2.1.

Microscopic Defocusing Digital Particle Image Velocimetry Principle

Our high-speed 3-D microscopic imaging system is based on the defocusing concept. In a standard 2-D imaging system, the light scattered by a point source is collected through a converging lens and a single aperture, forming the corresponding (and single) image. In contrast, DDPIV uses a mask with two or more apertures; hence a multiplicity of images is generated for any single point source. A simple DDPIV imaging system, consisting of a mask with two apertures separated by a distance $d$ , is shown in Fig. 1a . Point light source $A$ , located on the object focal plane (or reference plane), is imaged on the image plane as a focused point. As the point is moved out of the focal plane to position $B$ , it is focused on a point located out of the image plane. However, because of the two-aperture mask attached to the converging lens, two images $B^{\prime }$ and $B^{″}$ are generated on the image plane, separated by a distance $b$ .

Fig. 1

Principle and experimental setup. (a) Schematic of the defocusing principle (DDPIV). (b) Schematic of the 3-D high-speed microscopic imaging experimental setup.

As shown by Pereira and Gharib,²³ the depth coordinate $Z$ of point $B$ is related to the separation $b$ between its defocused image $B^{\prime }$ and $B^{″}$ . Given the focal distance $L$ between the focal plane and the aperture plane, the distance $d$ between the two apertures, and the optical magnification $M$ of the system, the relationship between coordinate $Z$ and distance $b$ can be determined by geometrical analysis as:

Consequently, the geometrical analysis of a general DDPIV system is not applicable for a 3-D microscopic imaging system.

Even though the focal distance $L$ in the imaging system cannot be directly measured, nor is $L$ dependent on known parameters, they can still be determined experimentally. One can rewrite Eq. 1 as:

Given that $M$ and $d$ are known parameters, the image separation $b$ can be easily measured at precisely controlled distances $w$ , and consequently the focal distance $L$ follows as per Eq. 3. In our experiments, the focal distance $L$ has been calibrated statistically by this means (see Sec. 4, Materials and Methods). Therefore, the complex microscopic system has been reduced to an equivalent general DDPIV optical system, for which Eq. 1 can now be used to calculate the depth coordinate $Z$ . Hence the 3-D coordinates of a point $B$ off the focal plane are now fully determined by this technique, since the remaining coordinates $X$ and $Y$ are easily resolved from the centroid of the multiple images $B^{\prime }$ and $B^{″}$ on the image plane.

In practice, a two-aperture mask introduces ambiguities, since the line pattern has no privileged orientation. For instance, two particles ( $C$ and $D$ ) aligned parallel to the line connecting the two apertures in the mask will generate two defocused image pairs ( $C^{\prime }\text{-}{C}^{″}$ and $D^{\prime }\text{-}{D}^{″}$ ) aligned in a straight line. However, matching the four defocused images produces six possible image pairs (i.e., $C^{\prime }\text{-}{C}^{″}$ , $D^{\prime }\text{-}{D}^{″}$ , $C^{\prime }\text{-}{D}^{\prime }$ , $C^{″}\text{-}{D}^{″}$ , $C^{\prime }\text{-}{D}^{″}$ , and $C^{″}\text{-}{D}^{\prime }$ ). Hence there is high probability to mismatch the defocused images and to generate particles that never existed. For this reason, a mask with three apertures forming an equilateral triangle was adopted, in which case the resulting triple-image pattern of a particle can be unambiguously identified.

2.2.

Experimental Setup

A schematic of the experimental setup of the 3-D imaging system is shown in Fig. 1b. The system is composed of an inverted microscope, a $Z$ -translation stage, a three-aperture mask, an argon-ion laser, and a high speed camera. An inverted microscope (Nikon, Eclipse TE2000-S) was utilized to enable microscale flow imaging. A mask with three apertures forming an equilateral triangle was attached to the back of a $20\times$ objective lens (Nikon Plan Apo, $\mathrm{NA}=0.75$ ). The diameter of the apertures is $2\mathrm{mm}$ , and the separation $d$ between each two apertures is $7\mathrm{mm}$ . Fluorescence microscopy was adopted with $1\text{-}\mathrm{\mu }\mathrm{m}$ yellow-green fluorescent polystyrene microspheres (Molecular ${\mathrm{Probes}}^{™}$ , FluoSpheres, USA) as the flow tracers. The particles were excited, through the regular bright-field illumination path of the microscope, by the 488-nm spectral line of a 5-W argon-ion laser, their emission line being at 515 nm. Time series of DDPIV images were recorded using a high-speed, complementary metal oxide semiconductor camera with a resolution of $1024\times 1024$ ${\mathrm{pixel}}^2$ capable of a recording frame rate of up to 2000 frames per second (fps) at full resolution (Photron, APX Ultima).

2.3.

Calibration

$1\text{-}\mathrm{\mu }\mathrm{m}$ fluorescent microspheres were chosen as the calibration targets, as they were also the working tracer particles for the zebrafish blood flow imaging experiments. Targets were mounted on a cover glass and placed underneath the microscope (see Materials and Methods in Sec. 4 for details). In video microscopy,²⁴ diffraction and spherical aberration are a source of image distortion. To account for this in the calibration, targets were spread over the whole field of view, allowing us to have ensemble statistics for the image separation $b$ and consequently for $L$ . These measurements are repeated at various distances $w$ . Sample defocused images of calibration targets at different relative distances are shown in Fig. 2 . The resolution of the image is $0.85\mathrm{\mu }\mathrm{m}$ per pixel.

Fig. 2

Defocused images of $1\text{-}\mathrm{\mu }\mathrm{m}$ microspheres calibrated at various locations. (a) Focal plane $w=0$ ; (b) defocusing plane $w=10\mathrm{\mu }\mathrm{m}$ ; and (c) defocusing plane $w=22\mathrm{\mu }\mathrm{m}$ . Sample defocusing patterns are linked as triangles in (b) and (c).

Figure 3 shows the calibrated mean values of the image separation $b$ and the corresponding focal distance $L$ , as a function of the relative position $w$ . The total depth range of the current 3-D imaging system is $40\mathrm{\mu }\mathrm{m}$ . The sensitivity of depth position is defined as the image separation change for a given spatial displacement, which is 0.944 pixel for $1\text{-}\mathrm{\mu }\mathrm{m}$ $Z$ displacement in the present imaging system. A subpixel detection algorithm²³ was used to calculate the image coordinates of particle centroids, thereby the depth resolution of the 3-D imaging system is in the order of $100\mathrm{nm}$ .²⁵ It can also be observed that the calibrated values of the focal distance $L$ are found to be independent on the depth coordinates as expected.

Fig. 3

Calibration results. (a) Mean separation $b$ at different relative distances $w$ . (b) Mean focal distance $L$ at different relative distances $w$ .

2.4.

In Vivo Imaging of Cardiac Cell Motions in Zebrafish

A very small volume (e.g., picoliters) of fluorescent microspheres was injected into the blood stream of a zebrafish embryo immobilized by a drop of 1% agarose gel at $32\mathrm{h}$ post-fertilization (hpf) and 48 hpf, respectively (see Materials and Methods in Sec. 4). The zebrafish embryonic heart is composed of two chambers, i.e., atrium and ventricle. Each chamber is about $80\mathrm{\mu }\mathrm{m}$ in diameter and $100\mathrm{\mu }\mathrm{m}$ longitudinally. A 32-hpf heart is a straight tube, with a two-part heartbeat indicating the formation of two chambers. At 48 hpf, the heart tube bends to separate the two chambers. Because direct injection of particles into the developing heart may cause serious damage to the cardiac muscle and impact the physics of normal cardiac cell motions, microinjection was performed at blood vessels in the tail of the embryo to minimize the potential influence. Tracer particles then entered the heart through the circulatory system. It has been observed that injected polystyrene microspheres are likely to aggregate or to be trapped by the walls of blood vessels within minutes after injection. In heart-wall motions imaging experiments, the microspheres trapped in the heart were used as tracers to describe heart-wall motions. However, particle trapping by heart walls may significantly affect their ability to function optimally as cardiovascular flow tracers. In blood flow imaging experiments, the test zebrafish embryo was immediately moved to the modified microscopic setup for imaging after a successful injection. Defocused image patterns of in vivo cardiac cell motions were recorded by the high-speed camera.

Although microinjection was carefully performed, some particles already adhered to the surrounding tissues outside of the heart, generating a nonuniform background in the images. Therefore, images were processed prior to identifying the particles by their self-similar triangular image patterns. Figure 4 shows a sample of the image processing. A single image frame [Fig. 4a] among a raw image sequence showing blood flow in the yolk sac of a 32-hpf embryo included both particles in the blood stream and those trapped by the tissues. Fluorescent image patterns of the particles on the tissues were considered as the background, and needed to be separated from the particles of interest in the blood to measure the flow velocities. Since the surrounding tissues were stationary during the experiment, the background particles existed in the same positions of every frame among an image sequence, while the particles moving in the blood were present in different locations. Therefore, the average intensity of the entire sequence only contains the background [Fig. 4b], which was initially calculated using ImageJ (National Institutes of Health, Bethesda, Maryland) and was then subtracted from every single image frame. After background removal, an image frame only contains signals of particles in the blood stream. The typical signal-to-noise ratio of the recorded images was $12.6\mathrm{dB}$ . Particle image detection in our experiments was then performed, following the procedures of Pereira and Gharib²³: low-pass filtering, particle subpixel detection, particle 2-D Gaussian modeling, and noise removal were performed to further improve the image quality, followed by triangle pattern matching to find the 3-D coordinates of the tracer particles.

Fig. 4

Particle image processing. (a) Defocusing images acquired in experiments. (b) Background due to microspheres trapped by surrounding tissues. (c) Processed images after background removal, low-pass filtering, particle detection, 2-D Gaussian model, and noise removal.

2.5.

Particle Tracking

PTV, a commonly used method in fluid mechanics, was applied to obtain the velocity field after the 3-D particle position was resolved. It tracks individual particle images in consecutive image frames, and computes the directionally resolved vector for each matched particle. A tracking algorithm called a relaxation scheme²⁶ has been used in this study. The basic concept of particle tracking in this algorithm is to search for the most probable link of a given particle, while assuming similar displacements of its neighbor particles (the so-called quasi-rigidity condition), within a neighborhood of specified size. Thus, the search for the matching particle in the next image frame is made in the direction that the neighbor particles are most likely pointing to, since it is legitimate to believe that the corresponding matching particle lies preferably in that direction. In an iterative process, the correct link probability is gradually increased close to unity, while the other probabilities tend to zero. After successful termination of this iterative process, the correct particle link is the one with the highest probability.

There is a criterion in the PTV method that requires the distance between tracer particles to be larger than the displacement. Since the high-speed image acquisition was used in the system and particle density was low in the images, the distance of a particle traveled between two consecutive time frames is smaller than the separation between particles, even though the cell motions in a beating zebrafish heart are highly dynamic. Hence, the criterion was met and PTV is capable of obtaining 3-D velocity vectors from the images. In addition, the recorded image sequences described continuous movements of each particle (i.e., no strong displacement gradient exists between every two consecutive images), and most particle links can thereby be correctly found. So the complex velocity fields inside the zebrafish heart can be reconstructed.

3-D velocity fields of tracer particles in the blood stream and on the heart wall were resolved by PTV. Figure 5 shows the blood flow velocity fields of a 32-hpf embryo. The blood in the yolk sac is seen to be moving first upward, then following a downward direction as the yolk sac is approximately spherical [Fig. 5a]. Images of six tracer particles inside the cardiovascular flow were captured at the same instance [Fig. 5b]. Forward flow velocity in the atrium during diastole was about $300\mathrm{\mu }\mathrm{m}∕\mathrm{s}$ , which is consistent with the measurements of Forouhar ²⁷ Trajectories of tracer particles trapped in the wall of the atrium and the ventricle of a 32-hpf embryo were obtained [Fig. 6 ]. Three-dimensional positions of the two particles in a cardiac cycle demonstrated the phase difference between the atrium and the ventricle in a beating heart. One can also observe that the wall motion in the ventricle is more active and more complex than that in the atrium. The shape of ventricle-motion trajectories is similar with what Forouhar ²⁷ obtained as well. A 48-hpf zebrafish heart was labeled by a few tracer particles in the wall (Video 1). Defocusing image patterns were recorded at 125 fps. The reconstructed 3-D positions and velocities of the particles demonstrated the movement of the ventricle during a complete cardiac cycle (Video 2 ).

Fig. 5

3-D PTV results of blood flow. (a) Mean velocity field of blood flow in the yolk sac of an embryonic zebrafish. (b) Instantaneous velocities and spatial positions of six tracer particles in the atrium during diastole. Vector arrows indicate velocity directions, with colors representing velocity magnitudes. Defocusing images were recorded at 500 fps.

Fig. 6

3-D PTV results of heart-wall motions. (a) Defocusing images of the trapped tracer particles during a cardiac cycle. The two bright image triplets originated from the tracer particles of interest in the heart wall, with the ventricle on the left side and the atrium on the right. The heart rate was about $2\mathrm{Hz}$ . Images were recorded at 60 fps. (b) Trajectory of the tracer particle in the ventricle wall within two cardiac cycles. (c) Trajectory of the tracer particle in the atrium wall within two cardiac cycles. The spheres stand for positions of the injected tracer particles, with colors representing velocity magnitudes and vector arrows indicating velocity directions. The big sphere in each image indicates the particle position at the corresponding time instance.

Video 1

A beating 48-hpf zebrafish heart labeled by injected tracer particles. Defocusing image patterns form upright triangles (QuickTime, $1.2\mathrm{MB.}$ ). 1.

Video 2

Reconstruction of the beating ventricle in Video 1. Velocity magnitudes are color coded and vector arrows represent velocity directions (QuickTime, $2.5\mathrm{MB.}$ ). 2;.

4.1.

Target Preparation and Calibration

$1\text{-}\mathrm{\mu }\mathrm{m}$ yellow-green fluorescent microspheres (Molecular ${\mathrm{Probes}}^{™}$ , FluoSpheres, USA) were initially diluted into preheated 1% ultra-low melting temperature agarose (Sigma-Aldrich, Saint Louis, Missouri) in 30% Danieau solution. One small droplet of the diluted solution was placed on a $24\times 50\text{-}{\mathrm{mm}}^2$ cover glass. We then put another cover glass over the droplet, and pressed the two cover glasses tightly, allowing the agarose solution to cool within minutes. Afterward, fluorescent targets were spread over the cover glasses.

The cover glasses were moved in steps of $10\mathrm{\mu }\mathrm{m}$ along the $Z$ axis using a precision translation stage, starting from the focal plane $(w=0)$ , where the triangular image patterns are reduced to points $(b=0)$ up to a position where particle detection was not possible.

4.2.

Embryo Preparation and Microinjection

Wild-type zebrafish embryos were raised at 28.5°C in 30% Danieau solution. At 32 hpf (or 48 hpf), embryos were dechorionated, and anaesthetized with 0.02% tricaine in 30% Danieau solution.

The test embryos were transferred to 1% agarose, and then pipetted in a drop of agarose on a chilled cover glass to facilitate particles injection. The orientation of the embryos was manually adjusted to expose the blood vessels in the tail, through the use of a small capillary tube. Embryos were appropriately positioned and mounted on the cover glass within minutes. Fluidized $1\text{-}\mathrm{\mu }\mathrm{m}$ yellow-green fluorescent microspheres were loaded into a micropipette with an approximately $10\text{-}\mathrm{\mu }\mathrm{m}$ tip, connected with a Pico-Spritzer assembly (General Valve PLC-100). The test embryo was placed underneath a Zeiss Stemi SV 11 dissecting microscope for observation. The micropipette tip penetrated the skin to inject the tracer particles into the blood vessels.