Computational modeling of multicellular constructs with the material point method

Computational modeling of the mechanics of cells and multicellular constructs with standard numerical discretization techniques such as the finite element (FE) method is complicated by the complex geometry, material properties and boundary conditions that are associated with such systems. The objectives of this research were to apply the material point method (MPM), a meshless method, to the modeling of vascularized constructs by adapting the algorithm to accurately handle quasi-static, large deformation mechanics, and to apply the modified MPM algorithm to large-scale simulations using a discretization that was obtained directly from volumetric confocal image data. The standard implicit time integration algorithm for MPM was modified to allow the background computational grid to remain fixed with respect to the spatial distribution of material points during the analysis. This algorithm was used to simulate the 3D mechanics of a vascularized scaffold under tension, consisting of growing microvascular fragments embedded in a collagen gel, by discretizing the construct with over 13.6 million material points. Baseline 3D simulations demonstrated that the modified MPM algorithm was both more accurate and more robust than the standard MPM algorithm. Scaling studies demonstrated the ability of the parallel code to scale to 200 processors. Optimal discretization was established for the simulations of the mechanics of vascularized scaffolds by examining stress distributions and reaction forces. Sensitivity studies demonstrated that the reaction force during simulated extension was highly sensitive to the modulus of the microvessels, despite the fact that they comprised only 10.4% of the volume of the total sample. In contrast, the reaction force was relatively insensitive to the effective Poisson's ratio of the entire sample. These results suggest that the MPM simulations could form the basis for estimating the modulus of the embedded microvessels through a parameter estimation scheme. Because of the generality and robustness of the modified MPM algorithm, the relative ease of generating spatial discretizations from volumetric image data, and the ability of the parallel computational implementation to scale to large processor counts, it is anticipated that this modeling approach may be extended to many other applications, including the analysis of other multicellular constructs and investigations of cell mechanics.     

Bullet-derived lead in tissues of the wild boar (Sus scrofa) and red deer (Cervus elaphus)

Bullet-derived lead in game food products is an important source of human contamination. Careless removal of tissues from around the bullet pathway in the animal body results in elevated lead doses being ingested by humans. To assess bullet-derived lead contamination of soft game tissues, muscle tissue samples were collected from ten wild boars and ten red deer immediately after they had been shot. The samples were collected from around the entry and exit wounds, from around the bullet pathway at different sites along its length, and from a distance of about 5, 15, 25, and 30 cm from the bullet track. The individuals examined differed in the lead contents in their tissues surrounding the entry and exit wounds and at different sites along the bullet pathway. One of the animals showed as much lead as 1,095.9 mg kg⁻¹ wet weight in the tissue surrounding the bullet track near the entry wound, 736.0 mg kg⁻¹ being recorded around the exit wound.     

Ventricular mechanics in diastole: material parameter sensitivity

Models of ventricular mechanics have been developed over the last 20 years to include finite deformation theory, anisotropic and inhomogeneous material properties and an accurate representation of ventricular geometry. As computer performance and the computational efficiency of the models improve, clinical application of these heart mechanics models is becoming feasible. One such application is to estimate myocardial material properties by adjusting the constitutive parameters to match wall deformation from MRI or ultrasound measurements, together with a measurement (or estimate) of ventricular pressure. Pigs are now the principal large animal model for these studies and in this paper we present the development of a new three-dimensional finite element model of the heart based on measurements of the geometry and the fibre and sheet orientations of pig hearts. The end-diastolic deformation of the model is computed using the "pole-zero" constitutive law which we have previously used to model the mechanics of passive myocardial tissue specimens. The sensitivities of end-diastolic fibre-sheet material strains and heart shape to changes in the material parameters are computed for the parameters of the pole-zero law in order to assess the utility of the models for inverse material property determination.     

Human annulus fibrosus material properties from biaxial testing and constitutive modeling are altered with degeneration

The annulus fibrosus (AF) of the intervertebral disk undergoes large and multidirectional stresses and strains. Uniaxial tensile tests are limited for measuring AF material properties, because freely contracting edges can prevent fiber stretch and are not representative of in situ boundary conditions. The objectives of this study were to measure human AF biaxial tensile mechanics and to apply and validate a constitutive model to determine material properties. Biaxial tensile tests were performed on samples oriented along the circumferential–axial and the radial–axial directions. Data were fit to a structurally motivated anisotropic hyperelastic model composed of isotropic extra-fibrillar matrix, nonlinear fibers, and fiber–matrix interactions (FMI) normal to the fibers. The validated model was used to simulate shear and uniaxial tensile behavior, to investigate AF structure–function, and to quantify the effect of degeneration. The biaxial stress–strain response was described well by the model (R ²?>?0.9). The model showed that the parameters for fiber nonlinearity and the normal FMI correlated with degeneration, resulting in an elongated toe-region and lower stiffness with degeneration. The model simulations in shear and uniaxial tension successfully matched previously published circumferential direction Young’s modulus, provided an explanation for the low values in previously published axial direction Young’s modulus, and was able to simulate shear mechanics. The normal FMI were important contributors to stress and changed with degeneration, therefore, their microstructural and compositional source should be investigated. Finally, the biaxial mechanical data and constitutive model can be incorporated into a disk finite element model to provide improved quantification of disk mechanics.     

Modeling Soft Tissue Damage and Failure Using a Combined Particle/Continuum Approach

Biological soft tissues experience damage and failure as a result of injury, disease, or simply age; examples include torn ligaments and arterial dissections. Given the complexity of tissue geometry and material behavior, computational models are often essential for studying both damage and failure. Yet, because of the need to account for discontinuous phenomena such as crazing, tearing, and rupturing, continuum methods are limited. Therefore, we model soft tissue damage and failure using a particle/continuum approach. Specifically, we combine continuum damage theory with Smoothed Particle Hydrodynamics (SPH). Because SPH is a meshless particle method, and particle connectivity is determined solely through a neighbor list, discontinuities can be readily modeled by modifying this list. We show, for the first time, that an anisotropic hyperelastic constitutive model commonly employed for modeling soft tissue can be conveniently implemented within a SPH framework and that SPH results show excellent agreement with analytical solutions for uniaxial and biaxial extension as well as finite element solutions for clamped uniaxial extension in 2D and 3D. We further develop a simple algorithm that automatically detects damaged particles and disconnects the spatial domain along rupture lines in 2D and rupture surfaces in 3D. We demonstrate the utility of this approach by simulating damage and failure under clamped uniaxial extension and in a peeling experiment of virtual soft tissue samples. In conclusion, SPH in combination with continuum damage theory may provide an accurate and efficient framework for modeling damage and failure in soft tissues.     

Microplane constitutive model and computational framework for blood vessel tissue

This paper presents a nonlinearly elastic anisotropic microplane formulation in 3D for computational constitutive modeling of arterial soft tissue in the passive regime. The constitutive modeling of arterial (and other biological) soft tissue is crucial for accurate finite element calculations, which in turn are essential for design of implants, surgical procedures, bioartificial tissue, as well as determination of effect of progressive diseases on tissues and implants. The model presented is defined at a lower scale (mesoscale) than the conventional macroscale and it incorporates the effect of all the (collagen) fibers which are anisotropic structural components distributed in all directions within the tissue material in addition to that of isotropic bulk tissue. It is shown that the proposed model not only reproduces Holzapfel's recent model but also improves on it by accounting for the actual three-dimensional distribution of fiber orientation in the arterial wall, which endows the model with advanced capabilities in simulation of remodeling of soft tissue. The formulation is flexible so that its parameters could be adjusted to represent the arterial wall either as a single material or a material composed of several layers in finite element analyses of arteries. Explicit algorithms for both the material subroutine and the explicit integration with dynamic relaxation of equations of motion using finite element method are given. To circumvent the slow convergence of the standard dynamic relaxation and small time steps dictated by the stability of the explicit integrator, an adaptive dynamic relaxation technique that ensures stability and fastest possible convergence rates is developed. Incompressibility is enforced using penalty method with an updated penalty parameter. The model is used to simulate experimental data from the literature demonstrating that the model response is in excellent agreement with the data. An experimental procedure to determine the distribution of fiber directions in 3D for biological soft tissue is suggested in accordance with the microplane concept. It is also argued that this microplane formulation could be modified or extended to model many other phenomena of interest in biomechanics.     

Finite element implementation of a generalized Fung-elastic constitutive model for planar soft tissues

Numerical simulations of the anisotropic mechanical properties of soft tissues and tissue-derived biomaterials using accurate constitutive models remain an important and challenging research area in biomechanics. While most constitutive modeling efforts have focused on the characterization of experimental data, only limited studies are available on the feasibility of utilizing those models in complex computational applications. An example is the widely utilized exponential constitutive model proposed by Fung. Although present in the biomechanics literature for several decades, implementation of this model into finite element (FE) simulations has been limited. A major reason for limited numerical implementations are problems associated with inherent numerical instability and convergence. To address this issue, we developed and applied two restrictions for a generalized Fung-elastic constitutive model necessary to achieve numerical stability. These are (1) convexity of the strain energy function, and (2) the condition number of material stiffness matrix set lower than a prescribed value. These constraints were implemented in the nonlinear regression used for constitutive model parameter estimation to the experimental biaxial mechanical data. We then implemented the generalized Fung-elastic model into a commercial FE code (ABAQUS, Pawtucket, RI, USA). Single element and multi-element planar biaxial test simulations were conducted to verify the accuracy and robustness of the implementation. Results indicated that numerical convergence and accurate FE implementation were consistently obtained. The present study thus presents an integrated framework for accurate and robust implementation of pseudo-elastic constitutive models for planar soft tissues. Moreover, since our approach is formulated within a general FE code, it can be straightforwardly adopted across multiple software platforms.     

A novel micro-to-macro approach for cardiac tissue mechanics

For studying cardiac mechanics, hyperelastic anisotropic computational models have been developed which require the tissue anisotropic and hyperelastic parameters. These parameters are obtained by tissue samples mechanically testing. The validity of such parameters are limited to the specific tissue sample only. They are not adaptable for pathological tissues commonly associated with tissue microstructure alterations. To investigate cardiac tissue mechanics, a novel approach is proposed to model hyperelasticity and anisotropy. This approach is adaptable to various tissue microstructural constituent’s distributions in normal and pathological tissues. In this approach, the tissue is idealized as composite material consisting of cardiomyocytes distributed in extracellular matrix (ECM). The major myocardial tissue constituents are mitochondria and myofibrils while the main ECM’s constituents are collagen fibers and fibroblasts. Accordingly, finite element simulations of uniaxial and equibiaxial tests of normal and infarcted tissue samples with known amounts of these constituents were conducted, leading to corresponding tissue stress–strain data that were fitted to anisotropic/hyperelastic models. The models were validated where they showed good agreement characterized by maximum average stress-strain errors of 16.17 and 10.01% for normal and infarcted cardiac tissue, respectively. This demonstrate the effectiveness of the proposed models in accurate characterization of healthy and pathological cardiac tissues.     

Soft-cuticle biomechanics: A constitutive model of anisotropy for caterpillar integument

The mechanical properties of soft tissues are important for the control of motion in many invertebrates. Pressurized cylindrical animals such as worms have circumferential reinforcement of the body wall; however, no experimental characterization of comparable anisotropy has been reported for climbing larvae such as caterpillars. Using uniaxial, real-time fluorescence extensometry on millimeter scale cuticle specimens we have quantified differences in the mechanical properties of cuticle to circumferentially and longitudinally applied forces. Based on these results and the composite matrix-fiber structure of cuticle, a pseudo-elastic transversely isotropic constitutive material model was constructed with circumferential reinforcement realized as a Horgan-Saccomandi strain energy function. This model was then used numerically to describe the anisotropic material properties of Manduca cuticle. The constitutive material model will be used in a detailed finite-element analysis to improve our understanding of the mechanics of caterpillar crawling.     

Determination of material parameters of the two-dimensional Holzapfel–Weizsäcker type model based on uniaxial extension data of arterial walls

Soft tissues are anisotropic materials yet a majority of mechanical property tests have been uniaxial, which often failed to recapitulate the tensile response in other directions. This paper aims to study the feasibility of determining material parameters of anisotropic tissues by uniaxial extension with a minimal loss of anisotropic information. We assumed that by preselecting a certain constitutive model, we could give the constitutive parameters based on uniaxial extension data from orthogonal strip samples. In our study, the Holzapfel–Weizsäcker type strain energy density function (H–W model) was used to determine the material parameters of arterial walls from two fresh donation bodies. The key points we applied were the relationships between strain components in uniaxial tensile tests and the methods of stochastic optimisation. Further numerical experiments were taken. The estimate–effect ratio, defined by the number of data with the precision of estimation less than 0.5% over whole size of data, was calculated to demonstrate the feasibility of our method. The material parameters for Chinese aorta and pulmonary artery were given with the maximum root mean square (RMS) errors 0.042, and the minimal estimate–effect ratio in numerical experiments was 90.79%. Our results suggest that the constitutive parameters of arterial walls can be determined from uniaxial extension data, given the passive mechanical behaviour governed by H–W model. This method may apply to other tissues using different constitutive models.     

Numerical simulation of the failure of ventricular tissue due to deep penetration: The impact of constitutive properties

Lead perforation is a rare but serious clinical complication of pacemaker implantation, and towards understanding this malfunction, the present study investigated myocardial failure due to deep penetration by an advancing rigid punch. To this end, a non-linear Finite Element model was developed that integrates constitutive data published in the literature with information from in vitro tensile testing in cross-fibre direction of porcine myocardial tissue. The Finite Element model considered non-linear, isotropic and visco-elastic properties of the myocardium, and tissue failure was phenomenologically described by a Traction Separation Law. In vitro penetration testing of porcine myocardium was used to validate the Finite Element model, and a particular objective of the study was to investigate the impact of different constitutive parameters on the simulated results. Specifically, results demonstrated that visco-elastic properties of the tissue strongly determine the failure process, whereas dissipative effects directly related to failure had a minor impact on the simulation results. In addition, non-linearity of the bulk material did not change the predicted peak penetration force and the simulations did not reveal elastic crack-tip blunting. The performed study provided novel insights into ventricular failure due to deep penetration, and provided useful information with which to develop numerical failure models.     

Nanoindentation testing and finite element simulations of cortical bone allowing for anisotropic elastic and inelastic mechanical response

Anisotropy is one of the most peculiar aspects of cortical bone mechanical behaviour, and the numerical approach can be successfully used to investigate aspects of bone tissue mechanics that analytical methods solve in approximate way or do not cover. In this work, nanoindentation experimental tests and finite element simulations were employed to investigate the elastic-inelastic anisotropic mechanical properties of cortical bone. The model allows for anisotropic elastic and post-yield behaviour of the tissue. A tension-compression mismatch and direction-dependent yield stresses are allowed for. Indentation experiments along the axial and transverse directions were simulated with the purpose to predict the indentation moduli and hardnesses along multiple orientations. Results showed that the experimental transverse-to-axial ratio of indentation moduli, equal to 0.74, is predicted with a ～3% discrepancy regardless the post-yield material behaviour; whereas, the transverse-to-axial hardness ratio, equal to 0.86, can be correctly simulated (discrepancy ～6% w.r.t. the experimental results) only employing an anisotropic post-elastic constitutive model. Further, direct comparison between the experimental and simulated indentation tests evidenced a good agreement in the loading branch of the indentation curves and in the peak loads for a transverse-to-axial yield stress ratio comparable to the experimentally obtained transverse-to-axial hardness ratio. In perspective, the present work results strongly support the coupling between indentation experiments and FEM simulations to get a deeper knowledge of bone tissue mechanical behaviour at the microstructural level. The present model could be used to assess the effect of variations of constitutive parameters due to age, injury, and/or disease on bone mechanical performance in the context of indentation testing.     

A computationally efficient optimization kernel for material parameter estimation procedures

Estimating material parameters is an important part in the study of soft tissue mechanics. Computational time can easily run to days, especially when all available experimental data are taken into account. The material parameter estimation procedure is examplified on a set of homogeneous simple shear experiments to estimate the orthotropic constitutive parameters of myocardium. The modification consists of changing the traditional least-squares approach to a weighted least-squares. This objective function resembles a L(2)-norm type integral which is approximated using Gaussian quadrature. This reduces the computational time of the material parameter estimation by two orders of magnitude.     

A discrete-time approach to the formulation of constitutive models for viscoelastic soft tissues

This paper presents a novel approach to constitutive modeling of viscoelastic soft tissues. This formulation combines an anisotropic strain energy function, accounting for preferred material directions, to define the elastic stress–strain relationship, and a discrete time black-box dynamic model, borrowed from the theory of system identification, to describe the time-dependent behavior. This discrete time formulation is straightforwardly oriented to the development of a recursive time integration scheme that calculates the current stress state by using strain and stress values stored at a limited number of previous time instants. The viscoelastic model and the numerical procedure are assessed by implementing two numerical examples, the simulation of a uniaxial tensile test and the inflation of a thin tube. Both simulations are performed using parameter values based on previous experiments on preserved bovine pericardium. Parameters are then adjusted to investigate the sensitivity of the model. The hypotheses the model relies upon are discussed and the main limitations are stated.     

A rheological network model for the continuum anisotropic and viscoelastic behavior of soft tissue

The mechanical behavior of soft tissue demonstrates a number of complex features including nonlinearity, anisotropy, viscoelasticity, and growth. Characteristic features of the time-dependent and anisotropic behavior are related to the properties of various components of the tissue such as fibrous collagen and elastin networks, large proteins and sugars attached to these networks, and interstitial fluid. Attempts to model the elastic behavior of these tissues based on assumptions about the behavior of the underlying constituents have been reasonably successful, but the essential addition of viscoelasticity to these models has been met with varying success. Here, a new rheological network model is proposed using, as its basis, an orthotropic hyperelastic constitutive model for fibrous tissue and a viscoelastic reptation model for soft materials. The resulting model has been incorporated into numerical and computational models, and is shown to capture the mechanical behavior of soft tissue in various modes of deformation including uniaxial and biaxial tension and simple shear.