A mathematical model of water flux through aortic tissue

Water flux in porcine aortic segments produced by the sudden application of a hydrostatic pressure gradient has been described in a recent paper by Harrison and Massaro (1976). A mathematical model is developed here to explain the results obtained when pressure is applied to either covered or uncovered samples. The model predicts that the rate of exudation in both instances should be substantially identical for a period of time ∼ 0.2τ, where τ is the consolidation time. The consolidation time is proportional to the hydraulic resistance to liquid flow, and inversely proportional to the compressive stiffness of the artery. The existence of a time-dependent water flux in an arteryin vivo during periodic pressurization is predicted by the mathematical model if the resistance to water flow at the endothelium is not excessive. The pore pressure within the bulk of the media is predicted to pulsate in a highly unexpected fashion. These predictions follow naturally from the fact that the consolidation phenomenon in large arteries, as determined by the compression tests of Harrison and Massaro, is of long duration, much longer than the period of a heartbeat. Pressure gradientsin vivo in interstitial fluid are then confined to a very small fraction of the total arterial wall thickness. A potential for plasma “sloshing” across the endothelial junctions exists. The convective flux of water across an endothelial layer may therefore be of a pulsatile character in normal arteriesin vivo.     

A theoretical model for water flux through the artery wall

A two layer model for water flux through the artery is studied using a mathematical model based on the theory for the consolidation of water saturated soils. The matrix is considered to be constituted by two layers with different permeabilities and different elastic constants and the two systems of equations are coupled with the condition of continuity of pressure, total stress, solid displacement and fluid seepage velocity at the interface. The luminal pressure is considered to be harmonic in time. Exact solutions are obtained for displacements and pressures in both the layers. For large consolidation times, large pressure gradients are found to exist near the boundaries and at the interface. The heterogeneous model may not only be useful to understand the mechanics of transport in the physiological system but it will also help the bioengineers to choose proper implant materials to design artificial vascular organs for the purpose of prosthesis.     

Structural changes in rat aortic intima due to transmural pressure.

Huang et al. (1997) propose a new hypothesis and develop a mathematical model to explain rationally the in vitro and in situ measured changes (Tedgui and Lever, 1984; Baldwin and Wilson, 1993) in the hydraulic conductivity of the artery wall of rabbit aorta with transmural pressure. The model leads to the intriguing prediction that this hydraulic conductivity would decrease by one half if the thin intimal layer between the endothelium and the internal elastic lamina volume-compresses approximately fivefold. This paper presents the first measurements of the effect of transmural pressure on intimal layer thickness and shows that the intimal matrix is, indeed, surprisingly compressible. We perfusion-fixed rat thoracic aortas in situ with 2 percent glutaraldehyde solution at 0, 50, 100, or 150 mm Hg lumen pressure and sectioned for light and electron microscopic observations. Electron micrographs show a dramatic, nonlinear decrease in average intimal thickness, i.e., 0.62 +/- 0.26, 0.27 +/- 0.14, 0.15 +/- 0.10, and 0.12 +/- 0.07 (SD) micron for 0, 50, 100, and 150 mm Hg lumen pressure, respectively. The volume strain of the intima is more than 20 times greater than the radial strain of the artery wall due to hoop tension and two orders of magnitude greater than the consolidation of the artery wall as a whole assuming constant medial density (Chuong and Fung, 1984). Moreover, in both light and electron microscopic observations, it is easy to find numerous sites where the endothelium puckers into the fenestral pores at high lumen pressure, as predicted by the theory in Huang et al. (1997). In contrast, the average diameter of a fenestral pore increases only 10 percent as the lumen pressure is increased from 0 to 150 mm Hg. These results indicate that the thin intimal layer comprising less than 1 percent of the wall thickness can have a profound effect on the filtration properties of the wall due to the large change in Darcy permeability of the layer and the large reduction in the entrance area of the flow entering the fenestral pores, though the pores themselves experience only a minor enlargement due to hoop tension.     

Two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability

Since the transport of biological fluids through contracting or expanding vessels is characterized by low seepage Reynolds numbers, the current study focuses on the viscous flow driven by small wall contractions and expansions of two weakly permeable walls. The scope is limited to two-dimensional symmetrical solutions inside a simulated channel with moving porous walls. In seeking an exact solution, similarity transformations are used in both space and time. The problem is first reduced to a nonlinear differential equation that is later solved both numerically and analytically. The analytical procedure is based on double perturbations in the permeation Reynolds number R and the wall expansion ratio alpha. Results are correlated and compared via variations in R and alpha. Under the auspices of small [R] and [alpha], the analytical result constitutes a practical equivalent to the numerical solution. We find that, when suction is coupled with wall contraction, rapid flow turning is precipitated near the wall where the boundary layer is formed. Conversely, when injection is paired with wall expansion, the flow adjacent to the wall is delayed. In this case, the viscous boundary layer thickens as injection or expansion rates are reduced. Furthermore, the pressure drop along the plane of symmetry increases when the rate of contraction is increased and when either the rate of expansion or permeation is reduced. As nonlinearity is retained, our solutions are valid from a large cross-section down to the state of a completely collapsed system.     

A numerical and experimental investigation of transitional pulsatile flow in a stenosed channel

In the present paper, a closely coupled numerical and experimental investigation of pulsatile flow in a prototypical stenotic site is presented. Detailed laser Doppler velocimetry measurements upstream of the stenosis are used to guide the specification of velocity boundary conditions at the inflow plane in a series of direct numerical simulations (DNSs). Comparisons of the velocity statistics between the experiments and DNS in the post-stenotic area demonstrate the great importance of accurate inflow conditions, and the sensitivity of the post-stenotic flow to the disturbance environment upstream. In general, the results highlight a borderline turbulent flow that sequentially undergoes transition to turbulence and relaminarization. Before the peak mass flow rate, the strong confined jet that forms just downstream of the stenosis becomes unstable, forcing a role-up and subsequent breakdown of the shear layer. In addition, the large-scale structures originating from the shear layer are observed to perturb the near wall flow, creating packets of near wall hairpin vortices.     

The influence of radial RBC distribution, blood velocity profiles, and glycocalyx on coupled NO/O2 transport.

The purpose of this investigation was to study the effect of the presence of red blood cells (RBCs) in the plasma layer near the arteriole wall on nitric oxide (NO) and oxygen (O2) transport. To this end, we extended a coupled NO and O2 diffusion-reaction model in the arteriole, developed by our group, to include the effect of the presence of RBCs in the plasma layer and the effect of convection. Two blood flow velocity profiles (plug and parabolic) were tested. The average hematocrit in the bloodstream was assumed to be constant in the central core and decreasing to zero in the boundary layer next to the endothelial surface layer. The effect of the presence or absence of RBCs near the endothelium was studied while varying the endothelial surface layer and boundary layer thickness. With RBCs present in the boundary layer, the model predicts that 1) NO decreases significantly in the endothelium and vascular wall; 2) there is a very small increase in endothelial and vascular wall Po2; 3) scavenging of NO by hemoglobin decreases with increasing thickness of the boundary layer; 4) the shape of the velocity profile influences both NO and Po2 gradients in the bloodstream; and 5) the presence of RBCs in the boundary layer near the endothelium has a much larger effect on NO than on O2 transport.     

IB–LBM simulation of the haemocyte dynamics in a stenotic capillary

To study the behaviour of a haemocyte when crossing a stenotic capillary, the immersed boundary–lattice Boltzmann method was used to establish a quantitative analysis model. The haemocyte was assumed to be spherical and to have an elastic cell membrane, which can be driven by blood flow to adopt a highly deformable character. In the stenotic capillary, the spherical blood cell was stressed both by the flow and the wall dimension, and the cell shape was forced to be stretched to cross the stenosis. Our simulation investigated the haemocyte crossing process in detail. The velocity and pressure were anatomised to obtain information on how blood flows through a capillary and to estimate the degree of cell damage caused by excessive pressure. Quantitative velocity analysis results demonstrated that a large haemocyte crossing a small stenosis would have a noticeable effect on blood flow, while quantitative pressure distribution analysis results indicated that the crossing process would produce a special pressure distribution in the cell interior and to some extent a sudden change between the cell interior and the surrounding plasma.     

Non-Newtonian Blood Flow in Left Coronary Arteries with Varying Stenosis: A Comparative Study

This paper presents Computational fluid dynamic (CFD) analysis of blood flow in three different 3-D models of left coronary artery (LCA). A comparative study of flow parameters (pressure distribution, velocity distribution and wall shear stress) in each of the models is done for a non-Newtonian (Carreau) as well as the Newtonian nature of blood viscosity over a complete cardiac cycle. The difference between these two types of behavior of blood is studied for both transient and steady states of flow. Additionally, flow parameters are compared for steady and transient boundary conditions considering blood as non-Newtonian fluid. The study shows that the highest wall shear stress (WSS), velocity and pressure are found in artery having stenosis in all the three branches of LCA. The use of Newtonian blood model is a good approximation for steady as well as transient blood flow boundary conditions if shear rate is above 100 s-1. However, the assumption of steady blood flow results in underestimating the values of flow parameters such as wall shear stress, pressure and velocity.     

Steady flow and wall compression in stenotic arteries: a three-dimensional thick-wall model with fluid-wall interactions.

Severe stenosis may cause critical flow and wall mechanical conditions related to artery fatigue, artery compression, and plaque rupture, which leads directly to heart attack and stroke. The exact mechanism involved is not well understood. In this paper a nonlinear three-dimensional thick-wall model with fluid-wall interactions is introduced to simulate blood flow in carotid arteries with stenosis and to quantify physiological conditions under which wall compression or even collapse may occur. The mechanical properties of the tube wall were selected to match a thick-wall stenosis model made of PVA hydrogel. The experimentally measured nonlinear stress-strain relationship is implemented in the computational model using an incremental linear elasticity approach. The Navier-Stokes equations are used for the fluid model. An incremental boundary iteration method is used to handle the fluid-wall interactions. Our results indicate that severe stenosis causes considerable compressive stress in the tube wall and critical flow conditions such as negative pressure, high shear stress, and flow separation which may be related to artery compression, plaque cap rupture, platelet activation, and thrombus formation. The stress distribution has a very localized pattern and both maximum tensile stress (five times higher than normal average stress) and maximum compressive stress occur inside the stenotic section. Wall deformation, flow rates, and true severities of the stenosis under different pressure conditions are calculated and compared with experimental measurements and reasonable agreement is found.     

Numerical simulation of sinusoidal flow in a straight elastic tube: effects of phase angles

Numerical simulations of flow in straight elastic (moving wall) tubes subjected to a sinusoidal pressure gradient were performed for conditions prevailing in large and medium sized arteries. The effects of varying the phase angle between the pressure gradient and the tube radius, the amplitude of wall motion, and the unsteadiness parameter (alpha) on flow rate and wall shear stress were investigated. Mean and peak flow rates and shear stresses were found to be strongly affected by the phase angle between the pressure gradient and the tube radius with greater sensitivity at higher diameter variation and higher alpha. In large artery simulations (alpha = 12), means flow rate was found to be 60% higher and peak flow rate to be 73% higher than corresponding rigid tube values for certain phase angles, while a threefold increase in mean wall shear stress and sevenfold increase in peak wall shear stress were observed in a sensitive phase angle range. Significant reversal in the wall shear stress direction occurred in the sensitive phase angle range even when there was negligible flow rate reversal. All effects were greatly diminished in simulations of medium sized vessels (alpha = 4). Some experimental evidence to support the predictions of a strong effect of phase angle on wall shear stress in large vessels is presented. Finally, physiological implications of the present work are discussed from a basis of aortic input impedance data, and a physical explanation for the extreme sensitivity of the flow field to small amplitude wall motion at high alpha is given.     

An in vitro investigation of the acetabular labral seal in hip joint mechanics

Labrum pathology may contribute to early joint degeneration through the alteration of load transfer between, and the stresses within, the cartilage layers of the hip. We hypothesize that the labrum seals the hip joint, creating a hydrostatic fluid pressure in the intra-articular space, and limiting the rate of cartilage layer consolidation. The overall cartilage creep consolidation of six human hip joints was measured during the application of a constant load of 0.75 times bodyweight, or a cyclic sinusoidal load of 0.75+/-0.25 times bodyweight, before and after total labrum resection. The fluid pressure within the acetabular was measured. Following labrum resection, the initial consolidation rate was 22% greater (p=0.02) and the final consolidation displacement was 21% greater (p=0.02). There was no significant difference in the final consolidation rate. Loading type (constant vs. cyclic) had no significant effect on the measured consolidation behaviour. Fluid pressurisation was observed in three of the six hips. The average pressures measured were: for constant loading, 541+/-61kPa in the intact joint and 216+/-165kPa following labrum resection, for cyclic loading, 550+/-56kPa in the intact joint and 195+/-145kPa following labrum resection. The trends observed in this experiment support the predictions of previous finite element analyses. Hydrostatic fluid pressurisation within the intra-articular space is greater with the labrum than without, which may enhance joint lubrication. Cartilage consolidation is quicker without the labrum than with, as the labrum adds an extra resistance to the flow path for interstitial fluid expression. However, both sealing mechanisms are dependent on the fit of the labrum against the femoral head.     

Laser anemometry measurements of pulsatile flow past aortic valve prostheses

Experimental results are presented on physiological pulsatile flow past caged ball and tilting disc aortic valve prostheses mounted in an axisymmetric chamber incorporated in a mock circulatory system. The measurements of velocity profiles and turbulent normal stresses during several times in a cardiac cycle were obtained using laser-Doppler anemometry. Our results show that with increased angle of opening for the tilting disc valves, a large but locally confined vortex is observed along the wall in the minor flow region throughout most of the cardiac cycle. The turbulent normal stresses measured downstream to the tilting disc in the minor flow region parallel to the tilt axis were found to be larger than those measured downstream to the caged ball valves. Comparison of measurements with steady flow at flow rates comparable to peak pulsatile flow rate show that the turbulent normal stresses are larger by a factor of two in pulsatile flow with a frequency of 1.2 Hz.     

Effects of elastic property of the wall on flow characteristics through arterial stenoses

Hemodynamic characteristics of blood flow through arterial stenoses are numerically investigated in this work. The blood is assumed as a Newtonian fluid and the pulsatile nature of flow is modeled by using measured values of the flowrate and pressure for the canine femoral artery. An isotropic elastic and incompressible material is assumed for the wall at each axial section, but a non-uniform distribution of the shear modulus in axial direction is used to model the high stiffness of the wall at the stenosis location. Full Navier equations for a thick wall are used as the governing equations for the wall displacements. A continuous grid extending over the flow field and the wall is considered and governing equations are transformed for use in the computational domain. Discretized forms of the transformed wall and flow equations, which are coupled through the boundary conditions at their interface, are obtained by control volume method and simultaneously solved using the well-known SIMPLER algorithm. To study the effects of wall deformability, solutions are obtained for both rigid and elastic walls. The results indicate that deformability of the wall causes an increase in the time average of pressure drop, but a decrease in the maximum wall shear stress. Displacement and stress distributions in the wall are presented.     

Effects of coronary sinus pressure elevation on coronary blood flow distribution in dogs with normal preload

Coronary sinus pressure (Pcs) elevation shifts the diastolic coronary pressure-flow relation (PFR) of the entire left ventricular myocardium to a higher pressure intercept. This finding suggests that Pcs is one determinant of zero-flow pressure (Pzf) and challenges the existence of a vascular waterfall mechanism in the coronary circulation. To determine whether coronary sinus or tissue pressure is the effective coronary back pressure in different layers of the left ventricular myocardium, the effect of increasing Pcs was studied while left ventricular preload was low. PFRs were determined experimentally by graded constriction of the circumflex coronary artery while measuring flow using a flowmeter. Transmural myocardial blood flow distribution was studied (15-micron radioactive spheres) at steady state, during maximal coronary artery vasodilatation at three points on the linear portion of the circumflex PFR both at low and high diastolic Pcs (7 +/- 3 vs. 22 +/- 5 mmHg; p less than 0.0001) (1 mmHg = 133.322 Pa). In the uninstrumented anterior wall the blood flow measurements were obtained in triplicate at the two Pcs levels. From low to high Pcs, mean aortic (98 +/- 23 mmHg) and left atrial (5 +/- 3 mmHg) pressure, percent diastolic time (49 +/- 7%), percent left ventricular wall thickening (32 +/- 4%), and percent myocardial lactate extraction (15 +/- 12%) were not significantly changed. Increasing Pcs did not alter the slope of the PFR; however, the Pzf increased in the subepicardial layer (p less than 0.0001), whereas in the subendocardial layer Pzf did not change significantly. Similar slopes and Pzf were observed for the PFR of both total myocardial mass and subepicardial region at low and high Pcs. Subendocardial:subepicardial blood flow ratios increased for each set of measurements when Pcs was elevated (p less than 0.0001), owing to a reduction of subepicardial blood flow; however, subendocardial blood flow remained unchanged, while starting in the subepicardium toward midmyocardium blood flow decreased at high Pcs. This pattern was similar for the uninstrumented anterior wall as well as in the posterior wall. Thus as Pcs increases it becomes the effective coronary back pressure with decreasing magnitude from the subepicardium toward the subendocardium of the left ventricle. Assuming that elevating Pcs results in transmural elevation in coronary venous pressure, these findings support the hypothesis of a differential intramyocardial waterfall mechanism with greater subendo- than subepi-cardial tissue pressure.     

A model of pulsatile flow in a uniform deformable vessel.

Simulations of blood flow in natural and artificial conduits usually require large computers for numerical solution of the Navier-Stokes equations. Often, physical insight into the fluid dynamics is lost when the solution is purely numerical. An alternative to solving the most general form of the Navier-Stokes equations is described here, wherein a functional form of the solution is assumed in order to simplify the required computations. The assumed forms for the axial pressure gradient and velocity profile are chosen such that conservation of mass is satisfied for fully established pulsatile flow in a straight, deformable vessel. The resulting equations are cast in finite-difference form and solved explicitly. Results for the limiting cases of rigid wall and zero applied pressure are found to be in good agreement with analytical solutions. Comparison with the experimental results of Klanchar et al. [Circ. Res. 66, 1624-1635 (1990]) also shows good agreement. Application of the model to realistic physiological parameter values provides insight as to the influence of the pulsatile nature of the flow field on wall shear development in the presence of a moving wall boundary. Specifically, the model illustrates the dependence of flow rate and shear rate on the amplitude of the vessel wall motion and the phase difference between the applied pressure difference and the oscillations of the vessel radius. The present model can serve as a useful tool for experimentalists interested in quantifying the magnitude and character of velocity profiles and shearing forces in natural and artificial biologic conduits.     

Formation of a boundary layer in steady-state blood flow

This paper is devoted to a study of boundary layer formation in the steady flow of blood through the human aorta. Blood is treated as an incompressible fluid. Consideration is given to a flat-top velocity profile which combines the potential flow with the boundary layer; expressions for the displacement thickness, skin-friction and pressure in the entry region are derived.     

Wall shear stress estimates in coronary artery constrictions.

Wall shear stress estimates from laminar boundary layer theory were found to agree fairly well with the magnitude of shear stress levels along coronary artery constrictions obtained from solutions of the Navier Stokes equations for both steady and pulsatile flow. The relatively simple method can be used for in vivo estimates of wall shear stress in constrictions by using a vessel shape function determined from a coronary angiogram, along with a knowledge of the flow rate.     

A semi-empirical model for flow of blood and other particulate suspensions through narrow tubes

A semi-empirical model applicable to the flow of blood and other particulate suspensions through narrow tubes has been developed. It envisages a central core of blood surrounded by a wall layer of reduced hematocrit. With the help of this model the wall layer thickness and extent of plug flow may be calculated using pressure drop, flow rate and hematocrit reduction data. It has been found from the available data in the literature that for a given sample of blood the extent of plug flow increases with decreasing tube diameter. Also for a flow through a given tube it increases with hematocrit. The wall layer thickness is found to decrease with increase in blood hematocrit. A comparison between the results of rigid particulate suspensions and blood reveals that the thicker wall layer and smaller plug flow radius in the case of blood may be attributed to the deformability of the erythrocytes.     

Wall shear rate measurements in an elastic curved artery model

Since atherosclerotic lesions tend to be localized at bends and branching points, knowledge of wall shear rate patterns in models of these geometries may help elucidate the mechanism of atherogenesis. This study uses the photochromic method of flow visualization to determine both the mean and amplitude of the wall shear rate waveform in straight and curved elastic arterial models to demonstrate the effects of curvature, elasticity, and the phase angle between the flow and pressure waveforms (impedance phase angle). Under sinusoidal flow conditions characteristic of large arteries, the mean shear rate at the inner wall of the curved tube is reduced 40–56% from its steady flow value, depending on the phase angle. Wall shear rate amplitudes in the curved tube are significantly reduced by wall motion (36–55% of the Womersley amplitude for a straight rigid tube). The shear rate amplitude at the outer wall decreases 30% as the phase angle is reduced from −20° to −66°, while the shear rate amplitude at the inner wall increases 45%. As a result, the oscillatory nature of flow at the outer wall decreases with decreasing negative phase angle, but flow at the inner wall becomes much more oscillatory. At large negative phase angles, characteristic of hypertension or vasoactive agents, the shear rate at the inner wall has a small mean and cycles through positive and negative values; the shear rate at the outer wall remains positive throughout the flow cycle. Thus, the impedance phase angle could affect atherogenesis along the inner wall if temporal and directional changes in wall shear rate play a role.     

Slip-flow and heat transfer of a non-newtonian nanofluid in a microtube

The slip-flow and heat transfer of a non-Newtonian nanofluid in a microtube is theoretically studied. The power-law rheology is adopted to describe the non-Newtonian characteristics of the flow, in which the fluid consistency coefficient and the flow behavior index depend on the nanoparticle volume fraction. The velocity profile, volumetric flow rate and local Nusselt number are calculated for different values of nanoparticle volume fraction and slip length. The results show that the influence of nanoparticle volume fraction on the flow of the nanofluid depends on the pressure gradient, which is quite different from that of the Newtonian nanofluid. Increase of the nanoparticle volume fraction has the effect to impede the flow at a small pressure gradient, but it changes to facilitate the flow when the pressure gradient is large enough. This remarkable phenomenon is observed when the tube radius shrinks to micrometer scale. On the other hand, we find that increase of the slip length always results in larger flow rate of the nanofluid. Furthermore, the heat transfer rate of the nanofluid in the microtube can be enhanced due to the non-Newtonian rheology and slip boundary effects. The thermally fully developed heat transfer rate under constant wall temperature and constant heat flux boundary conditions is also compared.