Outflow Boundary Conditions Research Articles

Assessing the methodology used to study the ascending aorta haemodynamics in bicuspid aortic valve.

Modern imaging techniques provide evermore-detailed anatomical and physiological information for use in computational fluid dynamics to predict the behaviour of physiological phenomena. Computer modelling can help plan suitable interventions. Our group used magnetic resonance imaging and computational fluid dynamics to study the haemodynamic variables in the ascending aorta in patients with bicuspid aortic valve before and after isolated tissue aortic valve replacement. Computer modelling requires turning a physiological model into a mathematical one, solvable by equations that undergo multiple iterations in four dimensions. Creating these models involves several steps with manual inputs, making the process prone to errors and limiting its inter- and intra-operator reproducibility. Despite these challenges, we created computational models for each patient to study ascending aorta blood flow before and after surgery. Magnetic resonance imaging provided the anatomical and velocity data required for the blood flow simulation. Patient-specific in- and outflow boundary conditions were used for the computational fluid dynamics analysis. Haemodynamic variables pertaining to blood flow pattern and derived from the magnetic resonance imaging data were calculated. However, we encountered problems in our multi-step methodology, most notably processing the flow data. This meant that other variables requiring computation with computational fluid dynamics could not be calculated. Creating a model for computational fluid dynamics analysis is as complex as the physiology under scrutiny. We discuss some of the difficulties associated with creating such models, along with suggestions for improvements in order to yield reliable and beneficial results.

Smoothed Particle Hydrodynamics Simulations of Turbulent Flow in Curved Pipes With Different Geometries: A Comparison With Experiments

Abstract The swirling secondary flow in curved pipes is studied in three-space dimensions using a weakly compressible smoothed particle hydrodynamics (WCSPH) formulation coupled to new nonreflecting outflow boundary conditions. A large eddy simulation (LES) model for turbulence is benchmarked with existing experimental data. After validation of the present model against experimental results for a 90 deg pipe bend, a detailed numerical study aimed at reproducing experimental flow measurements for a wide range of Reynolds numbers has been performed for different pipe geometries, including U pipe bends, S-shaped pipes, and helically coiled pipes. In all cases, the SPH calculated behavior shows reasonably good agreement with the measurements across and downstream the bend in terms of streamwise velocity profiles and cross-sectional contours. Maximum mean-root-square deviations from the experimentally obtained profiles are always less than ∼1.8%. This combined with the very good matching between the SPH and the experimental cross-sectional contours shows the uprising capabilities of the present scheme for handling engineering applications with streamline curvature, such as flows in bends and manifolds.

Investigation on the effect of convective outflow boundary condition on the bubbles growth, rising and breakup dynamics of nucleate boiling

In this work, a phase-change lattice Boltzmann model was employed to simulate the effects of two computational domains with confined space and free outlet boundary considering the conjugate heat transfer on entire nucleate boiling. This model was validated by Laplace's law and droplet wetting process, especially in the improved convective outlet boundary for multiphase flow in detail. The film boiling was employed to verify the phase change model quantitatively, and the results exhibited a good agreement with a theoretical model so that the accuracy and robustness of the current model were ensured. The dynamic behaviors of the entire nucleate boiling including nucleation embryo, embryo growth, twin and triple vapors coalescence, departure, rise and subsequent rupture were investigated comprehensively. The temperature and velocity fields, as well as pressure propagations associated with nucleate boiling were investigated. The automatically sharp growth of nucleation vapor pushes away the vicinity of vapor outwardly, while the bottom fluid still flows into the root of the vapor contact line, which coincides with experimental observation. The flow dynamic behaviors of double and triple bubbles coalescence are almost the same, whereas more isolated bubbles merge is beneficial to the growth and departure dramatically. The sliding growth on the heating substrate was found clearly, which is mainly due to the non-uniform temperature gradient below the two side of the vapor bubble. The curvilinear rising trajectory and large deformation of rising vapor bubbles were observed, leading to a temperature wake in the rear of the rising bubble. The increasing volume of the rising bubbles occurred in the free outlet domain was found, which is mainly caused by the combined pressure influence between the inside and vicinity of the bubble. Moreover, the comparisons of nucleate boiling in both two computational domains were also studied.

On outflow boundary conditions in finite element simulations of non-Newtonian internal flows

The matter of appropriate boundary conditions for open or truncated outflow regions in internal flow is still focus of discussion and research. In most practical applications, one can at best estimate mean pressure values or flow rates at such outlets. In the context of finite element methods, it is known that enforcing mean pressures through the pseudo-tractions arising from the Laplacian Navier-Stokes formulation yields accurate, physically consistent solutions. Nevertheless, when generalised Newtonian fluid models are considered, the resulting non-uniform viscosity fields render the classical Laplacian formulation inadequate. Thus, it is common practice to use the socalled stress-divergence formulation with natural boundary conditions known for causing nonphysical outflow behaviour. In order to overcome such a limitation, this work presents a novel mixed variational formulation that can be seen as a generalisation of the Laplacian Navier-Stokes form to fluids with shear-rate-dependent viscosity, as appearing in hemodynamic and polymeric flows. By appropriately manipulating the viscous terms in the variational formulation and employing a simple projection of the constitutive law, it is possible to devise a formulation with the desired natural boundary conditions and low computational complexity. Several numerical examples are presented to showcase the potential of our method, revealing improved accuracy and robustness in comparison with the state of the art.

Design, Development, and Temporal Evaluation of a Magnetic Resonance Imaging-Compatible In Vitro Circulation Model Using a Compliant Abdominal Aortic Aneurysm Phantom.

Biomechanical characterization of abdominal aortic aneurysms (AAAs) has become commonplace in rupture risk assessment studies. However, its translation to the clinic has been greatly limited due to the complexity associated with its tools and their implementation. The unattainability of patient-specific tissue properties leads to the use of generalized population-averaged material models in finite element analyses, which adds a degree of uncertainty to the wall mechanics quantification. In addition, computational fluid dynamics modeling of AAA typically lacks the patient-specific inflow and outflow boundary conditions that should be obtained by nonstandard of care clinical imaging. An alternative approach for analyzing AAA flow and sac volume changes is to conduct in vitro experiments in a controlled laboratory environment. In this study, we designed, built, and characterized quantitatively a benchtop flow loop using a deformable AAA silicone phantom representative of a patient-specific geometry. The impedance modules, which are essential components of the flow loop, were fine-tuned to ensure typical intraluminal pressure conditions within the AAA sac. The phantom was imaged with a magnetic resonance imaging (MRI) scanner to acquire time-resolved images of the moving wall and the velocity field inside the sac. Temporal AAA sac volume changes lead to a corresponding variation in compliance throughout the cardiac cycle. The primary outcome of this work was the design optimization of the impedance elements, the quantitative characterization of the resistive and capacitive attributes of a compliant AAA phantom, and the exemplary use of MRI for flow visualization and quantification of the deformed AAA geometry.

Consistency, convergence and error estimates for a mixed finite element–finite volume scheme to compressible Navier–Stokes equations with general inflow/outflow boundary data

Abstract We study convergence of a mixed finite element-finite volume scheme for the compressible Navier–Stokes equations in the isentropic regime under the full range $1<\gamma <\infty $ of the adiabatic coefficients $\gamma $ for the problem with general nonzero inflow–outflow boundary conditions. We propose a modification of Karper’s scheme (2013, A convergent FEM-DG method for the compressible Navier–Stokes equations. Numer. Math., 125, 441–510) in order to accommodate the nonzero boundary data, prove existence of its solutions, establish the stability and uniform estimates, derive a convenient consistency formulation of the balance laws and use it to show the weak convergence of the numerical solutions to a dissipative solution with the Reynolds defect introduced in Abbatiello et al. (2021, Generalized solutions to models of compressible viscous fluids. Discrete Contin. Dyn. Syst., 41, 1--28). If the target system admits a strong solution then the convergence is strong towards the strong solution. Moreover, we establish the convergence rate of the strong convergence in terms of the size of the space discretization $h$ (which is supposed to be comparable with the time step $\varDelta t$). In the case of the nonzero inflow–outflow boundary data all results are new. The latter result is new also for the no-slip boundary conditions and adiabatic coefficients $\gamma $ less than the threshold $3/2$.

Challenges in hemodynamics assessment in complex neurovascular geometries using computational fluid dynamics and benchtop flow simulation in 3D printed patient specific phantoms.

Complex hemodynamics assessments, as those related to carotid stenosis, are not always easily straightforward due to multifaceted challenges presented by the collateral flow in the Circle of Willis (CoW) and brain flow autoregulation. Advanced computational and benchtop methods to investigate hemodynamics aspects related to such complex flows are often used, however both have limitations and could lead to results which may diverge. In this study we investigated these aspects by performing correlated computational fluid dynamics (CFD) simulations and benchtop experiments in patient specific 3D printed phantoms. To investigate the flow in patients with carotid stenosis, we built two patient specific phantoms which contained the arterial lesion of interest, all main arteries leading to the brain, the CoW and main arteries branching from it. Each phantom was connected to a generic aortic arch. A programmable pump was connected and flow parameters were measured proximal and distal to the lesion and the contralateral arteries. The patient 3D geometry was used to perform a set of CFD simulations where inflow boundary conditions matched the experimental ones. Flow conditions were recorded at the same locations as the experimental setup. Further exploration into the translation from experimental to CFD was also performed by customizing vascular segmentation and physically manipulating arterial compliance properties. We initially observed significant differences between the CFD recordings and the experimental setup. Most of the differences were due to changes in phantom geometry when subjected to physiological pressures and simplistic outflow boundary conditions in the CFD simulations which do not account for pulsatility and nonlinear phenomena. Further work confirms the need for dynamic mesh behavior within CFD simulations attempting to computationally mimic 3D-printed benchtop experiments. Additionally, CFD simulation may benefit from considering geometry specific to a 3D-printed vascular phantom.

Numerical Simulation of Blood Flows in Patient-specific Abdominal Aorta with Primary Organs.

The abdominal aorta is the largest artery in the abdominal cavity that supplies blood flows to vital organs through the complex visceral arterial branches, including the celiac trunk (the liver, stomach, spleen, etc.), the renal arteries (the kidneys) and the superior and inferior mesenteric arteries (the small and large intestine, pancreas, etc.). An accurate simulation of blood flows in this network of arteries is important for the understanding of the hemodynamics in various organs of healthy and diseased patients, but the computational cost is very high. As a result, most researchers choose to focus on a portion of the artery or use a low-dimensional approximation of the artery. In the present work, we introduce a parallel algorithm for the modeling of pulsatile flows in the abdominal aorta with branches to the primary organs, and an organ-based two-level method for calculating the resistances for the outflow boundary conditions. With this highly parallel approach, the simulation of the blood flow for a cardiac cycle of the anatomically detailed aorta can be obtained within a few hours, and the blood distribution to organs including liver, spleen and kidneys are also computed with certain accuracy. Moreover, we discuss the significant hemodynamic differences resulted from the influence of the peripheral branches. In addition, we examine the accuracy of the results with respect to the mesh size and time-step size and show the high parallel scalability of the proposed algorithm with up to 3000 processor cores.

ON THE SENSITIVITY OF THE NONLINEAR TERM IN THE OUTFLOW BOUNDARY CONDITION

This paper studies the artificial outflow boundary condition for the Navier-Stokes system. This type of condition is widely used and it is therefore very important to study its influence on a numerical solution of the corresponding boundary-value problem. We particularly focus on the role of the coefficient in front of the nonlinear term in the boundary condition on the outflow. The influence of this term is examined numerically, comparing the obtained results in a close neighbourhood of the outflow. The numerical experiment is carried out for a fluid flow through the channel with so called sudden extension. Presented numerical results are obtained by means of the OpenFOAM toolbox. They confirm that the kinetic energy of the flow in the channel can be controlled by means of the proposed boundary condition.

Pulsatile Aortic Blood Flow—A Critical Assessment of Boundary Conditions

Abstract Patient specific (PS) blood flow studies have become popular in recent years but have thus far had limited clinical impact. This is possibly due to uncertainties and errors in the underlying models and simulations setup. This study focuses on the sensitivity of simulation results due to in- and outflow boundary conditions (BCs). Nine different inlet- and seven different outlet BCs were applied to two variants of a healthy subject's thoracic aorta. Temporal development of the flow is essential for the formation and development of helical/spiraling flow where the commonly observed clockwise helical motion may change direction during the heart-cycle. The sensitivity to temporal and spatial variations in the inlet conditions is significant both when expressed in terms of mean and maximal wall shear stress (WSS) and its different averaged variables, e.g., time-averaged WSS (TAWSS), oscillatory shear index (OSI) and relative residence time (RRT). The simulation results are highly sensitive to BC. For example, the maximal WSS may vary over three-orders of magnitude (1–1000 Pa) depending on particular combinations of BCs. Moreover, certain formulations of outlet BCs may be inconsistent with the computed flow field if the underlying assumptions of the space–time dependence are violated. The results of this study show that computational fluid dynamics (CFD) simulations can reveal flow details that can enhance understanding of blood flows. However, the results also demonstrate the potential difficulties in mimicking blood flow in clinical situations.

Generation and Absorption of Periodic Waves Traveling on a Uniform Current in a Fully Nonlinear BEM-based Numerical Wave Tank

Accurate and efficient numerical wave generation and absorption of two-dimensional nonlinear periodic waves traveling on a steady, uniform current were carried out in a potential, fully nonlinear numerical wave tank. The solver is based on the Βoundary Εlement Μethod (ΒΕΜ) with linear singularity distributions and plane elements and on the mixed Eulerian–Lagrangian formulation of the free surface equations. Wave generation is implemented along the inflow boundary by imposing the stream function wave solution, while wave absorption at both end-boundaries is effectively treated by introducing absorbing layers. On the absorbing beach side, the outflow boundary condition is modified to ensure that the solution accurately satisfies the dispersion relation of the generated waves. The modification involves a free-parameter that depends on the mass flux through the domain and is determined through a feedback error-correction loop. The developed method provides accurate time domain wave solutions for shallow, intermediate, and deep water depths of high wave steepness (wave heights up to 80% of the maximum value) that remain stable for 150 wave periods. This also holds in case a coplanar or opposing uniform current of velocity up to 20% of the wave celerity interacts with the wave.

Numerical considerations for advection-diffusion problems in cardiovascular hemodynamics.

Numerical simulations of cardiovascular mass transport pose significant challenges due to the wide range of Péclet numbers and backflow at Neumann boundaries. In this paper we present and discuss several numerical tools to address these challenges in the context of a stabilized finite element computational framework. To overcome numerical instabilities when backflow occurs at Neumann boundaries, we propose an approach based on the prescription of the total flux. In addition, we introduce a "consistent flux" outflow boundary condition and demonstrate its superior performance over the traditional zero diffusive flux boundary condition. Lastly, we discuss discontinuity capturing (DC) stabilization techniques to address the well-known oscillatory behavior of the solution near the concentration front in advection-dominated flows. We present numerical examples in both idealized and patient-specific geometries to demonstrate the efficacy of the proposed procedures. The three contributions discussed in this paper successfully address commonly found challenges when simulating mass transport processes in cardiovascular flows.

A multiscale fluid structure interaction model derived from Koiter shell equations

In this work we propose the numerical simulation of fluid structure interaction (FSI) problem by using a membrane model, derived from the Koiter shell equations. With this approach the thickness of the solid wall can be neglected, with a meaningful reduction of the computational cost of the numerical problem. The fluid structure problem is then reduced to the fluid equations on a moving mesh together with a particular Robin boundary condition imposed on the surface corresponding to the solid moving wall. Furthermore an artificial absorbing outflow boundary condition has been implemented in order to reduce the damping and reflections of the pressure waves at the domain’s outlet. This model is implemented and solved with an in-house finite elements code, and tested through axisymmetric cases that show the robustness of the developed algorithm. Finally, we report a comparison of the implemented model with results of a FSI monolithic model, based on non-linear incompressible structure.

An ensemble Kalman filter approach to parameter estimation for patient-specific cardiovascular flow modeling

Many previous studies have shown that the fidelity of three-dimensional cardiovascular flow simulations depends strongly on inflow and outflow boundary conditions that accurately describe the characteristics of the larger vascular network. These boundary conditions are generally based on lower-dimensional models that represent the upstream or downstream flow behavior in some aggregated fashion. However, the parameters of these models are patient-specific, and no clear technique exists for determining them. In this work, an ensemble Kalman filter (EnKF) is implemented for the purpose of estimating parameters in cardiovascular models through the assimilation of specific patients’ clinical measurements. Two types of models are studied: a fully zero-dimensional model of the right heart and pulmonary circulation, and a coupled 0D–1D model of the lower leg. Model parameters are estimated using measurements from both healthy and hypertensive patients, and demonstrate that the EnKF is able to generate distinct parameter sets whose model predictions produce features unique to each measurement set. Attention is also given toward the quality of model predictions made in the absence of direct clinical counterparts, as well as techniques to improve filter robustness against shrinking ensemble covariance.

A coupled one dimension and transmission line model for arterial flow simulation.

A broad choice of numerical schemes and methods currently exists for blood flow simulations. The results rely critically on the prescription of boundary conditions. The outflow boundary condition for a one-dimensional (1D) flow solver is usually prescribed via a Windkessel or lumped parameter model. The weakness of such an approach is the determination of the parameters. In the present work, we use an alternative approach, that is, a reflection coefficient (RC), to lumped parameter models for distal boundary conditions. With such a RC, the number of parameters required is reduced to one. We derive the theoretical foundation for the RC. Specifically, we couple a transmission line theory for peripheral resistance with a 1D arterial flow solver. We apply this method to a healthy and a stenosed virtual aorta, and show this method can reproduce some subtle features in arterial pressure propagation, such as the steepened pressure waveform and the reflection from the stenosed site. In summary, the RC parameter has strong physical implications in the theory of wave propagation and may be used in flow simulations where reflections need to be explicitly modeled. NOVELTY STATEMENT: A novel coupled one-dimensional-transimission line model has been developed in this work with detailed implementations. Only one outflow boundary condition, that is, the refection coefficient is required in the model. Reflections for a pulse wave from aortic terminals as well as from a stenotic site are numerically simulated.

Pulse wave propagation along human aorta: a model study

The problem of the pulse wave propagation and reflection along human aorta and the diagnostic importance of the pressure and flow signals are studied. The geometrical model is based on the detailed postmortem measurements on the systemic arterial trees. The aortic model consists of 36 aortic segments and 57 side branches of the aorta including the larger and medium vessels. It was shown most of the branches have zero wave reflection coefficients but the large branches like celiac, renal and iliac arteries could produce noticeable wave reflections. The smaller branches possess negative wave reflection coefficient and, thus, contribute to the blood suction effect and lower aortic resistance to the blood flow. Mathematical model is based on the axisymmetric incompressible Navier-Stokes equations for blood and the momentum equations for incompressible viscoelastic arterial wall. The Windkessel and structured tree outflow boundary conditions at the outlets of the branches have been used. The solution has been found as superposition of the forward and backward running waves. The blood flow curves measured in vivo by Doppler ultrasound in the larger systemic arteries of healthy volunteers have been used for identification of the model parameters. It is shown, the individual geometry plays an essential role in the location of the positive and negative wave reflection sites along the aorta and, thus, in the pressure and flow patterns and blood distribution into the branches. The influence of occlusion of the iliac arteries, low/high wall rigidity, and total length of aorta are studied on different individual geometries. The model can be used for determination of the individual parameters for patient-specific cardiovascular models and further modeling of the outcomes of the surgical and therapeutic procedures.

On Fokker-Planck equations with In- and Outflow of Mass

Motivated by modeling transport processes in the growth of neurons, we present results on (nonlinear) Fokker-Planck equations where the total mass is not conserved. This is either due to in- and outflow boundary conditions or to spatially distributed reaction terms. We are able to prove exponential decay towards equilibrium using entropy methods in several situations. As there is no conservation of mass it is difficult to exploit the gradient flow structure of the differential operator which renders the analysis more challenging. In particular, classical logarithmic Sobolev inequalities are not applicable any more. Our analytic results are illustrated by extensive numerical studies.

Modeling the hepatic arterial flow in living liver donor after left hepatectomy and postoperative boundary condition exploration.

Preoperative and postoperative hepatic perfusion is modeled with one-dimensional (1-D) Navier-Stokes equations. Flow rates obtained from ultrasound (US) data and impedance resulted from structured trees are the inflow and outflow boundary condition (BC), respectively. Structured trees terminate at the size of the arterioles, which can enlarge their size after hepatectomy. In clinical studies, the resistance to pulsatile arterial flow caused by the microvascular bed can be reflected by the resistive index (RI), a frequently used index in assessing arterial resistance. This study uses the RI in a novel manner to conveniently obtain the postoperative outflow impedance from the preoperative impedance. The major emphasis of this study is to devise a model to capture the postoperative hepatic hemodynamics after left hepatectomy. To study this, we build a hepatic network model and analyze its behavior under four different outflow impedance: (a) the same as preoperative impedance; (b) evaluated using the RI and preoperative impedance; (c) computed from structured tree BC with increased radius of terminal vessels; and (d) evaluated using structured tree with both increased radius of root vessel, ie, the outlets of the postoperative hepatic artery, and increased radius of terminal vessels. Our results show that both impedance from (b) and (d) give a physiologically reasonable postoperative hepatic pressure range, while the RI in (b) allows for a fast approximation of postoperative impedance. Since hemodynamics after hepatectomy are not fully understood, the methods used in this study to explore postoperative outflow BC are informative for future models exploring hemodynamic effects of partial hepatectomy.

Verification of the coupled‐momentum method with Womersley's Deformable Wall analytical solution

In this paper, we perform a verification study of the Coupled-Momentum Method (CMM), a 3D fluid-structure interaction (FSI) model which uses a thin linear elastic membrane and linear kinematics to describe the mechanical behavior of the vessel wall. The verification of this model is done using Womersley's deformable wall analytical solution for pulsatile flow in a semi-infinite cylindrical vessel. This solution is, under certain premises, the analytical solution of the CMM and can thus be used for model verification. For the numerical solution, we employ an impedance boundary condition to define a reflection-free outflow boundary condition and thus mimic the physics of the analytical solution, which is defined on a semi-infinite domain. We first provide a rigorous derivation of Womersley's deformable wall theory via scale analysis. We then illustrate different characteristics of the analytical solution such as space-time wave periodicity and attenuation. Finally, we present the verification tests comparing the CMM with Womersley's theory.

On outflow boundary conditions for CT-based computation of FFR: Examination using PET images

CT-based computations of fractional flow reserve (FFR) have been widely utilized for evaluating functional severity of a coronary artery stenosis. Whilst this approach has been successful clinically, assumptions involved in the analysis still need to be investigated for further improvement in predictive accuracy. To better understand the sensitivity of computational FFRs on outflow boundary condition – typically reflecting patient's own physiology only through anatomical features – FFR computations for 10 patients with different degree of stenosis was conducted. The computations were based on 3D anatomical model reconstructed from CT images and patient-specific in/outflow boundary conditions (BC). Two outflow BCs were considered: (1) conventional morphology-based and (2) PET perfusion-based conditions. The results showed that the FFRs derived from the two boundary conditions agree in general. It was also found that the FFRs computed with the morphology-based BC tend to estimate higher functional severity, especially in patients with reduced vasodilatory response under hyperaemia – an essential physiological condition in FFR measurement. Further investigation was made by varying hyperaemic resistances (30%-90% of the baseline) in the morphology-based BC. The variation of FFR for the varied resistances was narrow for patients with mild stenosis and wider for those who have severe stenosis. This latter approach confirmed that variability of FFR due to outflow condition tends to come from overestimation of vasodilatory response, especially those who have abnormal myocardial perfusion. The results suggest that outflow conditions that are more representative of each patient could be an effective way to improve CT-based FFR computation.