Mathematical Model Of Blood Flow Research Articles

The Effect of Nanoparticles on Drug Distribution in The Mathematical Model of Blood Flow

This research examines the influence of nanoparticles in the distribution of drugs in healthy blood flow on linear, angular velocity and blood temperature. Construction and simplification of a blood flow model based on boundary layer equations, dimensionless variables, stream functions, and similarity variables. The initial step is to establish a dimensional blood flow model. Using dimensionless variables, the equation is simplified into a dimensionless equation. A similarity equation is generated by converting the non-dimensional equation. The nanoparticles used are . At the linear velocity and temperature of blood flow is in the highest position. At the angular velocity of blood flow, the position of blood flow with nanoparticles is in the uppermost position. This research is used to estimate the velocity and temperature of blood flow with the influence of nanoparticles as drug distribution.

Mathematical modeling of blood flow in an annulus porous region between two coaxial deformable tubes: An advancement to peristaltic endoscope

In the present work, the electromagnetohydrodynamic peristaltic flow of blood through an annulus region between two concentric circular tubes has been investigated for the advancement of a peristaltic endoscope. In this model, the viscous and incompressible blood is assumed as non-Newtonian Jeffery fluid, and tubes are inclined at an angle β with the horizontal. The governing equations for the considered problem are simplified under the assumption of a low Reynolds number and long wavelength approximations. An analytical expression for various flow variables such as axial velocity, radial velocity, time mean flow rate, volumetric flow rate, frictional force at the surface of the inner tube, and mechanical efficiency is obtained. In this work, we discussed the impact of various flow parameters like the Hartmann number M, electric field strength parameter H, Jeffery fluid parameter λ1, radius ratio n, occlusion ϕ=ba, angle of inclination β, and Darcy number Da on the above flow variables. The streamlines are also made to discuss the trapping phenomena of a fluid bolus inside the tubes. The comparison of pressure rise for the various endoscopes and the validation of the present result with the previously established results are discussed.

Variable separated physics-informed neural networks based on adaptive weighted loss functions for blood flow model

Physics-informed neural networks (PINN) architectures have been recently explored to accelerate hemodynamics simulations by leveraging mathematical models for blood flow and empirical data. In this paper, a variable separated physics-informed neural networks based on adaptive weighted loss functions (AW-vsPINN) is developed for blood flow model in arteries. In particular, sub-neural networks are proposed to separately predict the unknown scalar state variables by sharing the same input layer. The AW-vsPINN adaptively adjusts the weights of loss terms by the minmax algorithm, which will be updated synchronously along with the network parameters and can balance the contributions of different loss terms during training. The two-stage optimization is implemented to train the neural networks. Specifically, the Adam optimizer is iterated for initial steps with the learning rate generated by the inverse time decay scheduler, and then the L-BFGS optimizer continues to train until the loss converges. Numerical results illustrate that the AW-vsPINN can remarkably improve prediction accuracy and enhance the ability of generalization compared to the conventional PINN. The proposed AW-vsPINN framework has high potential in predicting the blood flow information in cardiovascular disease.

Numerical study of irreversibility analysis on pulsatile flow of blood through a ω-shaped stenotic artery

The mathematical modeling of blood flow in a stenotic blood vessel is very important for understanding the effects of various geometric and rheological parameters on the normal flow of blood. The present study is concerned with the heat and mass transfer in the blood flow of a non-Newtonian fluid through a ω-shaped stenotic arterial segment. The non-Newtonian behavior of blood is described mathematically by employing the constitutive equation of the Herschel-Bulkley fluid. The wall of the artery is considered to be rigid having ω-shaped constriction in its lumen. The mild stenosis assumption is used to simplify the fully coupled momentum, energy, and concentration equations as well as the constitutive equation of the Herschel-Bulkley fluid. Numerical solutions for fluid velocity, temperature, mass concentration, and entropy generation are evaluated by using the explicit finite difference scheme. The effects of various emerging parameters on the velocity, wall shear stress, flow rate, temperature, concentration, and entropy distributions have been analyzed in detail. A comparison between different fluid models has also been presented. It is found that the temperature in the Herschel-Bulkley fluid is marginally lower than that of the Newtonian, Power law, and Bingham fluids whereas an opposite trend is noticed in the case of mass concentration.

Mathematical Analysis of Hemodynamic Parameters of Blood Flow in an Artery

In blood rheology we study volume flow rate, blood pressure, velocity, viscosity and shear stress of blood. Cross-sectional area plays an important role for smooth flow of the blood. But some other parameter like composition of blood, length of vessel also affects in the flow rate and pressure of blood. Velocity and volume flow rate are derived by using Poiseuille’s equation.This work presents a mathematical model of blood flow that was created using the N-S equations and computer simulation. Graphs are used to analyze the results.

Optimal heat transport induced by magnetic nanoparticle delivery in vascularised tumours

We describe a novel mathematical model for blood flow, delivery of nanoparticles, and heat transport in vascularised tumour tissue. The model, which is derived via the asymptotic homogenisation technique, provides a link between the macroscale behaviour of the system and its underlying, tortuous micro-structure, as parametrised in Penta and Ambrosi (2015). It consists of a double Darcy’s law, coupled with a double advection–diffusion–reaction system describing heat transport, and an advection–diffusion–reaction equation for transport and adhesion of particles. Particles are assumed sufficiently large and do not extravasate to the tumour interstitial space but blood and heat can be exchanged between the two compartments. Numerical simulations of the model are performed using a finite element method to investigate cancer hyperthermia induced by the application of magnetic field applied to injected iron oxide nanoparticles. Since tumour microvasculature is more tortuous than that of healthy tissue and thus suboptimal in terms of fluid and drug transport, we study the influence of the vessels’ geometry on tumour temperature. Effective and safe hyperthermia treatment requires tumour temperature within certain target range, generally estimated between 42 °C and 46 °C, for a certain target duration, typically 0.5h to 2h. As temperature is difficult to measure in situ, we use our model to determine the ranges of tortuosity of the microvessels, magnetic intensity, injection time, wall shear stress rate, and concentration of nanoparticles required to achieve given target conditions.

Effects of electro-osmotic and double diffusion on nano-blood flow through stenosis and aneurysm of the subclavian artery: numerical simulation

Most computational models assumed that blood flow through the stenosis flow in a straight tube. Actually, arteries or blood vessels are rarely straight conduits and stenosis on bends or curves is not uncommon but occurs in the aorta, subclavian artery, and other curved arteries. Therefore, this paper's mathematical modeling of blood flow was done through curvilinear coordinates. The motivation of this paper is to study the effect of double-diffusion convection through curved artery having stenosis and aneurysm regions, the blood flow is modeled as nano-Bingham Papanastasiou fluid under the influence of the electro-osmotic field. Governing equations of the flow are derived using the mild stenosis approximation and solved numerically using the finite difference method. Velocity, temperature, concentration, nanoparticle volume fraction, flow rate, impedance, and shear stress at the throat of aneurysm and stenosis through curved artery are computed numerically and the results are represented graphically. The flow through straight arteries is included as a special case of curved arteries at . Finally, the flow rate in the straight arteries is more than in curved, but shear stress and impedance are greater in curved than the straight arteries. The blood flow velocity is improved in the presence of the electro-osmotic field.

Modeling of the cerebral blood circulation in a capillary network accounting for the influence of the endothelial surface layer

Background and objectiveThe paper describes a mathematical model of blood flow in capillaries with accounting for the endothelial surface layer (ESL). MethodThe influence of ESL is modeled by a boundary layer with zero flow velocity. Finite element modeling and an analytical approach based on the homogenization of the core region of blood flow occupied by erythrocytes are developed to describe the resistance of a capillary. The reliability of the results obtained is verified for different values of the discharge hematocrit and vessel diameter using known in vivo data. ResultsThe proposed approach is applied to the numerical simulation of blood circulation in a capillary network of the germinal matrix of infants born at 25 gestational weeks. The influence of the hematocrit level and effective thickness of ESL on the resistance of the capillary network of the germinal matrix of preterm infants is studied. It was found that a decrease in the effective thickness of ESL in the capillary network (and/or a decrease in the hematocrit) leads to reducing the resistance of the capillary network. ConclusionA decrease in the effective thickness of ESL in the capillary network leads to an increase in the pressure drop in arterioles, which may be considered as an additional risk factor for hemorrhages in fragile blood vessels within the germinal matrix.

A mathematical model of blood flow in a stenosed artery with post-stenotic dilatation and a forced field.

Arterial stenosis is a common cardiovascular disease that restricts blood flow. A stenotic blood vessel creates tangent stress pressure, which lessens the arterial side and causes an aneurysm. The primary purpose of this study is to investigate blood flowing via an inclination pipe with stricture and expansion after stricture (widening) underneath the influence of a constant incompressible Casson liquid flowing with the magnetism field. The relations for surface shearing stress, pressure drop, flow resistance, and velocity are calculated analytically by applying a mild stenosis approximation. The effect of different physical characteristics on liquid impedance to flowing, velocity, and surface shearing stress are studied. With a non-Newtonian aspect of the Casson liquid, the surface shearing stress declines, and an impedance upturn. Side resistivity and shear-stress increase with the elevations of stricture, whilst together decreasing with a dilatation height.

Mathematical Model of Blood Flow through Capillaries to Study Transport of Nanoparticles Using Power Law Fluid Model

A model of blood capillary flow that is axially symmetric with a peripheral layer and wall slip is provided mathematically in this study. Power law fluid was employed in the core area of suspension of all erythrocytes whereas Newtonian fluid was used in the periphery plasma layer to analyse longitudinal transport of nanoparticles within blood vessels. The capillary walls are impenetrable to nanoparticles in our study, and they are not absorbent. The formulas for the velocity profile, flow rate, mean velocity, and solute concentration were produced, and the findings were discussed using graphs.

Mathematical Modeling of Blood Flow Through a Stenosed Bifurcated Artery With Heat Source and Magnetic Effect

The present article illustrates the influence of heat source on MHD blood flow in a branched artery with small obstacle in its main artery by treating blood as micropolar fluid. Artery forming branch is presumed as straight cylinder of restricted length and is symmetric about its axis. The irregular shape of constricted branched artery is changed to a well defined shape by utilizing a adial coordinate transformations. Numerical solution is computed for interested physical quantities such as rate of flow, and shearing stress on one as well as the other sides of the apex.

Mathematical Modeling of Blood Flow to Evaluate the Hemodynamic Significance of Peripheral Vascular Lesions

Background: Evaluating the severity of peripheral artery lesions is challenging. Image-based blood flow modeling from peripheral Computed Tomographic Angiography (pCTA) may provide a non-invasive method to determine the hemodynamic significance of lesions. This pilot study evaluates the performance of pCTA-based blood flow modeling in diagnosing functionally significant peripheral lesions in comparison with Digital Subtraction Angiography (DSA). Methods: Ten patients undergoing DSA and pCTA were included. The peripheral arteries were divided into 8 segments per extremity and stenosis severity was graded by visual estimation from DSA. Each segment was graded 0 to IV (normal, mildly-stenotic, moderately-stenotic, severely-stenotic, occluded) or non-evaluable. Independent from DSA review, a Resting Pressure Drop (RPD) and an Exercise Pressure Drop (ExPD) for each segment was calculated from pCTA-based blood flow modeling. A functionally significant (FS) lesion was defined as grade III or IV by DSA and RPD > 5 mmHg from pCTA-based modeling. Analysis was repeated with an ExPD > 20 mmHg. Sensitivity, specificity and accuracy were calculated for RPD > 5 mmHg and ExPD > 20 mmHg using DSA as the standard. Results: Mean age was 52±16 years, 4 patients were male, 8 patients presented with critical limb ischemia, mean ankle brachial index was 0.60±0.29, and 66 arterial segments were available for both assessment methods. Twenty-two segments had FS lesions by DSA. Using an RPD > 5 mmHg, sensitivity was 80%, specificity was 85% and accuracy was 79%. Using an ExPD > 20 mmHg, sensitivity was 84%, specificity was 89% and accuracy was 88%. Conclusion: Use of a resting pressure drop > 5 mmHg and an exercise pressure drop > 20 mmHg, measured by blood flow modeling from CT angiography, can accurately identify functionally significant stenosis in patients with peripheral vascular disease. This information motivates the need for a larger-scale prospective imaging trial to further validate this novel non-invasive approach.

A 2D FSI mathematical model of blood flow to analyze the hyper-viscous effects in atherosclerotic COVID patients

Recent researches on COVID 19 has been extended to analyze the various morphological and anatomical changes in a patient's body due to the invasion of the virus. These latest studies have concluded that there happens a high rise in the viscosity of the blood in a COVID 19 patient, supported by the extensive analysis of the clinical data. In the present paper, a mathematical model in the form of a differential equation system has been proposed to disclose the various changes that occur in the flow across the stenosis of an arterial segment. The consequences of the hyperviscosity of blood on the blood flow characteristics in a stenosed artery are analyzed by solving the model using a finite element method (FEM) solver. A laminar flow coupled with solid mechanics through the Fluid-Structure Interaction (FSI) interface has been studied using an Arbitrary Lagrangian-Eulerian (ALE) method. For the first time, the mathematical model was used to analyze the hyper-viscous flow condition in COVID 19 patients. The present research is mainly based on the numerous clinical reports enlisting the various morphological, hematological, and rheological changes in the blood.

Prediction of Resthenosis After Carotid Endarterectomy by the Method of Computer Simulation

The article describes a computer modeling technique that allows predicting the development of restenosis of the internal carotid artery after carotid endarterectomy (CEE). A clinical case has been demonstrated that proves the effectiveness of the developed method. It is indicated that for the correct formation of the geometric model, data from multispiral computed tomography with angiography of the patient after CEE with a layer thickness of 0.6 mm and a current of 355 mA are required. To build a flow model, data of color duplex scanning in three sections are required: 1. In the proximal section of the common carotid artery (3 cm proximal to the bifurcation); 2. In the section of the external carotid artery, 2 cm distal to the carotid sinus; 3. In the section of the internal carotid artery, 2 cm distal to the carotid sinus. The result of computer calculations using specialized software (Sim Vascular, Python, Open Foam) is a mathematical model of blood flow in a vessel. It is an array of calculated data describing the speed and other characteristics of the flow at each point of the artery. Based on the analysis of RRT and TAWSS indicators, a computer model of bifurcation is formed, which makes it possible to predict areas of increased risk of restenosis development. Thus, the developed technique is able to identify a cohort of patients after CEE, subjected to a high probability of loss of the vessel lumen. Such an opportunity will provide a more precise supervision of these patients in the postoperative period with the aim of early diagnosis of restenosis and timely prevention of the development of adverse cardiovascular events.

A review of mathematical modelling of blood flow in human circulatory system

An interface between biology and mathematics has initiated and fostered new mathematical areas, where the ideas from biology and mathematics are synergistically applied. Study of fluid dynamics plays a significant role in fluid flow inside the human body, and modeling of blood flow is an important field in circulatory physics. However, models have been developed are very complex with three-dimensional analysis. This project presents a novel and simple mathematical model of blood flow. Assuming blood is a Newtonian fluid which is governed by the Navier-Stokes equations and continuity equation and with making use of the Navier-Stokes equation, a simple differential equation called as the circulatory system equation is derived. Then by applying the ogical assumptions on this model, the general mathematical model of the normal blood flow rate is developed. Using Poisuelli’s equation, the sirculatory system equation is also used to develop a model for blood pressure. These two models are then analyzed against surface, pressure gradient and the vessel’s length using MATLAB.

A Mathematical Model of Blood Flow in Merging Veins under a Magnetic Resonance Imaging Influence

Venous blood flow through the superior and inferior vena cavae and via a merger to the right atrium of the heart is investigated. Based on the Newtonian assumption for blood, the model is developed using Boussinesq’s approximation for the continuity, momentum, energy, and mass diffusion equations, which are non-linear partial differential equations. The governing equations are non-dimensionalized and solved by the regular perturbation technique. Expressions for the concentration, temperature, and velocity are obtained, analyzed, and presented graphically and quantitatively, and discussed. The results show that an increase in the magnetic field strength and conflux angle produce fluctuations in the flow velocity.

Study of Magnetohydrodynamic Pulsatile Blood Flow through an Inclined Porous Cylindrical Tube with Generalized Time-Nonlocal Shear Stress

The effects of pulsatile pressure gradient in the presence of a transverse magnetic field on unsteady blood flow through an inclined tapered cylindrical tube of porous medium are discussed in this article. The fractional calculus technique is used to provide a mathematical model of blood flow with fractional derivatives. The solution of the governing equations is found using integral transformations (Laplace and finite Hankel transforms). For the semianalytical solution, the inverse Laplace transform is found by means of Stehfest’s and Tzou’s algorithms. The numerical calculations were performed by using Mathcad software. The flow is significantly affected by Hartmann number, inclination angle, fractional parameter, permeability parameter, and pulsatile pressure gradient frequency, according to the findings. It is deduced that there exists a significant difference in the velocity of the flow at higher time when the magnitude of Reynolds number is small and large.

Mathematical Modeling of Blood Flow With the Suspension of Nanoparticles Through a Tapered Artery With a Blood Clot

In this study, some key new developments in nanoscience which highlight the problem of nanoparticles in blood flow through mild stenosis in the presence of a blood clot have been presented. The blood flow behavior through the stenosed artery is considered using the Prandlt fluid model and the flow of blood is considered as suspension of nanoparticles. An appropriate non-linear system of equations governing blood flow is represented in a cylindrical coordinate system and solved exactly under mild stenotic conditions using the geometry of the stenotic artery in the presence of a clot. Heat transfer phenomena have been examined for the physical features of the flow of blood through a stenosed artery, which is tapered in shape and with the presence of a clot. The temperature profile has been discussed with graphs for several different parameters of clot size, stenosis height, heat source, and sink parameter. Tapering phenomena has been analyzed for temperature profile. It is examined that in converging tapering the temperature provides greater values as estimated together with the non-tapered arteries and diverging tapering arteries. In this work, it is also analyzed that with a rise in the clot size (σ) the temperature (θ) increases, whereas the radius of the artery with stenosis h(z) decreases and heat source and sink parameter (D) increases.

Time-Dependent Flow of a Couple Stress Fluid in an Elastic Circular Cylinder with Application to the Human Circulatory System

In this paper, we developed a mathematical model for blood flow in the human circulatory system. This model presumes blood to be a couple stress fluid, its flow to be pulsatile, and the artery an elastic circular pipe whose radius is assumed to vary with transmural pressure. The governing differential equation for the flow velocity is time-dependent and has been solved using the homotopy perturbation method. This velocity has been used to estimate the elastic modulus E of the artery, which is a measure of its stiffness and an important metric used by clinical practitioners to understand the state of the cardiovascular system. In this work, the radial artery has been considered and a limited set of experimental data, available for four cases, has been taken from the published literature to validate the model. While the experimental values of elastic modulus reported in literature lie in the range 2.68 1.81 MPa.s, those estimated through the proposed model range from 3.05 to 5.98 MPa.s, appearing to be in close agreement.

Fluid Flow and Solute Transfer in a Permeable Tube with Influence of Slip Velocity

In this paper, the influence of slip velocity on the fluid flow and solute transfer in a tube with permeable boundary is studied as a mathematical model for blood flow in glomerular capillaries. The viscous incompressible fluid flow across the permeable tube wall, as a result of differences in both hydrostatic and osmotic pressure, is considered. The solutions of differential equations governing the fluid flow and solute transfer are obtained using analytical and Crank-Nicolson type numerical methods. It is observed that the effect of slip on the hydrostatic and osmotic pressures, velocity profiles, concentration profile, solute mass flux and total solute clearance is significant and the results are presented graphically.