Nonlinear viscoelastic material property estimation of lower extremity residual limb tissues

Axisymmetric nonlinear finite-element analysis was used to simulate force-relaxation and creep data obtained during in vivo indentation of the residual limb soft tissues of six individuals with trans-tibial amputation [1]. The finite-element models facilitated estimation of an appropriate set of nonlinear viscoelastic material coefficients of extended James-Green-Simpson material formulation for bulk soft tissue at discrete, clinically relevant test locations. The results indicate that over 90% of the experimental data can be simulated using the two-term viscoelastic Prony series extension of James-Green-Simpson material formulation. This phenomenological material formulation could not, however, predict the creep response from relaxation experiments, nor the relaxation response from creep experiments [2-5]. The estimated material coefficients varied with test location and subject indicating that these coefficients cannot be readily extrapolated to other sites or individuals.     

Non-ideal effects in indentation testing of soft tissues

Indentation has several advantages as a loading mode for determining constitutive behavior of soft, biological tissues. However, indentation induces a complex, spatially heterogeneous deformation field that creates analytical challenges for the calculation of constitutive parameters. As a result, investigators commonly assume small indentation depths and large sample thicknesses to simplify analysis and then restrict indentation depth and sample geometry to satisfy these assumptions. These restrictions limit experimental resolution in some fields, such as brain biomechanics. However, recent experimental evidence suggests that conventionally applied limits are in fact excessively conservative. We conducted a parametric study of indentation loading with various indenter geometries, surface interface conditions, sample compressibility, sample geometry and indentation depth to quantitatively describe the deviation from previous treatments that results from violation of the assumptions of small indentation depth and large sample thickness. We found that the classical solution was surprisingly robust to violation of the assumption of small strain but highly sensitive to violation of the assumption of large sample thickness, particularly if the indenter was cylindrical. The ramifications of these findings for design of indentation experiments are discussed and correction factors are presented to allow future investigators to account for these effects without recreating our finite element models.     

Methods for quasi-linear viscoelastic modeling of soft tissue: application to incremental stress-relaxation experiments

The quasi-linear viscoelastic (QLV) model was applied to incremental stress-relaxation tests and an expression for the stress was derived for each step. This expression was used to compare two methods for normalizing stress data prior to estimating QLV parameters. The first and commonly used normalization method was shown to be strain-dependent. Thus, a second normalization method was proposed and shown to be strain-independent and more sensitive to QLV time constants. These analytical results agreed with representative tendon data. Therefore, this method for normalizing stress data was proposed for future studies of incremental stress-relaxation, or whenever comparing stress-relaxation at different strains.     

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.     

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.     

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 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.     

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.     

Alterations in the biorheological features of some soft tissues after limb lengthening.

The aim of this study was to investigate the effects of a surgical limb lengthening procedure on the biorheological features of some lengthened soft tissues. In this procedure external fixators were applied to goats' right radius to stretch the tissues. The right forelegs of goats were lengthened by 2, 4 cm, respectively. After lengthening ceased, the goats were examined after different periods of time. The lengthened median nerves, arteries and veins were harvested and used to study their biorheological features. Tensile strength of lengthened and control specimens were measured and their stress relaxation features and stress-strain relationships were studied. Results showed that at the beginning of recovery, the stress-strain curves, relaxation curves and tensile strengths of the lengthened specimens began to deviate from those of their controls. However, with increasing recovery time, the curves and tensile strength of the lengthened specimens reverted to those of their controls. All the tissues studied exhibited the same behavior.     

Mechanical characterization of anisotropic planar biological soft tissues using finite indentation: experimental feasibility

Heart valve tissue engineering offers a promising alternative for current treatment and replacement strategies, e.g., synthetic or bioprosthetic heart valves. In vitro mechanical conditioning is an important tool for engineering strong, implantable heart valves. Detailed knowledge of the mechanical properties of the native tissue as well as the developing tissue construct is vital for a better understanding and control of the remodeling processes induced by mechanical conditioning. The nonlinear, anisotropic and inhomogeneous mechanical behavior of heart valve tissue necessitates a mechanical characterization method that is capable of dealing with these complexities. In a recent computational study we showed that one single indentation test, combining force and deformation gradient data, provides sufficient information for local characterization of nonlinear soft anisotropic tissue properties. In the current study this approach is validated in two steps. First, indentation tests with varying indenter sizes are performed on linear elastic PDMS rubbers and compared to tensile tests on the same specimen. For the second step, tissue constructs are engineered using uniaxial or equibiaxial static constrained culture conditions. Digital image correlation (DIC) is used to quantify the anisotropy in the tissue constructs. For both validation steps, material parameters are estimated by inverse fitting of a computational model to the experimental results.     

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.     

Efficacy of lower limb compression and combined treatment of manual massage and lower limb compression on symptoms of exercise-induced muscle damage in women

Strategies to manage the symptoms of exercise-induced muscle damage (EIMD) are widespread, though are often based on anecdotal evidence. The aim of this study was to determine the efficacy of a combination of manual massage and compressive clothing and compressive clothing individually as recovery strategies after muscle damage. Thirty-two female volunteers completed 100 plyometric drop jumps and were randomly assigned to a passive recovery (n = 17), combined treatment (n = 7), or compression treatment group (n = 8). Indices of muscle damage (perceived soreness, creatine kinase activity, isokinetic muscle strength, squat jump, and countermovement jump performance) were assessed immediately before and after 1, 24, 48, 72, and 96 hours of plyometric exercise. The compression treatment group wore compressive tights for 12 hours after damage and the combined treatment group received a 30-minute massage immediately after damaging exercise and wore compression stockings for the following 11.5 hours. Plyometric exercise had a significant effect on all indices of muscle damage (p < 0.05). The treatments significantly reduced decrements in isokinetic muscle strength, squat jump performance, and countermovement jump performance and reduced the level of perceived soreness in comparison with the passive recovery group (p < 0.05). The addition of sports massage to compression after muscle damage did not improve performance recovery, with recovery trends being similar in both treatment groups. The treatment combination of massage and compression significantly moderated perceived soreness at 48 and 72 hours after plyometric exercise (p < 0.05) in comparison with the passive recovery or compression alone treatment. The results indicate that the use of lower limb compression and a combined treatment of manual massage with lower limb compression are effective recovery strategies following EIMD. Minimal performance differences between treatments were observed, although the combination treatment may be beneficial in controlling perceived soreness.     

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.     

Cell poking: quantitative analysis of indentation of thick viscoelastic layers.

A recently introduced device, the cell poker, measures the force required to indent the exposed surface of a cell adherent to a rigid substratum. The cell poker has provided phenomenological information about the viscoelastic properties of several different types of cells, about mechanical changes triggered by external stimuli, and about the role of the cytoskeleton in these mechanical functions. Except in special cases, however, it has not been possible to extract quantitative estimates of viscosity and elasticity moduli from cell poker measurements. This paper presents cell poker measurements of well characterized viscoelastic polymeric materials, polydimethylsiloxanes of different degrees of polymerization, in a simple shape, a flat, thick layer, which for our purposes can be treated as a half space. Analysis of the measurements in terms of a linear viscoelasticity theory yields viscosity values for three polymer samples in agreement with those determined by measurements on a macroscopic scale. Theoretical analysis further indicates that the measured limiting static elasticity of the layers may result from the tension generated at the interface between the polymer and water. This work demonstrates the possibility of obtaining quantitative viscoelastic material properties from cell poker measurements and represents the first step in extending these quantitative studies to more complicated structures including cells.     

Influence of gait parameters on the loading of the lower limb

The ground reaction force which acts on the foot during normal walking consists of the sum of two components: the support of the weight of the body and the acceleration of the body. The relationships between the initial loading rate of the lower limb (ignoring the contribution of the heelstrike transient) and the general gait parameters — cadence, stride length, and velocity — have been examined. Plots of the resultant ground reaction force were used to calculate the loading rate of the limb. A sample of 13 normal male subjects, aged from 18 to 63 years, walked at five different self-selected speeds. Velocity showed the highest correlation with loading rate (r = 0.95) and stride length the lowest (r = 0.85). The relationship between cadence and loading rate was non-linear.     

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.