Comparison of axisymmetric and three-dimensional models for gas uptake in a single bifurcation during steady expiration

Reactive gas uptake is predicted and compared in a single bifurcation at steady expiratory flow in terms of Sherwood number using an axisymmetric single-path model (ASPM) and a three-dimensional computational fluid dynamics model (CFDM). ASPM is validated in a two-generation geometry by comparing the average gas-phase mass transfer coefficients with the experimental values. ASPM predicted mass transfer coefficients within 20% of the experimental values. The flow and concentration variables in the ASPM were solved using Galerkin finite element method and in the CFDM using commercial finite element software FIDAP. The simulations were performed for reactive gas flowing at Reynolds numbers ranging from 60 to 350 in both symmetric bifurcation for three bifurcation angles, 30 deg, 70 deg, and 90 deg, and in an asymmetric bifurcation. The numerical models compared with each other qualitatively but quantitatively they were within 0.4-8% due to nonfully developed flow in the parent branch predicted by the CFDM. The radially averaged concentration variation along the axial location matched qualitatively between the CFDM and ASPM but quantitatively they were within 32% due to differences in the flow field. ASPM predictions compared well with the CFDM predictions for an asymmetric bifurcation. These results validate the simplified ASPM and the complex CFDM. ASPM predicts higher Sherwood number with a flat velocity inlet profile compared to a parabolic inlet velocity profile. Sherwood number increases with the inlet average velocity, wall mass transfer coefficient, and bifurcation angle since the boundary layer grows slower in the parent and daughter branches.     

A numerical study of the shape of the surface separating flow into branches in microvascular bifurcations.

The shape of the separating surface formed by the streamlines entering the branches of microvascular bifurcations plays a major role in determining the distribution of red blood cells and other blood constituents downstream from the bifurcation. Using the finite element method, we determined the shape of the surface through numerical solution of three dimensional Navier-Stokes equations for fluid flow at low Reynolds numbers in a T-type bifurcation of circular tubes. Calculations were done for a wide range of daughter branch to parent vessel diameter ratios and flow ratios. The effect of Reynolds number was also studied. Our numerical results are in good agreement with previously reported experimental data of Rong and Carr (Microvascular Research, Vol. 39, pp. 186-202, 1990). The numerical results of this study will be used to predict the concentration of blood constituents downstream from microvascular bifurcations providing that the inlet concentration profile is known.     

Measurement of pharyngeal cross-sectional area by finite element analysis.

A noninvasive measurement of pharyngeal cross-sectional area (CSA) during sleep would be advantageous for research studies. We hypothesized that CSA could be calculated from the measured pharyngeal pressure and flow by finite element analysis (FEA). The retropalatal airway was visualized by using a fiber-optic scope to obtain the measured CSA (mCSA). Flow was measured with a pneumotachometer, and pharyngeal pressure was measured with a pressure catheter at the palatal rim. FEA was performed as follows: by using a three-dimensional image of the upper airway, a mesh of finite elements was created. Specialized software was used to allow the simultaneous calculation of velocity and area for each element by using the measured pressure and flow. In the development phase, 677 simultaneous measurements of CSA, pressure, and flow from one subject during non-rapid eye movement (NREM) and rapid eye movement (REM) sleep were entered into the software to determine a series of equations, based on the continuity and momentum equations, that could calculate the CSA (cCSA). In the validation phase, the final equations were used to calculate the CSA from 1,767 simultaneous measurements of pressure and flow obtained during wakefulness, NREM, and REM sleep from 14 subjects. In both phases, mCSA and cCSA were compared by Bland-Altman analysis. For development breaths, the mean difference between mCSA and cCSA was 0.0 mm2 (95% CI, -0.1, 0.1 mm2). For NREM validation breaths, the mean difference between mCSA and cCSA was 1.1 mm2 (95% CI 1.3, 1.5 mm2). Pharyngeal CSA can be accurately calculated from measured pharyngeal pressure and flow by FEA.     

On the paths of fluid particles in an axisymmetrical aneurysm

The aim of this study is the characterization of the pulsatile flow field by demonstration of the paths of single particles in a model of an axisymmetric aneurysm. The detailed analysis of the flow field can give additional information on the flow pattern and the time of transition of blood particles in the segment. The basis of the calculations is the system of the Navier-Stokes equations for incompressible Newtonian fluid flow. To solve these equations numerically the finite element method was used. The trajectory equations for a fluid particle were solved by use of a predictor-corrector procedure. The results of the computer simulation demonstrate the development, shift and disappearance of vortices in the excavation and give references to zones of stasis. This behavior can be an important factor in thrombogenesis.     

Spatial and temporal variation of secondary flow during oscillatory flow in model human central airways

Axial and secondary velocity profiles were measured in a model human central airway to clarify the oscillatory flow structure during high-frequency oscillation. We used a rigid model of human airways consisting of asymmetrical bifurcations up to third generation. Velocities in each branch of the bifurcations were measured by two-color laser-Doppler velocimeter. The secondary velocity magnitudes and the deflection of axial velocity were dependent not only on the branching angle and curvature ratio of each bifurcation, but also strongly depended on the shape of the path generated by the cascade of branches. Secondary flow velocities were higher in the left bronchus than in the right bronchus. This spatial variation of secondary flow was well correlated with differing gas transport rates between the left and right main bronchus.     

Effects of vessel compliance on flow pattern in porcine epicardial right coronary arterial tree

The compliance of the vessel wall affects hemodynamic parameters which may alter the permeability of the vessel wall. Based on experimental measurements, the present study established a finite element (FE) model in the proximal elastic vessel segments of epicardial right coronary arterial (RCA) tree obtained from computed tomography. The motion of elastic vessel wall was measured by an impedance catheter and the inlet boundary condition was measured by an ultrasound flow probe. The Galerkin FE method was used to solve the Navier–Stokes and Continuity equations, where the convective term in the Navier–Stokes equation was changed in the arbitrary Lagrangian–Eulerian (ALE) framework to incorporate the motion due to vessel compliance. Various hemodynamic parameters (e.g., wall shear stress—WSS, WSS spatial gradient—WSSG, oscillatory shear index—OSI) were analyzed in the model. The motion due to vessel compliance affects the time-averaged WSSG more strongly than WSS at bifurcations. The decrease of WSSG at flow divider in elastic bifurcations, as compared to rigid bifurcations, implies that the vessel compliance decreases the permeability of vessel wall and may be atheroprotective. The model can be used to predict coronary flow pattern in subject-specific anatomy as determined by noninvasive imaging.     

Shape optimal design of the stem of a cemented hip prosthesis to minimize stress concentration in the cement layer

An optimal shape of the metal stem of a cemented total hip prosthesis minimizing stress concentration in the cement layer was searched for. A gradient projection method of numerical optimization and a finite element method of stress analysis were employed. A two-dimensional model of the femoral part of a total hip prosthesis was derived equivalent to a simplified three-dimensional axisymmetric model. The result of the stress analysis of the two-dimensional model compared favorably with that of the three-dimensional axisymmetric model. Using this two-dimensional model, an optimal shape of the stem, minimizing stress concentration in the cement layer, was obtained by a gradient projection method and the shape was checked again by the three-dimensional finite element analysis. The resulting optimal shape of the stem profile was in good agreement with conventional ones, except in the proximal region where a significant amount of stress reduction in the cement layer was achieved by tapering the stem to the limit that the stem still could withstand the increased stem stress.     

Which diameter and angle rule provides optimal flow patterns in a coronary bifurcation?

The branching angle and diameter ratio in epicardial coronary artery bifurcations are two important determinants of atherogenesis. Murray's cubed diameter law and bifurcation angle have been assumed to yield optimal flows through a bifurcation. In contrast, we have recently shown a 7/3 diameter law (HK diameter model), based on minimum energy hypothesis in an entire tree structure. Here, we derive a bifurcation angle rule corresponding to the HK diameter model and critically evaluate the streamline flow through HK and Murray-type bifurcations. The bifurcations from coronary casts were found to obey the HK diameter model and angle rule much more than Murray's model. A finite element model was used to investigate flow patterns for coronary artery bifurcations of various types. The inlet velocity and pressure boundary conditions were measured by ComboWire. Y-bifurcation of Murray type decreased wall shear stress-WSS (10%-40%) and created an increased oscillatory shear index-OSI in atherosclerosis-prone regions as compared with HK-type bifurcations. The HK-type bifurcations were found to have more optimal flow patterns (i.e., higher WSS and lower OSI) than Murray-type bifurcations which have been traditionally believed to be optimized. This study has implications for changes in bifurcation angles and diameters in percutaneous coronary intervention.     

Breakthrough performance of plasmid DNA on ion-exchange membrane columns

Breakthrough performance of plasmid DNA adsorption on ion-exchange membrane columns was theoretically and experimentally investigated using batch and fixed-bed systems. System dispersion curves showed the absence of flow non-idealities in the experimental arrangement. Breakthrough curves (BTC) were significantly affected by inlet flow rate and solute concentration. In the theoretical analysis, a model was integrated by the serial coupling of the membrane transport model and the system dispersion model. A transport model that considers finite kinetic rate and column dispersed flow was used in the study. A simplex optimization routine, coupled to the solution of the partial differential model equations, was employed to estimate the maximum adsorption capacity constant, the equilibrium desorption constant, and the forward interaction rate constant, which are the parameters of the membrane transport model. The analysis shows that as inlet concentration or flow rate increases, the deviation of the model from the experimental behavior decreases. The BTCs displacement as inlet concentration increases was explained in terms of a greater degree of column saturation reached and more efficient operation accomplished. The degree of column saturation was not influenced by inlet flow rate. It was necessary to consider in the column model the slight variation in the BTC produced by the axial dispersion, in order to accomplish the experimental curve dispersion. Consequently, the design criteria that for Pe > 40 the column axial dispersion can be neglected should be taken with precaution.     

Fluid–structure interaction analysis of turbulent pulsatile flow within a layered aortic wall as related to aortic dissection

Turbulent pulsatile flow and wall mechanics were studied numerically in an axisymmetric three-layered wall model of a descending aorta. The transport equations were solved using the finite element formulation based on the Galerkin method of weighted residuals. A fully-coupled fluid–structure interaction (FSI) analysis was utilized in this investigation. We calculated Von Mises wall stress, streamlines and fluid pressure contours. The findings of this study show that peak wall stress and maximum shear stress are highest in the media layer. The difference in the elastic properties of contiguous layers of the wall of the aorta probably determines the occurrence of dissection in the media layer. Moreover, the presence of aortic intramural hematoma is found to have a significant effect on the peak wall stress acting on the inner layer.     

Pendelluft flow in symmetric airway bifurcations.

We propose a mathematical model for pendelluft flow in a single airway bifurcation. The model is motivated by an apparatus used in an experimental study of the pendelluft by Ultman et al. (1988). We derive differential equations governing the fluid flow, which directly connect physiological parameters to the variables determining the pendelluft; this approach allows us to include nonlinearity in the model. If nonlinearity is neglected, our model is identical to the R-I-C circuits used by previous investigators. If nonlinearity is retained, we show that pendelluft can occur even in perfectly symmetric airway bifurcations. For the specific apparatus used in the experiments of High et al. (1991), we demonstrate that two qualitatively different pendelluft flows can occur in the system.     

Leukocyte deformability: finite element modeling of large viscoelastic deformation.

An axisymmetric deformation of a viscoelastic sphere bounded by a prestressed elastic thin shell in response to external pressure is studied by a finite element method. The research is motivated by the need for understanding the passive behavior of human leukocytes (white blood cells) and interpreting extensive experimental data in terms of the mechanical properties. The cell at rest is modeled as a sphere consisting of a cortical prestressed shell with incompressible Maxwell fluid interior. A large-strain deformation theory is developed based on the proposed model. General non-linear, large strain constitutive relations for the cortical shell are derived by neglecting the bending stiffness. A representation of the constitutive equations in the form of an integral of strain history for the incompressible Maxwell interior is used in the formulation of numerical scheme. A finite element program is developed, in which a sliding boundary condition is imposed on all contact surfaces. The mathematical model developed is applied to evaluate experimental data of pipette tests and observations of blood flow.     

Examining axial diffusion of nitric oxide in the lungs using heliox and breath hold.

Exhaled nitric oxide (NO) is highly dependent on exhalation flow; thus exchange dynamics of NO have been described by multicompartment models and a series of flow-independent parameters that describe airway and alveolar exchange. Because the flow-independent NO airway parameters characterize features of the airway tissue (e.g., wall concentration), they should also be independent of the physical properties of the insufflating gas. We measured the total mass of NO exhaled (A(I,II)) from the airways after five different breath-hold times (5-30 s) in healthy adults (21-38 yr, n = 9) using air and heliox as the insufflating gas, and then modeled A(I,II) as a function of breath-hold time to determine airway NO exchange parameters. Increasing breath-hold time results in an increase in A(I,II) for both air and heliox, but A(I,II) is reduced by a mean (SD) of 31% (SD 6) (P < 0.04) in the presence of heliox, independent of breath-hold time. However, mean (SD) values (air, heliox) for the airway wall diffusing capacity [3.70 (SD 4.18), 3.56 pl.s(-1).ppb(-1) (SD 3.20)], the airway wall concentration [1,439 (SD 487), 1,503 ppb (SD 644>)], and the maximum airway wall flux [4,156 (SD 2,502), 4,412 pl/s (SD 2,906)] using a single-path trumpet-shaped airway model that considers axial diffusion were independent of the insufflating gas (P > 0.55). We conclude that a single-path trumpet model that considers axial diffusion captures the essential features of airway wall NO exchange and confirm earlier reports that the airway wall concentration in healthy adults exceeds 1 ppm and thus approaches physiological concentrations capable of modulating smooth muscle tone.     

Lake circulation and sediment transport in Lake Myvatn

Lake circulation and sediment transport in Lake Myvatn have been calculated using AQUASEA, a numerical model developed by Vatnaskil Consulting Engineers. The goal of the modelling was to calculate changes in sediment transport within the lake due to changes in lake bathymetry caused by diatomite mining. The model uses the Galerkin finite element method and consists of a hydrodynamic flow model and a transport-dispersion model. The flow model is based on the shallow water equations and the wave equation. The transport model is based on the conservation of mass for suspended sediment. The model was calibrated against measurements performed during the summer of 1992. These included measurements of water elevation, current velocity, wave height, and concentration of suspended sediment. After calibration, the model was run for different mining scenarios to determine their impact on the sediment transport in the lake.     

Breakthrough performance of linear-DNA on ion-exchange membrane columns

Breakthrough performance of linear-DNA adsorption on ion-exchange membrane columns was theoretically and experimentally investigated using batch and fixed-bed systems. System dispersion curves showed the absence of flow non-idealities in the experimental arrangement. Breakthrough curves were not significantly affected by flow-rate or inlet solution concentration. In the theoretical analysis a model was integrated by the serial coupling of the membrane transport model and the system dispersion model. A transport model that considers finite kinetic rate and column dispersed flow was used in the study. A simplex optimization routine coupled to the solution of the partial differential model equations was employed to estimate the maximum adsorption capacity constant, the equilibrium desorption constant and the forward interaction rate-constant, which are the parameters of the membrane transport model. Through this approach a good prediction of the adsorption phenomena is obtained for inlet concentrations and flow rates greater than 0.2 mg/ml and 0.16 ml/min.     

Analysis of hemodynamic fluid phase mass transport in a separated flow region

The mass transfer behavior in the recirculation region downstream of an axisymmetric sudden expansion was examined. The Reynolds number, 500, and Schmidt number, 3200, were selected to model the mass transfer of molecules, such as ADP, in the arterial system. In a first step the transient mass transport applying zero diffusive flux at the wall was analyzed using experiments and two computational codes. The two codes were FLUENT, a commercially available finite volume method, and FTSP, a finite element code developed at Graz University of Technology. The comparison of the transient wall concentration values determined by the three methods was excellent and provides a measure of confidence for computational mass transfer calculations in convection dominated, separated flows. In a second step the effect of the flow separation on the stationary mass transport applying a permeability boundary condition at the water-permeable wall was analyzed using the finite element code FTSP. The results show an increase of luminal ADP surface concentration in the upstream and in the downstream tube of the sudden expansion geometry in the range of six and twelve percent of the bulk flow concentration. The effect of flow separation in the downstream tube on the wall concentration is a decrease of about ten percent of the difference between wall concentration and bulk concentration occurring at nearly fully developed flow at the downstream region at a distance of 66 downstream tube diameters from the expansion. The decrease of ADP flux into the wall is in the range of three percent of the flux at the downstream region.     

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.     

Measurement of axial diffusivities in a model of the bronchial airways.

Values for the effective axial diffusivity D for laminar flow of a gas species in the bronchial airways have been obtained as a function of the mean axial gas velocity u by experiment measurements of benzene vapor dispersion in a five generation glass tube model of the bronchial tree. For both inspiration and expiration D is seen to be approximately a linear function of u over the range of Reynolds' numbers 30-2,000 corresponding to peak flows in bronchial generations 0-13 under resting breathing conditions. The diffusivity for expiration is seen to be approximately one-third that for inspiration due presumably to increased radial mixing at bifurcations during expiration. The effective diffusivities relative to the molecular diffusivity can be expressed by the formulas D/Dmol = 1 + 1.08 NPe for inspiration and D/Dmol = 1 + .37 N-Pe for expiration. These velocity dependent diffusivities help to explain the short transit times of gas boluses from mouth to alveoli and will aid in the analysis of airway gas mixing by mathematical transport equations.