Flow in Elliptical Vessels Calculated for a Physiological Waveform

Abstract

Understanding the nature of pulsatile flow is an important issue in haemodynamics, especially the initiation and progression of vascular disease. The geometry of a non-circular vessel was idealised to an elliptical cross-section, and the dynamic properties of the flow were calculated for a physiological waveform. The Fourier harmonics for a common carotid waveform were determined, and the velocity profile and wall shear stress were calculated from the superposition of the individual contributions from each harmonic. The effects of ellipticity on the flow pattern were found to be significant. The velocity profile along the major axis of the elliptical cross-section developed a flattened peak, which widened as the vessel became more elliptical. Wall shear stress demonstrated an angular dependence in elliptical vessels, where the point of minimum shear stress was located at the end of the major axis. Comparison with a cylindrical vessel demonstrated a 3% decrease in peak wall shear stress (τ = 2.96, N·m^–2) at the end of the major axis, and 10% in the mean wall shear stress (τ = 0.44 N· m^–2), for an elliptical vessel (Ε = 0.8). The temporal average wall shear stress, which has been associated with atherogenic sites, also displayed a minimum at the end of the major axis that decreased with more elliptical cross-sections.

Pulsatile Viscous Flows in Elliptical Vessels and Annuli: Solution to the Inverse Problem, with Application to Blood and Cerebrospinal Fluid Flow

“…In particular, by invoking separation of variables (in elliptical coordinates) we obtained a tridiagonal system of linear, ordinary differential equations and we obtained the sought solution by means of a Galerkin-type truncation. This method seem to theoretically corroborate the early attempts in [49] (in which proper references to the inverse problem are missing). Moreover, it represents an alternative to the numerical approach in [24], heavily based on a wise computation strategy of the Mathieu functions.…”

“…Indeed, vessel ellipticity affects such a stress [26], which in turn has been identified as a key player in the onset and development of relevant pathologies likes atherosclerosis, even if several acquisitions needs to be consolidated (see e.g. [49,10]). Furthermore, still within the context of bio-fluid dynamics, many investigations are being carried out for elucidating the role of CSF dynamics within human spine, in relation to the onset and progression of relevant diseases like syringomyelia.…”

“…We also assume a given flow rate f (t) having period T ; (S1) We determine an integer M such that the Fourier spectrum of f (t) is suitably approximated by considering M modes; popular metrics (e.g. the Pearson correlation coefficient used in [49,24]) can be used to the purpose. Practical constraints, due for instance to the experimental procedure implemented for measuring f , may come into play at this stage (for instance, the Nyquist limit is considered in [24]).…”

MRI measurement of time‐resolved wall shear stress vectors in a carotid bifurcation model, and comparison with CFD predictions

“…As a result, the WSS vectors are not identical all around the wall, even from the CFD calculations. Higher values are found on the front and rear surfaces, i.e., at the ends of the minor axis (26). We believe that the model section is genuinely elliptical, as we have observed this effect on several occasions and with independent segmentation techniques.…”

Simple Patient-Based Transmantle Pressure and Shear Estimate From Cine Phase-Contrast MRI in Cerebral Aqueduct

“…5 and Table II) showed that slightly noncircular models of the cerebral aqueduct were not relevant, and called for fully elliptic cross sections geometries. Also, a Fourier mode decomposition of the flow rate during the cardiac cycle, similar to [22], is proposed in the next sections.…”