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In-vivo characterization of blood viscosity and carotid wall stiffness using MRI
Avril, S., Huntley, J.M. & CUSACK, R.
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ABSTRACT: A method is proposed for deducing the blood viscosity and wall stiffness in the carotid from Phase-Contrast MRI data. The approach is based on Womersley’s model of blood flow derived from the resolution of the Navier-Stokes equations, assuming blood as a Newtonian fluid and the artery as a linear elastic cylindrical pipe. After presenting its principle, the approach is applied to the experimental data obtained on a single volunteer. Promising results are obtained. 1. INTRODUCTION Numerical models are increasingly commonplace in the field of biomechanics. The simulation of blood flow [1] or of artery deformations [2] is essential for understanding phenomena such as plaque formation and failure. For that, accurate estimates of the mechanical properties governing the artery deformations and blood flow are required. The aim of this paper is to present a novel method that can provide these properties in a non intrusive way from full-field time-resolved velocity data provided by phase contrast Magnetic Resonance Imaging (MRI). Classically, different ways exist for characterizing arterial wall stiffness [3]. The most common way is to use the pulse wave velocity (PWV), denoted c, which satisfies, within the framework of a few assumptions, the Moens-Korteweg equation: c = Eh/2ρR0, where E is the Young’s modulus of the arterial wall ; h is the wall thickness ; R0 is the arterial radius (at rest) ; ρ is the blood density [8]. The PWV is calculated from measurements of pulse transit time and the distance travelled by the pulse between two recording sites, using manometers. However, manometers can only be used on a limb, which prevents applications to large arteries like the carotids or the aorta. No experimental study has been found in the literature where the PWV and consequently the artery elastic modulus could be identified without using a non-intrusive system. We show in this study that it is possible to measure these quantities with only shape and velocity measurements provided by MRI. A more sophisticated model, still compatible with the Moens-Korteweg equation, is used. Moreover, a technique to identify the blood viscosity along with the elastic modulus of the artery is proposed. The non-intrusive identification of blood viscosity is original. Indeed, blood viscosity is a very important parameter, usually measured using viscometers [4]. This means that blood needs to be taken from the human body before being tested, requiring the use of anticoagulants for avoiding blood coagulation. The effects of the anticoagulants on blood viscosity are almost unknown [4]. Accordingly, there is a potential interest in characterizing blood viscosity in its real environment thanks to medical imaging. 2. THEORY The approach for identifying the viscosity is based on an analytical model of the flow derived from a particular solution of the Navier-Stokes equations, assuming blood as a Newtonian fluid and the artery as a linear elastic cylindrical pipe. It is called the Womersley’s solution [5]. This solution can be compared to the velocity measurements provided by MRI. The deviation depends on several parameters, among which only the blood viscosity is unknown. A cost function is built up, representing the average quadratic deviation between the modelled velocity profiles and the measured ones. The minimization of this cost function is achieved using the Nelder Mead algorithm, providing the blood viscosity in no more than 20 iterations in practice. The approach for identifying the wall elasticity is based on a particular relationship between the cross section area of the artery and the blood flow. Indeed, for elastic vessels in which a pulse wave travels, the time variations of the cross section area can be deduced directly from the time variations of the blood flow just by shifting and scaling the curves. The scaling constant only involves the PSW. Thus, it can easily be derived from the measurements. Eventually, the wall stiffness is deduced from the PSW using the Moens-Korteweg equation. 3. EXPERIMENTAL RESULTS AND CONCLUSIONS The approach requires time-resolved measurements of both the field of blood velocities and the variations of the artery radius during a heart beat. Both can be obtained using a Phase contrast MR angiography scanner. The scanner used in our study was a 3T Siemens Tim Trio system (MRC Cognition and Brain Sciences Unit, Cambridge, UK). A 2D spin-echo FLASH sequence was used to acquire a single 3 mm thick slice of a volunteer’s neck with a matrix size of 240×256 giving in-plane dimensions 0.39×0.39 mm2. A cine sequence, with one segment per cycle, was used to acquire the temporal evolution of the flow throughout the pulse. Heart beats were detected by the measurement of blood flow in a patient’s finger with near infrared spectrometry. The cine data were reconstructed to give 50 snapshots evenly distributed throughout the cardiac cycle, corresponding to a mean sampling frequency of 61.5 s−1. The magnitude images were used to characterise geometry and the phase images the velocity at each moment in time. The magnitude of the signal is well suited for determining the edges of the artery because, due to the large quantity of blood flowing in the artery, the magnitude of the signal is larger inside the artery than outside. Thanks to this profile, the contour of the artery is detected for each frame and the deduced masks are applied onto the velocity maps (Figure 1). The modelled profiles that best fit the data in the least squares sense have been derived. The corresponding viscosity parameter is μ = 0.0073 Pa.s. This is consistent with values found in the literature regarding blood viscosity: values of μ varying between 0.003 and 0.03 Pa.s have been reported [5]. The artery wall elastic modulus gave the value E = 99 kPa. This is somewhat lower than the values from other studies. However, there is a large uncertainty on the parameter h (wall thickness), as this parameter was not measured precisely during the MRI sequences. It is more relevant to compare the coefficient of elasticity k = Eh/R0 which took the value of 14.9 kPa for our volunteer. For comparison, Stephanis et al. [3] found a coefficient of elasticity of about 15 kPa. Therefore, it seems likely that the underestimation of E in our results is mainly induced by errors in the estimation of the wall thickness. An accurate measurement of this quantity will be required for further experiments. Our results should also be verified by repeating the measurement on the same volunteer, for characterizing the repeatability and the robustness of the method. Moreover, an application of our approach on several volunteers and also on different locations of the carotid would be required for a more complete validation. A possible extension of our approach to arteries with plaques is also targeted. Figure 1- (a) Measured velocity fields within the carotid during a heart beat and (b) comparison of the profiles with the identified model for a few frames. 4. REFERENCES 1. Liepsch,D. (2002) An introduction to biofluid mechanics - Basic models and applications. Journal of Biomechanics, 35, 415-435. 2. Li,Z.-Y., Howarth, S., Trivedi, R., U-King-Im, J., Graves, M., Brown, A., Wang, L. and Gillard J. (2006) Stress analysis of carotid plaque rupture based on in vivo high resolution MRI. Journal of Biomechanics, 39, 2611-2622. 3. Stephanis, C., Mourmouras, D. and Tsagadopoulos D. (2003) On the elastic properties of arteries. Journal of Biomechanics, 36, 1727-1731. 4. Lagrée, P.-Y. (2000) An inverse technique to deduce the elasticity of a large artery. The European Physical Journal - Applied Physics, 9, 153-163. 5. Shin, S. and Keum, D.-Y. (2002) Measurement of blood viscosity using mass-detecting sensor. Biosensors and Bioelectronics, 17, 383-388.