Current address: Department of Bioengineering, University of Illinois Chicago, Chicago, Illinois, United States of America
Conceived and designed the experiments: JMH DTE SNB LM. Performed the experiments: JMH. Analyzed the data: JMH LM. Contributed reagents/materials/analysis tools: JMH DTE SNB LM. Wrote the paper: JMH LM.
The authors have declared that no competing interests exist.
Blood is a dense suspension of soft non-Brownian cells of unique importance. Physiological blood flow involves complex interactions of blood cells with each other and with the environment due to the combined effects of varying cell concentration, cell morphology, cell rheology, and confinement. We analyze these interactions using computational morphological image analysis and machine learning algorithms to quantify the non-equilibrium fluctuations of cellular velocities in a minimal, quasi-two-dimensional microfluidic setting that enables high-resolution spatio-temporal measurements of blood cell flow. In particular, we measure the effective hydrodynamic diffusivity of blood cells and analyze its relationship to macroscopic properties such as bulk flow velocity and density. We also use the effective suspension temperature to distinguish the flow of normal red blood cells and pathological sickled red blood cells and suggest that this temperature may help to characterize the propensity for stasis in Virchow's Triad of blood clotting and thrombosis.
Viewed from a distance, flowing blood looks like a uniform fluid, but up close the cells in the blood change their position and speed somewhat heterogeneously. These individual cell movements may play a role in the physiology and pathophysiology of nutrient and gas transport, clotting, and diseases where normal processes go wrong. To characterize these random motions, we need to follow individual cells in a very crowded suspension—cells usually occupy more than one-third of the volume in blood. We have developed computer software that can separate individual cells in a crowd and track them as they flow. We use this software to analyze blood flow at the level of the cell and find new and possibly important differences between the blood from healthy patients and the blood from patients with sickle cell disease, a disorder in which blood cells become stiff and often stop flowing. We provide evidence that blood from patients with sickle cell disease shows decreased random cellular motions and suggest that this difference may provide a physical basis for the increased risk of occlusion in sickle cell disease.
Red blood cells are the major component of blood and with a radius of ∼4 µm and a thickness of ∼1–2 µm are sufficiently large that the effects of thermal fluctuations are typically negligible, i.e. their equilibrium diffusivity is very small (
To investigate the short-time dynamics of flowing red blood cells we develop and use computational image processing
The top panel shows a sample tracking image. Cells are segmented using morphological criteria and are tracked from frame to frame. The middle panel shows a subset of tracked cells, each with a bounding box. Each cell has a series of small color circles projecting from its centroid showing the subsequent trajectory. The black arrows represent that particular cell's velocity fluctuation relative to the median, with magnitude amplified by 4 for visualization. The bottom panel shows the experimental setup which is described in detail in
See
We utilized microfluidic devices with cross-sectional area of 250 µm×12 µm, similar to the devices used to characterize the phase diagram for vaso-occlusion in an
The top panel (a) shows average fluctuations in squared cellular displacement as a function of time (e.g., 〈Δr2(τ)〉) with x- and y-axes defined in the top panel of
A diffusive process has a characteristic length scale λ corresponding to the mean free path that a cell travels before an interaction, and a characteristic time scale corresponding to the time between these interactions, typically given by the inverse of local shear rate
The top panel (a) shows the hydrodynamic diffusion coefficient D as a function of the bulk flow velocity Vbulk for flows fast enough for the diffusive behavior to be recovered, i.e. Vbulk>∼50 µm/s based on
To assess the relative role of microscopic determinants such as cell shape and stiffness on this diffusive process, we investigated the behavior of blood cells from patients with sickle cell disease. Red blood cells from these patients become stiff in deoxygenated environments as a result of the polymerization of a variant hemoglobin molecule
Hydrodynamic interactions between red blood cells lead to velocity fluctuations and diffusive dynamics of the individual cells. Changes in
These two panels compare probability distribution functions for normalized squared velocity fluctuations from two different experiments with chi-squared distributions with 2 degrees of freedom.
In conclusion, we have identified random walk-like behavior for pressure-driven dense suspensions of soft particles in quasi-two-dimensional confinement which we quantify in terms of cellular velocity fluctuations as a function of blood flow rate, shape, and stiffness. Our results suggest that these fluctuations may be involved in the collective pathophysiological processes of occlusion and thrombosis, both of which are strongly heterogeneous in space and time. While simple scaling ideas are suggestive, a well-defined microscopic mechanism for this process remains to be established.
This study was conducted according to the principles expressed in the Declaration of Helsinki. The study was approved by the Institutional Review Board of Partners Healthcare Systems (2006-P-000066). All patients provided written informed consent for the collection of samples and subsequent analysis.
Videos were captured of blood flowing in microfluidic devices under controlled oxygen concentration. Microfluidic fabrication and blood sample collection and handling are described in detail elsewhere
We developed morphological image processing algorithms to identify a significant fraction of the cells in captured frames of video. See
Annular cells were defined as fillable holes not touching the border. Markers for these annuli were created by subtracting border-contacting high-intensity regions and performing morphologic reconstruction on the result. This reconstruction operation used a marker image that was morphologically opened with a 5 µm line segment oriented in increments of 45 degrees. The reconstruction was then subtracted from the border-cleared image. The final result was dilated using a disk with radius 0.2 µm. Filled cells were defined using granulometry with a circular structuring element of radius 2 µm. Markers for these cells were selected using two transformations of this opened image: the distance transformation of the thresholded binary image followed by the h-maxima transformation with a height of 3.
Background pixels were identified by the skeletonization of a thresholded binary image. Previously determined cell markers were added to the binary image. The result was eroded using a disk with radius 0.5 µm. The skeletonization of this erosion was the background marker image. Foreground and background markers were used to impose minima on the intensity gradient of the original image after background subtraction and histogram equalization. The watershed transformation was then applied to the gradient of the intensity image.
The watershed catchment basins, or blobs, were then filtered heuristically by size, shape, and orientation of the objects' convex hulls. First-pass thresholds were determined empirically by manually segmenting several video frames in Adobe Photoshop. Initial size limits were total convex hull area between 5 and 50 µm2. A measure of convex hull circularity was calculated by comparing the effective radius based on the object area to the effective radius based on the object's perimeter. A circle has a ratio of 1. All other objects have ratios less than 1. The initial circularity threshold was set at 0.6. After an initial filtering process, video frames were re-filtered using thresholds for all morphologic characteristics based on the mean convex hull metrics with allowed variation of twice the standard deviation.
We then developed machine learning algorithms to track these segmented cells from frame to frame and to compute velocities for individual cells. For each object segmented in each video frame, potential “child” cells were iteratively identified in the subsequent frame and ranked by changes in size, shape, and displacement. Child cells were reassigned if a better “parent” cell was identified. Maximum changes in x- and y-displacement were calculated based on apparent flow rates. Y displacement was limited to 600 µm/s in either direction, and x displacement was limited to 1200 µm/s. Maximum changes in area, perimeter length, and eccentricity were determined by manual tracking of several video frames in Adobe Photoshop as part of a validation check on the tracking algorithm. Area was initially allowed to vary by 50%, perimeter by 50%, and eccentricity by 60%.
After all cells in a frame were tracked or determined to be un-trackable, the median inter-frame displacement was computed for all tracked objects. Any tracking events representing displacements that were five times greater than the maximum of the median or the analytic sensitivity threshold (1 µm) were excluded, and the whole frame was retracked with this tighter displacement threshold.
Tracking events which represented the extension of existing trajectories were rejected if they represented a change in cell velocity greater than twice the maximum of the median frame displacement or an analytic sensitivity threshold. After excluding these inconsistent tracking events, the whole video frame was retracked iteratively until no trajectory extensions exceeded this threshold.
Our measured cell velocities were based on more than 25 million displacements calculated across more than 500,000 video frames. We improved and measured the accuracy of our cell velocity measurements a number of different ways, including manual segmentation by an observer of selected video frames and manual tracking by an observer of selected of cells from frame to frame. Inaccuracies in cell velocity measurements can be separated into two categories: errors in the location of a cell, and errors in the assignment of a tracking event for two identified cells. We took a series of steps to reduce the magnitude and bias of this noise and to ensure that it does not influence our results.
The first type of inaccuracy comes from the need to assign a single pixel location to each cell. We chose the centroid of each segmented pixel blob. Because of random variation in image intensity due to lighting and movement of cells out of the focal plane, the calculated centroid for a given cell cannot be established with absolute precision. We estimated this uncertainty first by optimizing our segmentation algorithm based on several video frames segmented manually in Adobe Photoshop, and second by manually tracking several dozen trajectories and comparing our manual cell positions with those of the segmentation and tracking algorithms. We determined that the analytic resolution of the combined segmentation and tracking algorithms was at least 5 pixels (∼1 µm), meaning that the true location of a cell identified in a video frame could vary at least 5 pixels in any direction from the calculated position. This analytic sensitivity was then used to form a tolerance in heuristic cell tracking.
The second type of inaccuracy comes from the false positive linking of segmented blobs in successive video frames. These false positive tracking events could involve actual but distinct cells as well as spurious cells. In any video frame, there is a chance that non-cellular regions (e.g., circular regions of plasma bordered by cells) will have an intensity pattern similar enough to that of a cell to be identified as a cell. If these regions persist from one frame to the next, they may be falsely identified as a tracked cell, and their “velocities” will degrade the accuracy of our results. We took steps to prevent the introduction of such false positives in our data, and we took further steps to reduce the impact these false positives.
To prevent the introduction of false positives in our data, we first optimized our segmentation algorithm to minimize the introduction of false positives. We manually segmented several video frames using Adobe Photoshop and optimized our cellular segmentation algorithms by comparing calculated segmentation functions to manual segmentation functions. We then filtered image blobs by morphologic characteristics including area, perimeter, orientation, eccentricity, and shape. Thresholds for this heuristic filtering process were developed from the analysis of several manually-segmented video frames.
To reduce the impact of false positives and to improve the accuracy of the tracking algorithm, we used several heuristics. Tracking events were identified by evaluating cells in two passes. In the first pass, wide tolerances were used to identify likely tracking events without bias. The median of these displacements was then used to form tighter tolerances for a second pass. This second pass removed any tracking events which required displacements greater than five times the median displacement and five times the analytic sensitivity in the x- or y-direction. If a tracking event was added to an existing trajectory, the integrity or consistency of that trajectory was assessed. We excluded any change in displacement relative to the median that was greater than twice the median in either direction and twice the analytic sensitivity.
Velocities calculated for all processed videos were then assessed by comparing with tracking results for random composite videos. We assembled videos with successive frames randomly stitched together from different videos or from the same video but sampled from time points such that no cell would appear on two consecutive frames. Any tracking events identified in these videos were false positives. We created dozens of these videos and used them to estimate the false positive rate of our tracking logic. These random videos rarely yielded more than 10 tracking events between successive frames. We doubled this number and used a conservative threshold of 20 tracking events. We excluded any video from our analysis if a single pair of successive frames yielded fewer than 20 tracking events.
False positive tracking events were less likely to persist in multiple-step trajectories. For well-tracked videos, most of the shorter trajectories would persist as longer trajectories. We established minimum quality thresholds for the number of longer trajectories as a proportion of shorter trajectories. If too few of the shorter trajectories were successfully tracked for more frames, the videos were excluded from the analysis.
The image processing errors for each frame are likely to be independent from one frame to the next. The true velocity fluctuations, however, are likely to be correlated from frame to frame over very short times. We can therefore look at these measured fluctuations in velocity between different cells over increasing time intervals and confirm that they decrease as they are averaged over more and more frames. We know that over long times, there is a well-defined bulk flow velocity. Individual cells do not zoom ahead of the bulk over long times, nor do they stop in the middle of the stream for significant periods of time. Over long times, the fluctuations of individual cell velocities must therefore regress to zero, and the coefficient of variation measured over these long times will tend to zero, as is the case for these instances of normal blood in steady flow. The decreasing coefficient of variation therefore supports the validity of these velocity measurements.
The segmentation and tracking algorithms work best for cells that are isolated, appear in the focal plane, and generate a sharp phase contrast in the microscope. Cells in this subset which retain these characteristics across several frames will contribute very accurate velocity measurements. One can therefore be very confident that the median cell velocities calculated for cells with long trajectories will be valid. We can then compare cumulative displacements of cells with long tracking trajectories to overall cumulative displacements to assess the validity of tracking information derived from a given video.
Finally, in our data analysis, we compared our overall results to those for subsets of our data consisting of velocities calculated only from longer trajectories as compared to velocities calculated for shorter trajectories. We reasoned that the noise in our data set remaining after data processing is more prevalent in the shorter trajectories. The effects of limited analytic sensitivity will average out over long trajectories, and false positive segmentation and tracking events are very unlikely to persist across several frames. We re-ran our analysis using these reduced data sets and confirmed our reported findings.
We measured projected cell density first by thresholding grayscale intensity images using the MATLAB graythresh function. We then combined this thresholded image with the foreground cell markers calculated by our segmentation algorithm. Under steady state conditions, we would expect this density calculation to be relatively stable.
Previous studies have reported a coefficient of variation for hematocrit of 3% due to biological variation, and another 3% due to analytic variation achieved with commonly used automated hematologic analyzers
The relationship between an actual three-dimensional volumetric density and a projected two-dimensional density depends on the orientation of the red blood cells and the depth of the flow chamber in the direction of the projection. Under steady state conditions, our density measure is stable over time with a coefficient of variation typically between 10% and 25%.
A 3-second video of sickle cell blood captured at 60 frames per second flowing in 10% oxygen at about 53 µm/s.
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A 3-second video of sickle cell blood captured at 60 frames per second flowing in 0% oxygen at about 59 µm/s.
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We thank David Dorfman and Alicia Soriano in the Brigham and Women's Hospital Clinical Hematology Lab for help acquiring the blood samples; Ricardo Paxson, Witek Jachimczyk, and Brett Shoelson for help with image processing; M. Brenner, A. Ladd, P. Nott and H. Stone for discussions.