Toward a biomechanical tool to evaluate rupture potential of abdominal aortic aneurysm: identification of a finite strain constitutive model and evaluation of its applicability

Knowledge of the wall stresses in an abdominal aortic aneurysm (AAA) may be helpful in evaluating the need for surgical intervention to avoid rupture. This must be preceded by the development of a more suitable finite strain constitutive model for AAA, as none currently exists. Additionally, reliable stress analysis of in vivo AAA for the purposes of clinical diagnostics requires patient-specific values of the material parameters, which are difficult to determine noninvasively. The purpose of this work, therefore, was three-fold: (1) to develop a finite strain constitutive model for AAA; (2) to estimate the variation of model parameters within a sample population; and (3) to evaluate the sensitivity of computed stress distribution in AAA due to this biologic variation. We propose here a two parameter, hyperelastic, isotropic, incompressible material model and utilize experimental data from 69 freshly excised AAA specimens to both develop the functional form of the model and estimate its material parameters. Parametric analyses were performed via repeated finite element computations to determine the effect of varying each of the two model parameters on the stress distribution in a three-dimensional AAA model. The agreement between experimental data and the proposed functional form of the constitutive law was very good (R2 > 0.9). Our finite element simulations showed that the computed AAA wall stresses changed by only 4% or less when both the parameters were varied within the 95% confidence intervals for the patient population studied. This observation indicates that in lieu of the patient-specific material parameters, which are difficult to determine the use of population mean values is sufficiently accurate for the model to be reasonably employed in a clinical setting. We believe that this is an important advancement toward the development of a computational tool for the estimation of rupture potential for individual AAA, for which there is great clinical need.     

Free body analysis,beam mechanics,and finite element modeling of the mandible of Alligator mississippiensis

The mechanical behavior of mammalian mandibles is well‐studied, but a comprehensive biomechanical analysis (incorporating detailed muscle architecture, accurate material properties, and three‐dimensional mechanical behavior) of an extant archosaur mandible has never been carried out. This makes it unclear how closely models of extant and extinct archosaur mandibles reflect reality and prevents comparisons of structure–function relationships in mammalian and archosaur mandibles. We tested hypotheses regarding the mechanical behavior of the mandible of Alligator mississippiensis by analyzing reaction forces and bending, shear, and torsional stress regimes in six models of varying complexity. Models included free body analysis using basic lever arm mechanics, 2D and 3D beam models, and three high‐resolution finite element models of the Alligator mandible, incorporating, respectively, isotropic bone without sutures, anisotropic bone with sutures, and anisotropic bone with sutures and contact between the mandible and the pterygoid flange. Compared with the beam models, the Alligator finite element models exhibited less spatial variability in dorsoventral bending and sagittal shear stress, as well as lower peak values for these stresses, suggesting that Alligator mandibular morphology is in part designed to reduce these stresses during biting. However, the Alligator models exhibited greater variability in the distribution of mediolateral and torsional stresses than the beam models. Incorporating anisotropic bone material properties and sutures into the model reduced dorsoventral and torsional stresses within the mandible, but led to elevated mediolateral stresses. These mediolateral stresses were mitigated by the addition of a pterygoid‐mandibular contact, suggesting important contributions from, and trade‐offs between, material properties and external constraints in Alligator mandible design. Our results suggest that beam modeling does not accurately represent the mechanical behavior of the Alligator mandible, including important performance metrics such as magnitude and orientation of reaction forces, and mediolateral bending and torsional stress distributions. J.Morphol. 2011. © 2011 Wiley‐Liss, Inc.     

A modified Holzapfel-Ogden law for a residually stressed finite strain model of the human left ventricle in diastole

In this work, we introduce a modified Holzapfel-Ogden hyperelastic constitutive model for ventricular myocardium that accounts for residual stresses, and we investigate the effects of residual stresses in diastole using a magnetic resonance imaging–derived model of the human left ventricle (LV). We adopt an invariant-based constitutive modelling approach and treat the left ventricular myocardium as a non-homogeneous, fibre-reinforced, incompressible material. Because in vivo images provide the configuration of the LV in a loaded state even in diastole, an inverse analysis is used to determine the corresponding unloaded reference configuration. The residual stress in this unloaded state is estimated by two different methods. One is based on three-dimensional strain measurements in a local region of the canine LV, and the other uses the opening angle method for a cylindrical tube. We find that including residual stress in the model changes the stress distributions across the myocardium and that whereas both methods yield qualitatively similar changes, there are quantitative differences between the two approaches. Although the effects of residual stresses are relatively small in diastole, the model can be extended to explore the full impact of residual stress on LV mechanical behaviour for the whole cardiac cycle as more experimental data become available. In addition, although not considered here, residual stresses may also play a larger role in models that account for tissue growth and remodelling.     

Development of an in vivo method for determining material properties of passive myocardium

Calculation of mechanical stresses and strains in the left ventricular (LV) myocardium by the finite element (FE) method relies on adequate knowledge of the material properties of myocardial tissue. In this paper, we present a model-based estimation procedure to characterize the stress-strain relationship in passive LV myocardium. A 3D FE model of the LV myocardium was used, which included morphological fiber and sheet structure and a nonlinear orthotropic constitutive law with different stiffness in the fiber, sheet, and sheet-normal directions. The estimation method was based on measured wall strains. We analyzed the method's ability to estimate the material parameters by generating a set of synthetic strain data by simulating the LV inflation phase with known material parameters. In this way we were able to verify the correctness of the solution and to analyze the effects of measurement and model error on the solution accuracy and stability. A sensitivity analysis was performed to investigate the observability of the material parameters and to determine which parameters to estimate. The results showed a high degree of coupling between the parameters governing the stiffness in each direction. Thus, only one parameter in each of the three directions was estimated. For the tested magnitudes of added noise and introduced model errors, the resulting estimated stress-strain characteristics in the fiber and sheet directions converged with good accuracy to the known relationship. The sheet-normal stress-strain relationship had a higher degree of uncertainty as more noise was added and model error was introduced.     

On the sensitivity of wall stresses in diseased arteries to variable material properties

Accurate estimates of stress in an atherosclerotic lesion require knowledge of the material properties of its components (e.g., normal wall, fibrous plaque, calcified regions, lipid pools) that can only be approximated. This leads to considerable uncertainty in these computational predictions. A study was conducted to test the sensitivity of predicted levels of stress and strain to the parameter values of plaque used in finite element analysis. Results show that the stresses within the arterial wall, fibrous plaque, calcified plaque, and lipid pool have low sensitivities for variation in the elastic modulus. Even a +/- 50% variation in elastic modulus leads to less than a 10% change in stress at the site of rupture. Sensitivity to variations in elastic modulus is comparable between isotropic nonlinear, isotropic nonlinear with residual strains, and transversely isotropic linear models. Therefore, stress analysis may be used with confidence that uncertainty in the material properties generates relatively small errors in the prediction of wall stresses. Either isotropic nonlinear or anisotropic linear models provide useful estimates, however the predictions in regions of stress concentration (e.g., the site of rupture) are somewhat more sensitive to the specific model used, increasing by up to 30% from the isotropic nonlinear to orthotropic model in the present example. Changes resulting from the introduction of residual stresses are much smaller.     

Mechanical properties and microstructure of intraluminal thrombus from abdominal aortic aneurysm.

Accurate estimation of the wall stress distribution in an abdominal aortic aneurysm (AAA) may prove clinically useful by predicting when a particular aneurysm will rupture. Appropriate constitutive models for both the wall and the intraluminal thrombus (ILT) found in most AAA are necessary for this task. The purpose of this work was to determine the mechanical properties of ILT within AAA and to derive a more suitable constitutive model for this material. Uniaxial tensile testing was carried out on 50 specimens, including 14 longitudinally oriented and 14 circumferentially oriented specimens from the luminal region of the ILT, and 11 longitudinally oriented and 11 circumferentially oriented specimens from the medial region. A two-parameter, large-strain, hyperelastic constitutive model was developed and used to fit the uniaxial tensile testing data for determination of the material parameters. Maximum stiffness and strength were also determined from the data for each specimen. Scanning electron microscopy (SEM) was conducted to study the regional microstructural difference. Our results indicate that the microstructure of ILT differs between the luminal, medial, and abluminal regions, with the luminal region stronger and stiffer than the medial region. In all cases, the constitutive model fit the experimental data very well (R2>0.98). No significant difference was found for either of the two material parameters between longitudinal and circumferential directions, but a significant difference in material parameters, stiffness, and strength between the laminal and medial regions was determined (p<0.01). Therefore, our results suggest that ILT is an inhomogeneous and possibly isotropic material. The two-parameter, hyperelastic, isotropic, incompressible material model derived here for ILT can be easily incorporated into finite element models for simulation of wall stress distribution in AAA.     

Epicardial suction: a new approach to mechanical testing of the passive ventricular wall

The lack of an appropriate three-dimensional constitutive relation for stress in passive ventricular myocardium currently limits the utility of existing mathematical models for experimental and clinical applications. Previous experiments used to estimate parameters in three-dimensional constitutive relations, such as biaxial testing of excised myocardial sheets or passive inflation of the isolated arrested heart, have not included significant transverse shear deformation or in-plane compression. Therefore, a new approach has been developed in which suction is applied locally to the ventricular epicardium to introduce a complex deformation in the region of interest, with transmural variations in the magnitude and sign of nearly all six strain components. The resulting deformation is measured throughout the region of interest using magnetic resonance tagging. A nonlinear, three-dimensional, finite element model is used to predict these measurements at several suction pressures. Parameters defining the material properties of this model are optimized by comparing the measured and predicted myocardial deformations. We used this technique to estimate material parameters of the intact passive canine left ventricular free wall using an exponential, transversely isotropic constitutive relation. We tested two possible models of the heart wall: first, that it was homogeneous myocardium, and second, that the myocardium was covered with a thin epicardium with different material properties. For both models, in agreement with previous studies, we found that myocardium was nonlinear and anisotropic with greater stiffness in the fiber direction. We obtained closer agreement to previously published strain data from passive filling when the ventricular wall was modeled as having a separate, isotropic epicardium. These results suggest that epicardium may play a significant role in passive ventricular mechanics.     

The effects of plaque morphology and material properties on peak cap stress in human coronary arteries

Heart attacks are often caused by rupture of caps of atherosclerotic plaques in coronary arteries. Cap rupture occurs when cap stress exceeds cap strength. We investigated the effects of plaque morphology and material properties on cap stress. Histological data from 77 coronary lesions were obtained and segmented. In these patient-specific cross sections, peak cap stresses were computed by using finite element analyses. The finite element analyses were 2D, assumed isotropic material behavior, and ignored residual stresses. To represent the wide spread in material properties, we applied soft and stiff material models for the intima. Measures of geometric plaque features for all lesions were determined and their relations to peak cap stress were examined using regression analyses. Patient-specific geometrical plaque features greatly influence peak cap stresses. Especially, local irregularities in lumen and necrotic core shape as well as a thin intima layer near the shoulder of the plaque induce local stress maxima. For stiff models, cap stress increased with decreasing cap thickness and increasing lumen radius (R = 0.79). For soft models, this relationship changed: increasing lumen radius and increasing lumen curvature were associated with increased cap stress (R = 0.66). The results of this study imply that not only accurate assessment of plaque geometry, but also of intima properties is essential for cap stress analyses in atherosclerotic plaques in human coronary arteries.     

Assessment of the aortic stress-strain relation in uniaxial tension

The passive elastic characteristics of the abdominal aorta were investigated in two experimental animal models, aiming at assessing the stress-strain relation of the aortic wall. Twenty porcine and 15 rabbit healthy abdominal aortas were subjected to uniaxial stress-strain analysis, performed on a tensile-testing device, while immersed in a physiologic saline bath at body temperature. Measured parameters included original length, width and thickness, as well as axial force and extension. Based on these data, Kirchhoff stress and Green-St.Venant strain were computed and one-dimensional constitutive equations were defined, comprising of a power function and two exponential ones, in turn, for the low, physiologic and high-stress regions. The stress-strain curves were plotted as elastic modulus versus stress, displaying nonlinear part I and linear parts II and III. These were regressed, yielding parameters k, q (part I), a, b (part II) and c, d (part III). A detailed comparison of these constitutive parameters was undertaken between the two species, demonstrating variations in d (p<0.05). No statistical differences were found in parameters a, b, c, k and q, implying that the two aortas were equally stiff under low and physiologic stresses, whereas the porcine aorta was stiffer at higher stresses. In conclusion, a bi-exponential in addition to a power law was established, relating stress and strain in the aorta, which is advantageous in comparison with previous constitutive equations. Under passive conditions, the nonlinear nature of this constitutive law may account for the low, part I, physiologic, part II, and high-stress, part III of the stress-strain relationship, supporting the concept of the aortic wall as a biphasic material.     

A transversely isotropic constitutive model of excised guinea pig spinal cord white matter

Narrowing of the spinal canal generates an amalgamation of stresses within the spinal cord parenchyma. The tissue’s stress state cannot be quantified experimentally; it must be described using computational methods, such as finite element analysis. The objective of this research was to propose a compressible, transversely isotropic constitutive model, an augmentation of the isotropic Mooney–Rivlin hyperelastic strain energy function, to describe the guinea pig spinal cord white matter. Model parameters were derived from a combination of inverse finite element analysis on transverse compression experiments and least squared error analysis applied to quasi-static longitudinal tensile tests. A comparison of the residual errors between the predicted response and the experimental measurements indicated that the transversely isotropic constitutive law that incorporates an offset stretch reduced the error by a factor of four when compared to other commonly used models.     

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

Background and aimsPartial 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.MethodsPartial 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.ResultsThe 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.ConclusionThe 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.     

Sensitivity of stress and strain calculations to passive material parameters in cardiac mechanical models using unloaded geometries

Cardiac stress (load) and strain (stretch) are widely studied indicators of cardiac function and outcome, but are difficult or impossible to directly measure in relation to the cardiac microstructure. An alternative approach is to estimate these states using computer methods and image-based measurements, but this still requires knowledge of the tissue material properties and the unloaded state, both of which are difficult to determine. In this work, we tested the sensitivity of these two interdependent unknowns (reference geometry and material parameters) on stress and strain calculations in cardiac tissue. Our study used a finite element model of the human ventricle, with a hyperelastic passive material model, and was driven by a cell model mediated active contraction. We evaluated 21 different published parameter sets for the five parameters of the passive material model, and for each set we optimised the corresponding unloaded geometry and contractility parameter to model a single pressure-volume loop. The resulting mechanics were compared, and calculated systolic stresses were largely insensitive to the chosen parameter set when an unloading algorithm was used. Meanwhile, material strain calculations varied substantially depending on the choice of material parameters. These results indicate that determining the correct material and unloaded configuration may be highly important to understand strain driven processes, but less so for calculating stress estimates.     

A model for the nonlinear elastic response of large arteries

The nonlinear elastic response of large arteries subjected to finite deformations due to action of biaxial principal stresses, is described by simple constitutive equations. Generalized measures of strain and stress are introduced to account for material nonlinearity. This also ensures the existence of a strain energy density function. The orthotropic elastic response is described via quasi-linear relations between strains and stresses. One nonlinear parameter which defines the measures of strain and stress, and three elastic moduli are assumed to be constants. The lateral strain parameters (equivalent to Poisson's ratios in infinitesimal deformations) are deformation dependent. This dependence is defined by empirical relations developed via the incompressibility condition, and by the introduction of a fifth material parameter. The resulting constitutive model compares well with biaxial experimental data of canine carotid arteries.     

Myocardial material parameter estimation: a comparison of invariant based orthotropic constitutive equations

This 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.

A number of orthotropic laws based on Green strain components and one microstructurally based law have previously been investigated to fit experimental measurements of stress-strain behaviour. Here we extend this investigation to include several recently proposed functional forms, i.e. invariant based orthotropic and transversely isotropic constitutive relations.

These laws were compared on the basis of (i) ‘goodness of fit’: how well they fit a set of six shear deformation tests, (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 laws.

It was found that a specific form of the polyconvex type as well as the exponential Fung-type law from the previous study were most suitable for modelling the orthotropic behaviour of myocardium under simple shear.     

The motion of the left ventricle—III

We use the concept of a layered wall, where each separate layer is to be homogeneous, isotropic, and incompressible, to derive stress-strain relations for the middle layer muscle ring at the transverse midsection of the left ventricle; a convenient method of formulation is that based on the elastic potential function. The hoop or circumferential stress in all three layers is found using dimensional and mechanical parameters derived earlier. The various parameters are expressed as Fourier series so that their behavior over a complete ventricular cycle is known analytically. The cases of simple elongation and what we termcurvilinear simple elongation are considered for the middle layer muscle ring strain, and the resulting stress-strain relations are derived. The results are compared with an incompressible rubber-like material known as a Mooney material.     

Mechanical stresses in abdominal aortic aneurysms: influence of diameter, asymmetry, and material anisotropy

Biomechanical studies suggest that one determinant of abdominal aortic aneurysm (AAA) rupture is related to the stress in the wall. In this regard, a reliable and accurate stress analysis of an in vivo AAA requires a suitable 3D constitutive model. To date, stress analysis conducted on AAA is mainly driven by isotropic tissue models. However, recent biaxial tensile tests performed on AAA tissue samples demonstrate the anisotropic nature of this tissue. The purpose of this work is to study the influence of geometry and material anisotropy on the magnitude and distribution of the peak wall stress in AAAs. Three-dimensional computer models of symmetric and asymmetric AAAs were generated in which the maximum diameter and length of the aneurysm were individually controlled. A five parameter exponential type structural strain-energy function was used to model the anisotropic behavior of the AAA tissue. The anisotropy is determined by the orientation of the collagen fibers (one parameter of the model). The results suggest that shorter aneurysms are more critical when asymmetries are present. They show a strong influence of the material anisotropy on the magnitude and distribution of the peak stress. Results confirm that the relative aneurysm length and the degree of aneurysmal asymmetry should be considered in a rupture risk decision criterion for AAAs.     

The nonlinear material properties of liver tissue determined from no-slip uniaxial compression experiments

The mechanical response of soft tissue is commonly characterized from unconfined uniaxial compression experiments on cylindrical samples. However, friction between the sample and the compression platens is inevitable and hard to quantify. One alternative is to adhere the sample to the platens, which leads to a known no-slip boundary condition, but the resulting nonuniform state of stress in the sample makes it difficult to determine its material parameters. This paper presents an approach to extract the nonlinear material properties of soft tissue (such as liver) directly from no-slip experiments using a set of computationally determined correction factors. We assume that liver tissue is an isotropic, incompressible hyperelastic material characterized by the exponential form of strain energy function. The proposed approach is applied to data from experiments on bovine liver tissue. Results show that the apparent material properties, i.e., those determined from no-slip experiments ignoring the no-slip conditions, can differ from the true material properties by as much as 50% for the exponential material model. The proposed correction approach allows one to determine the true material parameters directly from no-slip experiments and can be easily extended to other forms of hyperelastic material models.     

A new constitutive model for multi-layered collagenous tissues

Collagenous tissues such as the aneurysmal wall or the aorta are multi-layered structures with the mean fibre alignments distinguishing one layer from another. A constitutive representation of the multiple collagen layers is not yet developed, and hence the aim of the present study. The proposed model is based on the constitutive theory of finite elasticity and is characterized by an anisotropic strain-energy function which takes the material structure into account. The passive tissue behaviour is modelled and the related mechanical response is assumed to be dominated by elastin and collagen. While elastin is modelled by the neo-Hookean material the constitutive response of collagen is assumed to be transversely isotropic for each individual layer and based on an exponential function. The proposed constitutive function is polyconvex which ensures material stability. The model has five independent material parameters, each of which has a clear physical interpretation: the initial stiffnesses of the collagen fabric in the two principal directions, the shear modulus pertaining to the non-collagenous matrix material, a parameter describing the level of nonlinearity of the collagen fabric, and the angle between the principal directions of the collagen fabric and the reference coordinate system. An extension-inflation test of the adventitia of a human femoral artery is simulated by means of the finite element method and an error function is minimized by adjusting the material parameters yielding a good agreement between the model and the experimental data.     

A constitutive model for vascular tissue that integrates fibril, fiber and continuum levels with application to the isotropic and passive properties of the infrarenal aorta

A fundamental understanding of the mechanical properties of the extracellular matrix (ECM) is critically important to quantify the amount of macroscopic stress and/or strain transmitted to the cellular level of vascular tissue. Structural constitutive models integrate histological and mechanical information, and hence, allocate stress and strain to the different microstructural components of the vascular wall. The present work proposes a novel multi-scale structural constitutive model of passive vascular tissue, where collagen fibers are assembled by proteoglycan (PG) cross-linked collagen fibrils and reinforce an otherwise isotropic matrix material. Multiplicative kinematics account for the straightening and stretching of collagen fibrils, and an orientation density function captures the spatial organization of collagen fibers in the tissue. Mechanical and structural assumptions at the collagen fibril level define a piece-wise analytical stress-stretch response of collagen fibers, which in turn is integrated over the unit sphere to constitute the tissue's macroscopic mechanical properties. The proposed model displays the salient macroscopic features of vascular tissue, and employs the material and structural parameters of clear physical meaning. Likewise, the constitutive concept renders a highly efficient multi-scale structural approach that allows for the numerical analysis at the organ level. Model parameters were estimated from isotropic mean-population data of the normal and aneurysmatic aortic wall and used to predict in-vivo stress states of patient-specific vascular geometries, thought to demonstrate the robustness of the particular Finite Element (FE) implementation. The collagen fibril level of the multi-scale constitutive formulation provided an interface to integrate vascular wall biology and to account for collagen turnover.     

An approach for the stress analysis of transversely isotropic biphasic cartilage under impact load.

Stress analysis of contact models for isotropic articular cartilage under impacting loads shows high shear stresses at the interface with the subchondral bone and normal compressive stresses near the surface of the cartilage. These stress distributions are not consistent, with lesions observed on the cartilage surface of rabbit patellae from blunt impact, for example, to the patello-femoral joint. The purpose of the present study was to analyze, using the elastic capabilities of a finite element code, the stress distribution in more morphologically realistic transversely isotropic biphasic contact models of cartilage. The elastic properties of an incompressible material, equivalent to those of the transversely isotropic biphasic material at time zero, were derived algebraically using stress-strain relations. Results of the stress analysis showed the highest shear stresses on the surface of the solid skeleton of the cartilage and tensile stresses in the zone of contact. These results can help explain the mechanisms responsible for surface injuries observed during blunt insult experiments.