Conservation of Forces and Total Work at the Interface Using the Internodes Method.

The Internodes method is a general purpose method to deal with non-conforming discretizations of partial differential equations on 2D and 3D regions partitioned into disjoint subdomains. In this paper we are interested in measuring how much the Internodes method is conservative across the interface. If hp-fem discretizations are employed, we prove that both the total force and total work generated by the numerical solution at the interface of the decomposition vanish in an optimal way when the mesh size tends to zero, i.e., like O ( h p ) , where p is the local polynomial degree and h the mesh-size. This is the same as the error decay in the H 1-broken norm. We observe that the conservation properties of a method are intrinsic to the method itself because they depend on the way the interface conditions are enforced rather then on the problem we are called to approximate. For this reason, in this paper, we focus on second-order elliptic PDEs, although we use the terminology (of forces and works) proper of linear elasticity. Two and three dimensional numerical experiments corroborate the theoretical findings, also by comparing Internodes with Mortar and WACA methods.

Deep Neural Network to Accurately Predict Left Ventricular Systolic Function Under Mechanical Assistance.

Characterizing left ventricle (LV) systolic function in the presence of an LV assist device (LVAD) is extremely challenging. We developed a framework comprising a deep neural network (DNN) and a 0D model of the cardiovascular system to predict parameters of LV systolic function. DNN input data were systemic and pulmonary arterial pressure signals, and rotation speeds of the device. Output data were parameters of LV systolic function, including end-systolic maximal elastance (E max,lv ), a variable essential for adequate hemodynamic assessment of the LV. A 0D model of the cardiovascular system, including a wide range of LVAD settings and incorporating the whole spectrum of heart failure, was used to generate data for the training procedure of the DNN. The DNN predicted E max,lv with a mean relative error of 10.1%, and all other parameters of LV function with a mean relative error of <13%. The framework was then able to retrieve a number of LV physiological variables (i.e., pressures, volumes, and ejection fraction) with a mean relative error of <5%. Our method provides an innovative tool to assess LV hemodynamics under device assistance, which could be helpful for a better understanding of LV-LVAD interactions, and for therapeutic optimization.

Model order reduction of flow based on a modular geometrical approximation of blood vessels.

We are interested in a reduced order method for the efficient simulation of blood flow in arteries. The blood dynamics is modeled by means of the incompressible Navier-Stokes equations. Our algorithm is based on an approximated domain-decomposition of the target geometry into a number of subdomains obtained from the parametrized deformation of geometrical building blocks (e.g., straight tubes and model bifurcations). On each of these building blocks, we build a set of spectral functions by Proper Orthogonal Decomposition of a large number of snapshots of finite element solutions (offline phase). The global solution of the Navier-Stokes equations on a target geometry is then found by coupling linear combinations of these local basis functions by means of spectral Lagrange multipliers (online phase). Being that the number of reduced degrees of freedom is considerably smaller than their finite element counterpart, this approach allows us to significantly decrease the size of the linear system to be solved in each iteration of the Newton-Raphson algorithm. We achieve large speedups with respect to the full order simulation (in our numerical experiments, the gain is at least of one order of magnitude and grows inversely with respect to the reduced basis size), whilst still retaining satisfactory accuracy for most cardiovascular simulations.

Implementation and Calibration of a Deep Neural Network to Predict Parameters of Left Ventricular Systolic Function Based on Pulmonary and Systemic Arterial Pressure Signals.

The evaluation of cardiac contractility by the assessment of the ventricular systolic elastance function is clinically challenging and cannot be easily obtained at the bedside. In this work, we present a framework characterizing left ventricular systolic function from clinically readily available data, including systemic and pulmonary arterial pressure signals. We implemented and calibrated a deep neural network (DNN) consisting of a multi-layer perceptron with 4 fully connected hidden layers and with 16 neurons per layer, which was trained with data obtained from a lumped model of the cardiovascular system modeling different levels of cardiac function. The lumped model included a function of circulatory autoregulation from carotid baroreceptors in pulsatile conditions. Inputs for the DNN were systemic and pulmonary arterial pressure curves. Outputs from the DNN were parameters of the lumped model characterizing left ventricular systolic function, especially end-systolic elastance. The DNN adequately performed and accurately recovered the relevant hemodynamic parameters with a mean relative error of less than 2%. Therefore, our framework can easily provide complex physiological parameters of cardiac contractility, which could lead to the development of invaluable tools for the clinical evaluation of patients with severe cardiac dysfunction.

Analysis of morphological and hemodynamical indexes in abdominal aortic aneurysms as preliminary indicators of intraluminal thrombus deposition.

In this paper, we study the correlation between the wall shear stress, a hemodynamical index derived from numerical simulations, and an new index MFA-ILT for the characterization of intraluminal thrombus (ILT) in the presence of abdominal aortic aneurysms. Based on the processing of medical images, we define our index MFA-ILT by projecting onto lumen surface a measure of the ILT thickness. From the physical point of view, hemodynamical indexes describe the mechanical stimuli at which the luminal surface of the vessel wall is subject to, due to blood flow. Specifically, we consider the time-averaged wall shear stress and the oscillatory shear index. The first index provides a measurement of the averaged magnitude of the shear stress; the second index measures the rate of change of shear stress. To reconstruct the hemodynamical indexes, we build in silico three-dimensional models. We use the same physical parameters and boundary conditions for all the aneurysms in the sample. The computer simulations do not require any additional invasive patient examination. We consider eleven cases of abdominal aortic aneurysms spanning a wide range of different morphological features. All the cases are characterized by a thin intraluminal thrombus. We can, therefore, assume that the lumen we currently observe does not significantly differ from the one before the thrombus deposition. Our results suggest that the value of wall shear stresses and intraluminal thrombus deposition are correlated. Moreover, we conclude that in six cases time-averaged wall shear stress provides a preliminary indication of the area at risk of thrombus deposition.

Numerical modeling of fluid-structure interaction in arteries with anisotropic polyconvex hyperelastic and anisotropic viscoelastic material models at finite strains.

The accurate prediction of transmural stresses in arterial walls requires on the one hand robust and efficient numerical schemes for the solution of boundary value problems including fluid-structure interactions and on the other hand the use of a material model for the vessel wall that is able to capture the relevant features of the material behavior. One of the main contributions of this paper is the application of a highly nonlinear, polyconvex anisotropic structural model for the solid in the context of fluid-structure interaction, together with a suitable discretization. Additionally, the influence of viscoelasticity is investigated. The fluid-structure interaction problem is solved using a monolithic approach; that is, the nonlinear system is solved (after time and space discretizations) as a whole without splitting among its components. The linearized block systems are solved iteratively using parallel domain decomposition preconditioners. A simple - but nonsymmetric - curved geometry is proposed that is demonstrated to be suitable as a benchmark testbed for fluid-structure interaction simulations in biomechanics where nonlinear structural models are used. Based on the curved benchmark geometry, the influence of different material models, spatial discretizations, and meshes of varying refinement is investigated. It turns out that often-used standard displacement elements with linear shape functions are not sufficient to provide good approximations of the arterial wall stresses, whereas for standard displacement elements or F-bar formulations with quadratic shape functions, suitable results are obtained. For the time discretization, a second-order backward differentiation formula scheme is used. It is shown that the curved geometry enables the analysis of non-rotationally symmetric distributions of the mechanical fields. For instance, the maximal shear stresses in the fluid-structure interface are found to be higher in the inner curve that corresponds to clinical observations indicating a high plaque nucleation probability at such locations. Copyright © 2015 John Wiley & Sons, Ltd.

Numerical simulation of left ventricular assist device implantations: comparing the ascending and the descending aorta cannulations.

In this work we present numerical simulations of continuous flow left ventricle assist device implantation with the aim of comparing difference in flow rates and pressure patterns depending on the location of the anastomosis and the rotational speed of the device. Despite the fact that the descending aorta anastomosis approach is less invasive, since it does not require a sternotomy and a cardiopulmonary bypass, its benefits are still controversial. Moreover, the device rotational speed should be correctly chosen to avoid anomalous flow rates and pressure distribution in specific location of the cardiovascular tree. With the aim of assessing the differences between these two approaches and device rotational speed in terms of flow rate and pressure waveforms, we set up numerical simulations of network of one-dimensional models where we account for the presence of an outflow cannula anastomosed to different locations of the aorta. Then, we use the resulting network to compare the results of the two different cannulations for several stages of heart failure and different rotational speed of the device. The inflow boundary data for the heart and the cannulas are obtained from a lumped parameters model of the entire circulatory system with an assist device, which is validated with clinical data. The results show that ascending and descending aorta cannulations lead to similar waveforms and mean flow rate in all the considered cases. Moreover, regardless of the anastomosis region, the rotational speed of the device has an important impact on wave profiles; this effect is more pronounced at high RPM.

Physiological simulation of blood flow in the aorta: comparison of hemodynamic indices as predicted by 3-D FSI, 3-D rigid wall and 1-D models.

Interest in patient-specific blood-flow circulation modeling has increased substantially in recent years. The availability of clinical data for geometric and elastic properties together with efficient numerical methods has now made model rendering feasible. This work uses 3-D fluid-structure interaction (FSI) to provide physiological simulation resulting in modeling with a high level of detail. Comparisons are made between results using FSI and rigid wall models. The relevance of wall compliance in determining parameters of clinical importance, such as wall shear stress, is discussed together with the significance of differences found in the pressure and flow waveforms when using the 1-D model. Patient-specific geometry of the aorta and its branches was based on MRI angiography data. The arterial wall was created with a variable thickness. The boundary conditions for the fluid domain were pressure waveform at the ascending aorta and flow for each outlet. The waveforms were obtained using a 1-D model validated by in vivo measurements performed on the same person. In order to mimic the mechanical effect of surrounding tissues in the simulation, a stress-displacement relation was applied to the arterial wall. The temporal variation and spatial patterns of wall shear stress are presented in the aortic arch and thoracic aorta together with differences using rigid wall and FSI models. A comparison of the 3-D simulations to the 1-D model shows good reproduction of the pressure and flow waveforms.