Hyperelastic modelling of arterial layers with distributed collagen fibre orientations

3d Mechanical properties of the partially obstructed guinea pig small intestine

BACKGROUND AND AIMS

Partial obstruction of the small intestine results in severe hypertrophy of smooth muscle cells, dilatation and functional denervation. Hypertrophy of the small intestine is associated with alteration of the wall structure and the mechanical properties. The aims of this study were to determine three dimensional material properties of the obstructed small intestine in guinea pigs and to obtain the 3D stress-strain distributions in the small intestinal wall.

METHODS

Partial obstruction of mid-jejunum was created surgically in five guinea pigs that were euthanized 2 weeks after the surgery. Ten-cm-long segments proximal to the obstruction site were used for the stretch-inflation mechanical test using a tri-axial test machine. The outer diameter, longitudinal force and the luminal pressure during the test were recorded simultaneously. An anisotropic exponential pseudo-strain energy density function was used as the constitutive equation to fit the experimental loading curve and for computation of the stress-strain distribution.

RESULTS

The wall thickness and the wall area increased significantly in the obstructed jejunum (P<0.001). The pressure-outer radius curves in the obstructed segments were translated to the left of the normal segments, indicating wall stiffening after the obstruction. The circumferential stress and the longitudinal stress through the wall were higher in the obstructed segments (P<0.02). This was independent of whether the zero-stress state or the no-load states were used as the reference state.

CONCLUSION

The mechanical behaviour of the obstructed small intestine can be described using a 3D constitutive model. The obstruction-induced biomechanical properties change was characterized by higher circumferential and longitudinal stresses in the wall and altered material constants in the 3D constitutive model.

Prediction of fibre architecture and adaptation in diseased carotid bifurcations

Many studies have used patient-specific finite element models to estimate the stress environment in atherosclerotic plaques, attempting to correlate the magnitude of stress to plaque vulnerability. In complex geometries, few studies have incorporated the anisotropic material response of arterial tissue. This paper presents a fibre remodelling algorithm to predict the fibre architecture, and thus anisotropic material response in four patient-specific models of the carotid bifurcation. The change in fibre architecture during disease progression and its affect on the stress environment in the plaque were predicted. The mean fibre directions were assumed to lie at an angle between the two positive principal strain directions. The angle and the degree of dispersion were assumed to depend on the ratio of principal strain values. Results were compared with experimental observations and other numerical studies. In non-branching regions of each model, the typical double helix arterial fibre pattern was predicted while at the bifurcation and in regions of plaque burden, more complex fibre architectures were found. The predicted change in fibre architecture in the arterial tissue during plaque progression was found to alter the stress environment in the plaque. This suggests that the specimen-specific anisotropic response of the tissue should be taken into account to accurately predict stresses in the plaque. Since determination of the fibre architecture in vivo is a difficult task, the system presented here provides a useful method of estimating the fibre architecture in complex arterial geometries.

Modelling carotid artery adaptations to dynamic alterations in pressure and flow over the cardiac cycle

Motivated by recent clinical and laboratory findings of important effects of pulsatile pressure and flow on arterial adaptations, we employ and extend an established constrained mixture framework of growth (change in mass) and remodelling (change in structure) to include such dynamical effects. New descriptors of cell and tissue behavior (constitutive relations) are postulated and refined based on new experimental data from a transverse aortic arch banding model in the mouse that increases pulsatile pressure and flow in one carotid artery. In particular, it is shown that there was a need to refine constitutive relations for the active stress generated by smooth muscle, to include both stress- and stress rate-mediated control of the turnover of cells and matrix and to account for a cyclic stress-mediated loss of elastic fibre integrity and decrease in collagen stiffness in order to capture the reported evolution, over 8 weeks, of luminal radius, wall thickness, axial force and in vivo axial stretch of the hypertensive mouse carotid artery. We submit, therefore, that complex aspects of adaptation by elastic arteries can be predicted by constrained mixture models wherein individual constituents are produced or removed at individual rates and to individual extents depending on changes in both stress and stress rate from normal values.

Finite element modelling of cell wall properties for onion epidermis using a fibre-reinforced hyperelastic model

A combined finite element method and inverse modelling approach is used to model the mechanical deformation of onion epidermis. A fibre-reinforced hyperelastic composite material model considering the fibre distribution has been used to simulate the mechanical behaviour of samples under tension. The mechanical parameters of onion epidermis are determined using an inverse modelling approach. The simulated results show a good correlation with experimental observations.

Microstructural and biomechanical alterations of the human aorta as a function of age and location

While it is known that the aorta stiffens with location and age, little is known about the underlying mechanisms that govern these alterations. The purpose of this study was to investigate the relationship between the anisotropic biomechanical behavior and extracellular matrix microstructure of the human aorta and quantify how each changes with location and age. A total of 207 specimens were harvested from 5 locations (ascending n = 33, arch n = 38, descending n = 54, suprarenal n = 52, and abdominal n = 30) of 31 autopsy donor aortas (aged 3 days to 93 years). Each specimen underwent planar biaxial testing in order to derive quantitative biomechanical endpoints of anisotropic stiffness and compliance. Quantitative measures of fiber alignment and degree of fiber alignment were also generated on the same samples using a small-angle light scattering (SALS) technique. Circumferential and axial stiffening occurred with age and increased from the proximal to distal aorta, and the abdominal region was found to be more stiff than all others (p ≤ 0.006). Specimens from donors aged 61 and above were drastically more stiff than younger specimens (p < 0.001) and demonstrated greater circumferential compliance and axial stiffening (p < 0.001). Fiber direction for all ages and locations was predominantly circumferential (p < 0.001), and the degree of fiber alignment was found to increase with age (p < 0.001). Our results demonstrate that the aorta becomes more biomechanically and structurally anisotropic after age 60; with significant changes occurring preferentially in the abdominal aorta, these changes may play an important role in the predisposition of disease formation (e.g., aneurysm) in this region with age.

A biomechanical analysis of venous tissue in its normal and post-phlebitic conditions

Although biomechanical studies of the normal rat vein wall have been reported (Weizsacker, 1988; Plante, 2002), there are no published studies that have investigated the mechanical effects of thrombus formation on murine venous tissue. In response to the lack of knowledge concerning the mechanical consequences of thrombus resolution, distinct thrombus-induced changes in the biomechanical properties of the murine vena cava were measured via biaxial stretch experiments. These data served as input for strain energy function (SEF) fitting and modeling (Gasser et al., 2006). Statistical differences were observed between healthy and diseased tissue with respect to the structural coefficient that represents the response of the non-collagenous, isotropic ground substance. Alterations following thrombus formation were also noted for the SEF coefficient which describes the anisotropic contribution of the fibers. The data indicate ligation of the vena cava leads to structural alterations in the ground substance and collagen fiber network.

Characterization of the mechanical properties of the coronary sinus for percutaneous transvenous mitral annuloplasty

The coronary sinus (CS) vessel serves as a conduit for the deployment of percutaneous transvenous mitral annuloplasty (PTMA) devices for the treatment of functional mitral regurgitation. Characterization of the mechanical response of the CS is an important step towards an understanding of tissue-device interaction in PTMA intervention. The purpose of this study was to investigate the mechanical properties of the porcine CS using the pressure-inflation test and constitutively model the wall behavior using a four fiber family strain energy function (SEF). The results showed that the CS exhibited an S-shaped pressure-radius response and could be dilated up to 88% at a pressure of 80mmHg. Excellent results from model fitting indicated that the four fiber family SEF could capture the experimental data well and could be used in future numerical simulations of tissue-device interaction. In addition, a histological study was performed to identify the micro-structure of the CS wall. We found a high content of striated myocardial fibers (SMFs) surrounding the CS wall, which was also mainly composed of SMFs, while the content of smooth muscle cells was very low. Elastin and collagen fibers were highly concentrated in the luminal and outer layers and sparsely distributed in the medial layer of the CS wall. These structural and mechanical properties of the CS should be taken into consideration in future PTMA device designs.

Role of elastin anisotropy in structural strain energy functions of arterial tissue

The vascular wall exhibits nonlinear anisotropic mechanical properties. The identification of a strain energy function (SEF) is the preferred method to describe its complex nonlinear elastic properties. Earlier constituent-based SEF models, where elastin is modeled as an isotropic material, failed in describing accurately the tissue response to inflation-extension loading. We hypothesized that these shortcomings are partly due to unaccounted anisotropic properties of elastin. We performed inflation-extension tests on common carotid of rabbits before and after enzymatic degradation of elastin and applied constituent-based SEFs, with both an isotropic and an anisotropic elastin part, on the experimental data. We used transmission electron microscopy (TEM) and serial block-face scanning electron microscopy (SBFSEM) to provide direct structural evidence of the assumed anisotropy. In intact arteries, the SEF including anisotropic elastin with one family of fibers in the circumferential direction fitted better the inflation-extension data than the isotropic SEF. This was supported by TEM and SBFSEM imaging, which showed interlamellar elastin fibers in the circumferential direction. In elastin-degraded arteries, both SEFs succeeded equally well in predicting anisotropic wall behavior. In elastase-treated arteries fitted with the anisotropic SEF for elastin, collagen engaged later than in intact arteries. We conclude that constituent-based models with an anisotropic elastin part characterize more accurately the mechanical properties of the arterial wall when compared to models with simply an isotropic elastin. Microstructural imaging based on electron microscopy techniques provided evidence for elastin anisotropy. Finally, the model suggests a later and less abrupt collagen engagement after elastase treatment.

A large strain material model for soft tissues with functionally graded properties

The reaction of articular cartilage and other soft tissues to mechanical loads has been characterized by coupled hydraulic (H) and mechanical (M) processes. An enhanced biphasic material model is presented, which may be used to describe the load response of soft tissue. A large-strain numerical approach of HM coupled processes has been applied. Physical and geometrical nonlinearities, as well as anisotropy and intrinsic rate-dependency of the solid skeleton have been realized using a thermodynamically consistent approach. The presented material model has been implemented into the commercially available finite element code MSC MARC. Initial verification of the model has been conducted analytically in tendonlike structures. The poroelastic and intrinsic viscoelastic features of the model were compared with the experimental data of an unconfined compression test of agarose hydrogel. A recent example from the area of cartilage research has been modeled, and the mechanical response was compared with cell viability. All examples showed good agreement between numerical and analytical/experimental results.

Biomechanical structural stresses of atherosclerotic plaques

Atherosclerotic plaques may rupture without warning, causing fatal clinical events such as myocardial infarction and stroke. Degree of stenosis, which is the current criterion for assessment of atherosclerotic disease severity, has been observed to have poor correlation with plaque vulnerability. Under physiological conditions, plaque undertakes mechanical loadings due to blood pressure and flow. From the material view point, rupture possibly occurs when the extra loading exceeds the material strength of the plaque. Therefore, morphological and mechanical features should be considered in an integrated way for a more accurate assessment of plaque vulnerability and for identification of the at-risk patient. Biomechanical stress analysis is a technique that allows such comprehensive assessment. This article focuses on the mechanical stresses in the plaque structure, which are believed to be of greater magnitude than the associated wall shear stress and are thought to be more closely associated with plaque rupture. We discuss the basic mechanics that govern plaque behavior, the material properties of atherosclerotic tissues and the studies investigating the association between high biomechanical stresses and plaque rupture. Parameter studies investigating the effect of morphologic factors on the critical biomechanical stresses and limitations of current simulation models are also reviewed.

On constitutive descriptors of the biaxial mechanical behaviour of human abdominal aorta and aneurysms

The abdominal aorta (AA) in older individuals can develop an aneurysm, which is of increasing concern in our ageing population. The structural integrity of the ageing aortic wall, and hence aneurysm, depends primarily on effective elastin and multiple families of oriented collagen fibres. In this paper, we show that a structurally motivated phenomenological 'four-fibre family' constitutive relation captures the biaxial mechanical behaviour of both the human AA, from ages less than 30 to over 60, and abdominal aortic aneurysms. Moreover, combining the statistical technique known as non-parametric bootstrap with a modal clustering method provides improved confidence intervals for estimated best-fit values of the eight associated constitutive parameters. It is suggested that this constitutive relation captures the well-known loss of structural integrity of elastic fibres owing to ageing and the development of abdominal aneurysms, and that it provides important insight needed to construct growth and remodelling models for aneurysms, which in turn promise to improve our ability to predict disease progression.

An inverse finite element method for determining the anisotropic properties of the cornea

An inverse finite element method was developed to determine the anisotropic properties of bovine cornea from an in vitro inflation experiment. The experiment used digital image correlation (DIC) to measure the three-dimensional surface geometry and displacement field of the cornea at multiple pressures. A finite element model of a bovine cornea was developed using the DIC measured surface geometry of the undeformed specimen. The model was applied to determine five parameters of an anisotropic hyperelastic model that minimized the error between the measured and computed surface displacement field and to investigate the sensitivity of the measured bovine inflation response to variations in the anisotropic properties of the cornea. The results of the parameter optimization revealed that the collagen structure of bovine cornea exhibited a high degree of anisotropy in the limbus region, which agreed with recent histological findings, and a transversely isotropic central region. The parameter study showed that the bovine corneal response to the inflation experiment was sensitive to the shear modulus of the matrix at pressures below the intraocular pressure, the properties of the collagen lamella at higher pressures, and the degree of anisotropy in the limbus region. It was not sensitive to a weak collagen anisotropy in the central region.

Cyclically stretching developing tissue in vivo enhances mechanical strength and organization of vascular grafts

Tissue-engineered vascular grafts must have qualities that rival native vasculature, specifically the ability to remodel, the expression of functional endothelial components and a dynamic and functional extracellular matrix (ECM) that resists the forces of the arterial circulation. We have developed a device that when inserted into the peritoneal cavity, attracts cells around a tubular scaffold to generate autologous arterial grafts. The device is capable of cyclically stretching (by means of a pulsatile pump) developing tissue to increase the mechanical strength of the graft. Pulsed (n=8) and unpulsed (n=8) devices were implanted for 10 days in Lovenaar sheep (n=8). Pulsation occurred for a period of 5-8 days before harvest. Thick unadhered autologous tissue with cells residing in a collagen ECM was produced in all devices. Collagen organization was greater in the circumferential direction of pulsed tissue. Immunohistochemical labelling revealed the hematopoietic origin of >90% cells and a significantly higher coexpression with vimentin in pulsed tissue. F-actin expression, mechanical failure strength and strain were also significantly increased by pulsation. Moreover, tissue could be grafted as carotid artery patches. This paper shows that unadhered tissue tubes with increased mechanical strength and differentiation in response to pulsation can be produced with every implant after a period of 10 days. However, these tissue tubes require a more fine-tuned exposure to pulsation to be suitable for use as vascular grafts.

A finite element study of invariant-based orthotropic constitutive equations in the context of myocardial material parameter estimation

A previous study investigated a number of invariant-based orthotropic and transversely isotropic constitutive equations for their suitability to fit three-dimensional simple shear mechanics data of passive myocardial tissue. The study was based on the assumption of a homogeneous deformation. Here, we extend the previous study by performing an inverse finite element material parameter estimation. This ensures a more realistic deformation state and material parameter estimates. The constitutive relations were compared on the basis of (i) 'goodness of fit': how well they fit a set of six shear deformation tests and (ii) 'variability': how well determined the material parameters are over the range of experiments. These criteria were utilised to discuss the advantages and disadvantages of the constitutive relations. It was found that a specific form of the polyconvex type as well as the exponential Fung-type equations were most suitable for modelling the orthotropic behaviour of myocardium under simple shear.

In vivo characterization of the aortic wall stress-strain relationship

Arterial stiffness has been shown to be a good indicator of arterial wall disease. However, a single parameter is insufficient to describe the complex stress-strain relationship of a multi-component, non-linear tissue such as the aorta. We therefore propose a new approach to measure the stress-strain relationship locally in vivo noninvasively, and present a clinically relevant parameter describing the mechanical interaction between aortic wall constituents. The slope change of the circumferential stress-strain curve was hypothesized to be related to the contribution of elastin and collagen, and was defined as the transition strain (epsilon(theta)(T)). A two-parallel spring model was employed and three Young's moduli were accordingly evaluated, i.e., corresponding to the: elastic lamellae (E(1)), elastin-collagen fibers (E(2)) and collagen fibers (E(3)). Our study was performed on normal and Angiotensin II (AngII)-treated mouse abdominal aortas using the aortic pressure after catheterization and the local aortic wall diameters change from a cross-correlation technique on the radio frequency (RF) ultrasound signal at 30 MHz and frame rate of 8 kHz. Using our technique, the transition strain and three Young's moduli in both normal and pathological aortas were mapped in 2D. The slope change of the circumferential stress-strain curve was first observed in vivo under physiologic conditions. The transition strain was found at a lower strain level in the AngII-treated case, i.e., 0.029+/-0.006 for the normal and 0.012+/-0.004 for the AngII-treated aortas. E(1), E(2) and E(3) were equal to 69.7+/-18.6, 214.5+/-65.8 and 144.8+/-55.2 kPa for the normal aortas, and 222.1+/-114.8, 775.0+/-586.4 and 552.9+/-519.1 kPa for the AngII-treated aortas, respectively. This is because of the alteration of structures and content of the wall constituents, the degradation of elastic lamella and collagen formation due to AngII treatment. While such values illustrate the alteration of structure and content of the wall constituents related to AngII treatment, limitations regarding physical assumptions (isotropic, linear elastic) should be kept in mind. The transition strain, however, was shown to be a pressure independent parameter that can be clinically relevant and noninvasively measured using ultrasound-based motion estimation techniques. In conclusion, our novel methodology can assess the stress-strain relationship of the aortic wall locally in vivo and quantify important parameters for the detection and characterization of vascular disease.

Structural inhomogeneity and fiber orientation in the inner arterial media

The microstructural orientation of vascular wall constituents is of interest to scientists and clinicians because alterations in their native states are associated with various cardiovascular diseases. In the arterial media, the orientation of these constituents is often described as circumferential. However, it has been noted that, just below the endothelial surface, the vascular wall constituents are oriented axially. To further study this reported change in orientation, and to resolve previous observations (which were made under conditions of no load), we used nonlinear optical microscopy to examine the orientation of collagen and elastin fibers in the inner medial region of bovine common carotid arteries. Images were obtained from this part of the arterial wall under varying degrees of mechanical strain: 0%, 10% axial, 10% circumferential, and 10% biaxial. We observed that close to the endothelium these components are aligned in the axial direction but abruptly change to a circumferential alignment at a depth of approximately 20 mum from the endothelial surface. The application of mechanical strain resulted in a significantly greater degree of fiber alignment, both collagen and elastin, in the strain direction, regardless of their initial unloaded orientation. Furthermore, variations in strain conditions resulted in an increase or a decrease in the overall degree of fiber alignment in the subendothelial layer depending on the direction of the applied strain. This high-resolution investigation adds more detail to existing descriptions of complex structure-function relationships in vascular tissue, which is essential for a better understanding of the pathophysiological processes resulting from injury, disease progression, and interventional therapies.

Constitutive Modeling of Anisotropic Finite-Deformation Hyperelastic Behaviors of Soft Materials Reinforced by Tortuous Fibers

Many biological materials are composites composed of a soft matrix reinforced with stiffer fibers. These stiffer fibers may have a tortuous shape and wind through the soft matrix. At small material deformation, these fibers deform in a bending mode and contribute little to the material stiffness; at large material deformation, these fibers deform in a stretching mode and induce a stiffening effect in the material behavior. The transition from bending mode deformation to stretching mode deformation yields a characteristic J-shape stress-strain curve. In addition, the spatial distribution of these fibers may render the composite an anisotropic behavior. In this paper, we present an anisotropic finite-deformation hyperelastic constitutive model for such materials. Here, the matrix is modeled as an isotropic neo-Hookean material. "The behaviors of single tortuous fiber are represented by a crimped fiber model". The anisotropic behavior is introduced by a structure tensor representing the effective orientation distribution of crimped fibers. Parametric studies show the effect of fiber tortuosity and fiber orientation distribution on the overall stress-strain behaviors of the materials.

Constitutive modelling of arteries

This review article is concerned with the mathematical modelling of the mechanical properties of the soft biological tissues that constitute the walls of arteries. Many important aspects of the mechanical behaviour of arterial tissue can be treated on the basis of elasticity theory, and the focus of the article is therefore on the constitutive modelling of the anisotropic and highly nonlinear elastic properties of the artery wall. The discussion focuses primarily on developments over the last decade based on the theory of deformation invariants, in particular invariants that in part capture structural aspects of the tissue, specifically the orientation of collagen fibres, the dispersion in the orientation, and the associated anisotropy of the material properties. The main features of the relevant theory are summarized briefly and particular forms of the elastic strain-energy function are discussed and then applied to an artery considered as a thick-walled circular cylindrical tube in order to illustrate its extension–inflation behaviour. The wide range of applications of the constitutive modelling framework to artery walls in both health and disease and to the other fibrous soft tissues is discussed in detail. Since the main modelling effort in the literature has been on the passive response of arteries, this is also the concern of the major part of this article. A section is nevertheless devoted to reviewing the limited literature within the continuum mechanics framework on the active response of artery walls, i.e. the mechanical behaviour associated with the activation of smooth muscle, a very important but also very challenging topic that requires substantial further development. A final section provides a brief summary of the current state of arterial wall mechanical modelling and points to key areas that need further modelling effort in order to improve understanding of the biomechanics and mechanobiology of arteries and other soft tissues, from the molecular, to the cellular, tissue and organ levels.

DT-MRI based computation of collagen fiber deformation in human articular cartilage: a feasibility study

Accurate techniques for simulating the deformation of soft biological tissues are an increasingly valuable tool in many areas of biomechanical analysis and medical image computing. To model the complex morphology and response of articular cartilage, a hyperviscoelastic (dispersed) fiber-reinforced constitutive model is employed to complete two specimen-specific finite element (FE) simulations of an indentation experiment, with and without considering fiber dispersion. Ultra-high field Diffusion Tensor Magnetic Resonance Imaging (17.6 T DT-MRI) is performed on a specimen of human articular cartilage before and after indentation to approximately 20% compression. Based on this DT-MRI data, we detail a novel FE approach to determine the geometry (edge detection from first eigenvalue), the meshing (semi-automated smoothing of DTI measurement voxels), and the fiber structural input (estimated principal fiber direction and dispersion). The global and fiber fabric deformations of both the un-dispersed and dispersed fiber models provide a satisfactory match to that estimated experimentally. In both simulations, the fiber fabric in the superficial and middle zones becomes more aligned with the articular surface, although the dispersed model appears more consistent with the literature. In the future, a multi-disciplinary combination of DT-MRI and numerical simulation will allow the functional state of articular cartilage to be determined in vivo.

Characterizing the mechanical contribution of fiber angular distribution in connective tissue: comparison of two modeling approaches

Modeling of connective tissues often includes collagen fibers explicitly as one of the components. These fibers can be oriented in many directions; therefore, several studies have considered statistical distributions to describe the fiber arrangement. One approach to formulate a constitutive framework for distributed fibers is to express the mechanical parameters, such as strain energy and stresses, in terms of angular integrals. These integrals represent the addition of the contribution of infinitesimal fractions of fibers oriented in a given direction. This approach leads to accurate results; however, it requires lengthy calculations. Recently, the use of generalized structure tensors has been proposed to represent the angular distribution in the constitutive equations of the fibers. Although this formulation is much simpler and fewer calculations are required, such structure tensors can only be used when all the fibers are in tension and the angular distribution is small. However, the amount of error introduced in these cases of non-tensile fiber loading and large angular distributions have not been quantified. Therefore, the objective of this study is to determine the range of values of angular distribution for which acceptable differences (less than 10%) between these two formulations are obtained. It was found, analytically and numerically, that both formulations are equivalent for planar distributions under equal-biaxial stretch. The comparison also showed, for other loading conditions, that the differences decrease when the fiber distribution is very small. Differences of less than 10% were usually obtained when the fiber distribution was very low (κ ≈ 0.03; κ ranges between 0 and 1/3, for aligned and isotropic distributed fibers, respectively). This range of angular distribution greatly limits the types of tissue that can be accurately analyzed using generalized structure tensors. It is expected that the results from this study guide the selection of a proper approach to analyze a particular tissue under a particular loading condition.

Biaxial distension of precision-cut lung slices

The mechanical forces acting on lung parenchyma during (mechanical) ventilation and its (patho)physiological consequences are currently under intense scrutiny. Several in vivo and cell culture models have been developed to study the pulmonary responses to mechanical stretch. While providing extremely useful information, these models do also suffer from limitations in being either too complex for detailed mechanical or mechanistic studies, or in being devoid of the full complexity present in vivo (e.g., different cell types and interstitial matrix). Therefore in the present study it was our aim to develop a new model, based on the biaxial stretching of precision-cut lung slices (PCLS). Single PCLS were mounted on a thin and flexible carrier membrane of polydimethylsiloxane (PDMS) in a bioreactor, and the membrane was stretched by applying varying pressures under static conditions. Distension of the membrane-PCLS construct was modeled via finite element simulation. According to this analysis, lung tissue was stretched by up to 38% in the latitudinal and by up to 44% in the longitudinal direction, resulting in alveolar distension similar to what has been described in intact lungs. Stretch for 5 min led to increased cellular calcium levels. Lung slices were stretched dynamically with a frequency of 15/min for 4 h without causing cell injury {3-[4,5-dimethylthiazol-2-yl]-2,5-diphenyltetrazolium bromide (MTT) test; live/dead straining}. These findings suggest that stretching of PCLS on PDMS-membranes may represent a useful model to investigate lung stretch in intact lung tissue in vitro for several hours.

Nonlinear elasticity of biological tissues with statistical fibre orientation

The elastic strain energy potential for nonlinear fibre-reinforced materials is customarily obtained by superposition of the potentials of the matrix and of each family of fibres. Composites with statistically oriented fibres, such as biological tissues, can be seen as being reinforced by a continuous infinity of fibre families, the orientation of which can be represented by means of a probability density function defined on the unit sphere (i.e. the solid angle). In this case, the superposition procedure gives rise to an integral form of the elastic potential such that the deformation features in the integral, which therefore cannot be calculated a priori. As a consequence, an analytical use of this potential is impossible. In this paper, we implemented this integral form of the elastic potential into a numerical procedure that evaluates the potential, the stress and the elasticity tensor at each deformation step. The numerical integration over the unit sphere is performed by means of the method of spherical designs, in which the result of the integral is approximated by a suitable sum over a discrete subset of the unit sphere. As an example of application, we modelled the collagen fibre distribution in articular cartilage, and used it in simulating displacement-controlled tests: the unconfined compression of a cylindrical sample and the contact problem in the hip joint.

Mitral Valve Finite Element Modeling: Implications of Tissues’ Nonlinear Response and Annular Motion

Finite element modeling represents an established method for the comprehension of the mitral function and for the simulation of interesting clinical scenarios. However, current models still do not include all the key aspects of the real system. We implemented a new structural finite element model that considers (i) an accurate morphological description of the valve, (ii) a description of the tissues’ mechanical properties that accounts for anisotropy and nonlinearity, and (iii) dynamic boundary conditions that mimic annulus and papillary muscles’ contraction. The influence of such contraction on valve biomechanics was assessed by comparing the computed results with the ones obtained through an auxiliary model with fixed annulus and papillary muscles. At the systolic peak, the leaflets’ maximum principal stress contour showed peak values in the anterior leaflet at the strut chordae insertion zone (300 kPa) and near the annulus (200–250 kPa), while much lower values were detected in the posterior leaflet. Both leaflets underwent larger tensile strains in the longitudinal direction, while in the circumferential one the anterior leaflet experienced nominal tensile strains up to 18% and the posterior one experienced compressive strains up to 23% associated with the folding of commissures and paracommissures, consistently with tissue redundancy. The force exerted by papillary muscles at the systolic peak was equal to 4.11 N, mainly borne by marginal chordae (76% of the force). Local reaction forces up to 45 mN were calculated on the annulus, leading to tensions of 89 N/m and 54 N/m for its anterior and posterior tracts, respectively. The comparison with the results of the auxiliary model showed that annular contraction mainly affects the leaflets’ circumferential strains. When it was suppressed, no more compressive strains could be observed and peak strain values were located in the belly of the anterior leaflet. Computational results agree to a great extent with experimental data from literature. They provided insight into some of the features characterizing normal mitral function, such as annular contraction and leaflets’ tissue anisotropy and nonlinearity. Some of the computed results may be useful in the design of surgical devices and techniques. In particular, forces applied on the annulus by the surrounding tissues could be considered as an indication for annular prostheses design.

The Effect of Evolving Damage on the Finite Strain Response of Inelastic and Viscoelastic Composites

A finite strain micromechanical model is generalized in order to incorporate the effect of evolving damage in the metallic and polymeric phases of unidirectional composites. As a result, it is possible to predict the response of composites with ductile and brittle phases undergoing large coupled inelastic-damage and viscoelastic-damage deformations, respectively. For inelastic composites, both finite strain elastoplastic (time-independent) and viscoplastic (time-dependent) behaviors are considered. The ductile phase exhibits initially a hyperelastic behavior which is followed by an inelastic one, and its analysis is based on the multiplicative split of its deformation gradient into elastic and inelastic parts. The embedded damage mechanisms and their evolutions are based on Gurson’s (which is suitable for the modeling of porous materials) and Lemaitre’s finite strain models. Similarly, the polymeric phase exhibits large viscoelastic deformations in which the damage evolves according to a suitable evolution law that depends on the amount of accumulated deformation. Evolving damage in hyperelastic materials can be analyzed as a special case by neglecting the viscous effects. The micromechanical analysis is based on the homogenization technique for periodic multiphase materials, which establishes the strong form of the Lagrangian equilibrium equations. These equations are implemented together with the interfacial and periodic boundary conditions, in conjunction with the current tangent tensor of the phase. As a result, the instantaneous strain concentration tensor that relates the local deformation gradient of the phase to the externally applied deformation gradient is established. This provides also the instantaneous effective stiffness tangent tensor of the composite as well as its current response. Results are given that exhibit the effect of damage on the initial yield surfaces, response and possible failure of the composite.

Functional morphology of the sound-generating labia in the syrinx of two songbird species

In songbirds, two sound sources inside the syrinx are used to produce the primary sound. Laterally positioned labia are passively set into vibration, thus interrupting a passing air stream. Together with subsyringeal pressure, the size and tension of the labia determine the spectral characteristics of the primary sound. Very little is known about how the histological composition and morphology of the labia affect their function as sound generators. Here we related the size and microstructure of the labia to their acoustic function in two songbird species with different acoustic characteristics, the white-crowned sparrow and zebra finch. Histological serial sections of the syrinx and different staining techniques were used to identify collagen, elastin and hyaluronan as extracellular matrix components. The distribution and orientation of elastic fibers indicated that the labia in white-crowned sparrows are multi-layered structures, whereas they are more uniformly structured in the zebra finch. Collagen and hyaluronan were evenly distributed in both species. A multi-layered composition could give rise to complex viscoelastic properties of each sound source. We also measured labia size. Variability was found along the dorso-ventral axis in both species. Lateral asymmetry was identified in some individuals but not consistently at the species level. Different size between the left and right sound sources could provide a morphological basis for the acoustic specialization of each sound generator, but only in some individuals. The inconsistency of its presence requires the investigation of alternative explanations, e.g. differences in viscoelastic properties of the labia of the left and right syrinx. Furthermore, we identified attachments of syringeal muscles to the labia as well as to bronchial half rings and suggest a mechanism for their biomechanical function.

The effect of material model formulation in the stress analysis of abdominal aortic aneurysms

A reliable estimation of wall stress in Abdominal Aortic Aneurysms (AAAs), requires performing an accurate three-dimensional reconstruction of the medical image-based native geometry and modeling an appropriate constitutive law for the aneurysmal tissue material characterization. A recent study on the biaxial mechanical behavior of human AAA tissue specimens demonstrates that aneurysmal tissue behaves mechanically anisotropic. Results shown in this communication show that the peak wall stress is highly sensitive to the anisotropic model used for the stress analysis. In addition, the present investigation indicates that structural parameters (e.g., collagen fiber orientation) should be determined independently and not by means of non-linear fitting to stress-strain test data. Fiber orientation identified in this manner could lead to overestimated peak wall stresses.

Scleral biomechanics in the aging monkey eye

PURPOSE

To investigate the age-related differences in the inhomogeneous, anisotropic, nonlinear biomechanical properties of posterior sclera from old (22.9 +/- 5.3 years) and young (1.5 +/- 0.7 years) rhesus monkeys.

METHODS

The posterior scleral shell of each eye was mounted on a custom-built pressurization apparatus, then intraocular pressure (IOP) was elevated from 5 to 45 mm Hg while the 3D displacements of the scleral surface were measured with speckle interferometry. Each scleral shell's geometry was digitally reconstructed from data generated by a 3-D digitizer (topography) and 20-MHz ultrasound (thickness). An inverse finite element (FE) method incorporating a fiber-reinforced constitutive model was used to extract a unique set of biomechanical properties for each eye. Displacements, thickness, stress, strain, tangent modulus, structural stiffness, and preferred collagen fiber orientation were mapped for each posterior sclera.

RESULTS

The model yielded 3-D deformations of posterior sclera that matched well with those observed experimentally. The posterior sclera exhibited inhomogeneous, anisotropic, nonlinear mechanical behavior. The sclera was significantly thinner (P = 0.038) and tangent modulus and structural stiffness were significantly higher in old monkeys (P < 0.0001). On average, scleral collagen fibers were circumferentially oriented around the optic nerve head (ONH). No difference was found in the preferred collagen fiber orientation and fiber concentration factor between age groups.

CONCLUSIONS

Posterior sclera of old monkeys is significantly stiffer than that of young monkeys and is therefore subject to higher stresses but lower strains at all levels of IOP. Age-related stiffening of the sclera may significantly influence ONH biomechanics and potentially contribute to age-related susceptibility to glaucomatous vision loss.

A computational remodeling approach to predict the physiological architecture of the collagen fibril network in corneo-scleral shells

Organized collagen fibrils form complex networks that introduce strong anisotropic and highly nonlinear attributes into the constitutive response of human eye tissues. Physiological adaptation of the collagen network and the mechanical condition within biological tissues are complex and mutually dependent phenomena. In this contribution, a computational model is presented to investigate the interaction between the collagen fibril architecture and mechanical loading conditions in the corneo-scleral shell. The biomechanical properties of eye tissues are derived from the single crimped fibril at the micro-scale via the collagen network of distributed fibrils at the meso-scale to the incompressible and anisotropic soft tissue at the macro-scale. Biomechanically induced remodeling of the collagen network is captured on the meso-scale by allowing for a continuous re-orientation of preferred fibril orientations and a continuous adaptation of the fibril dispersion. The presented approach is applied to a numerical human eye model considering the cornea and sclera. The predicted fibril morphology correlates well with experimental observations from X-ray scattering data.

A Structural Multi-Mechanism Damage Model for Cerebral Arterial Tissue

Early stage cerebral aneurysms are characterized by the disruption of the internal elastic lamina. The cause of this breakdown is still not understood, but it has been conjectured to be due to fatigue failure and/or by a breakdown in homeostatic mechanisms in the wall arising from some aspect of the local hemodynamics and wall tension. We propose to model this disruption using a structural damage model. It is built on a previously introduced nonlinear, inelastic multi-mechanism model for cerebral arteries (2005, “An Inelastic Multi-Mechanism Constitutive Equation for Cerebral Arterial Tissue,” Biomech. Model. Mechanobiol., 4(4), pp. 235–248), as well as a recent generalization to include the wall anisotropy (2009, “A Structural Multi-Mechanism Constitutive Equation for Cerebral Arterial Tissue,” Int. J. Solids Struct., 46(14–15), pp. 2920–2928). The current model includes subfailure damage of the elastin, represented by changes in the tissue mechanical properties and unloaded reference length. A structural model is used to characterize the gradual degradation, failure of elastin, and recruitment of anisotropic collagen fibers. The collagen fibers are arranged in two helically oriented families with dispersion in their orientation. Available inelastic experimental data for cerebral arteries are used to evaluate the constitutive model. It is then implemented in a commercial finite element analysis package and validated using analytical solutions with representative values for cerebral arterial tissue.

Continuum model of epithelial morphogenesis during Caenorhabditis elegans embryonic elongation

The purpose of this work is to provide a biomechanical model to investigate the interplay between cellular structures and the mechanical force distribution during the elongation process of Caenorhabditis elegans embryos. Epithelial morphogenesis drives the elongation process of an ovoid embryo to become a worm-shaped embryo about four times longer and three times thinner. The overall anatomy of the embryo is modelled in the continuum mechanics framework from the structural organization of the subcellular filaments within epithelial cells. The constitutive relationships consider embryonic cells as homogeneous materials with an active behaviour, determined by the non-muscle myosin II molecular motor, and a passive viscoelastic response, related to the directional properties of the filament network inside cells. The axisymmetric elastic solution at equilibrium is derived by means of the incompressibility conditions, the continuity conditions for the overall embryo deformation and the balance principles for the embryonic cells. A particular analytical solution is proposed from a simplified geometry, demonstrating the mechanical role of the microtubule network within epithelial cells in redistributing the stress from a differential contraction of circumferentially oriented actin filaments. The theoretical predictions of the biomechanical model are discussed within the biological scenario proposed through genetic analysis and pharmacological experiments.

A microsphere-based remodelling formulation for anisotropic biological tissues

Biological tissues possess the ability to adapt according to the respective local loading conditions, which results in growth and remodelling phenomena. The main goal of this work is the development of a new remodelling approach that, on the one hand, reflects the alignment of fibrous soft biological tissue with respect to representative loading directions. On the other hand, the continuum approach proposed is based on a sound micro-mechanically motivated formulation. To be specific, use of a worm-like chain model is made to describe the behaviour of long-chain molecules as present in, for instance, collageneous tissues. The extension of such a one-dimensional constitutive equation to the three-dimensional macroscopic level is performed by means of a microsphere formulation. Inherent with the algorithmic treatment of this type of modelling approach, a finite number of unit vectors is considered for the numerical integration over the domain of the unit sphere. As a key aspect of this contribution, remodelling is incorporated by setting up evolution equations for the referential orientations of these integration directions. Accordingly, the unit vectors considered now allow interpretation as internal variables, which characterize the material's anisotropic properties. Several numerical studies underline the applicability of the model that, moreover, nicely fits into iterative finite element formulations so that general boundary value problems can be solved.

Scaffold design and manufacturing: from concept to clinic

Since Robert Langer and colleagues pioneered the concept of reconstructing tissue using cells transplanted on synthetic polymer matrices in the early 1990s, research in the field of tissue engineering and regenerative medicine has exploded. This is especially true in the development of new materials and structures that serve as scaffolds for tissue reconstruction. The basic tenet of the last two decades holds scaffolds as degradable materials providing temporary function while enhancing tissue regeneration through the delivery of biologics. Although a number of new scaffolding materials and structures have been developed in research laboratories, the application of such materials practice even has been extremely limited. This paper argues that better integration of all these factors is needed to bring scaffolds from "concept to clinic". It reviews current work in all these areas and suggests where future work and funding is needed.

A Structure-Based Model of Arterial Remodeling in Response to Sustained Hypertension

A novel structure-based mathematical model of arterial remodeling in response to a sustained increase in pressure is proposed. The model includes two major aspects of remodeling in a healthy matured vessel. First, the deviation of the wall stress and flow-induced shear stress from their normal physiological values drives the changes in the arterial geometry. Second, the new mass that is produced during remodeling results from an increase in the mass of smooth muscle cells and collagen fibers. The model additionally accounts for the effect of the average pulsatile strain on the recruitment of collagen fibers in load bearing. The model was used to simulate remodeling of a human thoracic aorta, and the results are in good agreement with previously published model predictions and experimental data. The model predicts that the total arterial volume rapidly increases during the early stages of remodeling and remains virtually constant thereafter, despite the continuing stress-driven geometrical remodeling. Moreover, the effects of a perfect or incomplete restoration of the arterial compliance on the remodeling outputs were analyzed. For instance, the model predicts that the pattern of the time course of the opening angle depends on the extent to which the average pulsatile strain is restored at the end of the remodeling process. Future experimental studies on the time course of compliance, opening angle, and mass fractions of collagen, elastin, and smooth muscle cells can validate and improve the introduced hypotheses of the model.

A Phenomenological Approach Toward Patient-Specific Computational Modeling of Articular Cartilage Including Collagen Fiber Tracking

To model the cartilage morphology and the material response, a phenomenological and patient-specific simulation approach incorporating the collagen fiber fabric is proposed. Cartilage tissue response is nearly isochoric and time-dependent under physiological pressure levels. Hence, a viscoelastic constitutive model capable of reproducing finite strains is employed, while the time-dependent deformation change is purely isochoric. The model incorporates seven material parameters, which all have a physical interpretation. To calibrate the model and facilitate further analysis, five human cartilage specimens underwent a number of tests. A series of magnetic resonance imaging (MRI) sequences is taken, next the cartilage surface is imaged, then mechanical indentation tests are completed at 2–7 different locations per sample, resulting in force/displacement data over time, and finally, the underlying bone surface is imaged. Imaging and mechanical testing are performed with a custom-built robotics-based testing device. Stereo reconstruction of the cartilage and subchondral bone surface is employed, which, together with the proposed constitutive model, led to specimen-specific finite element simulations of the mechanical indentation tests. The force-time response of 23 such indentation experiment simulations is optimized to estimate the mean material parameters and corresponding standard deviations. The model is capable of reproducing the deformation behavior of human articular cartilage in the physiological loading domain, as demonstrated by the good agreement between the experiment and numerical results (R2=0.95±0.03, mean±standard deviation of force-time response for 23 indentation tests). To address validation, a sevenfold cross-validation experiment is performed on the 21 experiments representing healthy cartilage. To quantify the predictive error, the mean of the absolute force differences and Pearson’s correlation coefficient are both calculated. Deviations in the mean absolute difference, normalized by the peak force, range from 4% to 90%, with 40±25%(M±SD). The correlation coefficients across all predictions have a minimum of 0.939, and a maximum of 0.993 with 0.975±0.013(M±SD), which demonstrates an excellent match of the decay characteristics. A novel feature of the proposed method is 3D sample-specific numerical tracking of the fiber fabric deformation under general loading. This feature is demonstrated by comparing the estimated fiber fabric deformation with recently published experimental data determined by diffusion tensor MRI. The proposed approach is efficient enough to enable large-scale 3D contact simulations of knee joint loading in simulations with accurate joint geometries.