Mathematical Model Of Blood Flow Research Articles

Mathematical Model of Blood Flow in an Anatomically Detailed Arterial Network of the Arm

oo 2,3 Abstract. A distributed-parameter (one-dimensional) anatomically detailed model for the arterial network of the arm is developed in order to carry out hemodynamics simulations. This work focuses on the specific aspects related to the model set-up. In this regard, stringent anatomical and physiological considerations have been pursued in order to construct the arterial topology and to provide a systematic estimation of the involved parameters. The model comprises 108 arterial segments, with 64 main arteries and 44 perforator arteries, with lumen radii ranging from 0.24 cm - axillary artery- to 0.018 cm - perforator arteries. The modeling of blood flow in deformable vessels is governed by a well-known set of hyperbolic partial differential equations that accounts for mass and momentum conservation and a constitutive equation for the arterial wall. The variational formulation used to solve the problem and the related numerical approach are described. The model rendered consistent pressure and flow rate outputs when compared with patient records already published in the literature. In addition, an application to dimensionally-heterogeneous modeling is presented in which the developed arterial network is employed as an underlying model for a three-dimensional geometry of a branching point to be embedded in order to perform local analyses.

Shear-thinning effects of hemodynamics in patient-specific cerebral aneurysms

Two different generalized Newtonian mathematical models for blood flow, derived for the same experimental data, are compared, together with the Newtonian model, in three different anatomically realistic geometries of saccular cerebral aneurysms obtained from rotational CTA. The geometries differ in size of the aneurysm and the existence or not of side branches within the aneurysm. Results show that the differences between the two generalized Newtonian mathematical models are smaller than the differences between these and the Newtonian solution, in both steady and unsteady simulations.

Трехмерная математическая модель гемодинамики с учетом распределенного сердца

Трехмерная математическая модель гемодинамики с учетом распределенного сердца

Comparative Analysis of Mathematical Models for Blood Flow in Tapered Constricted Arteries

Pulsatile flow of blood in narrow tapered arteries with mild overlapping stenosis in the presence of periodic body acceleration is analyzed mathematically, treating it as two‐fluid model with the suspension of all the erythrocytes in the core region as non‐Newtonian fluid with yield stress and the plasma in the peripheral layer region as Newtonian. The non‐Newtonian fluid with yield stress in the core region is assumed as (i) Herschel‐Bulkley fluid and (ii) Casson fluid. The expressions for the shear stress, velocity, flow rate, wall shear stress, plug core radius, and longitudinal impedance to flow obtained by Sankar (2010) for two‐fluid Herschel‐Bulkley model and Sankar and Lee (2011) for two‐fluid Casson model are used to compute the data for comparing these fluid models. It is observed that the plug core radius, wall shear stress, and longitudinal impedance to flow are lower for the two‐fluid H‐B model compared to the corresponding flow quantities of the two‐fluid Casson model. It is noted that the plug core radius and longitudinal impedance to flow increases with the increase of the maximum depth of the stenosis. The mean velocity and mean flow rate of two‐fluid H‐B model are higher than those of the two‐fluid Casson model.

Effects of magnetic field, porosity, and wall properties for anisotropically elastic multi-stenosis arteries on blood flow characteristics

A mathematical model for blood flow through an elastic artery with multistenosis under the effect of a magnetic field in a porous medium is presented. The considered arterial segment is simulated by an anisotropically elastic cylindrical tube filled with a viscous incompressible electrically conducting fluid representing blood. An artery with mild local narrowing in its lumen forming a stenosis is analyzed. The effects of arterial wall parameters represent viscoelastic stresses along the longitudinal and circumferential directions Tt and Tθ, respectively. The degree of anisotropy of the vessel wall γ, total mass of the vessel, and surrounding tissues M and contributions of the viscous and elastic constraints to the total tethering C and K respectively on resistance impedance, wall shear stress distribution, and radial and axial velocities are illustrated. Also, the effects of the stenosis shape m, the constant of permeability X, the Hartmann number Hα and the maximum height of the stenosis size δ on the fluid flow characteristics are investigated. The results show that the flow is appreciably influenced by surrounding connective tissues of the arterial wall motion, and the degree of anisotropy of the vessel wall plays an important role in determining the material of the artery. Further, the wall shear stress distribution increases with increasing Tt and γ while decreases with increasing Tθ, M, C, and K. Transmission of the wall shear stress distribution and resistance impedance at the wall surface through a tethered tube are substantially lower than those through a free tube, while the shearing stress distribution at the stenosis throat has inverse characteristic through totally tethered and free tubes. The trapping bolus increases in size toward the line center of the tube as the permeability constant X increases and decreases with the Hartmann number Ha increased. Finally, the trapping bolus appears, gradually in the case of non-symmetric stenosis, and disappears in the case of symmetric stenosis. The size of trapped bolus for the stream lines in a free isotropic tube (i.e., a tube initially unstressed) is smaller than those in a tethered tube.

A study on two-layered model (Casson–Newtonian) for blood flow through an arterial stenosis: Axially variable slip velocity at the wall

The present paper sheds some light on a mathematical model for blood flow through stenosed arteries with axially variable peripheral layer thickness and variable slip at the wall. The model consists of a core region of suspension of all the erythrocytes assumed to be a Casson fluid and a peripheral layer of plasma as a Newtonian fluid. For such models, in literature, the peripheral layer thickness and slip velocity are assumed a priori based on experimental observations. In the present analysis, new analytic expressions for the thickness of the peripheral layer, slip velocity and core viscosity have been obtained in terms of measurable quantities (flow rate ( Q), centerline velocity ( U), pressure gradient (− dp/ dz), plasma viscosity ( μ p ) and yield stress ( θ)). Using the experimental values of Q, U, (− dp/ dz), μ p and θ, the values of the peripheral layer thickness, core viscosity, and slip velocity at the wall have been computed. The theoretically obtained peripheral layer thickness has been compared with its experimental value. It is found that the agreement between the two is very good (error<1.4%). Further, a comparison between theoretical and experimental values of core viscosity is made and it is observed that the error between the two becomes 3.7465% in the case of two-layered model (Casson–Newtonian) for tube diameter 40 μm. The analysis developed here could be used to determine the more accurate values of the apparent viscosity of blood, agreeability, rigidity and deformability of red cells. This information of blood could be useful in the development of new diagnosis tools for many diseases.

Induced magnetic field influences on blood flow through an anisotropically tapered elastic artery with overlapping stenosis in an annulus

A mathematical model for blood flow through an elastic artery with overlapping stenosis under the effect of induced magnetic field is presented. The present theoretical model may be considered as a mathematical representation to the movement of conductive physiological fluid through coaxial tubes such that the inner tube is uniform and rigid, representing a catheter tube, while the outer tube is an anisotropically tapered elastic cylindrical tube filled with a viscous incompressible electrically conducting fluid, representing blood. The analysis is carried out for an artery with mild local narrowing in its lumen, forming a stenosis. Analytical expressions for the stream function, the magnetic force function, the axial velocity, the axial induced magnetic field, and the distribution of the current density are obtained. The results for the resistance impedance, the wall shear stress distribution, the axial velocity, the axial induced magnetic field, and distribution of the current density have been computed numerically, and the results were studied for various values of the physical parameters, such as the the Hartmann number Ha, the magnetic Reynolds number Rm, the taper angle ϕ, the maximum height of stenosis δ, the degree of anisotropy of the vessel wall n, and the contributions of the elastic constraints to the total tethering K.

Numerical Analysis of One-dimensional Mathematical Model of Blood Flow to Reproduce Fundamental Pulse Wave Measurement for Scientific Verification of Pulse Diagnosis

Pulse diagnosis in traditional Chinese medicine is said to be able to detect not only illness but also decline of health in the patients from tactile sense of the pulse in the radial artery at the wrists. This diagnosis, however, is not supported by concrete scientific evidence. The authors have proposed a non-linear spring model of subcutaneous tissue on the radial artery and one-dimensional arterial blood flow model in an arm for the purpose of scientific verification of the pulse diagnosis. They performed, in the former study, a numerical experiment with this mathematical model in which the radial artery was indented in a stepwise manner by a pressure sensor, which extract the fundamental mechanism of the pulse diagnosis, to validate the subcutaneous tissue model and to find the appropriate coefficient to fit the experimental result. They investigated, in this research, contribution of parameters of supply pressure of the blood and tube law of the artery on the change in the pressure pulse waves with the indentation steps with respect to mean value Poav and amplitude ΔPo of the pressure. It was shown that mean supply pressure affects both Poav and ΔPo, while amplitude of the supply pressure affects ΔPo. It was also shown that profile of ΔPo vs. distance of the indentation changes drastically as the artery becomes hard. Lastly, it was examined to reproduce the experimentally obtained pressure pulse waves during the indentation in their former work with the mathematical model by adjusting the parameters. The result showed better agreement than the former result, but it implied that ulnar artery had to be taken into consideration for quantitative fitting of the pulse waves to the range where the radial artery was nearly flattened by the indentation.

Mathematical Model of Blood Flow in Small Blood Vessel in the Presence of Magnetic Field

A mathematical model for blood flow in the small blood vessel in the presence of magnetic field is presented in this paper. We have modeled the two phase model for the blood flow consists of a central core of suspended erythrocytes and cell-free layer surrounding the core. The system of differential equations has been solved analytically. We have obtained the result for velocity, flow rate and effective viscosity in presence of peripheral layer and magnetic field .All the result has been obtained and discussed through graphs.

Sensitivity of hemodynamics in a patient specific cerebral aneurysm to vascular geometry and blood rheology

Newtonian and generalized Newtonian mathematical models for blood flow are compared in two different reconstructions of an anatomically realistic geometry of a saccular aneurysm, obtained from rotational CTA and differing to within image resolution. The sensitivity of the flow field is sought with respect to geometry reconstruction procedure and mathematical model choice in numerical simulations. Taking as example a patient specific intracranial aneurysm located on an outer bend under steady state simulations, it is found that the sensitivity to geometry variability is greater, but comparable, to the one of the rheological model. These sensitivities are not quantifiable a priori. The flow field exhibits a wide range of shear stresses and slow recirculation regions that emphasize the need for careful choice of constitutive models for the blood. On the other hand, the complex geometrical shape of the vessels is found to be sensitive to small scale perturbations within medical imaging resolution. The sensitivity to mathematical modeling and geometry definition are important when performing numerical simulations from in vivo data, and should be taken into account when discussing patient specific studies since differences in wall shear stress range from 3% to 18%.

Modelling of blood flow resistance for an atherosclerotic artery with multiple stenoses and poststenotic dilatations

Modelling of blood flow resistance for an atherosclerotic artery with multiple stenoses and poststenotic dilatations

Mathematical Model of Blood Flow Through a Tapered Artery with Mild Stenosis and Hematocrit

Mathematical Model of Blood Flow Through a Tapered Artery with Mild Stenosis and Hematocrit

Mathematical modeling of the response of a resistive vessel to pressure

The response of small arterial vessels to internal pressure makes an essential contribution to autoregulation in the vascular bed. It is believed that intracellular cytosolic Ca2+ concentration plays a pivotal role in the regulation of smooth muscle contractility and hence of vascular lumen. A simple mathematical model of blood flow in a resistive vessel is suggested. The model is based on the experimental data obtained for cerebral arteries, but may be used for any other resistive vessel. The model not only describes the regulation of the vascular lumer by transmural pressure but also shows realistic behaviour of the radius and intracellular cytosolic Ca2+ concentration at different velocities of pressure change. Possible variations of the radius along the vessel as a consequence of the Bayliss effect are considered.

Two-Fluid Mathematical Models for Blood Flow in Stenosed Arteries: A Comparative Study

The pulsatile flow of blood through stenosed arteries is analyzed by assuming the blood as a two-fluid model with the suspension of all the erythrocytes in the core region as a non-Newtonian fluid and the plasma in the peripheral layer as a Newtonian fluid. The non-Newtonian fluid in the core region of the artery is assumed as a (i) Herschel-Bulkley fluid and (ii) Casson fluid. Perturbation method is used to solve the resulting system of non-linear partial differential equations. Expressions for various flow quantities are obtained for the two-fluid Casson model. Expressions of the flow quantities obtained by Sankar and Lee (2006) for the two-fluid Herschel-Bulkley model are used to get the data for comparison. It is found that the plug flow velocity and velocity distribution of the two-fluid Casson model are considerably higher than those of the two-fluid Herschel-Bulkley model. It is also observed that the pressure drop, plug core radius, wall shear stress and the resistance to flow are significantly very low for the two-fluid Casson model than those of the two-fluid Herschel-Bulkley model. Hence, the two-fluid Casson model would be more useful than the two-fluid Herschel-Bulkley model to analyze the blood flow through stenosed arteries.

Non-linear mathematical models for blood flow through tapered tubes

In this paper, the steady flow of blood through tapered tube has been analyzed assuming blood as (i) Casson fluid and (ii) Herschel–Bulkley fluid. The expressions for pressure drop, wall shear stress and resistance to flow have been obtained. The effects of tapering of the tube and the non-Newtonian nature of the fluid on pressure drop, wall shear stress and resistance to flow are discussed. For all fluids, the pressure drop increases with increasing angle of taper from 0.5° to 1° for a given value of yield stress θ and tapered tube Reynolds number Re ψ . The resistance to flow as well as the wall shear stress increase with increasing yield stress for Herschel–Bulkley fluid and also for Casson’s fluid when the other parameters held constant. Both for Herschel–Bulkley fluid and Casson’s fluid, the wall shear stress as well as the resistance to flow increase with increasing axial distance for a given tapered tube Reynolds number Re ψ , angle of taper ψ and yield stress θ.

Mathematical modelling of blood flow and oxygen transport in the human placenta

Mathematical modelling of blood flow and oxygen transport in the human placenta

Mathematical modelling of blood flow in curved arteries

We study the mixed convection boundary layer heat transfer of power law fluid over a moving conveyor along an inclined plate. The effects of shear flow and power law viscosity on the temperature field are taken into account according to a modified Fourier law. Approximate analytical solutions are obtained by the homotopy analysis method (HAM). Results indicate that heat transfer is strongly dependent on the values of power law exponent, inclination angle, boundary velocity ratio and Prandtl number. Three distinct characteristics are found for power law exponents 0 < n < 1, n = 1 and n > 1, especially the nonlinear behavior due to skin friction and local Nusselt number shown in Figs. 4 and 17, which has never been reported before. The decrease of inclination angle causes the loss of velocity boundary layer but the gain of temperature boundary layer. Heat transfer efficiency is enhanced but skin friction is diminished with the increase in velocity ratio (the ratio of conveyor velocity/mean velocity of flow field). Critical ratio (with skin friction zero) is obtained which strongly depends on the power law exponent. The effects of involved parameters on the velocity and temperature fields are analyzed.

Mathematical modeling of blood flow in an arterial bypass anastomosis

Mathematical modeling of blood flow in an arterial bypass anastomosis

A mathematical model for blood flow in magnetic field

In the present study a mathematical model of biomagnetic fluid dynamics (BFD), suitable for the description of the Newtonian blood flow under the action of an applied magnetic field, is proposed. This model is consistent with the principles of ferrohydrodynamics and magnetohydrodynamics and takes into account both magnetization and electrical conductivity of blood. As a representative application the laminar, incompressible, three-dimensional, fully developed viscous flow of a Newtonian biomagnetic fluid (blood) in a straight rectangular duct is numerically studied under the action of a uniform or a spatially varying magnetic field. The numerical results are obtained using a finite differences numerical technique based on a pressure-linked pseudotransient method on a collocated grid. The flow is appreciably influenced by the application of the magnetic field and in particularly by the strength and the magnetic field gradient. A comparison of the derived results is also made with those obtained using the existing BFD model indicating the necessity of taking into account the electrical conductivity of blood.

Blood flow and permeability in microvessels

The mechanics of blood flow in microvessels and microvessel permeability are reviewed. In the first part, characteristics of blood flow in vivo and in vitro are described from a fluid-mechanical point of view, and mathematical models for blood flow in microvessels are presented. Possible causes of the increased flow resistance obtained in vivo compared to in vitro are examined, including the effects of irregularities of vessel lumen, the presence of endothelial surface glycocalyx and white blood cells. In the second part, the ultrastructural pathways and mechanisms whereby endothelial cells and the clefts between the cells modulate microvessel permeability to water and solutes are introduced. Previous and current models for microvessel permeability to water and solutes are reviewed. These models examine the role of structural components of interendothelial cleft, such as junction strands and surface glycocalyx, in the determination of water and solute transport across the microvessel walls. Transport models in the tissue space surrounding the microvessel are also described.