Biphasic Continuum Research Articles

Keynote lecture. Constitutive modelling of phase transitions in granular soils

The use of numerical codes for simulating the propagation phase of fast landslides requires the definition of constitutive relationships suitable for the description of the mechanical behaviour of granular soils, under both solid and fluid conditions. In this paper, the authors briefly illustrate a constitutive modelling approach interpreting the material as a bi-phasic continuum, according to which the solid phase may behave like a solid, under quasi-static conditions, and a fluid, under collisional/dynamic conditions. The model presented seems to satisfactorily reproduce a series of results obtained by using a discrete element numerical code on mono and polydisperse granular specimens. The constitutive model has been implemented in a Material Point Method code to solve boundary value problems, in which strain rate large values are expected to influence the material response.

Biphasic Properties of PVAH (Polyvinyl Alcohol Hydrogel) Reflecting Biomechanical Behavior of the Nucleus Pulposus of the Human Intervertebral Disc.

PVAH is a mixture of solid and fluid, but its mechanical behavior has usually been described using solid material models. The purpose of this study was to obtain material properties that can reflect the mechanical behavior of polyvinyl alcohol hydrogel (PVAH) using finite element analysis, a biphasic continuum model, and to optimize the composition ratio of PVAH to replace the nucleus pulposus (NP) of the human intervertebral disc. Six types of PVAH specimens (3, 5, 7, 10, 15, 20 wt%) were prepared, then unconfined compression experiments were performed to acquire their material properties using the Holmes–Mow biphasic model. With an increasing weight percentage of PVA in PVAH, the Young’s modulus increased while the permeability parameter decreased. The Young’s modulus and permeability parameter were similar to those of the NP at 15 wt% and 20 wt%. The range of motion, facet joint force, and NP pressures measured from dynamic motional analysis of the lumbar segments with the NP model also exhibited similar values to those with 15~20 wt% PVAH models. Considering the structural stability and pain of the lumbar segments, it appears that 20 wt% PVAH is most suitable for replacing the NP.

Numerical Solution of a Two Dimensional Tumour Growth Model with Moving Boundary

We consider a biphasic continuum model for avascular tumour growth in two spatial dimensions, in which a cell phase and a fluid phase follow conservation of mass and momentum. A limiting nutrient that follows a diffusion process controls the birth and death rate of the tumour cells. The cell volume fraction, cell velocity--fluid pressure system, and nutrient concentration are the model variables. A coupled system of a hyperbolic conservation law, \cred{a viscous fluid model}, and a parabolic diffusion equation governs the dynamics of the model variables. The tumour boundary moves with the normal velocity of the outermost layer of cells, and this time--dependence is a challenge in designing and implementing a stable and fast numerical scheme. We recast the model into a form where the hyperbolic equation is defined on a fixed extended domain and retrieve the tumour boundary as the interface at which the cell volume fraction decreases below a threshold value. This procedure eliminates the need to track the tumour boundary explicitly and the computationally expensive re--meshing of the time--dependent domains. A numerical scheme based on finite volume methods for the hyperbolic conservation law, Lagrange $\mathbb{P}_2 - \mathbb{P}_1$ Taylor--Hood finite element method for the \cred{viscous} system, and mass--lumped finite element method for the parabolic equations is implemented in two spatial dimensions, and several cases are studied. We demonstrate the versatility of the numerical scheme in catering for irregular and asymmetric initial tumour geometries. We present numerical simulations for both cases and the results are consistent with theoretical and heuristic expectations such as early linear growth rate and preservation of radial symmetry when the boundary conditions are symmetric.

A poro-hyper-viscoelastic rate-dependent constitutive modeling for the analysis of brain tissues

In this paper, the dynamic behavior of bovine brain tissue, measured from in-vitro unconfined compression tests, is examined and represented through a viscoelastic biphasic model. The experiments have been carried out under three compression speeds of 10, 100, and 1000 mm/s. The results exhibited significant rate-dependent behavior. The brain tissue is modeled as a biphasic continuum consisting of a compressible solid matrix, fully saturated with an incompressible interstitial fluid. The governing equations based on conservation of mass and momentum are used to describe the solid-fluid interactions. An inverse scheme is employed in which a finite element model runs iteratively to optimize constitutive constants. The obtained material parameters of the proposed biphasic model show relatively good agreement (R2 ≥ 0.96) with the experimental tissue mechanical responses at different rates. The model can successfully capture the key aspects of the rate-dependency for both solid and fluid phases under large strain deformation. This poro-hyper viscoelastic model can effectively estimate the global and local rate-dependent tissue deformations, the spatial variations in pore spaces, hydrostatic pressure as well as fluid diffusion through the tissue.

Estimation of depth-dependent material properties of biphasic soft tissues through finite element optimization and sensitivity analysis

A knowledge of material properties of soft tissue, such as articular cartilage, is essential to assess its mechanical function. It is also increasingly more evident that the inhomogeneity of the tissues plays a significant role in its in vivo functioning. Hence, efficient and reliable tools are needed to accurately characterize the inhomogeneity of the soft tissue mechanical properties. The objective of this research is to propose a finite element optimization procedure to determine depth-dependent material properties of articular cartilage by processing experimental data. Cartilage is modeled as a biphasic continuum with a linear elastic solid phase. The optimization method is based on a sensitivity analysis where the sensitivity of the finite element results to a variation in the material properties is analytically evaluated. The elastic modulus and permeability of the tissue are assumed to vary either linearly or quadratically through the thickness of the cartilage layer. After adopting some initial estimates, these material properties are updated iteratively based on their sensitivities to the current results, and the difference between the actual experimental data and computational experimental data. The optimization method has been tested in two common experimental configurations of cartilage and found to be efficient to estimate the material properties.

On the appropriateness of modelling brain parenchyma as a biphasic continuum

Computational methods originally developed for analysis in engineering have been applied to the analysis of biological materials for many years. One particular application of these engineering tools is the brain, allowing researchers to predict the behaviour of brain tissue in various traumatic, surgical and medical scenarios. Typically two different approaches have been used to model deformation of brain tissue: single-phase models which treat the brain as a viscoelastic material, and biphasic models which treat the brain as a porous deformable medium through which liquid can move. In order to model the brain as a biphasic continuum, the hydraulic conductivity of the solid phase is required; there are many theoretical values for this conductivity in the literature, with variations of up to three orders of magnitude.We carried out a series of simple experiments using lamb and sheep brain tissue to establish the rate at which cerebrospinal fluid moves through the brain parenchyma. Mindful of possible variations in hydraulic conductivity with tissue deformation, our intention was to carry out our experiments on brain tissue subjected to minimal deformation. This has enabled us to compare the rate of flow with values predicted by some of the theoretical values of hydraulic conductivity from the literature. Our results indicate that the hydraulic conductivity of the brain parenchyma is consistent with the lowest theoretical published values. These extremely low hydraulic conductivities lead to such low rates of CSF flow through the brain tissue that in effect the material behaves as a single-phase deformable solid.

Optimizing Learning and Quality of Life throughout the Lifespan

This overview surveys the new optimism about the aging mind/brain, focusing on the potential for self-regulation practices to advance research in stress-protection and optimal health. It reviews recent findings and offers a research framework. The review links the age-related biology of stress and regeneration to the variability of mind/brain function found under a range of conditions from trauma to enrichment. The framework maps this variation along a biphasic continuum from atrophic dysfunction to peak performance. It adopts the concept of allostatic load as a measure of the wear-and-tear caused by stress, and environmental enrichment as a measure of the use-dependent enhancement caused by positive reinforcement. It frames the dissociation, aversive affect and stereotyped reactions linked with stress as cognitive, affective and behavioral forms of allostatic drag; and the association, positive affect, and creative responses in enrichment as forms of allostatic lift. It views the human mind/brain as a heterarchy of higher intelligence systems that shift between a conservative, egocentric mode heightening self-preservation and memory and a generative, altruistic mode heightening self-correction and learning. Cultural practices like meditation and psychotherapy work by teaching the self-regulation of shifts from the conservative to the generative mode. This involves a systems shift from allostatic drag to allostatic lift, minimizing wear-and-tear and optimizing plasticity and learning. For cultural practices to speed research and application, a universal typology is needed. This framework includes a typology aligning current brain models of stress and learning with traditional Indo-Tibetan models of meditative stress-cessation and learning enrichment.

Tissue–fluid interface analysis using biphasic finite element method

Numerical studies on fluid–structure interaction have primarily relied on decoupling the solid and fluid sub-domains with the interactions treated as external boundary conditions on the individual sub-domains. The finite element applications for the fluid–structure interactions can be divided into iterative algorithms and sequential algorithms. In this paper, a new computational methodology for the analysis of tissue–fluid interaction problems is presented. The whole computational domain is treated as a single biphasic continuum, and the same space and time discretisation is carried out for the sub-domains using a penalty-based finite element model. This procedure does not require the explicit modelling of additional boundary conditions or interface elements. The developed biphasic interface finite element model is used in analysing blood flow through normal and stenotic arteries. The increase in fluid flow velocity when passing through a stenosed artery and the drop in pressure at the region are captured using this method.

A Biphasic Continuum Approach for Viscoelastic High-Porosity Foams: Comprehensive Theory, Numerics, and Application

This work aims to present a valuable and predictive model for the description of polymeric soft foams which are widely exploited as packaging, padding, or cushioning materials. Belonging to the class of cellular solids, polymeric foams and synthetic sponges can be regarded as permeable or impermeable fluid-saturated porous materials depending on whether their microstructure is open-celled or closed-celled. The pore space formed by the cellular polymer skeleton is filled with a pore fluid (e.g., liquid or gas) which can significantly affect the transient compressive and impact response making particularly air-filled foamed plastics a light-weight and shock-absorbing material par excellence.

Biomechanical modelling of normal pressure hydrocephalus

This study investigates the mechanics of normal pressure hydrocephalus (NPH) growth using a computational approach. We created a generic 3-D brain mesh of a healthy human brain and modelled the brain parenchyma as single phase and biphasic continuum. In our model, hyperelastic constitutive law and finite deformation theory described deformations within the brain parenchyma. We used a value of 155.77 Pa for the shear modulus ( μ) of the brain parenchyma. Additionally, in our model, contact boundary definitions constrained the brain outer surface inside the skull. We used transmantle pressure difference to load the model. Fully nonlinear, implicit finite element procedures in the time domain were used to obtain the deformations of the ventricles and the brain. To the best of our knowledge, this was the first 3-D, fully nonlinear model investigating NPH growth mechanics. Clinicians generally accept that at most 1 mm of Hg transmantle pressure difference (133.416 Pa) is associated with the condition of NPH. Our computations showed that transmantle pressure difference of 1 mm of Hg (133.416 Pa) did not produce NPH for either single phase or biphasic model of the brain parenchyma. A minimum transmantle pressure difference of 1.764 mm of Hg (235.44 Pa) was required to produce the clinical condition of NPH. This suggested that the hypothesis of a purely mechanical basis for NPH growth needs to be revised. We also showed that under equal transmantle pressure difference load, there were no significant differences between the computed ventricular volumes for biphasic and incompressible/nearly incompressible single phase model of the brain parenchyma. As a result, there was no major advantage gained by using a biphasic model for the brain parenchyma. We propose that for modelling NPH, nearly incompressible single phase model of the brain parenchyma was adequate. Single phase treatment of the brain parenchyma simplified the mathematical description of the NPH model and resulted in significant reduction of computational time.

Validation of the Temperament Evaluation of Memphis, Pisa, Paris and San Diego (TEMPS-A): Portuguese-Lisbon version

Background The TEMPS-A has been validated in 8 languages, the original being American English, and includes among others such Latin languages as Italian, French, and Spanish-Buenos Aires. This is the first Portuguese-Lisbon validation. Methods The sample included 1173 students from six different universities and representing most disciplines (such as medicine, law, humanities, engineering, etc.), both sexes (67% female), and ages between 17 and 58 ( x ± SD = 21 ± 4). Standard psychometric tests were used for internal consistency, validity, and factor analysis. Results The study upheld the 5 Factor proposed structure of TEMPS-A. Cronbach α varied from 0.67 for the depressive and 0.83 for the anxious, with the others in-between. We could retain all 110 items of the Interview Schedule. The highest mean scores were found for the hyperthymic, and the lowest for the irritable. As expected, depressive and anxious subscales had strong correlations, followed by the cyclothymic and anxious, and cyclothymic and irritable; in exploratory factor analysis, these subscales constituted Factor I, contrasted to the depressive and the hyperthymic as a biphasic continuum (Factor II). Females scored higher on the depressive, cyclothymic and anxious, and the males on hyperthymic and irritable. Overall, however, no temperament was “dominant” in this population, all temperaments z-scores being 3.3–4%! Limitations Study limited to university students of young age. Conclusions TEMPS-A Lisbon is a reliable and valid instrument. The only relatively weak factor is the depressive, which is similar to other language versions. Gender differences and correlations of temperaments are generally similar to other countries. What appears relatively special to the Portuguese is the relatively “balanced” mix of temperaments in this university student population.

Process-modeling of composites using two-phase porous media theory

A biphasic continuum model is proposed for the modeling of a family of forming processes for fiber composites. The processes considered involve deformation of a fiber bundle network, wetting by penetration of resin into fiber bundles, and resin flow through the fiber bundle network. The continuum model represents three (or more) actual micro constituents as two continuous phases: a solid phase being the fiber network plus any extra void within the fiber bundles and a fluid phase being the liquid matrix. The model framework thus comprises the continuum formulation of a nonlinear compressible porous solid saturated with an incompressible fluid phase. Guided by the entropy inequality, we specify constitutive relations concerning three different mechanisms pertinent to the forming process: effective stress response of the fiber bundle network, compaction of the solid phase, and Darcian interaction between the two phases. We are particularly concerned with the compaction of the fiber-bundles of the solid phase, consisting of elastic packing combined with a viscous wetting process driven by the fluid pressure. The paper is concluded with a couple of numerical examples, where the volumetric deformation-pressure response of a fluid saturated fiber bundle network at undrained conditions is considered. Both volumetric relaxation and volumetric creep tests are analyzed. In particular, the response for different loading rates is assessed. As a final example, a finite element analysis of a relaxation test for the macroscopically undrained compression of a fluid-filled fibernetwork specimen is carried out.

Finite element formulations for hyperelastic transversely isotropic biphasic soft tissues

This paper presents a finite element model for the three-dimensional (3-D) nonlinear analysis of soft hydrated tissues such as articular cartilage in diarthrodial joints under physiologically relevant loading conditions. A biphasic continuum description is used to represent the soft tissue as a two-phase mixture of incompressible inviscid fluid and a hyperelastic, transversely isotropic solid. Alternate mixed-penalty and velocity-pressure finite element formulations are used to solve the nonlinear biphasic governing equations, including the effects of a strain-dependent permeability and a hyperelastic solid phase under finite deformation. The resulting first-order nonlinear system of equations are discretized in time using an implicit finite difference scheme and solved using the Newton-Raphson method. A significant contribution of this work is the implementation and testing of a biphasic description with a transversely isotropic hyperelastic solid phase. This description considers a Helmholtz free energy function of five invariants of the Cauchy-Green deformation tensor and the preferred direction of the material, allowing for asymmetric behavior in tension and compression. An exponential form is suggested and a set of material parameters is identified to represent the response of soft tissues in ranges of deformation and stress observed experimentally. After demonstrating the behavior of this constitutive model in simple tension and compression, a sample problem of unconfined compression is used to further validate the finite element implementation.

Streaming potentials during the confined compression creep test of normal and proteoglycan-depleted cartilage.

The streaming potential response of cartilage in the confined compression creep configuration was assessed theoretically and measured experimentally in normal and proteoglycan-depleted tissue. The analytical solution, using the linear biphasic continuum model including electrokinetics and assuming homogeneous material properties, predicted that: (i) the peak streaming potential is delta V = ke x delta sigma, where ke is the electrokinetic coefficient and delta sigma is the change in compressive stress; (ii) the potential is maintained at 95 to 100% of the peak value for 0 < t < 0.10 tau, where tau is the gel diffusion time constant; and (iii) during short times, 0 < t < 0.01 tau, 90% of the peak streaming potential occurs over a region extending 23% into the tissue sample. Experimentally, adult bovine cartilage disks, 0.5 mm thick, were subjected to step changes of compressive stress. The measured changes in potential indicated a linear response for changes in stress up to 0.10 MPa. The ke of normal cartilage, estimated from the short time (0 < t < 2 sec) change in potential, was -1.65 +/- 1.25 mV/MPa. Digestion of cartilage by chondroitinase ABC resulted in an increased (less negative) ke of -0.75 +/- 0.70 mV/MPa and a 33 +/- 29% depletion of anionic glycosaminoglycan, whereas digestion with trypsin resulted in a further increase in ke to +1.74 +/- 0.95 mV/MPa and a 98 +/- 1 % depletion of glycosaminoglycan. The streaming potential measurement may be a useful addition to the widely used confined compression creep test to assess cartilage material properties.

Mixed and Penalty Finite Element Models for the Nonlinear Behavior of Biphasic Soft Tissues in Finite Deformation: Part II — Nonlinear Examples

This two-part paper addresses finite element-based computational models for the three-dimensional (3-D) nonlinear analysis of soft hydrated tissues, such as articular cartilage in diarthrodial joints, under physiologically relevant loading conditions. A biphasic continuum description is used to represent the soft tissue as a two-phase mixture of incompressible inviscid fluid and a hyperelastic, transversely isotropic solid. Alternate mixed-penalty and velocity-pressure finite element formulations are used to solve the nonlinear biphasic governing equations, including the effects of strain-dependent permeability and a hyperelastic solid phase under finite deformation. The resulting first-order, nonlinear system of equations is discretized in time using an implicit finite difference scheme, and solved using the Newton-Raphson method. Details of the formulations were presented in Part 1 [I], In Part II, the two formulations are used to develop two-dimensional (2-D) quadrilateral and triangular elements and three-dimensional (3-D) hexahedral and tetrahedral elements. Numerical examples, including those representative of soft tissue material testing and simple human joints, are used to validate the formulations and to illustrate their applications. A focus of this work is the comparison of the alternate formulations for nonlinear problems. While it is demonstrated that both formulations produce a range of converging elements, the velocity-pressure formulation is found to be more efficient computationally.

Mixed and Penalty Finite Element Models for the Nonlinear Behavior of Biphasic Soft Tissues in Finite Deformation: Part I — Alternate Formulations

This paper addresses finite element-based computational models for the three-dimensional, (3-D) nonlinear analysis of soft hydrated tissues, such as the articular cartilage in diarthrodial joints, under physiologically relevant loading conditions. A biphasic continuum description is used to represent the soft tissue as a two-phase mixture of incompressible, inviscid fluid and a hyperelastic solid. Alternate mixed-penalty and velocity-pressure finite element formulations are used to solve the nonlinear biphasic governing equations, including the effects of a strain-dependent permeability and a hyperelastic solid phase under finite deformation. The resulting first-order nonlinear system of equations is discretized in time using an implicit finite difference scheme, and solved using the Newton-Raphson method. Using a discrete divergence operator, an equivalence is shown between the mixed-penalty method and a penalty method previously derived by Suh et al. [1]. In Part II [2], the mixed-penalty and velocity-pressure formulations are used to develop two-dimensional (2-D) quadrilateral and triangular elements and 3-D hexahedral and tetrahedral elements. Numerical examples, including those representative of soft tissue material testing and simple human joints, are used to validate the formulations and to illustrate their applications. A focus of this work is the comparison of alternate formulations for nonlinear problems. While it is demonstrated that both formulations produce a range of converging elements, the velocity-pressure formulation is found to be more efficient computationally.

Hydrogel Stress-Relaxation

Hydrogels exhibit stress-relaxation when stretched. The decay in tensile stress can be ascribed primarily to strain-induced swelling of the polymer network. This assertion is formalized in a biphasic continuum model of the gel-solvent system. An equation of motion and a linear consti tutive law of the polymer network are solved with boundary and initial conditions appropriate for a stress-relaxation experiment The model predicts that the time constant of decay depends upon the square of the significant length of the sample (i.e., thickness of thin films, diameter of sphere or thin cylindrical fibers), a result which is confirmed by experiments. In addition, the values of the material constants ( μ, the shear modulus; k, the bulk modulus; and 1/f, the hydraulic permeability) which are estimated using a non-linear regression algorithm, agree well with measurements of the material constants made by other methods.

Automated adaptive analysis of the biphasic equations for soft tissue mechanics using a posteriori error indicators

AbstractHydrated soft tissues of the human musculoskeletal system, such as articular cartilage in diarthrodial joints or the annulus fibrosis of the inter vertebral disk, are accurately represented by a biphasic continuum model consisting of an incompressible solid phase (collagen and proteoglycan) and incompressible, inviscid fluid (interstitial fluid) and derive from the continuum theory of mixtures. These tissues exhibit a viscoelastic‐type response which is caused by the diffusive drag of the fluid phase as it flows past the solid phase. In this study an automated, adaptive, finite element solution of the governing biphasic equations is presented. The finite element formulation is based on a mixed‐penalty approach in which the penalty form of the continuity equation for the mixture is included in the weak form. Pressure, solid and fluid velocities are interpolated independently, and the coefficients of the C−1 pressure field are eliminated at the element level. The resulting matrix form is a system of first order differential equations which is solved via standard finite difference methods. Mesh generation and updating, including both refinement and coarsening, for the two‐dimensional examples examined in this study are performed using Finite Quadtree. The adaptive analysis is based on an error indicator which is the L2 norm of the difference between the finite element solution and a projected finite element solution. Total stress, calculated as the sum of the solid and fluid phase stresses, is used in the error indicator. Rezoning is accomplished by transferring the finite element solution for the primary variables onto the locally updated mesh using a projected field. These projected values allow the finite difference algorithm to proceed in time using the updated mesh. The accuracy and effectiveness of this adaptive finite element analysis is demonstrated using two‐dimensional axisymmetric problems corresponding to the uncomined compression of a cylindrical disk of soft tissue and the indentation of a thin sheet of soft tissue. The method is shown to effectively capture the steep gradients in both problems and to produce solutions in good agreement with independent, converged, numerical solutions.

A finite element analysis of the indentation stress-relaxation response of linear biphasic articular cartilage.

The indentation problem of a thin layer of hydrated soft tissue such as cartilage or meniscus by a circular plane-ended indenter is investigated. The tissue is represented by a biphasic continuum model consisting of a solid phase (collagen and proteoglycan) and a fluid phase (interstitial water). A finite element formulation of the linear biphasic continuum equations is used to solve an axisymmetric approximation of the indentation problem. We consider stress-relaxation problems for which analytic solution is intractable; where the indenter is impermeable (solid) and/or when the interface between the indenter and tissue is perfectly adhesive. Thicknesses corresponding to a thin and thick specimen are considered to examine the effects of tissue thickness. The different flow, pressure, stress and strain fields which are predicted within the tissue, over time periods typically used in the mechanical testing of soft tissues, will be presented. Results are compared with the case of a porous free-draining indenter with a perfectly lubricated tissue-indenter interface, for which an analytic solution is available, to show the effects of friction at the tissue-indenter interface, and the effects of an impermeable indenter. While these effects are present for both thin and thick tissues, they are shown to be more significant for the thin tissue. We also examine the effects of the stiffness of the subchondral bone on the response of the soft tissue and demonstrate that the subchondral bone substrate can be modeled as a rigid, impermeable boundary. The effects of a curved tissue-subchondral bone interface, and the early time response are also studied. For physiologically reasonable levels of curvature, we will show that the curved tissue-subchrondal bone interface has negligible influence on the tissue response away from the interface. In addition, the short-time stress-relaxation responses of the tissue (e.g., at times less than 1s) demonstrate the essential role of the fluid phase in supporting the load applied to the tissue, and by extrapolation to shorter times characteristics of normal joint motion, suggest the essential role of a biphasic model in representing soft tissue behavior in joint response.

Biphasic model of the trabecular meshwork in the eye

A biphasic continuum model for the flow of intraocular fluid, the aqueous humour, in the trabecular meshwork is proposed in the paper. The model considers the meshwork as a circular ring with uniform thickness of homogeneous, isotropic, viscoelastic material swollen with continuously percolating aqueous humour. The model further assumes the permeability of the meshwork as a function of dilation in its solid phase. The study considers steady and quasisteady states. Approximate solutions to the problem are obtained by Picard's type iterative procedure and computational results for the dimensionless solid displacement, permeability and pressure profiles are presented. It is concluded that intraocular pressure has a permeability-decreasing effect, whereas the elasticity of the meshwork shows a permeability-increasing effect. Thus, rising intraocular pressure develops a further rise in itself, whereas the elasticity tends to diminish the intraocular pressure through facilitating aqueous outflow.