Mathematical Modelling of Cardiovascular System

This study is motivated by a desire to understand blood flow through stenosed and aneurysmic arterial segments of varying cross-sections, particularly through coronary and aortal arterial segments. Several mathematical models considering the Newtonian and non-Newtonian models for blood are presented and their predictions in a variety of circumstances are investigated. This process is repeated for the seven different scenarios under study which vary in complexity and realism. The structures considered for each of the cases varied from rigid wall models to elastic wall models including two-layer configurations for the arterial walls.
Initially, the need for mathematical models to analyze the blood flow in a diseased artery is discussed and a brief recap of the available research in this field is presented. This is followed by a short note on the formation of atherosclerotic plaque and aneurysms is revealed. The details of the FEM solver, the steps involved in preparing the solver, the methods to reduce the error, and the flow of the process are discussed. As blood behaves as both Newtonian and non-Newtonian, depending on the flow conditions, the study considers both the properties of blood according to the convenience of the topic under study. The different non-Newtonian models studied are the Carreau model, Bingham model, Herschel Bulkley model, Casson model, and the power law model. The best-suited non-Newtonian model for blood is investigated and the results obtained using the Carreau model are more promising than the other models.Arteries carry several chemical components in the matrix of blood. The chemical species include oxygen molecules, LDL molecules, the Aspirin molecules depending on the nutrition intake as well as the pathological conditions. The transport patterns of these chemical species are discussed by considering blood to be a Newtonian fluid. The relative concentration profiles of LDL and oxygen molecules are investigated and an inverse interrelation is discovered.In recent times around the world, the most discussed and worried topic is the causes and effects of coronavirus. One of the prominent alterations made to the blood by the virus is the increase in its viscosity. The hyperviscosity of blood in atherosclerotic and aneurysmic patients is investigated by assuming the Newtonian behavior of blood. The study suggested that the hyper-viscous effects further worsen the condition of stenosis and aneurysm leading to accelerated progression of the plaque and sac. The rapid progression of plaque and sac approach a threshold condition subsequently leading to the rupture and hence the thrombosis.The results obtained on solving the mathematical models using the FEM solver are used to draw conclusions and drawbacks of the assumptions made in the current study. The clinical significance of the study is mentioned with the recommendations for future study.