Modeling pulsatile flow in aortic aneurysms: effect of non-Newtonian properties of blood

Pulsatile flow in an axisymmetric rigid-walled model of an abdominal aorta aneurysm was analyzed numerically for various aneurysm dilations using physiologically realistic resting waveform at time-averaged Reynolds number of 300 and peak Reynolds number of 1607. Discretization of the governing equations was achieved using a finite element scheme based on the Galerkin method of weighted residuals. Comparisons with previously published work on the basis of special cases were performed and found to be in excellent agreement. Our findings indicate that the velocity fields are significantly affected by non-Newtonian properties in pathologically altered configurations. Non-Newtonian fluid shear stress is found to be greater than Newtonian fluid shear stress during peak systole. Further, the maximum shear stress is found to occur near the distal end of AAA during peak systole. The impact of non-Newtonian blood flow characteristics on pressure compared to Newtonian model is found insignificant under resting conditions. Viscous and inertial forces associated with blood flow are responsible for the changes in the wall that result in thrombus deposition and dilation while rupture of AAA is more likely determined by much larger mechanical stresses imposed by pulsatile pressure on the wall of AAA.     

Newtonian and non-Newtonian blood flow in coiled cerebral aneurysms

Endovascular coiling aims to isolate the aneurysm from blood circulation by altering hemodynamics inside the aneurysm and triggering blood coagulation. Computational fluid dynamics (CFD) techniques have the potential to predict the post-operative hemodynamics and to investigate the complex interaction between blood flow and coils. The purpose of this work is to study the influence of blood viscosity on hemodynamics in coiled aneurysms. Three image-based aneurysm models were used. Each case was virtually coiled with a packing density of around 30%. CFD simulations were performed in coiled and untreated aneurysm geometries using a Newtonian and a Non-Newtonian fluid models. Newtonian fluid slightly overestimates the intra-aneurysmal velocity inside the aneurysm before and after coiling. There were numerical differences between fluid models on velocity magnitudes in coiled simulations. Moreover, the non-Newtonian fluid model produces high viscosity (>0.007$>0.007$ [Pa s]) at aneurysm fundus after coiling. Nonetheless, these local differences and high-viscous regions were not sufficient to alter the main flow patterns and velocity magnitudes before and after coiling. To evaluate the influence of coiling on intra-aneurysmal hemodynamics, the assumption of a Newtonian fluid can be used.     

Blood flow in abdominal aortic aneurysms: pulsatile flow hemodynamics

Numerical predictions of blood flow patterns and hemodynamic stresses in Abdominal Aortic Aneurysms (AAAs) are performed in a two-aneurysm, axisymmetric, rigid wall model using the spectral element method. Physiologically realistic aortic blood flow is simulated under pulsatile conditions for the range of time-averaged Reynolds numbers 50< or =Re(m)< or =300, corresponding to a range of peak Reynolds numbers 262.5< or =Re(peak) < or = 1575. The vortex dynamics induced by pulsatile flow in AAAs is characterized by a sequence of five different flow phases in one period of the flow cycle. Hemodynamic disturbance is evaluated for a modified set of indicator functions, which include wall pressure (p(w)), wall shear stress (tau(w)), and Wall Shear Stress Gradient (WSSG). At peak flow, the highest shear stress and WSSG levels are obtained downstream of both aneurysms, in a pattern similar to that of steady flow. Maximum values of wall shear stresses and wall shear stress gradients obtained at peak flow are evaluated as a function of the time-average Reynolds number resulting in a fourth order polynomial correlation. A comparison between predictions for steady and pulsatile flow is presented, illustrating the importance of considering time-dependent flow for the evaluation of hemodynamic indicators.     

Slowness of blood flow and resultant thrombus formation in cerebral aneurysms

It is not yet fully understood what causes cerebral aneurysms to rupture. Although no definitive conclusion has been reached, it is considered that there are haemodynamic, biochemical and physiological factors contributing to rupture. Numerical techniques seem promising for investigation of this multi-physical phenomenon. In fact, recent intensive numerical studies with computational fluid dynamics have revealed detailed haemodynamic features of the flow in cerebral aneurysms such as velocity, pressure and wall shear stress distributions. It is, therefore, expected that biochemical and physiological aspects of aneurysmal rupture will also be actively investigated using numerical approaches. Considering this background, the authors have been working on modelling of thrombus formation in cerebral aneurysms caused by stagnant blood flow, because many studies have suggested that slow blood flow and resulting low wall shear stress are connected with rupture. Firstly, in this review paper, slowness of the intra-aneurysmal flow is reviewed with an energy balance theory, and secondly, thrombus formation in cerebral bifurcation aneurysms is discussed from the viewpoint of numerical modelling. A computational result obtained by application of the authors’ platelet aggregation–adhesion model is also provided.     

Analysis of non-linear pulsatile blood flow in arteries

Some aspects of the problem of computation of non-linear pulsatile blood flow in large arteries are investigated, in the context of the computational method developed by Ling and Atabek (1972). As examples, the following aspects are considered: stability of the computations; representation of higher-frequency components of the flow; effects of keeping or omitting non-linear terms in the equations; effects of varying the dimensionless parameters of the problem. The computational method is extended to include effects of viscoelasticity of arterial walls.     

Structure of pulsatile flow in a model of elastic cerebral aneurysm

This study examines the effect of aneurysmal wall elasticity on the structure of flow within an elastic aneurysm during pulsatile flow. We visualized flow structure in a model of an elastic saccular aneurysm located at the bifurcation of the anterior cerebral artery and extending to the anterior communicating artery, and measured changes in the diameter of the aneurysm wall during pulsatile flow using particle imaging velocimetry (PIV). We similarly measured these features during steady flow by PIV and found that dilation of the aneurysmal wall absorbed the dynamic energy within the aneurysm. Accordingly, aneurysm wall elasticity functions as a biocompatible reaction that relieves wall shear stress acting on the vascular wall during pulsatile flow, and should thus inhibit the development and rupture of an aneurysm.     

Numerical and experimental studies on pulsatile flow in aneurysms arising laterally from a curved parent vessel at various angles

Both numerical and experimental studies have been performed to characterize the fluid flow inside the lateral aneurysms arising from the curved parent vessels at various angles gamma. The implicit solver was based on the time-dependent Navier-Stokes equations of incompressible laminar flow. Solutions were generated by a cell-center finite-volume method that used second order upwind and second order center flux difference splitting for the convection and diffusion term, respectively. The second order Crank-Nicolson method was used in the time integration term while the SIMPLEC algorithm was adopted to handle the pressure-velocity coupling. Complementarily, the particle tracking velocimetry (PTV) was used to measure the velocity fields. The conditions selected were to simulate an internal carotid artery with a diameter of 5 mm by similarity rules. The values of gamma explored were 0 degrees, 45 degrees, 90 degrees, and 135 degrees. Pulsatile flow with Wormersley number 3.9 and Reynolds numbers varying from 350 to 850 was considered. The computed results are firstly verified by the PTV measured ones. Discussion of the results is in terms of pulsatile main and secondary velocity vector fields, inflow rates into the aneurysm, and the distributions of wall shear stress and static pressure. It is found that among the angles examined gamma=45( composite function) is the riskiest angle from a fluid dynamics point of view and the aneurysmal dome is at risk.     

Analysis of pulsatile blood flow: A carotid siphon model

Numerical results for axial and secondary flow velocity and pressure in a three-dimensional model of the human carotid siphon have been calculated; the investigations were carried out under physiologically relevant pulsatile flow conditoins. Time-dependent, three-dimensional Navier-Stokes equations were solved numerically by using a special finite element method. The results of the computer simulation presented here concentrate on the secondary motion effect during the pulsatile flow cycle in multiple three-dimensional curvatures.     

The effect of asymmetry in abdominal aortic aneurysms under physiologically realistic pulsatile flow conditions

In the abdominal segment of the human aorta under a patient's average resting conditions, pulsatile blood flow exhibits complex laminar patterns with secondary flows induced by adjacent branches and irregular vessel geometries. The flow dynamics becomes more complex when there is a pathological condition that causes changes in the normal structural composition of the vessel wall, for example, in the presence of an aneurysm. This work examines the hemodynamics of pulsatile blood flow in hypothetical three-dimensional models of abdominal aortic aneurysms (AAAs). Numerical predictions of blood flow patterns and hemodynamic stresses in AAAs are performed in single-aneurysm, asymmetric, rigid wall models using the finite element method. We characterize pulsatile flow dynamics in AAAs for average resting conditions by means of identifying regions of disturbed flow and quantifying the disturbance by evaluating flow-induced stresses at the aneurysm wall, specifically wall pressure and wall shear stress. Physiologically realistic abdominal aortic blood flow is simulated under pulsatile conditions for the range of time-average Reynolds numbers 50 < or = Rem < or = 300, corresponding to a range of peak Reynolds numbers 262.5 < or = Repeak < or = 1575. The vortex dynamics induced by pulsatile flow in AAAs is depicted by a sequence of four different flow phases in one period of the cardiac pulse. Peak wall shear stress and peak wall pressure are reported as a function of the time-average Reynolds number and aneurysm asymmetry. The effect of asymmetry in hypothetically shaped AAAs is to increase the maximum wall shear stress at peak flow and to induce the appearance of secondary flows in late diastole.     

Approximating hemodynamics of cerebral aneurysms with steady flow simulations

Computational fluid dynamics (CFD) simulations can be employed to gain a better understanding of hemodynamics in cerebral aneurysms and improve diagnosis and treatment. However, introduction of CFD techniques into clinical practice would require faster simulation times. The aim of this study was to evaluate the use of computationally inexpensive steady flow simulations to approximate the aneurysm's wall shear stress (WSS) field. Two experiments were conducted. Experiment 1 compared for two cases the time-averaged (TA), peak systole (PS) and end diastole (ED) WSS field between steady and pulsatile flow simulations. The flow rate waveform imposed at the inlet was varied to account for variations in heart rate, pulsatility index, and TA flow rate. Consistently across all flow rate waveforms, steady flow simulations accurately approximated the TA, but not the PS and ED, WSS field. Following up on experiment 1, experiment 2 tested the result for the TA WSS field in a larger population of 20 cases covering a wide range of aneurysm volumes and shapes. Steady flow simulations approximated the space-averaged WSS with a mean error of 4.3%. WSS fields were locally compared by calculating the absolute error per node of the surface mesh. The coefficient of variation of the root-mean-square error over these nodes was on average 7.1%. In conclusion, steady flow simulations can accurately approximate the TA WSS field of an aneurysm. The fast computation time of 6 min per simulation (on 64 processors) could help facilitate the introduction of CFD into clinical practice.     

Ventilatory effects of high and pulsatile blood flow in the pulmonary circulation

The mechanism of ventilatory stimulation that accompanies increases in cardiac output is unknown. Previous studies addressing this issue have been inconclusive. However, only steady pulmonary blood flow was used. The effect of flow pulsatility merits consideration, because increasing cardiac output raises not only mean pulmonary arterial pressure but also pulse pressure; mechanoreceptors with an important dynamic component to their responses may cause a response to pulsatile, but not steady, flow. Studies were done on anesthetized cats (n = 4) and dogs (n = 4). The right pulmonary artery was cannulated within the pericardium, and systemic blood was pumped from the left atrium to the right pulmonary artery. The right pulmonary circulation was perfused at different levels of flow, which was either steady or pulsatile. Steady-state flow of up to 150 ml.kg-1.min-1 (270 ml.kg-1.min-1 when corrected for the proportion of lung tissue perfused) did not affect breathing pattern. When high pulmonary flow was made pulsatile (pulse pressure approximately 23 mmHg), breath duration decreased from 3.7 +/- 0.72 to 3.4 +/- 0.81 (SD) s (P less than 0.01), representing a change in frequency of only 9%. There was no change in peak inspiratory activity. It was concluded that pulmonary vascular mechanoreceptors are not likely to contribute significantly to the increase in ventilation in association with increases in cardiac output.     

Microcontinuum model for pulsatile blood flow through a stenosed tube

The effects of polar nature of blood and pulsatility on flow through a stenosed tube have been analysed by assuming blood as a micropolar fluid. Linearized solutions of basic equations are obtained through consecutive applications of finite Hankel and Laplace transforms. The analytical expressions for axial and particle angular velocities, wall shear stress, resistance to flow and apparent viscosity have been obtained. The axial velocity profiles for Newtonian and micropolar fluids have been compared. The interesting observation of this analysis is velocity, in certain parts of cycle, for micropolar fluid is higher than Newtonain fluid. Variation of apparent viscosity eta a with tube radius shows both inverse Fahraeus-Lindqvist and Fahraeus-Lindqvist effects. Finally, the resistance to flow and wall shear stress for normal and diseased blood have been computed and compared.     

Wall shear stress at the initiation site of cerebral aneurysms

Hemodynamics are believed to play an important role in the initiation of cerebral aneurysms. In particular, studies have focused on wall shear stress (WSS), which is a key regulator of vascular biology and pathology. In line with the observation that aneurysms predominantly occur at regions of high WSS, such as bifurcation apices or outer walls of vascular bends, correlations have been found between the aneurysm initiation site and high WSS. The aim of our study was to analyze the WSS field at an aneurysm initiation site that was neither a bifurcation apex nor the outer wall of a vascular bend. Ten cases with aneurysms on the A1 segment of the anterior cerebral artery were analyzed and compared with ten controls. Aneurysms were virtually removed from the vascular models of the cases to mimic the pre-aneurysm geometry. Computational fluid dynamics (CFD) simulations were created to assess the magnitude, gradient, multidirectionality, and pulsatility of the WSS. To aid the inter-subject comparison of hemodynamic variables, we mapped the branch surfaces onto a two-dimensional parametric space. This approach made it possible to view the whole branch at once for qualitative evaluation. It also allowed us to empirically define a patch for quantitative analysis, which was consistent among subjects and encapsulated the aneurysm initiation sites in our dataset. To test the sensitivity of our results, CFD simulations were repeated with a second independent observer virtually removing the aneurysms and with a 20 % higher flow rate at the inlet. We found that branches harboring aneurysms were characterized by high WSS and high WSS gradients. Among all assessed variables, the aneurysm initiation site most consistently coincided with peaks of temporal variation in the WSS magnitude.     

Numerical simulations of pulsatile blood flow using a new constitutive model

In the present paper we use a new constitutive equation for whole human blood [R.G. Owens, A new microstructure-based constitutive model for human blood, J. Non-Newtonian Fluid Mech. (2006), to appear] to investigate the steady, oscillatory and pulsatile flow of blood in a straight, rigid walled tube at modest Womersley numbers. Comparisons are made with the experimental results of Thurston [Elastic effects in pulsatile blood flow, Microvasc. Res. 9 (1975), 145-157] for the pressure drop per unit length against volume flow rate and oscillatory flow rate amplitude. Agreement in all cases is very good. In the presentation of the numerical and experimental results we discuss the microstructural changes in the blood that account for its rheological behaviour in this simple class of flows. In this context, the concept of an apparent complex viscosity proves to be useful.     

Analysis of pulmonary arterial pressure profile after occlusion of pulsatile blood flow

In isolated canine lung lobes perfused with a pulsatile pump, arterial occlusions were performed and the postocclusion arterial pressure profiles were analyzed to estimate the pulmonary capillary pressure. A solenoid valve interposed between the pump and the lobar artery was used to perform arterial occlusions at several instants equally distributed within a pressure cycle. Double occlusions were also accomplished by simultaneously activating the solenoid valve and clamping the venous outflow of the lung lobe. To analyze an arterial occlusion pressure profile, we computed the best monoexponential fit of the pressure decay over a short period of time after the occlusion maneuvers. Two estimates of the capillary pressure were derived from this analysis: 1) the extrapolation of the exponential fit to the instant of occlusion, and 2) the point at which the recorded pressure decay curve merges with the exponential fit. The pressures thus determined were compared with the double occlusion pressure that provided an independent estimate of the pulmonary capillary pressure. Our results show that, under a wide range of conditions, the estimates of the capillary pressure obtained from the arterial occlusion data are nearly equal to the double occlusion pressures. Additionally, we estimated the capillary pressure variations within a pressure cycle by examining the occlusion pressures sampled at different instants of the cycle. The pulsatility of the pulmonary microvascular pressure varied with the pump frequency as well as the state of arterial and venous vasoaction. These variations are consistent with the representation of the lung vasculature as a low-pass filter.