A MATHEMATICAL MODEL TO STUDY THE SIMILARITIES OF BLOOD FLUID MODELS THROUGH INCLINED MULTI-STENOSED ARTERY

A mathematical model is presented to comparative steady of the flow behavior of Casson’s and Bingham Plastic fluid model through an inclined tube of non-uniform cross-section with multiple stenoses. The equation describing the flow has been solved and the expressions parameters on flow variables have been studied. The present study may be helpful for better understanding the flow characteristics of blood having multiple stenoses. The graphical representations have been made to validate the analytical findings with a view of its applicability to stenotic diseases. It is found that the flow of resistance increases with the height of the stenosis but decreases with the angle of inclination. The flow characteristics namely, velocity, pressure gradient, flow rate, resistance to flow have been derived. It is shown that the resistance to flow increases with the height of the secondary stenosis as well as with the yield stress. The results are compared with the available data presented by previous researchers

Modeling of atherosclerotic plaque growth using fluid-structure interaction.

Blood flow through a narrow arterial tube has been a classical mathematical problem dating back to the 1840's, after the pioneer experimental work conducted by Jean Louis Marie Poiseuille, Observations of blood flow, published in Ann. Sci. Naturelles in 1836. However, numerical simulations on atherosclerosis only started to thrive in the 1990's, due to rapid advancement in computer technology. The transportation of low density lipoprotein (LDL) enables lipids like cholesterol and triglycerides to be carried within the water-based bloodstream. Numerous research papers have shown that LDL particles would adhere on the artery wall at the locations where the local wall shear stress (WSS) is too low to move them further, resulting in localised regions with high LDL concentrations. Atherosclerosis tends to develop at regions where LDL accumulation is high and consistent. It is well known in the medical community that high level of LDL is closely related to myocardial infarction and/or stroke due to rupture of atherosclerotic plaques. Some rupture would release highly concentrated lipids and thrombogenic material into the bloodstream, leading to lethal blood clots that result in sudden cardiovascular casualties. In order to explore the dominant phenomenological mechanism, the thesis hypothesizes that LDL accumulation has the sole influence on plaque growth. By applying the hypothesis as the rule of growth, the thesis investigates how LDL accumulation affects the plaque morphology during its growth, aiming to provide a better understanding of atherosclerosis development. In this research the advanced two-way fluid-structural interaction (FSI) method is applied to model the growth of three-dimensional atherosclerotic plaques. The mild 45% axis-asymmetric stenosis model with the bi-elliptical cross-sectional plaque morphology is used as the base model, and then the plaque morphology is updated to the non-elliptical arbitrarily shaped profile across the centre of the plaque in the direction of the flow and elliptical profiles at various cross-sections of the plaque that are perpendicular to the flow direction. The updated plaque morphology is determined according to the simulation results of WSS distribution in the vicinity of the previous plaque and the relationship between the WSS and LDL accumulation derived from the literature. The growth-updated model is then used as the new geometry for the next round of simulation. This process repeats until the stenosis severity is increased to or beyond 79%, which is 1% greater than the critical stenosis as reported in literature. The numerical results of these growth-updated models presented and discussed in the thesis are extensive, providing valuable insight into the plaque development

Blood flow in stenosed arteries using two way, fluid-structural interaction

Blood flow in a stenosed artery was simulated using the two way, fluid-structural interactions in ANSYS---a commercial finite element and finite volume software widely used in industry. Several stenosis models were investigated using a pulsatile flow condition. In these models the form of the stenosis geometry was either axisymmetric or axis-asymmetric. The blood was modelled both as a non-Newtonian, power law fluid, and as the Casson model. The artery tissue was assumed to be a linear elastic material for the axis-symmetric stenosis model and nonlinear material for the axis-asymmetric stenosis models. A fibrous cap and lipid pool were also added into the stenosis geometry of the axis-asymmetric models to account for complexities of the physiology. The effect of the stenosis severity on arteries was investigated by examining the blood flow velocity, pressure and wall shear stress at multiple locations. References K. C. Ang and J. N. Mazumdar, Mathematical modelling of three-dimensional flow through an asymmetric arterial stenosis, Mathematical and Computer Modelling, 25, No 1, 19--29, 1997, doi:10.1016/S0895-7177(96)00182-3. W. Y. Chan, Y. Ding, J. Y. Tu, Modeling of non-Newtonian blood flow through a stenosed artery incorporating fluid-structure interaction, ANZIAM Journal, 47, 507--523, 2007, http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/1059. Santabrata Chakravarty and Prashanta Kumar Mandal, Two-Dimensional blood flow through tapered arteries under stenotic conditions, Non-Linear Mechanics, 35, 779--793, August, 2006, doi:10.1016/S0020-7462(99)00059-1. Samuel A. Kock and Jens V. Nygaard. Nikolaj Eldrup,Ernst-Torben Frund and Anette Klaeke, William P.Pausk. Erling Falk and W.Yong Kim, Mechanical Stresses in carotid plaques using MRI-based fluid-structure interaction models, Journal of Biomechanics, 41, 1651--1658, March, 2008, doi:10.1016/j.jbiomech.2008.03.019. K. W. Lee and X. Y. Xu, Modelling of flow and wall behaviour in a mildly stenosed tube, Medical Engineering and Physics, 24, 575--586, May, 2002, doi:10.1016/S1350-4533(02)00048-6. M. X. Li and J. J. Beech-Brandt and L. R. John and P. R. Hoskins and W. J. Easson, Numerical analysis pulsatile blood flow and vessel wall mechanics in different degrees of stenoses, Journal of Biomechanics, 40, 3715--3724, June, 2007, doi:10.1016/j.jbiomech.2007.06.023. B. Pincombe and J. Mazumdar, The effects of post-stenotic dilatations on the flow of a blood analogue through stenosed coronary arteries, Mathematical and Computer Modelling, 25, No 6, 57--70, 1997, doi:10.1016/S0895-7177(97)00039-3. J. Poiseuille, Observations of blood flow, Ann. Sci. Naturelles Srie 5, 2, 1836. S. U. Siddiqui and N. K. Verma and Shailesh Mishra and R. S. Gupta, Mathematical modelling of pulsatile flow of Cassons fluid in arterial stenosis, Applied Mathematics and Computation, 2007, doi:10.1016/j.amc.2007.05.070. S. U. Siddiqui and Shailesh Mishra, A study of modified Casson's fluid in modelled normal and stenotic capillary-tissue diffusion phenomena, Applied Mathematics and Computation, 189, 1048--1057, 2007, doi:10.1016/j.amc.2006.11.151. Dalin Tang and Chun Yang and David N. Ku, A 3D thin-wall model with fluid-structure interactions for blood flow in carotid arteries with symmetric and asymmetric stenoses, Computers and Structures, 72, 357--377, 1999, doi:10.1016/S0045-7949(99)00019-X. Dalin Tang and Chun Yang and Yan Huang and David N. Ku, Wall stress and strain analysis using a three-dimensional thick-wall model with fluid-structure interactions for blood flow in carotid arteries with stenoses, Computers and Structures, 72, 341--356, May, 1999, doi:10.1016/S0045-7949(99)00009-7. Dalin Tang, Chun Yang, Jie Zheng, Pamela K. Woodard, Gregorio A. Sicard, Jeffrey E. Saffiz, Shunichi Kobayashi, Thomas K. Pilgram, and Chun Yuan, 3D computational mechanical analysis for human Atherosclerotic plaques using MRI-based models with fluid-structure interactions, MICCAI, 328--336, 2004. Dalin Tang and Chun Yang and Jie Zheng and Pamela K.Woodard and Gregorio A.Sicard and Jefferey E. Saffitz and Chun Yuan, 3D mri-based multicomponent fsi models for artherosclerotic plaques, Biomedical Engineering, 32, 947--960, July, 2004