A novel formulation for Neumann inflow boundary conditions in biomechanics

The effect of airway motion and breathing phase during imaging on CFD simulations of respiratory airflow

RATIONALE

Computational fluid dynamics (CFD) simulations of respiratory airflow can quantify clinically useful information that cannot be obtained directly, such as the work of breathing (WOB), resistance to airflow, and pressure loss. However, patient-specific CFD simulations are often based on medical imaging that does not capture airway motion and thus may not represent true physiology, directly affecting those measurements.

OBJECTIVES

To quantify the variation of respiratory airflow metrics obtained from static models of airway anatomy at several respiratory phases, temporally averaged airway anatomies, and dynamic models that incorporate physiological motion.

METHODS

Neonatal airway images were acquired during free-breathing using 3D high-resolution MRI and reconstructed at several respiratory phases in two healthy subjects and two with airway disease (tracheomalacia). For each subject, five static (end expiration, peak inspiration, end inspiration, peak expiration, averaged) and one dynamic CFD simulations were performed. WOB, airway resistance, and pressure loss across the trachea were obtained for each static simulation and compared with the dynamic simulation results.

RESULTS

Large differences were found in the airflow variables between the static simulations at various respiratory phases and the dynamic simulation. Depending on the static airway model used, WOB, resistance, and pressure loss varied up to 237%, 200%, and 94% compared to the dynamic simulation respectively.

CONCLUSIONS

Changes in tracheal size and shape throughout the breathing cycle directly affect respiratory airflow dynamics and breathing effort. Simulations incorporating realistic airway wall dynamics most closely represent airway physiology; if limited to static simulations, the airway geometry must be obtained during the respiratory phase of interest for a given pathology.

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.

Modern discontinuous Galerkin methods for the simulation of transitional and turbulent flows in biomedical engineering: A comprehensive LES study of the FDA benchmark nozzle model

This work uses high-order discontinuous Galerkin discretization techniques to simulate transitional and turbulent flows through medical devices. Flows through medical devices are characterized by moderate Reynolds numbers and typically involve different flow regimes such as laminar, transitional, and turbulent flows. Previous studies for the FDA benchmark nozzle model revealed limitations of Reynolds-averaged Navier-Stokes turbulence models when applied to more complex flow scenarios. Recent works based on large-eddy simulation approaches indicate that these limitations can be overcome but also highlight potential limitations due to a high sensitivity with respect to numerical parameters. The methodology presented in this work introduces two novel ingredients compared with previous studies. Firstly, we use high-order discontinuous Galerkin methods for discretization in space. The inherent dissipation and dispersion properties of high-order discontinuous Galerkin discretizations are expected to render this approach well suited for transitional and turbulent flow simulations. Secondly, to mimic blinded CFD studies, we propose to use a precursor simulation approach in order to predict the inflow boundary condition for laminar, transitional, and turbulent flow regimes instead of prescribing analytical velocity profiles at the inflow. We investigate the whole range of Reynolds numbers as suggested by the FDA benchmark nozzle problem and compare the numerical results to experimental data obtained by particle image velocimetry in order to critically assess the predictive capabilities of the solver on the one hand and the suitability of the FDA nozzle problem as a benchmark in computational fluid dynamics on the other hand.

Gas exchange mechanisms in preterm infants on HFOV - a computational approach

High-frequency oscillatory ventilation (HFOV) is a commonly used therapy applied to neonates requiring ventilatory support during their first weeks of life. Despite its wide application, the underlying gas exchange mechanisms promoting the success of HVOF in neonatal care are not fully understood until today. In this work, a highly resolved computational lung model, derived from Magnetic Resonance Imaging (MRI) and Infant Lung Function Testing (ILFT), is used to reveal the reason for highly efficient gas exchange during HFOV, in the preterm infant. In total we detected six mechanisms that facilitate gas exchange during HFOV: (i) turbulent vortices in large airways; (ii) asymmetric in- and expiratory flow profiles; (iii) radial mixing in main bronchi; (iv) laminar flow in higher generations of the respiratory tract; (v) pendelluft; (vi) direct ventilation of central alveoli. The illustration of six specific gas transport phenomena during HFOV in preterm infants advances general knowledge on protective ventilation in neonatal care and can support decisions on various modes of ventilatory therapy at high frequencies.

Benchmark problems for numerical treatment of backflow at open boundaries

In computational fluid dynamics, incoming velocity at open boundaries, or backflow, often yields unphysical instabilities already for moderate Reynolds numbers. Several treatments to overcome these backflow instabilities have been proposed in the literature. However, these approaches have not yet been compared in detail in terms of accuracy in different physiological regimes, in particular because of the difficulty to generate stable reference solutions apart from analytical forms. In this work, we present a set of benchmark problems in order to compare different methods in different backflow regimes (with a full reversal flow and with propagating vortices after a stenosis). The examples are implemented in FreeFem++, and the source code is openly available, making them a solid basis for future method developments.

A multiphysics approach for modeling early atherosclerosis

This work is devoted to the development of a mathematical model of the early stages of atherosclerosis incorporating processes of all time scales of the disease and to show their interactions. The cardiovascular mechanics is modeled by a fluid-structure interaction approach coupling a non-Newtonian fluid to a hyperelastic solid undergoing anisotropic growth and a change of its constitutive equation. Additionally, the transport of low-density lipoproteins and its penetration through the endothelium is considered by a coupled set of advection-diffusion-reaction equations. Thereby, the permeability of the endothelium is wall-shear stress modulated resulting in a locally varying accumulation of foam cells triggering a novel growth and remodeling formulation. The model is calibrated and applied to an murine-specific case study, and a qualitative validation of the computational results is performed. The model is utilized to further investigate the influence of the pulsatile blood flow and the compliance of the artery wall to the atherosclerotic process. The computational results imply that the pulsatile blood flow is crucial, whereas the compliance of the aorta has only a minor influence on atherosclerosis. Further, it is shown that the novel model is capable to produce a narrowing of the vessel lumen inducing an adaption of the endothelial permeability pattern.

Dynamic modeling of uteroplacental blood flow in IUGR indicates vortices and elevated pressure in the intervillous space - a pilot study

Ischemic placental disease is a concept that links intrauterine growth retardation (IUGR) and preeclampsia (PE) back to insufficient remodeling of uterine spiral arteries. The rheological consequences of insufficient remodeling of uterine spiral arteries were hypothesized to mediate the considerably later manifestation of obstetric disease. However, the micro-rheology in the intervillous space (IVS) cannot be examined clinically and rheological animal models of the human IVS do not exist. Thus, an in silico approach was implemented to provide in vivo inaccessible data. The morphology of a spiral artery and the inflow region of the IVS were three-dimensionally reconstructed to provide a morphological stage for the simulations. Advanced high-end supercomputing resources were used to provide blood flow simulations at high spatial resolution. Our simulations revealed turbulent blood flow (high-velocity jets and vortices) combined with elevated blood pressure in the IVS and increased wall shear stress at the villous surface in conjunction with insufficient spiral artery remodeling only. Post-hoc histological analysis of uterine veins showed evidence of increased trophoblast shedding in an IUGR placenta. Our data support that rheological alteration in the IVS is a relevant mechanism linking ischemic placental disease to altered structural integrity and function of the placenta.

Power loss mechanisms in pathological tracheas

The effort required to inhale a breath of air is a critically important measure in assessing airway function. Although the contribution of the trachea to the total flow resistance of the airways is generally modest, pathological alterations in tracheal geometry can have a significant negative effect. This study investigates the mechanisms of flow energy loss in a healthy trachea and in four geometries affected by retrosternal goitre which can cause significant distortions of tracheal geometry including constriction and deviation with abnormal curvature. By separating out the component of energy loss related to the wall shear (frictional loss), striking differences are found between the patterns of energy dissipation in the normal and pathological tracheas. Furthermore the ratio of frictional to total loss is dramatically reduced in the pathological geometries.

Airflow and particle deposition simulations in health and emphysema: from in vivo to in silico animal experiments

Image-based in silico modeling tools provide detailed velocity and particle deposition data. However, care must be taken when prescribing boundary conditions to model lung physiology in health or disease, such as in emphysema. In this study, the respiratory resistance and compliance were obtained by solving an inverse problem; a 0D global model based on healthy and emphysematous rat experimental data. Multi-scale CFD simulations were performed by solving the 3D Navier-Stokes equations in an MRI-derived rat geometry coupled to a 0D model. Particles with 0.95 μm diameter were tracked and their distribution in the lung was assessed. Seven 3D-0D simulations were performed: healthy, homogeneous, and five heterogeneous emphysema cases. Compliance (C) was significantly higher (p = 0.04) in the emphysematous rats (C = 0.37 ± 0.14 cm(3)/cmH2O) compared to the healthy rats (C = 0.25 ± 0.04 cm(3)/cmH2O), while the resistance remained unchanged (p = 0.83). There were increases in airflow, particle deposition in the 3D model, and particle delivery to the diseased regions for the heterogeneous cases compared to the homogeneous cases. The results highlight the importance of multi-scale numerical simulations to study airflow and particle distribution in healthy and diseased lungs. The effect of particle size and gravity were studied. Once available, these in silico predictions may be compared to experimental deposition data.

A stable approach for coupling multidimensional cardiovascular and pulmonary networks based on a novel pressure-flow rate or pressure-only Neumann boundary condition formulation

In many biomedical flow problems, reversed flows along with standard treatment of Neumann boundary conditions can cause instabilities. We have developed a method that resolves these instabilities in a consistent way while maintaining correct pressure and flow rate values. We also are able to remove the necessary prescription of both pressure and velocities/flow rates to problems where only pressure is known. In addition, the method is extended to coupled 3D/reduced-D fluid and fluid-structure interaction models. Numerical examples mainly focus on using Neumann boundary condition in cardiovascular and pulmonary systems, particularly, coupled with 3D-1D and 3D-0D models. Inflow pressure, traction, and impedance boundary conditions are first tested on idealized tubes for various Womersley numbers. Both pressure and flow rate are shown to match the analytical solutions for these examples. Our method is then tested on a coupled 1D-3D-1D artery example, demonstrating the power and simplicity of extending this method toward fluid-structure interaction. Finally, the proposed method is investigated for a coupled 3D-0D patient-specific full lung model during spontaneous breathing. All coupled 3D/reduced-D results show a perfect matching of pressure and flow rate between 3D and corresponding reduced-D boundaries. The methods are straight-forward to implement in contrast to using Lagrange multipliers as previously proposed in other studies.

Coupled and reduced dimensional modeling of respiratory mechanics during spontaneous breathing

In this paper, we develop a total lung model based on a tree of 0D airway and acinar models for studying respiratory mechanics during spontaneous breathing. This model utilizes both computer tomography-based geometries and artificially generated lobe-filling airway trees to model the entire conducting region of the lung. Beyond the conducting airways, we develop an acinar model, which takes into account the alveolar tissue resistance, compliance, and the intrapleural pressure. With this methodology, we compare four different 0D models of airway mechanics and determine the best model based on a comparison with a 3D-0D coupled model of the conducting airways; this methodology is possible because the majority of airway resistance is confined to the lower generations, that is, the trachea and the first few bronchial generations. As an example application of the model, we simulate the flow and pressure dynamics under spontaneous breathing conditions, that is, at flow conditions driven purely by pleural space pressure. The results show good agreement, both qualitatively and quantitatively, with reported physiological values. One of the key advantages of this model is the ability to provide insight into lung ventilation in the peripheral regions. This is often crucial because this is where information, specifically for studying diseases and gas exchange, is needed. Thus, the model can be used as a tool for better understanding local peripheral lung mechanics without excluding the upper portions of the lung. This tool will be also useful for in vitro investigations of lung mechanics in both health and disease.