In Vitro Flow Mapping of Regurgitant Jets
Systematic Description of Free Jet With Laser Doppler Velocimetry
Background Color Doppler and magnetic resonance imaging give pictures of abnormal jets within which the respective contribution of fluid mechanics and image artifacts are difficult to establish because of current technical limitations of these modalities. We conducted the present study to provide numerical descriptions of the velocity fields within regurgitant free jets.
Methods and Results Laser Doppler measurements were collected in rigid models with pulsatile flow conditions, giving several series of two-dimensional flow images. The data were studied with the use of two-dimensional or M-mode flow images as well as regular plots. Numerical descriptions validated in steady flow conditions were tested at the various times of the cycle. In these free jets, the momentum was conserved throughout the cycle. The transverse velocity profiles were approximately similar. A central laminar core was found at peak ejection and during the deceleration. Its length (l=4.08 d−0.036 mm, r=.99) and its diameter (d) were proportional to the orifice diameter. At peak ejection, the velocity decay was hyperbolic, and the transverse velocity profiles were clearly gaussian. The different relations that were tested could be combined in a single formula describing the velocity field:
Conclusions These in vitro measurements demonstrated the presence of a central laminar core and similar transverse velocity profiles in free turbulent jets. This allowed us to validate a series of numerical relations that can be combined to describe the velocity fields at peak ejection. On the other hand, further studies are needed to describe the various singularities often encountered in pathology.
Color Doppler imaging of the jet extension is currently accepted as a useful tool for evaluating the severity of valvular regurgitations with either the transthoracic1 2 3 4 5 or transesophageal approach.6 7 8 It relies on the apparent extension of turbulent regurgitant jets on two-dimensional color flow mapping. Nevertheless, ultrasound Doppler has several limitations that preclude the ability to make highly accurate measurements: aliasing; velocity estimators; image frequency; and longitudinal, lateral, and azimuthal resolutions.9 10 11 12 13 14 Color flow mapping has been used in vitro for validating clinical approaches,15 16 17 18 but the combination of the limiting factors has made a precise quantitative description of the fluid mechanics within the jet almost impossible.
Similarly, MRI analyses of regurgitant jets have been proposed,19 20 21 22 and flow mapping has been used to assess the severity of aortic or mitral regurgitations. The limitations depend in part on the modality, but in all cases, the results depend on the velocity, resolution, presence of turbulences, frame rate, and flight time within a given structure.
Precise descriptions of turbulent jets have been obtained in steady flow conditions, mainly with gaseous fluids.23 24 25 26 27 They provide a numerical approach, but their validity for intracardiac pulsatile jets has not been established. Data that apply directly to cardiac jets are, so far, very limited.28 29 30 31 32 This lack of a clear knowledge of the velocity fields prevents an analysis of the respective contributions of the imaging modality and the fluid mechanics to the construction of images and appearance of potential artifacts. The present study was designed to establish a link between the images of jets and physics by providing precise bidimensional images of in vitro models of regurgitant jets and by testing the validity of classic equations of fluid mechanics under pulsatile conditions.
This study has been conducted on an in vitro model of jet in which velocity values were collected with the use of a single-component laser Doppler anemometer.
A physiological pulsatile flow was generated that was made up of four components (see Fig 1⇑): (a) a fluid tank (15 L) (the fluid composition [70% water/30% glycerol] was that of a fluid in which the viscosity is almost that of blood [3.5 cp at 22°C], sodium chloride was added and served as the conducting electrolyte for the electromagnetic flowmeter, and starch particles [<1%] were added to the water and served as backscattering targets in laser Doppler measurements); (b) an oscillating pump that determined the ejected volume and flow (2 to 8 L/min) (its movement was sinusoidal-like, its frequency was set by the rotation speed of the piston [50 to 100 cycles/min], and it had an input/output bivalve that operated as a lift-and-force pump, transforming the sinusoidal-like flow into a physiological pulse flow toward the measurement tube); (c) a honeycomb that allowed dissipation of the turbulences before the fluid passed through the measurement chamber to which it was connected via a very short, 16-mm-diameter tube (this configuration resulted in a flat velocity profile at the orifice); and (d) a resistance/compliance system made of an acrylic plastic pipe connected through a resistor and an additional air volume, creating compliance.
The measurements were collected in a chamber made of transparent glass, in which the various orifices were positioned.
An electromagnetic flowmeter connected to an oscilloscope was used to continuously monitor the flow curves whose morphology, close to that of intra-aortic curves, has been modulated by the resistance/compliance system. It was also possible to identify the start of each cycle (ECG simulation) on the oscilloscope via a synchronization system linked to the pump.
Laser Doppler Velocimeter
The laser Doppler anemometer (Dantec) consisted of four components: (1) a 50-mW laser argon power source (λ=514.5 nm); (2) an optical system, including mirrors, prisms, and a converging lens, that was used to obtain two incidental laser beams that intersect with an angle of 20°, giving a sample volume of <0.3 mm3 (the bisector of the angle was perpendicular to this axis of the jet, giving access to the longitudinal component of the flow); (3) a photomultiplier that transformed the light signal into an electrical signal; and (4) a Doppler frequency detector that was used to analyze the signal from the photomultiplier (filtering, detection of 16 zero-crossings, measurements of time intervals). The frequency of data acquisition was >3 kHz, corresponding to an even higher sampling rate during the ejection, as the low “diastolic” velocities were associated with a lower sampling rate. The adaptation of the low and high pass filters gave a velocity resolution of 0.023 m/s for the individual measurements and 0.005 m/s for the mean values.
The accuracy of the measurement was confirmed by the very low normalized standard deviation (<3%) of the individual raw data on the center of the origin of the jet.
The device was attached to an adjustable holder by two micrometer screws, which allowed accurate positioning and displacement (resolution of 0.1 mm) of the measurement site along the longitudinal and transverse axes, with the coordinates controlled by digital display.
Velocity measurements were carried out with displacement of the sample volume in two directions: (1) longitudinal velocity profiles were acquired with steps of 2 to 5 mm, depending on the dimensions of the orifice, with the first profile acquired along the centerline; and (2) to obtain a bidimensional analysis, these measurements were repeated after a displacement of the laser device in the transverse direction, at increasing distances of this axis (steps of 1 mm).
The digital velocity values were directly written in excel format on a PC. The ejection was divided into 16 to 24 successive periods of 10 milliseconds. For each measurement site, the instantaneous velocity was averaged over 20 cycles. The velocity values were stored along with their time period in the cycle and their spatial position, thus allowing the reconstruction of a two-dimensional matrix. Because all of the configurations had an axial symmetry, the acquisitions were limited to one half of the equatorial plane, with later reconstruction of the other half.
The data were directly transferred to a Macintosh IIfx computer (Apple). They were processed with the use of specific macros written with excel software (Microsoft). The resulting tables of values were directly transformed into pictures with the use of deltagraph software (Deltapoint). Various types of charts were generated.
Two-dimensional color flow images (Fig 2a⇓) were obtained by plotting the longitudinal distance on the x axis, the transverse distance on the y axis, and the velocity values on the z axis with the use of a color code, thus giving a format very similar to the usual color Doppler imaging. A “hot” scale made of 12 different colors was used, ranging from yellow for high velocities (>3.5 m/s) to dark red for lower velocities and black for very low and negative velocities (<0 m/s). Several series of consecutive images were synthesized. Each image was a display of the adjacent velocity profiles at one time of the cycle, as in ultrasound color flow mapping. In all of the pictures, the origins were chosen to locate the orifice to the left.
In addition to this format, M-mode images (Fig 2b⇑) were created, displaying distance on the y axis, time on the x axis, and velocity on the z axis, with the same color scale. They presented the evolution of flow profiles throughout the cycle. In this example, the investigated flow profile has been acquired on the centerline, in the direction of the jet, and the distance is the longitudinal distance. In all the pictures shown below, the orifice is at the top of the image. With the same picture format, other parameters and computations were displayed so we could study their time variations. To compare our data with the descriptions made in fluid mechanics, plots were traced. Fig 2c⇑ shows a diagram of velocity versus distance. The values along the dotted line in Fig 2a⇑ are displayed in a format showing the centerline longitudinal velocity profile at peak velocity.
Similarly, velocity-versus-time diagrams were obtained, as shown in Fig 2d⇑. This particular example shows the waveform of the velocity at the orifice. The plotted data correspond to the top row of the table of values displayed in Fig 2b⇑. These formats were also used with processed data. In particular, to test the validity of various equations, we plotted, with M-mode color images, the ratio of observed to theoretical values.
Various Experimental Configurations
The oscillating pump of the hydraulic bench was set to produce peak velocities varying from 3 to 5 m/s. The jets were considered to be free when the velocity at the wall of the cavity was sufficiently low to prevent hydrodynamic intervention of the chamber walls.30 They were studied in a parallelepipedic box with sides of 7 cm. The orifices were cylindrical (with diameters [d] varying from 5.8 mm to 11.3) with a length equal to 0.25 times the orifice diameter.
Theoretical Model and Main Abbreviations
The only accepted description of turbulent jets corresponds to jets in steady flow conditions.23 24 25 It is clearly illustrated by the data reported by Moore26 regarding aeric subsonic cold jets. When the origin of a turbulent jet is laminar, it generates a central laminar core at which the velocity is uniform. This core is approximately conical. It is surrounded and progressively invaded by a diverging cone made of structured vortices. When studied at increasing distances from the orifice, these vortices develop and grow until reaching the tip of the central core. Farther downstream from this point, they are destroyed into turbulence. The corresponding formula combines several concepts: the symmetrical development of the boundary layer between the jet and the surrounding immobile fluid, linear widening of the jet, normalization of the transverse velocity profiles, and the principle of conservation of the momentum.
This formula suggests that when the flow at the orifice is laminar, the length of the core is independent of the initial velocity. k is an empirical constant. The velocity decay in the turbulent area, downstream with respect to the tip of the core, is hyperbolic: \mathit|<|V|>|(\mathit|<|x|>|,0)/\mathit|<|V|>|(0,0)|<|=|>|\mathit|<|V|>|(\mathit|<|x|>|)/\mathit|<|V_|<|o|>||>||<|=|>|\mathit|<|l|>|/\mathit|<|x|>||<|=|>|\mathit|<|k|>||<|\cdot|>|\mathit|<|d|>|/\mathit|<|x|>| where V(x,0) is the velocity measured on the axis of the jet at a distance x from the orifice [V(x,0)=V(x)], and V(0,0) is the velocity within the orifice [V(0,0)=Vo=Vorifice]. k has often been evaluated with the velocity decay used to extrapolate the hyperbole for V(x)/Vo=1. The k constant can also be obtained with the use of a linear regression between V(x,0)/V(0,0) and d/x. The successive transverse velocity profiles can be normalized; their shape is independent of the distance from the orifice: \mathit|<|V|>|(\mathit|<|x,y|>|)/\mathit|<|V|>|(\mathit|<|x|>|,0)|<|=|>|\mathit|<|f|>|(\mathit|<|y|>|/\mathit|<|x|>|) where y is the distance from the axis. This formula is used to describe the principle of similarity of the transverse velocity profiles when studied in the turbulent zone. Experimental data have shown that these profiles are gaussian, and they have allowed a more precise description: log|<|[|>|\mathit|<|V|>|(\mathit|<|x|>|,0)/\mathit|<|V|>|(\mathit|<|x,y|>|)|<|]|>||<|=|>|\mathit|<|k|>||<|^\prime|>|(\mathit|<|y|>|/\mathit|<|x|>|)^|<|2|>|
Equation 4 can be transformed into the following: \mathit|<|V|>|(\mathit|<|x|>|,\mathit|<|y|>|)|<|=|>|\mathit|<|V|>|(\mathit|<|x|>|,0)|<|\cdot|>|10^|<||<|-|>|\mathit|<|k|>||<|^\prime|>|(\mathit|<|y|>|/\mathit|<|x|>|)^|<|2|>||>| and, combined with Equation 2, provides the following: \mathit|<|V|>|(\mathit|<|x|>|,\mathit|<|y|>|)|<|=|>|\mathit|<|V|>|(0,0)|<|\cdot|>|\mathit|<|k|<|\cdot|>|d|>|/\mathit|<|x|>||<|\cdot|>|10^|<||<|-|>|\mathit|<|k|>||<|^\prime|>|(\mathit|<|y|>|/\mathit|<|x|>|)^|<|2|>||>|
Provided a series of assumptions (ie, steady flow, longitudinal velocities, and gradients far superior to the transverse velocities and gradients), the combination of the conservation of the momentum with the gaussian profile implies a hyperbolic velocity decay on the centerline.33 34 35
Because all of these equations apply to steady flow conditions, they do not include time (t). To adapt the definitions to the pulsatile conditions of the present study, we added time to the definitions: V(x,y,t).
Calculations and Normalizations
The laminar core was visualized on bidimensional color flow images, and its evolution during the ejection was studied with the use of M-mode color flow images. The measurements were conducted mainly on plots of the longitudinal centerline velocity profiles. We considered its length to be defined by the distance l on the x axis at which the velocity falls below 0.95-fold the velocity at the orifice. To compare various initial peak velocities and configurations with various orifice diameters (d), we calculated the ratios of V(x,0,t) to V(0,0,t) and of x to d and plotted V(x,0,t))/V(0,0,t) versus x/d.
To study the hyperbolic velocity decay, we plotted V(x,0,t)/V(0,0,t) versus d/x. With this type of display, according to Equation 2, the velocity values collected beyond the tip of the core are theoretically distributed along a single line with a slope equal to k.
At the orifice and in the central core, the design of the flow model was expected to induce a flat velocity profile (very short entry segment). Thus, for depicting transverse velocity profiles close to the orifice, we took into account the orifice, included its diameter in the formula, and plotted V(x,y,t)/V(x,0,t) versus [(y−d/2)/x]. Farther away, in the turbulent zone, we neglected the orifice diameter and plotted V(x,y,t)/V(x,0,t) versus y/x and log[V(x,0,t)/V(x,y,t)] versus (y/x)2, according to Equation 4. The use of (y/x)2 instead of [(y−d/2)/x]2 is necessary because [(y−d/2)/x] has negative values, resulting in ambiguous information, when calculating its square value.
The total momentum of the jet (along a transverse velocity profile) at a distance x from the orifice [M(x,t)] was calculated on circular orifices with the use of a cylindrical integration where Δy is the distance separating two adjacent sampling sites on the transverse velocity profile: \mathit|<|M|>|(\mathit|<|x|>|,\mathit|<|t|>|)|<|=|>||<|\int|>|_|<|\mathit|<|y|>||>|\mathit|<|V|>|(\mathit|<|x|>|,\mathit|<|y|>|,\mathit|<|t|>|)^|<|2|>||<|\cdot|>|\mathit|<|d|>|A and
Simulation of Ultrasound Color Doppler
The velocity ambiguities17 18 19 are due to the Nyquist limit. All velocity values exceeding this limit are aliased. This limit is often threefold to sixfold inferior to the maximum velocity of regurgitant jets. It induces a signal covering the entire bandwidth and, according to the usual behavior of the velocity estimators, creates some kind of threshold at a level (usually 0.5 to 1.5 m/s) that is far below the velocity (3 to 5 m/s) that is, at peak velocity, uniform within the central laminar core of the jet.20 21 22 We therefore simulated the ultrasound color flow images by recording all of the velocities exceeding the threshold and giving them a value equal to the threshold. In addition, bright green was used to display this special velocity value.
Bidimensional Images at Peak Velocity
Bidimensional images were constructed at each time of the ejection. At peak velocity (Fig 3⇓), the central laminar core is shown clearly. It appears as an approximately conical isovelocity area that contains the highest velocity values present in the jet. This type of visualization allows evaluation of its length, width, and shape. The progressive widening of the jet is clearly visible. Interestingly, on the corresponding simulation of an ultrasound color flow mapping, this core is no longer visible because it is obscured by the thresholdlike effect induced by aliasing. The same situation applies to several MRI modalities.
Origin of Jet and Diameter of Laminar Core at Orifice
The transverse velocity profile was systematically studied at the orifice. It appeared flat throughout the ejection (Fig 4⇓). In addition, the standard deviation of the velocity measurements divided by the corresponding mean velocity was plotted and considered to be an indicator of turbulence. We found, on the centerline, a very low turbulence (2%) separating two lateral peaks (30%). This low centerline value demonstrates the laminarity of the jet origin. The diameter of this laminar core was measured with the use of three modalities: length of the transverse plateau (dplateau), diameter at midvelocity (d1/2velocity), and distance separating the two peaks of turbulence (dturbulence). The correlation coefficients between the orifice diameter and these measurements were very high, with a slope close to 1 for the two first approaches: dplateau=0.92 dorifice−0.82 mm (r=.99), d1/2velocity=1.20 dorifice−0.27 mm (r=.99), and dturbulence=1.34 dorifice−0.32 mm (r=.99).
When plotted with an M-mode color flow format, these data show a clear-cut linear band at the site of the orifice, with this band demonstrating a very constant width.
Length of Central Core
The length of the central laminar core has been measured, at peak velocity, on plots of the centerline velocity profiles (Fig 5⇓). The various experimental values presented without normalization showed a very significant scatter. On the other hand, when presenting V(x,0,tpeak)/V(0,0,tpeak) versus x/d, the values appeared distributed along a single dimensionless curve. The length of the central core has been defined as the x distance at which the velocity falls below 0.95-fold the mean value in the core. This length was compared with the orifice diameter on a series of circular orifices. The results displayed in Fig 6⇓ show a linear relation with a very high correlation coefficient, in accordance with Equation 1. The slope (k constant) was in the region of 4, very close to the x/d value beyond which the normalized velocity falls below 0.95-fold the initial velocity.
Evolution of Central Core During Ejection
The evolution of the centerline velocity profiles during the ejection has been studied with the use of both plots and M-mode color images. The latter format allowed side-by-side presentation of the successive profiles. In Fig 2b⇑, the orifice is at the top of the image, and the time is on the abscissa. The progressive development and vanishing of the jet are clearly visible. The propagation of the front of the jet appears linear. The phase shift is minimal between the velocity at the orifice and the velocity farther away. In Fig 7⇓, the centerline M-mode color image of two orifices is shown during the acceleration [V(x,0,t) versus x and t, a small orifice to the left and a large orifice to the right]. These two images are very similar, demonstrating that the normalization validated at peak ejection can be used during other phases of the cycle. The clearly visible propagation of the front of the jet was used to draw tangents whose slopes are equal to speeds of propagation of the jet. These measurements were obtained for all of the orifices with the use of thresholds fixed at 1 and 2 m/s, and we found linear relations between these values and the orifice diameter (Fig 8⇓). At higher thresholds and during deceleration, the speeds of forward and backward motion of the jet were found to be less linear. At peak velocity, for the higher thresholds, the speed of propagation is close to 0.
To depict more precisely the evolution of the central laminar core during the ejection, M-mode color images were synthesized with a different computation on the ordinate: the velocity values of each instantaneous profile were divided by the value at the orifice [V(x,0,t)/V(0,0,t)], and the abscissa was kept the same. This approach will give a normalized value equal to 1 at all of the points that have a velocity equal to the velocity at the orifice at the same time. The colors provide a map of the isovelocity areas (Fig 9⇓). The results clearly show that the central core is stable during the majority of the deceleration but shorter during the beginning of the acceleration. At the end of the deceleration, normalized values exceeding 1 are found close to the tip of the core.
Analysis of Centerline Velocity Decay at Peak Velocity: Turbulent Zone
The centerline velocity decay was studied at peak velocity on plots in different formats. The plot [V(x,0,tpeak)/V(0,0,tpeak)] versus x (Fig 10⇓, left) is identical to that of Fig 5⇑ (left). In Fig 10⇓ (right), [V(x,0,tpeak)/V(0,0,tpeak)] was plotted versus 1/x instead of x. With this particular presentation, the points recorded away from the orifice appeared close to the origin of the graph and the points recorded close to the orifice were plotted away from the origin. The hyperbolic decay appeared as a linear relation between the abscissa (1/x) and the ordinate [V(x,0,tpeak)/V(0,0,tpeak)]. The collected values were clearly distributed along a straight line close to the origin of the graph (eg, far from the orifice). Its slope was equal to the constant of the hyperbole. This figure showed two different orifices; they have a very similar behavior. The constants of the various hyperbolic decays were the following: dorifice of 5.8 mm, k=4.04, r=.99; dorifice of 7.1 mm, k=4.37, r=.99; dorifice of 8.2 mm, k=4.52, r=.99; dorifice of 9.8 mm, k=4.44, r=.99; and dorifice of 11.3 mm, k=3.68, r=.95.
All of the slopes were close to 4, which is very similar to the value for the k constant found previously.
Evolution of Velocity Decay During Ejection
To identify the periods of the ejection during which the velocity decay was hyperbolic, we calculated the ratio [R(x,0,t)] of the actual values to the values given by the hyperbolic relation [4·d·V(0,0,t)/x]: \mathit|<|R|>|(\mathit|<|x|>|,0,\mathit|<|t|>|)|<|=|>||<|[|>|\mathit|<|V|>|(\mathit|<|x|>|,0,\mathit|<|t|>|)/\mathit|<|V|>|(0,0,\mathit|<|t|>|)|<|]|>|/|<|[|>|4|<|\cdot|>|\mathit|<|d|>|/\mathit|<|x|>||<|]|>| with this ratio equal to 1 at all of the points at which the formula validated at peak ejection was verified, thus depicting a kind of isovelocity surface area. An example of this approach is shown in Fig 11⇓, where R(x,0,t) was plotted versus x and t with use of a color code. At peak velocity, a vertical yellow band shows that this ratio was equal to 1 from the tip of the core to the end of the analyzed window, demonstrating that the hyperbolic decay was achieved. Later in the cycle, the yellow band is clearly shorter and the values of the ratio are >1.2 or >1.35 at the distance from the tip of the core, showing that the hyperbolic decay was no longer verified.
Normalization of Two-dimensional Color Flow Images: Transverse Velocity Profiles
The series of adjacent transverse velocity profiles correspond to all of the information given, building a single two-dimensional color flow image. With this format, it was impossible to test the validity of the theoretical equations. For that purpose, plots were used (Fig 12⇓). Without normalization (Fig 12⇓, left), the data exhibit a wide scatter. When normalizing the velocity values by the centerline velocity and the transverse distance y by the longitudinal distance x [V(x,y,t)/V(x,0,t) versus y/x; Fig 12⇓, middle], most of the points were distributed along a single dimensionless curve that appeared gaussian. Some points remained outside this curve; they corresponded to data collected close to the orifice, at sites at which the dimensions of the orifice still had a significant influence. This appeared clearly when the normalization was slightly modified to take into account the orifice diameter: (y−d/2)/x instead of y/x. With this calculation, all of the data were on the same curve. This normalization demonstrated the similarity of the adjacent transverse velocity profiles at peak velocity.
According to the mathematical formula of gaussian curves, the gaussian shape of the distribution was tested with the use of a log display of the normalized velocity plotted versus (y/x)2. It was not possible to include the orifice diameter in the formula because some y−d/2 values were negative and their square value would have been mixed with the square positive values. For this reason, the log display has been used only for the data collected beyond the tip of the core (x/d >5). With this type of calculation, the calculated points were found to be distributed very close to a single straight line. The constant of the gaussian curve (Equation 4) was obtained with linear regression. The various values measured at peak velocity and the correlation coefficients were the following: dorifice of 5.8 mm, slope=45.7, r=.97; dorifice of 7.1 mm, slope=47.6, r=.95; dorifice of 8.2 mm, slope=49.1, r=.95; dorifice of 9.8 mm, slope=41.0, r=.95; and dorifice of 11.3 mm, slope=42.6, r=.96.
The average value is in the region of 45. These slopes characterize the linear progressive widening of the jet, and they could be translated into angle values.
Evolution of Transverse Velocity Profiles During Ejection
The evolution of the widening of the jet during the ejection was studied with the use of this log display, the r value, and the slopes calculated on the successive images. A simultaneous display of these three parameters showed that their evolutions during the ejection are approximately parallel to the variations of the velocity at the orifice. The constantly high values of r show that the transverse profiles are close to gaussian throughout the ejection, but the best fit is obtained at peak velocity. On the other hand, the slope of the relation clearly varies and depends on the ratio of the instantaneous velocity to the peak velocity. We found a linear relation between the velocity at the orifice and the slope of the gaussian relation. With the 5.8-mm orifice, the equation was as follows (r=.91): Constant of Gaussian Curve|<|=|>|22.96\mathit|<|V|>|(0,0,\mathit|<|t|>|)/\mathit|<|V|>|(0,0)_|<|\mathit|<|peak|>||>||<|+|>|19.8
Influence of Peak Velocity on Various Parameters
The above measurements were repeated for various peak velocities (from 3 to 5 m/s) to verify that the various descriptions obtained were independent of the peak velocity, as expected from the theoretical equations. The collected data demonstrated a strict normalization; the points collected at various velocities were distributed along a single dimensionless curve. Thus, different peak velocities did not change the dimensions of the central core or the shape of the transverse or longitudinal velocity profile.
Calculations of Momentum
The momentum was calculated at increasing distances from the orifice with the use of Equation 5. These calculations provided a longitudinal profile of the momentum at each time of the cycle. The evolution of these profiles was studied with M-mode color flow images. The results demonstrated that, as expected, the momentum varies during the ejection. On the other hand, at each time, the value of the momentum appeared constant, giving the same color (and, thus, the same value) from the orifice to the end of the analyzed window. Therefore, in our series of experiments in pulsatile conditions, the fundamental principle of conservation of the momentum was verified.
A final formula could be established at peak velocity (tpeak) by combining all of these results: (1) the k constant found both at the central core and at the velocity decay could be included in Equation 2, and (2) the transverse variations obtained with Equation 4 and the value of the constant of the gaussian were included in Equation 6: \mathit|<|V|>|(\mathit|<|x|>|,\mathit|<|y|>|,\mathit|<|t|>|_|<|\mathit|<|peak|>||>|)|<|=|>|\mathit|<|V|>|(0,0,\mathit|<|t|>|_|<|\mathit|<|peak|>||>|)|<|\cdot|>|4|<|\cdot|>|(\mathit|<|d|>|/\mathit|<|x|>|)|<|\cdot|>|10^|<||<|-|>|45(\mathit|<|y|>|/\mathit|<|x|>|)^|<|2|>||>|
It was valid beyond the tip of the central laminar core. When fitting all of the data collected at x distance exceeding the orifice diameter by fivefold against this formula, we obtained r=.93. The same analysis on an orifice-by-orifice basis gave r=.91 to r=.94.
The present experiments were designed to study whether classic descriptions of fluid mechanics apply to intracardiac free jets. The latter are hydraulic, cold, and pulsatile, whereas most of the fundamental descriptions have been obtained on aeric jets in steady flow conditions with very high velocities.24 25 26 27 36 37 38 With this effort, we tried to establish a link between physics and events occurring within a patient.
In comparison with previous studies,28 29 30 31 32 the present experimental setting allowed the direct transfer of very large amounts of digital data. Routines could be written in excel with partially automatic processing, providing opportunities for analyzing three-dimensional (x,y,t) data sets with a high resolution.
When extrapolating from theory to the clinical setting, basic concepts of fluid mechanics could be verified in conditions very similar to the situations prevailing in patients.
The first finding (see Figs 2 through 6⇑⇑⇑⇑⇑) was the demonstration of the presence of a central laminar core during ejection. Despite conditions that induce turbulence when occurring in a long, straight, rigid tube, the standard deviation of the velocity was measured at <0.03-fold the velocity within this core. The results of the present study extend our previous conclusions30 32 and agree with the data reported by Cagniot et al,31 Lu et al29 (with a different model), and Clark28 (with a different methodology); these authors also found this hydraulic structure, within which the turbulence intensity is very low. This core progressively appears during the acceleration and is sustained during most of the deceleration (Fig 10⇑). Its origin has clear-cut limits, a diameter that does not vary during the ejection (Fig 4⇑), and a very thin boundary layer. The length of this core is proportional to the orifice diameter (Fig 6⇑). This information is very important for the ongoing developments in flow imaging because, so far, this core is not visible and the jets are often considered to be entirely turbulent.
Conservation of Momentum and k Constant
The conservation of momentum and the similarity between the transverse velocity profiles are the second and third significant findings. Their combination is justified by their impact on the other equations. Our data clearly show that at any time and in any position, the calculated momentum is equal to the momentum measured at the same time at the orifice. When compared with the data reported by Thomas et al,35 our experiments benefit from the very high accuracy of the laser Doppler anemometer. Furthermore, the type of acquisition that was selected allowed us to follow the distribution of the momentum throughout the cycle, showing its independence with regard to the longitudinal distance. The demonstration of a clear similarity of the transverse velocity profiles has been accomplished with dimensionless plots (Fig 12⇑); at any time of the ejection, the data normalized by the centerline velocity appeared to be distributed along a single curve. Furthermore, beyond the tip of the core, the log display demonstrated that this curve is gaussian.
From a theoretical point of view, the combination of the conservation of the momentum and the similarity of the transverse profiles predicts the hyperbolic decay of the centerline velocity, provided one makes the following assumptions33 34 : flow in almost steady conditions, limited viscous losses, longitudinal velocity components far exceeding the radial velocity components, very low gradients, and fluctuations of the velocity in the radial direction.
Our data did not allow us to test the validity of these assumptions, but our results clearly indicate that the hyperbolic decay as defined by the k constant is established at peak ejection. The analyses of the central core and the hyperbolic decay allowed two different calculations of the k constant: one was obtained by extrapolating the hyperbolic curve for a ratio V(x,0,t)/V(0,0,t) equal to 1 (ie, the slope of the linear distribution; Fig 9⇑), and the other one was obtained by relating the length of the core to the orifice diameter (Figs 5 and 6⇑⇑). The calculated values were very similar, giving the value of ≈4. This empirical constant is lower than the often-quoted value of 6.8.23 24 We also reported a constant of ≈630 with the use of an ultrasonic flowmeter, but this first study was achieved with a lower resolution and with a maximum velocity estimator that tended to overestimate the length of the core. In the present study, the resolution and accuracy of the velocity measurements were higher; the two approaches of the k constant gave the same results; and, interestingly, the data reported by Ko and Davies,36 Moore,26 Zaman and Hussain,37 Acton,38 and Lu et al29 clearly showed a tip at a distance equal to fourfold to fivefold the orifice diameter, thus giving additional confidence in our results.
The constants of the gaussian transverse velocity profiles that we found at peak velocity are close to the value mentioned by Davies24 and correspond to jet angles of ≈5°. This value is smaller than the data reported by Thomas et al.39 Indeed, the classic jet angle is obtained by measuring the jet width at a distance y where V(x,y,t)=V(x,0,t)/2, whereas in the study reported by Thomas et al, it was measured with ultrasonic color flow imaging with lower thresholds.
The influence of the pulsatility on the similarity of the transverse profiles and on the hyperbolic decay is clear but difficult to explain. One possible factor might be the empirical k constant, which could vary during the cycle. This situation would have induced a different but constant R(x,0,t) (Equation 8), giving vertical stripes on the Fig 11⇑. This situation was not seen. A less strict similarity of the profiles appears to be a more satisfactory answer because, during acceleration and deceleration, we found lower r values when calculating the constant of the approximated gaussian curve, and the distribution of the data did not suggest any other mathematical relation. This absence of normalization during the acceleration and deceleration phases led us to restrict the validity of Equation 10 to the peak of the ejection.
This study was conducted with a single component of the velocity, and the radial values were neglected. Despite these simplified measurements, the stability of the momentum was verified, suggesting that the radial values are low. Very recently, we were able to confirm this hypothesis with a two-component measurement on a limited number of configurations.
All of the data were acquired with a rigid model. This configuration is imposed by the transparency of the tubes necessary for high-fidelity laser analyses, and to the best of our knowledge, it has been very difficult to combine optical qualities and physiological compliance.
We used a single pump frequency because the influence of different durations of the cycle on the central core and the hyperbolic decay has been specifically addressed by Cagniot et al31 and found to be limited.
All of the reported configurations correspond to free jets with circular orifices. Pathological conditions are clearly more complex because (1) the orifices are often far from circular, and eccentrical orifices can modify the jets39 ; (2) a wall close to a jet (eg, a mitral regurgitation induced by a valvular prolapse) can lead the jet to adhere due to the Coanda effect40 41 ; (3) some jets are impinging42 ; and (4) jets occurring in a narrow chamber (eg, aortic regurgitation) are confined, and the confinement can modify the velocity fields.30 These various singularities require further studies, which we are presently analyzing, and our preliminary results suggest that the length of the core is preserved in the majority of these singularities.43 44 45
The hydraulic structure of the origin of the jet has a direct impact on clinical imaging. The laminar central core is a volume within which, at a given time, the blood cells have the same velocity, thus giving a strong narrow band signal during velocimetry. This structure accounts for the well-contoured tracings obtained with continuous-wave Doppler. Furthermore, this core is particularly suitable for any kind of velocity measurements, and, during flow imaging, the velocity estimators operate in optimal conditions if the modality is adapted in terms of resolution or velocity range.
The stability of the diameter during the ejection and its relation with the orifice diameter give theoretical support to the clinical applications of color flow imaging that we46 and others7 have proposed for quantifying aortic or mitral regurgitation. In terms of flow structure, the jet diameter and, when feasible and reproducible, the cross-sectional area of the jet origin are direct parameters that allow quantification of the regurgitations; they do not vary with the velocity and, thus, the load conditions. Their measurements are clearly feasible routinely as, due to the high velocities, they can be obtained with Doppler echocardiography with angles close to 80°.
The equation of the hyperbolic decay has been proposed as an approach for quantifying regurgitations with the “momentum equation.”33 34 35 In Equation 2, V(0,0,t) can be measured with continuous wave Doppler; V(x,0,t) can be obtained with pulsed Doppler as long as (1) the velocity measurements are accurate in the turbulent zone and (2) the axis of the jet could be defined with color flow mapping; and x is the distance between the sample volume in pulsed mode and the orifice. Therefore, |<|[|>|\mathit|<|V|>|(\mathit|<|x|>|,0)|<|\cdot|>|\mathit|<|x|>||<|]|>|/|<|[|>|\mathit|<|V|>|(0,0)|<|\cdot|>|\mathit|<|k|>||<|]|>||<|=|>|\mathit|<|d|>| and where Q is the volumic regurgitant flow in case of regurgitation. Our data demonstrate that the turbulent decay is clearly hyperbolic at peak velocity, and, thus, this equation can be used provided accurate velocity measurements are used. On the other hand, the analysis throughout the cycle clearly shows that this equation is accurate only during a very brief period of time at peak velocity. This result prompts very cautious use of this equation because, in addition to the above prerequisite, the duration of the strict hyperbolic decay appears shorter than the zenith of the ejection. In Fig 11⇑, the yellow display of the values equal to 1 outlines this domain of accuracy.
The clearly gaussian shape of the velocity profiles in the region of the turbulent velocity decay is important. It is visible neither in the apparently uniform mosaic zone given by Doppler echocardiography nor in the void of signal obtained so far with MRI. This underlines the qualitative character of the flow mapping modalities currently available for use in patients.
The final formula is a synthesis that has no direct clinical impact because it describes phenomena occurring in the turbulent zone, in which currently available velocity estimators are particularly sensitive to the signal-to-noise ratio. On the other hand, it provides precise knowledge of the velocity fields at a given time, which might contribute to a better understanding of the respective contribution of the imaging modality and the fluid mechanics to the construction of flow mapping images. In addition, as shown by Thomas et al35 with the theoretical equation, the final formula (Equation 10) that was validated in the present study allows us to calculate the area of any isovelocity surface in the jet. It can be demonstrated that this surface area (S) is proportional to the square of the velocity at the orifice and, based on the Bernoulli equation, to the pressure drop driving the jet (ΔP): \mathit|<|S|<|=|>|K|>||<|\cdot|>||<|\Delta|>|\mathit|<|P|>||<|\cdot|>|\mathit|<|d|>|^|<|2|>|/\mathit|<|V_|<|isovelocity|>||>|
Thus, fluid mechanics can be used to predict, at least in part, the sensitivity of color flow mapping to the load conditions reported in patients.
The length of the central laminar core has the potential to be of great clinical usefulness. This parameter is both proportional to the orifice diameter and independent of the velocity and, thus, allows evaluation of the effective regurgitant orifice. So far, it cannot be precisely imaged with the use of ultrasound due to the angle dependency and aliasing, which prevent the measurement of high velocities. On the other hand, color Doppler tends to have adequate resolution. MRI does not provide adequate images due to the ratio between the duration of the sequences and the flight time of the blood elements within the core and due to the vortices. Newer sequences might, at least in part, overcome this limitation. In the future, if the data collected on noncircular orifices or adhering or confined jets prove that this length is preserved, it will appear as an extremely interesting parameter and might prompt the development of antialiasing systems. The combination of the length of the core with a continuous-wave Doppler (or MRI) measurement of the orifice velocity could give a calculation of the regurgitant volume in a large variety of lesions.
With the use of laser Doppler anemometry in an vitro model of valvular regurgitation, velocity fields were acquired in pulsatile conditions. The collected data showed that in a free jet, the momentum is conserved throughout the cycle. The transverse velocity profiles are approximately similar, leading to a hyperbolic velocity decay that is established only at peak ejection. Finally, the dimensions of the central laminar core were shown to be proportional to the orifice diameter. These different relations can be combined in a single formula describing the velocity field and accounting for the sensitivity of ultrasound color Doppler to load conditions. On the other hand, further studies are needed to provide descriptions of the various singularities often encountered in pathology.
Reprint requests to Dr B. Diebold, Hoˆpital Broussais, 96 rue Didot, 75674 Paris, Cedex 14, France. E-mail email@example.com.
- Received October 31, 1995.
- Revision received January 3, 1996.
- Accepted January 15, 1996.
- Copyright © 1996 by American Heart Association
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