Mathematical Model Of Blood Flow Research Articles

A Multiscale Approach for Modelling Wave Propagation in an Arterial Segment

A mathematical model of blood flow through an arterial vessel is presented and the wave propagation in it is studied numerically. Based on the assumption of long wavelength and small amplitude of the pressure waves, a quasi-one-dimensional (1D) differential model is adopted. It describes the non-linear fluid-wall interaction and includes wall deformation in both radial and axial directions. The 1D model is coupled with a six compartment lumped parameter model, which accounts for the global circulatory features and provides boundary conditions. The differential equations are first linearized to investigate the nature of the propagation phenomena. The full non-linear equations are then approximated with a numerical finite difference method on a staggered grid. †For a viscous fluid, Some numerical simulations show the characteristics of the wave propagation. The dependence of the flow, of the wall deformation and of the wave velocity on the elasticity parameter has been highlighted. The importance of the axial deformation is evidenced by its variation in correspondence of the pressure peaks. The wave disturbances consequent to a local stiffening of the vessel and to a compliance jump due to prosthetic implantations are finally studied.

Biomathematical modeling and analysis of blood flow in an intracranial aneurysm

The basic hypothesis of this study is that the intracranial aneurysm may enlarge and rupture due to dynamic instabilities of the blood flow and pressure inside the aneurysm. The specific question we attempted to answer is: which parameter(s) of aneurysmal geometry can serve as a reliable predictor(s) for aneurysmal rupture? We consider an idealized cylindrical aneurysm of the human common carotid artery and develop a mathematical model of blood flow through a normal artery and aneurysm connected in series. The mathematical model is nonlinear. It comprises nonlinear rheological properties of the normal artery and aneurysmal materials, and the inertial and resistance properties of the blood flow. The model equations were solved numerically and analyzed by methods of nonlinear dynamics. The critical aneurysmal diameter (CAD) is defined as a boundary point between the stable and unstable states of the model equations. The results confirm that a limit point of flow stability can occur only for a certain difference between aneurysmal and artery radii which are pre-disposed from a difference in their material properties. It was shown that CAD is dependent on both aneurysmal length and age of patient. Finally, the results suggest that the ratio between aneurysmal and normal artery diameters is a more reliable predictor of the aneurysmal rupture than the diameter alone. We conclude that an aneurysm diameter twice that of the normal artery could be dangerous.

Flow of Blood in Eye Vessels with Allowance for Aggregation

A nonlinear mathematical model of blood flow in a vessel has been constructed where blood is considered as a suspension with aggregating and deformable particles and vessel walls possess elasticity according to the Hooke law. The blood viscosity, which is dependent on the gradient of the rate of shear on the vessel wall, becomes a function of the time in the case of pulsatory motion. The cases of normal blood flow in eye vessels and of that in diabetes mellitus have been considered as examples. It has been shown that in the latter case the contribution of the aggregability of erythrocytes to the value of the viscosity of blood substantially increases.

Pulse-wave velocity measured in one heartbeat using MR tagging.

A noninvasive method for measuring the aortic pulse-wave velocity (PWV) in a single heartbeat is introduced. The method sinusoidally tags a column of blood within the vessel, and rapidly acquires a series of 1D projections of the tags as they move (in practice, 64 projections at 4-ms intervals). From these projections, the relative motion of blood at different positions along the vessel is measured. The PWV is obtained by fitting a mathematical model of blood flow to the tag trajectories. Tests of this method in a pulsatile flow phantom are presented using latex and polyurethane tubes. The PWV measured in these tubes was (mean +/- standard deviation) 4.4 +/- 0.5 m/s and 2.3 +/- 0.2 m/s, respectively. The distensibility of each tube was calculated from the PWV (latex = (7 +/- 2) 10(-3) mm Hg(-1), poly. = (25 +/- 4) 10(-3)mmHg(-1)) and found to agree within error with distensibility measurements based on the change of tube area with pressure (latex = (6.3 +/- 0.3) 10(-3)mmHg(-1), poly. = (27 +/- 1) 10(-3) mmHg(-1)). To test its feasibility, the PWV measurement was applied to four normal volunteers. The measured PWV values were 3.9 +/- 0.8 m/s, 3.6 +/- 0.9 m/s, 3.9 +/- 0.5 m/s, and 5.3 +/- 0.8 m/s. By acquiring an independent PWV measurement each heartbeat, errors introduced by arrhythmia and trigger variability appear to be avoided with this method.

Mathematical Modeling of Some Applied Problems in Haemodynamics

The present study discusses some basic issues of mathematical modeling in medicine and describes examples of specific results of analytical and numerical investigations in haemodynamics. It briefly describes a method for constructing exact solutions of the linearized equations of haemodynamics on a graph, an analysis of these solutions, and a particular application of the proposed theory for solving practical medical problems. A number of nonspecific aortoarteritis syndromes are simulated. Examples of mathematical modeling of blood flow in the systemic circulatory circuit under the action of a periodically contracting heart are presented. These examples reproduce some of the main physiological regularities of the circulatory system as a whole. One of the applications of mathematical modeling in haemodynamics includes the “stealing” of cerebral blood supply during a temporary occlusion of the subclavian artery.

Hemodynamic role of the circle of Willis in stenoses of internal carotid arteries. An analytical solution of a linear model

A mathematical model of blood flow through the circle of Willis was developed, within a linear framework. Comprehensive analytical solutions, including a remarkably small number of parameters, were derived in the cases of obstructive lesions of extracranial carotid arteries. The influence of these lesions and the role of anterior and posterior communicating arteries on the blood pressure at the entry of the cerebral territories were quantified and analyzed emphasizing that the responses of the system of Willis to obstructive carotid lesions are extremely varied, depending on the communicating artery anatomy. Comparison with numerical results obtained by using a non-linear model showed no physiologically significant differences. Such a model might be an essential tool for an accurate assessment of the cerebral hemodynamics in carotid diseases.

Numerical simulation of the hemodynamics and biomechanics of the arterial system

A mathematical model of blood flow in arterial vessels with account for the nonstationary character of the laminar flow and the time-varying pressure drop is considered. The viscosity of blood is assumed to obey a power law. The rheological properties of the walls of arterial vessels are described by i) the Voigt—Kelvin model, ii) the standard-body model, and iii) the three-component Voigt—Kelvin model. As a result of numerical simulation, flow-pressure characteristics of various arteries are obtained.

A nonlinear mathematical model of blood flow in a constricted artery experiencing body acceleration

A nonlinear mathematical model is developed analytically to study the flow characteristics of blood through an artery in the presence of multistenoses when it is subjected to whole body acceleration. An appropriate shape of the time-dependent multistenoses which are overlapped in the realm of the formation of arterial narrowing caused by atheroma is constructed mathematically. The artery is simulated as a flexible cylindrical tube containing a nonhomogeneous Newtonian fluid with variable viscosity characterising blood, having a narrowed portion forming multistenoses in its lumen. To update resemblance to the in vivo situation, the viscosity of blood is taken to be a function of hematocrit in such a way that it varies radially more near the central region where most blood cells are aggregated and less towards the arterial wall, a cell-free region, and thus, the nonhomogeneity of blood is accounted for. The unsteady flow mechanism in the multistenosed artery is subjected to a pulsatile pressure gradient arising from the normal functioning of the heart as also the body acceleration. The present analysis bears the potential to calculate the flow-field with low computational complexity by using the appropriate boundary conditions. Finally, the extensive quantitative analysis helps estimating the significant effects of the severity of the multistenoses, the external body acceleration together with the vessel wall distensibility and nonhomogeneity of the flowing blood on the unsteady flow characteristics in order to justify the applicability of the present model.

Two-layered model of blood flow through stenosed arteries

The present investigation deals with a two-layered mathematical model of blood flow through an artery provided with a cosine-shaped constriction. The model consists of a peripheral plasma layer free from red cells and a core region represented by a Casson fluid. The geometry of the interface between the plasma layer and the core region has been determined and compared with that of the constriction along the length of the tube. The theoretical results obtained in this analysis are the expressions for wall shear stress and pressure drop for variable plasma layer thickness. The effect of the variable plasma layer thickness on the flow characteristics has been shown graphically for different parameter values to enable a better understanding of the biomechanical problem.

Effects of External Body Accelerations on Blood Flow

A mathematical model of blood flow in single arteries and arterioles subject to both the pulsatile pressure gradient due to normal heart action and a single cycle of body acceleration is presented. The body acceleration is such that it increases from zero to a maximum value at a uniform rate, remains constant at this maximum value for some time and thereafter returns to zero at a uniform rate. To account for the red cells in blood, blood is considered to be a two-phase fluid, that is, a suspension of red cells in plasma. Numerical results are presented for the effects of body acceleration on flow variables, such as, flow rate, velocity, acceleration and shear stress corresponding to the flow of blood in the human aorta and arterioles.

Mathematical modelling of blood flow through an overlapping arterial stenosis

Of concern in the paper is a theoretical study of blood flow in an arterial segment in the presence of a time-dependent overlapping stenosis using an appropriate mathematical model. A remarkably new shape of the stenosis in the realm of the formation of the arterial narrowing caused by atheroma is constructed mathematically. The artery is simulated as an elastic (moving wall) cylindrical tube containing a viscoelastic fluid representing blood. The unsteady flow mechanism of the present investigation is subjected to a pulsatile pressure gradient arising from the normal functioning of the heart. The equations governing the motion of the system are sought in the Laplace transform space and their relevant solutions supplemented by the suitable boundary conditions are obtained numerically in the transformed domain through the use of an appropriate finite difference technique. Laplace inversion is also carried out by employing numerical techniques. A thorough quantitative analysis is performed at the end of the paper for the flow velocity, the flux, the resistive impedances, and the wall shear stresses together with their variations with the time, the pressure gradient, and the severity of the stenosis in order to illustrate the applicability of the present mathematical model under consideration.

Mathematical model of blood flow in a coronary capillary

The coronary capillary flow is analyzed theoretically based on continuum mechanics. The capillary is a long, elastic, and permeable vessel loaded externally by tissue pressure, and it is subject to possible periodic length changes, together with adjacent myocytes. Capillary flow is driven by arteriolar-venular pressure difference. Ultrafiltration due to transmural hydrostatic and osmotic gradients is included, and consideration of mass conservation leads to a nonlinear flow equation. The results show that under physiological conditions ultrafiltration is of minor importance, and the analysis predicts regional differences in capillary flow. In regions with high tissue pressure (subendocardium), capillaries undergo significant periodic volume changes, giving rise to intramyocardial pumping. In those regions, capillary wall elasticity is of major importance. In regions with low tissue pressure (subepicardium), the possible periodic capillary length changes are predominant. The predicted flow patterns are in good qualitative agreement with measured epicardial phasic flow. In conclusion, the methodological advantage of a distributive analysis is demonstrated by its ability to elucidate and evaluate the role of flow determinants and their complex interactions.

Theory for flow of Casson and Herschel-Bulkley fluids in cone-plate viscometers.

Mathematical models for blood flow in cone-plate viscometer have been considered, by assuming blood as a Casson/Herschel-Bulkley fluid. Three different cases have been analyzed (i) when there is no shearing, (ii) partial shearing and (iii) full shearing. The relationships between the angular velocity and torque have been obtained for the above three cases. By assuming total shearing, the analytical expression for apparent viscosity has been obtained. Variation of apparent viscosity with yield stress, angular velocity, Casson co-efficient of viscosity, consistency index and flow behaviour index has been computed. It is observed that as the angular velocity increases, the apparent viscosity decreases for both fluids. Further, it is found that as the cone angle increases, the apparent viscosity increases. This behaviour of apparent viscosity in cone-plate viscometer is interesting and unexpected and is being reported first time.

A mathematical model for the study of flow through disc-type prosthetic heart valves.

In this study, computational fluid dynamic methods are used to construct a simple mathematical model of blood flow through a disc-type prosthetic heart valve. The vorticity-transport equation and Poisson's equation are solved cyclically on a computer using a finite difference method. Stream function, vorticity, velocity and shear stresses are computed and plots are presented for Reynolds numbers from 50 to 6000 to help identify the flow characteristics of the valve which may contribute to problems such as thrombus formation, endothelial damage, tissue overgrowth, and erythrocyte damage. Numerical methods together with digital computers can thus be used to locate problem areas in different prosthetic valve designs.