Porohyperelastic Model Research Articles

Imaging the intramuscular pressure of living muscles with shear waves

Shear wave elastography (SWE) is an innovative method that allows for the nondestructive and quantitative characterization of muscular mechanical properties. This method finds extensive utility in fields such as sports medicine, sports rehabilitation, and the diagnosis of muscle-related ailments. Existing studies have demonstrated the promise of SWE in probing intramuscular pressure (IMP), a factor intimately tied to muscular physiological functions and the onset of certain diseases. Nonetheless, there remains a lack of a SWE method grounded in an appropriate biomechanical model capable of effectively imaging IMP in vivo. Addressing this issue, we propose a shear wave imaging method relaying on a porohyperelastic model encompassing well-defined parameters for both muscular active behavior and intramuscular pressure. Drawing upon wave motion analysis, we establish a correlation between shear wave velocities and IMP in analytical form. This theoretical solution on one hand help understand the interplay between IMP and muscle active stress and the impact of muscle contraction on IMP. On the other hand, it enables us to develop an elastography method to assess IMP in vivo. We conducted a series of experiments to underscore the applicability of our theory and elastography method. Ex vivo experiments were performed on porcine muscles, while in vivo tests were carried out on human skeletal muscles. The results from the ex vivo tests validate the efficacy of our method. Meanwhile, the in vivo outcomes suggest that our approach holds potential to assess the variation of IMP with muscle fatigue and injuries, inspect intramuscular injections, and diagnose acute and chronic compartment syndrome.

Porosity-permeability tensor relationship of closely and randomly packed fibrous biomaterials and biological tissues: Application to the brain white matter

The constitutive model for the porosity-permeability relationship is a powerful tool to estimate and design the transport properties of porous materials, which has attracted significant attention for the advancement of novel materials. However, in comparison with other materials, biomaterials, especially natural and artificial tissues, have more complex microstructures e.g. high anisotropy, high randomness of cell/fibre dimensions/position and very low porosity. Consequently, a reliable microstructure-permeability relationship of fibrous biomaterials has proven elusive. To fill this gap, we start a mathematical derivation from the fundamental brain white matter (WM) formed by nerve fibres. This is augmented by a numerical characterisation and experimental validations to obtain an anisotropic permeability tensor of the brain WM as a function of the tissue porosity. A versatile microstructure generation software (MicroFiM) for fibrous biomaterial with complex microstructure and low porosity was built accordingly and made freely accessible here. Moreover, we propose an anisotropic poro-hyperelastic model enhanced by the newly defined porosity-permeability tensor relationship which precisely captures the tissues macro-scale permeability changes due to the microstructural deformation in an infusion scenario. The constitutive model, theories and protocols established in this study will both provide improved design strategies to tailor the transport properties of fibrous biomaterials and enable the non-invasive characterisation of the transport properties of biological tissues. This will lead to the provision of better patient-specific medical treatments, such as drug delivery. Statement of significanceDue to the microstructural complexity, a reliable microstructure-permeability relationship of fibrous biomaterials has proven elusive, which hinders our way of tuning the fluid transport property of the biomaterials by directly programming their microstructure. The same problem hinders non-invasive characterisations of fluid transport properties in biological tissues, which can significantly improve the efficiency of treatments e.g. drug delivery, directly from the tissues accessible microstructural information, e.g. porosity. Here, we developed a validated mathematical formulation to link the random microstructure to a fibrous material’s macroscale permeability tensor. This will advance our capability to design complex biomaterials and make it possible to non-invasively characterise the permeability of living tissues for precise treatment planning. The newly established theory and protocol can be easily adapted to various types of fibrous biomaterials.

Modeling large-volume subcutaneous injection of monoclonal antibodies with anisotropic porohyperelastic models and data-driven tissue layer geometries

Subcutaneous injection of therapeutic monoclonal antibodies (mAbs) has become one of the fastest-growing fields in the pharmaceutical industry. The transport and mechanical processes behind large volume injections are poorly understood. Here, we leverage a large-deformation poroelastic model to study high-dose, high-speed subcutaneous injection. We account for the anisotropy of subcutaneous tissue using of a fibril-reinforced porohyperelastic model. We also incorporate the multi-layer structure of the skin tissue, generating data-driven geometrical models of the tissue layers using histological data. We analyze the impact of handheld autoinjectors on the injection dynamics for different patient forces. Our simulations show the importance of considering the large deformation approach to model large injection volumes. This work opens opportunities to better understand the mechanics and transport processes that occur in large-volume subcutaneous injections of mAbs.

A poro-viscoelastic model for the subcutaneous injection of monoclonal antibodies

Subcutaneous delivery of therapeutic monoclonal antibodies (mAbs) has recently attracted exceptional interests in the pharmaceutical industry. The mechanical response of the tissue during the injection and the drug distribution after the injection are yet to be understood. We are motivated to study the subcutaneous injection using a nonlinear poro-elastic model with a viscoelastic constitutive law of the solid. The model is first validated against experimental data of relaxation tests. Then the poro-viscoelastic model is applied to subcutaneous injection, and is compared against a poro-hyperelastic model. The pressure at the injection site, which serves as an indicator of possible fracture initiation, differs significantly by taking into account the viscosity of the solid.

Mathematical analysis and numerical approximation of a general linearized poro-hyperelastic model

We describe the behavior of a deformable porous material by means of a poro-hyperelastic model that has been previously proposed in Chapelle and Moireau (2014) under general assumptions for mass and momentum balance and isothermal conditions for a two-component mixture of fluid and solid phases. In particular, we address here a linearized version of the model, based on the assumption of small displacements. We consider the mathematical analysis and the numerical approximation of the problem. More precisely, we carry out firstly the well-posedness analysis of the model. Then, we propose a numerical discretization scheme based on finite differences in time and finite elements for the spatial approximation; stability and numerical error estimates are proved.Particular attention is dedicated to the study of the saddle-point structure of the problem, that turns out to be interesting because velocities of the fluid phase and of the solid phase are combined into a single quasi-incompressibility constraint. Our analysis provides guidelines to select the componentwise polynomial degree of approximation of fluid velocity, solid displacement and pressure, to obtain a stable and robust discretization based on Taylor–Hood type finite element spaces. Interestingly, we show how this choice depends on the porosity of the mixture, i.e. the volume fraction of the fluid phase.

A combined experimental atomic force microscopy-based nanoindentation and computational modeling approach to unravel the key contributors to the time-dependent mechanical behavior of single cells

Cellular responses to mechanical stimuli are influenced by the mechanical properties of cells and the surrounding tissue matrix. Cells exhibit viscoelastic behavior in response to an applied stress. This has been attributed to fluid flow-dependent and flow-independent mechanisms. However, the particular mechanism that controls the local time-dependent behavior of cells is unknown. Here, a combined approach of experimental AFM nanoindentation with computational modeling is proposed, taking into account complex material behavior. Three constitutive models (porohyperelastic, viscohyperelastic, poroviscohyperelastic) in tandem with optimization algorithms were employed to capture the experimental stress relaxation data of chondrocytes at 5% strain. The poroviscohyperelastic models with and without fluid flow allowed through the cell membrane provided excellent description of the experimental time-dependent cell responses (normalized mean squared error (NMSE) of 0.003 between the model and experiments). The viscohyperelastic model without fluid could not follow the entire experimental data that well (NMSE = 0.005), while the porohyperelastic model could not capture it at all (NMSE = 0.383). We also show by parametric analysis that the fluid flow has a small, but essential effect on the loading phase and short-term cell relaxation response, while the solid viscoelasticity controls the longer-term responses. We suggest that the local time-dependent cell mechanical response is determined by the combined effects of intrinsic viscoelasticity of the cytoskeleton and fluid flow redistribution in the cells, although the contribution of fluid flow is smaller when using a nanosized probe and moderate indentation rate. The present approach provides new insights into viscoelastic responses of chondrocytes, important for further understanding cell mechanobiological mechanisms in health and disease.

Investigation of the mechanical behavior of kangaroo humeral head cartilage tissue by a porohyperelastic model based on the strain-rate-dependent permeability

Solid–interstitial fluid interaction, which depends on tissue permeability, is significant to the strain-rate-dependent mechanical behavior of humeral head (shoulder) cartilage. Due to anatomical and biomechanical similarities to that of the human shoulder, kangaroos present a suitable animal model. Therefore, indentation experiments were conducted on kangaroo shoulder cartilage tissues from low (10−4/s) to moderately high (10−2/s) strain-rates. A porohyperelastic model was developed based on the experimental characterization; and a permeability function that takes into account the effect of strain-rate on permeability (strain-rate-dependent permeability) was introduced into the model to investigate the effect of rate-dependent fluid flow on tissue response. The prediction of the model with the strain-rate-dependent permeability was compared with those of the models using constant permeability and strain-dependent permeability. Compared to the model with constant permeability, the models with strain-dependent and strain-rate-dependent permeability were able to better capture the experimental variation at all strain-rates (p<0.05). Significant differences were not identified between models with strain-dependent and strain-rate-dependent permeability at strain-rate of 5×10−3/s (p=0.179). However, at strain-rate of 10−2/s, the model with strain-rate-dependent permeability was significantly better at capturing the experimental results (p<0.005). The findings thus revealed the significance of rate-dependent fluid flow on tissue behavior at large strain-rates, which provides insights into the mechanical deformation mechanisms of cartilage tissues.

Stress relaxation analysis of single chondrocytes using porohyperelastic model based on AFM experiments

Based on atomic force microscopytechnique, we found that the chondrocytes exhibits stress relaxation behavior. We explored the mechanism of this stress relaxation behavior and concluded that the intracellular fluid exuding out from the cells during deformation plays the most important role in the stress relaxation. We applied the inverse finite element analysis technique to determine necessary material parameters for porohyperelastic (PHE) model to simulate stress relaxation behavior as this model is proven capable of capturing the non-linear behavior and the fluid-solid interaction during the stress relaxation of the single chondrocytes. It is observed that PHE model can precisely capture the stress relaxation behavior of single chondrocytes and would be a suitable model for cell biomechanics.

A one-dimensional mixed porohyperelastic transport swelling finite element model with growth

A one-dimensional, large-strain, mixed porohyperelastic transport and swelling (MPHETS) finite element model was developed in MATLAB and incorporated with a well-known growth model for soft tissues to allow the model to grow (increase in length) or shrink (decrease in length) at constant material density. By using the finite element model to determine the deformation and stress state, it is possible to implement different growth laws in the program in the future to simulate how soft tissues grow and behave when exposed to various stimuli (e.g. mechanical, chemical, or electrical). The essential assumptions needed to use the MPHETS model with growth are clearly identified and explained in this paper. The primary assumption in this work, however, is that the stress upon which growth acts is the stress in the solid skeleton, i.e. the effective stress, Seff. It is shown that significantly different amounts of growth are experienced for the same loading conditions when using a porohyperelastic model as compared to a purely solid model. In one particular example, approximately 51% less total growth occurred in the MPHETS model than in the solid model even though both problems were subjected to the same external loading. This work represents a first step in developing more sophisticated models capable of capturing the complex mechanical and biochemical environment in growing and remodeling tissues.

Hydro-mechanical coupling in the periodontal ligament: A porohyperelastic finite element model

Harmonic tension–compression tests at 0.1, 0.5 and 1 Hz on hydrated bovine periodontal ligament (PDL) were numerically simulated. The process was modeled by finite elements (FE) within the framework of poromechanics, with the objective of isolating the contributions of the solid- and fluid phases. The solid matrix was modeled as a porous hyperelastic material (hyperfoam) through which the incompressible fluid filling the pores flowed in accordance with the Darcy’s law. The hydro-mechanical coupling between the porous solid matrix and the fluid phase circulating through it provided an apparent time-dependent response to the PDL, whose rate of deformation depended on the permeability of the porous solid with respect to the interstitial fluid. Since the PDL was subjected to significant deformations, finite strains were taken into account and an exponential dependence of PDL permeability on void ratio – and therefore on the deformation state – was assumed. PDL constitutive parameters were identified by fitting the simulated response to the experimental data for the tests at 1 Hz. The values thus obtained were then used to simulate the tests at 0.1 and 0.5 Hz. The results of the present simulation demonstrate that a porohyperelastic model with variable permeability is able to describe the two main aspects of the PDL’s response: (1) the dependency on strain-rate—the saturated material can develop volumetric strains by only exchanging fluid and (2) the asymmetry between tension and compression, which is due to the effect of both the permeability and the elastic properties on deformation.

Influence of unilateral disc displacement on the stress response of the temporomandibular joint discs during opening and mastication.

The temporomandibular joint plays a crucial role in human mastication acting as a guide of jaw movements. During these movements, the joint is subjected to loads which cause stresses and deformations in its cartilaginous structures. A perfect balance between the two sides of the joint is essential to maintain the physiological stress level within the tissues. Therefore, it has been suggested that a derangement of the joint is a contributing factor in the development of mandibular asymmetry, especially if problems of the temporomandibular joint start in childhood or adolescence. To analyze the movement of the mandible and the stresses undergone by the discs, two finite element models of the human temporomandibular joint including the masticatory system were developed, one corresponding to a healthy joint and the other with a unilateral anterior disc displacement with their movement controlled by muscle activation. A fibre-reinforced porohyperelastic model was used to simulate the behaviour of the articular discs. The stress distribution was analyzed in both models during free opening and closing, and during the introduction of a resistant force between incisors or molars. It was found that a slight unilateral anterior disc displacement does not lead to mandibular asymmetry but to a slight decrease of the maximum gape. With the introduction of a restriction between incisors, the maximum stresses moved to the anterior band in contrast to what happened if the restriction was imposed between molars where maximum stresses were located more posteriorly. Finally, the presence of a unilateral displacement of the disc involved a strong change in the overall behaviour of the joint including also the healthy side, where the maximum stresses moved to the posterior part.

Multiphysics Simulation of Blood Flow and LDL Transport in a Porohyperelastic Arterial Wall Model

Atherosclerosis localizes at a bend andor bifurcation of an artery, and low density lipoproteins (LDL) accumulate in the intima. Hemodynamic factors are known to affect this localization and LDL accumulation, but the details of the process remain unknown. It is thought that the LDL concentration will be affected by the filtration flow, and that the velocity of this flow will be affected by deformation of the arterial wall. Thus, a coupled model of a blood flow and a deformable arterial wall with filtration flow would be invaluable for simulation of the flow field and concentration field in sequence. However, this type of highly coupled interaction analysis has not yet been attempted. Therefore, we performed a coupled analysis of an artery with multiple bends in sequence. First, based on the theory of porous media, we modeled a deformable arterial wall using a porohyperelastic model (PHEM) that was able to express both the filtration flow and the viscoelastic behavior of the living tissue, and simulated a blood flow field in the arterial lumen, a filtration flow field and a displacement field in the arterial wall using a fluid-structure interaction (FSI) program code by the finite element method (FEM). Next, based on the obtained results, we further simulated LDL transport using a mass transfer analysis code by the FEM. We analyzed the PHEM in comparison with a rigid model. For the blood flow, stagnation was observed downward of the bends. The direction of the filtration flow was only from the lumen to the wall for the rigid model, while filtration flows from both the wall to the lumen and the lumen to the wall were observed for the PHEM. The LDL concentration was high at the lumenwall interface for both the PHEM and rigid model, and reached its maximum value at the stagnation area. For the PHEM, the maximum LDL concentration in the wall in the radial direction was observed at the position of 3% wall thickness from the lumenwall interface, while for the rigid model, it was observed just at the lumenwall interface. In addition, the peak LDL accumulation area of the PHEM moved about according to the pulsatile flow. These results demonstrate that the blood flow, arterial wall deformation, and filtration flow all affect the LDL concentration, and that LDL accumulation is due to stagnation and the presence of filtration flow. Thus, FSI analysis is indispensable.

Anterior Displacement of the TMJ Disk: Repositioning of the Disk Using a Mitek System. A 3D Finite Element Study

In this paper the behaviors of the temporomandibular joint (TMJ) with an anteriorly displaced disk without reduction and with a surgically repositioned one were compared with the response of a healthy disk during jaw opening. The movement of each joint was obtained imposing the same opening path between incisors and assuming that the movement of the condyle is determined by the passive action of the masticatory muscles and the restrictions imposed by the articulating surfaces and the ligaments. A fiber-reinforced porohyperelastic model was used to simulate the behavior of the articular disk. The influence of the friction coefficient in the diseased joint was also analyzed, finding that the final displacement of the complex condyle-disk was smaller as the friction coefficient increased. On the other hand, its displacement in the repositioned joint was different than in the healthy case because the artificial sutures used in the surgery do not fully stabilize the disk posteriorly as the retrodiscal tissue does. The stress response of the disk changed in both pathologic cases: in the displaced joint the highest stresses moved from the intermediate zone (healthy case) to the posterior band, and in the reconstructed one the most loaded zone moved posteriorly at total opening. Besides, local stress concentrations appeared in the neighborhood of the artificial sutures and therefore damage of the disk and releasing of the sutures might be possible postoperatively.

On the numerical simulation of the mechanical behaviour of articular cartilage

In this paper, a fibre-reinforced porohyperelastic model is extensively presented and used for the simulation of soft hydrated tissues, like cartilage. The derivation of the classical governing equations for biphasic tissues is obtained from a complete and robust description for multiphasic systems. An Augmented Lagrangian formulation has been used to enforce the incompressibility condition in the whole tissue. Porohyperelasticity has been included in order to simulate the biphasic nature of the tissue associated to a porous solid matrix and interstitial fluid. Anisotropy in the hyperelastic behaviour has been formulated by means of a strain energy function that takes into account the influence of collagen fibres. Several numerical examples are shown in order to validate the proposed model. The confined compression problem has been solved to demonstrate the improvement that an Augmented Lagrangian Method provides with respect to a penalty approach. On the other hand, the unconfined compression problem allowed us to study the load distribution in each phase and its dependence on the strain rate. Finally, one realistic application of this model is briefly discussed. Copyright © 2006 John Wiley & Sons, Ltd.

3D Finite Element Simulation of the Opening Movement of the Mandible in Healthy and Pathologic Situations

One of the essential causes of disk disorders is the pathologic change in the ligamentous attachments of the disk-condyle complex. In this paper, the response of the soft components of a human temporomandibular joint during mouth opening in healthy and two pathologic situations was studied. A three-dimensional finite element model of this joint comprising the bone components, the articular disk, and the temporomandibular ligaments was developed from a set of medical images. A fiber reinforced porohyperelastic model was used to simulate the behavior of the articular disk, taking into account the orientation of the fibers in each zone of this cartilage component. The condylar movements during jaw opening were introduced as the loading condition in the analysis. In the healthy joint, it was obtained that the highest stresses were located at the lateral part of the intermediate zone of the disk. In this case, the collateral ligaments were subject to high loads, since they are responsible of the attachment of the disk to the condyle during the movement of the mandible. Additionally, two pathologic situations were simulated: damage of the retrodiscal tissue and disruption of the lateral discal ligament. In both cases, the highest stresses moved to the posterior part of the disk since it was displaced in the anterior-medial direction. In conclusion, in the healthy joint, the highest stresses were located in the lateral zone of the disk where perforations are found most often in the clinical experience. On the other hand, the results obtained in the damaged joints suggested that the disruption of the disk attachments may cause an anterior displacement of the disk and instability of the joint.

Finite element analysis of the temporomandibular joint during lateral excursions of the mandible

One of the most significant characteristics of the temporomandibular joint (TMJ) is that it is in fact composed of two joints. Several finite element simulations of the TMJ have been developed but none of them analysed the different responses of its two sides during nonsymmetrical movement. In this paper, a lateral excursion of the mandible was introduced and the biomechanical behaviour of both sides was studied. A three-dimensional finite element model of the joint comprising the bone components, both articular discs, and the temporomandibular ligaments was used. A fibre-reinforced porohyperelastic model was introduced to simulate the behaviour of the articular discs, taking into account the orientation of the fibres in each zone of these cartilage components. The mandible movement during its lateral excursion was introduced as the loading condition in the analysis. As a consequence of the movement asymmetry, the discs were subjected to different load distributions. It was observed that the maximal shear stresses were located in the lateral zone of both discs and that the lateral attachment of the ipsilateral condyle–disc complex suffered a large distortion, due to the compression of this disc against the inferior surface of the temporal bone. These results may be related with possible consequences of a common disorder called bruxism. Although it would be necessary to perform an exhaustive analysis of this disorder, including the contact forces between the teeth during grinding, it could be suggested that a continuous lateral movement of the jaw may lead to perforations of both discs in their lateral part and may damage the lateral attachments of the disc to the condyle.

The effect of collagen reinforcement in the behaviour of the temporomandibular joint disc

In this paper, the influence of collagen fibres in the behaviour of the temporomandibular joint disc is studied. A three-dimensional finite element model of the joint is developed from a set of medical images. The model comprises the mandible, part of the cranium and both temporomandibular joints. Joints have been considered to be composed of the articular discs and the temporomandibular ligaments. A fibre-reinforced porohyperelastic model was used to study the response under clenching of the fibrocartilage that composes the articular disc. This was divided in an intermediate zone, and two bands, an anterior and other posterior, in order to define the orientation of collagen fibres. The study demonstrates that the introduction of collagen fibres in the biphasic behaviour of the articular disc implies for a prescribed displacement not only an increase of the pressurization in the tissue, but also higher stresses in the anterior and posterior bands, as well as in the lateral zone of the disc. Thus, modelling the disc as an isotropic solid matrix leads in this case to an overestimation of the stresses in the intermediate zone, an underestimation of the pore pressure in this area, and an underestimation of the stresses in the rest of the disc.