Holger Schmid

Impact of transmural heterogeneities on arterial adaptation

Recent experimental and computational studies have shown that transmurally heterogeneous material properties through the arterial wall are critical to understanding the heterogeneous expressions of constituent degrading molecules. Given that expression of such molecules is thought to be intimately linked to local magnitudes of stress, modelling the transmural stress distribution is critical to understanding arterial adaption during disease. The aim of this study was to develop an arterial growth and remodelling framework that can incorporate both transmurally heterogeneous constituent distributions and residual stresses, into a 3-D finite element model. As an illustrative example, we model the development of a fusiform aneurysm and investigate the effects of elastinous and collagenous heterogeneities on the stress distribution during evolution. It is observed that the adaptive processes of growth and remodelling exhibit transmural variations. For physiological heterogeneous constituent distributions, a stress peak appears in the media towards the intima, and a stress plateau occurs towards the adventitia. These features can be primarily attributed to the underlying heterogeneity of elastinous constituents. During arterial adaption, the collagen strain is regulated to remain in its homoeostatic level; consequently, the partial stress of collagen has less influence on the total stress than the elastin. However, following significant elastin degradation, collagen plays the dominant role for the transmural stress profile and a marked stress peak occurs towards the adventitia. We conclude that to improve our understanding of the arterial adaption and the aetiology of arterial disease, there is a need to: quantify transmural constituent distributions during histopathological examinations, understand and model the role of the evolving transmural stress distribution.

Consistent formulation of the growth process at the kinematic and constitutive level for soft tissues composed of multiple constituents

Previous studies have investigated the possibilities of modelling the change in volume and change in density of biomaterials. This can be modelled at the constitutive or the kinematic level. This work introduces a consistent formulation at the kinematic and constitutive level for growth processes. Most biomaterials consist of many constituents and can be approximated as being incompressible. These two conditions (many constituents and incompressibility) suggest a straightforward implementation in the context of the finite element (FE) method which could now be validated more easily against histological measurements. Its key characteristic variable is the normalised partial mass change. Using the concept of homeostatic equilibrium, we suggest two complementary growth laws in which the evolution of the normalised partial mass change is governed by an ordinary differential equation in terms of either the Piola–Kirchhoff stress or the Green–Lagrange strain. We combine this approach with the classical incompatibility condition and illustrate its algorithmic implementation within a fully nonlinear FE approach. This approach is first illustrated for a simple uniaxial tension and extension test for pure volume change and pure density change and is validated against previous numerical results. Finally, a physiologically based example of a two-phase model is presented which is a combination of volume and density changes. It can be concluded that the effect of hyper-restoration may be due to the systemic effect of degradation and adaptation of given constituents.

Influence of differing material properties in media and adventitia on arterial adaptation — application to aneurysm formation and rupture

Experimental and computational studies suggest a substantial variation in the mechanical responses and collagen fibre orientations of the two structurally important layers of the arterial wall. Some observe the adventitia to be an order of magnitude stiffer than the media whilst others claim the opposite. Furthermore, studies show that molecular metabolisms may differ substantially in each layer. Following a literature review that juxtaposes the differing layer-specific results we create a range of different hypothetical arteries: (1) with different elastic responses, (2) different fibre orientations, and (3) different metabolic activities during adaptation. We use a finite element model to investigate the effects of those on: (1) the stress response in homeostasis; (2) the time course of arterial adaptation; and (3) an acute increase in luminal pressure due to a stressful event and its influence on the likelihood of aneurysm rupture. Interestingly, for all hypothetical cases considered, we observe that the adventitia acts to protect the wall against rupture by keeping stresses in the media and adventitia below experimentally observed ultimate strength values. Significantly, this conclusion holds true in pathological conditions.

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.

A computationally efficient optimization kernel for material parameter estimation procedures.

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

A Hybrid 1D and 3D Approach to Hemodynamics Modelling for a Patient-Specific Cerebral Vasculature and Aneurysm

In this paper we present a hybrid 1D/3D approach to haemodynamics modelling in a patient-specific cerebral vasculature and aneurysm. The geometric model is constructed from a 3D CTA image. A reduced form of the governing equations for blood flow is coupled with an empirical wall equation and applied to the arterial tree. The equation system is solved using a MacCormack finite difference scheme and the results are used as the boundary conditions for a 3D flow solver. The computed wall shear stress (WSS) agrees with published data.

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.

Computational modeling of cerebral aneurysm formation — framework for modeling the interaction between fluid dynamics, signal transduction pathways and arterial wall mechanics

Many phenomenological models of cerebral aneurysm formation have been proposed. Such studies have focused on modeling the structural adaption of the arterial wall. However, further development is required to accurately represent the underlying mechanobiology during growth and remodeling processes. Here, we present a general framework for modeling the interplay of fluid dynamics, molecular signaling pathways and arterial wall mechanics.