A structural viscoelastic model of soft tissues

A non-linear elastic model taking into account the microscopic structure of biological soft tissues is briefly presented and extended to quasi linear viscoelasticity. The modelling of the rheological behavior for near zero stress values is then discussed.     

A comparison between mechano-electrochemical and biphasic swelling theories for soft hydrated tissues

Biological tissues like intervertebral discs and articular cartilage primarily consist of interstitial fluid, collagen fibrils and negatively charged proteoglycans. Due to the fixed charges of the proteoglycans, the total ion concentration inside the tissue is higher than in the surrounding synovial fluid (cation concentration is higher and the anion concentration is lower). This excess of ion particles leads to an osmotic pressure difference, which causes swelling of the tissue. In the last decade several mechano-electrochemical models, which include this mechanism, have been developed. As these models are complex and computationally expensive, it is only possible to analyze geometrically relatively small problems. Furthermore, there is still no commercial finite element tool that includes such a mechano-electrochemical theory. Lanir (Biorheology, 24, pp. 173-187, 1987) hypothesized that electrolyte flux in articular cartilage can be neglected in mechanical studies. Lanir's hypothesis implies that the swelling behavior of cartilage is only determined by deformation of the solid and by fluid flow. Hence, the response could be described by adding a deformation-dependent pressure term to the standard biphasic equations. Based on this theory we developed a biphasic swelling model. The goal of the study was to test Lanir's hypothesis for a range of material properties. We compared the deformation behavior predicted by the biphasic swelling model and a full mechano-electrochemical model for confined compression and 1D swelling. It was shown that, depending on the material properties, the biphasic swelling model behaves largely the same as the mechano-electrochemical model, with regard to stresses and strains in the tissue following either mechanical or chemical perturbations. Hence, the biphasic swelling model could be an alternative for the more complex mechano-electrochemical model, in those cases where the ion flux itself is not the subject of the study. We propose thumbrules to estimate the correlation between the two models for specific problems.     

A discrete-time approach to the formulation of constitutive models for viscoelastic soft tissues

This paper presents a novel approach to constitutive modeling of viscoelastic soft tissues. This formulation combines an anisotropic strain energy function, accounting for preferred material directions, to define the elastic stress–strain relationship, and a discrete time black-box dynamic model, borrowed from the theory of system identification, to describe the time-dependent behavior. This discrete time formulation is straightforwardly oriented to the development of a recursive time integration scheme that calculates the current stress state by using strain and stress values stored at a limited number of previous time instants. The viscoelastic model and the numerical procedure are assessed by implementing two numerical examples, the simulation of a uniaxial tensile test and the inflation of a thin tube. Both simulations are performed using parameter values based on previous experiments on preserved bovine pericardium. Parameters are then adjusted to investigate the sensitivity of the model. The hypotheses the model relies upon are discussed and the main limitations are stated.     

Unconfined compression of hydrated viscoelastic tissues: a biphasic poroviscoelastic analysis

A biphasic poroviscoelastic theory was used to analyze the unconfined compression creep and stress relaxation of a hydrated viscoelastic tissue. The intrinsic shear properties of the tissue matrix was described by an integral-type viscoelastic constitutive law while the intrinsic bulk property of the matrix was assumed to be linearly elastic. Parametric data were presented to show how the two major energy dissipative mechanisms, namely the interstitial fluid flow and the intrinsic matrix viscoelasticity, may each contribute to the apparent viscoelastic behavior of the whole tissue under unconfined compression. The hydraulic permeability of the tissue was found to enter in only as a scaling factor for time.     

A constitutive model for the mechanical behaviour of soft connective tissues

A model of the mechanical behaviour of soft connective tissue has been developed by considering the role of the collagen and glycosaminoglycan (GAG) components within the tissue in order to examine the mechanism by which a variation in the GAG components may exert a control over the mechanical properties of the tissue. It is proposed that the strain energy stored within the collagen fibrils of the loaded tissue can be transferred into a potential field created by the charged GAG components and their electrostatic interaction with the collagen fibrils. A fundamental mechanical unit is described to simulate this energy transfer and a combination of such units is used to represent the tissue. The computer implementation of the proposed tissue model shows it to reproduce many features which have been recognised in the rate dependent mechanical behaviour of soft tissues. These include the characteristic non-linearity of the force-deformation behaviour and the approximate invariance of the stress relaxation behaviour with deformation. The model is also consistent with earlier constitutive representations of tissue behaviour.     

A recruitment-based rheological model for mechanical behavior of soft tissues

When lung tissue is subjected to finite deformations, phenomena appear that can only be described using nonlinear models. This paper considers the lung as a material composed of two elements, a continuous phase that acts uninterruptedly and a second phase composed of fiber elements that are recruited progressively into the mechanical process. Each individual fiber participates in the mechanical response of the set only when the deformation is above a certain value. A nine-parameter model was designed adopting standard viscoelastic elements both for the matrix and for each of the fibers. The mechanical behavior of the lung can be reproduced by a fitting process with standard numerical procedures in both dynamic-mechanical measurements and stress relaxation processes. Mechanical stress relaxation tests and dynamic-mechanical measurements have been carried out on subpleural parenchymal strips from rat lung. The model permits the reproduction of lung behavior in both types of measurements. The results show a recruitment ratio that decreases with deformation and the nonparticipation of the parallel matrix fraction in the lung's mechanical response so that a uniaxial transmission of force in the lung occurs via the recruited elements and the matrix series.     

A rheological network model for the continuum anisotropic and viscoelastic behavior of soft tissue

The mechanical behavior of soft tissue demonstrates a number of complex features including nonlinearity, anisotropy, viscoelasticity, and growth. Characteristic features of the time-dependent and anisotropic behavior are related to the properties of various components of the tissue such as fibrous collagen and elastin networks, large proteins and sugars attached to these networks, and interstitial fluid. Attempts to model the elastic behavior of these tissues based on assumptions about the behavior of the underlying constituents have been reasonably successful, but the essential addition of viscoelasticity to these models has been met with varying success. Here, a new rheological network model is proposed using, as its basis, an orthotropic hyperelastic constitutive model for fibrous tissue and a viscoelastic reptation model for soft materials. The resulting model has been incorporated into numerical and computational models, and is shown to capture the mechanical behavior of soft tissue in various modes of deformation including uniaxial and biaxial tension and simple shear.     

A theoretical model for the elastic properties of very soft tissues.

A theoretical model is developed to predict the elastic properties of very soft tissues such as glands, tumors and brain. Tissues are represented as regular arrays of polyhedral (cubic or tetrakaidecahedral) cells, surrounded by extracellular spaces of uniform width. Cells are assumed to be incompressible, with very low resistance to shear deformation. Tissue shear rigidity is assumed to result mainly from the extracellular matrix, which is treated as a compressible elastic mesh of interconnected fibers. Small-strain elastic properties of tissue are predicted using a finite-element method and an analytical method. The model can be used to estimate the compressibility of a very soft tissue based on its Young's modulus and extracellular volume fraction.     

Effect of preservation period on the viscoelastic material properties of soft tissues with implications for liver transplantation

The liver harvested from a donor must be preserved and transported to a suitable recipient immediately for a successful liver transplantation. In this process, the preservation period is the most critical, since it is the longest and most tissue damage occurs during this period due to the reduced blood supply to the harvested liver and the change in its temperature. We investigate the effect of preservation period on the dynamic material properties of bovine liver using a viscoelastic model derived from both impact and ramp and hold experiments. First, we measure the storage and loss moduli of bovine liver as a function of excitation frequency using an impact hammer. Second, its time-dependent relaxation modulus is measured separately through ramp and hold experiments performed by a compression device. Third, a Maxwell solid model that successfully imitates the frequency- and time-dependent dynamic responses of bovine liver is developed to estimate the optimum viscoelastic material coefficients by minimizing the error between the experimental data and the corresponding values generated by the model. Finally, the variation in the viscoelastic material coefficients of bovine liver are investigated as a function of preservation period for the liver samples tested 1 h, 2 h, 4 h, 8 h, 12 h, 24 h, 36 h, and 48 h after harvesting. The results of our experiments performed with three animals show that the liver tissue becomes stiffer and more viscous as it spends more time in the preservation cycle.     

Lubricated squeezing flow: a useful method for measuring the viscoelastic properties of soft tissues

Conducting experiments on very soft biological tissues can be difficult. Traditionally, unconfined compression and shear have been used. Here, an improved method of compression testing, lubricated squeezing flow is described. This gives a uniform compression along the squeezing axis and almost uniform equi-biaxial elongation at right angles to the squeezing axis, with minimal shear deformation due to the constant lubrication of the sample surfaces during testing. Sample results for porcine liver obtained using this method are described here.     

An extended biphasic model for charged hydrated tissues with application to the intervertebral disc

Finite element models for hydrated soft biological tissue are numerous but often exhibit certain essential deficiencies concerning the reproduction of relevant mechanical and electro-chemical responses. As a matter of fact, singlephasic models can never predict the interstitial fluid flow or related effects like osmosis. Quite a few models have more than one constituent, but are often restricted to the small-strain domain, are not capable of capturing the intrinsic viscoelasticity of the solid skeleton, or do not account for a collagen fibre reinforcement. It is the goal of this contribution to overcome these drawbacks and to present a thermodynamically consistent model, which is formulated in a very general way in order to reproduce the behaviour of almost any charged hydrated tissue. Herein, the Theory of Porous Media (TPM) is applied in combination with polyconvex Ogden-type material laws describing the anisotropic and intrinsically viscoelastic behaviour of the solid matrix on the basis of a generalised Maxwell model. Moreover, other features like the deformation-dependent permeability, the possibility to include inhomogeneities like varying fibre alignment and behaviour, or osmotic effects based on the simplifying assumption of Lanir are also included. Finally, the human intervertebral disc is chosen as a representative for complex soft biological tissue behaviour. In this regard, two numerical examples will be presented with focus on the viscoelastic and osmotic capacity of the model.     

Extraction of quasi-linear viscoelastic parameters for lower limb soft tissues from manual indentation experiment.

A manual indentation protocol was established to assess the quasi-linear viscoelastic (QLV) properties of lower limb soft tissues. The QLV parameters were extracted using a curve-fitting procedure on the experimental indentation data. The load-indentation responses were obtained using an ultrasound indentation apparatus with a hand-held pen-sized probe. Limb soft tissues at four sites of eight normal young subjects were tested in three body postures. Four QLV model parameters were extracted from the experimental data. The initial modulus E0 ranged from 0.22 kPa to 58.4 kPa. The nonlinear factor E1 ranged from 21.7 kPa to 547 kPa. The time constant tau ranged from 0.05 s to 8.93 s. The time-dependent materials parameter alpha ranged from 0.029 to 0.277. Large variations of the parameters were noted among subjects, sites, and postures.     

A constitutive model for two-dimensional soft tissues and its application to experimental data

A constitutive relation proposed by Shoemaker (Ph.D. dissertation, 1984) to model the mechanical behavior of membraneous or two-dimensional soft tissues is described. Experiments by Schneider (Ph.D. dissertation, 1982) on human skin and Lee et al. (Am. J. Physiol., 249, H222-H230, 1985) on canine pericardium, and the application of the constitutive model to biaxial stress-strain data from these experiments, are discussed. Some experimental data and predictions of the model obtained by curvefitting are presented for comparison. Values of material parameters are also presented. It is concluded that the constitutive model is well able to fit results of individual tests, and that its generality (judged by consistency of parameters from test to test of the same specimen), though not complete, does compare favorably with some other results presented in the literature.     

A unified multiscale mechanical model for soft collagenous tissues with regular fiber arrangement

In this paper the mechanical response of soft collagenous tissues with regular fiber arrangement (RSCTs) is described by means of a nanoscale model and a two-step micro–macro homogenization technique. The non-linear collagen constitutive behavior is modeled at the nanoscale by a novel approach accounting for entropic mechanisms as well as stretching effects occurring in collagen molecules. Crimped fibers are reduced to equivalent straight ones at the microscale and the constitutive response of RSCTs at the macroscale is formulated by homogenizing a fiber reinforced material. This approach has been applied to different RSCTs (tendon, periodontal ligament and aortic media), resulting effective and accurate as proved by the excellent agreement with available experimental data. The model is based on few parameters, directly related to histological and morphological evidences and whose sensitivity has been widely investigated. Applications to simulation of some physiopathological mechanisms are also proposed, providing confirmation of clinical evidences and quantitative indications helpful for clinical practice.     

Evaluation of viscoelastic property inhomogeneities in soft tissues exposed to low frequency perturbations

This work presents the methods of recognition of inhomogeneities of tissue shear viscoelastic properties using partial data on the internal displacements in an object exposed to low-frequency perturbation. An approach to detect tissue inhomogeneities using the single displacement component is presented.     

A quasi-linear, viscoelastic, structural model of the plantar soft tissue with frequency-sensitive damping properties

Little is known about the structural properties of plantar soft-tissue areas other than the heel; nor is it known whether the structural properties vary depending on location. Furthermore, although the quasi-linear viscoelastic (QLV) theory has been used to model many soft-tissue types, it has not been employed to model the plantar soft tissue. The structural properties of the plantar soft tissue were quantified via stress relaxation experiments at seven regions (subcalcaneal, five submetatarsal, and subhallucal) across eight cadaveric feet. The cadaveric feet were 36.9 +/- 17.4 (mean +/- S.D.) years of age, all free from vascular diseases and orthopedics disorders. All tests were performed at a constant environmental temperature of 35 degrees C. Stress relaxation experiments were performed; different loads were employed for different areas based on normative gait data. A modification of the relaxation spectrum employed within the QLV theory allowed for the inclusion of frequency-sensitive relaxation properties in addition to nonlinear elastic behavior. The tissue demonstrated frequency-dependent damping properties that made the QLV theory ill suited to model the relaxation. There was a significant difference between the elastic structural properties (A) of the subcalcaneal tissue and all other areas (p = 0.004), and a trend (p = 0.067) for the fifth submetatarsal to have less viscous damping (c1) than the subhallucal, or first, second, or third submetatarsal areas. Thus, the data demonstrate that the structural properties of the foot can vary across regions, but careful consideration must be given to the applied loads and the manner in which the loads were applied.     

Towards an analytical model of soft biological tissues

In the past years, soft-tissue modelling research has seen substantial developments, a significant part of which can be ascribed to the refinement of numerical techniques, such as Finite Element analysis. A large class of physico-mechanical properties can be effectively simulated and predictions can be made for a variety of phenomena. However, there is still much that can be conceptually explored by means of fundamental theoretical analysis. In the past few years, driven by our interest in articular cartilage mechanics, we have developed theoretical microstructural models for linear elasticity and permeability that accounted for the presence and arrangement of collagen fibres in cartilage. In this paper, we investigate analytically the non-linear elasticity of soft tissues with collagen fibres arranged according to a given distribution of orientation, a problem that, aside from the case of fibres aligned in a finite number of distinct directions, has been treated exclusively numerically in the literature. We show that, for the case of a tissue with complex fibre arrangement, such as articular cartilage, the theoretical framework commonly used leads to an integral expression of the elastic strain energy potential. The present model is a first attempt in the development of a unified analytical microstructural model for non-linear elasticity and permeability of hydrated, fibre-reinforced soft tissues.     

A physically motivated constitutive model for cell-mediated compaction and collagen remodeling in soft tissues

Collagen is the main load-bearing component of many soft tissues and has a large influence on the mechanical behavior of tissues when exposed to mechanical loading. Therefore, it is important to increase our understanding of collagen remodeling in soft tissues to understand the mechanisms behind pathologies and to control the development of the collagen network in engineered tissues. In the present study, a constitutive model was developed by coupling a recently developed model describing the orientation and contractile stresses exerted by cells in response to mechanical stimuli to physically motivated collagen remodeling laws. In addition, cell-mediated contraction of the collagen fibers was included as a mechanism for tissue compaction. The model appeared to be successful in predicting a range of experimental observations, which are (1) the change in transition stretch of periosteum after remodeling at different applied stretches, (2) the compaction and alignment of collagen fibers in tissue-engineered strips, (3) the fiber alignment in cruciform gels with different arm widths, and (4) the alignment of collagen fibers in engineered vascular grafts. Moreover, by changing the boundary conditions, the model was able to predict a helical architecture in the vascular graft without assuming the presence of two helical fiber families a priori. Ultimately, this model may help to increase our understanding of collagen remodeling in physiological and pathological conditions, and it may provide a tool for determining the optimal experimental conditions for obtaining native-like collagen architectures in engineered tissues.