Introducing mesoscopic information into constitutive equations for arterial walls

We propose a new elastic constitutive law for arterial tissue in which the limiting polymeric chain extensibility of both collagen and elastin fibres is accounted for. The elastic strain-energy function is separated additively into two parts: an isotropic contribution associated with the matrix (incorporating the elastin fibre network) and an anisotropic one associated with the collagen fibres. Information on the limiting extensibility in each case provides some mesoscopic input into the model. The (logarithm-based) model is compared with the Fung-Demiray exponential model and certain other recently proposed models. Some aspects of the elastic response under extension and inflation of a thin-walled circular cylindrical tube (the artery) are then examined and compared with the corresponding response of a rubber-like tube. We point out that our model, when both isotropic and anisotropic terms are included, can be developed to accommodate changing mechanical properties associated with degradation of the elastin and collagen by considering the material constants that define the limit of chain extensibility to evolve in time.     

A membrane model from implicit elasticity theory: application to visceral pleura

A Fungean solid is derived for membranous materials as a body defined by isotropic response functions whose mathematical structure is that of a Hookean solid where the elastic constants are replaced by functions of state derived from an implicit, thermodynamic, internal energy function. The theory utilizes Biot’s (Lond Edinb Dublin Philos Mag J Sci 27:468–489, 1939) definitions for stress and strain that, in one-dimension, are the stress/strain measures adopted by Fung (Am J Physiol 28:1532–1544, 1967) when he postulated what is now known as Fung’s law. Our Fungean membrane model is parameterized against a biaxial data set acquired from a porcine pleural membrane subjected to three, sequential, proportional, planar extensions. These data support an isotropic/deviatoric split in the stress and strain-rate hypothesized by our theory. These data also demonstrate that the material response is highly nonlinear but, otherwise, mechanically isotropic. These data are described reasonably well by our otherwise simple, four-parameter, material model.     

Constitutive modeling of an electrospun tubular scaffold used for vascular tissue engineering

In this study, we sought to model the mechanical behavior of an electrospun tubular scaffold previously reported for vascular tissue engineering with hyperelastic constitutive equations. Specifically, the scaffolds were made by wrapping electrospun polycaprolactone membranes that contain aligned fibers around a mandrel in such a way that they have microstructure similar to the native arterial media. The biaxial stress-stretch data of the scaffolds made of moderately or highly aligned fibers with three different off-axis fiber angles \(\alpha \) (30\(^\circ \), 45\(^\circ \), and 60\(^\circ \)) were fit by a phenomenological Fung model and a series of structurally motivated models considering fiber directions and fiber angle distributions. In particular, two forms of fiber strain energy in the structurally motivated model for a linear and a nonlinear fiber stress–strain relation, respectively, were tested. An isotropic neo-Hookean strain energy function was also added to the structurally motivated models to examine its contribution. The two forms of fiber strain energy did not result in significantly different goodness of fit for most groups of the scaffolds. The absence of the neo-Hookean term in the structurally motivated model led to obvious nonlinear stress-stretch fits at a greater axial stretch, especially when fitting data from the scaffolds with a small \(\alpha \). Of the models considered, the Fung model had the overall best fitting results; its applications are limited because of its phenomenological nature. Although a structurally motivated model using the nonlinear fiber stress–strain relation with the neo-Hookean term provided fits comparably as good as the Fung model, the values of its model parameters exhibited large within-group variations. Prescribing the dispersion of fiber orientation in the structurally motivated model, however, reduced the variations without compromising the fits and was thus considered to be the best structurally motivated model for the scaffolds. It appeared that the structurally motivated models could be further improved for fitting the mechanical behavior of the electrospun scaffold; fiber interactions are suggested to be considered in future models.     

Identification and determination of material properties for porohyperelastic analysis of large arteries.

A "porohyperelastic" (PHE) material model is described and the theoretical framework presented that allows identification of the necessary material properties functions for soft arterial tissues. A generalized Fung form is proposed for the PHE constitutive law in which the two fundamental Lagrangian material properties are the effective strain energy density function, W(e), and the hydraulic permeability, kij. The PHE model is based on isotropic forms using W(e) = Ue (phi) = 1/2C0(e phi - 1) and the radial component of permeability, kRR = kRR(phi), with phi = C1'(I1 - 3) + C2'(I2 - 3) + K'(J - 1)2. The methods for determination of these material properties are illustrated using experimental data from in situ rabbit aortas. Three experiments are described to determine parameters in Ue and kRR for the intima and media of the aortas, i.e., (1) undrained tests to determine C0, C1', and C2'; (2) drained tests to determine K'; and (3) steady-state pressurization tests of intact and de-endothelialized vessels to determine intimal and medial permeability (adventitia removed in these models). Data-reduction procedures are presented that allow determination of kRR for the intima and media and Ue for the media using experimental data. The effectiveness and accuracy of these procedures are studied using input "data" from finite element models generated with the ABAQUS program. The isotropic theory and data-reduction methods give good approximations for the PHE properties of in situ aortas. These methods can be extended to include arterial tissue remodeling and anisotropic behavior when appropriate experimental data are available.     

Network dynamics of the nonlinear power-law relaxation of cell cortex

Living cells are known to exhibit universal power-law rheological behaviors, but their underlying biomechanical principles are still not fully understood. Here, we present a network dynamics picture to decipher the nonlinear power-law relaxation of cortical cytoskeleton. Under step strains, we present a scaling relation between instantaneous differential stiffness and external stress as a result of chain reorientation. Then, during the relaxation, we show how the scaling law theoretically originates from an exponential form of cortical disorder, with the scaling exponent decreased by the imposed strain or crosslinker density in the nonlinear regime. We attribute this exponent variation to the molecular realignment along the stretch direction or the transition of network structure from in-series to in-parallel modes, both solidifying the network toward our one-dimensional theoretical limit. In addition, the rebinding of crosslinkers is found to be crucial for moderating the relaxation speed under small strains. Together with the disorder nature, we demonstrate that the structural effects of networks provide a unified interpretation for the nonlinear power-law relaxation of cell cortex, and may help to understand cell mechanics from the molecular scale.     

The role of fiber-matrix interactions in a nonlinear fiber-reinforced strain energy model of tendon

The objective of this study was to develop a nonlinear and anisotropic three-dimensional mathematical model of tendon behavior in which the structural components of fibers, matrix, and fiber-matrix interactions are explicitly incorporated and to use this model to infer the contributions of these structures to tendon mechanical behavior. We hypothesized that this model would show that: (i) tendon mechanical behavior is not solely governed by the isotropic matrix and fiber stretch, but is also influenced by fiber-matrix interactions; and (ii) shear fiber-matrix interaction terms will better describe tendon mechanical behavior than bulk fiber-matrix interaction terms. Model versions that did and did not include fiber-matrix interaction terms were applied to experimental tendon stress-strain data in longitudinal and transverse orientations, and the R2 goodness-of-fit was evaluated. This study showed that models that included fiber-matrix interaction terms improved the fit to longitudinal data (R2(toe) = 0.88, R2(Lin) = 0.94) over models that only included isotropic matrix and fiber stretch terms (R2(Toe) = 0.36, R2(Lin) = 0.84). Shear fiber-matrix interaction terms proved to be responsible for the best fit to data and to contribute to stress-strain nonlinearity. The mathematical model of tendon behavior developed in this study showed that fiber-matrix interactions are an important contributor to tendon behavior The more complete characterization of mechanical behavior afforded by this mathematical model can lead to an improved understanding of structure-function relationships in soft tissues and, ultimately, to the development of tissue-engineered therapies for injury or degeneration.     

Material Properties and Constitutive Modeling of Infant Porcine Cerebellum Tissue in Tension at High Strain Rate

Background

The mechanical characterization of infant porcine cerebellum tissue in tension at high strain rate is crucial for modeling traumatic cerebellum injury, which is in turn helpful for understanding the biomechanics of such injuries suffered in traffic accidents.

Material and Method

In this study, the infant porcine cerebellum tissue was given three loading velocities, ie, 2s^-1, 20s^-1 and 100s^-1 with up to 30% strain to investigate the tensile properties. At least six tensile tests for each strain rate were validly performed. Fung, Gent, Ogden and exponential models were applied to fit the constitutive equations, so as to obtain material parameters from the experimental data.

Results

The Lagrange stress of infant porcine cerebellum tissue in tension appeared to be no more than 3000Pa at each loading velocity. More specifically, the Lagrange stress at 30% strain was (393.7±84.4)Pa, (928.3±56.3)Pa and (2582.4±282.2)Pa at strain rates of 2s^-1, 20s^-1 and 100s^-1, respectively. Fung (0.833≤R²≤0.924), Gent (0.797≤R²≤0.875), Ogden (0.859≤R²≤0.944) and exponential (0.930≤R²≤0.972) models provided excellent fitting to experimental data up to 30% strain.

Conclusions

The infant cerebellum tissue shows a stiffer response with increase of the loading speed, indicating a strong strain-rate sensitivity. This study will enrich the knowledge on the material properties of infant brain tissue, which may augment the biofidelity of finite element model of human pediatric cerebellum.     

Artery buckling analysis using a four-fiber wall model

Artery bent buckling has been suggested as a possible mechanism that leads to artery tortuosity, which is associated with aging, hypertension, atherosclerosis, and other pathological conditions. It is necessary to understand the relationship between microscopic wall structural changes and macroscopic artery buckling behavior. To this end, the objectives of this study were to develop arterial buckling equations using a microstructure-based 4-fiber reinforced wall model, and to simulate the effects of vessel wall microstructural changes on artery buckling. Our results showed that the critical pressure increased nonlinearly with the axial stretch ratio, and the 4-fiber model predicted higher critical buckling pressures than what the Fung model predicted. The buckling equation using the 4-fiber model captured the experimentally observed reduction of critical pressure induced by elastin degradation and collagen fiber orientation changes in the arterial wall. These results improve our understanding of arterial stability and its relationship to microscopic wall remodeling, and the model provides a useful tool for further studies.     

Parameter sensitivity analysis and improvement of a two-layer arterial wall model

The nonlinear two-layer arterial wall model introduced by von Maltzahn, et al. [11] is subjected to a rigorous parameter sensitivity and range of validity analysis. The model is based on the assumption that in large muscular conduit arteries the two mechanically significant layers are media and adventitia. Using curve-fitting techniques, the media is determined to be isotropic and the adventitia to be anisotropic. As a result of the range of validity analysis, the polynomial relationship for the energy density function of the media is changed to an exponential relationship. This leads to new coefficients for the polynomial of the adventitia. All coefficients have specific mechanical meanings. The parameter sensitivity analysis demonstrates convincingly that all model parameters are significantly important.     

On the sensitivity of wall stresses in diseased arteries to variable material properties

Accurate estimates of stress in an atherosclerotic lesion require knowledge of the material properties of its components (e.g., normal wall, fibrous plaque, calcified regions, lipid pools) that can only be approximated. This leads to considerable uncertainty in these computational predictions. A study was conducted to test the sensitivity of predicted levels of stress and strain to the parameter values of plaque used in finite element analysis. Results show that the stresses within the arterial wall, fibrous plaque, calcified plaque, and lipid pool have low sensitivities for variation in the elastic modulus. Even a +/- 50% variation in elastic modulus leads to less than a 10% change in stress at the site of rupture. Sensitivity to variations in elastic modulus is comparable between isotropic nonlinear, isotropic nonlinear with residual strains, and transversely isotropic linear models. Therefore, stress analysis may be used with confidence that uncertainty in the material properties generates relatively small errors in the prediction of wall stresses. Either isotropic nonlinear or anisotropic linear models provide useful estimates, however the predictions in regions of stress concentration (e.g., the site of rupture) are somewhat more sensitive to the specific model used, increasing by up to 30% from the isotropic nonlinear to orthotropic model in the present example. Changes resulting from the introduction of residual stresses are much smaller.     

Accuracy of alternative representations for integrated biochemical systems

The Michaelis-Menten formalism often provides appropriate representations of individual enzyme-catalyzed reactions in vitro but is not well suited for the mathematical analysis of complex biochemical networks. Mathematically tractable alternatives are the linear formalism and the power-law formalism. Within the power-law formalism there are alternative ways to represent biochemical processes, depending upon the degree to which fluxes and concentrations are aggregated. Two of the most relevant variants for dealing with biochemical pathways are treated in this paper. In one variant, aggregation leads to a rate law for each enzyme-catalyzed reaction, which is then represented by a power-law function. In the other, aggregation produces a composite rate law for either net rate of increase or net rate of decrease of each system constituent; the composite rate laws are then represented by a power-law function. The first variant is the mathematical basis for a method of biochemical analysis called metabolic control, the latter for biochemical systems theory. We compare the accuracy of the linear and of the two power-law representations for networks of biochemical reactions governed by Michaelis-Menten and Hill kinetics. Michaelis-Menten kinetics are always represented more accurately by power-law than by linear functions. Hill kinetics are in most cases best modeled by power-law functions, but in some cases linear functions are best. Aggregation into composite rate laws for net increase or net decrease of each system constituent almost always improves the accuracy of the power-law representation. The improvement in accuracy is one of several factors that contribute to the wide range of validity of this power-law representation. Other contributing factors that are discussed include the nonlinear character of the power-law formalism, homeostatic regulatory mechanisms in living systems, and simplification of rate laws by regulatory mechanisms in vivo.     

Spherical indentation method for determining the constitutive parameters of hyperelastic soft materials

A comprehensive study on the spherical indentation of hyperelastic soft materials is carried out through combined theoretical, computational, and experimental efforts. Four widely used hyperelastic constitutive models are studied, including neo-Hookean, Mooney–Rivlin, Fung, and Arruda–Boyce models. Through dimensional analysis and finite element simulations, we establish the explicit relations between the indentation loads at given indentation depths and the constitutive parameters of materials. Based on the obtained results, the applicability of Hertzian solution to the measurement of the initial shear modulus of hyperelastic materials is examined. Furthermore, from the viewpoint of inverse problems, the possibility to measure some other properties of a hyperelastic material using spherical indentation tests, e.g., locking stretch, is addressed by considering the existence, uniqueness, and stability of the solution. Experiments have been performed on polydimethylsiloxane to validate the conclusions drawn from our theoretical analysis. The results reported in this study should help identify the extent to which the mechanical properties of hyperelastic materials could be measured from spherical indentation tests.     

Application of a fiber-reinforced continuum theory to multiple deformations of the annulus fibrosus.

Accurate tissue stress predictions for the annulus fibrosus are essential for understanding the factors that cause or contribute to disc degeneration and mechanical failure. Current computational models used to predict in vivo disc stresses utilize material laws for annular tissue that are not rigorously validated against experimental data. Consequently, predictions of disc stress resulting from physical activities may be inaccurate and therefore unreliable as a basis for defining mechanical-biologic injury criteria. To address this need we present a model for the annulus as an isotropic ground substance reinforced with two families of collagen fibers, and an approach for determining the material constants by simultaneous consideration of multiple experimental data sets. Two strain energy functions for the annulus are proposed and used in the theory to derive the constitutive equations relating the stress to pure stretch deformations. These equations are applied to four distinct experimental protocols and the material constants are determined from a simultaneous, nonlinear regression analysis. Good agreement between theory and experiment is achieved when the invariants are included within multiple, separate exponentials in the strain energy function.     

On the influence of wall properties and Poiseuille flow in peristalsis

The present study extends the two-dimensional analysis of peristaltic motion by Fung and Yih to include an elastic or viscoelastic wall, and a Poiseuille flow. This fluid-solid interaction problem is investigated by considering equations of motion of both the fluid and the deformable boundaries. The wall characteristics appear in their equations of motion, which are solved to represent boundary conditions of fluid motion. The influence of Poiseuille flow on pure peristalisis is also investigated.

The phenomenon of the ‘mean flow reversal’ is found to exist both at the center and at the boundaries of the channel. When the walls of the channel are elastic, pure peristalsis involves flow reversal only at the center. This position may shift drastically to the boundaries, if viscous damping forces are considered.     

Spherical indentation of soft matter beyond the Hertzian regime: numerical and experimental validation of hyperelastic models

The lack of practicable nonlinear elastic contact models frequently compels the inappropriate use of Hertzian models in analyzing indentation data and likely contributes to inconsistencies associated with the results of biological atomic force microscopy measurements. We derived and validated with the aid of the finite element method force-indentation relations based on a number of hyperelastic strain energy functions. The models were applied to existing data from indentation, using microspheres as indenters, of synthetic rubber-like gels, native mouse cartilage tissue, and engineered cartilage. For the biological tissues, the Fung and single-term Ogden models achieved the best fits of the data while all tested hyperelastic models produced good fits for the synthetic gels. The Hertz model proved to be acceptable for the synthetic gels at small deformations (strain < 0.05 for the samples tested), but not for the biological tissues. Although this finding supports the generally accepted view that many soft materials can be assumed to be linear elastic at small deformations, the nonlinear models facilitate analysis of intrinsically nonlinear tissues and large-strain indentation behavior.     

Nonlinear relativistic quantum theory of Cherenkov emission of longitudinal Langmuir waves in a plasma

A nonlinear relativistic quantum theory of stimulated Cherenkov emission of longitudinal waves by a relativistic monoenergetic electron beam in a cold isotropic plasma is presented. The theory makes use of a quantum model based on the Klein-Gordon equation. The instability growth rates are obtained in the linear approximation and are shown to go over to the familiar growth rates in the classical limit. The mechanisms for the nonlinear saturation of relativistic Cherenkov beam instabilities are described with allowance for quantum effects, and the corresponding analytic solutions are derived.     

A viscoelastic model for use in predicting arterial pulse waves

In nonlinear mathematical models of the arterial circulation, the viscoelasticity of the vessel walls has generally been neglected or only taken into account in a highly approximate manner. A new method is proposed to simulate the nonlinear viscoelastic properties of the wall material with the aid of a convolution integral of the creep function and the pressure history. With this simulation it is possible to properly describe the measured characteristics of arterial viscoelasticity. Moreover, it is utilized in a mathematical model of arterial pulse propagation to study the influence of the internal wall friction on the shape, amplitude and mean value of pressure and flow pulses. The corresponding predictions are in much better agreement with in-vivo measurements, especially for the distal part of the circulation, than those obtained without viscoelasticity.     

An approach to determine pressure profile generated by compression bandage using quasi-linear viscoelastic model

Understanding the stress relaxation behavior of the compression bandage could be very useful in determining the behavior of the interface pressure exerted by the bandage on a limb during the course of the compression treatment. There has been no comprehensive study in the literature to investigate the pressure profile (interface pressure with time) generated by a compression bandage when applied at different levels of strain. The present study attempts to describe the pressure profile, with the use of a quasi-linear viscoelastic model, generated by a compression bandage during compression therapy. The quasi-linear viscoelastic (QLV) theory proposed by Fung (Fung, 1972, "Stress Strain History Relations of Soft Tissues in Simple Elongation," Biomechanics: Its Foundations and Objectives, Y. C. Fung, N. Perrone, and M. Anliker, eds., Prentice-Hall, Englewood Cliffs, NJ, pp. 181-207). was used to model the nonlinear time- and history-dependent relaxation behavior of the bandage using the ramp strain approach. The regression analysis was done to find the correlation between the pressure profile and the relaxation behavior of the bandage. The parameters of the QLV model, describing the relaxation behavior of the bandage, were used to determine the pressure profile generated by the bandage at different levels of strain. The relaxation behaviors of the bandage at different levels of strain were well described by the QLV model parameters. A high correlation coefficient (nearly 0.98) shows a good correlation of the pressure profile with the stress relaxation behavior of the bandage.The prediction of the pressure profile using the QLV model parameters were in agreement with the experimental data. The pressure profile generated by a compression bandage could be predicted using the QLV model describing the nonlinear relaxation behavior of the bandage. This new application of the QLV theory helps in evaluating the bandage performance during compression therapy as scientific wound care management.