Modelling and simulation of porcine liver tissue indentation using finite element method and uniaxial stress–strain data

We hypothesize that both compression and elongation stress–strain data should be considered for modeling and simulation of soft tissue indentation. Uniaxial stress–strain data were obtained from in vitro loading experiments of porcine liver tissue. An axisymmetric finite element model was used to simulate liver tissue indentation with tissue material represented by hyperelastic models. The material parameters were derived from uniaxial stress–strain data of compressions, elongations, and combined compression and elongation of porcine liver samples. in vitro indentation tests were used to validate the finite element simulation. Stress–strain data from the simulation with material parameters derived from the combined compression and elongation data match the experimental data best. This is due to its better ability in modeling 3D deformation since the behavior of biological soft tissue under indentation is affected by both its compressive and tensile characteristics. The combined logarithmic and polynomial model is somewhat better than the 5-constant Mooney–Rivlin model as the constitutive model for this indentation simulation.     

A validated model of passive skeletal muscle to predict force and intramuscular pressure

The passive properties of skeletal muscle are often overlooked in muscle studies, yet they play a key role in tissue function in vivo. Studies analyzing and modeling muscle passive properties, while not uncommon, have never investigated the role of fluid content within the tissue. Additionally, intramuscular pressure (IMP) has been shown to correlate with muscle force in vivo and could be used to predict muscle force in the clinic. In this study, a novel model of skeletal muscle was developed and validated to predict both muscle stress and IMP under passive conditions for the New Zealand White Rabbit tibialis anterior. This model is the first to include fluid content within the tissue and uses whole muscle geometry. A nonlinear optimization scheme was highly effective at fitting model stress output to experimental stress data (normalized mean square error or NMSE fit value of 0.993) and validation showed very good agreement to experimental data (NMSE fit values of 0.955 and 0.860 for IMP and stress, respectively). While future work to include muscle activation would broaden the physiological application of this model, the passive implementation could be used to guide surgeries where passive muscle is stretched.     

Biphasic poroviscoelastic simulation of the unconfined compression of articular cartilage: I--Simultaneous prediction of reaction force and lateral displacement

This study investigated the ability of the linear biphasic poroelastic (BPE) model and the linear biphasic poroviscoelastic (BPVE) model to simultaneously predict the reaction force and lateral displacement exhibited by articular cartilage during stress relaxation in unconfined compression. Both models consider articular cartilage as a binary mixture of a porous incompressible solid phase and an incompressible inviscid fluid phase. The BPE model assumes the solid phase is elastic, while the BPVE model assumes the solid phase is viscoelastic. In addition, the efficacy of two additional models was also examined, i.e., the transversely isotropic BPE (TIBPE) model, which considers transverse isotropy of the solid matrix within the framework of the linear BPE model assumptions, and a linear viscoelastic solid (LVE) model, which assumes that the viscoelastic behavior of articular cartilage is solely governed by the intrinsic viscoelastic nature of the solid matrix, independent of the interstitial fluid flow. It was found that the BPE model was able to accurately account for the lateral displacement, but unable to fit the short-term reaction force data of all specimens tested. The TIBPE model was able to account for either the lateral displacement or the reaction force, but not both simultaneously. The LVE model was able to account for the complete reaction force, but unable to fit the lateral displacement measured experimentally. The BPVE model was able to completely account for both lateral displacement and reaction force for all specimens tested. These results suggest that both the fluid flow-dependent and fluid flow-independent viscoelastic mechanisms are essential for a complete simulation of the viscoelastic phenomena of articular cartilage.     

Dynamic measurement of internal solid displacement in articular cartilage using ultrasound backscatter

Mechanics of articular cartilage can be represented using poroelastic theories where fluid and solid displacements are viscously coupled to create a time-dependent spatially heterogeneous behavior. In recent models of this tissue, finite element methods have been used to predict tissue deformation as a function of time for adult articular cartilage bearing a characteristic depth-dependent structure and composition. However, current experimental methods are limited in providing verification of these predictions. The current study presents an apparatus for imaging the radial displacement profile of cartilage in unconfined compression using an ultrasound technique called elastography. We acquired ultrasound A-scans across the lateral diameter of full-thickness cartilage disks containing a thin layer of underlying bone, during axial compression. Elastography was then applied to correlate temporally sequential A-scans to estimate the solid radial displacement profile in articular cartilage while it undergoes compression and stress-relaxation. Both time-dependent and depth-dependent solid radial displacement profiles were obtained with a precision better than 0.2 micro The results generally agree with predictions of poroelastic models, demonstrating lateral expansion with an effective Poisson's ratio just after completion of the compression phase of the mechanical tests reaching values from 0.18 to 0.4 (depending on compression speed), followed by contraction to lower values. A more restricted movement was observed at both the articular surface and near to the subchondral bone than at regions midway between these two locations.     

Scaffold architecture determines chondrocyte response to externally applied dynamic compression

It remains unclear how specific mechanical signals generated by applied dynamic compression (DC) regulate chondrocyte biosynthetic activity. It has previously been suggested that DC-induced interstitial fluid flow positively impacts cartilage-specific matrix production. Modifying fluid flow within dynamically compressed hydrogels therefore represents a promising approach to controlling chondrocyte behavior, which could potentially be achieved by changing the construct architecture. The objective of this study was to first determine the influence of construct architecture on the mechanical environment within dynamically compressed agarose hydrogels using finite element (FE) modeling and to then investigate how chondrocytes would respond to this altered environment. To modify construct architecture, an array of channels was introduced into the hydrogels. Increased magnitudes of fluid flow were predicted in the periphery of dynamically compressed solid hydrogels and also around the channels in the dynamically compressed channeled hydrogels. DC was found to significantly increase sGAG synthesis in solid constructs, which could be attributed at least in part to an increase in DNA. DC was also found to preferentially increase collagen accumulation in regions of solid and channeled constructs where FE modeling predicted higher levels of fluid flow, suggesting that this stimulus is important for promoting collagen production by chondrocytes embedded in agarose gels. In conclusion, this study demonstrates how the architecture of cell-seeded scaffolds or hydrogels can be modified to alter the spatial levels of biophysical cues throughout the construct, leading to greater collagen accumulation throughout the engineered tissue rather than preferentially in the construct periphery. This system also provides a novel approach to investigate how chondrocytes respond to altered levels of biophysical stimulation.     

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.     

Numerical Modeling of Interstitial Fluid Flow Coupled with Blood Flow through a Remodeled Solid Tumor Microvascular Network

Modeling of interstitial fluid flow involves processes such as fluid diffusion, convective transport in extracellular matrix, and extravasation from blood vessels. To date, majority of microvascular flow modeling has been done at different levels and scales mostly on simple tumor shapes with their capillaries. However, with our proposed numerical model, more complex and realistic tumor shapes and capillary networks can be studied. Both blood flow through a capillary network, which is induced by a solid tumor, and fluid flow in tumor’s surrounding tissue are formulated. First, governing equations of angiogenesis are implemented to specify the different domains for the network and interstitium. Then, governing equations for flow modeling are introduced for different domains. The conservation laws for mass and momentum (including continuity equation, Darcy’s law for tissue, and simplified Navier–Stokes equation for blood flow through capillaries) are used for simulating interstitial and intravascular flows and Starling’s law is used for closing this system of equations and coupling the intravascular and extravascular flows. This is the first study of flow modeling in solid tumors to naturalistically couple intravascular and extravascular flow through a network. This network is generated by sprouting angiogenesis and consisting of one parent vessel connected to the network while taking into account the non-continuous behavior of blood, adaptability of capillary diameter to hemodynamics and metabolic stimuli, non-Newtonian blood flow, and phase separation of blood flow in capillary bifurcation. The incorporation of the outlined components beyond the previous models provides a more realistic prediction of interstitial fluid flow pattern in solid tumors and surrounding tissues. Results predict higher interstitial pressure, almost two times, for realistic model compared to the simplified model.     

Glycosaminoglycan network geometry may contribute to anisotropic hydraulic permeability in cartilage under compression.

Resistance to fluid flow within cartilage extracellular matrix is provided primarily by a dense network of rod-like glycosaminoglycans (GAGs). If the geometrical organization of this network is random, the hydraulic permeability tensor of cartilage is expected to be isotropic. However, experimental data have suggested that hydraulic permeability may become anisotropic when the matrix is mechanically compressed, contributing to cartilage biomechanical functions such as lubrication. We hypothesized that this may be due to preferred GAG rod orientations and directionally-dependent reduction of inter-GAG spacings which reflect molecular responses to tissue deformations. To examine this hypothesis, we developed a model for effects of compression which allows the GAG rod network to deform consistently with tissue-scale deformations but while still respecting limitations imposed by molecular structure. This network deformation model was combined with a perturbation analysis of a classical analytical model for hydraulic permeability based on molecular structure. Finite element analyses were undertaken to ensure that this approach exhibited results similar to those emerging from more exact calculations. Model predictions for effects of uniaxial confined compression on the hydraulic permeability tensor were consistent with previous experimental results. Permeability decreased more rapidly in the direction perpendicular to compression than in the parallel direction, for matrix solid volume fractions associated with fluid transport in articular cartilage. GAG network deformations may therefore introduce anisotropy to the permeability (and other GAG-associated matrix properties) as physiological compression is applied, and play an important role in cartilage lubrication and other biomechanical functions.     

Quantification of fluid shear stress in bone tissue engineering scaffolds with spherical and cubical pore architectures

Recent studies have shown that mechanical stimulation, in the form of fluid perfusion and mechanical compression, can enhance osteogenic differentiation of mesenchymal stem cells and bone cells within tissue engineering scaffolds in vitro. The precise nature of mechanical stimulation within tissue engineering scaffolds is not only dictated by the exogenously applied loading regime, but also depends on the geometric features of the scaffold, in particular architecture, pore size and porosity. However, the precise contribution of each geometric feature towards the resulting mechanical stimulation within a scaffold is difficult to characterise due to the wide range of interacting parameters. In this study, we have applied a fluid–structure interaction model to investigate the role of scaffold geometry (architecture, pore size and porosity) on pore wall shear stress (WSS) under a range of different loading scenarios: fluid perfusion, mechanical compression and a combination of perfusion and compression. It is found that scaffold geometry (spherical and cubical pores), in particular the pore size, has a significant influence on the stimulation within scaffolds. Furthermore, we observed an amplified WSS within scaffolds under a combination of fluid perfusion and mechanical compression, which exceeded that caused by individual fluid perfusion or mechanical compression approximately threefold. By conducting this comprehensive parametric variation study, an expression was generated to allow the design and optimisation of 3D TE scaffolds and inform experimental loading regimes so that a desired level of mechanical stimulation, in terms of WSS is generated within the scaffold.     

Prediction of internal pressure profile of compression bandages using stress relaxation parameters

The efficacy of compression therapy using compression bandages is highly dependent on the level of compression applied and the sustenance of the pressure during the course of treatment. This study attempts to predict the pressure profile generated by compression bandages using constitutive equations describing relaxation behavior of viscoelastic materials. It is observed that this pressure profile is highly correlated with the stress relaxation behavior of the bandage. To model the pressure profile, the stress relaxation behavior of compression bandages was studied and modeled using three mechanical models: the Maxwell model, the standard linear solid model and the two-component Maxwell model with a nonlinear spring. It was observed that the models with more component values explained the experimental relaxation curves better. The parameters used for modelling relaxation behavior were used to describe the pressure profile, which is significantly dependent on the longitudinal stress relaxation behavior of the bandage, using the modified Laplace's law equation. This approach thus helps in evaluating the bandage performance with time during compression therapy as novel wound care management.     

Application of the u-p finite element method to the study of articular cartilage.

The finite element method using the principle of virtual work was applied to the biphasic theory to establish a numerical routine for analyses of articular cartilage behavior. The matrix equations that resulted contained displacements of the solid matrix (mu) and true fluid pressure (p) as the unknown variables at the element nodes. Both small and large strain conditions were considered. The algorithms and computer code for the analysis of two-dimensional plane strain, plane stress, and axially symmetric cases were developed. The u-p finite element numerical procedure demonstrated excellent agreement with available closed-form and numerical solutions for the configurations of confined compression and unconfined compression under small strains, and for confined compression under large strains. The model was also used to examine the behavior of a repaired articular surface. The differences in material properties between the repair tissue and normal cartilage resulted in significant deformation gradients across the repair interface as well as increased fluid efflux from the tissue.     

Indirect determination of trabecular bone effective tissue failure properties using micro-finite element simulations

Trabecular bone strength is marked not only by the onset of local yielding, but also by post-yield behavior. To study and predict trabecular bone elastic and yield properties, micro-finite element (micro-FE) models were successfully applied. However, trabecular bone strength predictions require micro-FE models incorporating post-yield behavior of trabecular bone tissue. Due to experimental difficulties, such data is currently not available. Here we used micro-FE modeling to determine failure behavior of trabecular bone tissue indirectly, by iteratively fitting FE simulation to experimental results. Failure parameters were fitted to an isotropic plasticity model based on Hill's yield function, using materially and geometrically nonlinear micro-FE models of seven bovine trabecular bone specimens. The predictive value of the averaged effective tissue properties was subsequently tested. The results showed that compression softening had to be included on the tissue level in order to accurately describe the apparent-level behavior of the bone specimens. A sensitivity study revealed that the simulated response was less sensitive to variations in the post-yield properties of the bone tissue than variations in the elastic and yield properties. Due to fitting of the tissue properties, apparent-level behavior could be accurately reproduced for each specimen separately. Predictions based on the averaged and fixed tissue properties were less accurate, due to inter-specimen variations in the tissue properties.     

In silico study of bone tissue regeneration in an idealised porous hydrogel scaffold using a mechano-regulation algorithm

Mechanical stimulation, in the form of fluid perfusion or mechanical strain, enhances osteogenic differentiation and overall bone tissue formation by mesenchymal stems cells cultured in biomaterial scaffolds for tissue engineering applications. In silico techniques can be used to predict the mechanical environment within biomaterial scaffolds, and also the relationship between bone tissue regeneration and mechanical stimulation, and thereby inform conditions for bone tissue engineering experiments. In this study, we investigated bone tissue regeneration in an idealised hydrogel scaffold using a mechano-regulation model capable of predicting tissue differentiation, and specifically compared five loading cases, based on known experimental bioreactor regimes. These models predicted that low levels of mechanical loading, i.e. compression (0.5% strain), pore pressure of 10 kPa and a combination of compression (0.5%) and pore pressure (10 kPa), could induce more osteogenic differentiation and lead to the formation of a higher bone tissue fraction. In contrast greater volumes of cartilage and fibrous tissue fractions were predicted under higher levels of mechanical loading (i.e. compression strain of 5.0% and pore pressure of 100 kPa). The findings in this study may provide important information regarding the appropriate mechanical stimulation for in vitro bone tissue engineering experiments.     

A Poroelastic Model Describing Nutrient Transport and Cell Stresses Within a Cyclically Strained Collagen Hydrogel

In the creation of engineered tissue constructs, the successful transport of nutrients and oxygen to the contained cells is a significant challenge. In highly porous scaffolds subject to cyclic strain, the mechanical deformations can induce substantial fluid pressure gradients, which affect the transport of solutes. In this article, we describe a poroelastic model to predict the solid and fluid mechanics of a highly porous hydrogel subject to cyclic strain. The model was validated by matching the predicted penetration of a bead into the hydrogel from the model with experimental observations and provides insight into nutrient transport. Additionally, the model provides estimates of the wall-shear stresses experienced by the cells embedded within the scaffold. These results provide insight into the mechanics of and convective nutrient transport within a cyclically strained hydrogel, which could lead to the improved design of engineered tissues.     

The influence of the fixed negative charges on mechanical and electrical behaviors of articular cartilage under unconfined compression

Unconfined compression test has been frequently used to study the mechanical behaviors of articular cartilage, both theoretically and experimentally. It has also been used in explant and gel-cell-complex studies in tissue engineering. In biphasic and poroelastic theories, the effect of charges fixed on the proteoglycan macromolecules in articular cartilage is embodied in the apparent compressive Young's modulus and the apparent Poisson's ratio of the tissue, and the fluid pressure is considered to be the portion above the osmotic pressure. In order to understand how proteoglycan fixed charges might affect the mechanical behaviors of articular cartilage, and in order to predict the osmotic pressure and electric fields inside the tissue in this experimental configuration, it is necessary to use a model that explicitly takes into account the charged nature of the tissue and the flow of ions within its porous interstices. In this paper, we used a finite element model based on the triphasic theory to study how fixed charges in the porous-permeable soft tissue can modulate its mechanical and electrochemical responses under a step displacement in unconfined compression. The results from finite element calculations showed that: 1) A charged tissue always supports a larger load than an uncharged tissue of the same intrinsic elastic moduli. 2) The apparent Young's modulus (the ratio of the equilibrium axial stress to the axial strain) is always greater than the intrinsic Young's modulus of an uncharged tissue. 3) The apparent Poisson's ratio (the negative ratio of the lateral strain to the axial strain) is always larger than the intrinsic Poisson's ratio of an uncharged tissue. 4) Load support derives from three sources: intrinsic matrix stiffness, hydraulic pressure and osmotic pressure. Under the unconfined compression, the Donnan osmotic pressure can constitute between 13%-22% of the total load support at equilibrium. 5) During the stress-relaxation process following the initial instant of loading, the diffusion potential (due to the gradient of the fixed charge density and the associated gradient of ion concentrations) and the streaming potential (due to fluid convection) compete against each other. Within the physiological range of material parameters, the polarity of the electric potential depends on both the mechanical properties and the fixed charge density (FCD) of the tissue. For softer tissues, the diffusion effects dominate the electromechanical response, while for stiffer tissues, the streaming potential dominates this response. 6) Fixed charges do not affect the instantaneous strain field relative to the initial equilibrium state. However, there is a sudden increase in the fluid pressure above the initial equilibrium osmotic pressure. These new findings are relevant and necessary for the understanding of cartilage mechanics, cartilage biosynthesis, electromechanical signal transduction by chondrocytes, and tissue engineering.     

Stress-relaxation of human patellar articular cartilage in unconfined compression: prediction of mechanical response by tissue composition and structure

Mechanical properties of articular cartilage are controlled by tissue composition and structure. Cartilage function is sensitively altered during tissue degeneration, in osteoarthritis (OA). However, mechanical properties of the tissue cannot be determined non-invasively. In the present study, we evaluate the feasibility to predict, without mechanical testing, the stress-relaxation response of human articular cartilage under unconfined compression. This is carried out by combining microscopic and biochemical analyses with composition-based mathematical modeling. Cartilage samples from five cadaver patellae were mechanically tested under unconfined compression. Depth-dependent collagen content and fibril orientation, as well as proteoglycan and water content were derived by combining Fourier transform infrared imaging, biochemical analyses and polarized light microscopy. Finite element models were constructed for each sample in unconfined compression geometry. First, composition-based fibril-reinforced poroviscoelastic swelling models, including composition and structure obtained from microscopical and biochemical analyses were fitted to experimental stress-relaxation responses of three samples. Subsequently, optimized values of model constants, as well as compositional and structural parameters were implemented in the models of two additional samples to validate the optimization. Theoretical stress-relaxation curves agreed with the experimental tests (R=0.95-0.99). Using the optimized values of mechanical parameters, as well as composition and structure of additional samples, we were able to predict their mechanical behavior in unconfined compression, without mechanical testing (R=0.98). Our results suggest that specific information on tissue composition and structure might enable assessment of cartilage mechanics without mechanical testing.     

Finite deformation of soft tissue: analysis of a mixture model in uni-axial compression

The dynamic finite deformational behavior of a biphasic model for soft hydrated tissue is examined. In the case of uni-axial confined compression the displacement and stress fields are derived for steady-state permeation, creep, and stress-relaxation. It is shown how to use the results of this analysis to obtain the constitutive relations, as well as the associated material parameters, from the corresponding experiments. It is also shown that the solutions from the theory go much farther, giving a detailed account of the deformation and interaction of the fluid and solid phases in the tissue.     

Experimental verification of the roles of intrinsic matrix viscoelasticity and tension-compression nonlinearity in the biphasic response of cartilage

A biphasic-CLE-QLV model proposed in our recent study [2001, J. Biomech. Eng., 123, pp. 410-417] extended the biphasic theory of Mow et al. [1980, J. Biomech. Eng., 102, pp. 73-84] to include both tension-compression nonlinearity and intrinsic viscoelasticity of the cartilage solid matrix by incorporating it with the conewise linear elasticity (CLE) model [1995, J. Elasticity, 37, pp. 1-38] and the quasi-linear viscoelasticity (QLV) model [Biomechanics: Its foundations and objectives, Prentice Hall, Englewood Cliffs, 1972]. This model demonstrates that a simultaneous prediction of compression and tension experiments of articular cartilage, under stress-relaxation and dynamic loading, can be achieved when properly taking into account both flow-dependent and flow-independent viscoelastic effects, as well as tension-compression nonlinearity. The objective of this study is to directly test this biphasic-CLE-QLV model against experimental data from unconfined compression stress-relaxation tests at slow and fast strain rates as well as dynamic loading. Twelve full-thickness cartilage cylindrical plugs were harvested from six bovine glenohumeral joints and multiple confined and unconfined compression stress-relaxation tests were performed on each specimen. The material properties of specimens were determined by curve-fitting the experimental results from the confined and unconfined compression stress relaxation tests. The findings of this study demonstrate that the biphasic-CLE-QLV model is able to describe the strain-rate-dependent mechanical behaviors of articular cartilage in unconfined compression as attested by good agreements between experimental and theoretical curvefits (r2 = 0.966 +/- 0.032 for testing at slow strain rate; r2 = 0.998 +/- 0.002 for testing at fast strain rate) and predictions of the dynamic response (r2 = 0.91 +/- 0.06). This experimental study also provides supporting evidence for the hypothesis that both tension-compression nonlinearity and intrinsic viscoelasticity of the solid matrix of cartilage are necessary for modeling the transient and equilibrium responses of this tissue in tension and compression. Furthermore, the biphasic-CLE-QLV model can produce better predictions of the dynamic modulus of cartilage in unconfined dynamic compression than the biphasic-CLE and biphasic poroviscoelastic models, indicating that intrinsic viscoelasticity and tension-compression nonlinearity of articular cartilage may play important roles in the load-support mechanism of cartilage under physiologic loading.     

A Poroelastic Model Describing Nutrient Transport and Cell Stresses Within a Cyclically Strained Collagen Hydrogel

In the creation of engineered tissue constructs, the successful transport of nutrients and oxygen to the contained cells is a significant challenge. In highly porous scaffolds subject to cyclic strain, the mechanical deformations can induce substantial fluid pressure gradients, which affect the transport of solutes. In this article, we describe a poroelastic model to predict the solid and fluid mechanics of a highly porous hydrogel subject to cyclic strain. The model was validated by matching the predicted penetration of a bead into the hydrogel from the model with experimental observations and provides insight into nutrient transport. Additionally, the model provides estimates of the wall-shear stresses experienced by the cells embedded within the scaffold. These results provide insight into the mechanics of and convective nutrient transport within a cyclically strained hydrogel, which could lead to the improved design of engineered tissues.