Outflow boundary conditions for 3D simulations of non-periodic blood flow and pressure fields in deformable arteries

The simulation of blood flow and pressure in arteries requires outflow boundary conditions that incorporate models of downstream domains. We previously described a coupled multidomain method to couple analytical models of the downstream domains with 3D numerical models of the upstream vasculature. This prior work either included pure resistance boundary conditions or impedance boundary conditions based on assumed periodicity of the solution. However, flow and pressure in arteries are not necessarily periodic in time due to heart rate variability, respiration, complex transitional flow or acute physiological changes. We present herein an approach for prescribing lumped parameter outflow boundary conditions that accommodate transient phenomena. We have applied this method to compute haemodynamic quantities in different physiologically relevant cardiovascular models, including patient-specific examples, to study non-periodic flow phenomena often observed in normal subjects and in patients with acquired or congenital cardiovascular disease. The relevance of using boundary conditions that accommodate transient phenomena compared with boundary conditions that assume periodicity of the solution is discussed.     

Fractal network model for simulating abdominal and lower extremity blood flow during resting and exercise conditions

We present a one-dimensional (1D) fluid dynamic model that can predict blood flow and blood pressure during exercise using data collected at rest. To facilitate accurate prediction of blood flow, we developed an impedance boundary condition using morphologically derived structured trees. Our model was validated by computing blood flow through a model of large arteries extending from the thoracic aorta to the profunda arteries. The computed flow was compared against measured flow in the infrarenal (IR) aorta at rest and during exercise. Phase contrast-magnetic resonance imaging (PC-MRI) data was collected from 11 healthy volunteers at rest and during steady exercise. For each subject, an allometrically-scaled geometry of the large vessels was created. This geometry extends from the thoracic aorta to the femoral arteries and includes the celiac, superior mesenteric, renal, inferior mesenteric, internal iliac and profunda arteries. During rest, flow was simulated using measured supraceliac (SC) flow at the inlet and a uniform set of impedance boundary conditions at the 11 outlets. To simulate exercise, boundary conditions were modified. Inflow data collected during steady exercise was specified at the inlet and the outlet boundaries were adjusted as follows. The geometry of the structured trees used to compute impedance was scaled to simulate the effective change in the cross-sectional area of resistance vessels and capillaries due to exercise. The resulting computed flow through the IR aorta was compared to measured flow. This method produces good results with a mean difference between paired data to be 1.1 +/- 7 cm(3) s(- 1) at rest and 4.0 +/- 15 cm(3) s(- 1) at exercise. While future work will improve on these results, this method provides groundwork with which to predict the flow distributions in a network due to physiologic regulation.     

Finite element models of total shoulder replacement: Application of boundary conditions

We present a one-dimensional (1D) fluid dynamic model that can predict blood flow and blood pressure during exercise using data collected at rest. To facilitate accurate prediction of blood flow, we developed an impedance boundary condition using morphologically derived structured trees. Our model was validated by computing blood flow through a model of large arteries extending from the thoracic aorta to the profunda arteries. The computed flow was compared against measured flow in the infrarenal (IR) aorta at rest and during exercise. Phase contrast-magnetic resonance imaging (PC-MRI) data was collected from 11 healthy volunteers at rest and during steady exercise. For each subject, an allometrically-scaled geometry of the large vessels was created. This geometry extends from the thoracic aorta to the femoral arteries and includes the celiac, superior mesenteric, renal, inferior mesenteric, internal iliac and profunda arteries. During rest, flow was simulated using measured supraceliac (SC) flow at the inlet and a uniform set of impedance boundary conditions at the 11 outlets. To simulate exercise, boundary conditions were modified. Inflow data collected during steady exercise was specified at the inlet and the outlet boundaries were adjusted as follows. The geometry of the structured trees used to compute impedance was scaled to simulate the effective change in the cross-sectional area of resistance vessels and capillaries due to exercise. The resulting computed flow through the IR aorta was compared to measured flow. This method produces good results with a mean difference between paired data to be 1.1 ± 7 cm³ s^? 1 at rest and 4.0 ± 15 cm³ s^? 1 at exercise. While future work will improve on these results, this method provides groundwork with which to predict the flow distributions in a network due to physiologic regulation.     

FSI Analysis of a healthy and a stenotic human trachea under impedance-based boundary conditions

In this work, a fluid-solid interaction (FSI) analysis of a healthy and a stenotic human trachea was studied to evaluate flow patterns, wall stresses, and deformations under physiological and pathological conditions. The two analyzed tracheal geometries, which include the first bifurcation after the carina, were obtained from computed tomography images of healthy and diseased patients, respectively. A finite element-based commercial software code was used to perform the simulations. The tracheal wall was modeled as a fiber reinforced hyperelastic solid material in which the anisotropy due to the orientation of the fibers was introduced. Impedance-based pressure waveforms were computed using a method developed for the cardiovascular system, where the resistance of the respiratory system was calculated taking into account the entire bronchial tree, modeled as binary fractal network. Intratracheal flow patterns and tracheal wall deformation were analyzed under different scenarios. The simulations show the possibility of predicting, with FSI computations, flow and wall behavior for healthy and pathological tracheas. The computational modeling procedure presented herein can be a useful tool capable of evaluating quantities that cannot be assessed in vivo, such as wall stresses, pressure drop, and flow patterns, and to derive parameters that could help clinical decisions and improve surgical outcomes.     

Direct numerical simulation of a 2D-stented aortic heart valve at physiological flow rates

We study the nonlinear interaction of an aortic heart valve, composed of hyperelastic corrugated leaflets of finite density attached to a stented vessel under physiological flow conditions. In our numerical simulations, we use a 2D idealised representation of this arrangement. Blood flow is caused by a time-varying pressure gradient that mimics that of the aortic valve and corresponds to a peak Reynolds number equal to 4050. Here, we fully account for the shear-thinning behaviour of the blood and large deformations and contact between the leaflets by solving the momentum and mass balances for blood and leaflets. The mixed finite element/Galerkin method along with linear discontinuous Lagrange multipliers for coupling the fluid and elastic domains is adopted. Moreover, a series of challenging numerical issues such as the finite length of the computational domain and the conditions that should be imposed on its inflow/outflow boundaries, the accurate time integration of the parabolic and hyperbolic momentum equations, the contact between the leaflets and the non-conforming mesh refinement in part of the domain are successfully resolved. Calculations for the velocity and the shear stress fields of the blood reveal that boundary layers appear on both sides of a leaflet. The one along the ventricular side transfers blood with high momentum from the core region of the vessel to the annulus or the sinusoidal expansion, causing the continuous development of flow instabilities. At peak systole, vortices are convected in the flow direction along the annulus of the vessel, whereas during the closure stage of the valve, an extremely large vortex develops in each half of the flow domain.     

Tuning of pulmonary arterial circulation evidenced by MR phase mapping in healthy volunteers.

Magnetic resonance (MR) phase mapping was used to noninvasively assess both blood flow and cross-sectional area (CSA) in the main pulmonary artery (MPA) of 12 healthy volunteers. Flow and CSA patterns exhibited two positive peaks: high systolic and small diastolic. This finding can be explained using a simple "distributed" theoretical model that takes into account the role of a reflected pressure wave from pulmonary vascular impedance in generating a diastolic flow. The mean reflection coefficient of pressure wave, MPA input impedance, and pulmonary vascular impedance were assessed. We verified, in this series, that pressure wave velocity appears to be age-dependent. MR phase mapping has been used to observe the tuning (resonance) of the right cardiovascular system at rest under physiological conditions. MR phase mapping could be used to assess pathological modifications of the tuning that occurs in cases of pulmonary arterial hypertension.     

Impedance-based outflow boundary conditions for human carotid haemodynamics

In this study, we develop structured tree outflow boundary conditions for modelling the human carotid haemodynamics. The model geometry was reconstructed through computerised tomography scan. Unsteady-state computational fluid dynamic analyses were performed under different conditions using a commercial software package ADINA R&D, Inc., (Watertown, MA, USA) in order to assess the impact of the boundary conditions on the flow variables. In particular, the results showed that the peripheral vessels massively impact the pressure while the flow is relatively unaffected. As an example of application of these outflow conditions, an unsteady fluid-structure interaction (FSI) simulation was carried out and the dependence of the wall shear stress (WSS) on the arterial wall compliance in the carotid bifurcation was studied. In particular, a comparison between FSI and rigid-wall models was conducted. Results showed that the WSS distributions were substantially affected by the diameter variation of the arterial wall. In particular, even similar WSS distributions were found for both cases, and differences in the computed WSS values were also found.     

FSI analysis of a human trachea before and after prosthesis implantation

In this work we analyzed the response of a stenotic trachea after a stent implantation. An endotracheal stent is the common treatment for tracheal diseases such as stenosis, chronic cough, or dispnoea episodes. Medical treatment and surgical techniques are still challenging due to the difficulties in overcoming potential complications after prosthesis implantation. A finite element model of a diseased and stented trachea was developed starting from a patient specific computerized tomography (CT) scan. The tracheal wall was modeled as a fiber reinforced hyperelastic material in which we modeled the anisotropy due to the orientation of the collagen fibers. Deformations of the tracheal cartilage rings and of the muscular membrane, as well as the maximum principal stresses, are analyzed using a fluid solid interaction (FSI) approach. For this reason, as boundary conditions, impedance-based pressure waveforms were computed modeling the nonreconstructed vessels as a binary fractal network. The results showed that the presence of the stent prevents tracheal muscle deflections and indicated a local recirculatory flow on the stent top surface which may play a role in the process of mucous accumulation. The present work gives new insight into clinical procedures, predicting their mechanical consequences. This tool could be used in the future as preoperative planning software to help the thoracic surgeons in deciding the optimal prosthesis type as well as its size and positioning.     

Influence of model boundary conditions on blood flow patterns in a patient specific stenotic right coronary artery

Background

In literature, the effect of the inflow boundary condition was investigated by examining the impact of the waveform and the shape of the spatial profile of the inlet velocity on the cardiac hemodynamics. However, not much work has been reported on comparing the effect of the different combinations of the inlet/outlet boundary conditions on the quantification of the pressure field and flow distribution patterns in stenotic right coronary arteries.

Method

Non-Newtonian models were used to simulate blood flow in a patient-specific stenotic right coronary artery and investigate the influence of different boundary conditions on the phasic variation and the spatial distribution patterns of blood flow. The 3D geometry of a diseased artery segment was reconstructed from a series of IVUS slices. Five different combinations of the inlet and the outlet boundary conditions were tested and compared.

Results

The temporal distribution patterns and the magnitudes of the velocity, the wall shear stress (WSS), the pressure, the pressure drop (PD), and the spatial gradient of wall pressure (WPG) were different when boundary conditions were imposed using different pressure/velocity combinations at inlet/outlet. The maximum velocity magnitude in a cardiac cycle at the center of the inlet from models with imposed inlet pressure conditions was about 29% lower than that from models using fully developed inlet velocity data. Due to the fact that models with imposed pressure conditions led to blunt velocity profile, the maximum wall shear stress at inlet in a cardiac cycle from models with imposed inlet pressure conditions was about 29% higher than that from models with imposed inlet velocity boundary conditions. When the inlet boundary was imposed by a velocity waveform, the models with different outlet boundary conditions resulted in different temporal distribution patterns and magnitudes of the phasic variation of pressure. On the other hand, the type of different boundary conditions imposed at the inlet and the outlet did not have significant effect on the spatial distribution patterns of the PD, the WPG and the WSS on the lumen surface, regarding the locations of the maximum and the minimum of each quantity.

Conclusions

The observations from this study indicated that the ways how pressure and velocity boundary conditions are imposed in computational models have considerable impact on flow velocity and shear stress predictions. Accuracy of in vivo measurements of blood pressure and velocity is of great importance for reliable model predictions.

     

An experimental study of pressure losses in pulsatile flows through rigid and pulsating stenosis

The time-dependent pressure curves of a pulsatile flow across rigid and pulsating stenoses were investigated experimentally in a laboratory simulator of the outflow tract of the heart right ventricle. The experiments were performed within the range of physiological conditions of frequency and flow rate. The experimental setup consisted of a closed flow system which was operated by a pulsatile pump, and a test chamber which enabled checking different modes of stenosis. Rigid constrictions were simulated by means of axisymmetric blunt-ended annular plugs with moderate-to-severe area reductions. The pulsating stenosis consisted of a short starling resistor device operated by a pulsating external pressure which was synchronized by the pulsatile flow. It was found that the shape of the time-dependent pressure curve upstream of the stenosis was different in the case of rigid stenosis than in the pulsating one. Potential clinical applications of the work may relate to diagnosis of the type of stenosis in the congenital heart disease known as Tetralogy of Fallot.     

A three-dimensional computational analysis of fluid-structure interaction in the aortic valve

Numerical analysis of the aortic valve has mainly been focused on the closing behaviour during the diastolic phase rather than the kinematic opening and closing behaviour during the systolic phase of the cardiac cycle. Moreover, the fluid-structure interaction in the aortic valve system is most frequently ignored in numerical modelling. The effect of this interaction on the valve's behaviour during systolic functioning is investigated. The large differences in material properties of fluid and structure and the finite motion of the leaflets complicate blood-valve interaction modelling. This has impeded numerical analyses of valves operating under physiological conditions. A numerical method, known as the Lagrange multiplier based fictitious domain method, is used to describe the large leaflet motion within the computational fluid domain. This method is applied to a three-dimensional finite element model of a stented aortic valve. The model provides both the mechanical behaviour of the valve and the blood flow through it. Results show that during systole the leaflets of the stented valve appear to be moving with the fluid in an essentially kinematical process governed by the fluid motion.     

Comparative study of viscoelastic arterial wall models in nonlinear one-dimensional finite element simulations of blood flow

It is well known that blood vessels exhibit viscoelastic properties, which are modeled in the literature with different mathematical forms and experimental bases. The wide range of existing viscoelastic wall models may produce significantly different blood flow, pressure, and vessel deformation solutions in cardiovascular simulations. In this paper, we present a novel comparative study of two different viscoelastic wall models in nonlinear one-dimensional (1D) simulations of blood flow. The viscoelastic models are from papers by Holenstein et al. in 1980 (model V1) and Valdez-Jasso et al. in 2009 (model V2). The static elastic or zero-frequency responses of both models are chosen to be identical. The nonlinear 1D blood flow equations incorporating wall viscoelasticity are solved using a space-time finite element method and the implementation is verified with the Method of Manufactured Solutions. Simulation results using models V1, V2 and the common static elastic model are compared in three application examples: (i) wave propagation study in an idealized vessel with reflection-free outflow boundary condition; (ii) carotid artery model with nonperiodic boundary conditions; and (iii) subject-specific abdominal aorta model under rest and simulated lower limb exercise conditions. In the wave propagation study the damping and wave speed were largest for model V2 and lowest for the elastic model. In the carotid and abdominal aorta studies the most significant differences between wall models were observed in the hysteresis (pressure-area) loops, which were larger for V2 than V1, indicating that V2 is a more dissipative model. The cross-sectional area oscillations over the cardiac cycle were smaller for the viscoelastic models compared to the elastic model. In the abdominal aorta study, differences between constitutive models were more pronounced under exercise conditions than at rest. Inlet pressure pulse for model V1 was larger than the pulse for V2 and the elastic model in the exercise case. In this paper, we have successfully implemented and verified two viscoelastic wall models in a nonlinear 1D finite element blood flow solver and analyzed differences between these models in various idealized and physiological simulations, including exercise. The computational model of blood flow presented here can be utilized in further studies of the cardiovascular system incorporating viscoelastic wall properties.     

Particle Image Velocimetry Study of Pulsatile Flow in Bi-leaflet Mechanical Heart Valves with Image Compensation Method

Particle Image Velocimetry (PIV) is an important technique in studying blood flow in heart valves. Previous PIV studies of flow around prosthetic heart valves had different research concentrations, and thus never provided the physical flow field pictures in a complete heart cycle, which compromised their pertinence for a better understanding of the valvular mechanism. In this study, a digital PIV (DPIV) investigation was carried out with improved accuracy, to analyse the pulsatile flow field around the bi-leaflet mechanical heart valve (MHV) in a complete heart cycle. For this purpose a pulsatile flow test rig was constructed to provide the necessary in vitro test environment, and the flow field around a St. Jude size 29 bi-leaflet MHV and a similar MHV model were studied under a simulated physiological pressure waveform with flow rate of 5.2 l/min and pulse rate at 72 beats/min. A phase-locking method was applied to gate the dynamic process of valve leaflet motions. A special image-processing program was applied to eliminate optical distortion caused by the difference in refractive indexes between the blood analogue fluid and the test section. Results clearly showed that, due to the presence of the two leaflets, the valvular flow conduit was partitioned into three flow channels. In the opening process, flow in the two side channels was first to develop under the presence of the forward pressure gradient. The flow in the central channel was developed much later at about the mid-stage of the opening process. Forward flows in all three channels were observed at the late stage of the opening process. At the early closing process, a backward flow developed first in the central channel. Under the influence of the reverse pressure gradient, the flow in the central channel first appeared to be disturbed, which was then transformed into backward flow. The backward flow in the central channel was found to be the main driving factor for the leaflet rotation in the valve closing process. After the valve was fully closed, local flow activities in the proximity of the valve region persisted for a certain time before slowly dying out. In both the valve opening and closing processes, maximum velocity always appeared near the leaflet trailing edges. The flow field features revealed in the present paper improved our understanding of valve motion mechanism under physiological conditions, and this knowledge is very helpful in designing the new generation of MHVs.     

A new method to measure nasal impedance in spontaneously breathing adults

As an alternative to standard rhinomanometric methods, we applied forced oscillations at the mouth in five normal subjects and determined their nasal impedance with a novel method involving flow subtraction. Pressure oscillations of constant amplitude were applied at the mouth of a subject both when the nostrils were open and when they were closed with a noseclip. The airflows measured under the two conditions were subtracted to yield the oscillating nasal airflow at the imposed pressure. The resultant pressure-flow relation defined the nasal impedance of the subject. For frequencies between 3 and 15 Hz, the transnasal pressure-flow relation was well described by a linear lumped parameter model consisting of a resistive and inertial element. Nasal resistance obtained with flow subtraction did not differ significantly from control measurements obtained while the subjects performed the Valsalva maneuver. In contrast, nasal inertance obtained with flow subtraction was approximately twice that obtained with the Valsalva method. The difference between inertances may reflect structural changes in nasopharyngeal dimensions that occur with the Valsalva maneuver. We conclude that the mechanical impedance of the nasal passage may be determined during spontaneous breathing from the response to imposed forced oscillations at the mouth. The noninvasive nature of this method suggests that it may be simpler to implement than traditional rhinomanometric methods.     

Proposition of an outflow boundary approach for carotid artery stenosis CFD simulation

The purpose of this study was to propose an innovative approach of setting outlet boundary conditions for the computational fluid dynamics (CFD) simulation of human common carotid arteries (CCAs) bifurcation based on the concept of energy loss minimisation at flow bifurcation. Comparisons between this new approach and previously reported boundary conditions were also made. The results showed that CFD simulation based on the proposed boundary conditions gave an accurate prediction of the critical stenosis ratio of carotid arteries (at around 65%). Other boundary conditions, such as the constant external pressure (P = 0) and constant outflow ratio, either overestimated or underestimated the critical stenosis ratio of carotid arteries. The patient-specific simulation results furthermore indicated that the calculated internal carotid artery flow ratio at CCA bifurcation (61%) coincided with the result obtained by clinical measurements through the use of Colour Doppler ultrasound.     

Pressure drop and flow rate measurements in a human aortic bifurcation cast for steady and pulsatile flow

Pressure drop and flow rate measurements in a rigid cast of a human aortic bifurcation under both steady and physiological pulsatile flow conditions are reported. Integral momentum and mechanical energy balances are used to calculate impedance, spatially averaged wall shear stress and viscous dissipation rate from the data. In the daughter branches, steady flow impedance is within 30% of the Poiseuille flow prediction, while pulsatile flow impedance is within a factor of 2 of fully developed, oscillatory, straight tube flow theory (Womersley theory). Estimates of wall shear stress are in accord with measurements obtained from velocity profiles. Mean pressure drop and viscous dissipation rate are elevated in pulsatile flow relative to steady flow at the mean flow rate, and the exponents of their Reynolds number dependence are in accord with available theory.     

Osmotic stimuli induce epithelial-dependent relaxation in the guinea pig trachea

Epithelium in airways, like endothelium in blood vessels, may regulate responses of adjacent smooth muscle. To study the intact trachea from guinea pigs we developed an in vitro preparation that permits independent stimulation from either the inner epithelial surface or the outer serosal surface. The whole guinea pig trachea was excised, cannulated, and perfused at a constant flow with Krebs-Henseleit (KH) solution that was in direct contact with the inner epithelial-lined surface. The outer serosal surface of the trachea was immersed in a separate system (bath) containing KH solution. Tracheal responses were assessed by measuring the pressure drop between the tracheal inlet and the outlet under conditions of constant flow. When the trachea was precontracted with carbachol or KCl, hyperosmolar stimuli (KCl, mannitol, urea, or NaCl) produced concentration-dependent relaxation when applied to the inner epithelial surface. Relaxation was not produced when the hyperosmolar stimulus was applied to the serosal surface and was markedly reduced or abolished when the epithelial surface had been physically damaged or removed. These results indicate that hyperosmotic stimuli induce epithelial-dependent relaxation of trachea. A defect in this mechanism may be partially responsible for the bronchoconstriction seen in asthmatic subjects after exercise.     

Pulsed magnetohydrodynamic blood flow in a rigid vessel under physiological pressure gradient

Blood flow in a steady magnetic field has been of great interest over recent years. Many researchers have examined the effects of magnetic fields on velocity profiles and arterial pressure, and major studies have focused on steady or sinusoidal flows. In this paper, we present a solution for pulsed magnetohydrodynamic blood flow with a somewhat realistic physiological pressure wave obtained using a Windkessel lumped model. A pressure gradient is derived along a rigid vessel placed at the output of a compliant module which receives the ventricle outflow. Then, velocity profile and flow rate expressions are derived in the rigid vessel in the presence of a steady transverse magnetic field. As expected, results showed flow retardation and flattening. The adaptability of our solution approach allowed a comparison with previously addressed flow cases and calculations presented a good coherence with those well established solutions.     

Hemodynamics analysis of a stenosed carotid bifurcation and its plaque-mitigating design

Considering transient two-dimensional laminar flow in a diseased carotid artery segment with realistic inlet and outflow conditions, detailed velocity profiles, pressure fields, wall shear stress distributions and coupled, localized plaque formations have been simulated. The type of outflow boundary condition influences to a certain degree the extent of plaque build-up, which in turn reduces "disturbed flow" phenomena such as flow separations, recirculation zones, and wavy flow patterns in the artery branches during portions of the pulse. Based on computer experiments varying key geometric factors, a plaque-mitigating design of a carotid artery bifurcation has been proposed. Elimination of the carotid bulb, a smaller bifurcation angle, lower area ratios, and smooth wall curvatures generated a design with favorable hemodynamics parameters, leading to reduced plaque build-up by factors of 10 and 2 in the internal carotid and in the external carotid, respectively.     

In vitro validation of finite-element model of AAA hemodynamics incorporating realistic outlet boundary conditions

The purpose of this study is to validate numerical simulations of flow and pressure in an abdominal aortic aneurysm (AAA) using phase-contrast magnetic resonance imaging (PCMRI) and an in vitro phantom under physiological flow and pressure conditions. We constructed a two-outlet physical flow phantom based on patient imaging data of an AAA and developed a physical Windkessel model to use as outlet boundary conditions. We then acquired PCMRI data in the phantom while it operated under conditions mimicking a resting and a light exercise physiological state. Next, we performed in silico numerical simulations and compared experimentally measured velocities, flows, and pressures in the in vitro phantom to those computed in the in silico simulations. There was a high degree of agreement in all of the pressure and flow waveform shapes and magnitudes between the experimental measurements and simulated results. The average pressures and flow split difference between experiment and simulation were all within 2%. Velocity patterns showed good agreement between experimental measurements and simulated results, especially in the case of whole-cycle averaged comparisons. We demonstrated methods to perform in vitro phantom experiments with physiological flows and pressures, showing good agreement between numerically simulated and experimentally measured velocity fields and pressure waveforms in a complex patient-specific AAA geometry.