Nanometer precise red blood cell sizing using a cost-effective quantitative dark field imaging system.

Because of the bulk, complexity, calibration requirements, and need for operator training, most current flow-based blood counting devices are not appropriate for field use. Standard imaging methods could be much more compact, inexpensive, and with minimal calibration requirements. However, due to the diffraction limit, imaging lacks the nanometer precision required to measure red blood cell volumes. To address this challenge, we utilize Mie scattering, which can measure nanometer-scale morphological information from cells, in a dark-field imaging geometry. The approach consists of a custom-built dark-field scattering microscope with symmetrically oblique illumination at a precisely defined angle to record wide-field images of diluted and sphered blood samples. Scattering intensities of each cell under three wavelengths are obtained by segmenting images via digital image processing. These scattering intensities are then used to determine size and hemoglobin information via Mie theory and machine learning. Validation on 90 clinical blood samples confirmed the ability to obtain mean corpuscular volume (MCV), mean corpuscular hemoglobin concentration (MCHC), and red cell distribution width (RDW) with high accuracy. Simulations based on historical data suggest that an instrument with the accuracy achieved in this study could be used for widespread anemia screening.


Introduction
Blood provides a snapshot of mammalian health. Many diseases, such as anemia, leukemia, hypersplenism, thyroid dysfunction etc., yield marked changes in blood values [1,2]. Therefore, the complete blood count (CBC), which provides red blood cell count (RBC), white blood cell count (WBC), platelet count (PLT), mean corpuscular volume (MCV), mean corpuscular hemoglobin concentration (MCHC), red cell distribution width (RDW), and other calculated indices, is one of the most common laboratory tests. Due to the relative ease of capillary blood sampling, it particularly can play a role in screening, such as using red blood cell morphology to indentify anemia [3,4]. However, clinically important variations in MCV are typically on the scale of 10-15 fL, corresponding to a difference in radius of the red blood cells of just 100-200nm, beyond the scale of all but the most advanced optical microscopes. Thus, typical clinical systems utilize the Coulter effect or laser light scattering of cells in flow to obtain nanometer-scale precise measurements of cell volume.
The use of Mie scattering to analyze the morphology and refractive index of cells has a long history. In 1969, Mullaney et al. theoretically predicted small angle light scattering (0.5 • to 2 • ) can be used to detect the size of micrometer-sized spherical particles (diameter from 5 to 20 µm) [5]. Later, in 1975, Salzman et al. accomplished an automated leukocyte differential with forward and side scattering [6]. In 1985 Tycko and Mohandas et al. developed a light scattering technique to determine volume and hemoglobin concentration of individual sphered red blood cells simultaneously by mapping between scattering intensities in two angle ranges and volume and hemoglobin concentration [7,8]. As instrumentation improves, this has been extended to platelet scattering as well [9].
This flow method, using either light scattering or the Coulter effect for volume measurements, along with fluorescence for cell type identification, has long been the standard for clinical CBC enumeration [10]. Clinical automated blood analyzers are fully automated (blood is robotically loaded) or semi-automated (blood is manually loaded), but in either case requires daily maintenance and calibration by skilled technicians to check flow rates and other parameters using standard samples. They furthermore are bulky, with even point-of-care-oriented systems being ∼ 30 L in volume and 30 pounds in weight (c.f. pocH-100i, Sysmex), which makes these systems more challenging to deploy in field settings. Meanwhile a large portion of the world's population lives far away from centralized medical infrastructure [11]. Therefore, there is a need for low-cost CBC instrumentation appropriate for use in these low-resource and remote areas and other settings distant from centralized medical infrastructure. In the past few decades, researchers have developed many low-cost microfluidic cytometers for blood cell counting and leukocyte differentials [12][13][14][15][16], as well as for cancer or other biomarker diagnosis [17,18]. Particular attention should be given to those systems that utilize light scattering [19] or the bio-lensing effect of red blood cells [20] to develop novel optical biomarkers of anemia [21].
However, lab-on-a-chip flow cytometry systems analyzing complex fluids such as blood require substantial complexity, such as valves and mixing channels, to add diluents and staining reagents [22], and to focus the cells to a single stream for cell-by-cell interrogation. To simplify the chip design, several groups have developed CBC systems based on microscopic imaging with single-step sample preparation that can be performed prior to inserting the blood into the chip [23][24][25][26][27][28][29]. These systems can thus enumerate cells without complex flow handling. However, despite the success of these systems, and their improved robustness and ease of use [27], their resolutions are still limited by the diffraction limit. As mentioned above, clinically relevant differences in MCV are on the 100-200nm scale, beyond the reach of these systems. Thus, they cannot obtain accurate measures of MCV, MCHC or RDW. However, these are the most important parameters for diagnosing anemia, which is endemic in resource-limited settings. Meanwhile, several groups have shown that light scattering within a microscopic geometry can accurately determine cell size with nanometer scale accuracy and precision. For example, in 1996 Doornbos et al. developed an instrument with two lasers to determine the angular light scattering of single bead or biological cell held in an optical trap, and tested angular scattering of a human lymphocyte from 20 to 60 degrees [30]. From 2001, Boustany and co-authors developed a microscope measuring the amount of light scattered into certain angular zones for each pixel on an image [31,32], showing the ability to detect differences in scattering from cells arising from changes in mitochondrial morphology whose size scale was below the diffraction limit. In 2007, Cottrell, Wilson and Foster designed a multifunctional imaging/scattering spectroscopy system based on a commercial microscope platform, which "enables coregistered brightfield, Fourier-filtered darkfield, and fluorescence imaging; monochromatic angle-resolved scattering measurements; and white-light wavelength-resolved scattering spectroscopy from the same field of view." [33]. In 2008, Smith and Berger developed a system for recording both Raman and angle-resolved elastic scattering from single cells and microparticles in a moderately focused laser beam [34,35], with nanometer-scale accuracy and precision for determining the diameter of sub-wavelength-sized objects. In 2012 Rothe et al. developed a darkfield scattering microscope measuring both angular and spectrally resolved scattering of single particles [36].
However, these aforementioned methods were developed primarily for biophysical applications, with sources, detectors, and optical designs optimized for assessing subcellular morphology of organelles such as mitochondria and lysosomes. They typically utilize highly monochromatic, single-spatial-mode laser illumination, and large numerical aperture objectives to obtain the maximum amount of information to recover sizes of hundred-nanometer-scale organelle objects. Due to the cost of such objectives and lasers, as well as the limited fields of view obtained by these systems, they would be impractical for a low cost device that needs to collect scattering from thousands of cells such that parameters such as MCV and RDW can be statistically precise. We have recently shown that light scattering of low-cost laser diode sources, detected by consumergrade cameras, can provide information on red blood cell morphology with enough precision to accurately diagnose different forms of microcytic anemia [3,37,38]. Yet these systems measure cell scattering in the Fourier plane of the optical system, thus they do not directly image the sample, and cannot provide RBC, WBC, or platelet counts. However, a complete blood count requires these numbers, and in the particular application of anemia screening, obtaining the RBC count can improve the screening accuracy. Therefore, in this paper we obtain nanoscale-accurate cell sizes using a low-cost imaging system, compatible with obtaining RBC and WBC counts, that utilizes multi-color, engineered illumination to obtain dark-field images of blood cells in a simply prepared solution. The intensity of the cell within each image is directly proportional to the amount of light scattered within a defined solid angle, represented by the numerical aperture of the imaging objective. We show below that for a specified illumination angle, using an inexpensive three-color LED, and for the range of sizes and refractive indices characteristic of human red blood cells, the scattered light from red blood cells fall onto a relatively flat manifold within the three dimensional space. Thus, by detecting the scattered light from these three LEDs for each cell, the cell's volume and hemoglobin concentration can be uniquely determined.
Combining this with automated image analysis and machine learning, we obtain nanometerprecise measures of the blood sample's mean cell volume, and highly accurate measurements of the mean cell hemoglobin concentration. The performance of the system is validated on NIST-traceable size standards, as well as a pilot clinical trial with 90 blood samples with values both within and beyond the normal range. Finally, using a historical chart review, we speculate about the ability of our system to diagnose different forms of microcytic anemia compared to a clinical system.

Theoretical analysis of dark field imaging via Mie theory
For a given sphere size and refractive index, the scattering intensity collected by a given optical system can be readily calculated. While red blood cells are biconcave in shape, we make use of the method of Kim and Ornstein [39] whereby adding a small amount of surfactant to the blood causes the cells to sphere while preserving their volume. Thus, in our system we consider red blood cells as spheres. Experimental details on how red blood cells are sphered are given later. The refractive index (n b ) is dependent upon the hemoglobin concentration (conc) according to the following Equation: where n m (λ) is the refractive index of water, given by Dooley [40], and β (λ) is tabulated by Friebel and Meinke [41]. From here, the intensity of scattered light versus scattering angle can be calculated using standard formulae found in, eg. the classic treatment of Bohren and Huffman [42].
Our dark-field imaging system, shown schematically in Fig. 1(A), utilizes an incoherent ring source placed in the Fourier plane of the optical system, such that each cell is illuminated at a well-defined polar angle θ with equal intensity at every azimuthal angle φ. Using Mie theory, we can compute the amount of scattered intensity a sphere of a given size and refractive index scatters into the numerical aperture of our microscope system. Due to the symmetric placement of the source, the assumption that the source ring is thin, and that the micron-sized scatterers considered here have azimuthally-symmetric scattering, the range of scattering angles collected by the system is identical for all positions around the ring. Thus, we can consider the scattering due to a single point on the ring source, and then presume that the total scattering is a linear multiple of this, reflecting the additional points on the source. Samples in our system are held in a plastic chamber with plane parallel faces, such that refraction from these faces can be neglected, and refraction can be considered to be occurring at an air/water interface, as shown in Fig. 1(B). According to Mie theory, the scattering intensity gathered by the objective is given by Eq. 2. Where, I 0 (λ) represents the normalized spectrum from a given LED, S 1 (cos θ) , S 2 (cos θ) are the scattering matrix elements of Mie theory, given in Eq. 4.74 of Bohren and Huffman [42].
θ and φ are the polar and azimuthal scattering angles, whose limits are described below. Due to dispersion, the minimum and maximum scattering angles collected by the objective are also wavelength dependent, according to Eq. 3: Further, due to the circular shape of the objective aperture, the limiting scattering angles φ min and φ max depends on the scattering angle θ, as shown in Eq. 4: Where θ = (θ max + θ min )/2 and ∆θ = (θ max − θ min )/2. The derivation of Eq. (4) can be found in Appendix A. The scattering intensity clearly varies when θ in or the wavelength changes. Thus, assuming the numerical aperture, and thus collection angles, are fixed by the objective (in our case with a NA of 0.2), then the incident angle and incident wavelengths are the key parameters we can vary to alter the sensitivity of the system to different cell sizes or refractive indices. Here, following the expected size ranges and concentration ranges for human red blood cells, we simulated scattered intensities with mean corpuscular volume (MCV) varying from 60 to 120 fL and mean corpuscular hemoglobin concentration (MCHC) varying from 28 to 36 g/dL. To ensure dark-field imaging, θ in varies in a range of NA +1 • to NA+30 • , with an interval of 0.5 • . Simulations of scattering intensities of red blood cells with different size and different hemoglobin concentrations under 3 illuminations at different incident angles (such that θ min = 3.5, 10, and 30 degrees) are shown in Fig. 2, demonstrating that the scattered intensities from blood cells of different sizes and hemoglobin concentrations form a manifold in that 3D space. Isolines of constant MCV or constant MCHC are plotted in red or blue, respectively. Spacing between isolines is 6 fL in MCV or 1 g/dL in MCHC. The illumination is presumed to come from a typical low-cost RGB LED, whose spectrum (measured by a Flame Spectrometer, Ocean Optics) is as shown in Fig. 1(C). For each color LED, each wavelength of scattering is calculated and the total scattered intensity is the weighted sum, where each wavelength is weighted according to its relative abundance in the LED spectrum. As shown in Fig. 2, when θ min increases, the manifold becomes tighter and more warped, to the point where the manifold can cross itself and thus destroying any possibility of one-to-one correspondence of intensity measurement and size or refractive index. Visualizations of Figs. 2(A)-(C) where the viewing angle is rotated, to better appreciate the manifold shapes, are provided in Visualization 1, Visualization 2, and Visualization 3.
To quantify how well each illumination angle performs, we use the average distance between two adjacent isolines as an empirical merit function. The result is shown in Fig. 2(D). Clearly, the smaller θ min , the better the performance of the system. However, the finite width of the illumination aperture and tolerances on positioning the ring limits θ min to ∼ 1 • . θ min is further limited by stray light generated inside of the objective. This stray light is generated by illumination light, which is much stronger than the scattering signal, striking interior surfaces in the objective and re-entering the optical path. In order to ensure complete isolation of the illumination and detection paths, θ min was chosen to be 3.5 • . Although this number is clearly not optimal according to Fig. 2(D), the manifold shown in Fig. 2(A) is still relatively flat and thus represents a practical balance between theoretical performance and minimal background light.

System construction and control
The system is a custom-built, automated and portable dark-field microscope, whose design and construction is based on prior work [27]. The schematic of the system is shown in Fig. 1(A). A 3-color LED under the control of an Arduino Mega2560 microcontroller was used for illumination. The LED's spectrum was measured using a compact spectrometer (Flame, Ocean Optics, USA) and is shown in Fig. 1(C), where each LED's spectrum has been normalized to have the same area under the curve. The illumination is diffused and then angularly filtered by passing the light through a custom-fabricated annular mask and condenser (ACA254-030-A, Thorlabs, USA, focal length f c = 30 mm), such that the mask is placed in the Fourier plane of the sample, ensuring nearly a single polar angle of illumination (angle range 15.04 ± 0.90 degrees relative to the optical axis). The sample is placed on an automated sample stage and automatically focused. Scattered light is collected by the objective (CFI Plan Apochromat Lambda 4X 0.2NA, Nikon, Japan). The range of scattering angles collected is thus 3.5 to 26.6 degrees. The dark-field image is captured by a low-cost, cooled CMOS camera (ASI1600MM-COOL, ZWO, China), which has a 4656 x 3520 pixel sensor with 3.8 micron pixels. Given the 4x magnification of our system, our Nyquist-limited resolution is 1.9 microns.
The camera, stages, and other system components are controlled through a single LabVIEW user interface. When a sample is placed on the stage, we wait at least one minute to allow the sample to settle to the bottom of the chamber prior to imaging. The autofocusing is then performed under green illumination using the method of Yazdanfar et al. [43]. In this method, four images are taken at different values of defocus. The Brenner gradient [44], a common image sharpness metric, is then calculated for each image. As described by Yazdanfar et al., near focus the Brenner gradient obeys a roughly Lorentzian functional form. Thus, the peak position of the Brenner gradient, and thus the position of the true focus, can be accurately estimated from these four images. Then three images will be taken under red, green and blue illumination successively with 5-second waiting time between images to allow the LED illumination intensity to stabilize.

Sample preparation
90 human blood were used to validate the feasibility of this system. These discarded and anonymized samples were collected under protocol number 2019KY-167, approved by the Ethical Review Board of the First Affiliated Hospital of the University of Science and Technology of China. The clinical samples were run on an automated, flow-based hematology analyzer (BC6800, mindray, China) in the First Affiliated Hosptial of University of Science and Technology of China, generating a CBC report whose values for MCV, MCHC, and RDW are regarded as ground truth. RDW is typically parameterized as the standard deviation (RDW-SD), expressed in femtoliters, or the coefficient of variation (RDW-CV), expressed as a percent. In this case, we utilize the RDW -CV from the clinical report. Following this, samples were stored in an incubator and transported to the laboratory for further testing using our method. During testing, approximately 5 µL of blood was taken from the tube with an Eppendorf pipette and diluted 200-fold in a premixed solution of phosphate buffered saline (PBS) containing Cocamidopropyl betaine (CAPB, 80 µg/mL final concentration) and gently shaken for about 30 seconds. CAPB is a zwitterionic surfactant that intercalates within the red cell membrane, lowering the surface tension and causing the cell to change its shape from a biconcave disk to a sphere, while preserving its volume. This method was pioneered by Kim and Ornstein [39], and expanded to zwitterionic surfactants by Fan et al. [45]. This method has been validated and is currently standard practice on all hematology analyzers that utilize light scattering for cell volume determination [46,47]. We have further validated the sphering in our own laboratory as described previously [27]. To ensure proper sphering, samples were allowed to stand about 3 min. Later, a 10 µL aliquot of each diluted sample was placed in a sample chamber (C10283, Thermo Fisher, USA, chamber height 100 ± 2 µm) and measured immediately.

Segmentation of dark-field blood images
As described above, for each blood sample a red, green, and blue scattering image was acquired. Due to the impact of field curvature distorting the cell images and potentially altering the measurement of scattered intensities, only the central part of each image (1200 pixels × 1200 pixels) was used for further analysis. Cells are identified using an image binarization routine, shown in Figs. 3(A)-(G). Figure 3(A) shows a zoomed ROI of a dark-field scattering image, where the mean image across the three illumination wavelengths was calculated. Figure 3(B) is the binary edge image after Fig. 3(A) was processed with the Canny edge detector [48]. Holes in the edge images were filled and the resulting binary image was then dilated using a 6-pixel width square dilation kernel, with the resulting binary mask shown in Fig. 3(C). The dilation allows us to accurately collect the weak tails of the scattered intensity due to blurring of the red blood cell images by the system PSF. However, careful examination of this image reveals that some cells were "lost" during this process. We found several cells in each image were not properly identified using the raw Canny edge, as they had 1-2 pixel "gaps" in their outline. These gaps are typically caused by the cells being in close proximity to another cell, such that their intensity gradient was reduced and the Canny edge filter did not recognize 1 or 2 pixels on the cell edge facing the nearby cell. When filling holes, these "broken" edges were not filled and the cells were lost. To correct this, these gaps were bridged using the MATLAB function "bwmorph", which can set a 0 pixel to 1 if it has 2 unconnected neighbors with a value of 1. The edge image following this bridging process is shown in Fig. 3(D). Note that while all cells have closed boundaries, some cell boundaries have been joined together. Following hole-filling and dilation (Fig. 3(E)), all cells have been identified following the bridging process, but some cells that were identified as single cells in Fig. 3(C) have been joined to create multi-cell clusters. While the multiple-cell clusters could be disaggregated using a watershed or other image segmentation method, in order to ensure accurate scattering intensity calculations, we analyze only single-cell regions and exclude any regions that contain multiple cells. Multiple-cell regions are differentiated from single-cell regions based on several characteristics. For each component in the binarized image, area, mean intensity and eccentricity are calculated, shown in Fig. 3(G), where each dot represents an individual object in a representative dark-field image of red blood cells. Thresholds of Area = 120, Intensity = 4000 and Eccentricity = 0.7 are used distinguish single-cell regions (shown in red in Figs. 3(G)) and multiple-cell regions (shown in blue). To capture the maximum number of single cells, we take the single-cell regions identified from Fig. 3(C) and the single-cell regions identified in Fig. 3(E) and combine them using a logical OR operation. The final segmented image is then shown in Fig. 3(F), where the greyscale image and binary masks are overlaid. Single-cell regions are shaded red, while multiple-cell regions are shaded blue. The single-cell binary masks can be used to extract the red, green, and blue scattering intensities for each cell within the image. The average number of single red blood cells analyzed in a given image based on this analysis was approximately 3000 in our study, demonstrating that despite not analyzing the multiple-cell clusters, we obtain a statistically relevant sampling for the purposes of calculating MCV, MCHC, and RDW. We assume that the single cells and multiple cells have identical characteristics, and thus not analyzing the multiple cell regions does not influence our results. The validity of this assumption is tested further in Appendix B, where we compare the eventual error in estimating mean cell volume versus RBC count. As the RBC count increases, the proportion of multiple cell regions to single cell regions is expected to increase. However, the error shows no correlation with RBC count, suggesting that not analyzing these multiple cell regions is not a significant contributor to the final error of our method. We further note that our image analysis makes no attempt to separate WBCs from RBCs. However, since WBC counts are typically 3 orders of magnitude lower than RBC counts, we expect that, if we sample approximately 3000 cells per image, only ∼ 3 of these would be WBCs. Thus, the influence these have on the final average size and refractive index values is negligible.

Clinical test using rigorous Mie theory
To calculate MCV, MCHC and RDW for red blood cell samples, a calibration must be performed to map observed scattering intensities with theoretical intensities. Samples are thus randomly divided into 10 groups and a 10-fold cross-validation was performed, where 9 groups were used as calibration samples for each fold, and the remaining group was used as validation. A linear relationship between theoretical Mie scattering intensities (I th ) and experimental scattering intensities (I ex ) can be estimated using average intensity of all cells in the images for each sample in the calibration group and their theoretical average scattering intensity based on their MCV and MCHC from the gold-standard analyzer. This mapping can then transform the experimental scattering intensities of the validation group to match the theoretical intensities.  The result of this 10-fold cross validation is shown in Fig. 4. The top row is the correlation analysis between clinical and image-based measurements, and the bottom is a Bland-Altman analysis [49]. In correlation plots, green circles represent individual samples, the red solid line represents the line of perfect prediction and blue lines represent the 95% confidence interval (CI) of the clinical value, as determined by precision specifications and comparisons with predicate instruments for the Mindray clinical analyzer [50]. In Bland-Altman plots, the red solid line represents the average disagreement between the two methods (the bias) and blue lines represent the 95% CI of the disagreement, or the range of errors likely to be observed in future measurements.
While the rigorous Mie theory analysis clearly shows the ability of our system to extract the size and refractive index of the red blood cells, we have found in previous work that rigorous Mie theory does not fully explain the observed scattering from red blood cells, due to the red blood cells being imperfect spheres, lying on a substrate, and given experimental uncertainties regarding the illumination and collection angles, as well as variability in the transmission efficiency for each angle within the optical system. This is particularly evident in the RDW estimation. This parameter is further complicated by the fact that manufacturers do not follow a standard guideline when calculating this value, leading to substantial differences between individual manufacturers [51]. In that reference, Lippi et al. particularly highlight that the Mindray analyzer used in our study reports an RDW-CV that is substantially larger than most other analyzers, and that this systematic bias is more severe at larger RDW-CV values. This matches with the observed discrepancies in our data, where our light-scattering measurements consistently underestimate In correlation plots, green circles represent each sample, the red solid line represents the perfect prediction and blue lines represent the 95% CI of the clinical value. In Bland-Altman plots, the red solid line represents the average disagreement between two methods and blue lines represent the 95% CI of the disagreement. the RDW-CV. However, these errors are largely systematic, suggesting that an empirical model could correct these static deviations. Thus, to overcome this issue, as well as static deviations from theory raised above, we created an empirical model to map scattering intensities directly to clinical values using a partial least squares (PLS) regression as described below.

Clinical test using empirical partial least squares model
As shown in Fig. 2(A), there is a roughly linear relationship between scattering intensity and MCV and MCHC (as the manifold is approximately planar). Thus, an empirical machine learning analysis method such as PLS, which is not hampered by minor deviations of the experimental settings from theoretical predictions, may provide enhanced performance. To explore this, we extracted 6 parameters from each sample: average red, green, and blue scattering intensities, as well as the standard deviations of each intensity (i.e. the diagonals of the covariance matrix) across all cells within the sample. These 6 parameters were then used to construct a PLS model to predict the MCV, MCHC, and RDW for each sample, using a 10-fold cross validation (some samples were measured multiple times). The results are shown in Fig. 5, where the top row and bottom row are correlation and Bland-Altman analysis, and lines and colors have the same meaning as in Fig. 4. There is a slight improvement in MCV and MCHC predictions compared with the rigorous Mie model, while RDW shows a substantial improvement. The close agreement between the Mie model and the PLS model gives a good indication that the PLS model is not being overtrained, despite the modest size of our dataset.

Analysis of potential application in anemia screening through historical chart review
A key use of the red blood cell morphology parameters determined in a complete blood count (CBC) is for screening for and diagnosis of various forms of anemia. We have previously demonstrated that healthy, IDA and TT samples could be discriminated with high accuracy via MCV, MCHC, and RDW using quadratic discriminants analysis (QDA) [3]. Acknowledging that our low-cost system has increased errors relative to the gold standard clinical system, we next assess the likely impact these errors might have in anemia screening. In our previous study, we acquired CBC data for 268 Chinese children, of whom 73 have iron deficiency anemia (IDA) or thalassemia trait (TT), two microcytic anemias with different treatment indications [52,53]. We then perturbed these data using Gaussian-distributed errors whose standard deviation matched the errors observed in our PLS results shown in Fig. 5. As shown in Figs. 6(A) and (B), a QDA analysis still provides excellent separation between healthy an anemic patients, and good separation between IDA and TT patients, despite the increased errors. These can be summarized in a ROC curve, shown in Figs. 6(C) and (D) and corresponding Sensitivity, Specificity and area under the curve (AUC) are tabulated in Table 1. Note that the ROC curves represent an average of 20 independent realizations of the perturbed data, to ensure that "lucky" perturbations did not unjustly favor our data.

Conclusion
In this paper, we demonstrate a simple quantitative dark-field imaging method to measure mean corpuscular volume (MCV), mean corpuscular hemoglobin concentration (MCHC) and red cell distribution width (RDW) using (i) only 5 µL blood; (ii) a single-step sample preparation; (iii) a large FOV, low-magnification and automated imaging compatible with measuring other cell count parameters [27]. The method described here utilizes small volumes of blood, making the blood collection appropriate for finger-stick or heel-prick blood sampling that does not require a trained phlebotomist or expert operator to perform. For MCV, MCHC and RDW, the method presented here showed excellent agreement with clinical results through measurement of 90 blood samples, including those outside the healthy range. MCV and MCHC could be accurately predicted through a simple regression between observed scattered intensities and those predicted by Mie theory. Using an empirical machine learning method, PLS, allowed us to slightly improve the prediction results for MCV and MCHC, while markedly improving those for RDW. Finally, combining our prediction error with historical data, we speculate that this system has adequate performance for potential future use in widespread microcytic anemia screening. However, we note that our method depends on an isovolumetric sphering protocol that reduces the red cell membrane surface tension. How this sphering protocol would affect anemias such as sickle cell anemia, where the red cell shape is of critical importance, has not been evaluated to the best of our knowledge, and may represent a limitation to this work.
Remaining work includes integrating the dark field imaging module with our previouslyreported fluorescence and bright field imaging blood counting platform. In our previous work we utilize acridine orange staining to separate red and white cells. By integrating these two methods, we can remove any negligible influence of white cells on our red cell sizing in pathologic cases, such as leukemia, where the white cell count is substantially increased. Further, the historical IDA and TT analysis presented here must be validated with actual clinical data. Nevertheless, the results demonstrated here give confidence that quantitative dark-field imaging integrated with a low-cost portable microscope can have a powerful impact on point-of-care measurements of CBC for healthcare screening in low-resource settings.

A. Derivation of extremal values of φ(θ)
The scattering geometry of our darkfield microscope is shown in Fig. 7, which depicts the scattering pattern of a 4.75 µm diameter polystyrene sphere illuminated by 517 nm light versus θ and φ. Also shown in black is the numerical aperture of our objective, which intercepts a portion of this pattern offset from the origin by the off-axis illumination utilized in the microscope. An isoline of constant φ is shown in red, while its intersections with the objective aperture (and thus defining φ min and φ max ) are shown as white dots.
To find the equation of φ min,max (θ) we begin by recognizing that the red dashed curve is a circle centered at the origin, whose equation in Cartesian coordinates is The equation of the circle defining the objective aperture, meanwhile, is: where θ is the center of the objective aperture and ∆θ is its width, and are defined as discussed in the main text, above. Solving these two equations for x yields: Finally, the angle φ min,max can be determined by trigonometry, using a triangle with one side equal to x, and hypotenuse of length θ, φ min,max (θ) = ±arccos which can be expanded to yield Eq. 4 in the main text.

B. MCV error vs. RBC count for 90 clinical samples
As described in the main text, Section 4.1, we purposely exclude regions of the image where multiple red-cells are touching. This is because, due to the limited resolution of our system (chosen to obtain a large FOV without needing to scan the sample), the scattering from cells in close proximity to each other cannot be rigorously separated. The underlying assumption, then, is that the single and multiple red cell regions are statistically identical, in terms of their cell volume and hemoglobin concentration. The reasonable agreement between our experimental data and rigorous Mie theory shown in Fig. 4 is one piece of evidence to support the validity of this assumption. Another piece of evidence can be obtained by comparing the final MCV error obtained Fig. 5 compared with the RBC count. As the RBC count increases, the density of cells in the chamber increases and the proportion of multiple cell regions to single cell regions increases, as the cells become more crowded. Figure 8 shows this comparison, along with a linear fit to the error vs. RBC count. As can be clearly seen, there is no correlation between the RBC count and the MCV error (R 2 = 0.02). Therefore, we conclude that our decision to discard the multiple cell regions from further analysis is not a major contributor to the observed error of our method.

Funding
Ministry of Science and Technology of the People's Republic of China (2016YFA0201300).

Disclosures
ZJS in a named inventor on US patent 10024858B2. The other authors declare no conflicts of interest.