Red Blood Cells Flowing through a Microchannel with a Hyperbolic Contraction: An Automatic Method

of human RBCs (i.e., hematocrit, Hct $\sim 2\,\%$ ). The blood was collected from a healthy adult volunteer, and EDTA (ethylenediaminetetraacetic acid) was added to the samples to prevent coagulation. The blood samples were washed by centrifugation and then stored hermetically at ${4^{\rm o}}\,\rm C$ until the experiments were performed at room temperature. For the RBCs exposed to chemicals, the cells were incubated for 10 mins at room temperature with 0.02 % diamide (Sigma-Aldrich). After the incubation time, RBCs exposed to chemicals were washed in physiological saline and re-suspended in Dextran 40 at 2 % Hct and then used immediately in our experiments.


The microchannels containing a hyperbolic contraction were produced in polydimethylsiloxane (PDMS) using a standard soft-lithography technique from a SU-8 photoresist mold. The molds were prepared in a clean room facility by photolithography using a high-resolution chrome mask. The geometry of the fabricated microchannel is shown in Fig. 1. The channel has a constant depth of 14  $\mu m$ throughout the PDMS device and the width of the upstream and downstream channels is 400  $\mu m$ . The minimum width in the hyperbolic contraction region is 20  $\mu m$ .

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Fig. 1
Geometry of the hyperbolic microchannel used in this study



2.2 Experimental Setup


For the microfluidic experiments, the device containing the microchannel was placed on the stage of an inverted microscope (IX71, Olympus). The flow rate of 0.5 μL/min was controlled using a syringe pump (PHD ULTRA). The images of the flowing RBCs were captured using a high speed camera (FASTCAM SA3, Photron) and transferred to the computer to be analyzed. An illustration of the experimental setup is shown in Fig. 2.

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Fig. 2
Experimental setup: inverted microscope, high speed camera and syringe pump


2.3 Image Analysis Algorithm


The proposed methodology has five major stages. First, we remove background, noise and some artifacts of the original movie, as a pre-processing stage, obtaining an image only with the RBCs. Next, we create an over-segmented image, based on the initial magnitude gradient image, using the watershed transform. The optical flow information of these regions is obtained by using the variational method proposed by Brox et al. [2]. After that, the cell tracking links the atomic regions in contiguous frames, according to their motion, to form the tracks by means of a keyhole model proposed by Reyes-Aldasoro et al. [21]. Finally, we measure the deformation index of each RBC.

Optical flow is defined as the 2D vector field that matches a pixel in one image to the warped pixel in the other image. In other words, optical flow estimation tries to assign to each pixel of the current frame a two-component velocity vector indicating the position of the same pixel in the reference frame. The segmentation of an image sequence based on motion is a problem that is loosely defined and ambiguous in certain ways. Optical flow estimation algorithms often generate an inaccurate motion field mainly at the boundaries of moving objects, due to reasons such as noise, aperture problem, or occlusion. Therefore, segmentation based on motion alone results in segments with inaccurate boundaries.

A hybrid framework is proposed to integrate differential optical flow approach and region-based spatial segmentation approach to obtain accurate RBC motion. For the task at hand we adopt a high accuracy optical flow estimation based on a coarse-to-fine warping strategy [2] which can provide dense optical flow information. Using atomic regions implicitly resolves the problem which requires smoothing of the optical flow field since the spatial (static) segmentation process will group together neighbouring pixels of similar intensity, so that all the pixels in a area of smooth intensity grouped in the same region will be labelled with the same motion. We thereby presume two basic assumptions: (i) it is assumed that all pixels inside a region of homogeneous intensity follow the same motion model, and (ii) motion discontinuities coincide with the boundaries of those regions. To ensure that our assumptions are met, we apply a strong over-segmentation method to the image.

Our goal is to assign a unique motion vector to each region. While the atomic region motion vector is computed from the optical flows, it is necessary to consider the real situation that some of the optical flows might have been contaminated with noise, causing the computation of the region motion vector deviate from its genuine motion vector. For each optical flow, its contribution to the deviation depends both on its magnitude and on its direction. Thus, we want to detect and exclude those optical flows which tend to cause large errors to the computation of the region motion vector. We achieve these goals by obtaining the dominant motion of the atomic region from the mode of each optical flow component in the region.


2.3.1 Pre-Processing Stage


At this stage, the image background is removed by subtracting the average of all movie images from each image. To improve the identification of the RBCs the image contrast is adjusted by histogram expansion.

Images taken with digital cameras will pick up noise from a variety of sources. As the watershed algorithm is very sensitive to noise it is desirable to apply a noise reduction filter in the pre-processing step. Several filters have been proposed in the literature to reduce the spurious boundaries created due to noise. However, most of these filters tend to blur image edges while they suppress noise. To prevent this effect we use the non-linear bilateral filter [25].

The basic idea underlying the bilateral filter is to replace the intensity of a pixel by taking a weighted average of the pixels within a neighbourhood (in a circle) with the weights depending on both the spatial and intensity difference between the central pixel and its neighbours. In smooth regions, pixel values in a small neighbourhood are similar to each other and the bilateral filter acts essentially as a standard domain filter, averaging away the small, weakly correlated, differences between pixel values caused by noise. Bilateral filter preserves image structure by only smoothing over those neighbours which form part of the “same region” as the central pixel.


2.3.2 Atomic Region Segmentation


An ideal over-segmentation should be easy and fast to obtain, and should not contain too many segmented regions and it should have its region boundaries as a superset of the true image region boundaries. In this section we present an algorithm step that groups pixels into “atomic regions”. The motivations of this preliminary grouping stage resemble the perceptual grouping task: (1) abandoning pixels as the basic image elements, we instead use small image regions of coherent structure to define the optical flow patches. In fact, since the real world does not consist of pixels, it can be argued that this is even a more natural image representation than pixels as those are merely a consequence of the digital image discretization.

Watershed transform is a classical and effective method for image segmentation in grey scale mathematical morphology. For images the idea of the watershed construction is quite simple. An image is considered as a topographic relief where for every pixel in position $\left(x,y \right)$ , its brightness level plays the role of the z-coordinate in the landscape. Local maxima of the activity image can be thought of as mountain tops, and minima can be considered as valleys.

In the flooding or immersion approach [26], single pixel holes are pierced at each regional minimum of the activity image which is regarded as topographic landscape. When sinking the whole surface slowly into a lake water leaks through the holes, rising uniformly and globally across the image, and proceeds to fill each catchment basin. Then, in order to avoid water coming from different holes merge, virtual dams are built at places where the water coming from two different minima would merge.

Figure 3 illustrates the immersion simulation approach. Fig. 3a shows a 1D function with five minima. Water rises in and fills the corresponding catchment basins, as in Figs. 3b–c. When water in basins b 3 and b 4 begin to merge a dam is built to prevent this overflow of water. Similarly, the other watershed lines are constructed. When the image surface is completely flooded the virtual dams or watershed lines separate the catchment basins from one another and correspond to the boundaries of the regions as shown in Fig. 3d.

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Fig. 3
Illustration of immersion watershed transform on a continuous 1D function interpreted as a landscape. The landscape is sequentially flooded from bottom to top. a Holes are pierced at each regional minimum. b At certain flooding height there are two regions with one dam between basin b 3 and basin b 4. c At intermediate flooding height there are three regions with two dams. d Final segmentation with five segments


2.3.3 Optical Flow


In many differential methods, the estimation of optical flow relies on the assumption that objects in an image sequence may change position but their appearance remains the same or nearly the same (brightness constancy assumption) [17] from time t to time $t + 1$ . Brox et al. [2] proposed a variational method that combines a brightness constancy assumption, a gradient constancy assumption and a discontinuity-preserving spatio-temporal smoothness constraint.

Estimating optical flow involves the solution of a correspondence problem. That is, what pixel in one frame corresponds to what pixel in the other frame. In order to find these correspondences one needs to define some assumptions that are not affected by the displacement. The combined variational approach [2] differs from usual variational approaches by the use of a gradient constancy assumption. This assumption provides the method with the capability to yield good estimation results even in the presence of small local or global variations of illumination.


Constancy Assumptions on Data

Given two successive images of a sequence $I\left( x,y,t \right)$ and $I\left( {x+u,y+v,t + 1} \right)$ we seek at each pixel ${\bf{x}}: = \left( x,y,t \right)^{\rm{T}}$ the optical flow vector ${\bf{v}}\left( {\bf{x}} \right): = \left( u,v,1 \right)^{\rm{T}}$ that describes the motion of the pixel at ${\bf{x}}$ to its new location $\left( {x + u, y + v, t + 1} \right)$ in the next frame.

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Jun 14, 2017 | Posted by in GENERAL SURGERY | Comments Off on Red Blood Cells Flowing through a Microchannel with a Hyperbolic Contraction: An Automatic Method

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