1
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Wajihah SA, Sankar DS. A review on non-Newtonian fluid models for multi-layered blood rheology in constricted arteries. ARCHIVE OF APPLIED MECHANICS = INGENIEUR-ARCHIV 2023; 93:1771-1796. [PMID: 36743075 PMCID: PMC9886544 DOI: 10.1007/s00419-023-02368-6] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 07/05/2022] [Accepted: 01/13/2023] [Indexed: 06/18/2023]
Abstract
Haemodynamics is a branch of fluid mechanics which investigates the features of blood when it flows not only via blood vessels of smaller/larger diameter, but also under normal as well as abnormal flow states, such as in the presence of stenosis, aneurysm, and thrombosis. This review aims to discuss the rheological properties of blood, geometry of constrictions, dilations and the emergence of single-layered fluid to four-layered fluid models. To discuss further the influence of the aforesaid parameters on the physiologically important flow quantities, the mathematical formulation and solution methodology of the two-layered and four layered arterial blood flow problems studied by the authors (Afiqah and Sankar in ARPN J Eng Appl Sci 15:1129--1143, 2020, Comput Methods Programs Biomed 199:105907, 2021. 10.1016/j.cmpb.2020.105907) are recalled. It should be pointed out that the increasing resistive impedance to flow in three distinct states encompassing healthy, anaemic, and diabetic demonstrates that the greater the restriction in the artery, very few blood is carried to the pathetic organs, leading to subjects' death. It is also discovered that the pulsatile nature of blood movement produces a dynamic environment that poses a slew of intriguing and unstable fluid mechanical state. It is hoped that the intriguing results gathered from this literature survey and review conducted may help the medical practitioners to forecast blood behaviour mobility in stenotic arteries. Furthermore, the physiological information gathered from the available clinical data from the literature on patients diagnosed with diabetes and anaemia may be beneficial to doctors in deciding the therapeutic procedure for treating some particular cardiovascular disease.
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Affiliation(s)
- S. Afiqah Wajihah
- Applied Mathematics and Economics Programme Area, School of Applied Sciences and Mathematics, Universiti Teknologi Brunei, Jalan Tungku Link, Gadong, Bandar Seri Begawan, BE1410 Brunei Darussalam
| | - D. S. Sankar
- Applied Mathematics and Economics Programme Area, School of Applied Sciences and Mathematics, Universiti Teknologi Brunei, Jalan Tungku Link, Gadong, Bandar Seri Begawan, BE1410 Brunei Darussalam
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2
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Nartsissov YR. Application of a multicomponent model of convectional reaction-diffusion to description of glucose gradients in a neurovascular unit. Front Physiol 2022; 13:843473. [PMID: 36072843 PMCID: PMC9444140 DOI: 10.3389/fphys.2022.843473] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/26/2021] [Accepted: 07/18/2022] [Indexed: 11/16/2022] Open
Abstract
A supply of glucose to a nervous tissue is fulfilled by a cerebrovascular network, and further diffusion is known to occur at both an arteriolar and a microvascular level. Despite a direct relation, a blood flow dynamic and reaction-diffusion of metabolites are usually considered separately in the mathematical models. In the present study they are coupled in a multiphysical approach which allows to evaluate the effects of capillary blood flow changes on near-vessels nutrient concentration gradients evidently. Cerebral blood flow (CBF) was described by the non-steady-state Navier-Stokes equations for a non-Newtonian fluid whose constitutive law is given by the Carreau model. A three-level organization of blood–brain barrier (BBB) is modelled by the flux dysconnectivity functions including densities and kinetic properties of glucose transporters. The velocity of a fluid flow in brain extracellular space (ECS) was estimated using Darcy’s law. The equations of reaction-diffusion with convection based on a generated flow field for continues and porous media were used to describe spatial-time gradients of glucose in the capillary lumen and brain parenchyma of a neurovascular unit (NVU), respectively. Changes in CBF were directly simulated using smoothing step-like functions altering the difference of intracapillary pressure in time. The changes of CBF cover both the decrease (on 70%) and the increase (on 50%) in a capillary flow velocity. Analyzing the dynamics of glucose gradients, it was shown that a rapid decrease of a capillary blood flow yields an enhanced level of glucose in a near-capillary nervous tissue if the contacts between astrocytes end-feet are not tight. Under the increased CBF velocities the amplitude of glucose concentration gradients is always enhanced. The introduced approach can be used for estimation of blood flow changes influence not only on glucose but also on other nutrients concentration gradients and for the modelling of distributions of their concentrations near blood vessels in other tissues as well.
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3
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Dhange M, Sankad G, Safdar R, Jamshed W, Eid MR, Bhujakkanavar U, Gouadria S, Chouikh R. A mathematical model of blood flow in a stenosed artery with post-stenotic dilatation and a forced field. PLoS One 2022; 17:e0266727. [PMID: 35776713 PMCID: PMC9249183 DOI: 10.1371/journal.pone.0266727] [Citation(s) in RCA: 7] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/09/2021] [Accepted: 03/28/2022] [Indexed: 11/28/2022] Open
Abstract
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.
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Affiliation(s)
- Mallinath Dhange
- Department of Mathematics, BLDEA’s VP Dr. PG Halakatti College of Engineering and Technology, Vijayapur, India
| | - Gurunath Sankad
- Department of Mathematics, BLDEA’s VP Dr. PG Halakatti College of Engineering and Technology, Vijayapur, India
| | - Rabia Safdar
- Department of Mathematics, Lahore College for Women University, Lahore, Pakistan
| | - Wasim Jamshed
- Department of Mathematics, Capital University of Science and Technology (CUST), Islamabad, Pakistan
| | - Mohamed R. Eid
- Department of Mathematics, Faculty of Science, New Valley University, Al-Kharga, Al-Wadi Al-Gadid, Egypt
- Department of Mathematics, Faculty of Science, Northern Border University, Arar, Saudi Arabia
- * E-mail:
| | - Umesh Bhujakkanavar
- Department of Science and Humanities, Rajarambapu Institute of Technology, Islampur, Maharashtra, India
| | - Soumaya Gouadria
- Department of physics, College of Science, Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia
| | - R. Chouikh
- Research Center for Advanced Materials Science (RCAMS), King Khalid University, Abha, Saudi Arabia
- Laboratory of Thermal Processes, Center for Energy Research and Technology, Borj-Cedria, Tunisia
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4
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Ali A, Bukhari Z, Umar M, Ismail MA, Abbas Z. Cu and Cu-SWCNT Nanoparticles' Suspension in Pulsatile Casson Fluid Flow via Darcy-Forchheimer Porous Channel with Compliant Walls: A Prospective Model for Blood Flow in Stenosed Arteries. Int J Mol Sci 2021; 22:ijms22126494. [PMID: 34204328 PMCID: PMC8234443 DOI: 10.3390/ijms22126494] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/24/2021] [Revised: 06/03/2021] [Accepted: 06/05/2021] [Indexed: 11/18/2022] Open
Abstract
The use of experimental relations to approximate the efficient thermophysical properties of a nanofluid (NF) with Cu nanoparticles (NPs) and hybrid nanofluid (HNF) with Cu-SWCNT NPs and subsequently model the two-dimensional pulsatile Casson fluid flow under the impact of the magnetic field and thermal radiation is a novelty of the current study. Heat and mass transfer analysis of the pulsatile flow of non-Newtonian Casson HNF via a Darcy–Forchheimer porous channel with compliant walls is presented. Such a problem offers a prospective model to study the blood flow via stenosed arteries. A finite-difference flow solver is used to numerically solve the system obtained using the vorticity stream function formulation on the time-dependent governing equations. The behavior of Cu-based NF and Cu-SWCNT-based HNF on the wall shear stress (WSS), velocity, temperature, and concentration profiles are analyzed graphically. The influence of the Casson parameter, radiation parameter, Hartmann number, Darcy number, Soret number, Reynolds number, Strouhal number, and Peclet number on the flow profiles are analyzed. Furthermore, the influence of the flow parameters on the non-dimensional numbers such as the skin friction coefficient, Nusselt number, and Sherwood number is also discussed. These quantities escalate as the Reynolds number is enhanced and reduce by escalating the porosity parameter. The Peclet number shows a high impact on the microorganism’s density in a blood NF. The HNF has been shown to have superior thermal properties to the traditional one. These results could help in devising hydraulic treatments for blood flow in highly stenosed arteries, biomechanical system design, and industrial plants in which flow pulsation is essential.
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Affiliation(s)
- Amjad Ali
- Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan 60800, Pakistan; (Z.B.); (M.U.)
- Correspondence:
| | - Zainab Bukhari
- Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan 60800, Pakistan; (Z.B.); (M.U.)
| | - Muhammad Umar
- Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan 60800, Pakistan; (Z.B.); (M.U.)
| | - Muhammad Ali Ismail
- Department of Computer and Information System Engineering, NED University of Engineering and Technology, Karachi 75270, Pakistan;
| | - Zaheer Abbas
- Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan;
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5
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Effect of Varying Viscosity on Two-Layer Model of Pulsatile Flow through Blood Vessels with Porous Region near Walls. Transp Porous Media 2019. [DOI: 10.1007/s11242-019-01302-1] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
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6
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Gayathri K, Shailendhra K. MRI and Blood Flow in Human Arteries: Are There Any Adverse Effects? Cardiovasc Eng Technol 2019; 10:242-256. [DOI: 10.1007/s13239-019-00400-x] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 12/28/2016] [Accepted: 01/07/2019] [Indexed: 10/27/2022]
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7
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Effect of Varying Viscosity on Two-Fluid Model of Blood Flow through Constricted Blood Vessels: A Comparative Study. Cardiovasc Eng Technol 2018; 10:155-172. [DOI: 10.1007/s13239-018-00379-x] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 04/04/2018] [Accepted: 09/18/2018] [Indexed: 10/28/2022]
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8
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Mukhopadhyay S, Mandal MS, Mukhopadhyay S. Numerical simulation of physiologically relevant pulsatile flow of blood with shear-rate-dependent viscosity in a stenosed blood vessel. INT J BIOMATH 2018. [DOI: 10.1142/s1793524518500821] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
Pulsatile flow of blood in a blood vessel having time-dependent shape (diameter) is investigated numerically in order to understand some important physiological phenomena in arteries. A smooth axi-symmetric cosine shaped constriction is considered. To mimic the realistic situation as far as possible, viscosity of blood is taken to be non-uniform, a shear-thinning viscosity model is considered and a physiologically relevant pulsatile flow is introduced. Taking advantage of axi-symmetry in the proposed problem, the stream function–vorticity formulation is used to solve the governing equations for blood flow. Effect of different parameters associated with the problem on the flow pattern has been investigated and disparities from the Newtonian case are discussed in detail.
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Affiliation(s)
- Subrata Mukhopadhyay
- Department of Mathematics, Acharya P. C. Roy Govt. College, Siliguri, W.B., India
| | - Mani Shankar Mandal
- Department of Mathematics, G.G.D.C at Kalna-I, Purba Bardhaman-713405, W.B., India
| | - Swati Mukhopadhyay
- Department of Mathematics, The University of Burdwan, Burdwan-713104, W.B., India
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9
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Neeraja G, Dinesh P, Vidya K, Raju C. Peripheral layer viscosity on the stenotic blood vessels for Herschel-Bulkley fluid model. INFORMATICS IN MEDICINE UNLOCKED 2017. [DOI: 10.1016/j.imu.2017.08.004] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022] Open
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10
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Abdollahzadeh Jamalabadi MY, Akbari Bidokhti AA, Khak Rah H, Vaezi S, Hooshmand P. Numerical Investigation of Oxygenated and Deoxygenated Blood Flow through a Tapered Stenosed Arteries in Magnetic Field. PLoS One 2016; 11:e0167393. [PMID: 27941986 PMCID: PMC5152821 DOI: 10.1371/journal.pone.0167393] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/07/2016] [Accepted: 11/14/2016] [Indexed: 11/19/2022] Open
Abstract
Current paper is focused on transient modeling of blood flow through a tapered stenosed arteries surrounded a by solenoid under the presence of heat transfer. The oxygenated and deoxygenated blood are considered here by the Newtonian and Non-Newtonian fluid (power law and Carreau-Yasuda) models. The governing equations of bio magnetic fluid flow for an incompressible, laminar, homogeneous, non-Newtonian are solved by finite volume method with SIMPLE algorithm for structured grid. Both magnetization and electric current source terms are well thought-out in momentum and energy equations. The effects of fluid viscosity model, Hartmann number, and magnetic number on wall shear stress, shearing stress at the stenosis throat and maximum temperature of the system are investigated and are optimized. The current study results are in agreement with some of the existing findings in the literature and are useful in thermal and mechanical design of spatially varying magnets to control the drug delivery and biomagnetic fluid flows through tapered arteries.
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Affiliation(s)
| | | | - Hamid Khak Rah
- Department of Mechanical Engineering, Shiraz Branch, Islamic Azad University, Shiraz, Iran
| | - Siavash Vaezi
- Department of Mechanical Engineering, Sharif University of Technology, Azadi St, Tehran, Iran
| | - Payam Hooshmand
- Department of Mechanical Engineering, Sanandaj Branch, Islamic Azad University, Sanandaj, Iran
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11
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Saleem N, Munawar S. A mathematical analysis of MHD blood flow of Eyring–Powell fluid through a constricted artery. INT J BIOMATH 2016. [DOI: 10.1142/s1793524516500273] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
The present study deals with the flow of blood through a stenotic artery in the presence of a uniform magnetic field. Different flow situations are taken into account by considering the regular and irregular shapes of stenosis lying inside the walls of artery. Blood inside the artery is assumed to be Eyring–Powell fluid. A mathematical model is developed and simplified under the physical assumptions of stenosis. The regular perturbation method is adopted to find the solutions for axial velocity and pressure gradient. The variations in pressure drop across the stenosis length, the impedance and the shear stress at the walls of stenotic artery are discussed in detail through graphs. It is observed that the Eyring–Powell fluid is helpful in reducing the resistance to the flow in stenotic artery. Moreover, symmetric form of stenosis is more hazardous as compared to asymmetric stenosis.
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Affiliation(s)
- Najma Saleem
- Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, Al Khobar 31952, Saudi Arabia
| | - Sufian Munawar
- Department of Quantitative Methods, College Business Administration, University of Dammam, Dammam 31441, Saudi Arabia
- Department of Informatics and Systems, School of Science and Technology, University of Management and Technology, Lahore 54000, Pakistan
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12
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Akbar NS. Metallic nanoparticles analysis for the blood flow in tapered stenosed arteries: Application in nanomedicines. INT J BIOMATH 2015. [DOI: 10.1142/s1793524516500029] [Citation(s) in RCA: 21] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
Blood flow model is recycled to study the influence of magnetic field and nanoparticles in tapered stenosed arteries. The metallic nanoparticles for the blood flow with water as base fluid are not explored so far. The representation for the blood flow is through an axially non-symmetrical but radially symmetric stenosis. Symmetry of the distribution of the wall shearing stress and resistive impedance and their growth with the developing stenosis is another important feature of our analysis. Exact solutions have been evaluated for velocity, resistance impedance, wall shear stress and shearing stress at the stenosis throat. The graphical results of different types of tapered arteries (i.e. converging tapering, diverging tapering, non-tapered artery) have been examined for different parameters of interest for pure water and Copper water ( Cu -water).
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Affiliation(s)
- Noreen Sher Akbar
- DBS&H, CEME, National University of Sciences and Technology, Islamabad, Pakistan
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13
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Vajravelu K, Awgichew G, Radhakrishnamacharya G. The effects of slip condition and multiple stenoses on micropolar fluid flow through a channel of non-uniform cross-section. INT J BIOMATH 2015. [DOI: 10.1142/s1793524515500552] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this paper, steady incompressible micropolar fluid flow through a non-uniform channel with multiple stenoses is considered. Assuming the stenoses to be mild and using the slip boundary condition, the equations governing the flow of the proposed model are solved, and closed-form expressions for the flow characteristics (resistance to flow and wall shear stress) are derived. The effects of different parameters on these flow characteristics are analyzed. It is observed that both the resistance to the flow and the wall shear stress increase with the heights of the stenoses and the slip parameter; but decrease with the Darcy number. Furthermore, the effects of the wall exponent parameter, the cross-viscosity coefficient and the micropolar parameter on the flow characteristics are discussed.
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Affiliation(s)
- K. Vajravelu
- Department of Mathematics, Department of Mechanical, Materials and Aerospace Engineering, University of Central Florida, Orlando FL 32816, USA
| | - Gurju Awgichew
- Department of Mathematics, National Institute of Technology, Warangal 506 004, Andhra Pradesh, India
| | - G. Radhakrishnamacharya
- Department of Mathematics, National Institute of Technology, Warangal 506 004, Andhra Pradesh, India
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14
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Prakash O, Makinde OD, Singh SP, Jain N, Kumar D. Effects of stenoses on non-Newtonian flow of blood in blood vessels. INT J BIOMATH 2015. [DOI: 10.1142/s1793524515500102] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this paper, a mathematical model for steady blood flow through blood vessels with uniform cross-section in stenoses arteries has been proposed. Blood is assumed to be non-Newtonian, incompressible and homogeneous fluid. Blood in human artery is represented as Bingham plastic fluid. Expressions for flow rate, wall shear stress, and resistance to flow against stenoses size have been obtained. Obtained results indicate that stenoses size decreases the flow rate and increases the wall shear stress as well as resistance to flow.
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Affiliation(s)
- Om Prakash
- School of Basic and Applied Sciences, Galgotias University Greater Noida, Uttar Pradesh, India
| | - O. D. Makinde
- Faculty of Military Science, Stellenbosch University, Private Bag X2, Saldanha 7395, South Africa
| | - S. P. Singh
- Department of Mathematics, DEI, Dayalbagh, Agra, Uttar Pradesh, India
| | - Nidhi Jain
- Department of Mathematics, DEI, Dayalbagh, Agra, Uttar Pradesh, India
| | - Devendra Kumar
- Department of Mathematics, Sachdeva Institute of Technology, Farah Mathura, Uttar Pradesh, India
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15
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SRINIVASACHARYA D, SRIKANTH D. FLOW OF MICROPOLAR FLUID THROUGH CATHETERIZED ARTERY — A MATHEMATICAL MODEL. INT J BIOMATH 2012. [DOI: 10.1142/s1793524511001611] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this paper, the flow of blood through catheterized artery with mild constriction at the outer wall is considered. The closed form solutions are obtained for velocity and microrotation components. The impedance (resistance to the flow) and wall shear stress are calculated. The effects of catheterization, coupling number, micropolar parameter, and height of the stenosis on impedance and wall shear stresses are discussed.
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Affiliation(s)
- D. SRINIVASACHARYA
- Department of Mathematics, National Institute of Technology, Warangal-506004, Andhra Pradesh, India
| | - D. SRIKANTH
- Department of Applied Mathematics, Defence Institute of Advanced Technology, Pune-411 025, India
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16
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MISRA JC, SHIT GC. ROLE OF SLIP VELOCITY IN BLOOD FLOW THROUGH STENOSED ARTERIES: A NON-NEWTONIAN MODEL. J MECH MED BIOL 2011. [DOI: 10.1142/s0219519407002303] [Citation(s) in RCA: 50] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
A mathematical model is developed in this paper for studying blood flow through a stenosed arterial segment by taking into account the slip velocity at the wall of the artery. Consideration of the non-Newtonian character of blood is made, where a constitutive relation of blood is described by the Herschel–Bulkley equation. The effect of slip at the arterial wall in the presence of mild, moderate, and severe stenosis growth at the lumen of an artery is investigated. Analytical expressions for skin friction, flow resistance, and the flow rate are derived by using the model. The derived expressions are computed numerically by considering an illustrative example. The study provides an insight into the effects of slip velocity on the volumetric flow rate of blood, flow resistance, and skin friction.
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Affiliation(s)
- J. C. MISRA
- Center for Theoretical Studies, Indian Institute of Technology, Kharagpur-721302, India
| | - G. C. SHIT
- Center for Theoretical Studies, Indian Institute of Technology, Kharagpur-721302, India
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17
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Liu M, Yang J. Electrokinetic effect of the endothelial glycocalyx layer on two-phase blood flow in small blood vessels. Microvasc Res 2009; 78:14-9. [DOI: 10.1016/j.mvr.2009.04.002] [Citation(s) in RCA: 36] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/11/2009] [Revised: 04/06/2009] [Accepted: 04/06/2009] [Indexed: 10/20/2022]
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18
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Wong K, Mazumdar J, Pincombe B, Worthley SG, Sanders P, Abbott D. Theoretical modeling of micro-scale biological phenomena in human coronary arteries. Med Biol Eng Comput 2006; 44:971-82. [PMID: 17048027 DOI: 10.1007/s11517-006-0113-6] [Citation(s) in RCA: 29] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/28/2005] [Accepted: 09/14/2006] [Indexed: 12/18/2022]
Abstract
This paper presents a mathematical model of biological structures in relation to coronary arteries with atherosclerosis. A set of equations has been derived to compute blood flow through these transport vessels with variable axial and radial geometries. Three-dimensional reconstructions of diseased arteries from cadavers have shown that atherosclerotic lesions spiral through the artery. The theoretical framework is able to explain the phenomenon of lesion distribution in a helical pattern by examining the structural parameters that affect the flow resistance and wall shear stress. The study is useful for connecting the relationship between the arterial wall geometries and hemodynamics of blood. It provides a simple, elegant and non-invasive method to predict flow properties for geometrically complex pathology at micro-scale levels and with low computational cost.
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Affiliation(s)
- Kelvin Wong
- Centre for Biomedical Engineering and School of Electrical & Electronic Engineering, The University of Adelaide, Adelaide, SA, Australia.
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19
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Pralhad RN, Schultz DH. Modeling of arterial stenosis and its applications to blood diseases. Math Biosci 2004; 190:203-20. [PMID: 15234617 DOI: 10.1016/j.mbs.2004.01.009] [Citation(s) in RCA: 51] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/30/2001] [Revised: 11/11/2002] [Accepted: 01/09/2004] [Indexed: 11/24/2022]
Abstract
Blood flow in a stenosed tube has been modeled in the present studies. Blood flow is assumed to be represented by a couple stress fluid. Flow parameters such as velocity, resistance to flow, and shear stress distribution have been computed for different suspension concentrations (haematocrit), and for the blood diseases; polycythemia, plasma cell dyscrasias, and for Hb SS (sickle cell). The results have been compared with the case of normal blood and for other theoretical models. The importance of size effects in blood flow studies has been highlighted.
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Affiliation(s)
- R N Pralhad
- Faculty of Applied Mathematics, Institute of Armament Technology, Girinagar, Pune 411 025, Maharashtra, India.
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20
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Pincombe B, Mazumdar J, Hamilton-Craig I. Effects of multiple stenoses and post-stenotic dilatation on non-Newtonian blood flow in small arteries. Med Biol Eng Comput 1999; 37:595-9. [PMID: 10723896 DOI: 10.1007/bf02513353] [Citation(s) in RCA: 26] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
Abstract
Fully-developed one-dimensional Casson flow through a single vessel of varying radius is proposed as a model of low Reynolds number blood flow in small stenosed coronary arteries. A formula for the resistance-to-flow ratio is derived, and results for yield stresses of tau 0 = 0, 0.005 and 0.01 Nm-2, viscosities of mu = 3.45 x 10(-3), 4.00 x 10(-3) and 4.55 x 10(-3) Pa.s and fluxes of 2.73 x 10(-6), x 10(-5) and x 10(-4) m3 s-1 are determined for segment of 0.45 mm radius and 45 mm length, with 15 mm abnormalities at each end where the radius varies by up to +/- 0.225 mm. When tau 0 = 0.005 N m-2, mu = 4 x 10(-3) Pa.s and Q = 1, the numerical values of the resistance-to-flow ratio vary from lambda = 0.525, when the maximum radii of the two abnormal segments are both 0.675 mm, to lambda = 3.06, when the minimum radii are both 0.225 mm. The resistance-to-flow ratio moves closer to unity as yield stress increases or as blood viscosity or flux decreases, and the magnitude of these alterations is greatest for yield stress and least for flux.
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Affiliation(s)
- B Pincombe
- Department of Applied Mathematics, University of Adelaide, South Australia.
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Srivastava VP. Two-phase model of blood flow through stenosed tubes in the presence of a peripheral layer: applications. J Biomech 1996; 29:1377-82. [PMID: 8884485 DOI: 10.1016/0021-9290(96)00037-1] [Citation(s) in RCA: 36] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/02/2023]
Abstract
The effects of red cell concentration and the peripheral layer on blood flow characteristics due to the presence of a mild stenosis, are studied. It is shown that the magnitudes of the flow characteristics significantly increase with cell concentration and the peripheral layer causes marked reduction in the magnitudes of the flow characteristics. Physiological relevance and the influence of various parameters are discussed.
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Affiliation(s)
- V P Srivastava
- Department of Mathematics and Astronomy, University of Lucknow, India
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22
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Abstract
The problem of blood flow through stenoses is solved using the incompressible generalized Newtonian model. The Herschel-Bulkley, Bingham and power-law fluids are incorporated. The geometry corresponds to a rigid circular tube with a partial occlusion. Calculations are performed by a Galerkin finite-element method. For the pulsatile case, a predictor-corrector time marching scheme is used with an adaptive time step. Results are obtained for steady and pulsatile physiological flows. Computations show that the memory effects taken into account in the model affect deeply the flow compared with Newtonian reference case. The disturbances are stronger by their vorticity intensity and persist after the geometrical obstacle. This is especially true for severe stenoses.
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Affiliation(s)
- C Tu
- Center for Systems Engineering and Applied Mechanics, Université Catholique de Louvain Bâtiment Euler, Louvain-la-Neuve, Belgium
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23
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Tandon PN, Rana UV. A new model for blood flow through an artery with axisymmetric stenosis. INTERNATIONAL JOURNAL OF BIO-MEDICAL COMPUTING 1995; 38:257-67. [PMID: 7774985 DOI: 10.1016/s0020-7101(05)80008-x] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/27/2023]
Abstract
Presented herein are the studies on the flow behavior of a blood type suspension through a circular tube with an axisymmetric stenosis. The suspension of the cells in plasma is represented by a layered fluid model, with a marginal cell-free layer of the suspending medium near the wall, a central core region and an annular layer of a biviscous fluid layer. It is understood that the proposed model may contribute to the inbuilt mechanism for drag reduction and prevention of the further development of the stenosis. The concept of lubricating pipe lining for transporting various industrial fluids is well represented through three-layered core-annular flows. The governing equations are solved numerically by using finite element method. The velocity fields, including separation and reattachment points, and the distribution of pressure and wall shear stresses have been brought out and discussed. The results of the analysis show that the presence of the marginal cell-free layer reduces the wall shear stresses and the length of the flow reversal zone. The non-Newtonian character of the suspension is helpful in reducing the abnormal effects of the stenosis. The model thus establishes the inbuilt character of blood for decreasing the stresses and this, in turn, reduces the load on the heart in propelling the blood.
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Affiliation(s)
- P N Tandon
- Department of Mathematics, University Brunei Darussalam, Negara, India
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24
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Srivastava VP. Particle-fluid suspension model of blood flow through stenotic vessels with applications. INTERNATIONAL JOURNAL OF BIO-MEDICAL COMPUTING 1995; 38:141-54. [PMID: 7729930 DOI: 10.1016/0020-7101(94)01046-4] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/26/2023]
Abstract
The present study deals with the problem of blood flow through stenotic vessels when blood is represented by a particle-fluid suspension model, i.e. a suspension of red blood cells in plasma. The expression for the dimensionless resistance to flow, the wall shear stress, and the shearing stress on the wall at the maximum height of the stenosis are derived. The results obtained in the analysis are discussed in brief, both qualitatively and quantitatively by comparison with other theories. It is observed that the magnitudes of the blood flow characteristics significantly increase with an increase in the red cell concentration. The importance of the decreasing vessel diameter is also pointed out. Finally, to observe the biological relevance of the analysis, the results obtained are used to compute the blood flow characteristics for normal and diseased blood using the experimental data from published literature and results are compared with those computed using the present theoretical approach.
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Affiliation(s)
- V P Srivastava
- Department of Mathematics and Astronomy, University of Lucknow, India
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25
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Srivastava VP, Saxena M. Two-layered model of Casson fluid flow through stenotic blood vessels: applications to the cardiovascular system. J Biomech 1994; 27:921-8. [PMID: 8063842 DOI: 10.1016/0021-9290(94)90264-x] [Citation(s) in RCA: 68] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/28/2023]
Abstract
The effects of peripheral layer viscosity on physiological characteristics of blood flow through the artery with mild stenosis have been investigated. Blood has been represented by a two-fluid model, consisting 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. The study is based on theoretical considerations and numerical evaluations and is restricted to the flow of blood through small arteries (130-1000 microns in diameter). It has been found that the resistance to flow and the wall shear stress decrease as the peripheral layer viscosity decreases. These characteristics are found to be decreasing as peripheral layer thickness increases. The numerical results show that the existence of the peripheral layer is helpful in functioning of the diseased arterial system. The analysis has been applied to calculate the resistance to flow and wall shear stress in different blood vessels.
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Affiliation(s)
- V P Srivastava
- Department of Mathematics and Astronomy, University of Lucknow, India
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26
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Chakravarty S, Mandal P. Mathematical modelling of blood flow through an overlapping arterial stenosis. ACTA ACUST UNITED AC 1994. [DOI: 10.1016/0895-7177(94)90116-3] [Citation(s) in RCA: 47] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/16/2022]
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27
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Abstract
A two-dimensional numerical model was developed to determine the effect of the non-Newtonian behavior of blood on a pulsatile flow at the aortic bifurcation. The blood rheology was described by a weak-form Casson equation. The successive-over-relaxation (SOR) method was used to solve both the vorticity and Poisson equations numerically. It was disclosed that the non-Newtonian property of blood did not drastically change the flow patterns, but caused an appreciable increase in the shear stresses and a slightly higher resistance to both flow separations and the phase shifts between flow layers.
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Affiliation(s)
- Z Lou
- Transportation Research Institute, University of Michigan, Ann Arbor 48109
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28
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29
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Chakravarty S, Datta A. Dynamic response of arterial blood flow in the presence of multi-stenoses. ACTA ACUST UNITED AC 1990. [DOI: 10.1016/0895-7177(90)90062-r] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/16/2022]
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30
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Hogan HA, Henriksen M. An evaluation of a micropolar model for blood flow through an idealized stenosis. J Biomech 1989; 22:211-8. [PMID: 2722892 DOI: 10.1016/0021-9290(89)90089-4] [Citation(s) in RCA: 31] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/02/2023]
Abstract
In this paper, the behavior of a viscous fluid described by Newtonian constitutive theory is compared with that predicted by a model based on micropolar continuum theory. The geometry chosen for this comparative analysis is a stenosis in which gradient effects should be pronounced. A range of boundary conditions for fluid microspin are considered. Although velocities and streamlines are found to be similar for the two continuum models, striking differences in shear stresses are revealed. These differences may be as high as 50% for vanishing microspin boundary conditions. Such significant discrepancies highlight the need for further study of higher order modeling of blood flow.
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Affiliation(s)
- H A Hogan
- Department of Mechanical Engineering, Texas A & M University, College Station 77843
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32
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33
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Majhi SN, Usha L. A mathematical note on the Fahraeus-Lindqvist effect in power law fluid. Bull Math Biol 1985; 47:765-9. [PMID: 4092111 DOI: 10.1007/bf02469303] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/08/2023]
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34
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Haldar K. Effects of the shape of stenosis on the resistance to blood flow through an artery. Bull Math Biol 1985; 47:545-50. [PMID: 4084691 DOI: 10.1007/bf02460013] [Citation(s) in RCA: 35] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/08/2023]
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35
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Abstract
The effects of an axially symmetric mild stenosis on the flow of blood, when blood is represented by a couple stress fluid model, have been studied. It is found that, for a fixed stenosis size, the resistance to flow and wall shear stress increase as the couple stress parameter eta decreases from unity. A comparison of the results with those of the Newtonian case shows that the magnitude of resistance to flow and wall shear under a given set of conditions, is greater in the case of the couple stress fluid model. It is seen that even in the case of a mild stenosis (19% area reduction), resistance to flow and wall shear values are increased over those for no stenosis by 60% and 62%, respectively, when compared with the case of a Newtonian fluid.
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36
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Devanathan R, Parvathamma S. Flow of micropolar fluid through a tube with stenosis. Med Biol Eng Comput 1983; 21:438-45. [PMID: 6888011 DOI: 10.1007/bf02442631] [Citation(s) in RCA: 24] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/22/2023]
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37
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Tandon PN, Misra JK. Microstructural and peripheral layer viscosity effects on the flow of blood through an artery with mild stenosis. Ann N Y Acad Sci 1983; 404:59-62. [PMID: 6575635 DOI: 10.1111/j.1749-6632.1983.tb19424.x] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/20/2023]
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38
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Perkkiö J, Keskinen R. On the effect of the concentration profile of red cells on blood flow in the artery with stenosis. Bull Math Biol 1983; 45:259-67. [PMID: 6882958 DOI: 10.1007/bf02462360] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/22/2023]
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