J Med Ultrasonics (2015) 42:323–330
DOI 10.1007/s10396-015-0620-x

ORIGINAL ARTICLE

Temporal averaging of two-dimensional correlation functions
for velocity vector imaging of cardiac blood flow
Hiroki Takahashi • Hideyuki Hasegawa •Hiroshi Kanai

Received: 20 October 2014 / Accepted: 12 February 2015 / Published online: 17 March 2015
Ó The Japan Society of Ultrasonics in Medicine 2015

Abstract
Purpose Ultrasonic imaging of blood flow in the cardiac
lumen is a very useful tool to evaluate the pumping function of the human heart. The speckle tracking techniquemakes it possible to estimate the blood velocity vector.
However, a stable estimation of the velocity vector ofblood flow is difficult because signal-to-noise ratios of
echoes from tiny blood particles are low. In this study, thespeckle tracking technique with averaging of multiple twodimensional correlation functions was employed for stableestimation of the blood velocity vector.
Methods Multiple two-dimensional correlation functions
can be averaged during a very short period by using the echodata acquired by high-frame-rate echocardiography withdiverging beam transmission. A steady flow experimentusing blood-mimicking fluid (mean fluid velocity 0.2 m/s,
flow angle 56° from the transducer surface) was implemented to investigate the effect of the averaging of twodimensional correlation functions at a frame rate of 6024 Hz.
Results First, to examine the averaging duration required
for stable estimation of the flow velocity vector, the accuracies of vector estimates were evaluated at different durations for averaging of two-dimensional correlation functionsin the steady flow measurement. It was found that the proposed averaging process with an averaging duration of over 8
frames could reduce the directional error in vector estimation

to almost half that of the conventional speckle tracking
technique. In subsequent experiments, the averaging duration was set at 12 frames corresponding to 2 ms. Measurements of steady flow at higher velocities were further
implemented. The steady flow measurements with higher
flow velocities of 0.4 and 0.6 m/s were simulated bychanging the frame interval of the echo data at a flow velocityof 0.2 m/s. Although the averaging duration was a mere
2 ms, directional errors at mean flow velocities of 0.2, 0.4,
and 0.6 m/s were reduced significantly. In an in vivo experiment of the healthy human heart, to produce a fineB-mode image, the diverging wave transmissions with different steered angles for compounding were interleaved in
the transmission sequence. From the in vivo experimental
result, the blood velocity vector of the left ventricular cavity
showed the flow getting into/out of the cavity in ejection andearly diastolic phases. Furthermore, estimated flow directions revealed rotating flow in the cavity in mid-diastole.
Conclusion Our proposed method has the feasibility tovisualize the vortex flow by velocity vector mapping
without a contrast agent.
Keywords Cardiac blood flow  Blood velocity vector 
Speckle tracking  High-frame-rate acquisition

Introduction
H. Takahashi (&)  H. Hasegawa  H. Kanai
Graduate School of Biomedical Engineering, Tohoku University,
6-6-05 Aramaki-aza-Aoba, Aoba-ku, Sendai, Miyagi 980-8579,
Japane-mail: h.takahashi@ecei.tohoku.ac.jp
H. Hasegawa  H. Kanai
Graduate School of Engineering, Tohoku University, 6-6-05
Aramaki-aza-Aoba, Aoba-ku, Sendai, Miyagi 980-8579, Japan

Ultrasonic imaging of blood flow in the cardiac lumen is avery useful tool to evaluate the pumping function of thehuman heart. This has been a major diagnostic modality toinvestigate problems of the human heart [1]. Color Doppler
flow imaging (CDI) is a typical method to obtain the information of intracardiac blood flow and is on-board in
most commercial systems. Recently, the frame rate of

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Doppler flow imaging was dramatically improved using
parallel beamforming with plane wave transmission [2, 3].
The high-frame-rate imaging with plane wave transmission
has been employed for measurement of the propagation
velocity of the pressure wave in a local region in the carotid artery [4, 5]. CDI, however, does not show the direction of blood flow because only velocity along an
ultrasonic beam is measured.
In a different approach, the transverse component (perpendicular to the ultrasonic beam) of blood flow velocity
has been mathematically calculated from the distribution of
the axial velocity obtained by CDI on the basis of hydrokinetic assumptions [6]. In addition, another methodwith the continuity equation using velocities of surrounding heart walls as boundary conditions has been proposed[7]. Recently, for improvement of the accuracy of themethod, optimal smoothing of the axial velocity distribution and weighting at the bilateral velocities of the heart
walls have been examined [8]. However, such assumptive
approaches cannot visualize real blood flow.
On the other hand, the blood velocity vector was estimated by combining Doppler measurements from two or
more directions [9–11]. This application, however, requiresa large angle formed by the intersection of beams; thus, it islimited to blood flow in a shallow region. A phased-arraytransducer with a small aperture, which is normally used in
cardiac measurement, is not suitable for such a combined
Doppler method. As a result, a linear-array transducer witha large aperture is being used in ongoing studies.
Echocardiographic particle image velocimetry (E-PIV)
realizes the mapping of velocity vectors and stream lines ofblood flow that are computed on the basis of motions ofultrasonic echoes from a contrast agent [12, 13]. However,
the intravenous injection of a contrast agent is required. On
the other hand, Løvstakken et al. [14] visualized motions ofechoes from blood particles utilizing CDI with customized
beam sweeping without the use of a contrast agent. To solve
the fatal issue of an insufficient frame rate with this method,
Hasegawa and Kanai [15] have developed a method for highframe-rate imaging of echoes from blood particles inside thecarotid lumen at a frame rate of several kilo-hertz with plane
wave transmission and parallel receive beamforming using alinear-array transducer. In cardiac ultrasound, the ultrasonicbeam is swept in a sector fashion. Hence, such plane wave
imaging significantly limits the frame rate in cardiac measurement because the widths of insonified plane waves do notincrease with the range distance from the transducer in thenear field, whereas lateral intervals of scan lines increase
with range distance.
For visualization of blood flow with a velocity of over
1 m/s and complex flow patterns such as vortex flow in the
cardiac lumen, high-frame-rate measurement is desirable.
In our previous study, we proposed a method for

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visualization of the direction of intracardiac blood flowusing high-frame-rate measurement with a spherically diverging beam [16, 17]. In that study, blood velocity vectors
in the cardiac lumen could be estimated by speckle tracking [18]. However, it was confirmed that the accuracy was
not sufficient even in steady flow experiments. In addition,
the quality of the B-mode image was much lower than thatof the conventional image due to the number of compounding per frame of a mere two or three.
In the present study, we proposed an improved estimatorof blood velocity vectors. In the proposed estimator, twodimensional correlation functions (2D-CFs) between consecutive frames were temporally averaged for stable estimation of blood velocity vectors, and the averaged
correlation function was up-sampled. High-frame-rate
measurement with transmission of diverging beam contributes to suppress degradation of the temporal resolution in
the vector estimation suffered from averaging of 2D-CFs
because multiple 2D-CFs during a very short period can beused owing to the high-frame-rate measurement. Furthermore, to acquire a B-mode image with better image quality, alarge number of steered diverging beams for compounded
B-mode imaging were interleaved before every transmitting
event for estimation of blood velocity vectors.

Materials and methods
Echo data acquisition scheme
The acquisition of echo signals at a high frame rate is
implemented with parallel receive beamforming withtransmission of spherically diverging beam (PBF-DB) [19,20]. With PBF-DB, the ultrasonic beam is diverged from avirtual point (sound) source (VPS) placed behind the arraysurface. Then, multiple receiving beams focusing on different spatial positions are generated using echo signals
received by transducer elements in parallel. Therefore,
PBF-DB can facilitate high-frame-rate measurement ofecho signals from the human heart by decreasing thenumber of transmissions per frame. The lateral spatial
resolution of PBF-DB is, however, degraded due to low
directivity of the transmitted ultrasound. To compensate forit, compounding of echo signals obtained by diverging
beam transmissions with different steering angles is effective [19]. Although the lateral resolution after compounding is further improved by increasing the number oftransmissions with different steering angles, the frame rateis decreased due to the increase in the number of transmissions per frame. For estimation of the velocity vector ofblood flow with a velocity of over 1 m/s and complex flow
like vortex flow in the cardiac lumen, high-frame-rate
measurement is desirable to suppress the signal

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decorrelation caused by the variation of relative distances
between scatterers, which corresponds to blood particles.
In this study, a large number of steered diverging
beams for B-mode imaging were interleaved before everytransmitting event for acquisition of blood velocity vectors. Hence, the transmission sequence comprised transmissions of steered diverging beams for acquisition ofcompounded B-mode images and single non-steered diverging beams per frame for estimation of the velocity
vector of blood flow, as illustrated in Fig. 1a. In thetransmission event for B-mode imaging, 15 diverging
beams with angular intervals of 6° were transmitted, andthe corresponding beamformed signals were compounded
to produce the B-mode image with better image quality,
as shown in Fig. 1b. On the other hand, in the transmission event for blood vector acquisition, a single non-steered diverging beam was transmitted and the
beamformed signals were generated without compounding, as shown in Fig. 1c. This diverging beam wastransmitted in the same direction in 40 frames for a robust
estimation of the velocity vector by 2D-CF averaging, as
described in next section. Finally, the acquisition frame
rate of the B-mode image overlaid with the distribution of
the blood velocity vector is defined as:
FR ¼

PRF
Ntx1 þ Ntx2

ð1Þ

where PRF, Ntx1, and Ntx2 are the pulse repetition frequency, the number of steered diverging beam transmissions for B-mode imaging, and the number of single
diverging beam transmissions for blood vector estimation,
respectively. In this study, PRF, Ntx1, and Ntx2 were set at6024 Hz, 15, and 40, respectively. The VPSs in transmission events of B-mode and blood velocity vector imaging

were placed at 100 and 50 mm behind the array surface,respectively.
Echo signals were measured by a modified diagnostic
ultrasound system (Hitachi-Aloka Medical Alpha 10). The
phased-array transducer was operated at a center frequencyof 3.75 MHz. Individual RF echo signals received by 96
elements were sampled at 15 MHz. Using these receive
echoes, echo amplitudes on 241 scan lines with an angularinterval of 0.375° were created by delay-and-sum focusing.
Figure 1d shows an illustration of the created scan lines.
These beamforming processes were performed in alltransmissions for B-mode imaging and mapping blood
velocity vectors. Note that the receive beamforming andthe proposed method were implemented with off-line
processing on a personal computer.
Estimation of blood velocity vector
At the transmission event for blood vector acquisition, thereceive-beamformed signals were high-pass filtered in the
direction of the frame (called moving target indication
filter) to visualize echoes from blood particles by suppressing clutter echoes from stationary and slowly moving
tissues except for blood such as the ribcage and myocardium. High-pass filtering was performed by a 12thorder Butterworth filter whose cutoff frequency was set to500 Hz corresponding to the axial velocity of almost
0.1 m/s at the ultrasound frequency of 3.75 MHz. The
amplitude of filtered signals was weighted by the coherence function of signals filtered in the direction of theframe to enhance echoes from blood particles coherently
moving [17]. The number of frames used for computationof the coherence was determined so that echoes from almost the same scatterers were used during the period. Here,Fig. 1 a Emission sequence of
spherically diverging ultrasoundand illustration of wavefront oftransmit ultrasound in events
for, b B-mode imaging andc estimation of blood velocity
vector. d Illustration of
positions of scan lines createdby receive beamforming inevery transmission

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we estimate the elapsed time in which blood cells passthrough the ultrasonic beam transversely. The width at
half-maximum of the lateral profile of an echo from a finewire (i.e., spatial resolution) is almost 1.9 mm. On theassumption that the transverse velocity (perpendicular to
the ultrasonic beam) of blood flow is 1 m/s, a blood cell isestimated to slip from the ultrasonic beam during 11 frames
at a frame rate of 6024 Hz. In the present study, the numberof frames for the coherence estimation was set at neighboring 6 frames to fill this necessary condition (\11frames) for the use of echoes from the same scatterers.
The speckle tracking technique with computation of 2DCF makes it possible to estimate the blood velocity vector
without depending on the angle between an ultrasonicbeam and the blood flow [18]. However, blood cells are
small and acoustic outputs of defocused ultrasonic beams
employed for high-frame-frame measurement are low.
Under such conditions, signal-to-noise ratios (SNRs) ofechoes from blood cells are low; therefore, it can bechallenging to accurately estimate the blood velocity vector. In this study, to compensate for this issue, the velocity
vector of blood flow was estimated by speckle tracking
with 2D-CF averaging for a stable estimation of the velocity vector.
There are several types of averaging operations throughmultiple frames for estimation of the velocity vector: (1)averaging of velocity vectors computed in each frame; (2)averaging of unnormalized 2D-CFs; (3) averaging of 2DCFs normalized by the variance. Low SNRs of echoes fromtiny blood particles can derive the peak at the wrong position(displacement) on the 2D-CF because the correlation coefficient at the true peak decreases due to low SNRs. The risein the correlation coefficient at the wrong position on the 2DCF should be suppressed to avoid errors in estimation of thevelocity vector. The correlation coefficient at the true position on the 2D-CF is considered to be high on average during
a short duration. On the other hand, deviation of the correlation coefficient caused by the noise component in echoes
can be assumed to be random with respect to each interFig. 2 Schematic illustration of
temporal averaging of
normalized 2D-CFs for stable
vector estimation

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frame, which causes the wrong peak on the 2D-CF. Hence,
the averaging operation of 2D-CFs for the short period iseffective to produce the peak at the true position on the 2DCF by suppressing the rise in the correlation coefficientcaused by low SNRs of echoes. In the present study, the 2DCF normalized by the variance was used for the averaging
operation to emphasize the temporal change in the phase of
echoes for estimation of the flow velocity vector.
Normalized 2D-CFs between two consecutive frames at
a specific spatial point are temporally averaged Ncf times(the number of inter-frame) as follows:
N =21
1 cfX
C ði; j; n þ kÞC ði; j; nÞ ¼
Ncf k¼N =2

ð2Þ

cf

where C (i, j, n) denotes the normalized 2D-CF of thelateral and axial lags of i and j, respectively, between the
nth and (n ? 1)th frames. Figure 2 illustrates the proposed
averaging process of normalized 2D-CFs. Then, the averaged 2D-CF is up-sampled with a reconstructive interpolation to follow a small displacement between consecutive
frames below the sampling interval [21].
The spatial resolution cell of the speckle tracking technique is regarded as the size of the spatial window forcomputation of 2D-CFs. Let us define this spatial window
as the kernel. The kernel sizes in the lateral and axial directions are set to 9° (corresponding to 24 lines) and
8.2 mm, respectively. The signal amplitude in the kernel isweighted by the two-dimensional Hann window function.
The effect on accuracy of blood vector estimation from thedifference in Ncf is examined in the following section.

Steady flow measurement
Experimental setup
The aim of this experiment was to examine the effect of
2D-CF averaging. Figure 3 illustrates the experimental

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327

Fig. 3 Experimental system formeasurement of steady flow.
The blood-mimicking fluid was
moved steadily through the flow
path by a screw pumpsetup. Echoes from a blood-mimicking fluid (ATS 707)
flowing in a cylindrical flow path surrounded by a tissuemimicking rubber (ATS 523A) were measured. The diameter of the flow path was 8 mm. The transducer surface
was placed at a distance of 7 cm from the outer surface ofthe flow path. The angle between the transducer surface
and the flow path was kept at 56°. The blood-mimicking
fluid was kept flowing steadily using a screw pump
(Heishin 2NL10PU). The flow rate was measured by amass flow meter (Keyence FD-SF8) connected to the flowpath.
The echo data were measured at a flow rate of 0.6
L/min. Higher flow rates of 1.2 and 1.8 L/min were
simulated by changing the frame interval for estimation of
the correlation function, i.e., twice and triple the originalframe interval. Therefore, a Reynolds number of 960 at aflow rate of 0.6 L/min was maintained across thesesimulated data. These flow rates of 0.6, 1.2, and 1.8 L/mincorresponded to the mean fluid velocities of 0.2, 0.4, and
0.6 m/s, respectively. The step size of the flow rate measured by the mass flow meter was 10 mL/min. The echodata were acquired at a frame rate of 6024 Hz with thesingle transmission of a non-steered diverging wave perframe. The VPS was located at 50 mm behind the arraysurface.

Fig. 4 Velocity vectors overlaid with B-mode image (gray scale) andecho speckles of blood-mimicking fluid (hot scale). A subset of
estimated velocity vectors along the direction perpendicular to theflow axis was visualized

Experimental results
Velocity vectors of the fluid inside the flow path wereobtained at an interval of 0.4 mm along the manually assigned line perpendicular to the flow axis, as shown in
Fig. 4.
To examine the averaging duration required for stableestimation of flow velocity vectors, directional errors of
estimated velocity vectors from the direction of the flow
path were evaluated at different durations of averaging.
The directional error edir was estimated as follows:
vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!)2 ﬃ
u Nj 1 (
X
1u
1
v

n
j
ð3Þ
cos1  edir ¼ t
p Nj j¼0
v j j nj
where vj and n represent the estimated velocity vector at
the jth spatial point and the unit vector parallel to the flow
direction, respectively. Figure 5 shows the directional

Fig. 5 Directional errors with respect to the various numbers offrames for averaging of normalized 2D-CFs at a flow rate of 0.6
L/min (a mean fluid velocity of 0.2 m/s). The standard deviation wascomputed from 50 frameserrors of estimated flow vectors at the different numbers offrames for averaging of 2D-CFs at a flow rate of 0.6 L/min(a mean fluid velocity of 0.2 m/s). The standard deviations

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in Fig. 5 were computed from 50 frames. The deviation of
directional errors in the vector estimation by the speckle
tracking technique with averaging of normalized 2D-CFs
was reduced to almost half that of the conventional speckle
tracking technique (non-averaging of 2D-CFs) at an averaging duration of over 8 frames. In the present study, Ncf
was set to 12, which corresponds to 2 ms, to prioritize thetemporal resolution in the estimation of velocity vectors.
Velocity vectors estimated with and without averaging
2D-CFs during 12 frames were compared based on flow
rates (mean flow velocities) and directional errors. Figure 6a shows flow rate Q estimated by the use of the velocity vectors as follows:

J Med Ultrasonics (2015) 42:323–330

Q¼p

N
j 1
X




rj Dr vj  n

ð4Þ

j¼0

where Dr and rj denote the interval between points at which
the vectors were estimated and the distance from the centerof flow path, respectively. Figure 6b shows directional
errors of the estimated velocity vectors from the directionof the flow path. The standard deviations in Fig. 6a, b werecomputed from 50 frames. In Fig. 6a, b, the red and green
crosses indicate estimates yielded by the speckle tracking
technique with and without the averaging operation of 2DCFs, respectively. Although the averaging duration of 2DCFs was a mere 2 ms, both errors were significantly reduced by 2D-CF averaging.

In vivo experimentFig. 6 a Flow rates (mean fluid velocities) of blood-mimicking fluid
estimated by the proposed method and a mass flow meter andb directional errors between the flow axis and velocity vectors
estimated by the proposed method. The red and green crosses indicate
estimates yielded by speckle tracking with and without averaging of
normalized 2D-CFs. The horizontal axis represents the flow rate,
which was increased by changing the frame interval in the echo dataacquired at a flow rate of 0.6 L/min (a mean fluid velocity of 0.2 m/s)
measured by the mass flow meter123

RF echo signals from the left ventricle were acquired in atransthoracic three-chamber view of a 27-year-old healthy
male. The healthy subject gave informed consent to thestudy. Ntx1 and Ntx2 were 15 and 40, respectively; therefore, the eventual frame rate was 109 Hz, as described in‘‘Echo data acquisition scheme’’. One blood vector image
was obtained using 40 successive transmissions of nonsteered diverging beams. The VPSs used for B-mode andblood velocity vector imaging were located at 100 and
50 mm behind the array surface.
Figure 7 shows the information obtained for blood flow
overlaid on compounded B-mode images in frames in earlysystole, early diastole, and mid-diastole. As for the bloodflow information, the cyan-colored arrow and the intensityof the hot color indicated the direction and the velocity
magnitude of the blood velocity vector, respectively. The
vector magnitudes at all pixels were obtained using reconstructive interpolation of the vector velocity field obtained by the speckle tracking technique with 2D-CF
averaging. The mitral valve and the aortic valve are locatedto the basal side of the left ventricle. As shown in Fig. 7a,b, the fluxes flowing into and out of the cardiac cavity werevisualized by blood flow vectors in early systolic and early
diastolic phases. In mid-diastole, the vortex-like flow appeared as shown in Fig. 7c. Recently, complex flow such asvortex formation has been verified and studied by magnetic
resonance imaging [22] and E-PIV. The proposed method
shows great potential for visualizing vortex flow using
contrast-free ultrasound.
Discussion
In the case of conventional E-PIV, it has been reported thatthe frame rate (below 200 Hz, corresponding to the

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in 21 frames weighted by the coherences were obtained,
which was greater than the 12 frames required for averaging of 2D-CFs. The number of acquired frames for bloodflow imaging can be reduced in optimization of the acquisition sequence.
With respect to cardiac blood flow imaging, the visualization of complex flow such as vortex flow has become
important. As shown in Fig. 6b, the directional error of theestimator for the velocity vector was reduced from 11.2 to
4.5 % at a mean fluid velocity of 0.2 m/s by averaging of
normalized 2D-CFs. A vector estimator with a better direction accuracy helps the visualization of vortex flow
patterns. Furthermore, the diverging wave transmissions
with different steered angles for compounding were interleaved in the transmission sequence to produce a fineB-mode image. Nevertheless, the frame rate in our study iscomparable to that of several tens of hertz of conventional
CDI; moreover, the blood velocity vector in the cardiac
lumen can be estimated, which is the technical merit of our
proposed method.

Conclusions
Fig. 7 Velocity vector images of the cardiac lumen in a systole,
b early diastole, and c mid-diastole. The direction and the velocity
magnitude of the blood velocity vector are indicated by the cyancolored arrow and the intensity of hot color

temporal resolution of over 5 ms) is not sufficient to estimate the flow velocity vector inside the cardiac lumen [23].
Normalized 2D-CF computing between a very short frameinterval (a couple of hundred microseconds) comes fromthe high-frame-rate measurement with diverging wave
transmission. In this study, speckle tracking with averaging
of normalized 2D-CFs was proposed for stable estimationof blood velocity vectors. The velocity vector obtained byaveraging of 2D-CFs means a coherent component of bloodflow vectors during the averaging duration. In the proposed
method, normalized 2D-CFs of 12 frame intervals could be
used for the averaging process during the very short duration of 2 ms owing to very high-frame-rate acquisition.
Although the averaging duration was only 2 ms, the accuracy of the velocity vector estimator was strongly improved, as shown in the experimental result of steady flow.
Therefore, fine blood vector estimation with the hightemporal resolution of 2 ms can be achieved with our
proposed method.
In the current sequence for mapping blood velocity
vectors, there were unnecessary transmissions for the averaging duration of 12 frame intervals. From echoes in 40
frames, echoes in 27 frames were obtained by 12th order
Butterworth filtering. Because the coherence was computed
using echoes in 6 frames after Butterworth filtering, echoes

In this study, velocity vector estimation by the speckle
tracking technique with averaging of normalized 2D-CFs
was proposed for stable estimation of the blood flow
vector in the cardiac lumen. The accuracies of estimatesof flow rate (corresponding to flow velocity) and direction
were evaluated by means of a steady flow experiment witha blood-mimicking fluid. The experimental results confirmed that the accuracy was improved by the proposed
averaging procedure for normalized 2D-CFs. Furthermore, an in vivo experiment of healthy human heart was
performed using an interleaved sequence of ultrasoundtransmission. This transmission sequence alternatedtransmissions of steered diverging beams for fine B-mode
imaging and transmissions of a single diverging beam perframe for estimation of the blood velocity vector bymeans of speckle tracking with averaging of normalized
2D-CFs. The distribution of intracardiac blood vectors
with the compounded B-mode image can be obtained at aframe rate of 109 Hz, which is comparable to conventional CDI. The blood flowing into/out of the cardiac
cavity in ejection and early diastolic phases, and thevortex flow in diastole, could be visualized in the distribution of blood velocity vectors.
Acknowledgments This work has been supported by the JSPS
Research Fellowship (25-5479) and the International Advanced Research and Education Organization of Tohoku University.
Conflict of interest There are no financial or other relations that
could lead to a conflict of interest.

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Ethical standard The in vivo experiment in this study was approved by the institutional ethical committee. The subject gave informed consent to this study.

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