Inverse material parameter estimation of patient-specific finite element models at the carotid bifurcation: The impact of excluding the zero-pressure configuration and residual stress

The carotid bifurcation experiences a complex loading environment due to its anatomical structure. Previous in-vivo material parameter estimation methods often use simplified model geometries, isotropic hyperelastic constitutive equations or neglect key aspects of the vessel, such as the zero-pressure configuration or residual stress, all of which have independently been shown to alter the stress environment of the vessel wall. Characterizing the location of high stress in the vessel wall has often been proposed as a potential indicator of structural weakness. However, excluding the afore-mentioned zero-pressure configuration, residual stress and patient-specific material parameters can lead to an incorrect estimation of the true stress values observed, meaning that stress alone as a risk indicator of rupture is insufficient. In this study, we investigate how the estimated material parameters and overall stress distributions in geometries of carotid bifurcations, extracted from in-vivo MR images, alter with the inclusion of the zero-pressure configuration and residual stress. This approach consists of the following steps: (1) geometry segmentation and hexahedral meshing from in-vivo magnetic resonance images (MRI) at two known phases; (2) computation of the zero-pressure configuration and the associated residual stresses; (3) minimization of an objective function built on the difference between the stress states of an "almost true" stress field at two known phases and a "deformed" stress field by altering the input material parameters to determine patient-specific material properties; and (4) comparison of the stress distributions throughout these carotid bifurcations for all cases with estimated material parameters. This numerical approach provides insights into the need for estimation of both the zero-pressure configuration and residual stress for accurate material property estimation and stress analysis for the carotid bifurcation, establishing the reliability of stress as a rupture risk metric.

Direct stress computations in arbitrarily shaped thin shells and elliptic bulge tests

Most of engineering and biological thin shell structures are characterized by non-axisymmetric and statically indeterminate configurations, which often require the knowledge of the constitutive material model to determine the stress state. In this work, a forward elastostatic method is introduced for the direct stress measurements in elastic homogeneous thin shells of arbitrary shape. The stress distribution is proven to be independent of the material properties for incompressible solids, while in compressible materials it depends only on the Poisson’s ratio, which is shown to have a negligible influence on the stress state. Hence, the proposed technique enables the direct assessment of the stress field in statically indeterminate thin shells without a known material model. The shell formulation is implemented using the finite difference method to independently measure the stresses during finite inflation of planar elliptical membranes and from the deformed shapes obtained through digital image correlation during bulge tests on a hyperelastic material, showing very good agreement with finite-element predictions and the applicability of the method to nonlinear elastic materials. Therefore, the procedure can be coupled with imaging techniques for the direct assessment of stresses in thin shell structures and in the identification of material parameters through non-axisymmetric bulge tests.

Combining Unique Planar Biaxial Testing with Full-Field Thickness and Displacement Measurement for Spatial Characterization of Soft Tissues

Soft tissues rely on the incredible complexity of their microstructure for proper function. Local variations in material properties arise as tissues develop and adapt, often in response to changes in loading. A barrier to investigating the heterogeneous nature of soft tissues is the difficulty of developing experimental protocols and analysis tools that can accurately capture spatial variations in mechanical behavior. In this article, we detail protocols enabling mechanical characterizations of anisotropic, heterogeneous soft tissues or tissue analogs. We present a series of mechanical tests designed to maximize inhomogeneous strain fields and in-plane shear forces. A customized, 3D-printable gripping system reduces tissue handling and enhances shear. High-resolution imaging and laser micrometry capture full-field displacement and thickness, respectively. As the equipment necessary to conduct these protocols is commercially available, the experimental methods presented offer an accessible route toward addressing heterogeneity. © 2022 The Authors. Current Protocols published by Wiley Periodicals LLC. Basic Protocol 1: Unique biaxial testing of soft tissues and tissue analogs Basic Protocol 2: Full-field thickness measurement of soft tissues and tissue analogs Support Protocol 1: Creating and speckling cruciform-shaped samples for mechanical testing Support Protocol 2: Creating custom gripping system to minimize sample handling.

Fluid-structure interaction simulation of tissue degradation and its effects on intra-aneurysm hemodynamics

Tissue degradation plays a crucial role in vascular diseases such as atherosclerosis and aneurysms. Computational modeling of vascular hemodynamics incorporating both arterial wall mechanics and tissue degradation has been a challenging task. In this study, we propose a novel finite element method-based approach to model the microscopic degradation of arterial walls and its interaction with blood flow. The model is applied to study the combined effects of pulsatile flow and tissue degradation on the deformation and intra-aneurysm hemodynamics. Our computational analysis reveals that tissue degradation leads to a weakening of the aneurysmal wall, which manifests itself in a larger deformation and a smaller von Mises stress. Moreover, simulation results for different heart rates, blood pressures and aneurysm geometries indicate consistently that, upon tissue degradation, wall shear stress increases near the flow-impingement region and decreases away from it. These findings are discussed in the context of recent reports regarding the role of both high and low wall shear stress for the progression and rupture of aneurysms.

Calibration of parameters for cardiovascular models with application to arterial growth

We present a computational framework for the calibration of parameters describing cardiovascular models with a focus on the application of growth of abdominal aortic aneurysms (AAA). The growth rate in this sort of pathology is considered a critical parameter in the risk management and is an essential indicator for the assessment of surveillance intervals. Parameters describing growth of AAAs are not measurable directly and need to be estimated from available data often given by medical imaging technologies. Registration procedures often applied in standard workflows of parameter identification to extract the image encoded information are a source of significant systematic error. The concept of surface currents provides means to effectively avoid this source of errors by establishing a mathematical framework to compare surface information, directly accessible from image data. By utilizing this concept it is possible to inversely estimate growth parameters using sophisticated numerical models of AAAs from measurements available as surface information. In this work we present a framework to obtain spatial distributions of parameters governing growth of arterial tissue, and we show how the use of surface currents can significantly improve the results. We further present the application to patient specific follow-up data resulting in a spatial map of volumetric growth rates enabling, for the first time, prediction of further AAA expansion.

Anisotropic linear elastic parameter estimation using error in the constitutive equation functional

A modified error in the constitutive equation-based approach for identification of heterogeneous and linear anisotropic elastic parameters involving static measurements is proposed and explored. Following an alternating minimization procedure associated with the underlying optimization problem, the new strategy results in an explicit material parameter update formula for general anisotropic material. This immediately allows us to derive the necessary constraints on measured data and thus restrictions on physical experimentation to achieve the desired reconstruction. We consider a few common materials to derive such conditions. Then, we exploit the invariant relationships of the anisotropic constitutive tensor to propose an identification procedure for space-dependent material orientations. Finally, we assess the numerical efficacy of the developed tools against a few parameter identification problems of engineering interest.

A nonlinear anisotropic inverse method for computational dissection of inhomogeneous planar tissues

Quantification of the mechanical behavior of soft tissues is challenging due to their anisotropic, heterogeneous, and nonlinear nature. We present a method for the 'computational dissection' of a tissue, by which we mean the use of computational tools both to identify and to analyze regions within a tissue sample that have different mechanical properties. The approach employs an inverse technique applied to a series of planar biaxial experimental protocols. The aggregated data from multiple protocols provide the basis for (1) segmentation of the tissue into regions of similar properties, (2) linear analysis for the small-strain behavior, assuming uniform, linear, anisotropic behavior within each region, (3) subsequent nonlinear analysis following each individual experimental protocol path and using local linear properties, and (4) construction of a strain energy data set W(E) at every point in the material by integrating the differential stress-strain functions along each strain path. The approach has been applied to simulated data and captures not only the general nonlinear behavior but also the regional differences introduced into the simulated tissue sample.

Automatic Segmentation of Mechanically Inhomogeneous Tissues Based on Deformation Gradient Jump

Variations in properties, active behavior, injury, scarring, and/or disease can all cause a tissue's mechanical behavior to be heterogeneous. Advances in imaging technology allow for accurate full-field displacement tracking of both in vitro and in vivo deformation from an applied load. While detailed strain fields provide some insight into tissue behavior, material properties are usually determined by fitting stress-strain behavior with a constitutive equation. However, the determination of the mechanical behavior of heterogeneous soft tissue requires a spatially varying constitutive equation (i.e., one in which the material parameters vary with position). We present an approach that computationally dissects the sample domain into many homogeneous subdomains, wherein subdomain boundaries are formed by applying a betweenness based graphical analysis to the deformation gradient field to identify locations with large discontinuities. This novel partitioning technique successfully determined the shape, size and location of regions with locally similar material properties for: (1) a series of simulated soft tissue samples prescribed with both abrupt and gradual changes in anisotropy strength, prescribed fiber alignment, stiffness, and nonlinearity, (2) tissue analogs (PDMS and collagen gels) which were tested biaxially and speckle tracked (3) and soft tissues which exhibited a natural variation in properties (cadaveric supraspinatus tendon), a pathologic variation in properties (thoracic aorta containing transmural plaque), and active behavior (contracting cardiac sheet). The routine enables the dissection of samples computationally rather than physically, allowing for the study of small tissues specimens with unknown and irregular inhomogeneity.

Physical factors effecting cerebral aneurysm pathophysiology

Many factors that are either blood-, wall-, or hemodynamics-borne have been associated with the initiation, growth, and rupture of intracranial aneurysms. The distribution of cerebral aneurysms around the bifurcations of the circle of Willis has provided the impetus for numerous studies trying to link hemodynamic factors (flow impingement, pressure, and/or wall shear stress) to aneurysm pathophysiology. The focus of this review is to provide a broad overview of such hemodynamic associations as well as the subsumed aspects of vascular anatomy and wall structure. Hemodynamic factors seem to be correlated to the distribution of aneurysms on the intracranial arterial tree and complex, slow flow patterns seem to be associated with aneurysm growth and rupture. However, both the prevalence of aneurysms in the general population and the incidence of ruptures in the aneurysm population are extremely low. This suggests that hemodynamic factors and purely mechanical explanations by themselves may serve as necessary, but never as necessary and sufficient conditions of this disease's causation. The ultimate cause is not yet known, but it is likely an additive or multiplicative effect of a handful of biochemical and biomechanical factors.

An anisotropic, hyperelastic model for skin: experimental measurements, finite element modelling and identification of parameters for human and murine skin

The mechanical characteristics of skin are extremely complex and have not been satisfactorily simulated by conventional engineering models. The ability to predict human skin behaviour and to evaluate changes in the mechanical properties of the tissue would inform engineering design and would prove valuable in a diversity of disciplines, for example the pharmaceutical and cosmetic industries, which currently rely upon experiments performed in animal models. The aim of this study was to develop a predictive anisotropic, hyperelastic constitutive model of human skin and to validate this model using laboratory data. As a corollary, the mechanical characteristics of human and murine skin have been compared. A novel experimental design, using tensile tests on circular skin specimens, and an optimisation procedure were adopted for laboratory experiments to identify the material parameters of the tissue. Uniaxial tensile tests were performed along three load axes on excised murine and human skin samples, using a single set of material parameters for each skin sample. A finite element model was developed using the transversely isotropic, hyperelastic constitutive model of Weiss et al. (1996) and was embedded within a Veronda-Westmann isotropic material matrix, using three fibre families to create anisotropic behaviour. The model was able to represent the nonlinear, anisotropic behaviour of the skin well. Additionally, examination of the optimal material coefficients and the experimental data permitted quantification of the mechanical differences between human and murine skin. Differences between the skin types, most notably the extension of the skin at low load, have highlighted some of the limitations of murine skin as a biomechanical model of the human tissue. The development of accurate, predictive computational models of human tissue, such as skin, to reduce, refine or replace animal models and to inform developments in the medical, engineering and cosmetic fields, is a significant challenge but is highly desirable. Concurrent advances in computer technology and our understanding of human physiology must be utilised to produce more accurate and accessible predictive models, such as the finite element model described in this study.

Characterizing heterogeneous properties of cerebral aneurysms with unknown stress-free geometry: a precursor to in vivo identification

Knowledge of elastic properties of cerebral aneurysms is crucial for understanding the biomechanical behavior of the lesion. However, characterizing tissue properties using in vivo motion data presents a tremendous challenge. Aside from the limitation of data accuracy, a pressing issue is that the in vivo motion does not expose the stress-free geometry. This is compounded by the nonlinearity, anisotropy, and heterogeneity of the tissue behavior. This article introduces a method for identifying the heterogeneous properties of aneurysm wall tissue under unknown stress-free configuration. In the proposed approach, an accessible configuration is taken as the reference; the unknown stress-free configuration is represented locally by a metric tensor describing the prestrain from the stress-free configuration to the reference configuration. Material parameters are identified together with the metric tensor pointwisely. The paradigm is tested numerically using a forward-inverse analysis loop. An image-derived sac is considered. The aneurysm tissue is modeled as an eightply laminate whose constitutive behavior is described by an anisotropic hyperelastic strain-energy function containing four material parameters. The parameters are assumed to vary continuously in two assigned patterns to represent two types of material heterogeneity. Nine configurations between the diastolic and systolic pressures are generated by forward quasi-static finite element analyses. These configurations are fed to the inverse analysis to delineate the material parameters and the metric tensor. The recovered and the assigned distributions are in good agreement. A forward verification is conducted by comparing the displacement solutions obtained from the recovered and the assigned material parameters at a different pressure. The nodal displacements are found in excellent agreement.

Identification of in vivo material and geometric parameters of a human aorta: toward patient-specific modeling of abdominal aortic aneurysm

Recent advances in computational modeling of vascular adaptations and the need for their extension to patient-specific modeling have introduced new challenges to the path toward abdominal aortic aneurysm modeling. First, the fundamental assumption in adaptation models, namely the existence of vascular homeostasis in normal vessels, is not easy to implement in a vessel model built from medical images. Second, subjecting the vessel wall model to the normal pressure often makes the configuration deviate from the original geometry obtained from medical images. To address those technical challenges, in this work, we propose a two-step optimization approach; first, we estimate constitutive parameters of a healthy human aorta intrinsic to the material by using biaxial test data and a weighted nonlinear least-squares parameter estimation method; second, we estimate the distributions of wall thickness and anisotropy using a 2-D parameterization of the vessel wall surface and a global approximation scheme integrated within an optimization routine. A direct search method is implemented to solve the optimization problem. The numerical optimization method results in a considerable improvement in both satisfying homeostatic condition and minimizing the deviation of geometry from the original shape based on in vivo images. Finally, the utility of the proposed technique for patient-specific modeling is demonstrated in a simulation of an abdominal aortic aneurysm enlargement.

Identifying heterogeneous anisotropic properties in cerebral aneurysms: a pointwise approach

The traditional approaches of estimating heterogeneous properties in a soft tissue structure using optimization-based inverse methods often face difficulties because of the large number of unknowns to be simultaneously determined. This article proposes a new method for identifying the heterogeneous anisotropic nonlinear elastic properties in cerebral aneurysms. In this method, the local properties are determined directly from the pointwise stress-strain data, thus avoiding the need for simultaneously optimizing for the property values at all points/regions in the aneurysm. The stress distributions needed for a pointwise identification are computed using an inverse elastostatic method without invoking the material properties in question. This paradigm is tested numerically through simulated inflation tests on an image-based cerebral aneurysm sac. The wall tissue is modeled as an eight-ply laminate whose constitutive behavior is described by an anisotropic hyperelastic strain energy function containing four parameters. The parameters are assumed to vary continuously in the sac. Deformed configurations generated from forward finite element analysis are taken as input to inversely establish the parameter distributions. The delineated and the assigned distributions are in excellent agreement. A forward verification is conducted by comparing the displacement solutions obtained from the delineated and the assigned material parameters at a different pressure. The deviations in nodal displacements are found to be within 0.2% in most part of the sac. The study highlights some distinct features of the proposed method, and demonstrates the feasibility of organ level identification of the distributive anisotropic nonlinear properties in cerebral aneurysms.

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.