Open Access
Add to BibSonomy
Abstract
Light scattering by single cells is widely applied for flow cytometric differentiation of cells. However, even for human red blood cells (RBCs), which can be modeled as homogeneous dielectric particles, the potential of light scattering is not yet fully exploited. We developed a dedicated flow cytometer to simultaneously observe the forward scattering cross section (FSC) of RBCs for orthogonal laser beams with incident wave vectors $\vec {k}_1$ and $\vec {k}_2$. At a wavelength $\lambda = 632.8\;\textrm{nm}$, bimodal distributions are observed in two-dimensional dot plots of FSC($\vec {k}_1$) vs. FSC($\vec {k}_2$), which result from the RBCs’ random orientation around the direction of flow, as well as from the distributions of their size and their optical properties. Typically, signals of $7.5\times 10^4$ RBCs were analyzed. We actively oriented the cells in the cytometer to prove that orientation is the main cause of bimodality. In addition, we studied the wavelength dependence of FSC($\vec {k}_1$) using $\lambda = 413.1\;\textrm{nm},\;457.9\;\textrm{nm},\;488\;\textrm{nm}$ and 632.8 nm, covering both weak and strong light absorption by the RBCs. Simulations of the light scattering by single RBCs were performed using the discrete dipole approximation (DDA) for a range of sizes, orientations and optical properties to obtain FSC distributions from RBC ensembles. Using the axisymmetric biconcave equilibrium shape of native RBCs, the experimentally observed distributions cannot be reproduced. If, however, an elongated shape model is employed that accounts for the stretching of the cell by hydrodynamic forces in the cytometer, the features of the strongly bimodal measured frequency distributions are reproduced by the simulation. Elongation ratios significantly greater than 1 in the range of 1.5 to 2.5 yield the best agreement between experiments and simulated data.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The shape and elastic properties of various body cells are known to change during disease. For example, changes in cell deformability have been reported for various cancerous body tissues [1]. Red blood cells (RBCs) exhibit remarkable rheological characteristics that are particularly important for their biological function of oxygen delivery to tissue and removal of carbon dioxide through the vascular system in a wide range of blood flow conditions [2]. Alterations of mechanical properties of human RBCs have been reported for peripheral vascular disease [3], sickle cell anemia [4], malaria [5,6], diabetes mellitus [7], sepsis and renal failure [8].
Optical methods for the diagnosis of blood diseases are a current topic of research. For example, it was recently shown that different anemias can be identified holographically by modeling RBCs as microlenses [9]. Other researchers reported in-flow measurements of light scattering patterns for the characterization of RBCs with data analysis based on rigorous wave-optical simulations [10–12].
Optical flow cytometry is a widely used tool to count and differentiate cell populations at high throughput of a few thousand cells per second [13,14]. In an optical flow cytometer, a cell suspension is injected through a steel capillary into a flow cell, where it is accelerated by a fast flowing laminar sheath flow of decreasing cross section. This stretches the sample stream and hydrodynamically focuses it to the center of the flow cell’s cross section. The sample stream intersects with one or several laser beams and the cells are identified by focusing scattered or fluorescence light due to specific labeling onto one or several detectors.
Standard flow cytometers measure the intensity of forward scatter and – at 90° to the laser excitation – side scatter and fluorescence. Light scattering serves to identify subpopulations of cells according to their physical properties and to preselect, e. g., lymphocytes for further immunological differentiation. For such “gating” procedures, the measurement of relative light scattering intensities is sufficient in standard applications. However, absolute scattering cross sections, which can be determined applying the calibration procedure described in Refs. [15,16] are still not available in routine instruments.
If native human RBCs are analyzed in a flow cytometer, one can often observe a bimodal histogram for the forward scattering cross section (FSC) [8,17]. This can be qualitatively explained by the fact that healthy RBCs of biconcave disk-like shape (discocytes) align with their axis of rotational symmetry perpendicular to the fluid flow in a channel much wider than the cell (250 µm channel width in this study compared to 2 µm–8 µm cell size). This leaves them with one angular degree of freedom, the rotation angle $\beta$ around the flow axis. The FSC of a “face-on” illuminated RBC ($\beta = 0^{\circ }$) is very different from a RBC illuminated from the side ($\beta = 90^{\circ }$). Depending on the laser wavelength and the solid angle of the detector, the FSC may increase in a “$\sin (\beta )^2$-like” fashion from $\beta = 0^{\circ }$ to $\beta = 90^{\circ }$. Since the circular cross section of the injection capillary (153 µm inner diameter) does not impose any particular cell orientation, all values of $\beta$ are equally likely and a bimodal histogram is observed in this case with peaks corresponding to the extrema of the FSC as a function of $\beta$. The quantitative properties of these histograms, such as inter-peak distance and height strongly depend on the microscopic details of the RBCs, such as shape as well as the distributions of cell size and intracellular hemoglobin (Hb) concentration. The use of these bimodal histograms as a clinical marker for altered RBC rheology in conditions such as terminal renal failure, diabetes mellitus, sepsis or acute inflammatory state has been proposed [8,17]. However, studies done so far employ empirical criteria to link the histograms to the RBC shape or to detect abnormal rheological properties. In our work, we employ a detailed simulation of the scattering of light from a deformed RBC shape model. This paves the way for a systematic analysis of these measurement quantities with the possibility to extract geometrical parameters of the RBCs. Determination of the elongation ratio of RBCs by routine flow cytometric analysis might be a useful clinical parameter to assess rheological properties in the future.
In this article, we present measurements of the FSC of native RBCs in a home-made flow cytometer. The cytometer features the simultaneous measurement of FSC in two orthogonal directions $\vec {k}_1$ and $\vec {k}_2$ of incident laser light at vacuum wavelengths $\lambda = 632.8\;\textrm{nm}$. In addition to these 2D FSC measurements we present 1D FSC measurements at $\lambda = 413.1\;\textrm{nm},\;457.9\;\textrm{nm},\;488\;\textrm{nm}$ and 632.8 nm. To examine the physical origin of the bimodal histograms of the 1D and 2D FSC, we perform simulations of the light scattering by single RBCs using the discrete dipole approximation (DDA) [18]. We introduce a simple shape model for a stretched RBC and compute the Mueller matrix elements in dependence on orientation, size and Hb concentration of the RBC. Integration of the corresponding combinations of Mueller matrix elements over the photodetector’s solid angle provides the FSC of the cells. FSC histograms are obtained by direct Monte Carlo (MC) sampling of the distributions of orientation angle, cell size, Hb concentration. We compare simulations with stretched and undeformed RBC shape models to measurement data to assess the effect of deformation on the observed bimodality of the frequency distributions of the FSC. Independently measured data from the complete blood count serve to set the distribution parameters of RBC volume and Hb concentration in the simulations. The effect of uniform random orientation of the RBCs around the flow axis is tested experimentally with a preferential orientation of the cells by means of a flattened sample injection capillary (20 µm opening in the narrow direction) and theoretically by simulations with non-uniform orientation distributions. The results for the RBC orientation and deformation are then compared to literature data for RBCs in different types of microfluidic flows and the implications of our findings for flow-cytometric analysis of mechanical properties of RBCs are discussed.
2. Materials and methods
2.1 Flow cytometric setup
For the systematic investigation of the impact of RBC orientation on the forward scatter we designed a dedicated flow cytometer [Fig. 1(a)], characterized by two orthogonal optical beam paths for the observation of forward scatter at 632.8 nm. To ensure symmetrical conditions for orientation measurements of native RBCs, for both optical pathways of the laser beams with wavevectors $\vec {k}^{633}_1$ and $\vec {k}^{633}_2$, identical optical components are selected behind the polarizing beam splitting cube. In addition, the alignment was optimized to minimize the difference in the lengths of both beams (1 m) to about 1 mm. Besides the ultra-stable HeNe-laser (Laboratory for Science, Model 210, Berkeley, CA, USA), the experimental setup for 1D forward scatter incorporates an Ar+-laser and a Kr+-laser (INNOVA 100 and INNOVA 302, Coherent, Santa Clara, CA, USA) [Fig. 1(c)]. The output beams of the lasers are shaped individually by spherical and cylindrical telescopes in order to form elliptical beams. The laser beams are then superimposed by dichroic beam splitters and focused via a microscope objective $4\times$/N.A. = 0.12 to a common spot of approximately the same size $10\;{\mu\textrm{m}} \times 42\;{\mu\textrm{m}}$ (full width of $1/\mathrm {e}^2$ points of intensity) in the flow cell at the intersection point with the blood cells. The minor axis of the elliptical focus and the polarization vectors are parallel to the direction of flow. A quadratic flow cell with $3.6\;\textrm{mm}\times 3.6\;\textrm{mm}$ outer dimensions, a length of 10 mm and a $ 250\;{\mu\textrm{m}}\times 250\;{\mu\textrm{m}}$ flow channel was used [Fig. 1(b)]. The HeNe-laser had a fixed wavelength of 632.8 nm, the Kr+-laser was operated at 413.1 nm while the Ar+-laser was tuned by means of a wavelength selecting prism to $\lambda = 488.0\;\textrm{nm}$ and 457.9 nm. The output powers of the ion-lasers were in the range of 50 mW–100 mW, the HeNe-laser had an output power of 4 mW. In the forward scatter measurements with two orthogonal HeNe-laser beams [Fig. 1(a)], two identical microscope objectives $7\times$ / N.A. = 0.19 for collimation of the scattered light were used. A stripe-shaped beam-stop parallel to the incident polarization vector was used resulting in an observation angle of $\vartheta _3 = 2.2^{\circ } \le \vartheta \le \vartheta _4 = 8.2^{\circ }$. For the 1D FSC($\vec {k}_1$) measurements at 4 laser wavelengths [Fig. 1(c)], the angle of observation $\vartheta _1 = 3.3^{\circ }\le \vartheta \le \vartheta _2 = 17.4^{\circ }$ is determined by a circular beam stop and the numerical aperture of the light collecting microscope objective $20\times$/N.A. = 0.4. When measuring the wavelength dependence of orientation effects of native RBCs, dichroic beam splitters [not shown in Fig. 1(c)] were used to separate the scattered light of the different wavelengths. Detection of the scattering signals was performed using photomultiplier tubes with laser line filters in front of the entrance windows. Signal analysis was done by a home-made data acquisition system based on plug-in boards for personal computers.
Fig. 1. Experimental setup to measure integrated scattering cross sections of forward scatter of RBCs at different wavelengths and solid angles of observation. (a) Setup for the detection of 2D FSC. The HeNe laser beam is divided by a polarizing beam splitter to allow simultaneous observation of forward scatter in two directions, characterized by orthogonal wavevectors $\vec {k}_1$ and $\vec {k}_2$. Two identical microscope objectives $7\times$ / N.A. = 0.19 to collimate scattered light were mounted. (b) Longitudinal section of the mounted flow cell with some characteristic measurements (in mm). Sheath fluid is indicated in blue, sample fluid in red. (c) For the 1D FSC($\vec {k}_1$) measurements at 4 laser wavelengths, an objective $20\times$ / N.A. = 0.4 was used. BS: beam splitter, IF: interference filter, PMT: photomultiplier tube, $\lambda /2$ and $\lambda /4$: retardation plates.
The flow cell of the cytometer was custom made by Hellma GmbH (Germany) and is characterized by a cylindrical part directly melted to the quadratic cuvette tube. The cylindrical part serves to mount the cell in a holder made from PMMA thus connecting it to the fluidic system, see Fig. 1(b). The inner cross section of the cylinder is 4 mm, which reduces along a conical part of length 8.4 mm finally ending inside the quadratic flow channel. The volume flow rate of the sheath flow used to hydrodynamically position the cells in the middle of the flow channel was adjusted to approximately $\dot {V} = 0.22\;\textrm{mL s}^{-1}$ by applying a driving pressure of 30 kPa. Consequently, the average fluid velocity in the $250\;{\mu\textrm{m}} \times 250\;{\mu\textrm{m}}$ of cross section $A = (1/16)\,{\textrm {mm}}^{2}$ amounted to $v_{\textrm {avg}} = \dot {V}/A = 3.56\;\textrm{m s}^{-1}$. A laminar flow profile is developed downstream in the flow channel of 10 mm length. At the point of measurement, i. e., at the point of intersection with the laser beams, a velocity of $v_{\textrm {max}} = 2v_{\textrm {avg}} = 7.1\;\textrm{m s}^{-1}$ should be achieved in the center, provided the parabolic Poiseuille profile is fully developed. Using two laser beams at a known distance (110 µm) the maximum velocity was determined to be 7 m s−1, indicating that the profile corresponds to a parabola and that at the point of measurement, the flow profile features only transverse gradients. To obtain a rough estimate of the longitudinal gradients in the conical part of the flow cell were the RBC suspension is injected, we assume a flat flow profile with $v(x,\;y,\;z) = v_{\textrm {avg}}(z) = \dot {V}/A(z)$ at every cross section of the cone, which has an apex angle $\psi = 25^{\circ }$. Furthermore, we assume that the circular cone transitions smoothly into the quadratic flow channel of area $A_{\textrm {min}} = (1/16)\;\textrm{mm}^{2}$, thus defining the minimal radius $R_{\textrm {min}} = \sqrt {A_{\textrm {min}}/\pi } = 141\;{\mu\textrm{m}}$. Since the radius reduces linearly with $z$ starting from $R_{\textrm {max}} = 2\;\textrm{mm}$ at the inlet, the area $A(z)$ is a quadratic polynomial of $z$. The longitudinal gradient of this extensional flow is given by $\dot {\varepsilon } = \mathrm {d} v_{\textrm {avg}}(z) /\mathrm {d} z$, which is highest at the narrowest part of the cone and amounts to
For verification of the stability of hydrodynamic positioning of particles or cells and the examination of the diameter of the sample flow we applied the established protocol in flow cytometry, i. e., when putting the flow cytometer into operation the width of the sample was measured using fluorescein solution as the sample fluid and the fluorescence was imaged on a CCD camera. Then, each day the performance of the cytometer was controlled by determining the coefficient of variation (CV) for monodisperse polystyrene microspheres by observing scattered or fluorescence light. Typically, the CV in scattered and fluorescence intensity amount to about 2% and correspond the values stated by the manufacturers.
Besides using the standard steel capillary with circular outlet, we actively oriented RBCs by mounting a capillary with flattened outlet cross section. For this purpose, several gauge 25s capillaries were carefully squeezed between two polished plates and selected by microscopic inspection. The minimum bending radius achieved amounted to 10 µm resulting in a rounded rectangular outlet cross section measuring 20 µm along the narrow direction. This allowed to orient the RBCs predominantly with their wide axis perpendicular to the short axis of the flattened capillary.
2.2 Blood preparation
Human blood was collected in a doctor’s office by venipuncture from a healthy individual in accordance with recommendations of medical associations [19] and with the ethical standards established by the Declaration of Helsinki. The volunteer agreed in written form that his/her samples will be used for research purposes. As anticoagulant ethylene-diaminetetraacetic acid (EDTA) was used, contained in 2.7 mL tubes (Monovette EDTA K, Sarstedt AG & Co., Germany).
For the blood samples a hematology analyzer (ABX Micros ES 60, Axonlab AG, Stuttgart, Germany) was used to determine the complete blood count (CBC) including the mean cellular volume (MCV) and the mean corpuscular Hb concentration (MCHC) of RBCs, needed as input for modeling their light scattering properties. The blood samples were diluted by a factor of about 100 in phosphate buffered saline (PBS, Carl Roth GmbH, Karlsruhe, Germany) when investigating native erythrocytes in our home made optical flow cytometer. For testing the device and for calibration, isovolumetric sphering of RBCs was applied using the procedure described by Kim and Ornstein [20,21]. To this end, the RBCs are suspended in a mixture of isotonic saline, bovine serum albumin (BSA) and sodium dodecyl sulfate (SDS), which causes their lipid membrane to contract while not changing the volume.
2.3 Calibration
Calibration of the FSC-axes to derive absolute scattering cross sections from the recorded light scatter intensities was performed by comparing measurements of isovolumetrically sphered RBCs [20,21] to Mie scattering computations [22,23]. The procedure, which is analogous to the method based on calibration using polystyrene microspheres [16], has the advantage that we use spheres for calibration with the same refractive index (RI) as the targeted particles, i. e., native RBCs. Plane wave incidence was assumed. The RIs and hematological parameters (see Tab. 1), the integration of angular scattering distributions and the MC sampling method for extracting probability distributions were the same as used for the DDA simulations (see sec. 2.4). The conversion factor between channel number of the analog-to-digital converter and absolute cross section was chosen by numerical optimization such that the scalar product between the measured and simulated frequency or probability distributions (i. e., the integral over their product) was maximal.
Table 1. Hematological parameters of the concentration distribution (normal) and size distribution (log-normal) of the RBC sample. $\mathrm {MCHC} = \mathbb {E}(c_\textrm {Hb})$, $\mathrm {MCV} = \mathbb {E}(V)$ and $\mathrm {RDW} = \mathrm {CV}(V)$ were obtained from the complete blood count (CBC). Here $\mathbb {E}$ denotes the expectation value (or mean) and $\mathrm {CV}$ denotes the coefficient of variation, i. e., the relative standard deviation. Since the hemoglobin concentration distribution width $\mathrm {HDW} = \mathrm {CV}(c_\textrm {Hb})$ is not a routinely measured parameter in impedance-based analyzers, we set it to a typical value that best fits the measurements of sphered RBCs.
2.4 Mathematical model and simulation
Because a RBC has minimal internal structure (if any at all) and its membrane is very thin compared to the laser wavelength and the RBC size [24], it can be modeled as a homogeneous dielectric particle. Hence the scattering of a laser beam by a single RBC is described by the Helmholtz equation for the electric field $\boldsymbol {E}$
with the vacuum wavevector $k = \omega /c = 2\pi /\lambda$ and the (space-dependent) complex refractive index (RI) that assumes a constant value $\mathfrak {n}_{\textrm {RBC}}\in \mathbb {C}$ within the cell and another constant value $n_{\textrm {m}}\in \mathbb {R}$ (RI of the host medium) outside. The total electric field is the sum of the incident and scattered fields Together with radiation conditions for $\boldsymbol {E}^s$ at $r\to \infty$ and continuity conditions for the tangential field components on the particle boundary this defines the scattering problem.To solve this problem for a given scatterer’s geometry and incident field, we use the discrete dipole approximation (DDA), where the volume of the scatterer is discretized into a cubic grid. Using the Green’s function of the Helmholtz operator, Eq. (2) can be re-written as a volume integral equation. Using the volume discretization, this integral equation is then approximated by a system of algebraic equations that can be solved by methods of numerical linear algebra. We use the ADDA 1.2 implementation of the DDA. The details of the method and implementation are given in Refs. [18,25]. The DDA has been tested against other numerical methods to compute light scattering by single RBCs and was found to be the method of choice for scatterers without rotational symmetry [26].
To describe the RBCs biconcave shape, we use the equation of Yurkin [27]:
This implicit equation describes the surface of the cell in cylindrical coordinates $(\rho,\;\varphi,\;z)$. This shape model is symmetric with respect to rotation around the $z$-axis, i. e., axisymmetric. It further has mirror-symmetry with respect to the $x y$-plane. As an alternative to the coefficients $R_1\dotsc R_4$ of Eq. (5), the model can be characterized by the diameter $D$ of the cell, its minimal thickness $b$ and maximal thickness $h$ and the diameter $c$ of the circle of maximal thickness, see Fig. 2(a).Fig. 2. (a) Cross section through the center of the undeformed shape model defined by Eq. (5). Surface triangulations of (b) the undeformed axisymmetric shape model and (c) the stretched model. Arrows indicate the orientation of the RBC relative to the flow axis and the two incident lasers with wavevectors $\vec {k}_1$ and $\vec {k}_2$. “Figure axis” denotes the symmetry axis before stretching.
To account for the hydrodynamic deformation of the RBCs in the flow cytometer, we extend the model by a non-uniform linear scaling. In Cartesian coordinates this is expressed by the mapping
Since $f_x$ is the direction of fluid flow along which the cell is stretched $f_x>1$. Along the other two directions, we expect a compression of the cell $f_y, f_z<1$. We chose $f_y = f_z = 1/\sqrt {f_x}$. In this way, the stretching is described by a single parameter $f_x$ and conserves the cell’s volume, since the determinant of the transformation matrix is then $f_x\, f_y\, f_z = 1$.The result is a shape model whose top-view is an ellipse of semi-major axis $f_x\,D/2$ and semi-minor axis $D/(2\sqrt {f_x})$. The minimum and maximum thickness are $b/\sqrt {f_x}$ and $h/\sqrt {f_x}$, respectively. For the stretched model we used $f_x = 9/4 = 2.25$. The parameters of the stretched and unstretched shape models are given in Tab. 2 and the corresponding shapes are depicted in Fig. 2(b) and (c). The parameters of the unstretched model describe a typical RBC at rest [28]. The parameters of the stretched model were tuned by hand in order to match the 1D and 2D FSC measurements. The RBC shape model is further allowed to rotate around its long axis, i. e., the direction of flow. The Euler angle $\beta$ is a free parameter of the model. Surface triangulations depicting the three-dimensional shape of both models are shown in Fig. 2(b) and (c). These surface triangulations serve only to illustrate the shape. They were not used for scattering simulations, where cubic volume discretization was employed.
Table 2. Parameters of the shape models used. The volume $V$ was varied by changing the diameter $D$ only. Parameters $D$, $c$, $h$ and $b$ refer to the axisymmetric shape before deformation. Values marked with an asterisk ($^*$) correspond to an average RBC with $V = 92.7\;{fL} = \mathrm {MCV}$. $S$ is the surface area and the sphericity index (SI) is defined as $\mathrm {SI} = {\sqrt [3]{36\pi \,V^2}}/{S}$.
For the optical properties of the fluid, in which the RBCs are suspended during measurement we assume those of water [29], since the sheath fluid is made up mostly of water. For the complex RI of the RBCs we assume wavelength- and concentration-dependent values [30]
Table 3. RI of water and RBCs [30] (at ${c_{\textrm {Hb}}} = 344\;\textrm{g L}^{-1} = {\textrm {MCHC}}$) assumed for simulation. $\mathfrak {m} = \mathfrak {n}/n_{\textrm {H}_2\textrm{O}}$ is the relative RI of the RBCs.
The incident beam is assumed to be a plane wave propagating along the $z$ axis, i. e., with wavevector (in the host medium) $\vec {k}_{\textrm {m}} = 2\pi \, n_{\textrm {H}_2\textrm{O}}/\lambda \, \boldsymbol {e}_z$. Among other quantities, DDA computations yield the Mueller matrix of the far field. At large distances from the scatterer ($k_{\textrm {m}}\,r\gg 1$) the electric field behaves like a spherical wave whose direction-dependent intensity is described by the Mueller matrix $\mathsf {S}(\vartheta ,\varphi )$. This real matrix describes the transformation of the Stokes vector $(I,Q,U,V)^T$, where $I$ is the total intensity, $Q$ and $U$ describe the linearly polarized portion of the light and $V$ describes circular polarization. The Stokes vector of the scattered field, relative to the scattering plane [spanned by the unit vectors $\boldsymbol {e}_z$ and $\boldsymbol {e}_r(\vartheta ,\varphi )$], is given by
Using a volume discretization with $ 45\;\textrm{nm}$ cubes, corresponding to 6.9 dipoles per wavelength at the shortest wavelength $\lambda = 413.1\;\textrm{nm}$ ($\lambda /n_{\textrm {H}_2\textrm{O}} = 307.7\;\textrm{nm}$) and 10.6 dipoles per wavelength at the longest wavelength $\lambda = 632.8\;\textrm{nm}$ ($\lambda /n_{\textrm {H}_2\textrm{O}} = 475.0\;\textrm{nm}$) we computed Mueller matrices in ADDA for $\vartheta = 0.0^{\circ }: 0.2^{\circ }: 28.0^{\circ }$ and $\varphi = 0^{\circ }: 1^{\circ }: 360^{\circ }$, from which the FSC was calculated for the respective aperture by numerical integration. In the $a:b:c$ expression $a$ and $c$ are the start and end values, respectively and $b$ is the step width of a uniform grid. A database of FSC values was thus computed for parameters
In order to obtain the probability distribution function (pdf) of the FSC, the pdfs of the input parameters $V, \beta , {c_{\textrm {Hb}}}$ need to be propagated through this nonlinear function $g$. This was achieved by a direct sampling MC technique: $N_{\textrm {MC}} = 10^5$ triples of pseudorandom numbers were drawn for $(V, \beta , {c_{\textrm {Hb}}})$ from their respective distributions: a uniform distribution for $\beta$, a normal distribution for ${c_{\textrm {Hb}}}$ and a log-normal distribution for $V$. The latter two are known to describe native and sphered RBCs well [21]. The FSC was computed for each of the $N_{\textrm {MC}}$ triples. To model the 2D FSC measurements, one sets
For a fixed size $D = 7.6\;{\mu\textrm{m}}$ (corresponding to $V = 91.5\;\textrm{fL}$, i. e., roughly the mean volume ${\textrm {MCV}}=(92.7\pm 1.9)\,{\textrm {fL}}$) and fixed concentration ${c_{\textrm {Hb}}} = 335\;\textrm{g L}^{-1}$, we computed $\left .g(V, \beta , {c_{\textrm {Hb}}})\right |_{V,{c_{\textrm {Hb}}} = {\textrm {const}}}$ and sampled histograms for $f_x = 1.75:0.25:3$. The best agreement between simulation and experiment was found for $f_x = 2.25$. Hence the full database was computed for $g(V, \beta , {c_{\textrm {Hb}}})$ with $f_x= 2.25$.
3. Results and discussion
Two-dimensional orthogonal forward light scatter measurements are illustrated in Fig. 3 (left column). Besides the scatter plots of $x = \textrm {FSC}(\vec {k}_1)$ vs. $y = \textrm {FSC}(\vec {k}_2)$, marginal histograms of $x$ and $y$ are shown individually. The cluster apparent in the experimental 2D-diagram at small scattering intensities close to the origin is caused by forward scattering of blood platelets. It should be noted that, even though the RBCs are illuminated with two lasers simultaneously, the intensities measured with the two detectors correspond to the FSCs, because the side-scattered intensity from the respective other laser is about three orders of magnitude lower than the forward-scattered intensity for RBCs [16]. In the top left panel of Fig. 3, where no preferential orientation was imposed on the RBCs, the distributions of the measurement data exhibit a pronounced bimodal distribution, which (in 2D) lies on a cross diagonal ($y = \textrm {const}-x$) of the plot. The latter indicates that the cells are oriented in a way, where they are asymmetric with respect to the direction of flow. If the RBCs were oriented with their rotational axis aligned to the flow, as it occurs, e. g., in microfluidic flows [32–34], or in any other way symmetric around the flow axis, they would “look the same” from both directions $\vec {k}_1$ and $\vec {k}_2$. This would result in a 2D distribution located on the main diagonal ($y=x$) of the plot. The fact that this is not observed allows to conclude that RBCs in the cytometer are not (or at least not all) axisymmetric with the direction of flow. As mentioned before, our interpretation is an orientation of the RBCs relative to the flow direction as depicted in Fig. 2 with a random orientation $\beta$ for each RBC. Since neither the cylindrical inlet of the cytometer, nor the circular injection capillary, nor the conical contraction of the cytometer impose a preferred direction, it seems reasonable to assume that $\beta$ follows a uniform distribution, i. e., that all angles are equally likely. On the other hand, this continuous rotational symmetry is broken in the quadratic flow channel, were the scattering signals are recorded. However, the channel ($250\;{\mu\textrm{m}}\times 250\;{\mu\textrm{m}}$) is much wider than the RBCs, which travel along its centerline. For a developed laminar flow profile in such a channel, it can be shown that the velocity gradients dominating on the length scale of the RBC exhibit a continuous rotational symmetry, too. Thus, there is no conceptual reason to assume $\beta$ to be non-uniformly distributed. Ideally one would expect a perfect symmetry between the upper and lower triangle of the plot in Fig. 3. In the experiment there are some small deviations, indicating a slight preference for cell orientation along $\vec {k}_1$. This may be caused by small imperfections, e. g., in the cross section of the injection capillary of the flow cytometer, its deviation from the coaxial position with respect to the flow channel or slight differences in the detector apertures, e. g., caused by variation of the beam stop’s position.
Fig. 3. Measurement data (left) and simulation (right) of the 2-direction orthogonal FSC for native RBCs. Each isolated dark blue dot in the 2D plot corresponds to a single cell. Color codes the density of the dots (bright=high density). The histograms on the top and side are projections of the dot plot to a single axis. The simulation was performed using the stretched shape model and the hematological parameters from the CBC (Tab. 1) and 1×105 random samples. Top row: RBCs injected through a circular injection capillary (7.5×104 events) and simulation with uniformly distributed orientation angle $\beta \in \mathcal {U}(-90^{\circ }, 90^{\circ })$. Bottom row: Flattened capillary (9.7×104 events) resulting in preferential orientation “face-on to $\vec {k}_1$” and simulation with normally distributed orientation $\beta \in \mathcal {N}(0, 36^{\circ })$.
The scatter plot for the simulation results, shown in the upper right panel of Fig. 3, and the marginal histograms reveal the qualitative features of the experiment (cross-diagonal, bimodal 2D distribution) very well. For detailed simulations, which were performed to interpret the measured bimodal distributions of the forward scattering cross section of native RBCs in flow cytometry, we applied the DDA to calculate the Mueller matrix and the FSC in dependence on different shapes, orientations and Hb concentrations of the RBCs. In particular, our systematic simulations proved that it is indispensable to use an elongated shape model for the RBCs for the description of experimental observations. To this end, an extension of an existing RBC shape model, in which the axisymmetric equilibrium shape (with figure axis perpendicular to the flow) is linearly stretched along the flow direction and compressed perpendicular to the flow direction was implemented in our analysis. Furthermore we accounted for the fact that intracellular Hb concentration varies within the cell population even of a single person by several percent (HDW = 5% for the blood sample examined) and consequently the RI varies from cell to cell, depending linearly on $c_\textrm {Hb}$. Even though not perfect, FSC histograms computed with our shape model and using measured hematological parameters (MCV, RDW and MCHC) are in remarkably good agreement with measurements of the 2D FSC.
Apart from our theoretical analysis, the cross-diagonal character of the 2D measurements already indicates that bimodality is a result of random orientation of the RBCs, described by the angle $\beta$. To experimentally validate our interpretation, we performed a measurement where the steel capillary injecting the RBCs in the flow channel of the cytometer was flattened to a cross section measuring about 20 µm along the narrow direction. The orientation was chosen in such a way that the short axis of the opening was parallel to $\vec {k}_1$. This results in preferential orientation of the cells along $\beta = 0^{\circ }$, i. e., the larger cross sections of the RBCs were perpendicular or “face on” to $\vec {k}_1$. The result is shown in the lower left panel of Fig. 3. A pronounced asymmetry occurs now in the heights (frequencies) of the formerly symmetric peaks: Compared to the peak with low FSC($\vec {k}_1$) and high FSC($\vec {k}_2$), the number of events in the cluster with high FSC($\vec {k}_1$) and low FSC($\vec {k}_2$) peak is significantly lower. This effect is reproduced if the orientation angle $\beta$ in the simulations is sampled from a normal distribution $\beta \in \mathcal {N}(0, 36^{\circ })$ like shown in the lower right panel of Fig. 3 instead of a uniform distribution $\beta \in \mathcal {U}( -90^{\circ }, 90^{\circ })$ used to calculate the results depicted in the top right panel. At first glance, the results in Fig. 3 may seem unexpected, since although the geometrical cross section is larger for a RBC in the “face on to $\vec {k}_1$” orientation, i. e., $\beta = 0^{\circ }$, their corresponding forward scattering cross section FSC($\vec {k}_1$) is smaller. This observation highlights the importance of interference effects, which dominate the observed intensity distributions. In particular, in the “face-on” orientation ($\beta = 0^{\circ }$) the oscillating dipoles (excited by the incident laser) at the outermost points of the RBC are further apart than in the perpendicular orientation ($\beta = 90^{\circ }$). For the elementary waves emitted by these dipoles and seen from the far-away detector, a larger geometrical diameter of the scatterer, i. e., a larger distance between the dipoles results in relatively large phase differences already at small angles $\vartheta$ to the forward direction. Hence, destructive and constructive interference occurs already at small angles and the far-field intensity distribution has a narrow, high-intensity forward lobe in this case. On the other hand, if the RBC is oriented with its narrow side to the beam ($\beta = 90^{\circ }$), the far-field intensity distribution is less concentrated around the forward direction and more light is scattered at higher near-forward angles. Because the forward lobe of the scattered intensity distribution is blocked by the beam stop in all cases, the RBC with perpendicular orientation has a higher FSC than the one with “face-on” orientation. To demonstrate the pronounced sensitivity of the FSC on the RBCs’ shape, we used the non-stretched, axisymmetric shape model in the simulation and the identical hematological parameters (see Tab. 1) as in Fig. 3. Such axisymmetric shape models have been used in many simulation studies in the context of RBCs in flow cytometry [10,11,35,36]. The result of our calculations, shown in Fig. 4, looks very different from the experiment and the simulations based on the elongated shape model shown in Fig. 3. Rather than describing a bimodal distribution, the maxima of the 2D FSC plot form a “loop” with four distinct maxima. Projected to the marginal 1D histograms this results in a trimodal distribution. It follows that the experimentally observed bimodal distribution cannot be reproduced when neglecting the elongated shape of the RBCs caused by the hydrodynamic forces in the flow cytometer.
Fig. 4. Simulation of the 2D FSC in using the undeformed, axisymmetric shape model and the hematological parameters from the CBC. The bimodal distribution shown in Fig. 3 cannot be reproduced when neglecting the deformation of RBCs in flow.
Alternatively, the impact of the elongation was assessed by simulating the dependence of FSC on $\beta$ for varying stretching factor $f_x=1.75,\dotsc , 3$. The result for monodisperse samples with constant size and Hb concentration are shown in Fig. 5. These simulated curves may help to understand why and under which conditions a bimodal distribution is observed. Since all angles $\beta$ are equally likely, a maximum occurs in the 1D- and 2D-FSC histograms whenever the derivative $\partial g(V, c_\textrm {Hb}, \beta )/\partial \beta$ is low, i. e., near the extrema of the curves in Fig. 5. As can be seen, the curves are almost monotonically increasing for high stretching factor $f_x$ and develop a pronounced intermediate maximum at $\beta \approx 60^{\circ }$ for lower values of $f_x$, which then results in multimodal histograms for 1D-FSC measurements and intricate distributions like shown in Fig. 4 for 2D-FSC measurements. However, the curves in Fig. 5 are just calculated for a fixed cellular volume and a Hb concentration of 335 g L−1 and do not give the full picture, for which polydispersity and variation of Hb concentration needs to be considered. It should be noted that if simulations of the 2D-FSC plots shown in Fig. 3 and 4 are performed with monodisperse ensembles, then all calculated points are arranged on curved lines instead of forming point clouds. The effects of the biological variations of cell size and intracellular Hb concentration (i. e., variable RI) turn out to have effects of similar magnitude as the change in RBCs shape in “smearing out” these curves.
Fig. 5. Dependence of the simulated FSC on orientation angle $\beta$ and elongation factor $f_x$ for fixed volume and Hb concentration.
Besides the influence of the orientation of native RBCs, their volume, shape and intracellular Hb concentration on forward scattering intensities, the laser wavelength strongly affects the total scattering cross section and the observed distribution. To investigate the wavelength dependence of the bimodal distributions we used three different lasers [see Fig. 1(c)] at vacuum wavelengths $\lambda = 413.1\;\textrm{nm}, 457.9\;\textrm{nm}, 488\;\textrm{nm}$ and 632.8 nm. The analysis was restricted to 1D histograms of the FSC. Due to the higher N.A. of the microscope objective used for these measurements and the different beam stop, for 632.8 nm the absolute values for the FSC differ slightly from the 2D measurements discussed above. The experimental results are shown in Fig. 6 in comparison with simulation data using the stretched shape model. As the laser wavelength is decreased from 632.8 nm down to 413.1 nm, the imaginary part of the refractive index of the RBCs increases, i. e., the RBCs become significantly absorbing due to the increased absorption of Hb. The change of the imaginary part of the RI causes a decrease of both, the inter-peak distance and the intensity of the FSC population at high intensities. These tendencies are reproduced in the simulation with the same shape model for all wavelengths and the Hb concentration-dependent RI dispersion of RBCs from Ref. [30]. A specific feature is observed at 413.1 nm, where the imaginary part of the RI of the RBCs is highest: the distribution is very narrow and approximately unimodal with only a slight shoulder remaining at high intensities of the FSC distribution. This characteristic, i. e., the reduced scattering cross section and the narrow distribution was exploited to differentiate red and white blood cells in diluted whole blood samples just by light scattering avoiding staining procedures [16].
Fig. 6. Comparison between measurements and simulations of the cross section for forward scatter FSC $(3.3^{\circ }\le \vartheta \le 17.4^{\circ })$ at four different laser vacuum wavelengths $\lambda$ with 2.9×105, 2.6×105, 2.1×105 and 2.8×105 measured events, respectively, from top to bottom. The $y$-axes show the probability density function. The simulated histograms (1×105 events) were smoothed using the 0.5 µm2 Gaussian noise that was also applied to the 2D histograms.
Whereas our simulations reproduce the observed bimodal distributions, absolute scattering cross sections were found to differ by about a factor of 1.8. As discussed above, deviations might be caused by a non-optimized shape of the RBCs (see Fig. 5) and by uncertainties in the solid angles of observation for forward scatter. Furthermore, we assumed in the simulations that the incident laser beam can be described as a plane wave. However, in the cytometric setup, the incident beam is shaped to an elliptical focus with semi-axes ($\textrm {e}^{-2}$ points of intensity) $w_{0 \perp } = 21\;{\mu\textrm{m}}$ perpendicular to the flow direction and $w_{0\parallel } = 5\;{\mu\textrm{m}}$ along the flow direction. To derive absolute scattering cross sections, sphered RBCs of approximately 5.6 µm diameter were used, significantly smaller compared to the stretched native cells which reach lengths of around 17 µm. Hence, the intensity is higher for sphered RBCs compared to elongated RBCs thus causing a systematic deviation to larger cross sections in our calculations. We estimated the impact of this effect by performing additional simulations with Gaussian laser beams of circular waist radii $w_0 \in [5, 30]\;{\mu\textrm{m}}$ at a fixed ${c_\textrm {Hb}} = 335\;\textrm{g L}^{-1}$. The results of these computations indicate that the main effect of the Gaussian beam compared to the plane wave consists in a reduction of the FSC in total with decreasing beam waist. As an example, the results for an average RBC are shown in Fig. 7. Apart from this, the general shape of the function $\left .g(V, \beta , {c_\textrm {Hb}})\right |_{{c_\textrm {Hb}} = \textrm {const}}$ does not change drastically with beam waist. An elliptical Gaussian beam [37] is expected to have a similar effect with a magnitude somewhere in between the cases of $w_0 = w_{0\parallel } = 5\;{\mu\textrm{m}}$ and $w_0 = w_{0 \perp } = 21\;{\mu\textrm{m}}$. Hence simulation with a realistic Gaussian beam is expected to yield the same qualitative agreement, while at the same time providing better quantitative agreement for the absolute values of the FSC.
Fig. 7. Simulated dependence of the FSC of the stretched shape model on the waist diameter $2w_0$ of a (circular) Gaussian beam for an average-volume RBC and three different orientations $\beta$ at fixed Hb concentration. Dashed lines indicate the short ($2\,w_{0\parallel } = 10\;{\mu\textrm{m}}$) and long ($2\,w_{0\perp } = 42\;{\mu\textrm{m}}$) axis, respectively, of the elliptical focus employed in the experiment. A plane wave corresponds to $w_0\to \infty$.
We have demonstrated that significant deformation of native healthy human RBCs occurs during measurement and have attributed them to the hydrodynamic forces present in the cytometer’s flow cell of $ 250\;{\mu\textrm{m}}\times 250\;{\mu\textrm{m}}$ channel cross section with cell velocities of about 7 m s−1. The deformation of single RBCs in flow has been extensively studied for microfluidic scenarios [32,34], such as pipe flow in microcapillaries of few µm width, i. e., comparable to the RBC’s size. Flow velocities in these experiments typically are in the µm s−1 to cm s−1 range. In such a stationary microcapillary flow the deformation of the RBC is caused by transverse velocity gradients and the proximity to the capillary walls. In contrast, the developed laminar flow profile in the flow cell of our cytometer rather corresponds to an infinite Poiseuille flow since the walls are macroscopically far away. Due to the much higher flow velocity, the transverse gradients and the curvature of the flow profile at the center (in absence of the RBC) are still comparable to microcapillary flows [32,34]. However, due to the long distance to the walls the RBC is not “squeezed through” the channel like in a microcapillary and the transverse gradients may actually be too weak to cause the strong elongation of the RBCs in the flow cytometer. To plug in some numbers, the velocity profile of a Poiseuille flow is given by
where $v_\textrm {max} = 7\;\textrm{m s}^{-1}$ is the maximum flow velocity and $\rho$ is the distance from the center line of the channel. For simplicity we assume a circular cross section for the flow channel of effective radius $R_\textrm {eff} = 250\;{\mu\textrm{m}}/\sqrt {\pi } = 141\;{\mu\textrm{m}}$. Hence for a deformed RBC measuring $2\Delta \rho = 5\;{\mu\textrm{m}}$ across, a typical transverse velocity gradient (shear rate) isIn our flow cytometer the RBC is measured after it was exposed to the extensional stress. The viscoelastic relaxation time constants of stretched RBCs are known to be about 100 ms–250 ms [41,42]. In this time a RBC flowing in the cytometer at 7 m s−1 covers a distance $\ge 700\;\textrm{mm}$, i. e., has long left the flow cell of 10 mm length. Hence the effect of the extensional stress is still well visible in the RBC shape at the point of measurement a few millimeters downstream of the entrance to the flow channel. However, it has been pointed out that elastic properties of RBCs in high-strain extensional flows may differ from those measured with low-strain techniques [40], thus possibly affecting the relaxation dynamics, too. Because the shape and orientation, as well as the quantitative measures of deformation found by us compare well to measurements of RBCs in extensional flows [38,39], we conclude that the measured RBC shape in the cytometer is an impression of the extensional stresses occurring further upstream during injection and not so much a steady state shape caused by the laminar flow profile in the narrow flow channel.
4. Conclusion
We have developed a novel flow cytometer set-up capable to simultaneously observe forward scatter in two orthogonal directions. Absolute scattering cross sections of native RBCs were determined by calibrating the instrument with measurements of iso-volumetrically sphered erythrocytes and calculations of their scattering cross section for the instrument specific solid angles of observation. Pronounced bimodal distributions were observed and are caused by the orientation of RBCs in the flow channel of the cytometer and their elongated shape due to hydrodynamic forces. This interpretation was experimentally validated by actively orienting the erythrocytes using a steel capillary for the injection of RBCs in the sheath flow whose outlet was changed from a circular cross section to a flattened, oval cross section. For modeling and simulation, we applied the DDA to compute the light scattering cross sections of native, oriented RBCs for the specific instrumental conditions and hematological RBC indices, i. e., mean cellular volume, red cell distribution width and mean cellular Hb concentration, obtained using a hematology analyzer. Our results clearly indicate that the elongations of RBCs, and hence their rheological properties, are essential to explain the experimentally observed bimodal distributions. Numerical simulations based on a biconcave shape model with an axis of rotational symmetry that corresponds to the RBC shape in the absence of external forces are in strong disagreement with the experimental observations reported here. On the other hand, we were able to reproduce the main features of the measured bimodal distributions by accounting for the elongation of RBCs in the flow cell. We found that a length of about 17 µm corresponding to an elongation factor of 2.25 yielded the best match to measurements at velocities of 7 m s−1 in the flow cytometer. Our results give a proof of principle that rheological properties of native erythrocytes can be derived from flow cytometric measurements. For this purpose, it is necessary to know the relevant characteristics of the instrument concerning the optical layout and the fluidic system in detail, i. e., the solid angle of observation, shape of the laser beam at the intersection point as well as the forces exerted on the cell during hydrodynamic focusing and while passing through the flow channel. The presented results provide an incentive to compare and combine the mathematical approach here that is based on accurate optical modeling with a modeling of rheological and mechanical properties of cells in a macroscopic flow geometry. To this end, our studies will be extended by systematic investigations of the influence of hydrodynamic forces when changing the velocity of the sheath flow. Furthermore, the proposed ad hoc shape model for deformed RBCs has some unrealistic properties. For example, it is known that the RBC volume and surface area remain quasi-constant during deformation of the cell [43]. While the distribution of RBC volumes was measured independently in a CBC and sampled accordingly in the simulations, the surface areas of the RBCs in the blood sample are unknown. However, the correlation between volume and surface area reported in the literature suggest that an average RBC in the blood sample considered here (92.7 fL volume) should have a surface area of about 125 µm2 [44]. The deformed shape model predicts a surface area of 169 µm2 (compare Tab. 2), which is not plausible. A more realistic description of the shape is required for a quantitative determination of mechanical properties and could possibly be found by simulations using mathematical models for the viscoelastic behavior of RBCs in fluid flows [33,43,45]. The numerical methods for such mechanical modeling are described, e. g., in Ref. [46] and simulations of RBCs in an impedance flow cytometer were already carried out in Ref. [47]. At the moment, approaches combining theoretical and numerical techniques from both areas – optics and cell mechanics – are sparse in the literature and either lack a rigorous treatment of the mechanics (like the present article) or of the optics [48,49]. However, by a rigorous treatment of both, it may be possible to extract quantitative elastic information about cells employing widely used high throughput flow cytometers as a complementary approach to novel flow-through measurement techniques like deformation cytometry [50], optical stretching [51] or scanning flow cytometry [12]. Although modeling of light scattering by elongated and oriented erythrocytes is computationally expensive, a real time analysis of 2D- or multiple angle forward scatter may be feasible by using pre-computed databases spanning the multidimensional parameter grid. This enterprise, however, will require to go substantially beyond the simple table-lookup method for the analysis of two-angle forward scatter sphered RBCs that was employed by Tycko et al. [21] and is implemented in some hematology analyzers. Whereas the lookup table for sphered RBCs has just 2 parameters, i. e., size and RI, our model currently has already 7 parameters: Orientation, size and shape (5 parameters) and Hb concentration. Since for a realistic shape model elastic properties should be included, possibly resulting in more parameters, development of new mathematical approaches like surrogate modeling or tools of large-scale data analysis is needed. On the other hand, a transfer of our method to present flow cytometers does not require preparation of sphered erythrocytes and would allow to understand in more detail the altered RBC rheology in the above mentioned diseases, i. e., renal failure, diabetis mellitus, sepsis or acute inflammatory state [8]. The method is particularly appealing for diseases in which the overall RBC morphology remains the same, but the elastic properties of the cells are altered, as this is the class of shapes covered by our shape model and the theoretical models for RBC mechanics. Furthermore, clinical research might be stimulated to improve diagnosis of patients suffering from sickle cell disease [4], anemia or malaria [5,6]. However, for these diseases data analysis may need to be improved by a more comprehensive modeling of RBC shapes and possible internal structures and the sensitivity of the FSC histograms needs to be assessed.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
References
1. S. Suresh, “Biomechanics and biophysics of cancer cells,” Acta Mater. 55(12), 3989–4014 (2007). [CrossRef]
2. M. Diez-Silva, M. Dao, J. Han, C.-T. Lim, and S. Suresh, “Shape and biomechanical characteristics of human red blood cells in health and disease,” MRS Bull. 35(5), 382–388 (2010). [CrossRef]
3. H. Reid, J. Dormandy, A. Barnes, P. Lock, and T. Dormandy, “Impaired red cell deformability in peripheral vascular disease,” Lancet 307(7961), 666–668 (1976). Originally published as Volume 1, Issue 7961. [CrossRef]
4. S. R. Goodman and C. Joiner, “Damage to the red blood cell membrane in sickle cell disease,” in Renaissance of Sickle Cell Disease Research in the Genome Era, P. Betty, ed. (Imperial College Press, 2007), p. 156.
5. H. A. Cranston, C. W. Boylan, G. L. Carroll, S. P. Sutera, J. R. Williamson, I. Y. Gluzman, and D. J. Krogstad, “Plasmodium falciparum maturation abolishes physiologic red cell deformability,” Science 223(4634), 400–403 (1984). [CrossRef]
6. J. P. Mills, M. Diez-Silva, D. J. Quinn, M. Dao, M. J. Lang, K. S. W. Tan, C. T. Lim, G. Milon, P. H. David, O. Mercereau-Puijalon, S. Bonnefoy, and S. Suresh, “Effect of plasmodial RESA protein on deformability of human red blood cells harboring Plasmodium falciparum,” Proc. Natl. Acad. Sci. 104(22), 9213–9217 (2007). [CrossRef]
7. H. Schmid-Schönbein, J. Weiss, and H. Ludwig, “A simple method for measuring red cell deformability in models of the microcirculation,” Blut 26(6), 369–379 (1973). [CrossRef]
8. M. Piagnerelli, K. Zouaoui Boudjeltia, D. Brohee, A. Vereerstraeten, P. Piro, J.-L. Vincent, and M. Vanhaeverbeek, “Assessment of erythrocyte shape by flow cytometry techniques,” J. Clin. Pathol. 60(5), 549–554 (2006). [CrossRef]
9. M. Mugnano, P. Memmolo, L. Miccio, F. Merola, V. Bianco, A. Bramanti, A. Gambale, R. Russo, I. Andolfo, A. Iolascon, and P. Ferraro, “Label-free optical marker for red-blood-cell phenotyping of inherited anemias,” Anal. Chem. 90(12), 7495–7501 (2018). [CrossRef]
10. D. Dannhauser, D. Rossi, F. Causa, P. Memmolo, A. Finizio, T. Wriedt, J. Hellmers, Y. Eremin, P. Ferraro, and P. A. Netti, “Optical signature of erythrocytes by light scattering in microfluidic flows,” Lab Chip 15(16), 3278–3285 (2015). [CrossRef]
11. K. V. Gilev, M. A. Yurkin, E. S. Chernyshova, D. I. Strokotov, A. V. Chernyshev, and V. P. Maltsev, “Mature red blood cells: from optical model to inverse light-scattering problem,” Biomed. Opt. Express 7(4), 1305–1310 (2016). [CrossRef]
12. K. Gilev, E. Yastrebova, D. Strokotov, M. Yurkin, N. Karmadonova, A. Chernyshev, V. Lomivorotov, and V. Maltsev, “Advanced consumable-free morphological analysis of intact red blood cells by a compact scanning flow cytometer,” Cytometry 91(9), 867–873 (2017). [CrossRef]
13. J. P. Robinson, “Flow cytometry,” in Encyclopedia of Biomaterials and Biomedical Engineering, G. E. Wnek and G. L. Bowlin, eds. (informa healthcare, 2004), pp. 630–640.
14. H. M. Shapiro, Practical Flow Cytometry (John Wiley & Sons, 2005).
15. V. Ost, “Aufbau und Charakterisierung eines Laser-Durchflusszytometers zur Zahlung und Differenzierung von Blutzellen <ger.> [Setup and characterization of a laser flow cytometer for counting and differentiation of blood cells],” Ph.D. thesis, Freie Universitat Berlin, Germany (1993).
16. V. Ost, J. Neukammer, and H. Rinneberg, “Flow cytometric differentiation of erythrocytes and leukocytes in dilute whole blood by light scattering,” Cytometry 32(3), 191–197 (1998). [CrossRef]
17. C. Ahlgrim, T. Pottgiesser, T. Sander, Y. O. Schumacher, and M. W. Baumstark, “Flow cytometric assessment of erythrocyte shape through analysis of FSC histograms: Use of kurtosis and implications for longitudinal evaluation,” PLoS One 8(3), e59862 (2013). [CrossRef]
18. M. A. Yurkin and A. G. Hoekstra, “The discrete-dipole-approximation code ADDA: Capabilities and known limitations,” J. Quant. Spectrosc. Radiat. Transfer 112(13), 2234–2247 (2011). [CrossRef]
19. A. von Meyer, J. Cadamuro, T. Streichert, E. Gurr, G. M. Fiedler, A. Leichtle, A. Petersmann, K.-H. Pick, M. Orth, L. Risch, O. Sonntag, Y. Schmitt, B. Wiegel, T. Gottfried, and W. G. Guder, “Standard operating procedure for peripheral venous blood sampling,” Laboratoriumsmedizin 41(6), 333–340 (2017). [CrossRef]
20. Y. R. Kim and L. Ornstein, “Isovolumetric sphering of erythrocytes for more accurate and precise cell volume measurement by flow cytometry,” Cytometry 3(6), 419–427 (1983). [CrossRef]
21. D. H. Tycko, M. H. Metz, E. A. Epstein, and A. Grinbaum, “Flow-cytometric light scattering measurement of red blood cell volume and hemoglobin concentration,” Appl. Opt. 24(9), 1355–1365 (1985). [CrossRef]
22. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330(3), 377–445 (1908). [CrossRef]
23. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
24. R. Barer, “Refractometry and interferometry of living cells,” J. Opt. Soc. Am. 47(6), 545–556 (1957). [CrossRef]
25. M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transfer 106(1-3), 558–589 (2007). [CrossRef]
26. T. Wriedt, J. Hellmers, E. Eremina, and R. Schuh, “Light scattering by single erythrocyte: Comparison of different methods,” J. Quant. Spectrosc. Radiat. Transfer 100(1-3), 444–456 (2006), VIII Conference on Electromagnetic and Light Scattering by Nonspherical Particles. [CrossRef]
27. M. Yurkin, “Discrete dipole simulations of light scattering by blood cells,” Ph.D. thesis, University of Amsterdam, Netherlands (2007).
28. Y. Fung, W. C. Tsang, and P. Patitucci, “High-resolution data on the geometry of red blood cells,” Biorheology 18(3-6), 369–385 (1981). [CrossRef]
29. M. Daimon and A. Masumura, “Measurement of the refractive index of distilled water from the near-infrared region to the ultraviolet region,” Appl. Opt. 46(18), 3811–3820 (2007). [CrossRef]
30. J. Gienger, H. Groß, J. Neukammer, and M. Bär, “Determining the refractive index of human hemoglobin solutions by Kramers–Kronig relations with an improved absorption model,” Appl. Opt. 55(31), 8951–8961 (2016). [CrossRef]
31. R. Henson, “Flow cytometry data reader and visualization,” https://mathworks.com/matlabcentral/fileexchange/8430-flow-cytometry-data-reader-and-visualization.
32. T. W. Secomb, R. Skalak, N. Özkaya, and J. F. Gross, “Flow of axisymmetric red blood cells in narrow capillaries,” J. Fluid Mech. 163, 405–423 (1986). [CrossRef]
33. C. Pozrikidis, “Axisymmetric motion of a file of red blood cells through capillaries,” Phys. Fluids 17(3), 031503 (2005). [CrossRef]
34. G. Tomaiuolo, M. Simeone, V. Martinelli, B. Rotoli, and S. Guido, “Red blood cell deformation in microconfined flow,” Soft Matter 5(19), 3736–3740 (2009). [CrossRef]
35. M. A. Yurkin, K. A. Semyanov, P. A. Tarasov, A. V. Chernyshev, A. G. Hoekstra, and V. P. Maltsev, “Experimental and theoretical study of light scattering by individual mature red blood cells by use of scanning flow cytometry and a discrete dipole approximation,” Appl. Opt. 44(25), 5249–5256 (2005). [CrossRef]
36. Ö. Ergül, A. Arslan-Ergül, and L. Gürel, “Computational study of scattering from healthy and diseased red blood cells,” J. Biomed. Opt. 15(4), 045004 (2010). [CrossRef]
37. W. H. Carter, “Electromagnetic field of a Gaussian beam with an elliptical cross section,” J. Opt. Soc. Am. 62(10), 1195–1201 (1972). [CrossRef]
38. S. S. Lee, Y. Yim, K. H. Ahn, and S. J. Lee, “Extensional flow-based assessment of red blood cell deformability using hyperbolic converging microchannel,” Biomed. Microdevices 11(5), 1021–1027 (2009). [CrossRef]
39. T. Yaginuma, M. S. N. Oliveira, R. Lima, T. Ishikawa, and T. Yamaguchi, “Human red blood cell behavior under homogeneous extensional flow in a hyperbolic-shaped microchannel,” Biomicrofluidics 7(5), 054110 (2013). [CrossRef]
40. J. E. Mancuso and W. D. Ristenpart, “Stretching of red blood cells at high strain rates,” Phys. Rev. Fluids 2(10), 101101 (2017). [CrossRef]
41. R. M. Hochmuth, P. R. Worthy, and E. A. Evans, “Red cell extensional recovery and the determination of membrane viscosity,” Biophys. J. 26(1), 101–114 (1979). [CrossRef]
42. J. P. Mills, L. Qie, M. Dao, C. T. Lim, and S. Suresh, “Nonlinear elastic and viscoelastic deformation of the human red blood cell with optical tweezers,” Mech. & chemistry biosystems : MCB 1(3), 169–180 (2004).
43. R. Skalak, A. Tozeren, R. Zarda, and S. Chien, “Strain energy function of red blood cell membranes,” Biophys. J. 13(3), 245–264 (1973). [CrossRef]
44. P. B. Canham and A. C. Burton, “Distribution of size and shape in populations of normal human red cells,” Circ. Res. 22(3), 405–422 (1968). [CrossRef]
45. C. Misbah, “Vesicles, capsules and red blood cells under flow,” J. Phys.: Conf. Ser. 392, 012005 (2012). [CrossRef]
46. J. B. Freund, “Numerical simulation of flowing blood cells,” Annu. Rev. Fluid Mech. 46(1), 67–95 (2014). [CrossRef]
47. É. Gibaud, “Numerical simulation of red blood cells flowing in a cytometer,” Ph.D. thesis, Université de Montpellier, France (2015).
48. J. Mauer, “Shapes and dynamics of blood cells in Poiseuille and shear flows,” Ph.D. thesis, Universität zu Köln, Germany (2016).
49. J. Mauer, M. Peltomäki, S. Poblete, G. Gompper, and D. A. Fedosov, “Static and dynamic light scattering by red blood cells: A numerical study,” PLoS One 12(5), e0176799–19 (2017). [CrossRef]
50. N. Toepfner, C. Herold, O. Otto, P. Rosendahl, A. Jacobi, M. Kräter, J. Stächele, L. Menschner, M. Herbig, L. Ciuffreda, L. Ranford-Cartwright, M. Grzybek, Ü. Coskun, E. Reithuber, G. Garriss, P. Mellroth, B. Henriques-Normark, N. Tregay, M. Suttorp, M. Bornhauser, E. R. Chilvers, R. Berner, and J. Guck, “Detection of human disease conditions by single-cell morpho-rheological phenotyping of blood,” eLife 7, e29213 (2018). [CrossRef]
51. B. Lincoln, H. M. Erickson, S. Schinkinger, F. Wottawah, D. Mitchell, S. Ulvick, C. Bilby, and J. Guck, “Deformability-based flow cytometry,” Cytometry 59A(2), 203–209 (2004). [CrossRef]
