinetics. Crossbridges detach at a rate proportional to the rate of change of muscle length. The model accurately accounts for the experimentally observed transient and steady-state oscillatory force-length behavior of both passive and activated ASM. However, the model does not predict the sustained decrement in isometric force seen when activated strips of ASM are subjected briefly to large stretches. We speculate that this force decrement reflects some mechanism unrelated to the cycling of crossbridges, and which may be involved in the reversal of bronchoconstriction induced by a deep inflation of the lungs in vivo.
Until now, there has been no comprehensive mathematical model of the nonlinear viscoelastic stress-strain behavior of extraocular muscles (EOMs). The present study describes, with the use of a quasilinear viscoelastic (QLV) model, the nonlinear, history-dependent viscoelastic properties and elastic stress-strain relationship of EOMs.
Six oculorotary EOMs were obtained fresh from a local abattoir. Longitudinally oriented specimens were taken from different regions of the EOMs and subjected to uniaxial tensile, relaxation, and cyclic loading testing with the use of an automated load cell under temperature and humidity control. Twelve samples were subjected to uniaxial tensile loading with 1.7%/s strain rate until failure. Sixteen specimens were subjected to relaxation studies over 1500 seconds. Cyclic loading was performed to validate predictions of the QLV model characterized from uniaxial tensile loading and relaxation data.
Uniform and highly repeatable stress-strain behavior was observed for 12 specimens extracted from various regions of all EOMs. Results from 16 different relaxation trials illustrated that most stress relaxation occurred during the first 30 to 60 seconds for 30% extension. Elastic and reduced relaxation functions were fit to the data, from which a QLV model was assembled and compared with cyclic loading data. Predictions of the QLV model agreed with observed peak cyclic loading stress values to within 8% for all specimens and conditions.
Close agreement between the QLV model and the relaxation and cyclic loading data validates model quantification of EOM mechanical properties and will permit the development of accurate overall models of mechanics of ocular motility and strabismus.
We recently proposed an eight-parameter model of the respiratory system to account for its mechanical behavior when flow is interrupted during passive expiration. The model consists of two four-parameter submodels representing the lungs and the chest wall, respectively. The lung submodel consists of an airways resistance together with elements embodying the viscoelastic properties of the lung tissues. The chest wall submodel has similar structure. We estimated the parameters of the model from data obtained in four normal, anesthetized, paralyzed, tracheostomized mongrel dogs. This model explains why lung tissue and chest wall resistances should be markedly frequency dependent at low frequencies and also permits a physiological interpretation of resistance measurements provided by the flow interruption method.
To investigate the contribution of nonlinear tissue viscoelasticity to the dynamic behavior of lung, time and frequency responses of isolated parenchymal strips of degassed dog lungs were investigated. The strips were subjected to loading and unloading stretch steps for 60 s and to sinusoidal oscillations (0.03-3 Hz) of different stretch amplitudes (delta lambda = 0.05, 0.1, and 0.2) and at different operating stresses (T(o) = 0.5, 1, and 2 kPa). Elastance (E) increased linearly with the logarithm of frequency (approximately 10% per frequency decade), and resistance (R) decreased hyperbolically with frequency. Both E and R varied little with delta lambda but they increased proportionally with T(o). Hysteresivity (eta = R x 2 pi x frequency/E) ranged from 0.07 to 0.10. In agreement with the frequency response, the magnitude of the unit step response increased with T(o) and was higher when loading than when unloading, and the stress relaxation ratio (approximately 0.10) did not vary greatly with T(o) or with delta lambda. The time and frequency behavior of the strips were interpreted in terms of the quasilinear viscoelastic model of Navajas et al. (J. Appl. Physiol. 73:2681-2692, 1992). The model explains most of the dependencies of step and oscillatory responses on the measurement conditions, in particular the proportional dependence of E and R on T(o). According to the model, about two-thirds of energy dissipated during oscillation arises from tissue viscoelasticity. The remaining dissipated energy could be accounted for by plasticity. Thus the effect of nonlinear elasticity on the dynamic behavior of lung tissue can be empirically described by a simple quasilinear model characterized by only two parameters.
The mechanical properties of lung tissue are important contributors to both the elastic and dissipative properties of the entire organ at normal breathing frequencies. A number of detailed studies have shown that the stress adaptation in the tissue of the lung following a step change in volume is very accurately described by the functiont
−k for some small positive constantk. We applied step increases in length to lung parenchymal strips and found the ensuing stress recovery to be extremely accurately described byt
−k over almost 3 decades of time, despite the quasi-static stress-length characteristics of the strips being highly nonlinear. The corresponding complex impedance of lung tissue was found to have a magnitude that varied inversely with frequency. We note that this is highly reminiscent of a phenomenon known as 1/f noise, which has been shown to occur ubiquitously throughout the natural world. 1/f noise has been postulated to be a reflection of the complexity of the system that produces it, something like a central limit theorem for dynamic systems. We have therefore developed the hypothesis that thet
−k
nature of lung tissue stress adaptation follows from the fact that lung tissue itself is composed of innumerable components that interact in an extremely rich and varied manner. Thus, although the constantk is no doubt determined by the particular constituents of the tissue, we postulate that the actual functional form of the stress adaptation is not.
The rheological properties of lung tissue are complex and nonlinear and contribute significantly to both the elastic and dissipative mechanical properties of the lung. Nevertheless, there remain large gaps in our understanding of precisely how the bulk rheological behavior of lung tissue is linked to the properties of its constituents and their interactions. In this paper a model is developed that attempts to provide such a link. The model consists of a sheet of randomly aligned fibers whose orientations are constantly changing due to thermal motion. When the sheet is suddenly stretched uniaxially, the fibers align themselves preferentially in the direction of strain. However, as a result of the continual thermal motion of the fibers, there is a net transfer of momentum between the fibers and the rest of the tissue. This produces a restoring force in the tissue sheet. The thermal motion also makes the fibers gradually revert back to a random orientation, so that the strain-generated stress within the tissue decays asymptotically to zero. It is shown that the behavior of this model closely approximates quasilinear viscoelasticity, in which the static stress–strain behavior is separable from the dynamic stress relaxation behavior. © 1998 Biomedical Engineering Society.
PAC98: 8745-k, 8710+e, 8722Pg
While measurements of lung tissue mechanics have been made in several species, relatively little has been reported in the mouse. Moreover, whether in vivo measurements truly reflect tissue properties is somewhat controversial. We measured complex impedance of the mouse respiratory system in vivo using a ventilator, which applies a multiple frequency volume signal to the airway opening. A constant phase model was fit to the impedance data, yielding parameters for tissue damping (G) and elastance (H). Hysteresivity (eta) was calculated as G/H. Quasistatic pressure-volume (P-V) curves were obtained during deflation. In vitro measurements of complex impedance and stress-strain curves were made in lung tissue strips. Values of eta were significantly higher in vivo than in vitro (0.111 +/- 0.004 versus 0.042 +/- 0.003). The higher values of eta in vivo may represent the effects of airway heterogeneities, surfactant, or changes in alveolar geometry. Measurement of mechanics in the tissue strip offers a better assessment of pure tissue properties.
Knowledge of strain-rate sensitivity of soft tissue viscoelastic and nonlinear elastic properties is important for accurate predictions of biomechanical behavior and for quantitative assessment of the effects of disease or surgical/pharmaceutical intervention. Soft tissues are known to exhibit mild rate sensitivity, but experimental artifacts related to testing system control can confound estimation of these effects. "Perfect" ramp-and-hold stress-relaxation tests become difficult at high strain rates because of problems related to undershoot/overshoot error and vibrations. These errors can introduce unwanted bias into parameter estimation methods that rely on idealizations of the applied ramp-and-hold displacement. To address these problems, we describe a new method for estimating quasilinear viscoelastic (QLV) parameters that directly fits the QLV constitutive model to the actual point-wise stress-time history of the test, using an adaptive grid refinement (AGR) global optimization algorithm. This new method significantly improves the accuracy and predictivity of QLV parameter estimates for heart valve tissues, compared to traditional methods that use idealized displacement data. We estimated QLV parameters for aortic valve tissue over a range of physiologic displacement rates, finding that the viscoelastic content parameter (C) increased slightly with increasing strain rate, but the fast (tau1) and slow (tau2) time constants were strain rate insensitive.
When lung tissue is subjected to a step in strain, it exhibits a stress adaptation profile that is a power function of time. Furthermore, this power function is independent of the strain, even though the quasi-static stress-strain relationship of the tissue is highly nonlinear. Such behavior is known as quasi-linear viscoelasticity, but its mechanistic basis is unknown. We describe a model of soft tissue rheology based on the sequential recruitment of Maxwell bodies. The model is homogeneous in its elemental constitutive properties, yet predicts both power-law stress relaxation and quasi-linear viscoelasticity even when the stress-strain behavior of the model is nonlinear. The model suggests that stress relaxation in lung tissue could occur via a sequence of micro-rips that cause stresses to be passed from one local stress bearing region to another.
The fitting of quasi-linear viscoelastic (QLV) constitutive models to material data often involves somewhat cumbersome numerical convolution. A new approach to treating quasi-linearity in 1-D is described and applied to characterize the behavior of reconstituted collagen. This approach is based on a new principle for including nonlinearity and requires considerably less computation than other comparable models for both model calibration and response prediction, especially for smoothly applied stretching. Additionally, the approach allows relaxation to adapt with the strain history. The modeling approach is demonstrated through tests on pure reconstituted collagen. Sequences of "ramp-and-hold" stretching tests were applied to rectangular collagen specimens. The relaxation force data from the "hold" was used to calibrate a new "adaptive QLV model" and several models from literature, and the force data from the "ramp" was used to check the accuracy of model predictions. Additionally, the ability of the models to predict the force response on a reloading of the specimen was assessed. The "adaptive QLV model" based on this new approach predicts collagen behavior comparably to or better than existing models, with much less computation.
With every beat of the heart, inflation of the lung or peristalsis of the gut, cell types of diverse function are subjected to substantial stretch. Stretch is a potent stimulus for growth, differentiation, migration, remodelling and gene expression. Here, we report that in response to transient stretch the cytoskeleton fluidizes in such a way as to define a universal response class. This finding implicates mechanisms mediated not only by specific signalling intermediates, as is usually assumed, but also by non-specific actions of a slowly evolving network of physical forces. These results support the idea that the cell interior is at once a crowded chemical space and a fragile soft material in which the effects of biochemistry, molecular crowding and physical forces are complex and inseparable, yet conspire nonetheless to yield remarkably simple phenomenological laws. These laws seem to be both universal and primitive, and thus comprise a striking intersection between the worlds of cell biology and soft matter physics.
Descriptions of the mechanical behaviors of articular cartilage and their correlations with collagen, proteoglycan, water, and ions are summarized, with particular emphasis on understanding the osmotic effect inside the tissue. First, a descriptive explanation is presented of the biphasic theory required to understand how interstitial water contributes toward the viscoelastic behavior of any hydrated soft tissue. Then, the famous osmotic effect in charged, hydrated soft tissue is interpreted in light of the triphasic mixture theory framework. In the introduction of mechanical testing methods, our emphasis is on the popular indentation technique, which can determine the material properties of cartilage in situ or in vivo. The widely accepted indentation analysis solutions in cartilage biomechanics history are summarized and evaluated. At the end of this paper, a new generalized correspondence principle between charged, hydrated soft tissue and linear, isotropic, elastic material (i.e., elasticity theory) is introduced. This principle makes the employment of triphasic theory as straightforward as using an elasticity theory to solve any equilibrium problem where the elasticity theory can be used to model the material. By using this generalized correspondence principle, the fixed charge density of bovine cartilage has been simply and conveniently calculated from the indentation testing data. The results of proteoglycan content from this mechanical test are remarkably consistent with those from standard biochemical assay. This new correspondence principle significantly improves the power of indentation tests in the determination of mechanoelectrochemical properties of articular cartilage.
On the structural origin of the quaslinear viscoelasrtic behavior of tissue. In: Frontier in Biome-chanics
- Y Lanir
Lanir, Y. On the structural origin of the quaslinear viscoelasrtic behavior of tissue. In: Frontier in Biome-chanics, edited by E. W. Schmid-Schonbein, S. L. Y.
The quasi-linear viscoelastic properties of diabetic and non-diabetic plantar soft tissue Hierarchical organization of modularity in metabolic networks Mesoscopic pictures of the sol-gel transition: ladder models and fractal networks
- S Pai
- W R Ledoux
- A L Somera
- D A Mongru
- Z N Oltvai
- A L Barabasi
Pai, S., and W. R. Ledoux. The quasi-linear viscoelastic properties of diabetic and non-diabetic plantar soft tissue. Ann. Biomed. Eng. 39:1517–1527, 2011. 26 Ravasz, E., A. L. Somera, D. A. Mongru, Z. N. Oltvai, and A. L. Barabasi. Hierarchical organization of modularity in metabolic networks. Science 297:1551–1555, 2002. 27 Schiessel, H., and A. Blumen. Mesoscopic pictures of the sol-gel transition: ladder models and fractal networks. Macromolecules 28:4013–4019, 1995.
Non-parametric block-structured modeling of lung tissue strip mechanics Dynamic response of the isolated pas-sive rat diaphragm strip
- G N Maksym
- R E Kearney
- J H Bates
- S Mijailovich
- G M Glass
- D Stamenovic
- J J Fredberg
Maksym, G. N., R. E. Kearney, and J. H. Bates. Non-parametric block-structured modeling of lung tissue strip mechanics. Ann. Biomed. Eng. 26:242–252, 1998. 23 Navajas, D., S. Mijailovich, G. M. Glass, D. Stamenovic, and J. J. Fredberg. Dynamic response of the isolated pas-sive rat diaphragm strip. J. Appl. Physiol. 73:2681–2692, 1992.