Clinicians frequently study arterial Doppler velocimetric waveforms depicted by Doppler sonography of the kidneys, the heart, the brain, and the feto-maternal circulation to assess the well-being of the aforementioned vital organs. The waveform interpretation of the Doppler indices can be studied using a mathematical model. The developed models serve as teaching tools and for easy comprehension of the regulatory mechanism of the organs. It will also obtain accurate wall shear stress (WSS) and likely atherosclerotic sites can be predicted early. The aim of this review is to reveal the imperatives of mathematical models in the study of the physical interpretation of Doppler velocimetry. The models will explore sonographic Doppler velocimetry and computational fluid dynamics (CFD) in determining the segments of the arteries that are prone to the development of atheromatous plaque. It will be achieved by comparing and computing the measurement differences of the WSS. A thorough literature review was carried out between 1971 and 2021 on the mathematical modeling of blood dynamics and Doppler velocimetry of different blood vessels, across various electronic databases including NC AHEC Digital Library, PUBMED, ERIC, MEDLINE, Free Medical Journals, and EMBASE. The results of the literature search were presented using the PRISMA flow chat. The narrative review of the mathematical models of arterial blood dynamics is based on incompressible Navier-Stokes equations, the Windkessel model, and CFD. It was deduced that the blood flow velocity decreased with time across the varying frequency from 0.2Hz to 0.50Hz in the interlobar arterial channels. The review also revealed that adult humans' Doppler indices of the renal-interlobar artery agree with developed models of renal interlobar arterial blood dynamics. The mathematical model measurements of the great vessels matched the sonographic Doppler velocimetry with <15% variation. In our fast-paced world of epidemiological transition, the imperatives of mathematical modeling of arterial flow dynamics based on the Navier-Stokes equations to represent various physiologic and pathologic situations cannot be overstated. The practical consequences include the possibility of mathematical models to acquire precise WSS distribution and early detection of potential atherosclerotic sites during cardiovascular Doppler sonography.
Keywords: Blood dynamic; Doppler velocimetry; mathematical model.
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