Sources of error in maximum velocity estimation using linear phased-array Doppler systems with steady flow.

Using linear-array Doppler ultrasound (US) transducers, the measured maximum velocity may be in error and lead to incorrect clinical diagnosis. This study investigates the existence and cause of maximum velocity estimation errors for steady flow of a blood-mimicking fluid in a tissue-mimicking phantom. A specially designed system was used that enabled fine control of flow rate, transducer positioning and transducer angle relative to the flow phantom. Doppler machine settings (transducer aperture size, focal depth, beam-steering, gain) were varied to investigate a wide range of clinical applications. To estimate the maximum velocity, a new signal-to-noise ratio (SNR) independent method was developed to calculate the maximum frequency from an ensemble averaged Doppler power spectrum. This enabled the impact of each factor on the total Doppler error to be determined. When using the new maximum frequency estimator, it was found that the effect of transducer focal depth, intratransducer, intramachine, intermachine (that was tested) and beam-steering did not significantly contribute to maximum velocity estimation errors. Instead, it was the dependence of the maximum velocity on the Doppler angle that made, by far, the greatest contribution to the estimation error. Because our maximum frequency estimator took into account the effect of intrinsic spectral broadening, the degree of overestimation error was not as great as that previously published. Thus, the effects of Doppler angle and intrinsic spectral broadening are the chief sources of Doppler US error and should be the focus of future efforts to improve the accuracy.

Influence of boundary conditions on a one-dimensional ultrasound backscattering model of blood.

A simulation model of one-dimensional (1-D) ultrasound (US) propagation in blood was used to study the relation between the backscattering coefficient and hematocrit. In this model, an ultrasonic plane wave was propagated in plasma normal to randomly placed slabs of constant thickness whose acoustical properties are the same as red blood cells, and the corresponding intensity reflection coefficient was calculated. The simulation results were compared to the 1-D Percus-Yevick (P-Y) theory as presented in the literature. Previous investigators have reported a close agreement over a limited range of simulation parameters between their results and the P-Y theory. However, a more careful investigation using a wider range of parameters has revealed major discrepancies. It is shown that these arise from an inappropriate choice of boundary conditions. By averaging the material properties beyond the boundaries of the simulation, as suggested by earlier theoretical work, the results are now in excellent agreement with the P-Y theory over a wide range of simulation parameters.

New X-wave solutions of free-space scalar wave equation and their finite size realization.

Based on the method proposed by Donnelly and Ziolkowski [1], [2], a new general solution has been obtained for the isotropic/homogeneous scalar wave equation in cylindrical coordinates. It is shown that well-known limited diffraction beams such as Durnin's Bessel beams [4], Lu and Greenleaf's X-wave [15], localized waves of Donnelly and Ziolkowski [1], [2], and limited-diffraction, band-limited waves of Li and Bharath [19], [20] can be obtained from this generic solution as particular cases. In addition, we have obtained new X-wave solutions and have calculated the field characteristics for one of them using a finite aperture realization. It is shown that with a proper choice of the free parameter values, well-behaved X-waves with narrow beamwidths and large depths of field can be achieved. For similar source spectra, the results are compared with Lu and Greenleaf's zeroth-order X-wave, and it is shown that the depth of field and beamwidth are very comparable.

Ventriculoperitoneal shunt flow dependency on the number of patent holes in a ventricular catheter.

Minimally invasive recanalization of obstructed ventricular catheters is a new treatment modality currently under investigation. It has been shown that blocked ventricular catheters can be at least partly reopened with ultrasonic cavitation, fiberoptic delivery of laser energy and electrocautery. It is not known, however, how many of the holes must be reopened in order to establish acceptable pressure-flow characteristics. In this study, the pressure-flow relationship of partially obstructed ventricular catheters was determined using experimental and electrical circuit/computer models. It has been demonstrated in all cases that an additional pressure of no more than 0.5 cm of water is generated when only a single 500-microm-diameter hole in the ventricular catheter permits flow. Partial recanalization of a ventricular catheter provides acceptable CSF hydrodynamics.

Noninvasive cerebrospinal fluid shunt flow measurement by Doppler ultrasound using ultrasonically excited bubbles: a feasibility study.

Because normal cerebrospinal fluid (CSF) has almost no natural Doppler scatterers, patency testing of ventriculoperitoneal cerebrospinal fluid shunts (small silastic tubing with lumen diameter of approximately 1 mm draining excessive CSF from the brain) cannot be performed by Doppler ultrasound. We have developed a low-frequency bubble excitation system that generates microbubble scatterers in both distilled water and CSF. Doppler ultrasound can then be used for flow measurement in a ventriculoperitoneal shunt. By using low duty-cycle (approximately 10%), low-frequency (approximately 30 kHz), and low-amplitude (approximately 30 kPa) ultrasound, a population of microbubbles can be maintained for sufficiently long times (>10 min) for Doppler ultrasound measurement, although bubble initiation is inconsistent. The minimum pressure needed for bubble maintenance was found to decrease with increasing burst length and duty cycle. It has been possible to detect the presence of CSF shunt flow down to a mean flow rate of 3 mL/h (mean velocity approximately 0.6 mm/s). The bubble maintenance scheme developed satisfies the safety parameters specified by the American Institute of Ultrasound in Medicine (AIUM) and the US Food and Drug Administration (FDA). Results from both in vitro and in vivo (externalized shunts) experiments indicate the feasibility of this scheme for determining realistic CSF shunt flows, though some practical problems remain before the technique will be ready for clinical use.

On the relation between aggregation, packing and the backscattered ultrasound signal for whole blood.

Previous studies have shown that the backscattered ultrasound (US) power from blood depends on the manner in which red blood cells (RBCs) are packed and, in particular, on spatial variations in the red blood cell number density (i.e., the RBC concentration variance). Experimental measurements have also shown that the backscattered US power depends on the degree of RBC aggregation, and it has been hypothesized that this is primarily due to the effect of RBC aggregation on the concentration variance. An initial simulation study of the relationship between RBC aggregation and packing statistics is presented, in which the effects of hematocrit, aggregate size, shape and size distribution on concentration variance are investigated. Both two-dimensional (2-D) and 3-D samples of aggregated and disaggregated RBCs were simulated; these enabled the concentration variance to be calculated. In agreement with theoretical predictions and experimental US results, the concentration variance for disaggregated RBCs is shown to be lowest at low and high hematocrits, and to peak at intermediate hematocrits. The concentration variance is shown to be particularly sensitive to changes in aggregate size and size distribution, and less sensitive to the shape of small aggregates. The results of this study provide a foundation for relating the state of aggregation in a blood sample to the manner in which RBCs are packed and, therefore, to the backscattered US power.

Relation of the flow field distal to a moderate stenosis to the Doppler power.

An experimental investigation was undertaken to establish how different flow regimes affect the Doppler signal. A rigid tube model consisting of a 70% asymmetric area stenosis was used with steady and pulsatile flow conditions. The characteristics of the flow field at various sites was determined using a photochromic flow visualization method. Continuous-wave Doppler measurements were made using a 41% suspension of human red blood cells (RBCs) in saline as well as a dilute suspension of 4% fixed RBCs. For steady flow, the photochromic results indicated that for Reynolds numbers (Re) of 545 and 1410, turbulence was generated and the length of the turbulent region was found to increase with increasing Re. Under pulsatile flow conditions, turbulence was triggered around peak systole and began to dissipate in late deceleration, and by the end of diastole the flow field almost relaminarized. During the turbulent phase of the flow cycle, the poststenotic flow field was seen to consist of four distinct flow regimes similar to those observed for steady flow. For higher Womersley parameters and Reynolds numbers the turbulent zone was found to be larger and to occupy a greater fraction of the flow cycle. These flow visualization results were compared with the Doppler power measurements made at the same locations and under similar flow conditions. At physiological hematocrits (41%) the onset of turbulence for both steady and pulsatile flow increased the backscattered Doppler power. The location of the peak Doppler power coincided with the region of maximum turbulence observed using the photochromic technique.

Simulation of red blood cell aggregation in shear flow.

A simulation model has been developed for red blood cell (RBC) aggregation in shear flow. It is based on a description of the collision rates of RBC, the probability of particles sticking together, and the breakage of aggregates by shear forces. The influence of shear rate, hematocrit, aggregate fractal dimension, and binding strength on aggregation kinetics were investigated and compared to other theoretical and experimental results. The model was used to simulate blood flow in a long large diameter tube under steady flow conditions at low Reynolds numbers. The time and spatial distribution of the state of aggregation are shown to be in qualitative agreement with previous B-mode ultrasound studies in which a central region of low echogenicity was noted. It is suggested that the model can provide a basis for interpreting prior measurements of ultrasound echogenicity and may help relate them to the local state of aggregation.

Defining the limitations of measurements from Doppler spectral recordings.

PURPOSE: The purpose of this study was to determine whether Doppler measurements of peak velocity and four other quantitative measures of spectral shape are affected significantly by the site of the Doppler recording in relation to the location of the maximum stenosis. METHOD: Continuous-wave and pulsed Doppler recordings were made distal to a 70% (area reduction or 45% diameter reduction) asymmetric stenosis in an in vitro flow model under steady and pulsatile flow conditions. Recordings were taken at six different locations proximal and distal to the stenosis. A photochromic dye technique was used to visualize the actual flow field in the model. RESULTS: Distal to the stenosis, the flow visualization results demonstrated a strong radial and axial variation of the velocity field and thus explained why the Doppler measurements of peak frequency and spectral broadening were strongly dependent on the recording site. The peak frequency was maximum within the throat of the stenosis and returned to the prestenotic value five tube diameters distal to the stenosis. Other measurements of spectral broadening and spectral shape varied greatly depending on the location of the recording site in the poststenotic region. Higher order spectral moments such as the coefficient of kurtosis were found to exhibit large temporal variability, which makes them inappropriate as diagnostic indicators. CONCLUSIONS: Because of the complex nature of the poststenotic flow field, these results clearly demonstrate that no single Doppler measurement can accurately quantify the severity of a stenosis. Of the Doppler measurements only peak velocity is related to the severity of stenosis. Reproducible peak velocity measurements are obtained only if the Doppler sample volume is positioned at or very near the throat of the stenosis and at an appropriate radial site that may not necessarily be at the center of the vessel.

Origin of the Doppler ultrasound spectrum from blood.

Backscattering of an incident ultrasound beam by blood gives rise to a Doppler spectrum whose characteristics are affected not only by the velocity distribution, but also by certain basic aspects of the RBC's behavior. Starting from fundamental assumptions, an explicit expression is derived for the Doppler spectrum in terms of the variance in the scatterer number density and the backscattering cross section. This shows that the Doppler power at a given frequency is weighted by the backscattering cross section and the manner in which the red blood cells (RBC's) are packed (packing factor). Since spatial variations in the flow field can result in changes in the variance and backscattering cross section, the mean Doppler frequency will not necessarily be proportional to the mean flow through the sample volume. Experimental results for two different flow fields are used to illustrate these effects. The implications of these findings are discussed in relation to volumetric flow estimation and the power mode display used in some color Doppler flow imaging systems.

Particle and voxel approaches for simulating ultrasound backscattering from tissue.

In modeling the backscattering of pulsed ultrasound from tissue, variations in compressibility are frequently approximated by assuming that an appropriate distribution of particles will produce a similar signal. For high particle densities, the computational burden can be large if simulations are conducted on a sample volume of realistic size. The approximate voxel approach offers the potential for greatly reduced computational burden, especially for three-dimensional simulations. One- and two-dimensional computer simulations were performed using both approaches with a realistic wideband Gaussian-shaped transmit signal. The mean squared errors of the simulated backscattered signals were compared to determine how the accuracy of the voxel approach depended on the scatterer concentration and voxel size selection. It is shown that the voxel approach can be used in place of the particle method with a high degree of accuracy, while considerably reducing computation time.

On a fractal packing approach for understanding ultrasonic backscattering from blood.

A new theoretical model is proposed to explain the change in the backscattered Doppler power with flow conditions. It defines a fractional packing dimension to represent the manner in which red blood cells are packed and this in turn is related to the variance of the scatterer number density. An explicit expression for the packing factor in terms of the integral of the pair correlation function is presented. Comparison of published experimental results for the backscattered power versus hematocrit with the theoretical model suggests that turbulence reduces the packing dimension. This reduction reflects a modification of the pair correlation function caused by changes in the flow field.

A wave transmission model of the umbilicoplacental circulation based on hemodynamic measurements in sheep.

Electrical analog models of the umbilical circulation were developed based on hemodynamic measurements in fetal sheep. The umbilical artery was represented by a transmission line and the placenta by a resistive load. Model predictions of input impedance and pressure and flow waveforms agreed with in vivo measurements under baseline conditions, following placental embolization, and during angiotensin II infusion. A unique positive impedance phase observed at the heart rate frequency under baseline conditions was best explained by the unusual viscoelastic properties of the umbilical arterial wall and small load reflections. Furthermore, a short, less vasoactive segment of the umbilical artery in the retroperitoneal space had a large impact on the input impedance of the umbilical circulation, which was particularly apparent when the artery was constricted during angiotensin II infusion. The model indicated that reflections arising near the approximate location where the first arterial branches leave the main umbilical artery have a measurable impact on impedance spectra when load reflections are low.

Principles and design feasibility of a Doppler ultrasound intravascular volumetric flowmeter.

A new Doppler ultrasound intravascular method is described for the measurement of volumetric flow. Based on the principles described by Hottinger and Meindl [15], it uses a novel semispherical transducer mounted at the tip of a catheter, which generates sample volumes in the form of a thin semispherical shell. Volumetric flow is calculated by using the average velocity determined from the received Doppler spectrum and the area of intersection of a sample volume that completely intersects flow across the vessel. Although a catheter-size transducer was not developed, a larger version was tested using an in vitro steady flow model. Maximum average flow error was limited to 9% for steady flows of 2 to 7 L/min. This error is believed to be a result of the nonuniform intensity generated by the prototype transducer, as well as slight variations in the received power, rather than any fundamental limitations of the flow measurement method itself. Since this study has verified the design principles and feasibility of this new approach, we believe that more detailed experimental investigations are warranted.

Influence of angle on wall shear stress distribution for an end-to-side anastomosis.

PURPOSE: The purpose of this article was to study the effects of anastomotic angle on the wall shear stress distribution for end-to-side anastomosis models under pulsatile flow conditions. METHOD: The photochromic tracer technique was used to visualize the flow field and to determine the instantaneous wall shear stress at multiple locations simultaneously. Models with angles of 20, 30, 45, and 60 degrees were examined. RESULTS: For all angles, low shear stress was present at the heel and on the bed opposite the heel of the anastomosis apparently as a result of the complete occlusion of the proximal end of the host vessel. Near the toe, increased flow separation occurred with increasing angle. On the bed across from the toe, increasing the angle led to increased shear stress. In addition, in this region the anastomotic angle significantly altered other properties of the shear stress field such as the mean and peak-to-peak magnitudes and cycle-to-cycle fluctuations. CONCLUSIONS: This study provides quantitative data on the wall shear stress distribution within an end-to-side anastomosis and its relation to the anastomotic angle. The results are discussed in terms of possible roles of shear-induced intimal hyperplasia.

Two-dimensional velocity reconstruction strategies for color flow Doppler ultrasound images.

A method for calculating the two-dimensional (2-D) velocity vector flow field from color flow Doppler ultrasound images obtained from two or three steering angles has been recently demonstrated. Various strategies for calculating 2-D vectors from noise-corrupted images obtained at multiple angles are described and compared in the present work. This was achieved by using some simple computer flow simulation models to generate color flow images for various beam steering angles. Velocity vectors were calculated for the entire image, and the vectors are displayed at selected points by superimposing them on a magnitude image. It is shown that such displays help improve the qualitative understanding of images obtained from vessels with complex flow geometry. The bias and variance of four reconstruction methods are compared for stimulated laminar flow in a tube. Based on the simulation results, it was found that the dominant factor that affects the reconstruction accuracy is the angle between the observation directions, and that a simple two component method generally gives as good a performance as the more complex alternatives that were studied.

Flow imaging in an end-to-side anastomosis model using two-dimensional velocity vectors.

Colour flow Doppler ultrasound images from vessels of complex geometry can be difficult to interpret, thereby limiting the effectiveness of the technique to correctly assess abnormalities and to relate the images to the underlying flow field in a quantitative manner. This paper describes progress in calculating and displaying two-dimensional (2-D) velocity vectors from a 30 degrees end-to-side anastomosis model under steady flow conditions at various Reynolds numbers. Velocity vectors were computed from colour Doppler ultrasound images obtained with a linear array for several incident beam directions, and the results were displayed either as colour-encoded magnitude images or by superimposing the vectors on one of the original colour images. Results are discussed in relation to flow visualization observations and the behaviour of flow in curved vessels.

Ultrasound scattering from blood with hematocrits up to 100%.

The backscattering coefficient of saline suspensions of porcine red blood cells was measured for hematocrits up to about 90%. It was found that the coefficient peaks at approximately 15%, but then, contrary to what a simple "gap theory" might suggest, it decays smoothly to zero, without showing another peak at high hematocrits. A one-dimensional (1-D) slab scattering model, in which the number of slabs per unit length represents the hematocrit and whose thickness and acoustical properties are similar to red cells/plasma, was also used to investigate the relation between the backscattered power and hematocrit. Monte-Carlo simulations performed for randomized boundary conditions show a similar relation to that of the 3-D system. The experimental data is compared to the Percus-Yevick theory for the packing of hard spheres, and the simulated data is compared to the Percus-Yevick theory for infinite slabs.

Aggregation effects in whole blood: influence of time and shear rate measured using ultrasound.

Ultrasound B-mode imaging (7 MHz) was used to measure blood echogenicity and velocity profiles simultaneously as they developed with axial distance for a steady flow of 28% hematocrit whole blood flowing in a long (> 60 D) large diameter (D = 2.54 cm) tube. At selected sites along the flow axis, velocity profiles were measured using block matching (cross correlation) between successive digitized images with a known time separation; from these shear rate profiles were calculated. The corresponding echogenicity profiles were also determined by averaging the digitized images. It was found that over a range of low shear rates, the echogenicity is enhanced in a manner similar to the previously reported influence on aggregation. Evidence is presented confirming the important role of aggregation in controlling the echogenicity. The transient effects of abrupt flow stoppage were studied and shown to provide useful insights into aggregation kinetics. Based on the above results, a detailed explanation is provided of the echogenicity variations seen in B-mode ultrasound images of slow-moving blood.

Hemodynamics of a side-to-end proximal arterial anastomosis model.

PURPOSE: The purpose of this article was to analyze the fluid mechanical effects of a side-to-end proximal anastomosis and to compare the results with those from our earlier study on the end-to-side distal anastomosis. METHODS: The photochromic tracer technique was used to determine the instantaneous wall shear stress and to visualize the overall flow field under pulsatile flow conditions. The flow consisted of a sinusoid plus a steady component with mean and modulation Reynolds numbers of 355 and 565, respectively, and a Womersley number of 7.9. RESULTS: At the toe and heel of the junction, very high and positive wall shear stresses were seen together with substantial nonperiodic fluctuations. The peak wall shear stress was about four times higher at the toe and about seven times higher at the heel than the maximum values observed at about four tube diameters upstream from the junction. On the bed of the host vessel, nonperiodic fluctuations were also observed, but the shear stresses were mainly negative with magnitudes comparable to those seen upstream. With leakages of 11% and 28% of the mean flow through the blocked end of the host vessel, the shear stress pattern seemed to be significantly affected only at the toe for the higher leakage. Further, when the mean Reynolds number was reduced to 320, the magnitudes of the variations in the wall shear stress were reduced proportionately, except at the heel, where the reduction was much larger than expected. CONCLUSIONS: It appears that the preferential development of intimal hyperplasia at the distal end-to-side anastomosis may be promoted by low wall shear stress at the toe and heel, and probably by high shear stresses or shear stress gradients on the bed.