Localizing role of hemodynamics in atherosclerosis in several human vertebrobasilar junction geometries.

Atherosclerosis is a common finding in the vertebrobasilar junction and in the basilar artery. Several theories try to link the process of atherogenesis with the forces exerted by the flowing blood. An attractive relation has been found between the locations in vessels at which atherosclerotic plaques are often present and the locations in models where complicated flow patterns exist. Most of the studies provided data on bifurcations. Finding a similar relation in an arterial confluence would certainly add to the credibility of the (causal) relationship between hemodynamics and atherosclerosis. Further support can be provided if variations of the geometry result in changes of the location of the atherosclerotic lesions, corresponding to the changes of the flow force distribution. In our previous numerical and experimental work, the influence of geometric and hemodynamic parameters, such as asymmetrical inflow, confluence angle, and blunting of the apex, on the flow in vertebrobasilar junction models has been investigated in detail. Recirculation areas and distribution of the wall shear stress have been computed. In this anatomic study, the effect of modulation of these geometric and hemodynamic parameters on the flow pattern is compared with the size and location of plaques in human vertebrobasilar junctions and basilar arteries. In addition, a comparison is made between the preferential areas of atherosclerotic plaques in junctions and bifurcations to demonstrate the localizing role of hemodynamics in atherogenesis. The apex of the vertebrobasilar junction and the lateral walls of the basilar artery appeared to be prone to atherosclerosis. In 43 of 85 vertebrobasilar junctions, a plaque was found at the apex. Furthermore, the summed plaque thickness at both lateral walls differs significantly (paired t test, P=.03) from that at the walls facing the pons and the skull base. In contrast, several authors found that the lateral walls of the mother vessel and the apex in bifurcations are often spared. Modulation of the various parameters in the models changed the size of the regions with low wall shear stress and/or recirculation areas dramatically. A comparable effect was found in the occurrence of plaques in the human vertebrobasilar junction; eg, for an atherosclerotic plaque at the apex, a predicted probability larger than 0.5 was computed for blunted apexes and for sharp-edged apexes with a confluence angle exceeding 90 degrees. Apparently, two geometric risk factors for an atherosclerotic plaque at the apex can be distinguished: a blunted apex and a large confluence angle.

The influence of the blunting of the apex on the flow in a vertebro-basilar junction model.

The apex of human vertebro-basilar junctions can be sharp-edged or blunted. In the present study, the effect of blunted apex on the flow in vertebro-basilar junction models is investigated. We compared the flow phenomena in a series of junction models with blunted apices and confluence angles 45, 85, and 125 deg with the flow phenomena in a series of junction models with sharp-edged apices and the same range of confluence angles, studied in a previous paper (Ravensbergen et al., 1996b). The blunting of the apex appears to have an effect on the size of the local recirculation area near the apex and the prevailing low velocities. Large recirculation areas are found in the models with blunted apices, especially in those with small confluence angles. In addition, the blunting of the apex has no influence on the flow further downstream, nor on the structure and strength of the secondary flow field. Furthermore, a blunted apex appears to be a geometric risk factor for atherosclerosis. This supports the hypotheses that recirculation areas and low wall shear stress influence the development of atherosclerotic plaques.

Numerical simulation of blood flow in an artery with two successive bends.

Blood flow in an artery with two successive bends is simulated by a finite-element computation of steady flow of a Newtonian viscous fluid through a rigid tube having the same shape as a specific part of the femoral artery. Notwithstanding the fact that the bends in the model geometry are rather gentle, the axial and secondary flow patterns, computed for a range of values of the Reynolds number Re, show strong and complicated three-dimensional flow effects. In particular, the flow pattern in the second bend for relatively small values of Re (Re < 240) turns out to be drastically different from that for larger Re-values.

The influence of the angle of confluence on the flow in a vertebro-basilar junction model.

In earlier work, it was demonstrated that the flow in models of the vertebro-basilar junction is highly three-dimensional and the geometry exerts a strong influence on the hemodynamics. The morphology of the vertebro-basilar junction is very variable amongst individuals. In a study of 85 human vertebro-basilar junctions, the angle between the vertebral arteries varied between 10 and 160 degrees. To determine how the flow is influenced by this geometrical parameter, the flow is studied both experimentally, with laser Doppler velocimetry, and numerically, with a finite element package. A series of junction models is used with a range of confluence angles (45, 85 and 125 degrees). It appears that the angle of confluence has a strong influence on the structure and strength of the secondary flow field. The secondary velocities persist far downstream. Furthermore, near the apex, a region with low velocities is present. The larger the confluence angle is, the larger this region is, and even backflow may occur. In addition, the occurrence of atherosclerotic plaques in 85 human vertebro-basilar junctions is studied. Only one preferential location was found: the apex, the other plaques seem to be randomly distributed. The magnitude of the confluence angle of junctions with sharp-edged apices has a significant influence (p = 0.006) on the occurrence of a plaque at the apex. Apparently, a large confluence angle is a geometrical risk factor for atherosclerosis.

Computation of steady three-dimensional flow in a model of the basilar artery.

The flow in the basilar artery arises from the merging of the flows from the two vertebral arteries. To study the flow phenomena in the basilar artery, computations have been performed using a finite element (FE) method. We consider steady flow in a geometrically symmetric confluence. For simplicity, channels with a rectangular cross-section have been used. Both symmetric and asymmetric flow cases have been considered. The results show that for the Reynolds number of interest the flow downstream of the junction is highly three-dimensional, and that the flow at the end of the basilar artery, where it splits again, will not be fully developed. The computed phenomena have been confirmed by laser Doppler velocity measurements.

Functional anatomy of the circulus arteriosus cerebri (WillisII).

The requirements for understanding the role played by the circle of Willis in the cerebral circulation are two-fold: 1. The basic patterns of blood flow in the circle of Willis should be studied in relation to the interindividual variation of the circle itself and 2. The interindividual variation should be investigated in a quantitative way and possible hemodynamically induced relations should be clarified. With regard to the first question an extensive study on mathematical models of the cerebral circulation was performed, revealing that flow patterns can be understood by applying the principles of the Wheatstone bridge, known from electric circuit theory. With this knowledge the question of the variability was investigated. For this purpose 19 measurements of 100 circles of Willis were obtained from human cadavers. This data set was analysed using both bivariate and multivariate statistical techniques. The result of this analysis indicated that several sources of variation are involved. The first one is the size: some people have larger vessels than others, probably depending on age sex and genetic factors. Other sources of variation appeared to be of hemodynamic origin. The clear hemodynamic relation between vessel sizes in the circle of Willis indicate that they are adapted to the amount of flow, just like vessels elsewhere in the body. For instance, the size of the posterior communicating artery reflects the relative contribution of the anterior (carotid) and posterior (vertebral) vessels to the cerebral circulation. Furthermore, a relation could be established between the variations of the major cerebral vascular territories and the circle of Willis, both during development and in adults.(ABSTRACT TRUNCATED AT 250 WORDS)

Mathematical models of the flow in the basilar artery.

The flow in the basilar artery arises from the merging of the flows from the two vertebral arteries. This study deals with the question whether a parabolic (Poiseuille) profile will have been established before the basilar artery divides into both posterior cerebral arteries. The inlet length (that is, the downstream distance needed for the flow to become approximately equal to the limiting Poiseuille flow) and velocity profiles have been computed from two- and three-dimensional mathematical models in which flow pulsatility and vessel wall distensibility have been neglected and the complex geometry of the junction has been taken into account in a simplified form. The results show that the flow at the end of the basilar artery is far from being parabolic and that an asymmetry in the entrance flow will be carried along towards the end of the basilar artery, thus affecting flows in the circle of Willis.

Linear and nonlinear one-dimensional models of pulse wave transmission at high Womersley numbers.

The accuracy of nonlinear and linear one-dimensional models in describing pulse wave propagation in a uniform cylindrical viscoelastic tube, with Womersley's parameter alpha equal to 7.6 at 1 Hz, was evaluated. To this end calculations of wave propagation using these models were compared with the experimentally determined propagation of the pressure wave in the tube. The experimentally generated pressure pulse had an amplitude of 9.0 kPa and caused a relative radius change of about 17%. The static pressure vs cross-sectional area relation was found to be nonlinear for these pressure changes. Maximum fluid velocity was about 2.9 ms-1, while the phase velocity was about 5.4 ms-1. The radius change and the ratio of fluid and phase velocities violated the linear model assumptions. The nonlinear model with viscous fluid friction modelled on the basis of Poiseuille's law and treating the tube wall as purely elastic, underestimated the damping of the pulse wave and predicted the formation of shock waves, which were not found experimentally. In the linear models, the viscous friction of the blood was modelled according to either Poiseuille's law or Womersley's theory and the tube wall was treated as either linearly elastic or linearly viscoelastic. A description of the viscous friction of the blood based on Poiseuille's law underestimated damping. Disregarding the viscoelasticity of the tube wall resulted in an underestimation of both phase velocity and damping. In spite of the nonlinearity of the system, the linear viscoelastic Womersley model described the pulse wave propagation satisfactorily.

Analysis of flow and vascular resistance in a model of the circle of Willis.

A very simple model of the flow in the circle of Willis is described in this paper. Disregarding pulsatility and vessel wall elasticity, fluxes in all segments of the circle of Willis and its afferent and efferent vessels are calculated by applying the Poiseuille-Hagen formula. Comparison with the fluxes calculated numerically from a more sophisticated mathematical model, including pulsatility, vessel wall elasticity and nonlinear effects, revealed only very slight differences. In short, fluxes in the afferent vessels and the segments of the circle of Willis are influenced by any change of resistance within the network, whereas the fluxes in the efferent segments are dominated by the efferent resistance distribution. However, a great advantage of the present simple model is that it offers the possibility of an analytical approach which yields both an easy sensitivity analysis of parameters and an insight into the mechanisms that govern the flow in a network like the circle of Willis. It can be concluded that these mechanisms are similar to the principles of the Wheatstone bridge, known from electrical circuit theory.

Numerical methods for solving one-dimensional cochlear models in the time domain.

In this article, a robust numerical solution method for one-dimensional (1-D) cochlear models in the time domain is presented. The method has been designed particularly for models with a cochlear partition having nonlinear and active mechanical properties. The model equations are discretized with respect to the spatial variable by means of the principle of Galerkin to yield a system of ordinary differential equations in the time variable. To solve this system, several numerical integration methods concerning stability and computational performance are compared. The selected algorithm is based on a variable step size fourth-order Runge-Kutta scheme; it is shown to be both more stable and much more efficient than previously published numerical solution techniques.

A mathematical model of the flow in the circle of Willis.

A mathematical model of the flow in the circle of Willis has been designed and the effects of (a) the large anatomical variation of the communicating arteries and (b) physiological changes of the resistances of the vertebral arteries have been studied. The influence of the posterior perforating arteries on the flow in the posterior communicating arteries has been investigated as well, with special attention being paid to the possible occurrence of a 'dead point'. In the model, the influence of diameters of the communicating arteries on the flow in the afferent vessels and the segments of the circle turns out to be considerable, especially in the range of the anatomical variation of the diameters. Within this range flow reductions due to an increased resistance of the vertebral artery will be compensated for by the system. Assuming that the values and ratios of the peripheral resistances are within the physiological range, a dead point is not to be expected in the flow in the posterior communicating arteries.

A mathematical model of the flow in the posterior communicating arteries.

This paper reports on a mathematical model designed to study the hemodynamics of one posterior communicating artery and its afferent and efferent vessels. The variables in the model are the diameter of the posterior communicating artery, the resistance in the vertebral artery and the ratio of the two peripheral resistances. In the model, the "posterior communicating artery" exhibits a compensatory capacity, as defined in the introduction, which appears to be independent of its diameter. The fluxes in the efferent vessels are dominated by the peripheral resistances.