Determining cardiac velocity fields and intraventricular pressure distribution from a sequence of ultrafast CT cardiac images

On the numerical treatment of viscous and convective effects in relative pressure reconstruction methods

The mechanism of many cardiovascular diseases can be understood by studying the pressure distribution in blood vessels. Direct pressure measurements, however, require invasive probing and provide only single-point data. Alternatively, relative pressure fields can be reconstructed from imaging-based velocity measurements by considering viscous and inertial forces. Both contributions can be potential troublemakers in pressure reconstruction: the former due to its higher-order derivatives, and the latter because of the quadratic nonlinearity in the convective acceleration. Viscous and convective terms can be treated in various forms, which, although equivalent for ideal measurements, can perform differently in practice. In fact, multiple versions are often used in literature, with no apparent consensus on the more suitable variants. In this context, the present work investigates the performance of different versions of relative pressure estimators. For viscous effects, in particular, two new modified estimators are presented to circumvent second-order differentiation without requiring surface integrals. In-silico and in-vitro data in the typical regime of cerebrovascular flows are considered, allowing a systematic noise sensitivity study. Convective terms are shown to be the main source of error, even for flows with pronounced viscous component. Moreover, the conservation (often integrated) form of convection exhibits higher noise sensitivity than the standard convective description, in all three families of estimators considered here. For the classical pressure Poisson estimator, the present modified version of the viscous term tends to yield better accuracy than the (recently introduced) integrated form, although this effect is in most cases negligible when compared to convection-related errors.

Hemodynamic forces using four-dimensional flow MRI: an independent biomarker of cardiac function in heart failure with left ventricular dyssynchrony?

Patients with heart failure with left ventricular (LV) dyssynchrony often do not respond to cardiac resynchronization therapy (CRT), indicating that the pathophysiology is insufficiently understood. Intracardiac hemodynamic forces computed from four-dimensional (4-D) flow MRI have been proposed as a new measure of cardiac function. We therefore aimed to investigate how hemodynamic forces are altered in LV dyssynchrony. Thirty-one patients with heart failure and LV dyssynchrony and 39 control subjects underwent cardiac MRI with the acquisition of 4-D flow. Hemodynamic forces were computed using Navier-Stokes equations and integrated over the manually delineated LV volume. The ratio between transverse (lateral-septal and inferior-anterior) and longitudinal (apical-basal) forces was calculated for systole and diastole separately and compared with QRS duration, aortic valve opening delay, global longitudinal strain, and ejection fraction (EF). Patients exhibited hemodynamic force patterns that were significantly altered compared with control subjects, including loss of longitudinal forces in diastole (force ratio, control subjects vs. patients: 0.32 vs. 0.90, P < 0.0001) and increased transverse force magnitudes. The systolic force ratio was correlated with global longitudinal strain and EF ( P < 0.01). The diastolic force ratio separated patients from control subjects (area under the curve: 0.98, P < 0.0001) but was not correlated to other dyssynchrony measures ( P > 0.05 for all). Hemodynamic forces by 4-D flow represent a new approach to the quantification of LV dyssynchrony. Diastolic force patterns separate healthy from diseased ventricles. Different force patterns in patients indicate the possible use of force analysis for risk stratification and CRT implantation guidance.

NEW & NOTEWORTHY In this report, we demonstrate that patients with heart failure with left ventricular dyssynchrony exhibit significantly altered hemodynamic forces compared with normal. Force patterns in patients mechanistically reflect left ventricular dysfunction on the organ level, largely independent of traditional dyssynchrony measures. Force analysis may help clinical decision making and could potentially be used to improve therapy outcomes.

Noninvasive Estimation of Pressure Changes Using 2-D Vector Velocity Ultrasound: An Experimental Study With In Vivo Examples

A noninvasive method for estimating intravascular pressure changes using 2-D vector velocity is presented. The method was first validated on computational fluid dynamic (CFD) data and with catheter measurements on phantoms. Hereafter, the method was tested in vivo at the carotid bifurcation and at the aortic valve of two healthy volunteers. Ultrasound measurements were performed using the experimental scanner SARUS, in combination with an 8 MHz linear array transducer for experimental scans and a carotid scan, whereas a 3.5-MHz phased array probe was employed for a scan of an aortic valve. Measured 2-D fields of angle-independent vector velocities were obtained using synthetic aperture imaging. Pressure drops from simulated steady flow through six vessel geometries spanning different degrees of diameter narrowing, running from 20%-70%, showed relative biases from 0.35% to 12.06%, depending on the degree of constriction. Phantom measurements were performed on a vessel with the same geometry as the 70% constricted CFD model. The derived pressure drops were compared to pressure drops measured by a clinically used 4F catheter and to a finite-element model. The proposed method showed peak systolic pressure drops of -3 kPa ± 57 Pa, while the catheter and the simulation model showed -5.4 kPa ± 52 Pa and -2.9 kPa, respectively. An in vivo acquisition of 10 s was made at the carotid bifurcation. This produced eight cardiac cycles from where pressure gradients of -227 ± 15 Pa were found. Finally, the aortic valve measurement showed a peak pressure drop of -2.1 kPa over one cardiac cycle. In conclusion, pressure gradients from convective flow changes are detectable using 2-D vector velocity ultrasound.

Left and right ventricular hemodynamic forces in healthy volunteers and elite athletes assessed with 4D flow magnetic resonance imaging

Intracardiac blood flow is driven by hemodynamic forces that are exchanged between the blood and myocardium. Previous studies have been limited to 2D measurements or investigated only left ventricular (LV) forces. Right ventricular (RV) forces and their mechanistic contribution to asymmetric redirection of flow in the RV have not been measured. We therefore aimed to quantify 3D hemodynamic forces in both ventricles in a cohort of healthy subjects, using magnetic resonance imaging 4D flow measurements. Twenty five controls, 14 elite endurance athletes, and 2 patients with LV dyssynchrony were included. 4D flow data were used as input for the Navier-Stokes equations to compute hemodynamic forces over the entire cardiac cycle. Hemodynamic forces were found in a qualitatively consistent pattern in all healthy subjects, with variations in amplitude. LV forces were mainly aligned along the apical-basal longitudinal axis, with an additional component aimed toward the aortic valve during systole. Conversely, RV forces were found in both longitudinal and short-axis planes, with a systolic force component driving a slingshot-like acceleration that explains the mechanism behind the redirection of blood flow toward the pulmonary valve. No differences were found between controls and athletes when indexing forces to ventricular volumes, indicating that cardiac force expenditures are tuned to accelerate blood similarly in small and large hearts. Patients’ forces differed from controls in both timing and amplitude. Normal cardiac pumping is associated with specific force patterns for both ventricles, and deviation from these forces may be a sensitive marker of ventricular dysfunction. Reference values are provided for future studies.

NEW & NOTEWORTHY Biventricular hemodynamic forces were quantified for the first time in healthy controls and elite athletes (n = 39). Hemodynamic forces constitute a slingshot-like mechanism in the right ventricle, redirecting blood flow toward the pulmonary circulation. Force patterns were similar between healthy subjects and athletes, indicating potential utility as a cardiac function biomarker.

Aortic relative pressure components derived from four-dimensional flow cardiovascular magnetic resonance

Purpose

To describe the assessment of the spatiotemporal distribution of relative aortic pressure quantifying the magnitude of its three major components.

Methods

Nine healthy volunteers and three patients with aortic disease (bicuspid aortic valve, dissection, and Marfan syndrome) underwent 4D-flow CMR. Spatiotemporal pressure maps were computed from the CMR flow fields solving the pressure Poisson equation. The individual components of pressure were separated into time-varying inertial (“transient”), spatially varying inertial (“convective”), and viscous components.

Results

Relative aortic pressure is primarily caused by transient effects followed by the convective and small viscous contributions (64.5, 13.6, and 0.3 mmHg/m, respectively, in healthy subjects), although regional analysis revealed prevalent convective effects in specific contexts, e.g., Sinus of Valsalva and aortic arch at instants of peak velocity. Patients showed differences in peak transient values and duration, and localized abrupt convective changes explained by abnormalities in aortic geometry, including the presence of an aneurysm, a pseudo-coarctation, the inlet of a dissection, or by complex flow patterns.

Conclusion

The evaluation of the three components of relative pressure enables the quantification of mechanistic information for understanding and stratifying aortic disease, with potential future implications for guiding therapy. Magn Reson Med 72:1162–1169, 2014. © 2013 The Authors. Magnetic Resonance in Medicine published by Wiley Periodicals, Inc. on behalf of International Society for Magnetic Resonance in Medicine. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Noninvasive 3D pressure calculation from PC-MRI via non-iterative harmonics-based orthogonal projection: constant flow experiment

Use of phase-contrast (PC) MRI in assessment of hemodynamics has significant clinical importance. In this paper we develop a novel approach to determination of hemodynamic pressures. 3D gradients of pressure obtained from Navier-Stokes equation are expanded into a series of orthogonal basis functions, and are subsequently projected onto an integrable subspace. Before the projection step however, a scheme is devised to eliminate the discontinuity at the vessel and image boundaries. In terms of the computation time, the proposed approach significantly improves on previous iterative methods for pressure calculations. The method has been validated using computational fluid dynamic simulations and in-vitro MRI studies of stenotic flows.

Understanding the need of ventricular pressure for the estimation of diastolic biomarkers

The diastolic function (i.e., blood filling) of the left ventricle (LV) is determined by its capacity for relaxation, or the decay in residual active tension (AT) generated during systole, and its constitutive material properties, or myocardial stiffness. The clinical determination of these two factors (diastolic residual AT and stiffness) is thus essential for assessing LV diastolic function. To quantify these two factors, in our previous work, a novel model-based parameter estimation approach was proposed and successfully applied to multiple cases using clinically acquired motion and invasively measured ventricular pressure data. However, the need to invasively acquire LV pressure limits the wide application of this approach. In this study, we address this issue by analyzing the feasibility of using two kinds of non-invasively available pressure measurements for the purpose of inverse mechanical parameter estimation. The prescription of pressure based on a generic pressure-volume (P-V) relationship reported in literature is first evaluated in a set of 18 clinical cases (10 healthy and 8 diseased), finding reasonable results for stiffness but not for residual active tension. We then investigate the use of non-invasive pressure measures, now available through imaging techniques and limited by unknown or biased offset values. Specifically, three sets of physiologically realistic synthetic data with three levels of diastolic residual active tension (i.e., impaired relaxation capability) are designed to quantify the percentage error in the parameter estimation against the possible pressure offsets within the physiological limits. Maximum errors are quantified as 11 % for the magnitude of stiffness and 22 % for AT, with averaged 0.17 kPa error in pressure measurement offset using the state-of-the-art non-invasive pressure estimation method. The main cause for these errors is the limited temporal resolution of clinical imaging data currently available. These results demonstrate the potential feasibility of the estimation diastolic biomarkers with non-invasive assessment of pressure through medical imaging data.

A finite-element approach to the direct computation of relative cardiovascular pressure from time-resolved MR velocity data

The evaluation of cardiovascular velocities, their changes through the cardiac cycle and the consequent pressure gradients has the capacity to improve understanding of subject-specific blood flow in relation to adjacent soft tissue movements. Magnetic resonance time-resolved 3D phase contrast velocity acquisitions (4D flow) represent an emerging technology capable of measuring the cyclic changes of large scale, multi-directional, subject-specific blood flow. A subsequent evaluation of pressure differences in enclosed vascular compartments is a further step which is currently not directly available from such data. The focus of this work is to address this deficiency through the development of a novel simulation workflow for the direct computation of relative cardiovascular pressure fields. Input information is provided by enhanced 4D flow data and derived MR domain masking. The underlying methodology shows numerical advantages in terms of robustness, global domain composition, the isolation of local fluid compartments and a treatment of boundary conditions. This approach is demonstrated across a range of validation examples which are compared with analytic solutions. Four subject-specific test cases are subsequently run, showing good agreement with previously published calculations of intra-vascular pressure differences. The computational engine presented in this work contributes to non-invasive access to relative pressure fields, incorporates the effects of both blood flow acceleration and viscous dissipation, and enables enhanced evaluation of cardiovascular blood flow.

Polynomial regularization for robust MRI-based estimation of blood flow velocities and pressure gradients

In cardiovascular diagnostics, phase-contrast MRI is a valuable technique for measuring blood flow velocities and computing blood pressure values. Unfortunately, both velocity and pressure data typically suffer from the strong image noise of velocity-encoded MRI. In the past, separate approaches of regularization with physical a-priori knowledge and data representation with continuous functions have been proposed to overcome these drawbacks. In this article, we investigate polynomial regularization as an exemplary specification of combining these two techniques. We perform time-resolved three-dimensional velocity measurements and pressure gradient computations on MRI acquisitions of steady flow in a physical phantom. Results based on the higher quality temporal mean data are used as a reference. Thereby, we investigate the performance of our approach of polynomial regularization, which reduces the root mean squared errors to the reference data by 45% for velocities and 60% for pressure gradients.

Haptic rendering of dynamic volumetric data

With current methods for volume haptics in scientific visualization, features in time-varying data can freely move straight through the haptic probe without generating any haptic feedback the algorithms are simply not designed to handle variation with time but consider only the instantaneous configuration when the haptic feedback is calculated. This article introduces haptic rendering of dynamic volumetric data to provide a means for haptic exploration of dynamic behaviour in volumetric data. We show how haptic feedback can be produced that is consistent with volumetric data moving within the virtual environment and with data that, in itself, evolves over time. Haptic interaction with time-varying data is demonstrated by allowing palpation of a CT sequence of a beating human heart.

Wall shear stress: theoretical considerations and methods of measurement

In arterial blood flow, the wall shear stress expresses the force per unit area exerted by the wall on the fluid in a direction on the local tangent plane. There is substantial evidence that the wall shear stress induced by the pulsatile blood flow in the arterial system affects the atherogenic process. It is now widely accepted that the vessel segments that appear to be at the highest risk for development of atherosclerosis are those with low wall shear stress or oscillating wall shear stress. The purpose of this article is to define wall shear stress, to introduce relevant concepts of fluid mechanics to nonexperts, and to critically review the various methods that have been used for the assessment of wall shear stress in animal and human blood circulation, paying special attention to the case of coronary arteries.

Integrable pressure gradients via harmonics-based orthogonal projection

In the past, several methods based on iterative solution of pressure-Poisson equation have been developed for measurement of pressure from phase-contrast magnetic resonance (PC-MR) data. In this paper, a non-iterative harmonics-based orthogonal projection method is discussed which can keep the pressures measured based on the Navier-Stokes equation independent of the path of integration. The gradient of pressure calculated with Navier-Stokes equation is expanded with a series of orthogonal basis functions, and is subsequently projected onto an integrable subspace. Before the projection step however, a scheme is devised to eliminate the discontinuity at the vessel boundaries. The approach was applied to velocities obtained from computational fluid dynamics (CFD) simulations of stenotic flow and compared with pressures independently obtained by CFD. Additionally, MR velocity data measured in in-vitro phantom models with different degree of stenoses and different flow rates were used to test the algorithm and results were compared with CFD simulations. The pressure results obtained from the new method were also compared with pressures calculated by an iterative solution to the pressure-Poisson equation. Experiments have shown that the proposed approach is faster and is less sensitive to noise.

BEM performance in calculation of pressure distribution in spline based segmented medical images

Conventional methods for non-invasively estimation of pressure distribution in the cardiovascular flow domain use the differential form of governing equations. This study evaluates the advantages of using integral form of governing equations. The concepts provided with the Boundary Element Method (BEM) together with the boundary based image segmentation tools are used to develop a fast calculation algorithm. Boundary based segmentation provides facility for BEM with domain pixel extraction, boundary meshing, wall normal vector calculation and accurate calculation of boundary element length. The integral form of governing equation reviewed in detail. Both the differential and integral based formulations are evaluated using mathematical test flow image.

Calculation of left ventricle relative pressure distribution in MRI using acceleration data

Measurements of pressure variations within the cardiac chambers could provide important information for clinical assessments of cardiovascular function. In this work an MRI method for evaluating spatial distributions of intracardiac relative pressure is presented. We first calculated pressure gradients from MR maps of blood acceleration by applying the NS equation. We then used an original algorithm to compute pressure distribution in a region of interest (ROI) by minimizing the pressure gradient curl so that the result in a given pixel is independent of the integration path. The method was assessed in five healthy volunteers by means of MR 2D maps of the blood acceleration in the left ventricle (LV) during ejection and filling phases. The pressure variations calculated from acceleration mapping fit the known physiological variations better than those based on velocity maps acquired in the same volunteers. Furthermore, the optimization algorithm presented here produced the same results as iterative algorithms proposed by other authors, but in much less time and without requiring adjustable parameters or boundary conditions.

Factors affecting the accuracy of pressure measurements in vascular stenoses from phase-contrast MRI

In this work the effects of noise, resolution, and velocity (flow) on the measurement of intravascular pressure from phase-contrast (PC) MRI are discussed. To elucidate these effects, we employed an axisymmetric geometry that enabled us to calculate pressures in <2 min on a Sun Ultra SPARC 10 workstation. To determine the effects of vascular stenoses, we fabricated several stenotic phantom geometries (with 50%, 75%, and 90% area stenoses), and performed both MRI and computational fluid dynamics (CFD) simulations for various flow rates for these phantom geometries. Noise with Gaussian statistics was added to the velocity field obtained from the CFD simulations. The pressure maps obtained directly from CFD simulations for our phantom geometries were compared with pressure maps derived by our algorithm when 1) the input was noise-corrupted velocity data from CFD, and 2) the input was PC-MRI data collected from the phantoms. The quantitative effects of noise, resolution, and flow rate on the accuracy of pressure measurements were determined. We found that for flow rates below the Reynolds number for turbulent flow, resolution is a more significant determinant of accuracy than SNR. Furthermore, if other parameters remain constant, increased flow rates may result in decreased accuracy.

Noninvasive measurement of time-varying three-dimensional relative pressure fields within the human heart

Understanding cardiac blood flow patterns is important in the assessment of cardiovascular function. Three-dimensional flow and relative pressure fields within the human left ventricle are demonstrated by combining velocity measurements with computational fluid mechanics methods. The velocity field throughout the left atrium and ventricle of a normal human heart is measured using time-resolved three-dimensional phase-contrast MRI. Subsequently, the time-resolved three-dimensional relative pressure is calculated from this velocity field using the pressure Poisson equation. Noninvasive simultaneous assessment of cardiac pressure and flow phenomena is an important new tool for studying cardiac fluid dynamics.

Estimation of relative cardiovascular pressures using time-resolved three-dimensional phase contrast MRI

Accurate, easy-to-use, noninvasive cardiovascular pressure registration would be an important addition to the diagnostic armamentarium for assessment of cardiac function. A novel noninvasive and three-dimensional (3D) technique for estimation of relative cardiovascular pressures is presented. The relative pressure is calculated using the Navier-Stokes equations along user-defined lines placed within a time-resolved 3D phase contrast MRI dataset. The lines may be either straight or curved to follow an actual streamline. The technique is validated in an in vitro model and tested on in vivo cases of normal and abnormal transmitral pressure differences and intraaortic flow. The method supplements an intuitive visualization technique for cardiovascular flow, 3D particle trace visualization, with a quantifiable diagnostic parameter estimated from the same dataset.

Combined MR imaging and CFD simulation of flow in the human descending aorta

A combined MR and computational fluid dynamics (CFD) study is made of flow in the upper descending thoracic aorta. The aim was to investigate further the potential of CFD simulations linked to in vivo MRI scans. The three-dimensional (3D) geometrical images of the aorta and the 3D time-resolved velocity images at the entry to the domain studied were used as boundary conditions for the CFD simulations of the flow. Despite some measurement uncertainties, comparisons between simulated and measured flow structures at the exit from the domain demonstrated encouraging levels of agreement. Moreover, the CFD simulation allowed the flow structure throughout the domain to be examined in more detail, in particular the flow separation region in the distal aortic arch and its influence on the downstream flow during late systole. Additional information such as relative pressure and wall shear stress, which could not be measured via MRI, were also extracted from the simulation. The results have encouraged further applications of the methods described. J. Magn. Reson. Imaging 2001;13:699-713.

Three-dimensional modeling for functional analysis of cardiac images: a review

Three-dimensional (3-D) imaging of the heart is a rapidly developing area of research in medical imaging. Advances in hardware and methods for fast spatio-temporal cardiac imaging are extending the frontiers of clinical diagnosis and research on cardiovascular diseases. In the last few years, many approaches have been proposed to analyze images and extract parameters of cardiac shape and function from a variety of cardiac imaging modalities. In particular, techniques based on spatio-temporal geometric models have received considerable attention. This paper surveys the literature of two decades of research on cardiac modeling. The contribution of the paper is three-fold: 1) to serve as a tutorial of the field for both clinicians and technologists, 2) to provide an extensive account of modeling techniques in a comprehensive and systematic manner, and 3) to critically review these approaches in terms of their performance and degree of clinical evaluation with respect to the final goal of cardiac functional analysis. From this review it is concluded that whereas 3-D model-based approaches have the capability to improve the diagnostic value of cardiac images, issues as robustness, 3-D interaction, computational complexity and clinical validation still require significant attention.

X-ray videodensitometric methods for blood flow and velocity measurement: a critical review of literature

Blood flow rate and velocity are important parameters for the study of vascular systems, and for the diagnosis, monitoring and evaluation of treatment of cerebro- and cardiovascular disease. For rapid imaging of cerebral and cardiac blood vessels, digital x-ray subtraction angiography has numerous advantages over other modalities. Roentgen-videodensitometric techniques measure blood flow and velocity from changes of contrast material density in x-ray angiograms. Many roentgen-videodensitometric flow measurement methods can also be applied to CT, MR and rotational angiography images. Hence, roentgen-videodensitometric blood flow and velocity measurement from digital x-ray angiograms represents an important research topic. This work contains a critical review and bibliography surveying current and old developments in the field. We present an extensive survey of English-language publications on the subject and a classification of published algorithms. We also present descriptions and critical reviews of these algorithms. The algorithms are reviewed with requirements imposed by neuro- and cardiovascular clinical environments in mind.

Estimation of pressure gradients in pulsatile flow from magnetic resonance acceleration measurements

A method for estimating pressure gradients from MR images is demonstrated. Making the usual assumption that the flowing medium is a Newtonian fluid, and with appropriate boundary conditions, the inertial forces (or acceleration components of the flow) are proportional to the pressure gradients. The technique shown here is based on an evaluation of the inertial forces from Fourier acceleration encoding. This method provides a direct measurement of the total acceleration defined as the sum of the velocity derivative vs. time and the convective acceleration. The technique was experimentally validated by comparing MR and manometer pressure gradient measurements obtained in a pulsatile flow phantom. The results indicate that the MR determination of pressure gradients from an acceleration measurement is feasible with a good correlation with the true measurements (r = 0.97). The feasibility of the method is demonstrated in the aorta of a normal volunteer. Magn Reson Med 44:66-72, 2000.

Regularization of blood motion fields by modified Navier-Stokes equations

A technique for the regularization of incompressible fluid motion fields based on the use of modified Navier-Stokes equations is presented. It is shown that the technique belongs to the class of Tikhonov-type regularization methods. The technique was applied to an analytically known fluid-dynamic problem (Couette flow). Noisy and scattered versions of the analytical velocity field were generated, and the accuracy in reconstructing the analytical field was evaluated. It was found that accuracy depends on the value of a regularization parameter, and its optimal value depends on the entity of noise and scattering. When the optimal value is selected, the accuracy of the regularization technique is excellent, even for consistent noise and scattering levels. The technique was finally applied to echocardiographic data in order to estimate the blood velocity field within the left ventricle.

MRI monitoring of laser ablation using optical flow

The optical flow method is used for visualizing and quantifying the dynamics of tissue changes observed by MRI during thermal ablations. An approach was implemented for parallel two-dimensional optical flow calculations including the replacement of spurious velocities. Velocity magnitude results were found to be accurate in low-noise cases in tests using series of synthetic images. Optical flow results are presented from thermal ablation experiments utilizing a homogeneous polyacrylamide gel phantom and heterogeneous rabbit liver tissue in vivo, exhibiting heating and cooling with the accompanying quantitative characterization of the dilation and contraction of the thermally affected region. Results demonstrate that optical flow is capable of noninvasive real-time monitoring and control of interstitial laser therapy (ILT).

Definition of a four-dimensional continuous planispheric transformation for the tracking and the analysis of left-ventricle motion

Cardiologists assume that analysis of the motion of the heart (especially the left ventricle) can provide useful information about the health of the myocardium. A 4-D polar transformation is defined to describe the left-ventricle (LV) motion and a method is presented to estimate it from sequences of 3-D images. The transformation is defined in 3-D planispheric coordinates (3PC) by a small number of parameters involved in a set of simple linear equations. It is continuous and regular in time and space, and periodicity in time can be imposed. The local motion can be easily decomposed into a few canonical motions (radial motion, rotation around the long-axis, elevation). To recover the motion from original data, the 4-D polar transformation is calculated using an adaptation of the iterative closest-point algorithm. We present the mathematical framework and a demonstration of its feasability on a series of gated SPECT sequences.

In vivo validation of MR pulse pressure measurement in an aortic flow model: preliminary results

MR imaging experiments were conducted to investigate the feasibility of estimating vascular pulse pressure waveforms from measurements of blood flow rates and vessel cross-sectional area. Blood flow waveforms were measured in the aorta's of three 25-30-kg pigs at multiple imaging sections using phase-contrast velocity imaging. Estimates of pulse pressure were derived from these data by evaluating a model characterizing the relationship between pressure, flow, and the cross-sectional area of a vessel segment. Comparisons between the MR-derived estimates of pressure and those obtained from a micromanometer pressure catheter indicate that accurate measurements (mean error +/- SD = 8.2 +/- 3.4, n = 6) can be obtained using conventional velocity imaging techniques. Optimization of the method will require the application of rapid imaging techniques and the development of strategies for obtaining a more localized measurement. With these improvements, our results suggest that MR-based measurement of pulse pressure and related elastic parameters is feasible.

Estimation of three-dimensional cardiac velocity fields: assessment of a differential method and application to three-dimensional CT data

We have investigated an optical flow method for the estimation of the three-dimensional endocardial wall motion from high-resolution X-ray CT data. This method was originally proposed by Song and Leahy. It is based on the optical flow, the divergence-free and the smoothness constraints. Due to the characteristics of the imaging modality, we studied the restriction of this approach to the boundary of the left ventricular (LV) cavity. The behaviour of the method is quantified through simulations approximating the overall motion of the LV cavity through an affine transform involving a dilation and a rotation. The method implies the choice of three parameters weighting the constraints. The results show a weak dependence of the velocity field on the weighting of the optical flow constraint. The accuracy of the method relies more heavily on the relative weighting of the smoothness and divergence-free constraints. In our experiments, the best results were obtained for a largely predominant divergence-free constraint. The results also show that the accuracy of the method is reasonable for low values of the rotation angle (minimum mean error of 1.1 voxel for 5 degrees). This is compatible with values reported in other studies for the overall rotation of the LV. We provide a qualitative description of the results obtained in vivo on a canine heart by visualizing the distribution of the estimated velocity vector magnitudes over the endocardial surface. These results (evolution of the field over time, maximum velocities) are in agreement with the known physiological behaviour of the heart.

Computation of flow pressure fields from magnetic resonance velocity mapping

Magnetic resonance phase velocity mapping has unrivalled capacities for acquiring in vivo multi-directional blood flow information. In this study, the authors set out to derive both spatial and temporal components of acceleration, and hence differences of pressure in a flow field using cine magnetic resonance velocity data. An efficient numerical algorithm based on the Navier-Stokes equations for incompressible Newtonian fluid was used. The computational approach was validated with in vitro flow phantoms. This work aims to contribute to a better understanding of cardiovascular dynamics and to serve as a basis for investigating pulsatile pressure/flow relationships associated with normal and impaired cardiovascular function.

Tracking myocardial deformation using phase contrast MR velocity fields: a stochastic approach

The authors propose a new approach for tracking the deformation of the left-ventricular (LV) myocardium from two-dimensional (2-D) magnetic resonance (MR) phase contrast velocity fields. The use of phase contrast MR velocity data in cardiac motion problems has been introduced by others (N.J. Pelc et al., 1991) and shown to be potentially useful for tracking discrete tissue elements, and therefore, characterizing LV motion. However, the authors show here that these velocity data: 1) are extremely noisy near the LV borders; and 2) cannot alone be used to estimate the motion and the deformation of the entire myocardium due to noise in the velocity fields. In this new approach, the authors use the natural spatial constraints of the endocardial and epicardial contours, detected semiautomatically in each image frame, to help remove noisy velocity vectors at the LV contours. The information from both the boundaries and the phase contrast velocity data is then integrated into a deforming mesh that is placed over the myocardium at one time frame and then tracked over the entire cardiac cycle. The deformation is guided by a Kalman filter that provides a compromise between 1) believing the dense field velocity and the contour data when it is crisp and coherent in a local spatial and temporal sense and 2) employing a temporally smooth cyclic model of cardiac motion when contour and velocity data are not trustworthy. The Kalman filter is particularly well suited to this task as it produces an optimal estimate of the left ventricle's kinematics (in the sense that the error is statistically minimized) given incomplete and noise corrupted data, and given a basic dynamical model of the left ventricle. The method has been evaluated with simulated data; the average error between tracked nodes and theoretical position was 1.8% of the total path length. The algorithm has also been evaluated with phantom data; the average error was 4.4% of the total path length. The authors show that in their initial tests with phantoms that the new approach shows small, but concrete improvements over previous techniques that used primarily phase contrast velocity data alone. They feel that these improvements will be amplified greatly as they move to direct comparisons in in vivo and three-dimensional (3-D) datasets.

Phase unwrapping of MR phase images using Poisson equation

The authors have developed a technique based on a solution of the Poisson equation to unwrap the phase in magnetic resonance (MR) phase images. The method is based on the assumption that the magnitude of the inter-pixel phase change is less than pi per pixel. Therefore, the authors obtain an estimate of the phase gradient by "wrapping" the gradient of the original phase image. The problem is then to obtain the absolute phase given the estimate of the phase gradient. The least-squares (LS) solution to this problem is shown to be a solution of the Poisson equation allowing the use of fast Poisson solvers. The absolute phase is then obtained by mapping the LS phase to the nearest multiple of 2 K from the measured phase. The proposed technique is evaluated using MR phase images and is proven to be robust in the presence of noise. An application of the proposed method to the 3-point Dixon technique for water and fat separation is demonstrated.

MR measurements of pulsatile pressure gradients

A magnetic resonance (MR) imaging method for evaluating pulsatile pressure gradients in laminar blood flow is presented. The technique is based on an evaluation of fluid shear and inertial forces from cardiac-gated phase-contrast velocity measurements. The technique was experimentally validated by comparing MR and manometer pressure gradient measurements performed in a pulsatile flow phantom. Analyses of random noise propagation and sampling error were performed to determine the precision and accuracy of the method. The results indicate that a precision of 0.01-0.03 mmHg/cm and an accuracy of better than 8% can be achieved by using standard clinical pulse sequences in tubes exceeding 6 mm in diameter. The authors conclude that MR measurement of pressure gradients is feasible and that additional hemodynamic information may be derived from conventional phase-contrast imaging studies.