Numerical simulation of pulsatile flow in a compliant curved tube model of a coronary artery

The endothelial cells (ECs) lining a blood vessel wall are exposed to both the wall shear stress (WSS) of blood flow and the circumferential strain (CS) of pulsing artery wall motion. These two forces and their interaction are believed to play a role in determining remodeling of the vessel wall and development of arterial disease (atherosclerosis). This study focused on the WSS and CS dynamic behavior in a compliant model of a coronary artery taking into account the curvature of the bending artery and physiological radial wall motion. A three-dimensional finite element model with transient flow and moving boundaries was set up to simulate pulsatile flow with physiological pressure and flow wave forms characteristic of the coronary arteries. The characteristic coronary artery curvature and flow conditions applied to the simulation were: aspect ratio (lambda) = 10, diameter variation (DV) = 6 percent, mean Reynolds number (Re) = 150, and unsteadiness parameter (alpha) = 3. The results show that mean WSS is about 50 percent lower on the inside wall than the outside wall while WSS oscillation is stronger on the inside wall. The stress phase angle (SPA) between CS and WSS, which characterizes the dynamics of the mechanical force pattern applied to the endothelial cell layer, shows that CS and WSS are more out of phase in the coronaries than in any other region of the circulation (-220 deg on the outside wall, -250 deg on the inside wall). This suggests that in addition to WSS, SPA may play a role in localization of coronary atherosclerosis.     

Influence of a downstream narrowing on the flow profile in a tube

The distance over which the upstream flow conditions in a tube are disturbed by a stenosis downstream, i.e. the outlet length, was investigated for Reynolds numbers in the range 210-2900. Two methods were used, the Navier-Stokes equations were solved with a computer and a physical model was constructed and maximal velocities were measured with an ultrasound Doppler system. The computer model showed that Re number does not influence the outlet length, varying the stenosis area from 25% to 90% has an effect. However, the outlet length remained small, below 70% of the diameter of the tube. The physical model confirmed for a 75% stenosis that the outlet length is small, this method set the limit at not more than 1.2 times the tube diameter.     

Numerical simulation of peristaltic flow of a biorheological fluid with shear-dependent viscosity in a curved channel

Peristaltic motion of a non-Newtonian Carreau fluid is analyzed in a curved channel under the long wavelength and low Reynolds number assumptions, as a simulation of digestive transport. The flow regime is shown to be governed by a dimensionless fourth-order, nonlinear, ordinary differential equation subject to no-slip wall boundary conditions. A well-tested finite difference method based on an iterative scheme is employed for the solution of the boundary value problem. The important phenomena of pumping and trapping associated with the peristaltic motion are investigated for various values of rheological parameters of Carreau fluid and curvature of the channel. An increase in Weissenberg number is found to generate a small eddy in the vicinity of the lower wall of the channel, which is enhanced with further increase in Weissenberg number. For shear-thinning bio-fluids (power-law rheological index, n < 1) greater Weissenberg number displaces the maximum velocity toward the upper wall. For shear-thickening bio-fluids, the velocity amplitude is enhanced markedly with increasing Weissenberg number.     

A numerical calculation of flow in a curved tube model of the left main coronary artery

The flow pattern in the left main coronary artery has been calculated using an idealized geometry and by numerically solving the full Navier-Stokes equations for a Newtonian fluid. Two different forms for the entrance velocity profile were used, one a time-varying, flat profile and the other a time-varying, less flat velocity profile. The results obtained demonstrate the presence of secondary motions for conditions simulating flow in the left main coronary artery, with maximum secondary flow velocities being on the order of three to four percent of the maximum axial velocity. This secondary flow phenomenon has an important influence on the wall shear stress distribution, in spite of the fact that there is virtually no alteration in the axial velocity profile. The maximum ratio of the outer wall shear stress to that on the inner wall is 1.4 at a Reynolds number of Re = 270, and it increases with increasing Reynolds number, reaching a value of 1.7 at Re = 810. Although there are significant differences in the results in the immediate vicinity of the inlet for the two different forms of the entrance velocity profile used, this difference does not persist far into the tube. Independent of the choice of the entrance velocity profile, it appears that there will be significant secondary flow effects on the wall shear stress.     

The influence of the non-Newtonian properties of blood on the flow in large arteries: unsteady flow in a 90 degrees curved tube.

A numerical and experimental investigation of unsteady entry flow in a 90 degrees curved tube is presented to study the impact of the non-Newtonian properties of blood on the velocity distribution. The time-dependent flow rate for the Newtonian and the non-Newtonian blood analog fluid were identical. For the numerical computation, a Carreau-Yasuda model was employed to accommodate the shear thinning behavior of the Xanthan gum solution. The viscoelastic properties were not taken into account. The experimental results indicate that significant differences between the Newtonian and non-Newtonian fluid are present. The numerical results for both the Newtonian and the non-Newtonian fluid agree well with the experimental results. Since viscoelasticity was not included in the numerical code, shear thinning behavior of the blood analog fluid seems to be the dominant non-Newtonian property, even under unsteady flow conditions. Finally, a comparison between the non-Newtonian fluid model and a Newtonian fluid at a rescaled Reynolds number is presented. The rescaled Reynolds number, based on a characteristic rather than the high-shear rate viscosity of the Xanthan gum solution, was about three times as low as the original Reynolds number. Comparison reveals that the character of flow of the non-Newtonian fluid is simulated quite well by using the appropriate Reynolds number.     

The effect of a barrier to gene flow on patterns of geographic variation.

Explicit formulae are given for the effects of a barrier to gene flow on random fluctuations in allele frequency; these formulae can also be seen as generating functions for the distribution of coalescence times. The formulae are derived using a continuous diffusion approximation, which is accurate over all but very small spatial scales. The continuous approximation is confirmed by comparison with the exact solution to the stepping stone model. In both one and two spatial dimensions, the variance of fluctuations in allele frequencies increases near the barrier; when the barrier is very strong, the variance doubles. However, the effect on fluctuations close to the barrier is much greater when the population is spread over two spatial dimensions than when it occupies a linear, one-dimensional habitat: barriers of strength comparable with the dispersal range (B approximately equal to sigma) can have an appreciable effect in two dimensions, whereas only barriers with strength comparable with the characteristic scale (B approximately equal to L=sigma/sqrt{2mu}) are significant in one dimension (mu is the rate of mutation or long-range dispersal). Thus, in a two-dimensional population, barriers to gene flow can be detected through their effect on the spatial pattern of genetic marker alleles.     

On the steady laminar flow of a viscous incompressible fluid in an elastic tube

The conditions are examined under which an approximate relation between the radius of the tube and the distance along the axis, as obtained by N. Rashevsky (1945), is consistent with the assumptions made in the solution. These conditions are reduced to relations between two dimensionless parameters of the system. Second approximations are found for three different ranges of values of these parameters.     

Influence of interstitial bone microcracks on strain-induced fluid flow

It is well known that microcracks act as a stimulus for bone remodelling, initiating resorption by osteoclasts and new bone formation by osteoblasts. Moreover, microcracks are likely to alter the fluid flow and convective transport through the bone tissue. This paper proposes a quantitative evaluation of the strain-induced interstitial fluid velocities developing in osteons in presence of a microcrack in the interstitial bone tissue. Based on Biot theory in the low-frequency range, a poroelastic model is carried out to study the hydro-mechanical behaviour of cracked osteonal tissue. The finite element results show that the presence of a microcrack in the interstitial osteonal tissue may drastically reduce the fluid velocity inside the neighbouring osteons. This fluid inactive zone inside osteons can cover up to 10% of their surface. Consequently, the fluid environment of bone mechano-sensitive cells is locally modified.     

Detection of differential gene flow from patterns of quantitative variation

A major goal in anthropological genetics is the assessment of the effects of different microevolutionary forces. Harpending and Ward (1982) developed a model that aids in this effort by comparing observed and expected heterozygosity within populations in a local region. The expected heterozygosity within a population is a function of the total heterozygosity of the entire region and the distance of the population from the regional mean centroid of allele frequencies. Greater than average gene flow from an external source will result in a population having greater heterozygosity than expected. Less than average gene flow from an external source will result in a population having less heterozygosity than expected. We extend the Harpending-Ward model to quantitative traits using an equal and additive effects model of inheritance. Here the additive genetic variance within a population is directly proportional to heterozygosity, and its expectation is directly proportional to the genetic distance from the centroid. Under certain assumptions the expectations for phenotypic variances are similar. Observed and expected genetic or phenotypic variance can then be compared to assess the effects of differential external gene flow. When the additive genetic covariance matrix or heritabilities are not known, the phenotypic covariance matrix can be used to provide a conservative application of the model. In addition, we develop new methods for estimation of the genetic relationship matrix (R) from quantitative traits. We apply these models to two data sets: (1) six principal components derived from twenty dermatoglyphic ridge count measures for nine villages in Nepal and (2) ten anthropometric measurements for seven isolated populations in western Ireland. In both cases both the univariate and multivariate analyses provide results that can be directly interpreted in terms of historically known patterns of gene flow.     

Investigation of fluid flow patterns in a hollow fiber module using magnetic resonance velocity imaging

Summary Magnetic Resonance Imaging (MRI) was used to obtain new information about fluid flow patterns in hollow fiber reactors. Significant changes in inlet flow distribution were observed as a function of Reynolds number. Images taken at the tube bundle entrance and exit showed that maldistribution of flow persists throughout the module. Furthermore, the results suggest that individual fibers act in a mixed degree as feeders or collectors depending upon radial position. These effects must be considered when modelling or designing hollow fiber reactors.     

Blood flow in small curved tubes

Blood flow in small curved tubes is modeled by the two-fluid model where a relatively cell-free fluid layer envelops a fluid core of higher viscosity. The parameters in the model are successfully curve fitted to experimental data for straight tubes. The curved tube equations are then solved by perturbation theory. It was found that curvature in general lowers the tube resistance, but increases the shear stress near the inside wall.     

Influence of flow rate variation on the development of Escherichia coli biofilms

This work investigates the effect of flow rate variation on mass transfer and on the development of Escherichia coli biofilms on a flow cell reactor under turbulent flow conditions. Computational fluid dynamics (CFD) was used to assess the applicability of this reactor for the simulation of industrial and biomedical biofilms and the numerical results were validated by streak photography. Two flow rates of 374 and 242 L h^?1 (corresponding to Reynolds numbers of 6,720 and 4,350) were tested and wall shear stresses between 0.183 and 0.511 Pa were predicted in the flow cell reactor. External mass transfer coefficients of 1.38 × 10^?5 and 9.64 × 10^?6 m s^?1 were obtained for the higher and lower flow rates, respectively. Biofilm formation was favored at the lowest flow rate because shear stress effects were more important than mass transfer limitations. This flow cell reactor generates wall shear stresses that are similar to those found in some industrial and biomedical settings, thus it is likely that the results obtained on this work can be used in the development of biofilm control strategies in both scenarios.     

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.

     

The influence of reduced flow during a drought on patterns of variation in macroinvertebrate assemblages across a spatial hierarchy in two lowland rivers

1. With the aim of determining whether patterns of variation in macroinvertebrate assemblage composition across the hierarchy of spatial units in two lowland rivers changed during a supra‐seasonal drought (1997–2000), patterns during a reduced flow season (1999–2000) were compared with those during two preceding higher flow seasons (1997–98 and 1998–99) using samples from the Glenelg and Wimmera Rivers, two lowland regulated rivers in western Victoria, Australia. 2. We hypothesised that (i) differences between reaches would increase during the reduced flow season owing to decreased hydrological connectivity, (ii) differences between the habitats would decrease because the cessation of flow in run habitats should cause them to become more similar to pool habitats and (iii) differences between microhabitats would also decrease because of reduced scour of inorganic substrata and large woody debris. 3. During each season, macroinvertebrates were sampled from three microhabitats (sand/silt substratum, large woody debris and macrophytes) that were hierarchically nested within a run or pool habitat and within one of three reaches within each river. A range of physico‐chemical variables was also sampled. 4. Analysis of similarity showed that assemblage composition in both rivers during the higher flow seasons differed more among microhabitats than other spatial units. However, during the reduced flow season, assemblage composition in the Wimmera River differed most among reaches. This change in pattern was associated with the combined effects of decreased flow and longitudinal increases in salinity. In contrast, the fauna of the Glenelg River appeared to be resistant to the effects of the reduced flow season, owing to limited decline in water quality despite lower river discharge. 5. As salinisation and poor water quality in the Wimmera River result from human activities in the catchment, these results support the idea that human impacts on rivers can change macroinvertebrate scaling patterns and exacerbate the effects of drought beyond the tolerance of many riverine macroinvertebrates.