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 new constitutive formulation for characterizing the mechanical behavior of soft tissues.

We present a new constitutive formulation that combines certain desirable features of two previously used approaches (phenomenological and microstructural). Specifically, we assume that certain soft tissues can be idealized as composed of various families of noninteracting fibers and a homogeneous matrix. Both the fibers and the matrix are assumed to follow the gross deformation. Within the usual framework of pseudoelasticity, incompressibility, homogeneity, and the continuum hypothesis, a pseudostrain-energy function (W) is proposed wherein W is expressed in terms of matrix and fibrous contributions. Unlike phenomenological approaches where a W is usually chosen in an ad hoc manner, the present approach can be used to postulate reasonable forms of W based on limited structural information and multiaxial stress-strain data. Illustrative applications of the theory are discussed for visceral pleura and myocardium. Concise structurally motivated constitutive relations result, wherein load-dependent anisotropy, nonlinear material behavior, finite deformations, and incompressibility are accounted for.     

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 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.     

A variational constitutive model for soft biological tissues

In this paper, a fully variational constitutive model of soft biological tissues is formulated in the finite strain regime. The model includes Ogden-type hyperelasticity, finite viscosity, deviatoric and volumetric plasticity, rate and microinertia effects. Variational updates are obtained via time discretization and pre-minimization of a suitable objective function with respect to internal variables. Genetic algorithms are used for model parameter identification due to their suitability for non-convex, high dimensional optimization problems. The material behavior predicted by the model is compared to available tests on swine and human brain tissue. The ability of the model to predict a wide range of experimentally observed behavior, including hysteresis, cyclic softening, rate effects, and plastic deformation is demonstrated.     

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 finite nonlinear hyper-viscoelastic model for soft biological tissues

Soft tissues exhibit highly nonlinear rate and time-dependent stress-strain behaviour. Strain and strain rate dependencies are often modelled using a hyperelastic model and a discrete (standard linear solid) or continuous spectrum (quasi-linear) viscoelastic model, respectively. However, these models are unable to properly capture the materials characteristics because hyperelastic models are unsuited for time-dependent events, whereas the common viscoelastic models are insufficient for the nonlinear and finite strain viscoelastic tissue responses. The convolution integral based models can demonstrate a finite viscoelastic response; however, their derivations are not consistent with the laws of thermodynamics. The aim of this work was to develop a three-dimensional finite hyper-viscoelastic model for soft tissues using a thermodynamically consistent approach. In addition, a nonlinear function, dependent on strain and strain rate, was adopted to capture the nonlinear variation of viscosity during a loading process. To demonstrate the efficacy and versatility of this approach, the model was used to recreate the experimental results performed on different types of soft tissues. In all the cases, the simulation results were well matched ($R^2⩾0.99$) with the experimental data.     

A model of fracture testing of soft viscoelastic tissues

Fracture, or tear, toughness of soft tissues can be computed from the work of fracture divided by the area of new crack surface. For soft tissues without significant plastic deformation, total work, which can be measured experimentally, is composed of the sum of fracture and viscoelastic work. In order to deduce fracture work, a method is needed to estimate viscoelastic work.Two different methods (Ph.D. Dissertation, University of Minnesota, 2000; J. Mater. Sci.: Mater. Med. 12 (2001) 327) have been proposed to estimate viscoelastic work in a fracture test of a soft tissue. The relative merits of these methods are unknown because the true viscoelastic work in an experiment is unknown. In order to characterize the accuracy of these methods, a theoretical model of crack propagation of viscoelastic soft tissue in a tensile test is presented, from which the exact viscoelastic work is calculated. The material is assumed to obey the standard linear solid model.The "exact" solution for the viscoelastic work during the fracture is computed from the model and compared with the work estimated by the two methods. It was found that both methods tend to underestimate the viscoelastic work done, and thus overestimate the fracture work and fracture toughness, although the errors were greater with the Fedewa method. It was further found that low displacement rates can give rise to a "snap" effect, where rapid crack growth can cause a disproportionate amount of viscoelastic energy to be dissipated during unloading. This modeling approach may be useful in evaluating other experimental methods of soft tissue fracture.     

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.     

A structural model for the mechanical behavior of trabecular bone

A quantitative model is developed for trabecular bone by approximating the trabecular geometry with a hypothetical network of compact bone. For the region immediately beneath the articular cartilage in the distal end of the femur, finite element analyses were performed with a high speed computer, assuming a physiological static load. The results indicate that bending and buckling of trabeculae are considerable in any elastic deformation of the bone; that fatigue fracture in some fraction of suitably oriented trabeculae is inevitable in normal ambulation; and that the stiffness varies considerably with lateral position across the subchondral plate. The latter depends totally on trabecular arrangement and may play a role in joint function and degeneration. The adjustments necessary to bring the gross stiffness into agreement with experiment imply that the intertrabecular soft tissues are of no consequence to the mechanical properties and that the compact bone of which trabeculae are made is probably not as stiff as cortical bone.     

Inexpensive, semi-automated system for measuring mechanical properties of soft tissues.

Stiffness and strength are important properties of many tissues, but standard material-testing equipment is expensive, often ill-suited for testing soft tissues, and rarely accessible to biologists. We describe a system built around a microcomputer and an electronic balance which is particularly well-suited for measuring stress and strain in small samples of soft tissue. We use a discarded floppy disk drive as a linear actuator to strain the sample, while an electronic balance measures the tension (used to calculate stress). We give an algorithm for a program to drive a microcomputer which controls the floppy disk drive via its parallel port and records the balance measurements via its serial port. We used this system to obtain stress-strain curves from a sample of latex rubber and a sample of soft insect cuticle. Three tests of the rubber sample gave nearly identical results, with smooth, J-shaped stress-strain curves. The stress-strain curves gave a modulus elasticity value of 1.72 Mpa over the steep, straight region, well within the range for natural latex rubber. We also tested a sample of abdominal cuticle from a caterpillar (Manduca sexta). The caterpillar cuticle had a J-shaped stress-strain curve with a modulus of elasticity of 2.11 Mpa over the steep part of the curve. J. Exp. Zool. 284:374-378, 1999.     

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 mechanical model to compute elastic modulus of tissues for harmonic motion imaging

Numerous experimental and computational methods have been developed to estimate tissue elasticity. The existing testing techniques are generally classified into in vitro, invasive in vivo and non-invasive in vivo. For each experimental method, a computational scheme is accordingly proposed to calculate mechanical properties of soft biological tissues. Harmonic motion imaging (HMI) is a new technique that performs radio frequency (RF) signal tracking to estimate the localized oscillatory motion resulting from a radiation force produced by focused ultrasound. A mechanical model and computational scheme based on the superposition principle are developed in this paper to estimate the Young's modulus of a tissue mimicking phantom and bovine liver in vitro tissue from the harmonic displacement measured by HMI. The simulation results are verified by two groups of measurement data, and good agreement is shown in each comparison. Furthermore, an inverse function is observed to correlate the elastic modulus of uniform phantoms with amplitude of displacement measured in HMI. The computational scheme is also implemented to estimate 3D elastic modulus of bovine liver in vitro.     

A poroviscohyperelastic model for numerical analysis of mechanical behavior of single chondrocyte

The aim of this paper is to use a poroviscohyperelastic (PVHE) model, which is developed based on the porohyperelastic (PHE) model to explore the mechanical deformation properties of single chondrocytes. Both creep and relaxation responses are investigated by using finite element analysis models of micropipette aspiration and atomic force microscopy experiments, respectively. The newly developed PVHE model is compared thoroughly with the standard neo-Hookean solid and PHE models. It has been found that the PVHE can accurately capture both creep and stress relaxation behaviors of chondrocytes better than other two models. Hence, the PVHE is a promising model to investigate mechanical properties of single chondrocytes.     

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.     

Identifying a minimal rheological configuration: a tool for effective and efficient constitutive modeling of soft tissues

We describe a modeling methodology intended as a preliminary step in the identification of appropriate constitutive frameworks for the time-dependent response of biological tissues. The modeling approach comprises a customizable rheological network of viscous and elastic elements governed by user-defined 1D constitutive relationships. The model parameters are identified by iterative nonlinear optimization, minimizing the error between experimental and model-predicted structural (load-displacement) tissue response under a specific mode of deformation. We demonstrate the use of this methodology by determining the minimal rheological arrangement, constitutive relationships, and model parameters for the structural response of various soft tissues, including ex vivo perfused porcine liver in indentation, ex vivo porcine brain cortical tissue in indentation, and ex vivo human cervical tissue in unconfined compression. Our results indicate that the identified rheological configurations provide good agreement with experimental data, including multiple constant strain rate load/unload tests and stress relaxation tests. Our experience suggests that the described modeling framework is an efficient tool for exploring a wide array of constitutive relationships and rheological arrangements, which can subsequently serve as a basis for 3D constitutive model development and finite-element implementations. The proposed approach can also be employed as a self-contained tool to obtain simplified 1D phenomenological models of the structural response of biological tissue to single-axis manipulations for applications in haptic technologies.