Mathematical and Numerical Modeling of Healthy and Unhealthy Cerebral Arterial Tissues

Over the last two decades, we have witnessed an increasing application of mathematical models and numerical simulations for the study of the cardiovascular system. Indeed, both tools provide an important contribution to the analysis of the functioning of the different components of the cardiovascular system (i.e. heart, vessels and blood) and of their interactions either in physiological and pathological conditions. For this reason, reliable constitutive models for the cardiac, arterial and venous tissues as well as for the blood are an essential prerequisite for a number of different objectives that range from the improved diagnostic to the study of the onset and development of cardiovascular diseases (e.g atherosclerosis or aneurysms). This work focuses on the mathematical and numerical modeling of healthy and unhealthy cerebral arterial tissue. In particular, it presents a detailed analysis of different constitutive models for the arterial tissue by means of finite element numerical simulations of arterial wall mechanics and fluid-structure interaction problems occurring in hemodynamics. Hyperelastic isotropic and anisotropic constitutive laws are considered for the description of the passive mechanical behavior of the vessels. An anisotropic multi-mechanism model, specifically proposed for the cerebral arterial tissue, for which the activation of the collagen fibers occurs at finite strains is employed. Firstly, the constitutive laws are numerically validated by considering numerical simulations of static inflation tests on a cylindrical geometry representing a specimen of anterior cerebral artery. With this regard, the material parameters for the constitutive law are obtained from the data fitting of experimental measurements obtained on the same vessel. The constitutive models are critically discussed according to their capability of describing the physiogical highly nonlinear behavior of arteries and on other numerical aspects related to the computational simulation of arterial wall mechanics. Afterwards, simulations of the blood flow and vessel wall interactions are carried out on idealized blood vessels in order to analyze the influence of the modeling choice for the arterial wall on hemodynamic and mechanical quantities that are commonly considered as indicators of physiological or pathological conditions of arteries. We also consider the numerical simulations of unhealthy cerebral arterial tissues by taking into account the mechanical weakening of the vessel wall that occurs during early development stages of cerebral aneurysms by means of static inflation and FSI simulations. We employ both isotropic and anisotropic models study the effects of the mechanical degradation on hemodynamic and mechanical quantities of interest. The FSI simulations are carried out both on idealized geometries of blood vessels and on domains representing idealized and anatomically realistic cerebral aneurysms.

Numerical modeling and neural networks to identify constitutive parameters from in situ tests

Rafal Filip Obrzud

This study proposes a new advanced algorithm for determining material parameters based on in situ tests. In situ testing gives an opportunity to perform soil characterization in natural stress conditions on a representative soil mass. Most field techniques reduce soil disturbances to minimum, allowing investigating the response of virgin soil. Self-boring pressuremeter tests (SBPT) and standard piezocone tests (CPTU) are widely used to deduce properties of clayey soils through analytical and empirical correlations between soil properties and experimental measurements. Empirical correlations usually require some tuning based on reference laboratory data because first-order estimates for typical correlation coefficients may give unreliable evaluation of soil properties. Analytical correlations are mostly based on cavity expansion methods which are restricted to either fully drained or perfectly undrained problems, so that inverse closed-form solutions for relatively simple constitutive models can be derived. In practice, however, depending on physical and consolidation properties of the soil, partially drained conditions may occur during field testing, leading to an erroneous estimation of clay characteristics. Therefore, elaborating a generic parameter identification framework, which is based on the artificial neural network (NN) technique and which may improve the reliability of soil properties derived from in situ testing, is the main goal of this research. This study explores the possibility of using NNs to solve complex inverse problems including partially drained conditions. In other words, NNs are used to map experimental measurements onto set of soil properties. The development of NN-based inverse models is based on a training data sets which consists of pseudo-experimental measurements derived from numerical simulations of both the SBPT and the CPTU test in normally- and lightly overconsolidated clay type material. The study presents a generic two-level procedure designed for the calibration of constitutive models of soils. It is demonstrated that NN inverse models can be easily integrated into the classical back-analysis. At the first level, the NN approach is applied to achieve the first approximation of parameters. This technique is used to avoid potential pitfalls related to the conventional gradient-based optimization (GBO) technique, considered here as a corrector that improves predicted parameters. Trained NNs as parallel operating systems can provide output variables instantly and without a costly GBO iterative scheme. The proposed framework is verified for the elasto-plastic Modified Cam Clay (MCC) model that can be calibrated based on standard triaxial laboratory tests, i.e. the isotropic consolidation test and the consolidated isotropic drained compression test. The study presents formulations of the input data for the NN predictors, enhanced by a dimensional reduction of experimental data using principal component analysis (PCA). The determination of model characteristics is demonstrated, first on numerical pseudo-experiments and then on the experimental data. Furthermore, the efficiency of the proposed approach in terms of accuracy and computational effort is also discussed. The verified two-level strategy is applied to a numerical procedure of parameter identification for the boundary value problem (BVP) of the SBPT. The coupled hydro-mechanical finite element (FE) formulation allows the generated excess pore water pressure to be dissipated during simulations of the expansion test, followed then by a holding test. Numerical simulations demonstrate that volume changes that may occur in clay during the expansion test due to partial drainage, can cause local soil hardening near the cavity wall and affect parameter interpretation for pressuremeter tests. Therefore, the NN technique is applied to obtain an initial guess for model parameters, taking into account the possible partially drained conditions during the expansion test. Parameter identification based on measurements obtained through the pressuremeter expansion test and two types of holding tests is illustrated on the MCC model. NNs are trained using a set of synthetic test samples, which are generated by means of FE simulations based on constrained random permutations of input model parameters. The measurements obtained through expansion and consolidation tests are normalized by the proposed normalizing formulas so that NN predictors operate independently of the testing depth. Examples of parameter determination are demonstrated on both numerical data and field measurements from the Fucino clay deposit. The efficiency of the combined parameter identification in terms of accuracy, effectiveness and computational effort is also discussed. Finally, an application of NN predictors as a stand-alone support for soil profiling is presented for the piezocone test. By similarity to the SBPT problem, a number of NN inverse models are developed based on the results derived from rigorous FE analyzes. The FE model of piezocone penetration involves numerical formulation for the two-phase material obeying the MCC law and including the large strain theory, as well as the large deformation formulation for contact interface. It is demonstrated that a considerable computational effort related to the generation of the training database can be reduced by optimizing the mesh size and "steady-state" depth in function of soil rigidity index. Due to a severe loss of measurement accuracy observed in the finite elements adhered to the "rough" interface, an equivalent semi-numerical approach is proposed to account for frictional effects in different drainage conditions which are delineated from a number of numerical simulations. The validity of the developed penetration model is verified in detail by means of comparisons with other theoretical solutions and parametric studies synthesized from literature, as well as experimental evidence for both undrained and partially drained scenarios. An extended parametric study including influence analyzes of strength and stress anisotropy, rigidity index and cone roughness on two cone factors provides new insight into the analysis of cone penetration. The shortcomings of the FE model due to the limitations of the applied constitutive model are also discussed. As regards NN models, different configurations of input variables, including standard normalized piezocone metrics and other available soil characteristics are investigated in terms of feasibility of effective NN training. The post-training regression analyzes are performed for numerical data allowing the assessment of the influence of specific input variables on accuracy of parameters predictions. Finally, the developed NN models are applied to predict parameters based on field measurements for a number of characterization sites. Provided examples demonstrate that NN inverse models may constitute an effective complementary support during the first-order quantification of the MCC parameters from piezocone measurements.

EPFL2009

Modeling the fluid dynamics of the heart: from blood flows in idealized left ventricles to patient-specific aortic valve simulations

Simone Deparis, Elena Faggiano, Marco Fedele, Davide Forti, Alfio Quarteroni, Anna Tagliabue

The computational fluid dynamics of the heart represents a challenging task both in terms of mathematical and numerical modeling; this is mainly due to the pulsatile nature of the blood flow, to its complex interaction with the valves, and, more in general, to the reciprocal action of the components responsible for the heart functioning. Even if one focuses on the study of the left ventricle, the blood flow patterns result to be significantly dependent on the mechanical contraction and relaxation of the muscle, on the conformation of the chamber, and on the interaction with the valves, which define a complex fluid-structure interaction problem. In this respect, also the blood flow in the aorta, and hence in the downstream circulation, is strongly affected by the aortic valve, whose behavior should be suitably mathematically and numerically modeled. In this work, we firstly focus on the study of the fluid dynamics inside the left ventricle in idealized configurations, for which we propose a mathematical model based on the Navier-Stokes equations endowed with mixed, time-dependent boundary conditions, which allow a simplified treatment of the aortic and mitral valves’ behavior. In this idealized setting, we perform numerical simulations which highlight the role and influence of modeling the valves to study and characterize the blood flow patterns inside the ventricle, as well as other parameters of clinical relevance. In addition, we consider a reduced, patient-specific fluid-structure-interaction model for the simulation of the blood flow through the aortic valve. Specifically, we propose an efficient coupled model which represents the valve dynamics by means of a zero-dimensional (0D) equation with the opening angle as primitive variable, while the blood flow by means of the full 3D Navier-Stokes equations. In this coupled model, the valve’s leaflets, which are reconstructed from MRI data of the open and closed configurations for a specific patient, influence the Navier-Stokes equations by means of resistive immersed surfaces, whose position depends on the opening angle of the valve. Moreover, the dynamics of the valve described by the 0D model is dependent on the velocity and pressure variables, specifically on the pressure jump and the flow rate through the valve itself. We per form patient-specific numerical simulations of the aortic valve based on this reduced 3D-0D model, for which we highlight its ability to correctly capture the fluid dynamics indicators expected for the patient.

2015

A numerical study of isotropic and anisotropic constitutive models with relevance to healthy and unhealthy cerebral arterial tissues

Alfio Quarteroni, Adélia Sequeira, Paolo Tricerri

This paper presents an analysis of hyperelastic constitutive models for continuous bodies both from a modeling and numerical point of view. Contributions are made within the context of finite element numerical simulations. Numerical results with relevance to flows in the cardiovascular system are outlined in the case of a sophisticated fluid–structure interaction problem, in specific complex geometries of anatomically accurate cerebral arteries in diseased state. In this regard, the work carefully outlines the numerical validation of constitutive models for healthy and unhealthy cerebral arterial tissues by means of simulations of static inflation tests on an idealized specimen of anterior cerebral artery (ACA). The healthy tissue is described by means of isotropic and anisotropic models that, are fitted with respect to experimental data describing the mechanical behavior of the ACA; the numerical results are presented highlighting the most important numerical aspects influencing the correct and efficient simulation of the mechanics of continuous bodies such as, for instance, the arterial wall. We further consider numerical simulations of unhealthy conditions of the tissue by taking into account different levels of weakening of its mechanical properties. Taking the cerebral cardiovascular system as a challenging test problem, we focus on the study of the effects of the imposed mechanical levels of degradation on kinematic quantities of interest by simulating static inflation tests for the different models. This work does not aim to propose a new mathematical model for the mechanical damage occurring at the onset of cardiovascular diseases such as cerebral aneurysms. The modeling and numerical techniques presented may be applied to a wide range of problems, equally challenging to that of the cardiovascular system with complex structural models and fluid-structure coupling.

Elsevier2016