Effect of turbulent models on left ventricle diastolic flow patterns simulation

Abstract

Vortex structures, as one of the most important features of cardiac flow, have a crucial impact on the left ventricle function and pathological conditions. These swirling flows are closely related to the presence of turbulence in left ventricle which is investigated in the current study. Using an extended model of the left heart, including a fluid-structure interaction (FSI) model of the mitral valve with a realistic geometry, the effect of using two numerical turbulent models, k-ε and Spalart-Allmaras (SA), on diastolic flow patterns is studied and compared with results from laminar flow model. As a result of the higher dissipation rate in turbulent models (k-ε and SA), vortices are larger and stronger in the laminar flow model. Comparing E/A ratio in the three models (Laminar, k-ε, and SA) with experimental data from healthy subjects, it is concluded that the results from k-ε model are more accurate.     

Ventricular septal defect with bidirectional shunting: Mathematical considerations

Previously proposed formulae for the quantitative estimation of bidirectional shunts across ventricular septal defects require determination of the oxygen contents of mixed venous, pulmonary artery, pulmonary venous, and aortic blood. Because these formulae do not take into account the mixing of oxygenated with unoxygenated blood within the ventricles, their use must result in underestimation of shunt flows in each direction. A mathematical model for a ventricular defect is examined, in which it is assumed that mixing of blood occurs in each of six sites in the venae cavae or right atrium, right ventricle, pulmonary artery, left atrium, left ventricle, and aorta. A total of fourteen streams of blood can flow from one to another of these mixing sites. As long as complete mixing occurs in the six specified mixing sites, any degree of mixing or non-mixing of the various streams is permitted. From the equations characterizing the model, formulae are derived in which the shunt flow in each direction is expressed in terms of the oxygen contents in the six mixing sites and the fractions of blood which enter the shunt from either side without prior mixing in a ventricular mixing site. The previously reported formulae, which apply when no ventricular mixing is allowed to occur, lead to theoretical minimum values for the shunt flows in each direction. At the opposite extreme where all the shunting blood is required to mix in a ventricle before entering the shunt, formulae for maximum possible shunt flows are also obtained. The absolute values for the left-to-right and right-to-left shunt flows, which must lie somewhere between the theoretical maximum and minimum values, cannot be computed from blood gas data alone. This work was supported in part by grant HE-07563 from the National Heart Institute of the National Institutes of Health and grants-in-aid from the American and North Carolina Heart Associations and the Life Insurance Medical Research Fund. Work completed during tenure as U.S.P.H.S. post-doctoral fellow.     

Parallel-plate flow chamber for studies of 3D flow-endothelium interaction

Flow induced shear stress influences vascular cellular biology and pathophysiology in numerous ways. Previous in vitro studies on interactions between flow and endothelial cells using parallel-plate flow chambers involve two-dimensional flows, whereas flows in larger vessels are commonly three-dimensional. We have constructed a parallel plate flow chamber with a backward facing step aligned oblique to the axis of the chamber. Flow visualisation by steady injection of ink through a hypodermic tube reveals swirling flow in the recirculation region downstream of the step. At given angles of the step, theta; (to the axis of the chamber), the pitch of the swirl and the width of the separation region, as measured in the direction perpendicular to the step, increase with the Reynolds number (Re). On the other hand, at given values of Re, reduction of theta; results in increases in the swirl pitch but decreases in the width of the separation zone. Furthermore, clearance time of ink from the separation region is shorter with an oblique step than a perpendicular one at given Re. Computer simulation confirms the 3D swirling flow created by the oblique step and provides detailed distribution of wall shear stresses in the flow chamber.     

Blood flow measurements by the reference sample method with microsphere injection in to the aorta: an accurate and easy approach.

The validity of hemodynamic measurements by the reference sample method with microspheres injection into the aorta, via a carotid artery catheter, was evaluated in rats and compared with the results obtained after left ventricle injection. In the aorta injection group, a good mix of microspheres was observed in 83% of the animals. Moreover, a symmetrical distribution of microspheres was observed in 10 out of 12 rats (83%). An excellent correlation between right and left kidney-testes blood flows was observed (r = 0.93 and 0.96, respectively; P less than 0.01). Mean arterial pressure was not modified during microspheres injection into the aorta. Cardiac output (104 +/- 26 vs 101 +/- 23 ml/min, NS) and portal blood flow (14.2 +/- 3.3 vs 13.5 +/- 2.2 ml/min, NS) were similar after aorta and left ventricle injections series, respectively. Our results indicate that the injection of microspheres into the aorta is an adequate and easy approach to systemic and splanchnic hemodynamic measurements. This approach could be a good alternative to left ventricle injection of microspheres in experimental studies in rats.     

Effect of prosthetic mitral valve geometry and orientation on flow dynamics in a model human left ventricle

Pulsatile flow dynamics through bileaflet (St Jude and Duromedics), tilting disc (Bjork-Shiley and Omniscience), caged ball (Starr-Edwards), pericardial (Edwards) and porcine (Carpentier-Edwards) mitral valves in a model human left ventricle (LV) were studied. The model human ventricle, obtained from an in situ diastolic casting, was incorporated into a mock circulatory system. Measurements were made at various heart rates and flow rates. These included the transvalvular pressure drop and regurgitation in percent and cm3 beat-1. The effect of valve geometry and the orientation of the valve with respect to the valve annulus was analyzed using a flow visualization technique. Qualitative flow visualization study indicates certain preferred orientations for the tilting disc and bileaflet valve prostheses in order to obtain a smooth washout of flow in the LV chamber.     

Computational analysis of blood flow in an integrated model of the left ventricle and the aorta

To study the effects of intraventricular flow dynamics on the aortic flow, we created an integrated model of the left ventricle and aorta and conducted a computer simulation of diastolic and systolic blood flow within this model. The results demonstrated that the velocity profile at the aortic annulus changed dynamically, and was influenced by the intraventricular flow dynamics. The profile was almost flat in early systole but became nonuniform as systole progressed, and was skewed toward the posterior side in midsystole and toward the anterior side in later systole. At a distance from the aortic annulus, a different velocity profile was induced by the twisting and torsion of the aorta. In the ascending aorta, the fastest flow was initially located in the posteromedial sector, and it moved to the posterior section along the circumference as systole progressed. The nonuniformity of the aortic inflow gave rise to a complex wall shear stress (WSS) distribution in the aorta. A comparison of the WSS distribution obtained in this integrated analysis with that obtained in flow calculations using an isolated aorta model with Poiseuille and flat inlet conditions showed that intraventricular flow affected the WSS distribution in the ascending aorta. These results address the importance of an integrated analysis of flow in the left ventricle and aorta.     

Swirling flow implementation in a photobioreactor for batch and continuous cultures of Porphyridium cruentum

Light is the main limiting factor in photoautotrophic-intensive production of microorganisms, and improvement of its use is an important concern for photobioreactor design and operation. Swirling flows, which are known to improve mass and photon transfers, were applied to annular light chambers of a photobioreactor and studied by simulation and microalgal culture. Two hydrodynamic conditions were compared: axial flow generating poor radial mixing, and tangential flow generating three-dimensional swirling motion. Batch and continuous cultures of the Rhodophyte Porphyridium cruentum were performed in a 100-L, 1.5-m(2), fully controlled photobioreactor with eight light chambers. The inlet design of these chambers was modified to create the hydrodynamic conditions for comparison. Various intensities of swirling motion were used, characterized by the velocity factor (VF), defined as the ratio between annular chamber flow and inlet aperture sections. Experiments were performed within the range of photon flux densities (PFD) optimizing the yield of light energy transformation into living substance for the species and the temperature used. Culture kinetics with swirling flows generated by apertures of VF = 2, 4, and 9 were compared with pseudoaxial VF = 2 chosen as reference. Batch cultures with VF = 4 swirling flow showed no significant difference, whereas continuous cultures proved more discriminating. Although no significant difference was obtained for VF = 2, a 7% increase of steady-state productivity and a 26% decrease in time required to reach this steady state were obtained with VF = 4 swirling flow. This beneficial effect of swirling flow could have accounted for increased mixing. Conversely, VF = 9 swirling flow resulted in a 9% decrease of steady-state productivity and a 9% increase in the time required to reach this steady state, a negative effect that could have accounted for increased shear stress. CO(2) bioconversion yield at steady state showed a 34% increase for VF = 4. These results suggest that swirling motion makes microalgal cultures more efficient, provided that the resulting adverse effects remain acceptable. Experimental investigation was completed by a theoretical approach in which simulation of continuous cultures of P. cruentum was based on the hydrodynamic conditions achieved in the photobioreactor. Although the results obtained with pseudoaxial flow were correctly predicted, simulations with swirling flow showed a marked enhancement of productivity not observed experimentally. The influence of side effects induced by increased mixing (particularly hydrodynamic shear stress) was considered with respect to modeling assumptions. Comparison of experimental results with theoretical simulation provided a better understanding of the mixing effect, a key factor in improving the efficiency of such bioprocesses.     

Computer simulation of intraventricular flow and pressure gradients during diastole

A two-dimensional axisymmetric computer model is developed for the simulation of the filling flow in the left ventricle (LV). The computed results show that vortices are formed during the acceleration phases of the filling waves. During the deceleration phases these are amplified and convected into the ventricle. The ratio of the maximal blood velocity at the mitral valve (peak E velocity) to the flow wave propagation velocity (WPV) of the filling wave is larger than 1. This hemodynamic behavior is also observed in experiments in vitro (Steen and Steen, 1994, Cardiovasc. Res., 28, pp. 1821-1827) and in measurements in vivo with color M-mode Doppler echocardiography (Stugaard et al., 1994, J. Am. Coll. Cardiol., 24, 663-670). Computed intraventricular pressure profiles are similar to observed profiles in a dog heart (Courtois et al., 1988, Circulation, 78, pp. 661-671). The long-term goal of the computer model is to study the predictive value of noninvasive parameters (e.g., velocities measured with Doppler echocardiography) on invasive parameters (e.g., pressures, stiffness of cardiac wall, time constant of relaxation). Here, we show that higher LV stiffness results in a smaller WPV for a given peak E velocity. This result may indicate an inverse relationship between WPV and LV stiffness, suggesting that WPV may be an important noninvasive index to assess LV diastolic stiffness, LV diastolic pressure and thus atrial pressure (preload).     

Regional heterogeneity of myocardial blood flow within the rabbit left ventricle during haemorrhagic hypotension.

Several studies have reported an extensive regional heterogeneity in myocardial blood flow. The reported coefficients of variation for regional myocardial perfusion range from about 0.2 to 0.4 in normotensive animals. The spatial distribution of myocardial perfusion during haemorrhagic hypotension seems not to have been assessed. The goal of the present study was to determine the regional heterogeneity in myocardial blood flow within the rabbit left ventricle during normal conditions and after haemorrhagic hypotension. Radioactive microspheres were infused into the left ventricle in barbiturate anaesthetized rabbits over either 30 or 120 sec. The haemorrhagic hypotension was induced by bleeding, so that mean arterial blood pressure was reduced to about 50% of control. The left ventricles were divided into samples of about 0.025 g each. Regional heterogeneity in the blood flow was expressed as the coefficient of variation corrected for the Poisson distribution of microspheres (CVc). The CVc was 0.37 +/- 0.09 (mean +/- SD) during control and 0.41 +/- 0.11 after bleeding, the CVc obtained after bleeding being somewhat higher than during control (P < 0.05). We obtained a high correlation coefficient (tau about 0.68) between regional perfusion values at control and after bleeding which indicates a stable perfusion pattern within the myocardium. We conclude that the regional distribution of coronary blood flow within the left ventricle is markedly heterogenous during control condition and that this pattern is not changed during haemorrhagic hypotension.     

Measurement of pulsatile limb and finger blood flow by electrical impedance plethysmography: Criteria for the diagnosis of abnormal flow

The range of pulsatile arm and finger blood flow, measured by electrical impedance plethysmography, has been investigated in a hospital ward. The range of absolute blood flows, in ml min⁻¹, was found to be too wide to be used as a standard for identifying single blood flow readings as being abnormal. A blood flow ratio was calculated by dividing the blood flow in the right forearm or middle finger by the blood flow in the left forearm or middle finger. This ratio was found to have a clearly defined range. A blood flow in a unilaterally injured or otherwise abnormal arm or finger was considered to be significantly altered if the blood flow ratio fell outside the previously defined normal range. The diagnosis of significantly altered arm and finger blood flow from abnormalities in the blood flow ratio was tested in a series of experiments, in which artificial changes in upper limb flow were created by high elevation of the right hand. The ratio was measured in 11 patients with unilateral upper limb injuries and in 3 patients who required an urgent assessment of the upper limb circulation. Abnormalities in the ratio were identified in 12 out of 18 subjects after high elevation of the hand and in 8 out of the 14 patients.     

Effects of transients on pulsatile flow in arteries

Analytical solutions to the model problem of unsteady Newtonian fluid flow in straight, elastic-walled vessels can provide: theoretical insights into the flow of blood in arteries; a theoretical basis for clinical measurements in diagnoses of arterial flow rates; and guidance for boundary conditions in numerical simulations of flow in finite computational domains. However, while Womersley's analyses of blood flow assume solution forms that treat the flow as periodic and continuously unsteady, many flow variables in the smaller arteries are not continuously unsteady at all. They are characterized more accurately as rapid transient motions followed by a period of recovery to a stationary state, repeated in successive cycles. These flows are not continually unsteady ones described by Womersley's solutions but unsteady flows restarted from rest in each cycle, characterized as initial-boundary value problems. In this paper, we compare the Womersley and initial-boundary value solutions for model transients that stop then restart, explain these previously unreported limitations of Womersley's solutions, and demonstrate how the initial-boundary value solutions provide excellent agreement with measurements of blood flow in the anterior tibial and popliteal arteries of patients. Some consequences of these findings for understanding and interpreting measurements of blood flow, and for prescribing boundary conditions in computer simulations of arterial blood flow are discussed.     

Redistribution of myocardial blood flow in chronic ischemia under the effects of pharmacological agents

In the experiments with anesthetized dogs under chronic myocardial ischemia the effect of propranolol, diltiazem, lithium and sodium hydroxybutyrate on the myocardial blood flow redistribution was studied with the help of ultrasonic method. The redistribution was estimated by the ratio change of blood flows in veins which drain blood directly from the focus of myocardial ischemia and total myocardial of left ventricular (v cardiac magna). It was established that propranolol increases the ratio and diltiazem decreases it. Some differences in the effect of antihypoxic drugs were revealed. Sodium hydroxybutyrate redistributed the blood flow in favour of the focus of myocardial ischemia and lithium hydroxybutyrate increased the blood flow both in the focus of myocardial ischemia and in the conditionally-intact region of myocardium of left ventricular.     

Changes of regional myocardial blood flow during and after acute focal experimental ischaemia.

The regional blood flow through the myocardium of the left ventricle was measured in 11 dogs after ligation of the left anterior descending coronary artery, by means of a local injection of 133Xe depot and precordial detection of its washout 2 hours after ligation. Immediately after ligation the blood flow in the ischaemic area declined considerably but at the same time there was a significant increase of blood flow in the non-ischaemic left ventricular myocardium. The regional flow in the ischaemic and non-ischaemic area increased insignificantly for 2 hrs. These changes were not due to alterations in coronary artery pressure, as the mean arterial pressure declined significantly during the first hour. After temporary ischaemia by ligation of the left anterior descending coronary artery for 2--4 minutes, an intensive reactive hyperaemia developed in the ischaemic region (the blood flow reached 221% of control values on the average) which was the more intensive, the greater the drop of blood flow in the ischaemic area after ligation.     

Coupling of left ventricular and aortic volume elasticity in the rabbit

Changes in volume elasticity (VE) of the left ventricle and aorta could be important for blood flow. A procedure is presented to rapidly assess VE of the left ventricle and aorta by analyzing changes in the eigenfrequency. Six control rabbits and 11 rabbits with atheromatosis (12 wk of high-cholesterol feeding) were studied. In control rabbits, during the first half of the systole, left ventricular VE continuously increased to +43% (P < 0.05). Then VE gradually declined to an end-diastolic minimum (20% of the average systolic levels, P < 0.05). Aortic VE changes were in the opposite direction to the ventricle. Aortic VE continuously decreased throughout the systole; the last value was 20% lower than at the beginning of the systole (P < 0.05). Conversely, diastolic VE of the aorta took on greater values. This inverse time course between ventricle and aorta may reduce energy requirements for conveying blood. High cholesterol-fed rabbits did not reveal the inverse behavior of ventricular and aortic VE, e.g., aortic VE increased during the systole (119%, P < 0.05).     

Simulated pathline visualization of computed periodic blood flow patterns

Improvements in computer hardware and software have made it possible to model pulsatile blood flow in realistic arterial geometries. Such studies produce enormous amounts of velocity data, which are often difficult to interpret and communicate using traditional contour and/or vector field plots. Inspired by in vitro flow visualization techniques such as particle image velocimetry (PIV), we describe a simple and effective method for visualizing periodic three-dimensional velocity data, based on the subdivision and sequential display of computed particle trajectories. Analogous to a PIV experiment, the length and spacing of such simulated particle pathlines are controlled by user-specified shutter-speed and frame rate variables. Strategies for color-coding pathlines to highlight important hemodynamic features such as recirculation zones and branch flow division are presented.     

PIV measurements of flow in a centrifugal blood pump: time-varying flow

Measurements of the time-varying flow in a centrifugal blood pump operating as a left ventricular assist device (LVAD) are presented. This includes changes in both the pump flow rate as a function of the left ventricle contraction and the interaction of the rotating impeller and fixed exit volute. When operating with a pulsing ventricle, the flow rate through the LVAD varies from 0-11 L/min during each cycle of the heartbeat. Phase-averaged measurements of mean velocity and some turbulence statistics within several regions of the pump, including the inlet, blade passage, exit volute, and diffuser, are reported at 20 phases of the cardiac cycle. The transient flow fields are compared to the constant flow rate condition that was reported previously in order to investigate the transient effects within the pump. It is shown that the quasi-steady assumption is a fair treatment of the time varying flow field in all regions of this representative pump, which greatly simplifies the comprehension and modeling of this flow field. The measurements are further interpreted to identify the effects that the transient nature of the flow field will have on blood damage. Although regions of recirculation and stagnant flow exist at some phases of the cardiac cycle, there is no location where flow is stagnant during the entire heartbeat.     

Influence of dilated cardiomyopathy and a left ventricular assist device on vortex dynamics in the left ventricle

Together with new developments in mechanical cardiac support, the analysis of vortex dynamics in the left ventricle has become an increasingly important topic in literature. The aim of this study was to develop a method to investigate the influence of a left ventricular assist device (LVAD) on vortex dynamics in a failing ventricle. An axisymmetric fluid dynamics model of the left ventricle was developed and coupled to a lumped parameter model of the complete circulation. Simulations were performed for healthy conditions and dilated cardiomyopathy (DCM). Vortex structures in these simulations were analysed by means of automated detection. Results show that the strength of the leading vortex ring is lower in a DCM ventricle than in a healthy ventricle. The LVAD further influences the maximum strength of the vortex and also causes the vortex to disappear earlier in time with increasing LVAD flows. Understanding these phenomena by means of the method proposed in this study will contribute to enhanced diagnostics and monitoring during cardiac support.     

Influence of dilated cardiomyopathy and a left ventricular assist device on vortex dynamics in the left ventricle

Together with new developments in mechanical cardiac support, the analysis of vortex dynamics in the left ventricle has become an increasingly important topic in literature. The aim of this study was to develop a method to investigate the influence of a left ventricular assist device (LVAD) on vortex dynamics in a failing ventricle. An axisymmetric fluid dynamics model of the left ventricle was developed and coupled to a lumped parameter model of the complete circulation. Simulations were performed for healthy conditions and dilated cardiomyopathy (DCM). Vortex structures in these simulations were analysed by means of automated detection. Results show that the strength of the leading vortex ring is lower in a DCM ventricle than in a healthy ventricle. The LVAD further influences the maximum strength of the vortex and also causes the vortex to disappear earlier in time with increasing LVAD flows. Understanding these phenomena by means of the method proposed in this study will contribute to enhanced diagnostics and monitoring during cardiac support.     

Investigation of left heart flow using a numerical correlation to model heart wall motion

A three-dimensional computational fluid dynamics (CFD) method has been developed to model the flow in the left heart including atrium and ventricle. Since time resolution of the medical scans does not fit the requirements of the CFD calculations, the main challenge in a numerical simulation of heart chambers is wall motion modeling. This study employs a novel three-dimensional approximation scheme to correlate the wall boundary and grid movement in systole and diastole. It uses a geometry extracted from medical images in the literature and deformed based on the reported flow rates. The opening and closing of the mitral (MV) and the aortic valve (AV) considered as simultaneous events. Unstructured tetragonal grids were used for the meshing of the domain. The calculation was performed by a Navier–Stokes solver using the arbitrary Lagrange–Euler (ALE) formulation. Results show that the proposed correlation for the wall motion could predict the main features of heart flows.     

Dynamical relations for left ventricular ejection: flow rate, momentum, force and impulse

An investigation was carried out to quantitatively evaluate left ventricular volume flow rate, momentum, force and impulse derived from application of conservation principles for mass and momentum of blood within the ventricle during the ejection phase. An automated digital image processing system was developed and applied to left ventricular angiograms which are computer processed and analyzed frame by frame to determine the dynamical relations by numerical methods. Our initial experience with force and impulse has indicated that neither quantity seemed to be a sensitive indicator of coronary artery disease as evaluated by qualitative angiography for the particular patient group studied. Utilization of the dynamical relations in evaluating human left ventricular performance requires improved means of measurement and interpretation of clinical studies.