Shape optimization in unsteady blood flow: a numerical study of non-Newtonian effects

This paper presents a numerical study of non-Newtonian effects on the solution of shape optimization problems involving unsteady pulsatile blood flow. We consider an idealized two dimensional arterial graft geometry. Our computations are based on the Navier-Stokes equations generalized to non-Newtonian fluid, with the modified Cross model employed to account for the shear-thinning behavior of blood. Using a gradient-based optimization algorithm, we compare the optimal shapes obtained using both the Newtonian and generalized Newtonian constitutive equations. Depending on the shear rate prevalent in the domain, substantial differences in the flow as well as in the computed optimal shape are observed when the Newtonian constitutive equation is replaced by the modified Cross model. By varying a geometric parameter in our test case, we investigate the influence of the shear rate on the solution.     

Shape optimization in unsteady blood flow: A numerical study of non-Newtonian effects

This paper presents a numerical study of non-Newtonian effects on the solution of shape optimization problems involving unsteady pulsatile blood flow. We consider an idealized two dimensional arterial graft geometry. Our computations are based on the Navier–Stokes equations generalized to non-Newtonian fluid, with the modified Cross model employed to account for the shear-thinning behavior of blood. Using a gradient-based optimization algorithm, we compare the optimal shapes obtained using both the Newtonian and generalized Newtonian constitutive equations. Depending on the shear rate prevalent in the domain, substantial differences in the flow as well as in the computed optimal shape are observed when the Newtonian constitutive equation is replaced by the modified Cross model. By varying a geometric parameter in our test case, we investigate the influence of the shear rate on the solution.     

Design optimization of vena cava filters: an application to dual filtration devices

Pulmonary embolism (PE) is a significant medical problem that results in over 300,000 fatalities per year. A common preventative treatment for PE is the insertion of a metallic filter into the inferior vena cava that traps thrombi before they reach the lungs. The goal of this work is to use methods of mathematical modeling and design optimization to determine the configuration of trapped thrombi that minimizes the hemodynamic disruption. The resulting configuration has implications for constructing an optimally designed vena cava filter. Computational fluid dynamics is coupled with a nonlinear optimization algorithm to determine the optimal configuration of a trapped model thrombus in the inferior vena cava. The location and shape of the thrombus are parametrized, and an objective function, based on wall shear stresses, determines the worthiness of a given configuration. The methods are fully automated and demonstrate the capabilities of a design optimization framework that is broadly applicable. Changes to thrombus location and shape alter the velocity contours and wall shear stress profiles significantly. For vena cava filters that trap two thrombi simultaneously, the undesirable flow dynamics past one thrombus can be mitigated by leveraging the flow past the other thrombus. Streamlining the shape of the thrombus trapped along the cava wall reduces the disruption to the flow but increases the area exposed to low wall shear stress. Computer-based design optimization is a useful tool for developing vena cava filters. Characterizing and parametrizing the design requirements and constraints is essential for constructing devices that address clinical complications. In addition, formulating a well-defined objective function that quantifies clinical risks and benefits is needed for designing devices that are clinically viable.     

Role of cytoskeleton and deformability in laminin-mediated cell rolling

Recent mathematical models show that molecular events that mediate rolling interactions also have an impact on the stochastic features of rolling. In spherical cells, statistical fluctuations in cell displacement were shown to be an indication that only a few adhesion bonds are involved in rolling interactions. In this study, we investigated whether cell shape and cell deformability could also modulate the stochastic features of rolling. As an experimental model we considered the flow-initiated rolling of MCF-10 breast epithelial cells on laminin. The dynamic adhesion of MCF-10A cells to laminin, which involves integrin alpha 6 beta 4, occurs slow enough to allow for an accurate determination of the trajectories of rolling cells. The data from high-magnification videomicroscopy showed that cell shape, cell deformability, and the level of fluid shear stress were all strong determinants of the rolling velocity and the extent of fluctuations in the trajectory of rolling cells. MCF-10A cells with large surface projections rolled faster and wobbled more extensively than spherical cells under the same flow conditions. The extent of wobbling decreased and the variation of rolling velocity increased with increasing fluid shear stress. MCF-10A cells treated with cytochalasin B, which increased cell deformability and caused extensive blebbing without significantly altering surface expression of laminin integrins, reduced mean rolling velocity and increased its variance. Because leukocytes change shape as they roll in postcapillary blood venules at high shear rates, results indicate the need for further expanding the present biophysical models of rolling to the case of deformable cells.     

Flow rates in perfusion bioreactors to maximise mineralisation in bone tissue engineering in vitro

In bone tissue engineering experiments, fluid-induced shear stress is able to stimulate cells to produce mineralised extracellular matrix (ECM). The application of shear stress on seeded cells can for example be achieved through bioreactors that perfuse medium through porous scaffolds. The generated mechanical environment (i.e. wall shear stress: WSS) within the scaffolds is complex due to the complexity of scaffold geometry. This complexity has so far prevented setting an optimal loading (i.e. flow rate) of the bioreactor to achieve an optimal distribution of WSS for stimulating cells to produce mineralised ECM. In this study, we demonstrate an approach combining computational fluid dynamics (CFD) and mechano-regulation theory to optimise flow rates of a perfusion bioreactor and various scaffold geometries (i.e. pore shape, porosity and pore diameter) in order to maximise shear stress induced mineralisation. The optimal flow rates, under which the highest fraction of scaffold surface area is subjected to a wall shear stress that induces mineralisation, are mainly dependent on the scaffold geometries. Nevertheless, the variation range of such optimal flow rates are within 0.5–5 mL/min (or in terms of fluid velocity: 0.166–1.66 mm/s), among different scaffolds. This approach can facilitate the determination of scaffold-dependent flow rates for bone tissue engineering experiments in vitro, avoiding performing a series of trial and error experiments.     

Model for the alignment of actin filaments in endothelial cells subjected to fluid shear stress

Cultured vascular endothelial cells undergo significant morphological changes when subjected to sustained fluid shear stress. The cells elongate and align in the direction of applied flow. Accompanying this shape change is a reorganization at the intracellular level. The cytoskeletal actin filaments reorient in the direction of the cells' long axis. How this external stimulus is transmitted to the endothelial cytoskeleton still remains unclear. In this article. we present a theoretical model accounting for the cytoskeletal reorganization under the influence of fluid shear stress. We develop a system of integro-partial-differential equations describing the dynamics of actin filaments, the actin-binding proteins, and the drift of transmembrane proteins due to the fluid shear forces applied on the plasma membrane. Numerical simulations of the equations show that under certain conditions, initially randomly oriented cytoskeletal actin filaments reorient in structures parallel to the externally applied fluid shear forces. Thus, the model suggests a mechanism by which shear forces acting on the cell membrane can be transmitted to the entire cytoskeleton via molecular interactions alone.     

Micromotion-induced peri-prosthetic fluid flow around a cementless femoral stem

Micromotion-induced interstitial fluid flow at the bone-implant interface has been proposed to play an important role in aseptic loosening of cementless implants. High fluid velocities are thought to promote aseptic loosening through activation of osteoclasts, shear stress induced control of mesenchymal stem cells differentiation, or transport of molecules. In this study, our objectives were to characterize and quantify micromotion-induced fluid flow around a cementless femoral stem using finite element modeling. With a 2D model of the bone-implant interface and full-factorial design, we first evaluated the relative influence of material properties, and bone-implant micromotion and gap on fluid velocity. Transverse sections around a femoral stem were built from computed tomography images, while boundary conditions were obtained from experimental measurements on the same femur. In a second step, a 3D model was built from the same data-set to estimate the shear stress experienced by cells hosted in the peri-implant tissues. The full-factorial design analysis showed that local micromotion had the most influence on peak fluid velocity at the interface. Remarkable variations in fluid velocity were observed in the macrostructures at the surface of the implant in the 2D transverse sections of the stem. The 3D model predicted peak fluid velocities extending up to 2.2 mm/s in the granulation tissue and to 3.9 mm/s in the trabecular bone. Peak shear stresses on the cells hosted in these tissues ranged from 0.1 to 12.5 Pa. These results offer insight into mechanical stimuli encountered at the bone-implant interface.     

Optimization of cardiovascular stent design using computational fluid dynamics

Coronary stent design affects the spatial distribution of wall shear stress (WSS), which can influence the progression of endothelialization, neointimal hyperplasia, and restenosis. Previous computational fluid dynamics (CFD) studies have only examined a small number of possible geometries to identify stent designs that reduce alterations in near-wall hemodynamics. Based on a previously described framework for optimizing cardiovascular geometries, we developed a methodology that couples CFD and three-dimensional shape-optimization for use in stent design. The optimization procedure was fully-automated, such that solid model construction, anisotropic mesh generation, CFD simulation, and WSS quantification did not require user intervention. We applied the method to determine the optimal number of circumferentially repeating stent cells (N(C)) for slotted-tube stents with various diameters and intrastrut areas. Optimal stent designs were defined as those minimizing the area of low intrastrut time-averaged WSS. Interestingly, we determined that the optimal value of N(C) was dependent on the intrastrut angle with respect to the primary flow direction. Further investigation indicated that stent designs with an intrastrut angle of approximately 40 deg minimized the area of low time-averaged WSS regardless of vessel size or intrastrut area. Future application of this optimization method to commercially available stent designs may lead to stents with superior hemodynamic performance and the potential for improved clinical outcomes.     

Analysis of Flow Phenomena in Gastric Contents Induced by Human Gastric Peristalsis Using CFD

This paper uses computational fluid dynamics to simulate and analyze intragastric fluid motions induced by human peristalsis. We created a two-dimensional computational domain of the distal stomach where peristalsis occurs. The motion of the gastric walls induced by an antral contraction wave (ACW) on the wall of the computational domain was well simulated using a function defined in this study. Retropulsive flow caused by ACW was observed near the occluded region, reaching its highest velocity of approximately 12 mm/s in the narrowest region. The viscosity of the model gastric contents applied in this study hardly affected the highest velocity, but greatly affected the velocity profile in the computational domain. The shear rate due to gastric fluid motion was calculated using the numerical output data. The shear rate reached relatively high values of approximately 20 s⁻¹ in the most occluded region. The shear rate profile was almost independent of the fluid viscosity. We also simulated mass transfer of a gastric digestive enzyme (pepsin) in model gastric content when peristalsis occurs on the gastric walls. The visualized simulation results suggest that gastric peristalsis is capable of efficiently mixing pepsin secreted from the gastric walls with an intragastric fluid.     

The influence of the non-Newtonian properties of blood on the flow in large arteries: steady flow in a carotid bifurcation model.

Laser Doppler anemometry experiments and finite element simulations of steady flow in a three dimensional model of the carotid bifurcation were performed to investigate the influence of non-Newtonian properties of blood on the velocity distribution. The axial velocity distribution was measured for two fluids: a non-Newtonian blood analog fluid and a Newtonian reference fluid. Striking differences between the measured flow fields were found. The axial velocity field of the non-Newtonian fluid was flattened, had lower velocity gradients at the divider wall, and higher velocity gradients at the non-divider wall. The flow separation, as found with the Newtonian fluid, was absent. In the computations, the shear thinning behavior of the analog blood fluid was incorporated through the Carreau-Yasuda model. The viscoelastic properties of the fluid were not included. A comparison between the experimental and numerical results showed good agreement, both for the Newtonian and the non-Newtonian fluid. Since only shear thinning was included, this seems to be the dominant non-Newtonian property of the blood analog fluid under steady flow conditions.     

Determinants of friction in soft elastohydrodynamic lubrication

Elastohydrodynamic lubrication (EHL) protects soft tissues from damage and wear in many biological systems (e.g. synovial joints, cornea of the eye, and pleural surfaces of the lung and chest wall). Among studies of lubrication of deformable solids, few have examined the effects of external loads, geometry, and material properties on EHL of soft tissues. To examine these effects, we studied the tribology of soft tissues in a two-dimensional finite element simulation of a thin layer of fluid separating a sliding rigid surface from a soft asperity or bump with an initial sinusoidal shape. We computed the frictional force, deformation of the solid, and change in fluid thickness as functions of independent variables: sliding velocity, normal load, material properties, and bump amplitude and length. Double-logarithmic regression was used to determine the exponents of the scaling relationships of friction coefficient and minimum fluid thickness to the independent variables. The analysis showed that frictional shear force is strongly dependent on velocity, viscosity, and load, moderately dependent on bump length and elasticity, and only weakly dependent on the bump amplitude. The minimum fluid thickness is strongly dependent on velocity and viscosity, and changes moderately with load, elasticity, amplitude, and length. The shape of the bump has little effect. The results confirm that the shear-induced deformation of an initially symmetrical shape, including generalizations to other symmetrical geometries such as quadratic or piecewise linear bumps, leads to load-supporting behavior.     

Microfluidic and computational study of structural properties and resistance to flow of blood clots under arterial shear

The ability of a blood clot to modulate blood flow is determined by the clot’s resistance, which depends on its structural features. For a flow with arterial shear, we investigated the characteristic patterns relating to clot shape, size, and composition on the one hand, and its viscous resistance, intraclot axial flow velocity, and shear distributions on the other. We used microfluidic technology to measure the kinetics of platelet, thrombin, and fibrin accumulation at a thrombogenic surface coated with collagen and tissue factor (TF), the key clot-formation trigger. We subsequently utilized the obtained data to perform additional calibration and validation of a detailed computational fluid dynamics model of spatial clot growth under flow. We then ran model simulations to gain insights into the resistance of clots formed under our experimental conditions. We found that increased thrombogenic surface length and TF surface density enhanced the bulk thrombin and fibrin generation in a nonadditive, synergistic way. The height of the platelet deposition domain—and, therefore, clot occlusivity—was rather robust to thrombogenic surface length and TF density variations, but consistently increased with time. Clot viscous resistance was non-uniform and tended to be higher in the fibrin-rich, inner “core” region of the clot. Interestingly, despite intraclot structure and viscous resistance variations, intraclot flow velocity variations were minor compared to the abrupt decrease in flow velocity around the platelet deposition region. Our results shed new light on the connection between the structure of clots under arterial shear and spatiotemporal variations in their resistance to flow.

     

Pulsatile flow in an end-to-side vascular graft model: comparison of computations with experimental data

Various hemodynamic factors have been implicated in vascular graft intimal hyperplasia, the major mechanism contributing to chronic failure of small-diameter grafts. However, a thorough knowledge of the graft flow field is needed in order to determine the role of hemodynamics and how these factors affect the underlying biological processes. Computational fluid dynamics offers much more versatility and resolution than in vitro or in vivo methods, yet computations must be validated by careful comparison with experimental data. Whereas numerous numerical and in vitro simulations of arterial geometries have been reported, direct point-by-point comparisons of the two techniques are rare in the literature. We have conducted finite element computational analyses for a model of an end-to-side vascular graft and compared the results with experimental data obtained using laser-Doppler velocimetry. Agreement for velocity profiles is found to be good, with some clear differences near the recirculation zones during the deceleration and reverse-flow segments of the flow waveform. Wall shear stresses are determined from velocity gradients, whether by computational or experimental methods, and hence the agreement for this quantity, while still good, is less consistent than for velocity itself from the wall shear stress numerical results, we computed four variables that have been cited in the development of intiimal hyperplasia-the time-averaged wall shear stress, an oscillating shear index, and spatial and temporal wall shear stress gradients in order to illustrate the versatility of numerical methods. We conclude that the computational approach is a valid alternative to the experimental approach for quantitative hemodynamic studies. Where differences in velocity were found by the two methods, it was generally attributed to the inability of the numerical method to model the fluid dynamics when flow conditions are destabilizing. Differences in wall shear, in the absence of destabilizing phenomena, were more likely to be caused by difficulties in calculating wall shear from relatively low resolution in vitro data.     

Effects of transmural pressure and wall shear stress on LDL accumulation in the arterial wall: a numerical study using a multilayered model

The accumulation of low-density lipoprotein (LDL) is recognized as one of the main contributors in atherogenesis. Mathematical models have been constructed to simulate mass transport in large arteries and the consequent lipid accumulation in the arterial wall. The objective of this study was to investigate the influences of wall shear stress and transmural pressure on LDL accumulation in the arterial wall by a multilayered, coupled lumen-wall model. The model employs the Navier-Stokes equations and Darcy's Law for fluid dynamics, convection-diffusion-reaction equations for mass balance, and Kedem-Katchalsky equations for interfacial coupling. To determine physiologically realistic model parameters, an optimization approach that searches optimal parameters based on experimental data was developed. Two sets of model parameters corresponding to different transmural pressures were found by the optimization approach using experimental data in the literature. Furthermore, a shear-dependent hydraulic conductivity relation reported previously was adopted. The integrated multilayered model was applied to an axisymmetric stenosis simulating an idealized, mildly stenosed coronary artery. The results show that low wall shear stress leads to focal LDL accumulation by weakening the convective clearance effect of transmural flow, whereas high transmural pressure, associated with hypertension, leads to global elevation of LDL concentration in the arterial wall by facilitating the passage of LDL through wall layers.     

Natural Convection Flow near a Vertical Plate that Applies a Shear Stress to a Viscous Fluid

The unsteady natural convection flow of an incompressible viscous fluid near a vertical plate that applies an arbitrary shear stress to the fluid is studied using the Laplace transform technique. The fluid flow is due to both the shear and the heating of the plate. Closed-form expressions for velocity and temperature are established under the usual Boussinesq approximation. For illustration purposes, two special cases are considered and the influence of pertinent parameters on the fluid motion is graphically underlined. The required time to reach the steady state in the case of oscillating shear stresses on the boundary is also determined.     

The effect of blood viscoelasticity on pulsatile flow in stationary and axially moving tubes

An analytical solution for pulsatile flow of a generalized Maxwell fluid in straight rigid tubes, with and without axial vessel motion, has been used to calculate the effect of blood viscoelasticity on velocity profiles and shear stress in flows representative of those in the large arteries. Measured bulk flow rate Q waveforms were used as starting points in the calculations for the aorta and femoral arteries, from which axial pressure gradient ▿P waves were derived that would reproduce the starting Q waves for viscoelastic flow. The ▿P waves were then used to calculate velocity profiles for both viscoelastic and purely viscous flow. For the coronary artery, published ▿P and axial vessel acceleration waveforms were used in a similar procedure to determine the separate and combined influences of viscoelasticity and vessel motion.Differences in local velocities, comparing viscous flow to viscoelastic flow, were in all cases less than about 2% of the peak local velocity. Differences in peak wall shear stress were less than about 3%.In the coronary artery, wall shear stress differences between viscous and viscoelastic flow were small, regardless of whether axial vessel motion was included. The shape of the wall shear stress waveform and its difference, however, changed dramatically between the stationary and moving vessel cases. The peaks in wall shear stress difference corresponded with large temporal gradients in the combined driving force for the flow.     

Coupling curvature-dependent and shear stress-stimulated neotissue growth in dynamic bioreactor cultures: a 3D computational model of a complete scaffold

The main challenge in tissue engineering consists in understanding and controlling the growth process of in vitro cultured neotissues toward obtaining functional tissues. Computational models can provide crucial information on appropriate bioreactor and scaffold design but also on the bioprocess environment and culture conditions. In this study, the development of a 3D model using the level set method to capture the growth of a microporous neotissue domain in a dynamic culture environment (perfusion bioreactor) was pursued. In our model, neotissue growth velocity was influenced by scaffold geometry as well as by flow- induced shear stresses. The neotissue was modeled as a homogenous porous medium with a given permeability, and the Brinkman equation was used to calculate the flow profile in both neotissue and void space. Neotissue growth was modeled until the scaffold void volume was filled, thus capturing already established experimental observations, in particular the differences between scaffold filling under different flow regimes. This tool is envisaged as a scaffold shape and bioprocess optimization tool with predictive capacities. It will allow controlling fluid flow during long-term culture, whereby neotissue growth alters flow patterns, in order to provide shear stress profiles and magnitudes across the whole scaffold volume influencing, in turn, the neotissue growth.     

The influence of inlet velocity profile on predicted flow in type B aortic dissection

In order for computational fluid dynamics to provide quantitative parameters to aid in the clinical assessment of type B aortic dissection, the results must accurately mimic the hemodynamic environment within the aorta. The choice of inlet velocity profile (IVP) therefore is crucial; however, idealised profiles are often adopted, and the effect of IVP on hemodynamics in a dissected aorta is unclear. This study examined two scenarios with respect to the influence of IVP—using (a) patient-specific data in the form of a three-directional (3D), through-plane (TP) or flat IVP; and (b) non-patient-specific flow waveform. The results obtained from nine simulations using patient-specific data showed that all forms of IVP were able to reproduce global flow patterns as observed with 4D flow magnetic resonance imaging. Differences in maximum velocity and time-averaged wall shear stress near the primary entry tear were up to 3% and 6%, respectively, while pressure differences across the true and false lumen differed by up to 6%. More notable variations were found in regions of low wall shear stress when the primary entry tear was close to the left subclavian artery. The results obtained with non-patient-specific waveforms were markedly different. Throughout the aorta, a 25% reduction in stroke volume resulted in up to 28% and 35% reduction in velocity and wall shear stress, respectively, while the shape of flow waveform had a profound influence on the predicted pressure. The results of this study suggest that 3D, TP and flat IVPs all yield reasonably similar velocity and time-averaged wall shear stress results, but TP IVPs should be used where possible for better prediction of pressure. In the absence of patient-specific velocity data, effort should be made to acquire patient’s stroke volume and adjust the applied IVP accordingly.

     

Numerical investigation of the non-Newtonian pulsatile blood flow in a bifurcation model with a non-planar branch

The pulsatile flow of non-Newtonian fluid in a bifurcation model with a non-planar daughter branch is investigated numerically by using the Carreau-Yasuda model to take into account the shear thinning behavior of the analog blood fluid. The objective of this study is to deal with the influence of the non-Newtonian property of fluid and of out-of-plane curvature in the non-planar daughter vessel on wall shear stress (WSS), oscillatory shear index (OSI), and flow phenomena during the pulse cycle. The non-Newtonian property in the daughter vessels induces a flattened axial velocity profile due to its shear thinning behavior. The non-planarity deflects flow from the inner wall of the vessel to the outer wall and changes the distribution of WSS along the vessel, in particular in systole phase. Downstream of the bifurcation, the velocity profiles are shifted toward the flow divider, and low WSS and high shear stress temporal oscillations characterized by OSI occur on the outer wall region of the daughter vessels close to the bifurcation. Secondary motions become stronger with the addition of the out-of-plane curvature induced by the bending of the vessel, and the secondary flow patterns swirl along the non-planar daughter vessel. A significant difference between the non-Newtonian and the Newtonian pulsatile flow is revealed during the pulse cycle; however, reasonable agreement between the non-Newtonian and the rescaled Newtonian flow is found. Calculated results for the pulsatile flow support the view that the non-planarity of blood vessels and the non-Newtonian properties of blood are an important factor in hemodynamics and may play a significant role in vascular biology and pathophysiology.     

Numerical investigation of the non-Newtonian blood flow in a bifurcation model with a non-planar branch

The non-Newtonian fluid flow in a bifurcation model with a non-planar daughter branch is investigated by using finite element method to solve the three-dimensional Navier–Stokes equations coupled with a non-Newtonian constitutive model, in which the shear thinning behavior of the blood fluid is incorporated by the Carreau–Yasuda model. The objective of this study is to investigate the influence of the non-Newtonian property of fluid as well as of curvature and out-of-plane geometry in the non-planar daughter vessel on wall shear stress (WSS) and flow phenomena. In the non-planar daughter vessel, the flows are typified by the skewing of the velocity profile towards the outer wall, creating a relatively low WSS at the inner wall. In the downstream of the bifurcation, the velocity profiles are shifted towards the flow divider. The low WSS is found at the inner walls of the curvature and the lateral walls of the bifurcation. Secondary flow patterns that swirl fluid from the inner wall of curvature to the outer wall in the middle of the vessel are also well documented for the curved and bifurcating vessels. The numerical results for the non-Newtonian fluid and the Newtonian fluid with original Reynolds number and the corresponding rescaled Reynolds number are presented. Significant difference between the non-Newtonian flow and the Newtonian flow is revealed; however, reasonable agreement between the non-Newtonian flow and the rescaled Newtonian flow is found. Results of this study support the view that the non-planarity of blood vessels and the non-Newtonian properties of blood are an important factor in hemodynamics and may play a significant role in vascular biology and pathophysiology.