Quantitative analysis of PC MRI velocity maps: pulsatile flow in cylindrical vessels

Abstract

The accuracy of MR phase contrast (PC) velocity mapping, and the subsequent derivation of wall shear stress (WSS) values, has been quantitatively assessed. Using a retrospectively gated PC gradient-echo technique, the temporal-spatial velocity fields were measured for pulsatile flow in a rigid cylindrical vessel. The experimental data were compared with values derived from the Womersley solution of the Navier-Stokes equations. For a sinusoidal waveform, the overall root-mean-square (rms) difference between the measured and analytical velocities corresponded to 13% of the peak fluid velocity. The WSS derived from the data displayed a 14% rms difference with the analytical model. As an example of a more complicated flow, a triangular saw-tooth waveform was deconstructed into its Fourier components. Velocity maps and the WSS were calculated by the superposition of the individual solutions, weighted by the Fourier series coefficient, for each harmonic. The velocity and experimentally derived WSS agreed with the analytical results (4% and 12% rms difference, respectively). Evaluation of the analytical models allowed an estimate of the inherent accuracy in the measurement of velocity maps and WSS values.

Introduction

Atherosclerosis is the progressive condition that leads to a thickening of the arterial walls, corresponding to a reduction in the vessel’s elasticity. In severe cases, this process may eventually lead to the total occlusion of the artery, but in general any alterations in the innermost layer of the artery has a major influence on the disease of the vessel. Important indicators for the clinical diagnosis of atherosclerosis are the blood velocity and associated physical properties of the flow field [1], [2], [3]. In this particular application, magnetic resonance imaging has become a valuable non-invasive technique [4], [5].

Phase contrast (PC) velocity mapping has been used for both the qualitative and quantitative assessment of flow [4], [6], [7], [8]. The ability to obtain a temporal-spatial velocity map in this way permits a direct examination of blood velocities and flow rate throughout the cardiac cycle, which in turn are potentially useful in the diagnosis of vascular disease. Improvement on the basic experiment has been achieved through the introduction of novel methods of data acquisition [9], [10], [11], [12], [13] and the refinement of post-processing techniques that may be used to determine additional information about the velocity field [14], [15], [16], [17], [18].

Properties such as flow-rate [19], pressure [14], [15] and wall shear stress (WSS) [16], [17], [18] can be derived from the velocity data. The basic internal mechanical forces acting on a blood vessel are pressure and shear stress. These forces are related to the pulsatile flow of blood through the arteries. It is generally accepted that arterial WSS patterns influence the development of arterial disease [1], [2], [3]. The development of atherosclerosis, at least in its initial stages, may be related to the physiological response of arteries to the WSS on a localised basis. However, the inherent accuracy of specific PC MRI acquisitions, and post-processing techniques, has not been fully quantified for pulsatile flows.

Previous studies have demonstrated a close correlation between PC MRI and Doppler ultrasound, an established clinical technique, for both in vitro [19], [20] and in vivo [21] cases. However, caution is necessary when comparing different modalities such as these. The MRI technique provides an estimate of the mean velocity, while Doppler ultrasound measures the peak velocity within a voxel. The discrepancies between the quantities measured must be quantitatively addressed before inter-modality comparisons are feasible. Validation with an analytical model, although providing a reliable alternative have rarely been performed [22], [23]. Velocity maps for a pulsatile flow were obtained by Frayne et al [22], where the flow rate was measured directly from the MRI data, which was subsequently used to calculate the velocity with an analytical equation of motion. However, this procedure did not test the data for the presence of systematic errors in the velocity measurement, it only verified the analytical equation of motion. A more comprehensive examination would determine the flow rate from the flow simulator’s command waveform.

With the increased interest in information extracted from the flow field, the inherent accuracy of the measured velocities is of intrinsic importance. In particular, the WSS are extremely sensitive to errors in the velocity maps. As a result of the relatively limited spatial resolution of PC MRI data, the WSS is usually calculated from an empirical fit to the velocity map close to the vessel wall [16], [18], [24], [25]. However, this approach requires a phenomenological model that satisfactorily represents the velocity maps, and an accurate estimate for the position of the vessel wall. The calculation is subject to several potential errors, and therefore any new method must ideally be verified with an analytical model.

The aim of the present study was to investigate the agreement between an analytical calculation of the flow field, defined from a flow rate waveform, and a retrospectively gated PC MRI velocity map, where the flow was generated with a computer controlled flow simulator. A technique is developed in which the flow field at the measurement site is known for an arbitrary waveform, without the need for a separate flow rate calibration. Hence, the accuracy of a PC velocity measurement was established with an in vitro flow simulation in order to determine the inherent accuracy of this particular technique.