Womersley Number-Based Estimates of Blood Flow Rate in Doppler Analysis: In Vivo Validation by Means of Phase-Contrast MRI
14 Jun 2010-IEEE Transactions on Biomedical Engineering (IEEE Trans Biomed Eng)-Vol. 57, Iss: 7, pp 1807-1815
TL;DR: In this article, a Womersley number-based formula was proposed for the peak instant linking the maximum velocity and the flow rate by including a well-established dimensionless fluid-dynamics parameter (Womersley Number) in order to account for the hemodynamics conditions.
Abstract: A common clinical practice during single-point Doppler analysis is to measure the centerline maximum velocity and to recover the time-averaged flow rate by exploiting an assumption on the shape of velocity profile (a priori formula), either a parabolic or a flat one. In a previous study, we proposed a new formula valid for the peak instant linking the maximum velocity and the flow rate by including a well-established dimensionless fluid-dynamics parameter (the Womersley number), in order to account for the hemodynamics conditions (Womersley number-based formula). Several in silico tests confirmed the reliability of the new formula. Nevertheless, an in vivo confirmation is missing limiting the clinical applicability of the formula. An experimental in vivo protocol using cine phase-contrast MRI (2-D PCMRI) technique has been designed and applied to ten healthy young volunteers in three different arterial districts: the abdominal aorta, the common carotid artery, and the brachial artery. Each PCMRI dataset has been used twice: 1) to compute the value of the blood flow rate used as a gold standard and 2) to estimate the flow rate by measuring directly the maximum velocity and the diameter (i.e., emulating the intravascular Doppler data acquisition) and by applying to these data the a priori and the Womersley number-based formulae. All the in vivo results have confirmed that the Womersley number-based formula provides better estimates of the flow rate at the peak instant with respect to the a priori formula. More precisely, mean performances of the Womersley number-based formula are about three times better than the a priori results in the abdominal aorta, five times better in the common carotid artery, and two times better in the brachial artery.
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TL;DR: This review article will address the two principal components of the cardiovascular system: arterial circulation and heart function, and systematically describe all aspects of the problem, ranging from data imaging acquisition to the development of reduced-order models that are of paramount importance when solving problems with high complexity, which would otherwise be out of reach.
Abstract: Mathematical and numerical modelling of the cardiovascular system is a research topic that has attracted remarkable interest from the mathematical community because of its intrinsic mathematical difficulty and the increasing impact of cardiovascular diseases worldwide. In this review article we will address the two principal components of the cardiovascular system: arterial circulation and heart function. We will systematically describe all aspects of the problem, ranging from data imaging acquisition, stating the basic physical principles, analysing the associated mathematical models that comprise PDE and ODE systems, proposing sound and efficient numerical methods for their approximation, and simulating both benchmark problems and clinically inspired problems. Mathematical modelling itself imposes tremendous challenges, due to the amazing complexity of the cardiocirculatory system, the multiscale nature of the physiological processes involved, and the need to devise computational methods that are stable, reliable and efficient. Critical issues involve filtering the data, identifying the parameters of mathematical models, devising optimal treatments and accounting for uncertainties. For this reason, we will devote the last part of the paper to control and inverse problems, including parameter estimation, uncertainty quantification and the development of reduced-order models that are of paramount importance when solving problems with high complexity, which would otherwise be out of reach.
176 citations
Cites methods from "Womersley Number-Based Estimates of..."
...…profile is not prescribed a priori, this technique has been used to improve the parabolic-based law implemented in the Doppler technology for the estimation of the flow rate starting from the peak velocity (Ponzini, Vergara, Redaelli and Veneziani 2006, Vergara et al. 2010, Ponzini et al. 2010)....
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TL;DR: This review paper addresses the so called geometric multiscale approach for the numerical simulation of blood flow problems, from its origin (that the authors can collocate in the second half of '90s) to their days, and details the most popular numerical algorithms for the solution of the coupled problems.
163 citations
Patent•
07 Sep 2011TL;DR: In this article, an anatomical model of a coronary artery is generated from the medical image data, based on a spatio-temporal representation of contrast agent propagation, and the estimated velocity of the blood in the coronary artery as a boundary condition.
Abstract: In order to assess coronary artery disease from medical image data, an anatomical model of a coronary artery is generated from the medical image data. A velocity of blood in the coronary artery is estimated based on a spatio-temporal representation of contrast agent propagation in the medical image data. Blood flow is simulated in the anatomical model of the coronary artery using a computational fluid dynamics (CFD) simulation using the estimated velocity of the blood in the coronary artery as a boundary condition.
126 citations
TL;DR: These findings support the hypothesis that helical flow in CCA might reduce the likelihood of flow disturbances at the bifurcation and confirm the physiological role of CCA in transporting and enforcing helicals flow structures into the b ifurcation, giving further contribution to the helicity-driven suppression of disturbed shear.
Abstract: The rationale for this study lies in the well-known predilection for vascular disease of the carotid bifurcation, attributed to an altered shear stress distribution at the luminal surface and mitigated by helical fluid structures establishing inside the bifurcation. Here we investigate the mechanistic role played by the common carotid artery (CCA) in promoting complex intravascular flow and in influencing the hemodynamics at the distal carotid bifurcation. Fifty-five image-based computational hemodynamic models of eleven right carotid geometries were reconstructed from its brachiocephalic origin to above the bifurcation to assess how five different CCA reconstruction length affects intravascular fluid structures entering the bifurcation. A quantitative description of helical flow is adopted, in parallel to the description of disturbed shear at the bifurcation luminal surface. Our findings support the hypothesis that helical flow in CCA might reduce the likelihood of flow disturbances at the bifurcation. This confirms the physiological role of CCA in transporting and enforcing helical flow structures into the bifurcation, giving further contribution to the helicity-driven suppression of disturbed shear. A quantitative analysis of CCA geometry highlights the beneficial effect of proximal CCA curvature on helical flow and shows the complex interlacement among CCA geometry, helical flow, and disturbed shear at the bifurcation. Since helicity-based descriptors and geometric descriptors relative to the bifurcation have been shown to be significant predictors of disturbed shear, in principle they may be augmented by factors related to CCA geometry and hemodynamics.
68 citations
TL;DR: A novel model to calculate the cerebral venous return, normalized to the arterial inflow, in the different segments of the IJV suggests the pivotal role of the collateral network in draining the blood into the superior vena cava under CCSVI condition.
Abstract: The quantification of the flow returning from the head through the cervical veins and the collaterals of the internal jugular vein (IJV), is becoming of prominent interest in clinical practice. We developed a novel model to calculate the cerebral venous return, normalized to the arterial inflow, in the different segments of the IJV. We assessed, by established Echo Colour Doppler (ECD) methodology, the head inflow (HBinF) defined as the sum of common carotids and vertebral arteries, as well as the cerebral flow (CBF) defined as the sum of internal carotid and vertebral arteries. We also assessed the head outflow (HBoutF) defined as the sum of the measurements at the junction of the IJV and the vertebral veins. In addition, we also calculated the collateral flow index (CFI) by estimating the flow which re-enters directly into the superior vena cava as the amount of blood extrapolated by the difference between the HBinF and the HBoutF. We preliminarily tested the model by comparing ten healthy controls (HC) with ten patients affected by chronic cerebral spinal venous insufficiency (CCSVI), a condition characterized by some blockages in the IJV which are bypassed by collateral circulation. In HC the HBinF was 956+-105ml/min, whereas the HBoutF was > 90% of the HBinF, leading to a final CFI value of 1%. The last result shows that a very small amount of blood is drained by the collaterals. In upright we confirmed a reduction of the outflow through the IJV which increased CFI to 9%. When we applied the model to CCSVI, the HBinF was not significantly different from controls. In supine, the flow of CCSVI patients in the IJV junction was significantly lower (p < 0.001) while the correspondent CFI value significantly increased (61%, p < 0.0002). Our preliminary application of the novel model in the clinical setting suggests the pivotal role of the collateral network in draining the blood into the superior vena cava under CCSVI condition.
54 citations
Cites methods from "Womersley Number-Based Estimates of..."
...5 mm has been adopted for three reasons: 1) the small sample volume assures that the Doppler angle is constant over the whole sample volume, 2) the use of constant sample volume simplifies very much the on-line work of the Doppler operator and thus assuring a more accurate measurement 3) the use of small sample volume minimizes the vessel wall artifacts [8-12]....
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References
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TL;DR: An alternative approach, based on graphical techniques and simple calculations, is described, together with the relation between this analysis and the assessment of repeatability.
43,884 citations
TL;DR: The experiments of McDonald and his co-workers have shown that in the larger arteries of the rabbit and the dog there is a reversal of the flow, and the simple mathematical treatment has strong similarities with the theory of the distribution of alternating current in a conductor of finite size.
Abstract: The experiments of McDonald and his co-workers (McDonald, 1952, 1955; Helps & McDonald, 1953) have shown that in the larger arteries of the rabbit and the dog there is a reversal of the flow. Measurements of the pressure gradient (Helps & McDonald, 1953) showed a phase-lag between pressure gradient and flow somewhat analogous with the phase-lag between voltage and current in a conductor carrying alternating current, and the simple mathematical treatment given below has strong similarities with the theory of the distribution of alternating current in a conductor of finite size.
1,675 citations
"Womersley Number-Based Estimates of..." refers background or methods in this paper
...The Womersley theory has been developed by Womersley [56] for a singlesinusoidal pressure drop applied over a cylindrical rigid-wall geometry for the calculation of velocity, rate of flow, and viscous drag in arteries when the pressure gradient is known....
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...Other approaches are based on the application of the Womersley theory (that is on the analytical expression of the flow rate in terms of velocity, see [4], [26], and [56])....
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...It is well known that the velocity profile depends on the Womersley number (see [56] for a theoretical discussion), which increases when the heart rate increases....
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...Nevertheless, in the work of Womersley [56], it is also clearly mentioned that a unique and representative value of W can be calculated for a specific arterial district (in the paper, where the case of the human femoral artery is discussed), using the heart rate as reference frequency....
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...diameter in vascular districts (see [33], [48], and [56])....
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TL;DR: The Doppler guide wire measures phasic flow velocity patterns and linearly tracks changes in flow rate in small, straight coronary arteries and should facilitate measurement ofphasic coronary flow velocity during coronary angiography and angioplasty.
Abstract: BACKGROUNDAn improved intravascular ultrasonic Doppler device could aid the clinical assessment of coronary hemodynamics. We evaluated a new device consisting of a 12-MHz piezoelectric transducer integrated onto the tip of a 0.018-in. flexible, steerable angioplasty guide wire.METHODS AND RESULTSDoppler spectra were recorded in model tubes with pulsatile blood flow and in-line electromagnetic flowmeter. In four straight tubes (i.d., 0.79-4.76 mm), the time average of spectral peak velocity (APV) was linearly related to blood flow (QEMF) (r2 greater than or equal to 0.98 for each tube). A Doppler-derived quantitative flow estimate (QD) was calculated as the product of vessel cross-sectional area and mean velocity, with mean velocity estimated as 0.5 x APV. The slope of QD versus QEMF for the four tubes was near unity. APV was less accurate in a 7.94-mm straight tube and in tortuous segments. In four dogs, the left circumflex coronary artery (LCx) was perfused from the femoral artery via a cannula with in-l...
1,024 citations
TL;DR: A flow sensitizing zeugmatographic phase-modulation interlace for NMR-imaging which is exactly analogous to Lauterbur's spatial-location-sensitizing magnetic field gradients and enables up to 6-D imaging of the joint density of spins delta (r,v).
771 citations
"Womersley Number-Based Estimates of..." refers methods in this paper
...Second, it allows to perform simultaneously both geometrical and fluid-dynamics data measurements [2], [7]–[9], [12], [16], [30]–[32], [34], [35], [50], without the usage of any endovascular device or contrast agent that can potentially alter the blood flow....
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Book•
15 Jan 1994
TL;DR: Part 1 Physical principles, instrumentation, and routine examination: physical principles of ultrasound cross-sectional scanning, technical principles and instrumentation digital image processingCross-sectional echocardiographic examination standard plane positions - standard imaging planes three-dimensional reconstruction of echOCardiographic images principles of Doppler flow measurement, principles of flow, and signal processing.
Abstract: Part 1 Physical principles, instrumentation, and routine examination: physical principles of ultrasound cross-sectional scanning - technical principles and instrumentation digital image processing cross-sectional echocardiographic examination standard plane positions - standard imaging planes three-dimensional reconstruction of echocardiographic images principles of Doppler flow measurement Doppler instrumentation principles of flow Doppler signal processing principles of colour flow mapping fluid dynamics of regurgitant jets and their imaging by colour Doppler the routine Doppler examination M-mode echocardiography - principles and examination techniques miscellaneous echocardiographic techniques I - contrast echocardiography miscellaneous echocardiographic techniques II - transesophageal, epicardial, intraoperative, and intravascular/intracardiac echocardiography, and sonomicrometry left ventricular inflow tract I - the mitral valve left ventricular inflow tract II - the left atrium, pulmonary veins and coronary sinus left ventricular outflow tract - the aortic valve, aorta and subvalvular outflow tract left ventricle I - general considerations, assessment of chamber size and function left ventricle II - segmental dysfunction - concept and techniques of measurement left ventricle III - coronary artery disease - clinical manifestations and complications left ventricle IV - assessment of myocardial perfusion with contrast two-dimensional echocardiography left ventricle V - diastolic function - its principles and evaluation echocardiographic assessment of the cardiomyopathies right ventricular inflow tract right ventricular outflow tract right ventricle interatrial and interventricular septa Doppler estimation of volumetric flow complex congenital heart disease I - a diagnostic approach complex congenital heart disease II - a pathologic approach foetal echocardiography coronary arteries pericardial diseases cardiac tumours and masses echocardiographic findings in infective endocarditis echo-Doppler assessment of prosthetic heart valves echocardiography in cardiac transplantation the echocardiogram in disorders of cardiac rhythm and conduction tissue characterizations Appendix A: normal cross-sectional echocardiographic measurements in adults Appendix B: application of Fourier analysis in echocardiography
674 citations
"Womersley Number-Based Estimates of..." refers background in this paper
...a selected point of the section is available [55]....
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