Quasi-linear Viscoelastic Theory Research Articles

Single-cell mechanical characterization in constriction-based cytometry

Mechanical characterization of suspended cells by constriction-based microfluidic devices has currently various limitations related to the available analysis models. In this work, we propose a new methodology to analyze the experiments. This approach is based on numerical simulations to describe fluid forces and cell deformation and on an extension of the quasi-linear viscoelasticity theory developed by Fung. The cells are considered visco-hyperelastic, homogeneous, and isotropic. The approach allows for assessing the mechanical parameters of individual cells, which is not possible using previous approaches, notably increasing the power of the constriction-based microfluidic technique. A practical procedure to compute mechanical parameters is proposed and demonstrated by analyzing experiments with suspended cells. The methodology developed in this work provides a convenient tool to overcome critical limitations of the state of the art and to leverage the potential of these microfluidic devices.

Automated model discovery for muscle using constitutive recurrent neural networks

The stiffness of soft biological tissues not only depends on the applied deformation, but also on the deformation rate. To model this type of behavior, traditional approaches select a specific time-dependent constitutive model and fit its parameters to experimental data. Instead, a new trend now suggests a machine-learning based approach that simultaneously discovers both the best model and best parameters to explain given data. Recent studies have shown that feed-forward constitutive neural networks can robustly discover constitutive models and parameters for hyperelastic materials. However, feed-forward architectures fail to capture the history dependence of viscoelastic soft tissues. Here we combine a feed-forward constitutive neural network for the hyperelastic response and a recurrent neural network for the viscous response inspired by the theory of quasi-linear viscoelasticity. Our novel rheologically-informed network architecture discovers the time-independent initial stress using the feed-forward network and the time-dependent relaxation using the recurrent network. We train and test our combined network using unconfined compression relaxation experiments of passive skeletal muscle and compare our discovered model to a neo Hookean standard linear solid, to an advanced mechanics-based model, and to a vanilla recurrent neural network with no mechanics knowledge. We demonstrate that, for limited experimental data, our new constitutive recurrent neural network discovers models and parameters that satisfy basic physical principles and generalize well to unseen data. We discover a Mooney–Rivlin type two-term initial stored energy function that is linear in the first invariant I1 and quadratic in the second invariant I2 with stiffness parameters of 0.60 kPa and 0.55 kPa. We also discover a Prony-series type relaxation function with time constants of 0.362s, 2.54s, and 52.0s with coefficients of 0.89, 0.05, and 0.03. Our newly discovered model outperforms both the neo Hookean standard linear solid and the vanilla recurrent neural network in terms of prediction accuracy on unseen data. Our results suggest that constitutive recurrent neural networks can autonomously discover both model and parameters that best explain experimental data of soft viscoelastic tissues. Our source code, data, and examples are available at https://github.com/LivingMatterLab.

Tensor decomposition for modified quasi-linear viscoelastic models: Towards a fully non-linear theory

We discuss the decomposition of the tensorial relaxation function for isotropic and transversely isotropic (TI) modified quasi-linear viscoelastic (MQLV) models. We show how to formulate the constitutive equation using a convenient decomposition of the relaxation tensor into scalar components and tensorial bases. We show that the bases must be symmetrically additive, i.e., they must sum up to the symmetric fourth-order identity tensor. This is a fundamental property both for isotropic and anisotropic bases that ensures the constitutive equation is consistent with the elastic limit. We provide two robust methods to obtain such bases. Furthermore, we show that, in the TI case, the bases are naturally deformation-dependent for deformation modes that induce rotation or stretching of the fibres. Therefore, the MQLV framework allows to capture the non-linear phenomenon of strain-dependent relaxation, which has always been a criticised limitation of the original quasi-linear viscoelastic theory. We illustrate this intrinsic non-linear feature, unique to the MQLV model, with two examples (uni-axial extension and perpendicular shear).

Constituent-based quasi-linear viscoelasticity: a revised quasi-linear modelling framework to capture nonlinear viscoelasticity in arteries

Arteries exhibit fully nonlinear viscoelastic behaviours (i.e. both elastically and viscously nonlinear). While elastically nonlinear arterial models are well established, effective mathematical descriptions of nonlinear viscoelasticity are lacking. Quasi-linear viscoelasticity (QLV) offers a convenient way to mathematically describe viscoelasticity, but its viscous linearity assumption is unsuitable for whole-wall vascular applications. Conversely, application of fully nonlinear viscoelastic models, involving deformation-dependent viscous parameters, to experimental data is impractical and often reduces to identifying specific solutions for each tested loading condition. The present study aims to address this limitation: By applying QLV theory at the wall constituent rather than at the whole-wall level, the deformation-dependent relative contribution of the constituents allows to capture nonlinear viscoelasticity with a unique set of deformation-independent model parameters. Five murine common carotid arteries were subjected to a protocol of quasi-static and harmonic, pseudo-physiological biaxial loading conditions to characterise their viscoelastic behaviour. The arterial wall was modelled as a constrained mixture of an isotropic elastin matrix and four families of collagen fibres. Constituent-based QLV was implemented by assigning different relaxation functions to collagen- and elastin-borne parts of the wall stress. Nonlinearity in viscoelasticity was assessed via the pressure dependency of the dynamic-to-quasi-static stiffness ratio. The experimentally measured ratio increased with pressure, from 1.03 pm 0.03 (mean pm standard deviation) at 80–40 mmHg to 1.58 pm 0.22 at 160–120 mmHg. Constituent-based QLV captured well this trend by attributing the wall viscosity predominantly to collagen fibres, whose recruitment starts at physiological pressures. In conclusion, constituent-based QLV offers a practical and effective solution to model arterial viscoelasticity.

Growth mechanics of the viscoelastic membranes

Living membranes can grow as a response to their environment, and they are also known as viscoelastic materials. However, the growth phenomenon and the viscoelastic behavior of these membranes have been studied separately up to now. The main goal of this paper is to develop a general formulation that can account for growth and viscoelastic behaviors in membranes, simultaneously. Therefore, we propose a formulation for growing quasi-linear viscoelastic membranes. The growth model begins with a multiplicative decomposition of the gradient deformation tensor into elastic and growth tensor, where the growth of the membrane is considered as a single scalar-valued parameter. Furthermore, the quasi-linear viscoelasticity theory is utilized in the elastic configuration as additive decomposition of the second Piola–Kirchhoff stress tensor. To solve the evolution equations for both growth and quasi-linear viscoelastic relations, two implicit second order accuracy integration methods are employed. Moreover, strain-driven and stress-driven growth phenomena are accounted for based on a Total Lagrangian (TL) finite element formulation. The developed formulations can capture growth phenomena and viscoelastic properties of the membranes, individually, and it also enables to model co-existence of growth and viscoelasticity. Eventually, to evaluate the applicability of the developed formulation, several examples are investigated and compared to those available in the literature.

Effect of viscoelasticity on inversion in axial deformation and on the volume of pressurized thin-walled tubes

Axial prestretching can significantly affect the mechanical response of an inflated tube. This phenomenon is known, for example in the biomechanics of the cardiovascular system, where arteries, if excised from a body, manifest non-negligible shortening. Previous studies have shown that axially prestretched tubes are more easily distensible by internal pressure than their non-prestretched counterparts. It has even been found that, in the case of the material described by the Gent strain energy density function, the volume attained by a tube at a given internal pressure can be maximized by the initial elongation of the tube to the prestretch point referred to as inversion axial prestretching. However, to the best of our knowledge, all studies dealing so far with the effect of axial prestretching on the mechanical response of a pressurized tube have assumed elastic behavior. Nevertheless, soft tissues as well as elastomer materials are always more or less viscoelastic. The present study shifts our attention from purely theoretical elastic models to a more realistic assumption of viscoelastic behavior, which is modeled by means of the Quasilinear Theory of Viscoelasticity (QLV). The adopted model assumes the elastic part of the material response to be given by the Gent model. Results suggest that the viscoelastic tube loses the property of inversion in axial deformation but can still preserve the potential to optimize the inflation volume by means of axial prestretching. This result can be helpful in designing elastomer pipelines and pumps, as well as contributing to the understanding of the principals of blood circulation physiology.

Acoustoelastic analysis of soft viscoelastic solids with application to pre-stressed phononic crystals

The effective dynamic properties of specific periodic structures involving rubber-like materials can be adjusted by pre-strain, thus facilitating the design of custom acoustic filters. While nonlinear viscoelastic behaviour is one of the main features of soft solids, it has been rarely incorporated in the study of such phononic media. Here, we study the dynamic response of nonlinear viscoelastic solids within a ‘small-on-large’ acoustoelasticity framework, that is we consider the propagation of small amplitude waves superimposed on a large static deformation. Incompressible soft solids whose behaviour is described by the Fung–Simo quasi-linear viscoelasticity theory (QLV) are considered. We derive the incremental equations using stress-like memory variables governed by linear evolution equations. Thus, we show that wave dispersion follows a strain-dependent generalised Maxwell rheology. Illustrations cover the propagation of plane waves under homogeneous tensile strain in a QLV Mooney–Rivlin solid. The acoustoelasticity theory is then applied to phononic crystals involving a lattice of hollow cylinders, by making use of a dedicated perturbation approach. In particular, results highlight the influence of viscoelastic dissipation on the location of the first band gap. We show that dissipation shifts the band gap frequencies, simultaneously increasing the band gap width. These results are relevant to practical applications of soft viscoelastic solids subject to static pre-stress.

Shear shock formation in incompressible viscoelastic solids

Experiments have shown that shear waves induced in brain tissue can develop into shock waves, thus providing a possible explanation of deep traumatic brain injuries. Here, we study the formation of shock waves in soft viscoelastic solids subject to an imposed velocity at their boundary. We consider the plane shearing motion of a semi-infinite half-space, which corresponds to a spatially one-dimensional problem. Incompressible soft solids whose behaviour is described by the Fung–Simo quasi-linear viscoelasticity theory (QLV) are considered, where the elastic response is either exponential or polynomial of Mooney–Rivlin–Yeoh type. Waveform breaking can occur at the blow-up of acceleration waves, leading to one sufficient condition for the formation of shocks. A slow scale analysis based on a small amplitude parameter yields a damped Burgers-like equation, thus leading to another shock formation condition. Numerical experiments performed using a dedicated finite volume scheme show that these estimates have limited accuracy. Their validity is restricted in the elastic limit too, where exact shock formation conditions are known.

Region-Dependent Viscoelastic Properties of Human Brain Tissue Under Large Deformations.

This study characterizes the mechanical properties of human brain tissue resected during the course of surgery under multistep indentation loading up to 30% strain. The experimental characterization using fresh, post-operative, human brain tissue is highly advantageous since postmortem times can affect its biomechanical behavior. Although the quasilinear theory of viscoelasticity (QLV) approach has been widely used to model brain tissue mechanical properties, our analysis concluded that the linear viscoelastic approach provided a better fit to the experimental data overall. The only statistically significant regional difference in observed stiffness was between the cortex gray and dentate gyrus. There were no statistically significant age or sex dependent differences, although the data suggested that the cortex white matter in males was stiffer than that in females. Our results can help improve the accuracy of finite element models of brain tissue deformation to predict its response to traumatic brain injury.

Quasi-linear viscoelastic behavior of fresh porcine ureter.

To evaluate the viscoelastic properties of the fresh porcine ureter. Prove the QLV theory can sufficiently model the stress relaxation function of porcine ureters, and determine the QLV model constants which may provide insight into the synthesis of ureteral scaffolds with biomimetic viscoelastic properties in tissue engineering. Hysteresis tests were applied to study the differences in dissipated energy ratio for each different strain group. In stress relaxation tests, samples were sub-grouped and quickly ramping up to 5%, 20%, and 30% strain in each group and gradually relaxed to a corresponding level. Bringingtogether the quasi-linear viscoelasticity (QLV) theory and stress relaxation function to determine the eight constants of the ureteral tissue, and fitting the raw data with the model via MATLAB. The hysteresis response measurement results revealed that the porcine ureter was a highly dissipative material and there were differences between toe and linear region in stress-stain curve. The stress relaxation results revealed ureters responded with time-dependent decay of stress. The eight constants of the ureteral QLV model were determined for three different strain groups, and we proved that the QLV model can sufficiently adapt the experimental data of the ureter stress relaxation. This study investigated the time-dependent properties of the porcine ureter, and demonstrated the QLV theory could be used to evaluate the viscoelastic properties of the porcine ureteral tissue.

Time-Dependent Response of Incompressible Membranes Based on Quasi-Linear Viscoelasticity Theory

In this work, large deformation of incompressible, hyperelastic membranes based on the quasi-linear viscoelasticity (QLV) theory is formulated. Time integration algorithm and the expression for consistent fourth-order tangent tensor are presented. The formulation covers isotropic as well as anisotropic polymeric and biological materials. To solve numerical examples, a nonlinear finite element formulation in the Lagrangian framework is developed. Finally, several examples are provided to investigate the applicability of the present formulation. It is found that the numerical results are in good agreement with the analytical and experimental results available in the literature.

Quasi-Linear Viscoelastic Characterization of Soft Tissue-Mimicking Materials.

We present a novel method based on the quasi-linear viscoelastic (QLV) theory to describe the time-dependent behavior of soft materials. Unlike previous methods for deriving QLV parameters, we characterize the elastic and viscous behavior of materials separately by using two different sets of experiments. To model the nonlinear elastic behavior, we fit the elastic stress response with a one-term Ogden model. Then, we model the relaxation behavior with a Prony series to compare the stress relaxation of the material at different timescales. This new method allows us to characterize materials with narrow confidence intervals (high accuracy), independently from the loading conditions. We validate our model using samples made of phantom materials that mimic normal and cancerous prostate tissues in terms of Young's modulus. Our model is shown to distinguish materials with similar elastic (viscous) properties but different viscous (elastic) properties. Drawing a precise distinction between the phantoms, this method could be useful for prostate cancer (PCa) diagnosis; but significant clinical studies will be needed in the future.

Viscoelastic characterization of porcine brain tissue mechanical properties under indentation loading

The goal of this study was to measure the mechanical properties of porcine brain tissue and determine if they were dependent on anatomical region or direction. Multistep stress relaxation indentations with a cylindrical probe were performed at 10, 20, and 30% nominal strain on multiple regions in the sagittal, horizontal, and coronal planes. Linear and nonlinear (using the quasilinear theory of viscoelasticity [QLV]) constitutive formulations were applied to extract parameters to capture the mechanical behavior of brain tissue. The linear viscoelastic analytic approach provided the best fit to the experimental data of the models tested. Within each directional plane there were region-dependent differences. The cerebellum was the softest region within each loading direction. Although the majority of the regions were isotropic, the cerebellum white matter and thalamus were anisotropic. The characterization of these mechanical properties can be used to inform finite element models of the pig brain to help predict a more biofidelic response in animal models of traumatic brain injury. Statement of SignificanceFinite element models been developed to predict brain tissue response to traumatic brain injury (TBI) to advance protective and preventative strategies. In order to improve the accuracy of these computational models, appropriate mechanical experimentation is required to identify brain viscoelasticity, heterogeneity, and anisotropy. Our custom indentation design allows for high spatial resolution to characterize mechanical properties based on anatomical region and loading direction. Due to the challenges in procuring human brain tissue, porcine brain models are a suitable substitute to study TBI based on its structural similarities to that of human brains. This study will further illuminate the complexity of brain tissue mechanics in response to injury loading.

Stress relaxation of porcine tendon under simulated biological environment: experiment and modeling

The aim of the study was to investigate the viscoelastic response in the low and high physiological strain with the use of experimental and modeling approach. Viscoelastic response in the low, transition and high physiologic strain (3, 6 and 9%) with consideration of simulated biological environment (0.9% saline solution, 37 °C) was measured in relaxation tests. Preconditioning of tendons was considered in the testing protocol and the applied range of load was obtained from tensile testing. The quasi-linear viscoelasticity theory was used to fit experimental data to obtain constants (moduli and times of relaxation), which can be used for description of the viscoelastic behavior of tendons. The exponential non-linear elastic representation of the stress response in ramp strain was also estimated. Differences between stress relaxation process can be seen between tendons stretched to the physiological strain range (3%) and exceeding this range (6 and 9%). The strains of 6% and 9% showed a similar stress relaxation trend displaying relatively rapid relaxation for the first 70 seconds, whereas the lowest strain of 3% displayed relatively slow relaxation. Results of the model fitting showed that the quasi-linear viscoelastic model gives the best fit in the range of low physiological strain level.

Instantaneous and Equilibrium Responses of the Brain Tissue by Stress Relaxation and Quasi-Linear Viscoelasticity Theory

Human brain and brainstem tissues have viscoelastic characteristics and their behaviours are functions of strains, as well as strain rates. Determination of the equilibrium and instantaneous stresses happening at low and high strain rates provide insights into a better understanding of the behaviour of such tissues. In this manuscript we present the results of a series of stress relaxation tests, at six different values of strains conducted on porcine brainstem tissue samples to indirectly measure the equilibrium and instantaneous stresses. The equilibrium stresses at low strain rates are measured from long-term responses of the stress relaxation test. The instantaneous stresses at high strain rates are determined using Quasi-Linear Viscoelasticity (QLV) theory at six strains. The results show that the instantaneous stresses are much larger (almost 11 times) than the equilibrium stresses and across all the strains. It can be concluded that the instantaneous response can be reasonably estimated from the long-term response which can be easily measured experimentally. The experimental results also show that the reduced relaxation moduli, estimated from the QLV theory, vary for the six strains tested.

A modified formulation of quasi-linear viscoelasticity for transversely isotropic materials under finite deformation.

The theory of quasi-linear viscoelasticity (QLV) is modified and developed for transversely isotropic (TI) materials under finite deformation. For the first time, distinct relaxation responses are incorporated into an integral formulation of nonlinear viscoelasticity, according to the physical mode of deformation. The theory is consistent with linear viscoelasticity in the small strain limit and makes use of relaxation functions that can be determined from small-strain experiments, given the time/deformation separability assumption. After considering the general constitutive form applicable to compressible materials, attention is restricted to incompressible media. This enables a compact form for the constitutive relation to be derived, which is used to illustrate the behaviour of the model under three key deformations: uniaxial extension, transverse shear and longitudinal shear. Finally, it is demonstrated that the Poynting effect is present in TI, neo-Hookean, modified QLV materials under transverse shear, in contrast to neo-Hookean elastic materials subjected to the same deformation. Its presence is explained by the anisotropic relaxation response of the medium.

A hybrid method to study the mechanical information induced by needle rotating

Acupuncture has been used in China for more than 2000 years; its curative effect is acknowledged wildly in the world, but the underlying mechanisms are not revealed. The study focuses on mechanical effect of manual acupuncture. In this paper, a mathematical model based on viscoelastic theory and quasilinear viscoelastic theory of soft tissue is constructed; a hybrid method was used to study the tissue displacement and energy dissipation induced by needle rotation. It shows that (1) needle rotation will induce the extracellular matrix deformation and generate a mechanical stress field in the interstitium, (2) the deformation depends on rotation frequency and is mainly restrained in the vicinity of the needle, and (3) energy dissipation is affected by rotation frequency and very sensitive at low frequency. These results are consistent with experimental observations, and may help to explain some acupuncture theories and phenomena. Furthermore, this work not only contributes to reveal the mechanical effect of acupuncture (needle rotation) by modern scientific methods but also provides a hybrid method to study the strain and stress deriving from mechanical stimuli of connective tissue.

On the variation in maternal birth canal in vivo viscoelastic properties and their effect on the predicted length of active second stage and levator ani tears

The pubovisceral muscles (PVM) help form the distal maternal birth canal. It is not known why 13% of vaginal deliveries end in PVM tears, so insights are needed to better prevent them because their sequelae can lead to pelvic organ prolapse later in life. In this paper we provide the first quantification of the variation in in vivo viscoelastic properties of the intact distal birth canal in healthy nulliparous women using Fung’s Quasilinear Viscoelastic Theory and a secondary analysis of data from a clinical trial of constant force birth canal dilation to 8 cm diameter in the first stage of labor in 26 nullipara. We hypothesized that no significant inter-individual variation would be found in the long time constant, τ2, which characterizes how long it takes the birth canal to be dilated by the fetal head. That hypothesis was rejected because τ2 values ranged 20-fold above and below the median value. These data were input to a biomechanical model to calculate how such variations affect the predicted length of the active second stage of labor as well as PVM tear risk. The results show there was a 100-fold change in the predicted length of active second stage for the shortest and longest τ2 values, with a noticeable increase for τ2 values over 1000 s. The correlation coefficent between predicted and observed second stage durations was 0.51. We conclude that τ2 is a strong theoretical contributor to the time a mother has to push in order to deliver a fetal head larger than her birth canal, and a weak predictor of PVM tear risk.

Rate-dependent behavior of connective tissue through a micromechanics-based hyper viscoelastic model

In this paper, a micromechanical study on rate-dependent behavior of connective tissues is performed. To this end, a hyper viscoelastic constitutive model consisting a hyperelastic part for modeling equilibrium response of tissues and a viscous part using a hereditary integral is proposed to capture the time-dependent behavior of the tissues. With regard to the hierarchical structure of the tissue, strain energy function are developed for modeling elastic response of the tissue constituents i.e. collagen fibers and ground matrix. The rate-dependency is incorporated into the model using a viscous element with rate-dependent relaxation time. The proposed constitutive model is implemented into ABAQUS using the UMAT subroutine. Results of the presented micromechanics model are compared and validated with the available experimental data and also the quasi-linear viscoelastic (QLV) theory at different strain rates. It was found that the proposed model could well predict the behavior of the connective tissues in wide range of strain rates. Results show that, the presented model contains less constitutive parameters and leads to more accurate results in comparison with the QLV.

A comparison of the material properties of natural and synthetic vascular walls

Characterization of the mechanical properties of native and synthetic vascular grafts is an essential task in the process of designing novel vascular constructs. The aim in this study was to compare the mechanical behavior of ovine left Subclavian artery with that of POSS-PCU (a commercial biomaterial which is currently under clinical investigation. ClinicalTrials.gov Identifier: NCT02301312). We used Delfino's strain energy potential within the framework of quasilinear viscoelasticity theory to capture the viscoelastic response of the considered materials. The material parameters of the quasilinear viscoelastic constitutive equation were determined through a combination of experimental and computational method. First, a uniaxial tensile testing device was used to perform a series of stress relaxation tests on ring samples. Then, the derived quasilinear viscoelastic models were implemented into finite element system. With the aid of mechanical experimentation and finite element simulation, the material parameters were obtained, modified and used for comparison of the mechanical properties of vascular walls. The results showed that the stiffness and the long term viscoelastic parameters of POSS-PCU may lead to different stress responses of the vascular walls. These two factors can be improved by modifications in manufacturing parameters of the synthetic vessel.