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Abstract
Red blood cells in small capillaries exhibit a wide variety of deformations that reflect their true physiological conditions at these important locations. By applying a technique for the high-speed microscopy of flowing cells, termed spectrally encoded flow cytometry (SEFC), we image the light reflected from the red blood cells in human capillaries, and propose an analytical slipper-like model for the cell morphology that can reproduce the experimental in vivo images. The results of this work would be useful for studying the unique flow conditions in these vessels, and for extracting useful clinical parameters that reflect the true physiology of the blood cells in situ.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
High resolution in vivo imaging of blood cells in flow is highly desirable for the biomedical research, and could contribute to numerous applications in clinical diagnosis [1]. While in vivo flow cytometry has been demonstrated useful for counting [2] and imaging [3] blood cells, most of these techniques often rely on fluorescence labeling of the cells, which is problematic for applications in human subjects. While using low fluorophore concentrations may help in reducing the overall toxicity [4] and cell modifications [5], it may often result in rapid photobleaching and fluorescence blinking, reducing cell detection and counting accuracy. For most clinical applications, then, reflectance confocal microscopy (RCM) is often the preferred approach for noninvasive imaging with subcellular resolution [6]. By rejecting the out-of-focus light with a small pinhole [6] or a single-mode fiber [7], RCM allows effective optical sectioning in thick tissues with high lateral and axial resolutions. In vivo RCM was effectively applied in ophthalmology [8,9], dermatology [10–12] and for early detection of cancer [12,13].
For in vivo microscopy of blood cells, however, the high mechanical scan rates required for RCM often limit its ability to image the rapidly moving cells. This challenge was recently addressed using a variant of spectrally encoded confocal microscopy [14], termed spectrally encoded flow cytometry (SEFC) [15], a method capable of high-resolution confocal microscopy of rapidly flowing cells within small blood vessels [16]. Using a diffraction grating and a high numerical-aperture (NA) objective lens, SEFC uses wavelength to encode a single lateral dimension, allowing rapid single-shot measurements across an entire line, with imaging speeds limited only by the line rate of the spectrometer camera. SEFC has been demonstrated promising for in vivo imaging of blood cells [16], for measuring hematocrit [16,17] and for in vivo differentiation of white blood cells [18].
Imaging and precise counting of the red blood cells, however, is challenging in SEFC, and in RCM in general, mainly because these cells lack light-scattering organelles such as nuclei, mitochondria and lipid granules. The only reflections come from their smooth plasma membranes, which result in images that exhibit high-contrast interference patterns of rings and arc-like shapes. We have previously suggested a numerical model for explaining the confocal images of the red blood cells in a laminar flow within a flow chamber [19]; however, these images were inconsistent with our in vivo SEFC images in humans, which appear much less structured and include asymmetric interference fringes across the entire vessel depth [16,18].
In this paper we propose an analytical model for describing the morphology of some of the red blood cells that flow in small capillaries, and numerically generate interference patterns that match their in vivo appearance in numerous experiments with healthy human subjects. Using wave propagation analysis of both the illumination and collection optical paths in a confocal microscope configuration, we simulate the reflection patterns produced by the red blood cells and adjust the cells parameters to fit the real in vivo SEFC images. The results may allow to accurately assess the shapes of individual red blood cells within patients, which could help to better understand their various in situ characteristics during capillary flow.
2. Methods
2.1 In vivo SEFC
Reflectance confocal images of flowing blood cells were obtained using an SEFC system similar to that reported in Ref. [18]. Briefly, the lower lip of a healthy volunteer was positioned against a cover glass in front of a high-NA objective lens, and the spectrally encoded focal line was placed across selected capillary loops within the field of view. An example of a raw SEFC image is presented in Fig. 1(a), showing reflections from a single line (vertical y-axis) as a function of time (horizontal t-axis). The raw image can be converted into a conventional x-y image by transforming the time axis into a spatial x-axis using x = v·t, where v denotes the estimated average flow velocity (Fig. 1(b)) across the vessel. In practically all SEFC imaging experiments conducted in human subjects so far, the streaming data included a consistent, dense collection of complex patterns of high-contrast stripes and arcs, which were oriented mainly along the y-axis. Three magnified views of selected regions of interest in the raw image are shown in Fig. 1(c), with their horizontal axis expanded according to the average local velocity, and presented so that the apparent flow direction is from left to right. Realizing that any sort of high-frequency mechanical vibrations are impossible under the low Reynolds numbers characteristic of blood flow, our main hypothesis was that these patterns can originate only from interference between reflections from two or more non-parallel surfaces of the cells. As a rule of thumb, the apparent fringe density of 8-12 fringes per 5 µm would roughly correspond to a pair of planar surfaces with relative angles of 30°-55° between them.
Fig. 1. (a) A typical in vivo SEFC raw image of flowing blood cells within an approximately 10-µm-diameter human capillary. (b) The estimated parabolic flow profile across the capillary, reaching a maximum velocity of 3 mm/s. (c) Three magnified views of selected red cells (dashed-line rectangles in (a)) that exhibit the characteristic fringe pattern. Scale bar represents 1 µm.
2.2 Point-spread function simulation
In order to reproduce the fringe patterns visible in the in vivo images, we started by simulating the point-spread function of a high-NA reflection confocal microscope. Unlike our previous work that relied on the Fresnel approximation [19], here we have used the explicit equation for the optical transfer function of free space given by:
After passing through the lens, the converging wave reaches the upper and lower surfaces of the cell, and the reflected wave around each lateral point (xc,yc) on the cell was assumed to have a uniform unit amplitude, and a phase-front given by:
where k = 2π / λ and the function zi denote the surfaces of the upper (i = 1) or the lower (i = 2) membranes of the cell, embodying the exact shapes of the two faces of the membranes within the phase of the two reflected waves. After passing through a 4-f system with 1:1 magnification, the intensity of the two interfering waves was detected through a small pinhole with a diameter of 0.75 µm, which was simulated using simple linear summation of the field intensity at a central circular 10-pixel-diameter region. The simulated lateral field-of-view at the focal plane was 8×8 µm2, with sampling resolution of 0.075 µm. The FWHM of the illumination spot was 0.55 µm and 2.5 µm in the lateral and axial dimensions, respectively, and the respective confocal resolution (FWHM) was 0.44 µm (lateral) and 1.8 µm (axial).2.3 Modeling the red blood cell shape
At equilibrium, the red blood cells exhibit a biconcave shape [22], and these shapes are generally maintained in large vessels and at low flow velocities [19,23]. Under non-equilibrium conditions typical of viscosity-dominant flow, however, the cells are continuously distorting [22,24] and presenting a wide variety of morphologies. In our SEFC images obtained from small capillary loops, the relatively uniform arc patterns may suggest a typical cell morphology in which the two faces of the cell membrane exhibit a relatively moderate, constant curvature that is dominated by a relative linear slope between them.
We therefore propose an asymmetric biconvex model for the observed cells, comprising two oblate spheroid surfaces that are both convex relative to the cell’s center point (Fig. 2). The top membrane surface z1 is given by:
Fig. 2. Analytical model of a red blood cell within a small capillary vessel. (a) x-z cross section of the cell, where the positive x-axis corresponds to the direction of flow. The illumination intensity is shown in greyscale to illustrate its axial position relative to the cell surfaces. (b) The three-dimensional shape of the top and bottom spheroidal surfaces representing the top and bottom cell membranes. Regions of the membranes roughly parallel to the optical axis are not shown as they are not expected to backscatter the illumination light.
The bottom membranal surface z2 has a similar spheroid structure with spheroid axes c2 and a2, but with an additional linear y-axis slope with angle θ :
3. Results
Simulated reflectance confocal images of the modeled red blood cells (Eqs. (4)–(6)) are shown in Fig. 3 for angles θ between 10° and 20° and membrane separations Δz between 2 µm and 2.4 µm. The simulated spheroidal membranes have a lateral radius Rcell = 4 µm, similar lateral spheroid axes a1 = a2 = 4.5 µm, and different axial spheroid axes c1 = 1 µm and c2 = 0.75 µm. All simulated cells had corpuscular volumes within the normal range (80–100 fl). Clearly, larger angles between the membranes have led to denser arc patterns caused by the transitions between constructive and destructive interference of the reflected waves. Note also that the higher slope of the bottom surface had slightly reduced the image intensity due to the limited numerical aperture (NA = 0.7 that corresponds to a maximum collection angle of 44.4°). The distance Δz between the two membranes, which was simulated using small increments of 0.2 µm (half-wavelength roundtrip), affected mainly the apparent modulation phase of the interference arcs; note however that the resulting shift in the fringe pattern is not uniform across the image, due to the Gouy phase shift that creates a slight increase in the effective wavelength near the focal point. Some reduction in image intensity is also visible with the increase in Δz due to the limited 1.8 µm axial resolution.
Fig. 3. Numerically simulated SEFC images of a slipper-like red blood cell model with different membrane angles θ and axial separations Δz. Note the increasing fringe density as θ increases. Scale bar represents 1 µm.
Selection of in vivo SEFC images that display interference patterns similar to those produced by our cell model are shown in Fig. 4(a). By using our analytical cell model (Eqs. (4)–(6)), and by selecting specific angles θ and membrane separations Δz, we attempted to match between the simulated and the experimental images (Fig. 4(b)-(d)). Overall, when compared to the experimental images (right-hand panels in Fig. 4(b)-(d)), the simulated confocal images (middle panels in b-d) display similar numbers of the interference fringes with comparable fringe densities. Obviously, our relatively simplistic cell model could not fully account for the asymmetries found practically in all SEFC images, which are most likely caused by optical and mechanical distortions induced by the dense cell flow.
Fig. 4. Comparison between in vivo and simulated images of red blood cells. (a) Selected in vivo SEFC images of red cells that show interference patterns of periodic arcs and rings. (b)-(d) Matching the simulated cell parameters to three exemplary in vivo SEFC images using the parameters θ and Δz. The corresponding simulated corpuscular volumes are (b) 95 fl, (c) 80 fl, and (d) 85 fl. Scale bars represents 1 µm.
4. Discussion
Numerous attempts to understand the deformation of red blood cells during flow have often relied on various flow models that were implemented by numerical simulations and compared to in vivo microscopy [25], as well as to in vitro experiments with blood cells flowing through microchannels mimicking physiological conditions [26,27].
Within small capillaries in the human oral mucosa, the blood flow is characterized by high cell densities and a wide range of velocities, an environment that give rise to several typical cell morphologies, including the well-known “slipper” shape [27]. Other known cell shapes include the “disk” shape at low cell densities and low velocities, the “parachute” shape at low densities and high velocities, and a fast highly-aligned or a “zigzag” collective flow within large vessels [23]. Another interesting collective behavior of blood cells seen in several medical conditions and in large vessels is the formation of highly ordered flow that results in complex three dimensional structures [28].
Our in vivo SEFC studies were conducted on medium-sized capillaries having diameters up to 10 µm in the oral mucosa, where the confocal plane was positioned slightly below (1-4 µm) the front vessel wall. In this region, where the cell population is dense and flow velocities are relatively high, the above-mentioned periodic patterns of arcs and rings were regularly observed in almost all imaging sessions. The slipper-like shape deformation suggested in this paper is expected to be common in such conditions (Fig. 5), in which a substantial volume of the cell precedes the rest of the cell due to the friction forces exerted by the vessel endothelium.
Fig. 5. An illustration of cell flow within a capillary vessel based on the SEFC images. The relatively uniform angle θ between the cell top and bottom membranes is the result of friction forces exerted by the vessel wall, generating the characteristic interference fringes visible in the SEFC images.
By using relatively simple equations for describing two opposing oblate membrane spheroids (Eqs. (4)-(6)) we have numerically generated the fringe patterns that were visible in most of the SEFC images. These results could provide important information on the flowing cells, and could potentially be used in the future for extracting key clinical parameters such as mean corpuscular volume (MCV) and the red cell distribution width (RDW). All cell morphologies simulated in this work showed corpuscular volumes within the normal range, as expected from measurement in healthy volunteers. The resulting morphologies could also assist the computation of various mechanical parameters of the red cells [29,30], including membrane elasticity of the cells and the flow viscosity [31,32].
Nevertheless, our approach for estimating the exact shapes of the flowing cells has several limitations that stem mainly from the process of reconstructing a 3D shape based on 2D interference data. Note that the tight focusing of the high-NA objective lens allows to address the phase-unwrapping problem, as it alters the brightness and the contrast of the interference patterns, making them depth dependent and therefore reducing depth ambiguity. The effect of the Gouy phase shift [33,34] also helps in reconstructing the cell shape by introducing an axial π-phase shift that further reduce depth ambiguity. Thus, in principle, once an analytical model fully matches an experimental image of a cell, the accuracies in determining the distance between the top and bottom membranes could be within a few tens of nanometers, which is far more accurate then typically required for assessing any clinically relevant parameter such as the MCV. Yet the very few individual cells that could be reconstructed numerically are obviously a small fraction of the cells that flow across the vessel, as most of the cells often generate a wide variety of chaotic interference patterns that would be extremely difficult to reconstruct. Furthermore, in cases where the cells are in close contact with each other the measurement of individual cells could be difficult. While we did not encounter any type of highly ordered flow or cell aggregation during our in vivo experiments, such structures would probably produce periodic patterns that could be easy to detect and analyze using specialized cell grouping models. Additional solutions to these problems may include reducing the fringe contrast by modifying the PSF [35], by using more sophisticated 3D models of the cells, and by employing adaptive learning algorithms for correlating the recorded SEFC images with the desired blood parameters.
In summary, we have established an analytical model for describing the 3D shapes of selected red blood cells in capillary flow. Based on the reflective confocal images provided by in vivo SEFC, our cell model is able to reproduce the periodic patterns visible in most of the in vivo images obtained from healthy human volunteers. The red blood cell model presented in this work, which comprised of two opposing oblate spheroidal surfaces, may lead to better understanding of the cells behavior within human capillaries, and perhaps could help in extracting important clinical parameters from the SEFC raw images.
Funding
Israel Science Foundation (990/19).
Acknowledgements
This work was supported in part by the Lorry I. Lokey Interdisciplinary Center for Life Sciences and Engineering.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper may be obtained from the authors upon reasonable request.
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