We consider the fully developed flow of an incompressible Newtonian fluid in a cylindrical vessel with elliptical cross section, and in the annulus between two confocal ellipses. Since flow rate can actually be derived from measurements, we address the inverse problem, namely computing the velocity field associated with a given time-periodic flow rate. We propose a novel numerical strategy, which is nonetheless grounded on several analytical relations and which leads to the solution of systems of ordinary differential equations. We also report numerical results based on measured data for human blood flow in the internal carotid artery, and cerebrospinal fluid flow in the upper cervical region of the human spine. Our method holds promise to be more amenable to implementation than previous ones, based on challenging computation of Mathieu functions, especially for strongly elliptical cross sections. The main goal of this study is to provide an improved source of initial/boundary data, as well as a benchmark solution for pulsatile flows in elliptical sections. In addition to bio-fluid dynamics investigations, the proposed method can be applied to many problems in the biomedical field.1. Introduction. Pulsatile flows are driven by a time-periodic force, which is generally the pressure gradient. A remarkable case is that of heartbeat-driven, human physiological flows, including blood circulation [30] and, even if less directly, cerebrospinal fluid (CSF) flow [18]. In addition to bio-fluid dynamics, time-periodic flows are widely studied with regard to chemical-physics applications, mass and heat transfer problems, and peristaltic pumping [26]. However, in many cases of practical interest the pressure gradient is unknown or hardly measurable, while the fluxhereafter understood as a synonym of the flow rate-can actually be estimated through measurements. For instance, blood and CSF flow are commonly obtained by phasecontrast magnetic resonance imaging (MRI) or Doppler ultrasonography: Flow rate is obtained by somehow integrating low-space-resolution velocity measurements, which, however, are not resolved enough for determining the velocity profile. Anyway, measurement issues are outside the scope of this paper; hence the flux is assumed to be known with reasonable accuracy. Our study is mainly motivated by the fact that many portions of the vasculature are characterized by a rather elliptical cross section, due to the presence of surrounding organs. Moreover, the spinal subarachnoid space can be well approximated by an elliptical annulus [18]; CSF dynamics in such a domain is affected by pulsatility and plays a major role in the (still poorly understood)