Biphasic Mixture Theory Research Articles

Nonlinear plane waves in saturated porous media with incompressible constituents

We consider the propagation of nonlinear plane waves in porous media within the framework of the Biot–Coussy biphasic mixture theory. The tortuosity effect is included in the model, and both constituents are assumed incompressible (Yeoh-type elastic skeleton, and saturating fluid). In this case, the linear dispersive waves governed by Biot’s theory are either of compression or shear-wave type, and nonlinear waves can be classified in a similar way. In the special case of a neo-Hookean skeleton, we derive the explicit expressions for the characteristic wave speeds, leading to the hyperbolicity condition. The sound speeds for a Yeoh skeleton are estimated using a perturbation approach. Then we arrive at the evolution equation for the amplitude of acceleration waves. In general, it is governed by a Bernoulli equation. With the present constitutive assumptions, we find that longitudinal jump amplitudes follow a nonlinear evolution, while transverse jump amplitudes evolve in an almost linearly degenerate fashion.

Mathematical modeling of elastodynamics and cell growth inside a deformable scaffold fitted to the periphery of a bioreactor

A mathematical model is developed for fluid flow, nutrient transport, and cell growth inside a bioreactor with a scaffold at the periphery and lumen at the centerline. The scaffold material is assumed to be deformable. In order to deal with the deformation of the solid phase and fluid phase inside the scaffold, biphasic mixture theory equations are adopted, which are derived from the theory of mixtures. The flow inside the lumen is governed by Stokes equation. Advection–diffusion–reaction equation is used for the mass balance of the nutrient within the scaffold region. Cell growth depends on the nutrient concentration and is expressed by the Contois equation that accounts for contact inhibition. We use lubrication approximation to reduce the system of hydrodynamic and nutrient transport equations. This leads to a coupled system of partial differential equations (PDEs) with time‐dependent variables. Laplace transformation is used to deal with time‐dependent terms, and Durbin's algorithm is used to retrieve the time dependency. We investigate the outreach of nutrients inside the scaffold region, which regulates the growth of cells at a particular time. Based on the available experimental data, we consider relevant reaction kinematics of the cells and observe the corresponding nutrient distribution inside the bioreactor. The factors that affect the nutrient concentration are lumen radius, porosity, and permeability of the scaffold, Thiele modulus, pressure gradient, and so forth. The total mass transfer rate is computed to understand the nutrient distribution across various regions of the bioreactor.

Compressive stress relaxation behavior of articular cartilage and its effects on fluid pressure and solid displacement due to non-Newtonian flow

In this study, we investigate the effects of the power-law index and permeability parameter on the deformation of soft tissue (articular cartilage) which is bathed in the non-Newtonian fluid under stress-relaxation in compression. Ramp displacement is imposed on the surface of hydrated soft tissue. Deformation of the tissue and the fluid pressure is examined for the fast and slow rate of compression. We have employed a linear biphasic mixture theory to develop a mathematical model for compressive stress-relaxation behavior of articular cartilage for non-Newtonian flow. Numerical results indicate that shear-thinning fluids induce less solid deformation and exhibit more fluid pressure as compared to shear-thickening fluids for fast and slow rate of compression. The results also show that linear permeability induces more deformation as compared to strain-dependent nonlinear permeability due to viscoelastic nature of articular cartilage.

Elastohydrodynamics of a deformable porous packing in a channel competing under shear and pressure gradient

Elastohydrodynamics of a deformable porous medium sandwiched between two parallel plates is investigated under the influence of an externally applied pressure gradient as well as an induced shear due to the movement of the upper plate. Biphasic mixture theory is used to describe the macroscopic governing equations for the fluid velocity and the solid displacement, assuming the deformable porous medium as a continuum space. The corresponding reduced mathematical model is a coupled system of elliptic partial differential equations. It is assumed that the fluid at the lower plate experiences slip due to the surface roughness of the plate. The exact solution for unidirectional fluid velocity and solid deformation resembling plain Poiseuille–Couette flow are presented for steady and unsteady states. Asymptotic analysis of the biphasic mixture in the case of low and high Darcy numbers is performed to validate the obtained solution using Prandtl’s matching technique. It is observed that the Womersley number dictates whether the fluid is trapped inside the channel or escapes the channel. The competition between the shear and the pressure gradient is analyzed, and a critical criterion is established that dictates the dominant factor. A mathematical analysis of the current problem is invaluable in understanding the mechanical behavior of biomass under pressure-driven flow in applications such as tissue engineering or shear driven flow inside endothelial glycocalyx layers, which are discussed in brief. In this context, our analysis on the extent of tissue deformation in response to frequency variations is expected to give useful insights to identify the right diagnosis.

The effect of magnetic field on flow induced-deformation in absorbing porous tissues.

In order to understand the interaction between magnetic field and biological tissues in a physiological system, we present a mathematical model of flow-induced deformation in absorbing porous tissues in the presence of a uniform magnetic field. The tissue is modeled as a deformable porous material in which high cavity pressure drives fluid through the tissue where it is absorbed by capillaries and lymphatics. A biphasic mixture theory is used to develop the model under the assumptions of small solid deformation and strain-dependent linear permeability. A spherical cavity formed during injection of fluid in the tissue is used to find fluid pressure and solid displacement as a function of radial distance and time. The governing nonlinear PDE for fluid pressure is solved numerically using method of lines whereas tissue solid displacement is computed by employing trapezoidal rule. The effect of magnetic parameter on fluid pressure, solid displacement and tissue permeability is illustrated graphically.

In vivo mimicking model for solid tumor towards hydromechanics of tissue deformation and creation of necrosis.

The present work addresses transvascular and interstitial fluid transport inside a solid tumor surrounded by normal tissue (close to an in vivo mimicking setup). In general, biological tissues behave like a soft porous material and show mechanical behavior towards the fluid motion through the interstitial space. In general, forces like viscous drag that are associated with the fluid flow may compress the tissue material. On the macroscopic level, we try to model the motion of fluids and macromolecules through the interstitial space of solid tumor and the normal tissue layer. The transvascular fluid transport is assumed to be governed by modified Starling's law. The poroelastohydrodynamics (interstitial hydrodynamics and the deformation of tissue material) inside the tumor and normal tissue regions is modeled using linearized biphasic mixture theory. Correspondingly, the velocity distribution of fluid is coupled to the displacement field of the solid phase (mainly cellular phase and extracellular matrix) in both the normal and tumor tissue regions. The corresponding velocity field is used within the transport reaction equation for fluids and macromolecules through interstitial space to get the overall solute (e.g., nutrients, drug, and other macromolecules) distribution. This study justifies that the presence of the normal tissue layer plays a significant role in delaying/assisting necrosis inside the tumor tissue. It is observed that the exchange process of fluids and macromolecules across the interface of the tumor and normal tissue affects the effectiveness factor corresponding to the tumor tissue.

On the use of biphasic mixture theory for investigating the linear stability of viscous flow through a channel lined with a viscoelastic porous bio-material

Linear stability of viscous fluids flowing through a two-dimensional channel lined with a poroelastic layer which is saturated with the same viscous fluid is numerically studied in this work. Having assumed that the solid matrix of the poroelastics layer obeys the standard linear solid (SLS) model, the basic flow/deformation was obtained for this fluid–solid-interaction (FSI) problem using the “biphasic mixture theory”. The vulnerability of the basic solution so-obtained to infinitesimally-small, normal-mode perturbations was then investigated using a temporal, normal-mode, linear stability analysis. An eigenvalue problem was obtained which was solved numerically using the Chebyshev pseudo-spectral collocation method. The main objective of the present work was to investigate the role played by the inhomogeneity/anisotropy of the poroelastic layer on the critical Reynolds number for the core flow. From the obtained results, we have reached the conclusion that anisotropy has no significant effect on the stability picture of the main flow. The effect of inhomogeneity on the critical Reynolds number, however, was found to be significant and highly dependent on the permeability number being smaller or larger than a threshold.

Nutrient transport through deformable cylindrical scaffold inside a bioreactor: An application to tissue engineering

Tissue engineering deals with the development and growth of cells to obtain tissue structure or organs. In this process, a porous material known as a scaffold is designed to support the cells to grow in the desired form. This designed structure is incubated inside a bioreactor which provides adequate nutrient supply for the cells to grow. The major challenge in tissue engineering is to model a bioreactor and the scaffold which supports high-density cells so that adequate nutrient can be delivered in inner regions to get the significant cell growth. Mathematical modeling of such processes is essential to understand the entire mechanism. Correspondingly, a mathematical model for fluid flow and nutrient transport is developed inside a cylindrical bioreactor containing tubular scaffold which is degradable as well as deformable due to its elastic property. Living biological cells are assumed to adhere to the solid matrix of scaffold firmly. The volume fraction of the solid phase which includes cells and scaffold matrix is assumed to be constant despite the fact that proliferation of cells and degradation of scaffold matrix are undergoing simultaneously. Fluid flow and deformation of the tissue (scaffold) are modeled based on biphasic mixture theory. Navier–Stokes equation accounts for the free fluid in the annular region of the bioreactor. Advection–diffusion-reaction equation governs the concentration of a nutrient in the scaffold region whereas advection–diffusion equation governs the nutrients in the free lumen. Semi-analytical treatment on the mathematical model includes Laplace transformation to deal with the time dependency of the governing fluid flow equation and consequently, Durbin’s algorithm is used to retrieve the time-dependent variables. The model allows the uniform nutrient transport inside the scaffold region so that cells get sufficient amount of nutrient to grow and reduce the cell death. Moreover, Sherwood number has been calculated to analyse the mass transport of solute concentration from scaffold region to the free fluid region and vice-versa through the interface.

Mathematical analysis of hydrodynamics and tissue deformation inside an isolated solid tumor

In this article, we present a biphasic mixture theory based mathematical model for the hydrodynamics of interstitial fluid motion and mechanical behavior of the solid phase inside a solid tumor. The tumor tissue considered here is an isolated deformable biological medium. The solid phase of the tumor is constituted by vasculature, tumor cells, and extracellular matrix, which are wet by a physiological extracellular fluid. Since the tumor is deformable in nature, the mass and momentum equations for both the phases are presented. The momentum equations are coupled due to the interaction (or drag) force term. These governing equations reduce to a one-way coupled system under the assumption of infinitesimal deformation of the solid phase. The well-posedness of this model is shown in the weak sense by using the inf-sup (Babuska?Brezzi) condition and Lax?Milgram theorem in 2D and 3D. Further, we discuss a one-dimensional spherical symmetry model and present some results on the stress fields and energy of the system based on ??2 and Sobolev norms. We discuss the so-called phenomena of ?necrosis? inside a solid tumor using the energy of the system.

Pressure-driven flows of Quemada fluids in a channel lined with a poroelastic layer: A linear stability analysis

Laminar flow of a thixotropic fluid obeying the Quemada model is numerically investigated in a channel lined with a poroelastic layer saturated with a Newtonian fluid. Having assumed that the solid matrix in the poroelastic layer obeys the linear elastic model, basic flow/deformation were obtained in the main channel and also in the poroelastic layer using the biphasic mixture theory. The basic state was then subjected to infinitesimally-small, normal-mode perturbations and their vulnerability to poroelastic instability was studied based on the linear, temporal stability analysis. The eigenvalue problem so obtained was then solved numerically using the spectral collocation method. The main objective of the work was to investigate the role played by the pororelastic layer's parameters (i.e., porosity, flexibility, permeability, density, and thickness) on the stability picture. The roles played by the rheological behavior of the core fluid and the fluid flowing through the poroelastic layer were also investigated. Numerical results were obtained at low-permeability limit, typical of physiological systems, demonstrating that the effect of the layer's properties can be stabilizing or destabilizing depending on the rheological properties of the two fluids involved in the problem. In general, thixotropy was found to have a stabilizing effect on the core flow. The same was found to be true as to the effect of a fluid's shear-thinning. The analysis also served to show that the effect of a fluid's yield stress might also be stabilizing.

Modeling the impact of prostate edema on LDR brachytherapy: a Monte Carlo dosimetry study based on a 3D biphasic finite element biomechanical model

Prostate volume changes due to edema occurrence during transperineal permanent brachytherapy should be taken under consideration to ensure optimal dose delivery. Available edema models, based on prostate volume observations, face several limitations. Therefore, patient-specific models need to be developed to accurately account for the impact of edema. In this study we present a biomechanical model developed to reproduce edema resolution patterns documented in the literature. Using the biphasic mixture theory and finite element analysis, the proposed model takes into consideration the mechanical properties of the pubic area tissues in the evolution of prostate edema. The model’s computed deformations are incorporated in a Monte Carlo simulation to investigate their effect on post-operative dosimetry. The comparison of Day1 and Day30 dosimetry results demonstrates the capability of the proposed model for patient-specific dosimetry improvements, considering the edema dynamics. The proposed model shows excellent ability to reproduce previously described edema resolution patterns and was validated based on previous findings. According to our results, for a prostate volume increase of 10–20% the Day30 urethra D10 dose metric is higher by 4.2%–10.5% compared to the Day1 value. The introduction of the edema dynamics in Day30 dosimetry shows a significant global dose overestimation identified on the conventional static Day30 dosimetry. In conclusion, the proposed edema biomechanical model can improve the treatment planning of transperineal permanent brachytherapy accounting for post-implant dose alterations during the planning procedure.

A theoretical study on the elastic deformation of cellular phase and creation of necrosis due to the convection reaction process inside a spherical tumor

Several biological processes, such as convective nutrient transport and convective drug delivery in biological tissues involves the transvascular and interstitial movement of biofluids. This work addresses transvascular and interstitial transport of nutrient inside a spherical tumor. Most of the biological tissues behave like deformable porous material and show mechanical behavior towards the fluid motion, due to the fact, that the forces like the drag, which is associated with fluid flow may compress the tissue material. On the macroscopic level, transport of solutes like nutrients, drug molecules, etc. within the tumor interstitial space is modeled. The hydrodynamic problem is treated with biphasic mixture theory under steady state and spherically symmetry situation. The transvascular transport of nutrient is modeled with the modified Sterling’s equation. The present model describes the overall nutrient distribution and predicts various criteria for the necrosis formation inside the tumor. Present study justifies that the parameters, which controls the nutrient supply to the tumor interstitial space through the blood vessel network inside the tumor, competes with reversible nutrient consumption kinetics of the tumor cells. This study also finds the role of some of those parameters on the deformation of cellular phase of the tumor as a consequence of interstitial fluid flow.

Effect of endothelial glycocalyx layer redistribution upon microvessel poroelastohydrodynamics

The endothelial glycocalyx layer (EGL) is a macromolecular layer that lines the inner surface of blood vessels. It is believed to serve a number of physiological functions in the microvasculature, including protection of the vessel walls from potentially harmful levels of fluid shear, as a molecular sieve that acts to regulate transendothelial mass transport, and as a transducer of mechanical stress from the vessel lumen. To best fulfil some of its roles, it has been suggested that the EGL redistributes, so that it is thickest at the cell–cell junctions. It has also been suggested that the majority of mechanotransduction occurs through the solid phase of the EGL, rather than via its fluid phase. The difficulties associated with measuring the distribution of the EGL in vivo make these hypotheses difficult to confirm experimentally. Consequently, to gauge the impact of EGL redistribution from a theoretical standpoint, we compute the flow through a porous-lined microvessel, the endothelial surface of which has been informed by confocal microscopy images of a postcapillary venule. Following earlier studies, we model the poroelastohydrodynamics of the EGL using biphasic mixture theory, taking advantage of a recently developed boundary integral representation of these equations to solve the coupled poroelastohydrodynamics using the boundary element method. However, the low permeabilities of the EGL mean that viscous effects are confined to thin layers, thereby also enabling an asymptotic treatment of the dynamics in this limit. In this asymptotic regime, we also consider a two-layer Stokes flow model for the lumen flow to approximate the effect of red blood cells within the lumen. We demonstrate that redistribution of the EGL can have a substantial impact upon microvessel haemodynamics. We also confirm that the bulk of the mechanical stress is indeed carried through the solid phase of the EGL.

Hydrodynamics and convection enhanced macromolecular fluid transport in soft biological tissues: Application to solid tumor

This work addresses a theoretical framework for transvascular exchange and extravascular transport of solute macromolecules through soft interstitial space inside a solid tumor. Most of the soft biological tissues show materialistic properties similar to deformable porous material. They exhibit mechanical behavior towards the fluid motion since the solid phase of the tumor tissue gets compressed by the drag force that is associated with the extracellular fluid flow. This paper presents a general view about the transvascular and interstitial transport of solute nutrients inside a tumor in the macroscopic level. Modified Starling׳s equation is used to describe transvascular nutrient transport. On the macroscopic level, motion of extracellular fluid within the tumor interstitium is modeled with the help of biphasic mixture theory and a spherical symmetry solution is given as a simpler case. This present model describes the average interstitial fluid pressure (IFP), extracellular fluid velocity (EFV) and flow rate of extracellular fluid, as well as the deformation of the solid phase of the tumor tissue as an immediate cause of extracellular fluid flow. When the interstitial transport is diffusion dominated, an analytical treatment of advection–diffusion–reaction equation finds the overall nutrient distribution. We propose suitable criteria for the formation of necrosis within the tumor interstitium. This study introduces some parameters that represent the nutrient supply from tumor blood vessels into the tumor extracellular space. These transport parameters compete with the reversible nutrient metabolism of the tumor cells present in the interstitium. The present study also shows that the effectiveness factor corresponding to a first order nutrient metabolism may reach beyond unity if the strength of the distributive solute source assumes positive non-zero values.

An analytical study on hydrodynamics of an unsteady flow and mass transfer through a channel asymmetrically lined with deformable porous layer

The present study is motivated by its possible relevance to the rheology of blood and the transport of nutrient and drug molecules (macro–microscopic level) through capillaries, venules and digestive tract under the influence of endothelial-cell glycocalyx (EG). As an approximation, we study unsteady fluid flow through a rectangular channel lined with asymmetric porous lining. In order to model a more realistic situation, we adopt biphasic mixture theory for the unsteady hydrodynamics of fluid transport in poroelastic layer(s) (EG). The flow within the lumen region between the deformable porous layers is governed by unsteady Stoke’s equation. The unsteady nature of the flow is being taken care with the Fourier series expansion analysis. Shear stress is calculated at the interface of free lumen and porous layers along with volumetric flow rate through the free lumen region. An expression for Sherwood number is derived at the said interface from the pseudo steady state solution of the mass transfer equation that quantifies the transport of solute nutrient through the lumenal space. The corresponding convection–diffusion equation is solved within the mass transfer boundary layer using similarity transformation method. It is observed that the rate of volumetric flow through the free lumen increases and interfacial shear stress (i.e., shear stress at the interface of at the porous and free lumen) decreases with stress jump coefficient at the mentioned interface. Length averaged Sherwood number deceases with increase in depth of the porous layer and increases with the constant permeation velocity when the other parameters characterizing the mass transfer process remain constant.

A boundary-integral representation for biphasic mixture theory, with application to the post-capillary glycocalyx.

We describe a new boundary-integral representation for biphasic mixture theory, which allows us to efficiently solve certain elastohydrodynamic–mobility problems using boundary element methods. We apply this formulation to model the motion of a rigid particle through a microtube which has non-uniform wall shape, is filled with a viscous Newtonian fluid, and is lined with a thin poroelastic layer. This is relevant to scenarios such as the transport of small rigid cells (such as neutrophils) through microvessels that are lined with an endothelial glycocalyx layer (EGL). In this context, we examine the impact of geometry upon some recently reported phenomena, including the creation of viscous eddies, fluid flux into the EGL, as well as the role of the EGL in transmitting mechanical signals to the underlying endothelial cells.

Modeling the nonlinear large deformation kinetics of volume phase transition for the neutral thermosensitive hydrogels

By regarding the neutral thermosensitive hydrogel as a biphasic mixture media, an incompressible deformable porous solid skeleton, and an incompressible interstitial fluid phase saturating the interconnected space, a novel biphasic mixture theory for modeling the nonlinear large deformation kinetics of the volume phase transition was developed from the first law of thermodynamics and the continuum mixture theory. The governing equations consist of the saturated continuity equation, the momentum equations of the fluid, and the momentum equations of the bulk hydrogel. The explicit expressions of the constitutive relationships, which are capable of characterizing the nonlinear large deformation kinetics of the volume phase transition of the N-isopropylacrylamide thermosensitive hydrogels, were derived from the Helmholtz free energy formed by the elastic and mixing contributions. Both the governing equations and the constitutive relationships are in the Lagrangian framework, and special attention was paid to the deformable porous solid skeleton.

505 2相混合体理論に基づく脳の変形解析に関する検討(OS8-1:デジタルヒューマン,オーガナイズドセッション8:デジタルヒューマン)

505 2相混合体理論に基づく脳の変形解析に関する検討(OS8-1:デジタルヒューマン,オーガナイズドセッション8:デジタルヒューマン)

A Conewise Linear Elasticity mixture model for the analysis of tension-compression nonlinearity in articular cartilage.

A biphasic mixture model is developed that can account for the observed tension-compression nonlinearity of cartilage by employing the continuum-based Conewise Linear Elasticity (CLE) model of Curnier et al. (J. Elasticity, 37, 1-38, 1995) to describe the solid phase of the mixture. In this first investigation, the orthotropic octantwise linear elasticity model was reduced to the more specialized case of cubic symmetry, to reduce the number of elastic constants from twelve to four. Confined and unconfined compression stress-relaxation, and torsional shear testing were performed on each of nine bovine humeral head articular cartilage cylindrical plugs from 6 month old calves. Using the CLE model with cubic symmetry, the aggregate modulus in compression and axial permeability were obtained from confined compression (H-A = 0.64 +/- 0.22 MPa, k2 = 3.62 +/- 0.97 x 10(-16) m4/N.s, r2 = 0.95 +/- 0.03), the tensile modulus, compressive Poisson ratio, and radial permeability were obtained from unconfined compression (E+Y = 12.75 +/- 1.56 MPa, v- = 0.03 +/- 0.01, kr = 6.06 +/- 2.10 x 10(-16) m4/N.s, r2 = 0.99 +/- 0.00), and the shear modulus was obtained from torsional shear (mu = 0.17 +/- 0.06 MPa). The model was also employed to predict the interstitial fluid pressure successfully at the center of the cartilage plug in unconfined compression (r2 = 0.98 +/- 0.01). The results of this study demonstrate that the integration of the CLE model with the biphasic mixture theory can provide a model of cartilage that can successfully curve-fit three distinct testing configurations while producing material parameters consistent with previous reports in the literature.

A new formulation and computation of the triphasic model for mechano-electrochemical mixtures

From the generalized first law of thermodynamics for an irreversible thermodynamical system, a new set of governing equations for the mixture theory is derived based on the triphasic model for mechano-electrochemical mixtures. It is shown that, in the case of electroneutral solution, a new biphasic mixture theory including the electrochemical effects can be derived from the new governing equations. The chemical-expansion stress representing both the influences of deformation on the fixed charge density and the electric potential of fixed charge field is given. For comparison and verification purposes, the numerical solution for a confined compression problem of a charged hydrated soft tissue is computed using the multiquadric method.