Determination of pressure data from velocity data with a view toward its application in cardiovascular mechanics. Part 1. Theoretical considerations

The non-invasive determination of the pressure (mean normal stress) in a flowing fluid has ramifications in a variety of important problems: the flow of blood in blood vessels, flows taking place in inaccessible locations in complex internal geometries that occur in mechanical systems, etc. In this paper we discuss a rigorous new mathematical procedure for the determination of the pressure (mean normal stress) field, from data for the velocity field that can be obtained through imaging procedures such as 4D magnetic resonance imaging or echocardiography. We then use the procedure to demonstrate its efficacy by considering flows in an idealized geometry with a symmetric and asymmetric obstruction. We delineate the superiority of the method with regard to the methods that are currently in place. In Part 2 of this two part paper, we study the loss of pressure and the dissipation that occurs due to the flow of blood across a diseased valve (the pressure loss being an important indicator of the extent of the valvular disease) as well as the flow taking place in a realistic cerebral aneurysm.