A Numerical Approach to a Parabolic Equation: The Finite-Difference Method for Blood Vessels

This paper considers a mathematical model of a vascular system at rest. The model is described by the pressure transfer equation in a vessel, taking into account such parameters as blood viscosity, vessel permeability coefficient, vessel length, and elasticity coefficient. The model includes initial and boundary conditions for determining the pressure in a vessel in time and space. The results show the dynamics of pressure changes in a vessel and help to understand the processes of vascular autoregulation. The research focusses on the movement and interaction of the pulse wave with various areas of the vascular system. Our model incorporates authentic boundary conditions, such as heart valve operation and relationships with tiny blood vessels (microvasculature). This model is beneficial for evaluating the cardiovascular system under both healthy and unhealthy conditions, helping us understand how diseases such as atherosclerosis and hypertension affect blood flow. Furthermore, the model can assist in testing and improving medical equipment, such as stents and vascular prostheses. The main objective of the current investigation was to develop a physiological model to simulate blood flow dynamics in the human cardiovascular system taking into account the aorta, arteries, arterioles, capillaries, veins, and vena cava under physiological conditions that involve 24-hour monitoring. From this given mathematical model, it shows that it can be used to learn different diseases, including the impact of atherosclerosis and hypertension within the circulatory system. Therefore, the developed model has become effective in assessing and forecasting hemodynamic actions in the circulatory system.