Material characterization of the pig kidney in relation with the biomechanical analysis of renal trauma

The objective of this study was an investigation of the material properties of the fresh pig kidney and parametric characterization of its elastic and inelastic material behavior. The material investigation included density measurements, uniaxial as well as three-dimensional compression tests, tensile tests. and shear tests on the samples extracted from the fresh pig kidney. For comparison, density measurements on a number of soft synthetic materials were also performed. Compression tests on the radial and the tangential specimens from the cortex tissue were performed at various loading rates. Three-axial compression tests were performed on the cortex tissues placed in a compression chamber. Shear tests were performed by punching a cylinder into a slice of the cortex. Tensile tests were carried out on the outer capsule. For characterization of the material behavior, a non-linear theoretical simulation based on a two parameter Blatz model was used. For characterization of the time-dependent behavior of the pig kidney cortex, a four-parameter linear viscoelastic model was employed. From the present experimental and theoretical studies, a number of conclusions were derived: (1) The general behavior of the pig kidney cortex samples under compression showed the general non-linear features typical of the soft tissues; the stress strain diagram was composed of a very flat part at very low stress level to about 30% relative deformation which was followed by a steeply rising stiffening leading to the radial rupture of samples marked by a maximum nominal rupture strain of about 50%. (2) The uniaxial compression tests on the radial and the tangential samples from the cortex tissue showed an increase of the rupture stress with the increase in the loading rate, but a decrease in the related rupture strain. (3) The long-term uniaxial compression tests on the cortex specimens under sustained constant load showed an instantaneous deformation followed by a creep response which eventually approached an asymptote. (4) Simulation of the non-linear material behavior of the cortex tissue under uniaxial compression by the Blatz model gave two pairs of material parameters for the cortex in the radial and the tangential directions. Furthermore, fitting of the assumed four-parameter linear viscoelastic model with the experimental data resulted in the viscoelastic material parameters.     

Anomalous rate dependence of the preconditioned response of soft tissue during load controlled deformation

It has been observed in load controlled laboratory tests of myocardium and skin that the tissues can exhibit a decrease in nonlinear stiffness with an increase in loading rate: the faster a test is performed, the more compliant is the preconditioned material behavior. This response seems to conflict with what is generally expected of soft tissues based on stretch or strain controlled tests, in which an increased rate of deformation results in a stiffer material response. It is hypothesized that this anomalous behavior has not been observed previously due to the small number of cyclic load controlled mechanical characterization tests that are geared specifically towards viscoelastic tissue response. The goal of this paper is to examine the preconditioned response of soft tissue to load controlled deformation using nonlinear viscoelastic material models including quasi-linear viscoelasticity, and to determine under what conditions this anomalous behavior becomes apparent. Results from this study suggest that this behavior is a true phenomenon unique to load controlled deformations that results from the interplay of nonlinear effects and creep behavior. These results call for increased attention to experimental parameters when testing and modeling nonlinear viscoelastic material behavior.     

Nonlinear viscoelastic, thermodynamically consistent, models for biological soft tissue

The mechanical behavior of most biological soft tissue is nonlinear viscoelastic rather than elastic. Many of the models previously proposed for soft tissue involve ad hoc systems of springs and dashpots or require measurement of time-dependent constitutive coefficient functions. The model proposed here is a system of evolution differential equations, which are determined by the long-term behavior of the material as represented by an energy function of the type used for elasticity. The necessary empirical data is time independent and therefore easier to obtain. These evolution equations, which represent non-equilibrium, transient responses such as creep, stress relaxation, or variable loading, are derived from a maximum energy dissipation principle, which supplements the second law of thermodynamics. The evolution model can represent both creep and stress relaxation, depending on the choice of control variables, because of the assumption that a unique long-term manifold exists for both processes. It succeeds, with one set of material constants, in reproducing the loading-unloading hysteresis for soft tissue. The models are thermodynamically consistent so that, given data, they may be extended to the temperature-dependent behavior of biological tissue, such as the change in temperature during uniaxial loading. The Holzapfel et al. three-dimensional two-layer elastic model for healthy artery tissue is shown to generate evolution equations by this construction for biaxial loading of a flat specimen. A simplified version of the Shah-Humphrey model for the elastodynamical behavior of a saccular aneurysm is extended to viscoelastic behavior.     

A model for the compressible,viscoelastic behavior of human amnion addressing tissue variability through a single parameter

A viscoelastic, compressible model is proposed to rationalize the recently reported response of human amnion in multiaxial relaxation and creep experiments. The theory includes two viscoelastic contributions responsible for the short- and long-term time-dependent response of the material. These two contributions can be related to physical processes: water flow through the tissue and dissipative characteristics of the collagen fibers, respectively. An accurate agreement of the model with the mean tension and kinematic response of amnion in uniaxial relaxation tests was achieved. By variation of a single linear factor that accounts for the variability among tissue samples, the model provides very sound predictions not only of the uniaxial relaxation but also of the uniaxial creep and strip-biaxial relaxation behavior of individual samples. This suggests that a wide range of viscoelastic behaviors due to patient-specific variations in tissue composition can be represented by the model without the need of recalibration and parameter identification.     

Viscoelastic characterization of mesenchymal gap tissue and consequences for tension accumulation during distraction

Nonlinear viscoelastic analysis was used to characterize the time-dependent behavior of mesenchymal gap tissue generated during distraction osteogenesis. Six (n = 6) lengthened tibiae were harvested from New Zealand white rabbits at 18 days. This gap tissue was subjected to a series of step displacement tests of increasing magnitude, and force relaxation behavior was monitored. Isochrones in stress-strain space were fit to odd cubic functions of strain. An analytic expression, linear in both e and e3, was developed to predict stress accumulation within the gap tissue as a function of time during distraction. Stress relaxation functions were described well by two-term Prony series. The two time constants determined from mechanical testing results were consistent, suggesting the presence of two fundamental physiologic relaxation processes. Gap tissue stresses were predicted to rise considerably during early stages of lengthening when distraction magnitudes exceeded the clinical norm of 0.25 mm. These differences in tension accumulation were less pronounced by the time lengthening was completed. Specifically, these results may in part explain clinical observations of decreased bone regeneration and altered tissue proliferation and differentiation at higher distraction rates. More generally, this work provides a framework for the rigorous characterization of the viscoelastic properties of biologic tissues ordinarily exposed to step strains.     

The apparent viscoelastic behavior of articular cartilage--the contributions from the intrinsic matrix viscoelasticity and interstitial fluid flows

Articular cartilage was modeled rheologically as a biphasic poroviscoelastic material. A specific integral-type linear viscoelastic model was used to describe the constitutive relation of the collagen-proteoglycan matrix in shear. For bulk deformation, the matrix was assumed either to be linearly elastic, or viscoelastic with an identical reduced relaxation spectrum as in shear. The interstitial fluid was considered to be incompressible and inviscid. The creep and the rate-controlled stress-relaxation experiments on articular cartilage under confined compression were analyzed using this model. Using the material data available in the literature, it was concluded that both the interstitial fluid flow and the intrinsic matrix viscoelasticity contribute significantly to the apparent viscoelastic behavior of this tissue under confined compression.     

Design and numerical implementation of a 3-D non-linear viscoelastic constitutive model for brain tissue during impact

Finite Element (FE) head models are often used to understand mechanical response of the head and its contents during impact loading in the head. Current FE models do not account for non-linear viscoelastic material behavior of brain tissue. We developed a new non-linear viscoelastic material model for brain tissue and implemented it in an explicit FE code. To obtain sufficient numerical accuracy for modeling the nearly incompressible brain tissue, deviatoric and volumetric stress contributions are separated. Deviatoric stress is modeled in a non-linear viscoelastic differential form. Volumetric behavior is assumed linearly elastic. Linear viscoelastic material parameters were derived from published data on oscillatory experiments, and from ultrasonic experiments. Additionally, non-linear parameters were derived from stress relaxation (SR) experiments at shear strains up to 20%. The model was tested by simulating the transient phase in the SR experiments not used in parameter determination (strains up to 20%, strain rates up to 8s(-1)). Both time- and strain-dependent behavior were predicted accurately (R2>0.96) for strain and strain rates applied. However, the stress was overestimated systematically by approximately 31% independent of strain(rate) applied. This is probably caused by limitations of the experimental data at hand.     

A visco-hyperelastic constitutive approach for modeling polyvinyl alcohol sponge

This study proposes the quasi-linear viscoelastic (QLV) model to characterize the time dependent mechanical behavior of poly(vinyl alcohol) (PVA) sponges. The PVA sponges have implications in many viscoelastic soft tissues, including cartilage, liver, and kidney as an implant. However, a critical barrier to the use of the PVA sponge as tissue replacement material is a lack of sufficient study on its viscoelastic mechanical properties. In this study, the nonlinear mechanical behavior of a fabricated PVA sponge is investigated experimentally and computationally using relaxation and stress failure tests as well as finite element (FE) modeling. Hyperelastic strain energy density functions, such as Yeoh and Neo-Hookean, are used to capture the mechanical behavior of PVA sponge at ramp part, and viscoelastic model is used to describe the viscose behavior at hold part. Hyperelastic material constants are obtained and their general prediction ability is verified using FE simulations of PVA tensile experiments. The results of relaxation and stress failure tests revealed that Yeoh material model can define the mechanical behavior of PVA sponge properly compared with Neo-Hookean one. FE modeling results are also affirmed the appropriateness of Yeoh model to characterize the mechanical behavior of PVA sponge. Thus, the Yeoh model can be used in future biomechanical simulations of the spongy biomaterials. These results can be utilized to understand the viscoelastic behavior of PVA sponges and has implications for tissue engineering as scaffold.     

Rheology of gastric mucin exhibits a pH-dependent sol-gel transition

Gastric mucin, a high molecular weight glycoprotein, is responsible for providing the gel-forming properties and protective function of the gastric mucus layer. Bulk rheology measurements in the linear viscoelastic regime show that gastric mucin undergoes a pH-dependent sol-gel transition from a viscoelastic solution at neutral pH to a soft viscoelastic gel in acidic conditions, with the transition occurring near pH 4. In addition to pH-dependent gelation behavior in this system, further rheological studies under nonlinear deformations reveal shear thinning and an apparent yield stress in this material which are also highly influenced by pH.     

Biphasic indentation of articular cartilage--I. Theoretical analysis

A mathematical solution has been obtained for the indentation creep and stress-relaxation behavior of articular cartilage where the tissue is modeled as a layer of linear KLM biphasic material of thickness h bonded to an impervious, rigid bony substrate. The circular (radius = a), plane-ended indenter is assumed to be rigid, porous, free-draining, and frictionless. Double Laplace and Hankel transform techniques were used to solve the partial differential equations. The transformed equations and boundary conditions yielded an integral equation of the Fredholm type which was analyzed asymptotically and solved numerically. Our asymptotic analyses showed that the linear KLM biphasic material behaves like an incompressible (v = 0.5) single-phase elastic solid at t = 0+; the instantaneous response of the material is governed by the shear modulus (mu s) of the solid matrix. The linear KLM biphasic material behaves like a compressible elastic solid with material properties defined by those of the solid matrix, i.e. (lambda s, mu s) or (mu s, v s) as t----infinity. The transient viscoelastic creep and stress-relaxation behavior, 0 less than t less than infinity, of this material is controlled by the frictional drag (which is inversely proportional to the permeability k) associated with the flow of the interstitial fluid through the porous-permeable solid matrix. For given values of the Poisson's ratio of the solid matrix v s and the aspect ratio a/h, where a is the radius of the indenter and h is the thickness of the layer, the creep behavior with respect to the dimensionless time H Akt/a2 is completely controlled by the load parameter P/2 mu sa2 and the stress relaxation behavior is completely controlled by the rate of compression parameter R0 = kH A/V0h where H A = lambda s + 2 mu s and the equilibrium strain u0/h. This mathematical solution may now be used to describe an indentation experiment on articular cartilage to determine the intrinsic material properties of the tissue, i.e. permeability k, and the elastic coefficients of the solid phase (lambda s, mu s) or (mu s, v s).     

Characterizing viscoelastic properties of breast cancer tissue in a mouse model using indentation

Breast cancer is one of the leading cancer forms affecting females worldwide. Characterizing the mechanical properties of breast cancer tissue is important for diagnosis and uncovering the mechanobiology mechanism. Although most of the studies were based on human cancer tissue, an animal model is still describable for preclinical analysis. Using a custom-build indentation device, we measured the viscoelastic properties of breast cancer tissue from 4T1 and SKBR3 cell lines. A total of 7 samples were tested for each cancer tissue using a mouse model. We observed that a viscoelastic model with 2-term Prony series could best describe the ramp and stress relaxation of the tissue. For long-term responses, the SKBR3 tissues were stiffer in the strain levels of 4–10%, while no significant differences were found for the instantaneous elastic modulus. We also found tissues from both cell lines appeared to be strain-independent for the instantaneous elastic modulus and for the long-term elastic modulus in the strain level of 4–10%. In addition, by inspecting the cellular morphological structure of the two tissues, we found that SKBR3 tissues had a larger volume ratio of nuclei and a smaller volume ratio of extracellular matrix (ECM). Compared with prior cellular mechanics studies, our results indicated that ECM could contribute to the stiffening the tissue-level behavior. The viscoelastic characterization of the breast cancer tissue contributed to the scarce animal model data and provided support for the linear viscoelastic model used for in vivo elastography studies. Results also supplied helpful information for modeling of the breast cancer tissue in the tissue and cellular levels.     

Biphasic poroviscoelastic simulation of the unconfined compression of articular cartilage: II--Effect of variable strain rates

This study investigated the abilities of the linear biphasic poroviscoelastic (BPVE) model and the linear biphasic poroelastic (BPE) model to simulate the effect of variable ramp strain rates on the unconfined compression stress relaxation response of articular cartilage. Curve fitting of experimental data showed that the BPVE model was able to successfully account for the ramp strain rate-dependent viscoelastic behavior of articular cartilage under unconfined compression, while the BPE model was able to account for the complete viscoelastic response at a slow strain rate, but only the long-term viscoelastic response at faster strain rates. We concluded that the short-term viscoelastic behavior of articular cartilage, when subjected to a fast ramp strain rate, is primarily governed by a fluid flow-independent (intrinsic) viscoelastic mechanism, whereas the long-term viscoelastic behavior is governed by a fluid flow-dependent (biphasic) viscoelastic mechanism. Furthermore, a linear viscoelastic representation of the solid stress was found to be a valid model assumption for the simulation of ramp strain rate-dependent relaxation behaviors of articular cartilage within the range of ramp strain rates investigated.     

Analysis of lung parenchyma as a parametric porous medium.

The dynamic behavior of the lung in health and disease depends on its viscoelastic properties. To better understand these properties, several mathematical models have been utilized by many investigators. In the present work, we present a new approach that characterizes the dynamics of gas flow into a viscoelastic porous medium that models the lung structure. This problem is considered in terms of the lung input impedance on a macro level and parenchymal tissue impedance on the level of an alveolar wall. We start from a basic theoretical analysis in which macroscopic tissue deformations are represented in accordance with the linearized Navier-Stokes equations. This approach has strong theoretical underpinnings in other situations but has not been applied to analyze the impedance of the inflated lung. Our analysis provides a theoretical basis for analyzing the interaction between flow into the lungs as a biophysical diffusion process and parenchymal viscoelasticity described phenomenologically, within the frameworks of standard viscoelasticity and structural damping. This lung impedance incorporates parameters of porosity, permeability, and viscoelasticity on micro and macro levels of parenchymal tissue. The analysis shows the theoretical basis of the transformation from the impedance of alveolar walls or isolated tissue strips to that of the intact parenchyma. We also show how the loading impedance at the lung boundary may have a significant impact on the dynamic behavior of whole lung viscoelasticity. Our analysis may be useful in directing specific tests of different models and for analyzing experimental measurements of viscoelastic parameters of lung material under normal and pathological conditions.     

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.     

Some viscoplastic characteristics of bovine and human cortical bone

Multiple cycle tensile creep tests were performed on human and bovine cortical bone specimens. The tests enabled total strain to be decomposed into elastic, linear viscoelastic, creep and permanent plastic components. The results indicate that a stress threshold exists; above which time dependent effects dominate material response and below which the behavior is primarily linear viscoelastic, with time effects playing only a secondary role. A constant stress above the threshold produces a constant steady state creep rate, with the magnitude of the creep rate being an exponential function of the stress magnitude. Additionally, it was found that a major portion of the inelastic strain is always recovered on unloading and that the accumulation of creep strain increases the material compliance on subsequent loadings below the threshold. These two factors suggest that a damage mechanism is responsible for the nonlinear behavior.     

The possible role of poroelasticity in the apparent viscoelastic behavior of passive cardiac muscle

This paper investigates the contribution of extracellular fluid flow to the apparent viscoelastic behavior of passive cardiac muscle. The muscle is modeled as an incompressible, isotropic, poroelastic solid saturated by an incompressible viscous fluid. Based on Biot's linear and nonlinear consolidation theories, solutions are presented for general time-dependent uniaxial loading of unconfined cylindrical muscle specimens. The nonlinear analysis includes the effects of large strain, material nonlinearity, and strain-dependent permeability. The computed results show that, for axial stretch ratios greater than 1.1, the changing permeability and the loading rate strongly affect the total stress relaxation and the short-time relaxation rate. Comparisons of theoretical and published experimental results show that extracellular fluid flow can account for several observed biomechanical features of passive myocardium, including the insensitivity of stress-strain curves to loading rate and of stress-relaxation curves to the amount of stretch. Theoretical hysteresis loops, however, are too small. Thus, both poroelastic and tissue viscoelastic effects must be considered in studies of passive cardiac muscle.     

Rheological characterization of culture broth containing the exopolysaccharide PS-EDIV from Sphingomonas pituitosa

Sphingomonas pituitosa excretes the capsular exopolysaccharide PS-EDIV into the culture broth augmenting considerably its fluid viscosity. Since this change particularly affects key processes like mixing and transport during the microbial production, this work was aimed at the rheological characterization of the polymer-containing culture broth of S. pituitosa. The study included investigations on basic properties of the culture broth, but also on the dependence of the biomass–polymer-solution properties on different physicochemical post-cultivation treatment steps like variations of temperature, pH-value or concentration of salts. The essential result is the characterization of the viscoelastic behavior of the culture broth, which was more gel-like than sol-like and exhibited slight elastic properties. This rheological behavior showed that the PS-EDIV culture broth formed non-Newtonian fluids, indicating that it is a pseudoplastic biopolymer, with yield stress appearance and exhibits thixotropic properties. Rheograms were fitted to the Herschel–Bulkley model. The amplitude sweep revealed a deformation of 21% as the limiting value of the linear viscoelastic interval. Furthermore, the PS-EDIV culture broth showed a high viscosity which was strongly influenced by salt type and concentration but weakly influenced by temperature and pH-value within the investigated experimental boundaries.     

On the viscoelastic character of liver tissue: experiments and modelling of the linear behaviour

This paper presents the results of recent investigations into the viscoelastic character of liver tissue, especially the linear response. The linear domain of liver tissue has been determined by a strain sweep oscillation test. The fluid-like viscoelastic behaviour of liver tissue was confirmed by stress relaxation experiments. To demonstrate the consistency and quality of the experimental data, the relaxation spectrum was used to reconstruct the relaxation modulus, as well as the storage and loss moduli. The generalised Maxwell model was successfully employed to model the mechanical behaviour in the linear viscoelastic domain.