Finite element analysis of biological soft tissue surrounded by a deformable membrane that controls transmembrane flow

Background

Many biological soft tissues are hydrated porous hyperelastic materials, which consist of a complex solid skeleton with fine voids and fluid filling these voids. Mechanical interactions between the solid and the fluid in hydrated porous tissues have been analyzed by finite element methods (FEMs) in which the mixture theory was introduced in various ways. Although most of the tissues are surrounded by deformable membranes that control transmembrane flows, the boundaries of the tissues have been treated as rigid and/or freely permeable in these studies. The purpose of this study was to develop a method for the analysis of hydrated porous hyperelastic tissues surrounded by deformable membranes that control transmembrane flows.

Results

For this, we developed a new nonlinear finite element formulation of the mixture theory, where the nodal unknowns were the pore water pressure and solid displacement. This method allows the control of the fluid flow rate across the membrane using Neumann boundary condition. Using the method, we conducted a compression test of the hydrated porous hyperelastic tissue, which was surrounded by a flaccid impermeable membrane, and a part of the top surface of this tissue was pushed by a platen. The simulation results showed a stress relaxation phenomenon, resulting from the interaction between the elastic deformation of the tissue, pore water pressure gradient, and the movement of fluid. The results also showed that the fluid trapped by the impermeable membrane led to the swelling of the tissue around the platen.

Conclusions

These facts suggest that our new method can be effectively used for the analysis of a large deformation of hydrated porous hyperelastic material surrounded by a deformable membrane that controls transmembrane flow, and further investigations may allow more realistic analyses of the biological soft tissues, such as brain edema, brain trauma, the flow of blood and lymph in capillaries and pitting edema.

     

Finite element simulation of skeletal muscular structures obtained from images of histological serial sections

In this study, we present a method for the three-dimensional reconstruction of objects obtained from histological serial sections (exemplified by those of a pennate striated skeletal muscle) and its application to the finite element method. A hyperelastic material model is used for modeling biological soft tissue. The reconstruction process relies on the direct construction of a volumetric mesh using an octree approach which leads to a stable finite element method. Stability can be expressed in the spectral matrix condition number. To visualize stress patterns within the underlying anatomy the simulation results are projected onto images of the histological scenario.     

A sub-domain inverse finite element characterization of hyperelastic membranes including soft tissues

Quantification of the mechanical behavior of hyperelastic membranes in their service configuration, particularly biological tissues, is often challenging because of the complicated geometry, material heterogeneity, and nonlinear behavior under finite strains. Parameter estimation thus requires sophisticated techniques like the inverse finite element method. These techniques can also become difficult to apply, however, if the domain and boundary conditions are complex (e.g. a non-axisymmetric aneurysm). Quantification can alternatively be achieved by applying the inverse finite element method over sub-domains rather than the entire domain. The advantage of this technique, which is consistent with standard experimental practice, is that one can assume homogeneity of the material behavior as well as of the local stress and strain fields. In this paper, we develop a sub-domain inverse finite element method for characterizing the material properties of inflated hyperelastic membranes, including soft tissues. We illustrate the performance of this method for three different classes of materials: neo-Hookean, Mooney Rivlin, and Fung-exponential.     

Dynamic finite element implementation of nonlinear,anisotropic hyperelastic biological membranes

We present a novel method for the implementation of hyperelastic finite strain, non-linear strain-energy functions for biological membranes in an explicit finite element environment. The technique is implemented in LS-DYNA but may also be implemented in any suitable non-linear explicit code. The constitutive equations are implemented on the foundation of a co-rotational uniformly reduced Hughes-Liu shell. This shell is based on an updated-Lagrangian formulation suitable for relating Cauchy stress to the rate-of-deformation, i.e. hypo-elasticity. To accommodate finite deformation hyper-elastic formulations, a co-rotational deformation gradient is assembled over time, resulting in a formulation suitable for pseudo-hyperelastic constitutive equations that are standard assumptions in biomechanics. Our method was validated by comparison with (1) an analytic solution to a spherically-symmetric dynamic membrane inflation problem, incorporating a Mooney-Rivlin hyperelastic equation and (2) with previously published finite element solutions to a non-linear transversely isotropic inflation problem. Finally, we implemented a transversely isotropic strain-energy function for mitral valve tissue. The method is simple and accurate and is believed to be generally useful for anyone who wishes to model biologic membranes with an experimentally driven strain-energy function.     

Region-specific constitutive modeling of the plantar soft tissue

Recent research has shown that hyperelastic properties of the plantar soft tissue consisting of adipose tissue and fibrous septa change from region to region. However, relatively little research has been conducted to develop analytical or computational models to describe the region-specific behavior of the plantar soft tissue. The objective of the research is to develop a region-specific constitutive model of the plantar soft tissue. Plantar soft tissue specimens were dissected from six regions [subcalcaneal (CA), sublateral (LA), subnavicular (Nav), 1st, 3rd, and 5th submetatarsal (M1, M3, M5)] from cadaveric foot samples, and a picrosirius red staining technique was used to visualize the collagen fibers in fibrous septa. The volume fractions of adipose tissue and fibrous septa and the volume fractions of the principal orientations of the fibrous septa were calculated with the intensity gradient method. Region-specific constitutive models were then developed in finite element analysis considering the microstructure of the plantar soft tissue. The hyperelastic region specific material properties of the plantar soft tissue were validated with experimental unconfined compression tests and indentation tests from the literature. The results show that the models give reasonable predictions of the stiffness of the soft tissue within a standard deviation of the tests. The region-specific constitutive models help to explain how changes in the constituents are related to mechanical behavior of the soft tissue on a region specific basis.     

A finite nonlinear hyper-viscoelastic model for soft biological tissues

Soft tissues exhibit highly nonlinear rate and time-dependent stress-strain behaviour. Strain and strain rate dependencies are often modelled using a hyperelastic model and a discrete (standard linear solid) or continuous spectrum (quasi-linear) viscoelastic model, respectively. However, these models are unable to properly capture the materials characteristics because hyperelastic models are unsuited for time-dependent events, whereas the common viscoelastic models are insufficient for the nonlinear and finite strain viscoelastic tissue responses. The convolution integral based models can demonstrate a finite viscoelastic response; however, their derivations are not consistent with the laws of thermodynamics. The aim of this work was to develop a three-dimensional finite hyper-viscoelastic model for soft tissues using a thermodynamically consistent approach. In addition, a nonlinear function, dependent on strain and strain rate, was adopted to capture the nonlinear variation of viscosity during a loading process. To demonstrate the efficacy and versatility of this approach, the model was used to recreate the experimental results performed on different types of soft tissues. In all the cases, the simulation results were well matched ($R^2⩾0.99$) with the experimental data.     

A penetration-based finite element method for hyperelastic 3D biphasic tissues in contact. Part II: finite element simulations

The penetration method allows for the efficient finite element simulation of contact between soft hydrated biphasic tissues in diarthrodial joints. Efficiency of the method is achieved by separating the intrinsically nonlinear contact problem into a pair of linked biphasic finite element analyses, in which an approximate, spatially and temporally varying contact traction is applied to each of the contacting tissues. In Part I of this study, we extended the penetration method to contact involving nonlinear biphasic tissue layers, and demonstrated how to derive the approximate contact traction boundary conditions. The traction derivation involves time and space dependent natural boundary conditions, and requires special numerical treatment. This paper (Part II) describes how we obtain an efficient nonlinear finite element procedure to solve for the biphasic response of the individual contacting layers. In particular, alternate linearization of the nonlinear weak form, as well as both velocity-pressure, v-p, and displacement-pressure, u-p, mixed formulations are considered. We conclude that the u-p approach, with linearization of both the material law and the deformation gradients, performs best for the problem at hand. The nonlinear biphasic contact solution will be demonstrated for the motion of the glenohumeral joint of the human shoulder joint.     

Quantifying nonlinear anisotropic elastic material properties of biological tissue by use of membrane inflation

Determination of material parameters for soft tissue frequently involves regression of material parameters for nonlinear, anisotropic constitutive models against experimental data from heterogeneous tests. Here, parameter estimation based on membrane inflation is considered. A four parameter nonlinear, anisotropic hyperelastic strain energy function was used to model the material, in which the parameters are cast in terms of key response features. The experiment was simulated using finite element (FE) analysis in order to predict the experimental measurements of pressure versus profile strain. Material parameter regression was automated using inverse FE analysis; parameter values were updated by use of both local and global techniques, and the ability of these techniques to efficiently converge to a best case was examined. This approach provides a framework in which additional experimental data, including surface strain measurements or local structural information, may be incorporated in order to quantify heterogeneous nonlinear material properties.     

OSAHS患者上气道和软腭的流固耦合有限元模型的建立

目的:重建OSAHS患者上气道和软腭的流固耦合有限元模型,研究OSAHS患者上气道及软腭气流动力学特征,为进一步探讨OSAHS的的发病机制奠定基础。方法:对一名中度OSAHS患者的上气道及周围组织进行MRI扫描,将以DICOM格式存储的扫描数据导入Mimics15.0软件中进行预处理,得到上气道和软腭的模型;再利用逆向工程软件Geomagic Studio 2013建立了2 mm气道壁;然后在3-D重建软件NX中,生成气道壁和气道以及软腭之间的组合模型;最后将该组合模型导入ANSYS Workbench13.0软件中,通过网格划分、定义材料属性、设定模型的边界条件操作建立了上气道和软腭的流固耦合有限元模型。结果:利用Mimics、Ansys等软件建立了完整的上气道和软腭的流固耦合有限元模型。共得到气道:2806835单元和529281个节点;气道壁:2304348单元和3487609个节点;软腭:131855单元和204784个节点。结论:本研究建立的上气道及软腭的流固耦合有限元模型符合人体的生物力学特点,为下一步的数值模拟实验提供了一个更真实、可靠的模型。     

An evaluation of three-dimensional diarthrodial joint contact using penetration data and the finite element method

We have developed an approximate method for simulating the three-dimensional contact of soft biphasic tissues in diarthrodial joints under physiological loading. Input to the method includes: (i) kinematic information describing an in vitro joint articulation, measured while the cartilage is deformed under physiological loads, (ii) geometric properties for the relaxed (undeformed) cartilage layers, obtained for the analyses in this study via stereophotogrammetry, and (iii) material parameters for the biphasic constitutive relations used to represent cartilage. Solid models of the relaxed tissue layers are assembled in physiological positions, resulting in a mathematical overlap of the cartilage layers. The overlap distribution is quantified and converted via the biphasic governing equations into applied traction boundary conditions for both the solid and fluid phases for each of the contacting layers. Linear, biphasic, three-dimensional, finite element analysis is performed using the contact boundary conditions derived for each of the contacting layers. The method is found to produce results consistent with the continuity requirements of biphasic contact. Comparison with results from independent, biphasic contact analyses of axisymmetric problems shows that the method slightly underestimates the contact area, leading to an overestimation of the total traction, but yields a good approximation to elastic stress and solid phase displacement.     

Modeling of time-dependent force response of fingertip to dynamic loading

An extended exposure to repeated loading on fingertip has been associated to many vascular, sensorineural, and musculoskeletal disorders in the fingers, such as carpal tunnel syndrome, hand-arm vibration syndrome, and flexor tenosynovitis. A better understanding of the pathomechanics of these sensorineural and vascular diseases in fingers requires a formulation of a biomechanical model of the fingertips and analyses to predict the mechanical responses of the soft tissues to dynamic loading. In the present study, a model based on finite element techniques has been developed to simulate the mechanical responses of the fingertips to dynamic loading. The proposed model is two-dimensional and incorporates the essential anatomical structures of a finger: skin, subcutaneous tissue, bone, and nail. The skin tissue is assumed to be hyperelastic and viscoelastic. The subcutaneous tissue was considered to be a nonlinear, biphasic material composed of a hyperelastic solid and an invicid fluid, while its hydraulic permeability was considered to be deformation dependent. Two series of numerical tests were performed using the proposed finger tip model to: (a) simulate the responses of the fingertip to repeated loading, where the contact plate was assumed to be fixed, and the bone within the fingertip was subjected to a prescribed sinusoidal displacement in vertical direction; (b) simulate the force response of the fingertip in a single keystroke, where the keyboard was composed of a hard plastic keycap, a rigid support block, and a nonlinear spring. The time-dependent behavior of the fingertip under dynamic loading was derived. The model predictions of the time-histories of force response of the fingertip and the phenomenon of fingertip separation from the contacting plate during cyclic loading agree well with the reported experimental observations.     

A nonlinear biphasic viscohyperelastic model for articular cartilage

Experiments on articular cartilage have shown nonlinear stress-strain curves under finite deformations as well as intrinsic viscous effects of the solid phase. The aim of this study was to propose a nonlinear biphasic viscohyperelastic model that combines the intrinsic viscous effects of the proteoglycan matrix with a nonlinear hyperelastic constitutive equation. The proposed equation satisfies objectivity and reduces for uniaxial loading to a solid type viscous model in which the actions of the springs are represented by the hyperelastic function proposed by Holmes and Mow [1990. J. Biomechanics 23, 1145-1156.]. Results of the model, that were efficiently implemented in an updated Lagrangian algorithm, were compared with experimental infinitesimal data reported by DiSilverstro and Suh [2001. J. Biomechanics 34, 519-525.] and showed acceptable fitting for the axial force (R(2)=0.991) and lateral displacement (R(2)=0.914) curves in unconfined compression as well as a good fitting of the axial indentation force curve (R(2)=0.982). In addition, the model showed an excellent fitting of finite-deformation confined compression stress relaxation data reported by Ateshian et al. [1997. J. Biomechanics 30, 1157-1164.] and Huang et al. [2005. J. Biomechanics 38, 799-809.] (R(2)=0.993 and R(2)=0.995, respectively). The constitutive equation may be used to represent the mechanical behavior of the proteoglycan matrix in a fiber reinforced model of articular cartilage.     

Finite element implementation of mechanochemical phenomena in neutral deformable porous media under finite deformation

Biological soft tissues and cells may be subjected to mechanical as well as chemical (osmotic) loading under their natural physiological environment or various experimental conditions. The interaction of mechanical and chemical effects may be very significant under some of these conditions, yet the highly nonlinear nature of the set of governing equations describing these mechanisms poses a challenge for the modeling of such phenomena. This study formulated and implemented a finite element algorithm for analyzing mechanochemical events in neutral deformable porous media under finite deformation. The algorithm employed the framework of mixture theory to model the porous permeable solid matrix and interstitial fluid, where the fluid consists of a mixture of solvent and solute. A special emphasis was placed on solute-solid matrix interactions, such as solute exclusion from a fraction of the matrix pore space (solubility) and frictional momentum exchange that produces solute hindrance and pumping under certain dynamic loading conditions. The finite element formulation implemented full coupling of mechanical and chemical effects, providing a framework where material properties and response functions may depend on solid matrix strain as well as solute concentration. The implementation was validated using selected canonical problems for which analytical or alternative numerical solutions exist. This finite element code includes a number of unique features that enhance the modeling of mechanochemical phenomena in biological tissues. The code is available in the public domain, open source finite element program FEBio (http:∕∕mrl.sci.utah.edu∕software).     

Spherical indentation of soft matter beyond the Hertzian regime: numerical and experimental validation of hyperelastic models

The lack of practicable nonlinear elastic contact models frequently compels the inappropriate use of Hertzian models in analyzing indentation data and likely contributes to inconsistencies associated with the results of biological atomic force microscopy measurements. We derived and validated with the aid of the finite element method force-indentation relations based on a number of hyperelastic strain energy functions. The models were applied to existing data from indentation, using microspheres as indenters, of synthetic rubber-like gels, native mouse cartilage tissue, and engineered cartilage. For the biological tissues, the Fung and single-term Ogden models achieved the best fits of the data while all tested hyperelastic models produced good fits for the synthetic gels. The Hertz model proved to be acceptable for the synthetic gels at small deformations (strain < 0.05 for the samples tested), but not for the biological tissues. Although this finding supports the generally accepted view that many soft materials can be assumed to be linear elastic at small deformations, the nonlinear models facilitate analysis of intrinsically nonlinear tissues and large-strain indentation behavior.