The three-dimensional flow field generated by a feeding calanoid copepod measured using digital holography
1 Johns Hopkins University, Department of Mechanical Engineering, N. Charles
Street, Baltimore, MD 21218, USA
2 Great Lakes WATER Institute, University of Wisconsin-Milwaukee, Milwaukee,
Wisconsin 53204, USA
* Author for correspondence (e-mail: Katz{at}jhu.edu)
Accepted 7 July 2003
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Summary |
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Key words: digital in-line holography, particle image velocimetry, copepod, Diaptomus minutus, flow
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Introduction |
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Current knowledge is largely based on video observations, including a
Schlieren system for visualizing wakes
(Strickler, 1977), high-speed
cinematography for understanding how food particles are captured and feeding
currents are generated (Koehl and
Strickler, 1981
; Strickler,
1984
), and three-dimensional (3-D) video tracking of free-swimming
copepods in particle fields
(Paffenhöfer et al.,
1995
; Strickler,
1985
). The latter combines particle tracks at different times to
observe two-dimensional (2-D) flow fields in lateral and dorsal views. More
recently, Particle Image Velocimetry (PIV) has been used to map instantaneous
2-D flow fields around tethered specimens
(van Duren et al., 1998
).
However, since flow fields around copepods are 3-D, unsteady and vary with
swimming speeds and orientations (Bundy and
Paffenhöfer, 1996
;
Strickler, 1982
;
van Duren et al., 1998
), 3-D
measurements are essential. Considering the importance of copepods in the
aquatic and marine food webs, and 30 years of related hypotheses, it seems
worthwhile to investigate whether digital cinematographic holography may break
barriers and allow us to test these hypotheses.
Unlike video microscopy, holography maintains the same lateral resolution
over a substantial depth (Vikram,
1992). This advantage has led to the development of several
submersible holography systems for studying plankton, starting with a sample
volume of a few ml (Carder et al.,
1982
), to samples of one liter and above
(Katz et al., 1999
;
Malkiel et al., 1999
;
O'Hern et al., 1988
;
Watson et al., 2001
). The
latter utilize pulsed lasers and emulsions as recording media. Cinematographic
holography was introduced for laboratory research decades ago
(Heflinger et al., 1978
;
Knox and Brooks, 1969
), but
difficulties in acquiring and processing data resulted in limited
applications. Recent development in digital imaging, which simplifies the
acquisition, and computing power, which enables numerical reconstruction, has
led to renewed interest in cinematic holography
(Kebbel et al., 1999
;
Owen and Zozulya, 2000
;
Xu et al., 2001
). The limited
resolution of digital imaging, which is at least an order of magnitude lower
than that of holographic emulsions, restricts us to in-line holography. This
technique (details follow) maximizes the fringe spacing. Because the reference
beam typically passes through the sample volume, it becomes increasingly
degraded with increasing particle concentration. Consequently, the
reconstructed images are noisier and the maximum particle concentration is
lower compared to off-axis holograms
(Zhang et al., 1997
).
Holographic PIV is the only technique to date that can measure a 3-D
instantaneous velocity distribution over a finite volume
(Barnhart et al., 1994;
Pu and Meng, 2000
;
Sheng et al., 2003
;
Tao et al., 2002
;
Zhang et al., 1997
) at a
resolution of millions of vectors. The velocity is obtained by recording two
exposures of a flow field seeded with microscopic particles, and measuring the
displacement of these particles. However, the depth coordinate of a
reconstructed particle is less accurate than the lateral coordinates, severely
reducing the ability to estimate the corresponding velocity component. This
problem has been solved by recording two inclined holograms, using each for
determining a 3-D distribution of two velocity components, and matching the
two sets to obtain the 3-D velocity. Utilizing emulsion and off-axis
holography, Tao et al. (2002
)
measured 136x130x128 3-D velocity vectors in a cubic sample with
sides of about 45 mm. By inserting an inclined mirror in the path of the
illuminating beam inside the test facility, the incident and reflected beams
create two views that can be recorded on the same emulsion
(Sheng et al., 2003
). Using
off-axis holography, this method enables measurement of particle locations to
within 7 µm, and resolves about 200 particles mm-3. We adopt
this approach, but use digital in-line holography to record a time series of a
free-swimming copepod and the flow field surrounding this animal. To our
knowledge, this is the first time that digital holographic PIV has been
implemented as a tool for simultaneous observation of the animal's behavior
and measurement of the complex flow around it.
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Materials and methods |
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The expanded beam illuminates an 80 ml sample volume constructed of antireflection-coated windows attached to two supporting prisms. First-surface mirrors are attached to the inclined side walls of the test chamber to direct the illumination beam through the test section. These mirrors provide two perpendicular views of objects located in regions where the incident and reflected beams overlap (cf. Fig. 2). One view is created by light that is first reflected from the mirror and then incident on the object, while the other (mirrored view) is generated by light incident on the object and then reflected off the mirror. The two views are recorded onto a single recording plane, but they are laterally separated. During reconstruction, the first view appears in focus at its original location, while the second perpendicular view is a mirror image (right side of Fig. 2). Use of two mirrors (Fig. 1) doubles the triangular overlapping volume without necessitating an increase in recording area.
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The interference patterns are recorded on a lensless Megaplus ES 4.0
digital camera, which has a 2048x2048 pixel CCD sensor with a 7.4 µm
pitch, providing a 15 mmx15 mm field of view. The in-line system
minimizes the angle between the light scattered from the objects and the
remaining reference beam. Consequently, the fringe spacing is maximized
(Vikram, 1992), overcoming the
limited resolution of the CCD. The CCD's interline transfer capability
eliminates the need for a shutter. Because the experiment involves recording
moving objects (a copepod and seed particles), the exposure time is adjusted
to be short enough to prevent the smallest resolvable fringe spacing from
smearing. Here, the velocity does not exceed 4 mm s-1 (540 pixels
s-1), and the electronic shuttering of the camera (0.1 ms exposure
time) is sufficient to reduce the movement during exposure to much less than 1
pixel. Faster flows (
1 m s-1) would require higher power, even
pulsed lasers, to generate the energy required for recording a hologram.
Images have been recorded at the maximum frame rate of the present camera, 15
frames per second, buffering 10 s segments (150 frames) to the RAM of the
computer hosting the image acquisition card.
Our subject animal, a female Diaptomus minutus Lilljeborg 1889, was freshly caught in Lake Michigan and transported to Baltimore in a Dewar's jar. The test section was filled with artesian well water from the Pryor Street well in Milwaukee. It was seeded with monodisperse, 20 µm diameter polystyrene spheres (specific gravity 1.05) at a concentration of 4 mm-3. Although denser seeding would improve the spatial resolution of the flow, it would also reduce the signal-to-noise ratio of reconstructed images. Image acquisition to RAM was started when the copepod appeared in the field of view. Successful image sequences, which captured the copepod in a region with two views, and were sufficiently far from the wall (>4 mm), were stored. Fig. 3 is a sample digital in-line hologram containing two interference patterns (`shadows') of the same copepod, along with the very faint diffraction patterns of the seed particles.
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Digital reconstruction regenerates the original light intensity
distribution at any desired distance from the recording plane. The amplitude
of light U(x,y,z) at any point in the reconstruction volume
is a superposition of light reaching this point from an array of source
wavelets distributed over the entire recording plane (the hologram). Using the
Fresnel-Huygens Principle (Hecht,
2002) with a paraxial approximation,
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The purpose of the next phase of the analysis is to match the in-focus
perpendicular views of particles in order to determine their exact locations
in space. Recall that a single view can provide lateral positional accuracy to
within a pixel (7.4 µm) in the lateral plane, but is substantially less
accurate in the axial direction (0.5 mm). These views are laterally separated
(as in Fig. 3). An automated
procedure for matching the two views in densely seeded flows recorded using
off-axis holography (on emulsion) is outlined in Sheng et al.
(2003). There, the array of
reconstructed images is thresholded, the particles are segmented into 3-D
volumes, and then reduced to line segments that pass through the barycenters
of these volumes and have lengths corresponding to their axial extent. The
intersection of two perpendicular line segments determines the 3-D coordinates
of the particle centroid. The analysis requires the calibration of the
location and orientation of the mirror, achieved by manually matching
corresponding views of several reference particles. The calibrated mirror
orientation is used to transform (flip) the mirrored views onto their original
locations in space. For two views of the same particle to be matched, we
require that the perpendicular segments are less than a particle diameter away
from each other. The 3-D coordinates of the particle centroid are approximated
as the midpoint of the shortest line segment connecting the views.
This approach did not work while analyzing the present in-line digital holograms. The primary reason was excessive background noise resulting from the reference (illuminating) beam passing through the sample volume. It should work with less heavily seeded flows and/or an optical setup with a separate reference beam that does not pass through the sample volume. The limited resolution of the digital camera prevented the implementation of off-axis holography, which is characterized by much finer fringe spacing. Consequently, we have used two semi-manual techniques to obtain the 3-D locations of particles and velocity fields.
The first task is to identify particles and distinguish them from background noise. Three successive exposures are superposed, and elongated traces, or three closely spaced spots resulting from the displacement of particles, are identified. This approach has turned out to be a very effective tool for distinguishing between real particles and speckle noise. Based on the location of these traces, we estimate (computationally) the most likely location of the mirrored view. The linear transformation used for this calculation is calibrated by matching corresponding views of the copepod's extremities, which can be easily identified. The mirrored view lies along an almost horizontal line that corresponds to the depth uncertainty of the first view. If we find three spots (or elongated traces) along this line, the two views of the same particle are matched. At the present particle concentration, having more than one match is very unlikely. If we do not find the second view of a particle along the expected line, for example, when it is located in the shadow of the copepod, the unmatched trace is disregarded.
When the second view is found, the program proceeds to measure the particle
displacement in each of the two views, using cross-correlation of the
intensity distribution, generated by the three exposures. Details on the
cross-correlation procedures, including methods of achieving sub-pixel
accuracy, are discussed by Roth and Katz
(2001), Roth et al.
(1999
) and Sridhar and Katz
(1995
). The lateral
displacement in each of the views and the average vertical displacements are
combined into a 3-D velocity vector, positioned at the particle mean location.
The uncertainty in velocity of individual particles is about 0.05 mm
s-1.
Extended particle tracks in the vicinity of the copepod are more easily identified. Superposing sets of ten exposures near the plane of the copepod in each view, and placing them side-by-side, allows identification of corresponding tracks based on their elevation at a given time. This identification is verified by comparing subsequent sets of exposures. Combining the sets generates the 3-D trajectory of the particle. Even when one of the views is partially blocked by the copepod, it is usually possible to match the segments that are not blocked and then interpolate them to estimate the trajectory in the blocked section. The endpoints of segments are stored and used for measuring the velocity along the path of the particle. In regions with relatively high velocity, the tracks appear as a series of dots. In this case, individual traces are used for measuring the velocity as discussed before.
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Results |
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The reconstructed images from 15 sequences consistently show that the copepod sinks for several seconds. It then executes a short hop upwards and resumes sinking (see supporting movie). As it sinks, the copepod generates a feeding current by moving its feeding appendages. The present 15 Hz recording rate may not be sufficient for following the (high frequency) motion of the appendages, but different phases of their motion are discernible (Fig. 4). The `high'-speed flow generated by the appendages (Fig. 5A) is most evident in the ventral region of the copepod (z<79.5 mm), extending to its anterior and posterior regions. The feeding current generates a reaction force that propels the copepod. This vertically directed force acts against the copepod's excess weight (weight minus buoyancy) and drag, reducing its sinking rate compared to the terminal speed (speed with no feeding current).
The recirculating flow pattern generated by the copepod is clearly demonstrated in Fig. 6 by combining 130 reconstructed images, shifted in time, so that the image of the copepod is fixed. Since the copepod velocity is constant during this period, the shift applied to each image is a linear function of time. Blurring of the copepod occurs due to slight variations in sinking speed (<5%), and motion of the appendages. Streaks generated by the seed particles are clearly evident in both views. Above the copepod, several particles are drawn towards the center of the copepod. Below and to the sides of the copepod, particles are ejected downwards, subsequently looping around and migrating upward (relative to the copepod), some of them touching its antennules. Quantitative data on the trajectory and velocity along the path of selected particles are presented in Figs 7 and 8. The velocity peaks in a narrow domain near the tips of the appendages, reaching 3.6 mm s-1, i.e. 12.5 times the sinking velocity. We use these data to estimate the propulsive force generated by the feeding appendages.
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Excess weight and propulsive force
The particles slightly beyond the reach of the antennules circumvent the
copepod, creating a closed streamline pattern, without a separated wake. This
pattern is characteristic of low Reynolds number flows
(ReL=UL/, U and L being
characteristic velocity and length scales, respectively, and
, the
kinematic viscosity of the liquid), such as Stokes flows with
Re<<1, or Oseen flows with R
1
(Pozrikidis, 1992
). The
present Re, based on the sinking velocity and prosome length, is
0.29. Based on the diameter of the recirculation zone, Re increases
to 1.2. Thus, the present flow lies in the transition region between Stokes
and Oseen flows. Here we use the simpler Stokes flow in order to estimate the
magnitude of the forces produced by the copepod.
The horizontal (u) and vertical (v) velocity components
induced by a vertical point force (Stokeslet) in Stokes flow
(Jiang et al., 2002c;
Pozrikidis, 1992
) are:
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The flow pattern generated by a Stokeslet in an absolute frame of reference
is illustrated in Fig. 9A, with
Uref=wexcess/8µL. A
recirculating flow pattern appears in a frame of reference moving downward at
a fraction of Uref
(Fig. 9B). For the
recirculation to form, there must be a residual downward velocity component
(jet) below the object in its own reference frame. This flow can only be
generated by a propulsive force. Thus, in reducing its sinking speed and
generating a propulsive feeding current, the copepod creates a recirculating
pattern that extends slightly beyond its antennae. This combination of sinking
and feeding is not discussed in detailed conceptual and numerical analyses of
the forces acting on a swimming copepod under various conditions (Jiang et
al.,
2002b
,2002c
).
They include cases of stationary bodies producing feeding currents (hovering
or conceivably tethered) and freely sinking bodies, both of which do not
generate recirculating patterns.
|
The data can be used for estimating wexcess. In a
reference frame sinking at vsink, the vertical velocity
component vanishes at
y0=±wexcess/4µvsink.
Estimating y0 as half the distance between the points with
zero velocity in Fig. 6
(y0=2 mm), and since vsink= 0.29 mm
s-1, one obtains
wexcess=7.2x10-9 N. With a volume of
1.1x10-10 m3 (determined following
Chojnacki, 1983
), the estimated
excess density of the copepod is 6.7 kg m-3 (1006.7 kg
m-3). This value falls in the range measured by Svetlichny
(1980
) and Knutsen et al.
(2001
). The same analysis can
be performed in any frame, including one without recirculation, by measuring
the velocity and distance between points above and below the copepod with the
same velocity.
One can also estimate the propulsive force, P, generated by the
feeding appendages. Averaging of a Stokeslet at y=0 over a
certain radius, R, one obtains
. Based on Figs
7,
8, there are two regions with a
radius of 200 µm and characteristic peak velocity of
mm s-1 in the
vicinity of the feeding appendages. Combining the two regions,
, i.e.
P=1.8x10-8 N, 2.5 times higher than
wexcess. At a constant sinking velocity, P must
balance the sum of wexcess and the drag force
(Jiang et al., 2002c
). Since
the relative velocity around the copepod is downward (Figs
5,
6,
7,
8), so is the drag, requiring
P to be larger than wexcess. Thus, the estimated
drag is about 1.5 times the excess weight. The propulsive force also generates
a moment (negative x direction), since it is applied at about 30
µm in front of the centerline of the prosome, based on the location of
maximum
. The magnitude of this
moment is about 5.4x10-13 Nm. To overcome this moment and
avoid tumbling, the copepod turns its tail, creating a velocity bias in the
positive z direction (see insert in
Fig. 8). Turning the flow
creates a reaction force and a moment in the positive x direction.
Based on the average velocity bias (
0.1 mm s-1,
Fig. 8), over a radius of 250
µm (half the tail length), the force is 0.6x10-9 N.
Multiplying it by the distance to the center of mass of the copepod (1 mm),
the estimated moment is 6x10-13 Nm. Thus, the moment
generated by the tail's redirection of fluid is of sufficient magnitude to
compensate against the moment of the propulsive force, allowing the copepod to
maintain a relatively vertical orientation.
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Discussion |
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Comparison with other techniques
It is useful to compare the capabilities and difficulties of digital
in-line holography with other 3-D particle tracking techniques, such as Stereo
PIV and 3-D particle tracking with multiple cameras. 2-D PIV
(Bartol et al., 2003;
Drucker and Lauder, 2001
;
van Duren et al., 1998
;
Wilga and Lauder, 2002
)
provides high density data, but only in a plane. Typical stereo PIV
(Nauen and Lauder, 2002
;
Prasad, 2000
) provides all
three velocity components, but still only in a plane. It also requires
elaborate calibration procedures. Existing scanning PIV
(Brucker, 1997
) and potentially
future stereo-scanning PIV, provide data in multiple planes at different times
at the cost of added complexity. All are unsuitable for following the 3-D
trajectories of swimming organisms.
Alternatively, multiple pinhole photography
(Kieft et al., 2002;
Maas et al., 1993
;
Moroni et al., 2003
;
Ott and Mann, 2000
;
Stuer et al., 1999
;
Virant and Dracos, 1997
) and
its variants (Pereira and Gharib,
2002
) can be used for 3-D tracking of particles. The increased
depth of focus required by this method is inherently coupled to reduced
resolution and the need for bright illumination. Conversely, in holography the
image resolution does not have to be compromised. Fine details, e.g. setae and
swimming appendages can be imaged at any depth. Furthermore, in pinhole
images, the elongated traces of particles extend in depth through the entire
volume of interest. Consequently, matching of perpendicular views can only be
performed at low particle concentrations. Multiple camera systems partially
overcome this effect and can provide as many as 1600 vectors per time step,
and may track as many as several hundred particles over extended periods
(Virant and Dracos, 1997
).
Such systems require extensive calibration and relatively complex processing
algorithms. In in-line holography, the elongated traces extend in depth less
than 1 mm, enabling matching of views at concentrations as high as several
particles per mm3 (i.e. 13 500 in the present overlapping volumes).
The associated calibration process is also straightforward. On the other hand,
one of the shortcomings of in-line holography, as shown in this paper, is the
noise generated in a great part by deterioration of the reference beam. This
problem can be partially resolved by using a separate reference beam that does
not pass through the sample volume. A separate reference beam should also
allow a much higher seeding concentration, but at the cost of added complexity
to the optical setup. When using high-resolution emulsions, instead of a
digital camera, we have successfully reconstructed 200 particles per
mm3. Another shortcoming of holography is inherent to the use of
coherent light, which makes the system more sensitive to the quality of
windows and variations of density within the sample volume.
As shown in this paper, digital holographic PIV is a relatively simple, but powerful tool for analysis of 3-D copepod (or other organism) dynamics and its interaction with its local environment, be it with other organisms or the local 3-D flow field.
Copepod behavior
While generating the feeding current (Strickler,
1982,
1984
) the mouthparts, studded
with chemoreceptors (Friedman and
Strickler, 1975
), scan the water flowing by them. When the
presence of a food particle is perceived, additional movements of the
mouthparts capture the particle and bring it to the mouth
(Koehl and Strickler, 1981
;
Strickler, 1984
,
1985
). Sensory setae on the
stretched out antennules (Fig.
4; Huys and Boxshall,
1991
) are mechanoreceptors
(Strickler and Bal, 1973
), as
well as chemoreceptors (Bundy and
Paffenhöfer, 1993
). These receptors enlarge the volume of
water scanned for food (Jiang et al.,
2002a
; Strickler,
1985
). Particles that do not smell `good enough' are either
actively rejected after capture by the mouthparts, or passively rejected
(allowed to pass without being captured). For a stationary hovering copepod,
these rejected particles remain in the water below the copepod, and are not
entrained into the feeding current again
(Strickler, 1982
).
The recirculating pattern in Figs 5, 6, 7, 8 is, to our knowledge, the first report of a multi-encounter feeding pattern in calanoid copepods. The combination of feeding and sinking results in particles being drawn toward the copepod with its feeding current, passively rejected and then recirculated. The recirculation is interrupted aperiodically, after 8.7 s in the present example, when the copepod hops. During a hop, the copepod jumps about 0.5 mm upward to a position where its mouthparts are just below the stagnation point of the previous recirculation zone. If it were not for the hop, the copepod would never encounter new particles, due to the closed recirculation pattern. The recirculation allows the copepod to taste some of the particles that have passed near the mouthparts once more, this time with different, perhaps even more sensitive chemoreceptors on its antennules. Considering that the 20 µm polystyrene, spherical particles are mechanically desirable but chemically undesirable, this additional sensing by the sensors on the antennules ensures that the animal does not forfeit potentially good food. The available sequences of holograms suggest that the timing between hops is sufficient for a rejected particle to reach the antennule (about half the recirculation cycle). We speculate that once the copepod senses, using its antennules, a particle that has already been tasted and rejected (and is still not acceptable), it hops to another volume to look for different food. These assertions require testing by altering the properties of the particles, e.g. replacing them with desirable food, and observing the behavior of the copepod.
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Acknowledgments |
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Footnotes |
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