A computational model for collagen fibre remodelling in the arterial wall

As the interaction between tissue adaptation and the mechanical condition within tissues is complex, mathematical models are desired to study this interrelation. In this study, a mathematical model is presented to investigate the interplay between collagen architecture and mechanical loading conditions in the arterial wall. It is assumed that the collagen fibres align along preferred directions, situated in between the principal stretch directions. The predicted fibre directions represent symmetrically arranged helices and agree qualitatively with morphometric data from literature. At the luminal side of the arterial wall, the fibres are oriented more circumferentially than at the outer side. The discrete transition of the fibre orientation at the media-adventitia interface can be explained by accounting for the different reference configurations of both layers. The predicted pressure-radius relations resemble experimentally measured sigma-shaped curves. As there is a strong coupling between the collagen architecture and the mechanical loading condition within the tissue, we expect that the presented model for collagen remodelling is useful to gain further insight into the processes involved in vascular adaptation, such as growth and smooth muscle tone adaptation.     

Improved prediction of the collagen fiber architecture in the aortic heart valve

Living tissues show an adaptive response to mechanical loading by changing their internal structure and morphology. Understanding this response is essential for successful tissue engineering of load-bearing structures, such as the aortic valve. In this study, mechanically induced remodeling of the collagen architecture in the aortic valve was investigated. It was hypothesized that, in uniaxially loaded regions, the fibers aligned with the tensile principal stretch direction. For biaxial loading conditions, on the other hand, it was assumed that the collagen fibers aligned with directions situated between the principal stretch directions. This hypothesis has already been applied successfully to study collagen remodeling in arteries. The predicted fiber architecture represented a branching network and resembled the macroscopically visible collagen bundles in the native leaflet. In addition, the complex biaxial mechanical behavior of the native valve could be simulated qualitatively with the predicted fiber directions. The results of the present model might be used to gain further insight into the response of tissue engineered constructs during mechanical conditioning.     

Nonlinear anisotropic stress analysis of anatomically realistic cerebral aneurysms

BACKGROUND: Static deformation analysis and estimation of wall stress distribution of patient-specific cerebral aneurysms can provide useful insights into the disease process and rupture. METHOD OF APPROACH: The three-dimensional geometry of saccular cerebral aneurysms from 27 patients (18 unruptured and nine ruptured) was reconstructed based on computer tomography angiography images. The aneurysm wall tissue was modeled using a nonlinear, anisotropic, hyperelastic material model (Fung-type) which was incorporated in a user subroutine in ABAQUS. Effective material fiber orientations were assumed to align with principal surface curvatures. Static deformation of the aneurysm models were simulated assuming uniform wall thickness and internal pressure load of 100 mm Hg. RESULTS: The numerical analysis technique was validated by quantitative comparisons to results in the literature. For the patient-specific models, in-plane stresses in the aneurysm wall along both the stiff and weak fiber directions showed significant regional variations with the former being higher. The spatial maximum of stress ranged from as low as 0.30 MPa in a small aneurysm to as high as 1.06 MPa in a giant aneurysm. The patterns of distribution of stress, strain, and surface curvature were found to be similar. Sensitivity analyses showed that the computed stress is mesh independent and not very sensitive to reasonable perturbations in model parameters, and the curvature-based criteria for fiber orientations tend to minimize the total elastic strain energy in the aneurysms wall. Within this small study population, there were no statistically significant differences in the spatial means and maximums of stress and strain values between the ruptured and unruptured groups. However, the ratios between the stress components in the stiff and weak fiber directions were significantly higher in the ruptured group than those in the unruptured group. CONCLUSIONS: A methodology for nonlinear, anisotropic static deformation analysis of geometrically realistic aneurysms was developed, which can be used for a more accurate estimation of the stresses and strains than previous methods and to facilitate prospective studies on the role of stress in aneurysm rupture.     

Mechano-regulation of mesenchymal stem cell differentiation and collagen organisation during skeletal tissue repair

A number of mechano-regulation theories have been proposed that relate the differentiation pathway of mesenchymal stem cells (MSCs) to their local biomechanical environment. During spontaneous repair processes in skeletal tissues, the organisation of the extracellular matrix is a key determinant of its mechanical fitness. In this paper, we extend the mechano-regulation theory proposed by Prendergast et al. (J Biomech 30(6):539–548, 1997) to include the role of the mechanical environment on the collagen architecture in regenerating soft tissues. A large strain anisotropic poroelastic material model is used in a simulation of tissue differentiation in a fracture subject to cyclic bending (Cullinane et al. in J Orthop Res 20(3):579–586, 2002). The model predicts non-union with cartilage and fibrous tissue formation in the defect. Predicted collagen fibre angles, as determined by the principal decomposition of strain- and stress-type tensors, are similar to the architecture seen in native articular cartilage and neoarthroses induced by bending of mid-femoral defects in rats. Both stress and strain-based remodelling stimuli successfully predicted the general patterns of collagen fibre organisation observed in vivo. This provides further evidence that collagen organisation during tissue differentiation is determined by the mechanical environment. It is envisioned that such predictive models can play a key role in optimising MSC-based skeletal repair therapies where recapitulation of the normal tissue architecture is critical to successful repair.     

A remodelling metric for angular fibre distributions and its application to diseased carotid bifurcations

Many soft biological tissues contain collagen fibres, which act as major load bearing constituents. The orientation and the dispersion of these fibres influence the macroscopic mechanical properties of the tissue and are therefore of importance in several areas of research including constitutive model development, tissue engineering and mechanobiology. Qualitative comparisons between these fibre architectures can be made using vector plots of mean orientations and contour plots of fibre dispersion but quantitative comparison cannot be achieved using these methods. We propose a 'remodelling metric' between two angular fibre distributions, which represents the mean rotational effort required to transform one into the other. It is an adaptation of the earth mover's distance, a similarity measure between two histograms/signatures used in image analysis, which represents the minimal cost of transforming one distribution into the other by moving distribution mass around. In this paper, its utility is demonstrated by considering the change in fibre architecture during a period of plaque growth in finite element models of the carotid bifurcation. The fibre architecture is predicted using a strain-based remodelling algorithm. We investigate the remodelling metric's potential as a clinical indicator of plaque vulnerability by comparing results between symptomatic and asymptomatic carotid bifurcations. Fibre remodelling was found to occur at regions of plaque burden. As plaque thickness increased, so did the remodelling metric. A measure of the total predicted fibre remodelling during plaque growth, TRM, was found to be higher in the symptomatic group than in the asymptomatic group. Furthermore, a measure of the total fibre remodelling per plaque size, TRM/TPB, was found to be significantly higher in the symptomatic vessels. The remodelling metric may prove to be a useful tool in other soft tissues and engineered scaffolds where fibre adaptation is also present.     

IVUS-based computational modeling and planar biaxial artery material properties for human coronary plaque vulnerability assessment

Image-based computational modeling has been introduced for vulnerable atherosclerotic plaques to identify critical mechanical conditions which may be used for better plaque assessment and rupture predictions. In vivo patient-specific coronary plaque models are lagging due to limitations on non-invasive image resolution, flow data, and vessel material properties. A framework is proposed to combine intravascular ultrasound (IVUS) imaging, biaxial mechanical testing and computational modeling with fluid-structure interactions and anisotropic material properties to acquire better and more complete plaque data and make more accurate plaque vulnerability assessment and predictions. Impact of pre-shrink-stretch process, vessel curvature and high blood pressure on stress, strain, flow velocity and flow maximum principal shear stress was investigated.     

An efficient two-stage approach for image-based FSI analysis of atherosclerotic arteries

Patient-specific biomechanical modeling of atherosclerotic arteries has the potential to aid clinicians in characterizing lesions and determining optimal treatment plans. To attain high levels of accuracy, recent models use medical imaging data to determine plaque component boundaries in three dimensions, and fluid–structure interaction is used to capture mechanical loading of the diseased vessel. As the plaque components and vessel wall are often highly complex in shape, constructing a suitable structured computational mesh is very challenging and can require a great deal of time. Models based on unstructured computational meshes require relatively less time to construct and are capable of accurately representing plaque components in three dimensions. These models unfortunately require additional computational resources and computing time for accurate and meaningful results. A two-stage modeling strategy based on unstructured computational meshes is proposed to achieve a reasonable balance between meshing difficulty and computational resource and time demand. In this method, a coarsegrained simulation of the full arterial domain is used to guide and constrain a fine-scale simulation of a smaller region of interest within the full domain. Results for a patient-specific carotid bifurcation model demonstrate that the two-stage approach can afford a large savings in both time for mesh generation and time and resources needed for computation. The effects of solid and fluid domain truncation were explored, and were shown to minimally affect accuracy of the stress fields predicted with the two-stage approach.     

Patient-Specific Artery Shrinkage and 3D Zero-Stress State in Multi-Component 3D FSI Models for Carotid Atherosclerotic Plaques Based on <i>In Vivo</i> MRI Data

Image-based computational models for atherosclerotic plaques have been developed to perform mechanical analysis to quantify critical flow and stress/strain conditions related to plaque rupture which often leads directly to heart attack or stroke. An important modeling issue is how to determine zero stress state from in vivo plaque geometries. This paper presents a method to quantify human carotid artery axial and inner circumferential shrinkages by using patient-specific ex vivo and in vivo MRI images. A shrink-stretch process based on patient-specific in vivo plaque morphology and shrinkage data was introduced to shrink the in vivo geometry first to find the zero-stress state (opening angle was ignored to reduce the complexity), and then stretch and pressurize to recover the in vivo plaque geometry with computed initial stress, strain, flow pressure and velocity conditions. Effects of the shrink-stretch process on plaque stress/strain distributions were demonstrated based on patient-specific data using 3D models with fluid-structure interactions (FSI). The average artery axial and inner circumferential shrinkages were 25% and 7.9%, respectively, based on a data set obtained from 10 patients. Maximum values of maximum principal stress and strain increased 349.8% and 249% respectively with 33% axial stretch. Influence of inner circumferential shrinkage (7.9%) was not very noticeable under 33% axial stretch, but became more noticeable under smaller axial stretch. Our results indicated that accurate knowledge of artery shrinkages and the shrink-stretch process will considerably improve the accuracy of computational predictions made based on results from those in vivo MRI-based FSI 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.     

Improving the Efficiency of Abdominal Aortic Aneurysm Wall Stress Computations

An abdominal aortic aneurysm is a pathological dilation of the abdominal aorta, which carries a high mortality rate if ruptured. The most commonly used surrogate marker of rupture risk is the maximal transverse diameter of the aneurysm. More recent studies suggest that wall stress from models of patient-specific aneurysm geometries extracted, for instance, from computed tomography images may be a more accurate predictor of rupture risk and an important factor in AAA size progression. However, quantification of wall stress is typically computationally intensive and time-consuming, mainly due to the nonlinear mechanical behavior of the abdominal aortic aneurysm walls. These difficulties have limited the potential of computational models in clinical practice. To facilitate computation of wall stresses, we propose to use a linear approach that ensures equilibrium of wall stresses in the aneurysms. This proposed linear model approach is easy to implement and eliminates the burden of nonlinear computations. To assess the accuracy of our proposed approach to compute wall stresses, results from idealized and patient-specific model simulations were compared to those obtained using conventional approaches and to those of a hypothetical, reference abdominal aortic aneurysm model. For the reference model, wall mechanical properties and the initial unloaded and unstressed configuration were assumed to be known, and the resulting wall stresses were used as reference for comparison. Our proposed linear approach accurately approximates wall stresses for varying model geometries and wall material properties. Our findings suggest that the proposed linear approach could be used as an effective, efficient, easy-to-use clinical tool to estimate patient-specific wall stresses.     

Stress distribution in cylindrical and conical implants under rotational micromovement with different boundary conditions and bone properties: 3-D FEA

Factors related to micromovements at bone-implant interface have been studied because they are considered adverse to osseointegration. Simplifications are commonly observed in these FEA evaluations. The aim of this study was to clarify the influence of FEA parameters (boundary conditions and bone properties) on the stress distribution in peri-implant bone tissue when micromovements are simulated in implants with different geometries. Three-dimensional models of an anterior section of the jaw with cylindrical or conical titanium implants (4.1 mm in width and 11 mm in length) were created. Micromovement (50, 150, or 250 μm) was applied to the implant. The FEA parameters studied were linear vs. non-linear analyses, isotropic vs. orthogonal anisotropic bone, friction coefficient (0.3) vs. frictionless bone-implant contact. Data from von Mises, shear, maximum, and minimum principal stresses in the peri-implant bone tissue were compared. Linear analyses presented a relevant increase of the stress values, regardless of the bone properties. Frictionless contact reduced the stress values in non-linear analysis. Isotropic bone presented lower stress than orthogonal anisotropic. Conical implants behave better, in regard to compressive stresses (minimum principal), than cylindrical ones, except for nonlinear analyses when micromovement of 150 and 250 μm were simulated. The stress values raised as the micromovement amplitude increased. Non-linear analysis, presence of frictional contact and orthogonal anisotropic bone, evaluated through maximum and minimum principal stress should be used as FEA parameters for implant-micromovement studies.     

The use of small angle light scattering in assessing strain induced collagen degradation in arterial tissue ex vivo

Collagen is the predominant load bearing component in many soft tissues including arterial tissue and is therefore critical in determining the mechanical integrity of such tissues. Degradation of collagen fibres is hypothesized to be a strain dependent process whereby the rate of degradation is affected by the magnitude of strain applied to the collagen fibres. The aim of this study is to investigate the ability of small angle light scattering (SALS) imaging to identify strain dependent degradation of collagen fibres in arterial tissue ex vivo, and determine whether a strain induced protection mechanism exists in arterial tissue as observed in pure collagen and other collagenous tissues. SALS was used in combination with histological and second harmonic generation (SHG) analysis to determine the collagen fibre architecture in arterial tissue subjected to strain directed degradation. SALS alignment analysis identified statistically significant differences in fibre alignment depending on the strain magnitude applied to the tissue. These results were also observed using histology and SHG. Our findings suggest a strain protection mechanism may exist for arterial collagen at intermediate strain magnitudes between 0% and 25%. These findings may have implications for the onset and progression of arterial disease where changes in the mechanical environment of arterial tissue may lead to changes in the collagen degradation rate.     

A microstructurally based continuum model of cartilage viscoelasticity and permeability incorporating measured statistical fiber orientations

The remarkable mechanical properties of cartilage derive from an interplay of isotropically distributed, densely packed and negatively charged proteoglycans; a highly anisotropic and inhomogeneously oriented fiber network of collagens; and an interstitial electrolytic fluid. We propose a new 3D finite strain constitutive model capable of simultaneously addressing both solid (reinforcement) and fluid (permeability) dependence of the tissue’s mechanical response on the patient-specific collagen fiber network. To represent fiber reinforcement, we integrate the strain energies of single collagen fibers—weighted by an orientation distribution function (ODF) defined over a unit sphere—over the distributed fiber orientations in 3D. We define the anisotropic intrinsic permeability of the tissue with a structure tensor based again on the integration of the local ODF over all spatial fiber orientations. By design, our modeling formulation accepts structural data on patient-specific collagen fiber networks as determined via diffusion tensor MRI. We implement our new model in 3D large strain finite elements and study the distributions of interstitial fluid pressure, fluid pressure load support and shear stress within a cartilage sample under indentation. Results show that the fiber network dramatically increases interstitial fluid pressure and focuses it near the surface. Inhomogeneity in the tissue’s composition also increases fluid pressure and reduces shear stress in the solid. Finally, a biphasic neo-Hookean material model, as is available in commercial finite element codes, does not capture important features of the intra-tissue response, e.g., distributions of interstitial fluid pressure and principal shear stress.     

Modelling skeletal muscle fibre orientation arrangement

Skeletal muscle tissues have complex geometries. In addition, the complex fibre orientation arrangement makes it quite difficult to create an accurate finite element muscle model. There are many possible ways to specify the complex fibre orientations in a finite element model, for example defining a local element coordinate system. In this paper, an alternative method using ABAQUS, which is combination of the finite element method and the non-uniform rational B-spline solid representation, is proposed to calculate the initial fibre orientations. The initial direction of each muscle fibre is specified as the tangent direction of the NURBS curve which the fibre lies on, and the directions of the deformed fibres are calculated from the initial fibre directions, the deformation gradients and the fibre stretch ratios. Several examples are presented to demonstrate the ability of the proposed method. Results show that the proposed method is able to characterise both the muscle complex fibre orientation arrangement and its complex mechanical response.     

Modelling skeletal muscle fibre orientation arrangement

Skeletal muscle tissues have complex geometries. In addition, the complex fibre orientation arrangement makes it quite difficult to create an accurate finite element muscle model. There are many possible ways to specify the complex fibre orientations in a finite element model, for example defining a local element coordinate system. In this paper, an alternative method using ABAQUS, which is combination of the finite element method and the non-uniform rational B-spline solid representation, is proposed to calculate the initial fibre orientations. The initial direction of each muscle fibre is specified as the tangent direction of the NURBS curve which the fibre lies on, and the directions of the deformed fibres are calculated from the initial fibre directions, the deformation gradients and the fibre stretch ratios. Several examples are presented to demonstrate the ability of the proposed method. Results show that the proposed method is able to characterise both the muscle complex fibre orientation arrangement and its complex mechanical response.     

A fibre reorientation model for orthotropic multiplicative growth

The main goal of this contribution consists in the development of a remodelling framework for orthotropic continua whereby the underlying symmetry group is incorporated via two fibre families. Special emphasis is placed on the modelling of biological tissues at finite deformations. Besides the incorporation of a referential mass source, anisotropic growth is addressed by means of a multiplicative decomposition of the overall deformation gradient into an elastic and a growth distortion. Projected quantities of a configurational growth stress tensor are advocated as driving forces for time-dependent saturation–type evolution of the principal values of the growth distortion. Moreover, the reorientation of both fibre families, which directly affects the strain energy as well as the growth distortion itself, is guided by analyzing critical energy points. In particular, a time-dependent formulation is developed which aligns the fibre directions according to the principal stretch directions. Finally, the proposed framework is embedded into a finite element context so that representative numerical examples, examining growth and resorption in volume and density together with fibre reorientation, close this study.     

Multi-view stereo in the operating room allows prediction of healing complications in a patient-specific model of reconstructive surgery

Excessive mechanical stress leads to wound healing complications following reconstructive surgery. However, this knowledge is not easily applicable in clinical scenarios due to the difficulty in measuring stress contours during complex tissue rearrangement procedures. Computational tools have been proposed as an alternative to address this need, but obtaining patient specific geometries with an affordable and flexible setup has remained a challenge. Here we present a methodology to determine the stress contours from a reconstructive procedure on a patient-specific finite element model based on multi-view stereo (MVS). MVS is a noninvasive technology that allows reconstruction of 3D geometries using a standard digital camera, making it ideal for the operating room. Finite element analysis can then be used on the patient-specific geometry to perform a virtual surgery and predict regions at risk of complications. We applied our approach to the case of a 7-year-old patient who was treated to correct a cranial contour deformity and resect two large areas of scalp scarring. The simulation showed several zones of high stress, particularly near the suture lines at the distal ends of the flaps. The patient did show delayed healing and partial flap tip necrosis at one of such predicted regions at the 30-day follow up visit. Our results further establish the application of computational tools in individualized medical scenarios to advance preoperative planing and anticipate regions of concern immediately after surgery.     

Flow pattern analysis in a highly stenotic patient-specific carotid bifurcation model using a turbulence model

The aim of this study is to investigate the blood flow pattern in carotid bifurcation with a high degree of luminal stenosis, combining in vivo magnetic resonance imaging (MRI) and computational fluid dynamics (CFD). A newly developed two-equation transitional model was employed to evaluate wall shear stress (WSS) distribution and pressure drop across the stenosis, which are closely related to plaque vulnerability. A patient with an 80% left carotid stenosis was imaged using high resolution MRI, from which a patient-specific geometry was reconstructed and flow boundary conditions were acquired for CFD simulation. A transitional model was implemented to investigate the flow velocity and WSS distribution in the patient-specific model. The peak time-averaged WSS value of approximately 73 Pa was predicted by the transitional flow model, and the regions of high WSS occurred at the throat of the stenosis. High oscillatory shear index values up to 0.50 were present in a helical flow pattern from the outer wall of the internal carotid artery immediately after the throat. This study shows the potential suitability of a transitional turbulent flow model in capturing the flow phenomena in severely stenosed carotid arteries using patient-specific MRI data and provides the basis for further investigation of the links between haemodynamic variables and plaque vulnerability. It may be useful in the future for risk assessment of patients with carotid disease.     

Determination of dynamically adapting anisotropic material properties of bone under cyclic loading

Because bone tissue adapts to loading conditions, finite element simulations of remodelling bone require a precise prediction of dynamically changing anisotropic elastic parameters. We present a phenomenological theory that refers to the tissue in terms of the tendency of the structure to align with principal stress directions. We describe the material parameters of remodelling bone. This work follows findings by the same research group and independently by Danilov (1971) in the field of plasticity, where the dependencies of the components of the stiffness tensor in terms of time are based on Hill's anisotropy. We modify such an approach in this novel theory that addresses bone tissue that can regenerate. The computational assumption of the theory is that bone trabeculae have the tendency to orient along one of the principal stress directions but during remodelling the principal stresses change continuously and the resulting orientation of the trabeculae can differ from the principal stress direction at any given time. The novelty of this work consists in the limited number of parameters needed to compute the twenty-one anisotropic material parameters at any given location in the bone tissue. In addition to the theory, we present here two cases of simplified geometry, loading and boundary conditions to show the effect of (1) time on the material properties; and (2) change of loading conditions on the anisotropic parameters. The long term goal is to experimentally verify that the predictions generated by theory provide a reliable simulation of cancellous bone properties.