1. Introduction

When healthy red blood cells are drawn from the human body and suspended in isotonic solution, they take on a biconcave disc shape with a diameter of approximately 8 µm [1]. However, when circulating through the vascular system the same cells are marshalled into narrow capillary lumens on the order of 5 µm wide [2]. The available channel for cellular passage is even less, due to the plasma layer and endothelial glycocalyx [3]. This mismatch in size between red cells and capillary lumens is handled by folding and narrowing of cells relative to their unconstrained biconcave disc shape [4]. Although this phenomenon is well known, it is not yet understood why it is that the oxygen carriers of our vascular system are by default too wide to pass through the principle sites of oxygen exchange.

There are three categories of explanation for this odd state of affairs. One is that elongated cell shapes facilitate exchange: the diffusion distance is reduced between the cell membrane and the capillary endothelium [5], and across the red cell cytoplasm due to its reduced radial extent [6], whilst contact surface area at the zone of highest oxygen gradient increases [7].

A second category of explanation is that the mismatch in size between red cells and capillaries does not really matter – cells are so deformable that narrow tubes (within the physiological range) do not significantly increase resistance to flow. For example, blood viscosity is effectively reduced in vessels below approximately 300 µm diameter as cells migrate towards the central flow axis, creating a cell-free layer of comparatively low resistance [8] (however, this is not a strong argument as viscosity rises sharply again in narrow vessels as described below). Furthermore, although there is significant variance in the volume and membrane surface area between red cells measured in vitro, the ratio of volume to surface area is tightly conserved. This supports the idea of a common narrowest diameter through which most red cells should be able to pass, by adopting an appropriate “hot dog” shape that does not require a change in either cellular volume or surface area [1]. This common minimum diameter is ∼3 to 3.5 µm, i.e. less than or equal to the smallest capillary lumens observed in human. It has therefore been proposed that the passage of red cells through capillaries is limited by geometrical considerations of volume and surface area alone, with cells freely taking on the required shape without any limitation imposed by the biomechanical properties of the cell or its membrane [9]. This hypothesis is supported in regards to surface area by experimental work showing that red cells, although highly amenable to shear deformation, do not undergo a significant increase in overall membrane surface area (i.e. stretching) without rupture [1,10]. Additionally, dramatic stiffening of the red cell membrane does not seem to alter the narrowest synthetic channel through which a red cell may pass [9,11]. Under this theory then, the vast majority of healthy red cells should easily meet the challenges imposed by even the narrowest of capillary channels encountered within healthy tissue; impaired flow in disease is proposed to result from changes to the volume and/or surface area of cells [9].

A third category of explanation is that changes in red cell shape in capillaries may in fact hamper gas exchange. This could be advantageous to avoid delivering a surplus of oxygen to tissue during times of “baseline” metabolic activity, resulting in oxidative pathway damage. Dynamic changes in vessel diameter would then facilitate an increased oxygen supply only when it is needed [12]. In support of this explanation is evidence that folding of the cell membrane and changes to plasma layer thickness, predicted in narrow channels, result in reduced oxygen flux compared with the canonical biconcave shape [4,13]. Moreover, recent experiments using cell-resolved oximetry in vivo suggest that gas exchange may not be limited by diffusion across the cell membrane, but by that across the red cell cytoplasm. This would significantly mitigate any purported benefits of cell deformation to gas exchange [6]. However, the applicability of this finding in the capillaries is uncertain as these experiments were carried out under comparatively low shear rates (e.g. as appropriate for the aorta, rather than for capillaries). The highest shear rates, and hence friction, in the cardiovascular system coincide with the moment-to-moment location of individual red cells within capillary tubes. This is so because the plasma layer surrounding each cell is very thin (on the order of 1 µm), necessitating a precipitous drop in velocity over a short distance (i.e. a high shear rate) to satisfy the boundary condition of zero flow at the vessel wall [2,14]. This means that cells encounter higher friction than plasma, manifesting as a sharp rise in viscosity within capillary tubes narrower than approx. 6-7 µm. The influence of this phenomenon is readily apparent at capillary bifurcations where it is well known that the branch with lower resistance typically receives more red cells than would be expected based on its share of the volumetric flow [15–17]. The logical extension of this is that, across a network of capillary bifurcations, fewer cells will enter the narrower capillaries which means that less oxygen is available for exchange through those capillaries.

To further differentiate between the three explanations described above requires knowledge of the shape and distribution of red cells observed within capillary networks. Current understanding is based largely on modelling work and on in vitro preparations in which blood cells are propelled through synthetic channels. Assessing cellular passage non-invasively and in humans is therefore of high interest, in particular within neural tissue where capillaries are narrow yet metabolic activity is greatest. Such investigations can now be achieved in the living human retina, using adaptive optics to overcome the natural optical imperfections of the eye. When sampled with sufficient speed, this permits individual red blood cells to be resolved and tracked in their passage across the capillary network [18,19]. This approach carries clear advantages in that the blood column is observed in its natural state without any changes in the state of tissue (for example changes in pH or temperature during intravital microscopy preparations), of blood vessels (for example preserving the endothelial glycocalyx), or of blood cells (for example the drawing, washing and fluorescent labelling of cells) which may produce unanticipated departures from natural flow patterns or cell rheology.

Previous work using adaptive optics in the living retina has established methods to visualize and measure the diameter of capillary flow channels [20], to quantify flow velocity [18,19,21–27], and to measure cellular flux [28–30]. To study the blood column specifically, one approach is to “freeze” a confocal scanning pattern in space, producing repeated cross-sections at high frequency at a single cross-section on a target vessel [27]. The natural flow of the blood column past this point then allows it to be rendered in high detail [25,29]. Recently, our group proposed a new method to study the blood column, using a full-field imaging approach which allows dozens of capillaries to be captured simultaneously in order to study connected, contiguous portions of the retinal microvasculature [31]. This is made possible by acquiring image data at high speed to allow the tracking of cells over many image frames. By observing a given cell for an extended period, and by observing large numbers of cells trafficked through a vessel, a high fidelity 2D profile of the “average cell” may be reconstructed within each capillary vessel. Here we have built upon this method, using 2D profile information together with certain assumptions to infer the shape of cells in 3D. This allowed us to estimate the volume and surface area of cells and relate these parameters to variations in capillary lumen diameter, cellular orientation within the capillary, and the separation between successive cells (i.e. cell density). This provides new information relevant to the differing perspectives presented above, regarding the ease with which red cells can move through capillary tubes. The results presented here support the general notion of high resistance to cellular passage through narrow capillaries.

2. Methods

2.1 Subjects

Data were collected from 3 young, healthy human subjects (2 female) with ages ranging from 23 to 36 years. There were no signs of ocular pathology upon detailed examination of the eye by a trained optometrist. The study was conducted in accord with the tenets of the Declaration of Helsinki with approval from the University of Melbourne Human Ethics Committee.

2.2 Selection of blood vessels

To minimize stratification of the laminar vascular network supplying the inner retina, images were acquired within the foveal region on the edge of the foveal avascular zone. Subject fixation was directed on a calibrated grid to locations ranging from 1.1 to 2.2° from the foveal centre. The imaged field was 1.25° in diameter. Regions featuring comparatively dense coverage by the capillary bed were targeted. Vessels were included for cell shape analysis where the internal lumen diameter (measured as described below) was 8 µm or less.

2.3 High-resolution optical imaging system

A flood-based adaptive optics ophthalmoscope was used to image the retina. Wavefront aberrations were measured by projecting a thin pencil of 835 nm light from a superluminescent diode (Hamamatsu, Japan) onto the retina and coupling the returned light to a Hartmann-Shack wavefront sensor. Wavefront information was used to drive a deformable mirror (HiSpeed DM97-15, Alpao, France) to correct the ocular aberrations at 20-30 Hz. When root-mean-square wavefront error reached an acceptable level (usually 0.06 µm or better over a 7.0 mm pupil), a 3 second video sequence was acquired with a scientific CMOS camera (Andor NEO sCMOS, Andor Technology, Belfast) operating at 300-400 fps, with 50% duty cycle and pixels corresponding to approximately 0.5 µm on the retina for an emmetropic eye. The light source was a supercontinuum laser passed through a tunable transmission filter (Fianium, Southampton, UK) to provide a band at 750 ± 25 nm for imaging. Light power measured at the cornea ranged from 0.3 mW to 1.3 mW, which is at least 10 times below the ANSI standards for maximum permissible exposure [32].

2.4 Initial image processing

Acquired video frames were dark-fielded (the average intensity in a region well outside the illuminated bright-field region was subtracted) and flat-fielded (divided by the average of several hundred frames collected when the subject was asked to shift their gaze erratically) to correct for scattered light and non-uniform delivery of light to the retina. Frames were then shifted (translated only; intrasequence rotation was negligible) to correct for eye motion relative to the first frame, using cross-correlation. A motion contrast image was generated from registered video data by computing the variance in intensity of each pixel over time (Fig. 1(A)) [20]. Centrelines of all visible blood vessels were then traced manually on this motion contrast image using Photoshop (Adobe, USA). For this work, vessel segments chosen for analysis were those with sufficient cellular contrast and which showed largely single-file cellular passage (i.e. minimal back-to-back cells) without interjection of the much larger white cells. Vessels chosen for analysis also had at least 1 second of contiguous video data available. All vessels meeting these criteria were included.

 

Fig. 1. Workflow to generate 2D binary cell profile. A) 2D en face motion contrast image of the retinal capillary network. The central flow axis is manually traced and transverse sections computed (examples in green). This leads to a straightened representation (B), where colored lines show example trajectories parallel to the flow axis. Cell velocity (C) is measured with pixel intensity cross-correlation [22]. Alternating cells and plasma can be visualized with a kymograph (D), where the image axes now correspond to space and time (see scale bars in E). The five panels here correspond to the colored lines in B. By successive shift of each image row according to the flow velocity, motion-stabilized kymographs are produced (E). Each stabilized panel corresponds to the original kymograph to its left. By averaging the stabilized kymographs in time, the blood column is rendered, (F). Image data are wrapped for visualization; epoch in green corresponds to panels shown in D and E. (G) Average intensity of the cells visible in F, in a small region of interest around the centroid of each cell. (H) Probability map computed over the same region as G, showing the likelihood that a pixel was determined to be part of a cell. Note that spatial scale is zoomed as indicated cf. F. Scale bar shown in E applies also to D; spatial scale in E applies also to F.

Download Full Size | PDF

2.5 Generation of blood column images

We recently described a novel method to render the capillary blood column in high speed video data [31]. From the manually traced vessel centrelines described above, a spline is fit to enable computation of perpendicular image cross-sections (Fig. 1(A), green). This leads to a straightened representation of the vessel and surrounding tissue (Fig. 1(B)). The vessel centreline and adjacent parallel axes are then considered (Fig. 1(B)), example parallel axes shown by coloured lines). Along each of these axes the recorded intensity varies through time, depending on whether cells or plasma are present within the tube [21,33] as well as whether a given pixel is inside or outside the vessel. This information can be rendered in a kymograph as shown in Fig. 1(D) (central panel corresponds to vessel centreline, and outer panels to the edge of the vascular lumen). Bright bars correspond to the trajectories of cells, dark bars to the intervening plasma, and the angle of inclination indicates the flow velocity.

Because blood is a fluid and therefore incompressible, within a vessel the cells and plasma gaps between them should maintain their relative positions through time (this assumes no change in vessel diameter). If the flow velocity is known, for example from pixel intensity cross-correlation [22] whose output is plotted in Fig. 1(C), the information from each frame can therefore be shifted (i.e. the rows of Fig. 1(D) are slid horizontally) to entirely compensate for motion due to blood flow [34]. Assuming that all parts of a cell travel together, this principle can be extended across the full width of the blood column (motion-corrected kymographs shown in Fig. 1(E)). This allows the relatively noisy single-frame data to then be averaged, by collapsing data in the time dimension. This permits detailed rendering of the blood column propagated through each vessel during the measurement period (Fig. 1(F)).

For each capillary, cells may appear brighter than plasma or vice versa depending on the axial separation between the focal plane and the vessel [21,33]; registered video data were reviewed manually to confirm cellular identity for each vessel, with vessels excluded where this could not be established conclusively. Regardless of their true appearance, for consistency the example images shown here are rendered such that cells appear brighter than plasma.

The method described above explicitly assumes that all parts of the blood column travel at the same speed (i.e. the cross-sectional flow profile has a top-hat shape). This is not strictly true, with notable departures including the differential motion of and within the plasma layer, as well as tank-treading revolutions of the cell membrane whereby the membrane cycles around without changing its position relative to the cell centroid [2]. However for the purpose of measuring cell shape alone as opposed to dynamics within a cell, these issues have no discernible impact.

2.6 Generating a binary profile for the average cell in a vessel

The appearance of the “average cell” is shown in Fig. 1(G). A shadowed appearance is evident whereby a signal is evident beyond the nominal bounds of the vascular lumen. We have described the appearance of this extravascular signal previously [22,31], and have attributed it to scattered light resulting from differences in the refractive index of cells relative to the surrounding retinal tissue. This mechanism was proposed in part because the same light distribution is noted for plasma: to a lesser degree in accord with its greater similarity in refractive index to the retinal tissue, and with opposite contrast since it has a lower refractive index than the retinal tissue (as opposed to red cells which have a higher refractive index).

As described recently [31], blood column images were automatically thresholded to localize individual cells, allowing a map to be produced which describes the probability of encountering a cell as a function of distance from the cell centre (Fig. 1(H)). The thresholding procedure first classified each position along the blood column as “cell” or “plasma” by principal component analysis carried out on the thousands of cross-sections in the blood column image; as described above and in our previous publication [31], the opposite sign of contrast for cells and plasma make this straightforward. Next, within the set of data classified as “cell”, we used a standard automatic thresholding approach to separate pixels that were inside versus outside the vessel [35]. By virtue of averaging typically hundreds of cells observed over several seconds, the fidelity of the probability maps significantly exceeded our instantaneous spatial resolution (∼2 µm through a dilated 8 mm pupil at 750 nm). Pixels in the probability map exceeding 50% are presumed to lie within the bounds of the “average cell”. The resulting binary mask was converted into a cellular outline, which was smoothed for shape analysis (Fig. 1(H), green).

Cell orientation was assessed based on the 2D information alone, by fitting an ellipse to the cell profile (i.e. to the green line shown in Fig. 1(H)). Orientation was expressed by the angle, in degrees, by which the long axis of the fitted ellipse was tilted with respect to the flow axis. Ellipticity (length / width) was also returned as a metric of cellular elongation, and to help differentiate meaningful measurements of obliquity from spurious obliquity associated with an almost spherical cell.

2.7 Estimating cell shape in 3D

To approximate a 3D shape for the average cell based on the 2D en face cell profile, the following logic was used:

 

Fig. 2. Conversion of binary cell profile to 3D shape. A) Vertical image slices are considered at each column of the cell profile image (example slices shown with vertical lines, extent of vascular lumen shown by horizontal dashed lines). B) At each slice the two points from the profile are used to determine an out-of-plane ellipse, with major axes matching the estimated width and depth at that position (see text). (C) Procedure repeated for all slices, creating a 3D point cloud. (D) A closed polygon is fit to encompass the point cloud, shown from an oblique perspective to emphasize the 3D shape. (E) The same 3D shape viewed side-on, matching the perspective of the profile from A. (F) The same 3D shape viewed top-down. (G) The same 3D shape viewed from down the capillary tube, showing the presumed circular cross-section.

Download Full Size | PDF

It should be noted that the procedure described above considers only convex shapes, which never have more than two points on the cell edge at a given column in the profile image. Simple concave shapes, such as an invagination of the cell membrane, result in four such points. This was handled by repeating Steps 2-4 for both the inner and outer pair of points, for cases where two pairs were found. No cell profiles were observed to have more complex concavities, however this may be expected based on the limited transverse resolution of approx. 2 µm.

The above procedure ensured that all parts of a cell remained within a circular cross-section when viewed “down the tube”, regardless of the orientation of the cell (e.g. Figure 2(G)). It also ensured that out-of-plane information is interpreted differently depending on the orientation of the cell. For example, a cell with a thin 2D profile with the long axis perpendicular to the flow would be interpreted as being disciform in shape, like a plate viewed from the side. Another potential way to interpret such a 2D profile would be as a vertical “sausage” which underfills the tube in the out-of-plane direction, as proposed by some earlier authors [4]. Under our procedure, such a sausage shape would only be inferred from such a 2D profile if the long axis were oriented horizontally; an obliquely oriented cell would be interpreted to lie somewhere between these two extremes.

It should be emphasized that estimating a 3D shape from a 2D en face image as described above is an inherently ambiguous exercise. First, no axially resolved information was utilized in our approach (although this is probably best given the inherent limitations to axial resolution for our imaging system, relative to transverse resolution). Second, the capillary cross section could vary significantly from a circular shape. This information would be beyond the axial resolution of our device, and could only be determined with sufficient precision from interferometric approaches such as OCT or holography [36,37]. Third, double-folds or other contortions of the cell membrane [4] are largely beyond the transverse resolution of our device due to the finite diameter of the eye’s pupil, a limitation not suffered with other methods in which the tissue is dissected to view it with a microscope to permit more detailed study of cell shape [4,37]. Fourth, such detailed nuances in shape would be lost in our approach anyway, because their form and orientation would vary from one cell to the next [4,37], whereas we consider the average cell to mitigate the high degree of noise resulting from the limited illumination power permissible for safe viewing of the retina in vivo. These limitations mean that the detailed variations in cell shape evident in some studies should not be expected here. For the purpose of studying capillaries of the human retina in vivo, the only avenue for improvement that we are aware of with currently available technology would potentially be the use of adaptive optics OCT to accurately characterize the cross-section of capillary lumens.

3. Results

3.1 Descriptive statistics

Measured parameters reported here are included in an accompanying Data File 1 [38]. Table 1 summarises estimated shape parameters for the average cell within 121 unique capillary segments across 3 subjects. Lumen diameters ranged from 3.2 to 7.9 µm. Although only a subset of vessels within each field were assessed, the mean diameter of 5 µm agrees well with previous in vivo and histological estimates [39]. The mean red cell volume of 99 fL agrees well with normative values measured in vitro, typically given around 90 fL [1,17]. Mean surface area was 110 µm²; this is somewhat lower than traditional estimates from in vitro work at around 130 µm² [1,17], but comparable to some recent estimates at 113 µm² for which it was not necessary to prepare or fix drawn blood [40].

Table 1. Measured parameters of the “average cell” observed within 121 capillary segments

View Table

Minimum cylindrical diameter (MCD) represents the narrowest tube through which a given cell could squeeze without any change to its volume or surface area, taking on a “hot dog” shape (it should be noted that this parameter is computed only from volume and surface area, not directly from the width of a cell; it is an idealized concept representing the smallest vessel that a cell could pass through). It is commonly held that cells do not change their volume or surface area upon passage through the cardiovascular system, including the narrowest capillary vessels [1,9,11]. Because MCD is defined entirely by just volume and surface area, it has been proposed that if MCD is measured for a given cell in any blood vessel, or even in vitro, its value can be inferred to be the same at any point throughout the cardiovascular system. However, our mean MCD of 4.3 µm and the 95% range of 3.0 to 5.7 µm are both significantly larger than previous in vitro reports, e.g. a mean of ∼3.3 µm and 95% range of ∼3.0 to 3.5 µm [1]. This implies that volume and surface area (the sole determinants of MCD) may not in fact remain constant at all positions within the vascular tree, as explored further below.

The “sphericity” index represents how closely the volume and surface area match a spherical shape. It is calculated as 4.84 V^2/3 / SA, with values ranging between 0 (not spherical) to 1.0 (spherical) [1]. Under in vitro conditions, freely suspended red cells take up the classic biconcave disc shape with average sphericity score of ∼0.76; our average of 0.93 indicates that cells have taken on a shape that is closer to spherical than observed in vitro. As evident in Fig. 2 as well as figures below, cells may still be elongated in the flow direction, however they are expected to become rounded in the radial direction to match the geometry of the capillary tube [1].

The separation observed between successive cells was heterogenous across vessels, with 95% range spanning ∼10 to 40 µm (centroid-to-centroid). Cellular orientation was also widely variable; in 35 of 121 vessels the long axis was inclined relative to the flow direction by more than 20°. Potential explanations for these observations are explored further below.

3.2 Filtration by cell size across the capillary network

Figure 3(A) shows a moderate dependence of cellular volume on lumen diameter (R² = 0.53, p < 10⁻²⁰), meaning that the chances of observing a large cell were greater in wider vessels. There are two potential physiological explanations for this: that larger cells tend to “prefer” larger vessels (i.e. a network-level filtration effect), and/or that cells lose volume to pass through smaller vessels. To help differentiate between these, Fig. 3(B) shows the relationship between cell surface area and vessel diameter (R² = 0.46, p < 10⁻¹⁶). Red cells are widely believed to be incapable of significant change in their surface area without rupture [1,4,10,41–44], which means that surface area should be more indicative of the “original” size of cells irrespective of their interaction with the microvascular system. The strong correlation retained between surface area and lumen diameter suggests a significant network-level filtration effect, i.e. that larger cells are more likely to enter larger vessels. However, see below for further consideration of whether changes in cell volume also occur, after controlling for differences in cell size due to this network filtration effect.

 

Fig. 3. Relationship between measured size of the average cell and diameter of the capillary lumen. Both cell volume (A) and cell surface area (B) were correlated with lumen diameter.

Download Full Size | PDF

3.3 Increased cellular separation for larger cells in narrower vessels

If large cells are more reluctant to enter narrow vessels as claimed above, the separation between successive cells should be greater for larger cells and/or narrower vessels. Our data showed only a weak correlation between cell separation (centroid-to-centroid) and either of these parameters alone (R² = 0.12 and R² = 0.08, respectively). However, a bivariate model combining both parameters gave strong predictive power for cell separation, as a function of cell size and lumen diameter (R² = 0.62, p ∼ 0). The bivariate fit is demonstrated in Fig. 4(A). Data are projected onto the model space to allow evaluation of the fit quality on a 2D graph (compared with the plotted straight line y = x). The significant increase in predictive power with the bivariate model suggests that it is more important to consider the size of cells relative to the width of the capillary lumen, rather than consider these parameters in absolute terms. It also suggests a masking effect: narrower vessels are expected to show increased cell separation, but they contain smaller cells which are expected to be more closely packed. The multivariate regression coefficient shown in Fig. 4(A) indicates that, for a cell of a given size, inter-cell separation increased by approx. 7.1 µm for every 1 µm reduction in lumen diameter.

 

Fig. 4. Larger cells in smaller vessels become elongated and separated by greater lengths of plasma. A linear bivariate model combining cell surface area and lumen diameter was used to predict (A) average inter-cellular separation and (B) average cell length. Red boxes indicate example cells in Fig. 5 (from left to right, corresponding to A, B, C, D, E and F respectively).

Download Full Size | PDF

Figure 5 shows some examples of the appearance of the blood column from across the horizontal scale plotted in Fig. 4 (red boxes from left-to-right match the letter order of the panels in Fig. 5). The probability map used to generate the 2D profile for each “average cell” is shown (top-left inset in grayscale panels), as are the inferred 3D cell shapes (red polygons plotted from varying perspectives adjacent to each blood column image). Supporting the correlations plotted on Fig. 4 (and Fig. 6 and Fig. 7 below), the examples show wider lumens with obliquely oriented and tightly packed cells transitioning towards narrower lumens with aligned and widely separated cells.

 

Fig. 5. Examples showing variance in cell shape in retinal capillaries. Grayscale: blood column image, wrapped for visualization. Insets: probability map used to demarcate 2D bounds of the average cell (x4 scale). Red shapes: 3D rendering; perspective for all panels is represented by axes in A and described further as follows: Top left: side-on (matching 2D image). Bottom left: top-down. Top right: view down capillary tube, showing differences in lumen diameter. Bottom right: perspective from behind, below and to the right (relative to flow axis). Letter order of panels corresponds to horizontal order of examples in Figs. 4, 6 and 7. Broadly, cells ordered from wide vessels and tightly packed, oblique geometries to narrower vessels with sparse packing aligned with the flow axis (measured lumen diameters printed on each panel).

Download Full Size | PDF

 

Fig. 6. Variation in cellular orientation relative to the flow axis. (A) Orientation plotted as a function of a bivariate model of surface area (SA) and lumen diameter (LD). A two-line model was used due to an apparent “kink” in the underlying data; beyond this tipping point cells are more likely to rotate such that they are obliquely oriented (tipping point for the average sized cell corresponds to a lumen ∼5 µm wide). (B) Cell orientation plotted against average inter-cellular separation. Again, a tipping point is seen whereby cells become more obliquely oriented to support packing closer than ∼15 µm. In both panels, red boxes indicate example cells shown in Fig. 5 (examples from right-to-left in panel A, and from left-to-right in panel B, correspond respectively to the letter ordering of examples in Fig. 5).

Download Full Size | PDF

 

Fig. 7. Evidence for a reduction in cellular volume, for larger cells in narrower vessels. (A) Bivariate model of volume loss as a function of cell size (SA) and lumen diameter (LD), for cells within the “tipping point” of Fig. 6(B); volume of the excluded cells (red) was less well constrained by SA and LD. Red boxes indicate example cells of Fig. 5 (from right-to-left, examples correspond to panels A, B, C, D, E and F respectively). (B) Minimum cylindrical diameter (MCD) for each cell, as a function of lumen diameter, for cells within the tipping point of Fig. 6(B). For these cells the MCD was only slightly less than the lumen diameter (dashed line); the shape of the remaining cells (red) was less strongly tied to lumen diameter.

Download Full Size | PDF

3.4 Elongation of larger cells in narrower vessels

It may be expected that when cells narrow their shape to enter a narrow capillary tube, there should be a commensurate increase in their length i.e. that overall size of the cell is maintained (for example, compare Fig. 5(A) with Fig. 5(F); estimated cell size is comparable with surface area at 113 and 120 µm² respectively, but the latter is more than twice as long as it is found within a narrower capillary tube, 6.3 and 4.0 µm wide respectively). To explore this, we expressed cell length as the longest axis of the best-fitting ellipse for the 2D binary cell profile, however we found no direct correlation between this parameter and lumen diameter (p = 0.19). The lack of a correlation may occur due to the filtration effect noted above (i.e. because larger cells are more likely to be observed in wider vessels), as well as due to a loss in cellular volume detailed further below. Therefore, as above, a simple bivariate model of cell surface area and lumen diameter was fit to the cell length data which gave an excellent fit as shown in Fig. 4(B) (R² = 0.84, p < 10⁻⁴⁷). The results show that larger cells passing through narrower tubes undergo greater changes in their shape. A univariate model of surface area alone gave only a modest correlation (confirming, as expected, that longer cells tend to have greater surface area; R² = 0.35, p < 10⁻¹¹). Again it is clear that the interplay between cell size and lumen diameter, rather than either alone, is an important predictor of the shape adopted by cells in narrow capillaries.

Similar to the modelling of cell length described above, we also considered cell “sphericity” as proposed by Canham and Burton [1]. This parameter was weakly correlated with lumen diameter (R² = 0.11, p < 10⁻³), and not correlated with cell size either by surface area or by volume (p = 0.15). However, as above, a bivariate model of surface area and lumen diameter significantly improved the prediction of cell sphericity (not plotted; R² = 0.34, p < 10⁻¹⁰).

3.5 Re-orientation towards the flow axis for larger cells in narrower vessels

As reported in our previous work, in many vessels the long axis of cells was not aligned with the axis of flow [31]. In some vessels this occurred for a minority of cells, whilst in others most cells appeared affected. Here we extended these observations, comparing cell orientation to lumen diameter and cell size in accord with the modelling approach described above.

As above, cell orientation was not strongly associated with cell size (p > 0.9) whilst there was a moderate positive correlation between cell orientation and lumen diameter (R² = 0.38, p < 10⁻¹³). However, again, a bivariate model combining cell surface area and lumen diameter gave a much stronger fit to the data (R² = 0.68, p < 10⁻³⁰). Data are projected onto the model space in Fig. 6(A). Large cells in narrow vessels were well-aligned with flow, becoming misaligned as cell size decreased and/or lumen diameter increased.

A biphasic relationship is evident in Fig. 6(A), with an initial flat portion consisting of well-aligned cells giving way to a steeper portion where cell orientation becomes progressively more oblique. We therefore also fit the data with a two-phase model with a variable transition point (straight line plotted on Fig. 6(A)), rather than assessing just the y = x fit in the model space as was done in Fig. 4. This model significantly improved predictive power for cellular orientation (R² = 0.83, p ∼0). The transition point of this two-phase model can perhaps be interpreted as a “tipping point” beyond which cells are compelled to align with the flow axis. We found no difference in cell size (surface area) between the group of cells located above the tipping point compared with those below it (p = 0.15), but did find a significant difference in lumen diameter (p < 10⁻¹⁸). This in turn suggests the concept of a “critical lumen diameter” which is important in dictating the orientation of cells. For a cell of average surface area (110 µm² in this dataset), the transition point in Fig. 6(A) corresponds to a lumen diameter of 5.1 µm (95% range from 4.9 to 5.6 µm, based on bootstrapping 10,000 times i.e. data resampled with replacement).

Given the tendency for cells to align with the flow axis below a certain lumen diameter, the question may be asked as to why cells do not simply always align with the flow axis. The answer may in part be that oblique cell orientations support an increased packing density. Figure 6(B) plots cell orientation against inter-cellular separation. Again, a biphasic nature is evident. The plotted two-line fit here suggests that beyond an inter-cell separation of ∼15 µm, cells transitioned from being misaligned with the flow axis to aligned with it (e.g. a change from a “parachute” to a “slipper” configuration). The former orientation evidently supports tighter cell packing (smaller intercell distance), whereas the latter is adopted in narrower vessels at the cost of a commensurate reduction in hematocrit (longer intercell distance).

3.6 Loss of cell volume for larger cells in narrower vessels

Here we present evidence supporting the notion that red cells may lose some portion of their cellular volume, presumably water, on passage through narrow capillaries. The arguments provided below, in order, are based on a) modelling cell volume as a function of surface area and lumen diameter; b) consideration of the minimum cylindrical diameter (MCD) for the average cell, compared with lumen size; c) consideration of the relationship between the volume and sphericity of cells, and d) consideration of the geometric scaling which relates volume to surface area.

Modelling cell volume: As mentioned above, cell surface area is thought to be largely immutable and so may serve as a proxy for the “original” size of a cell, independent of its interaction with the capillary network. This allows us to investigate whether cell volume is also conserved, or whether it might be reduced after entering a narrow vessel. However, big cells will tend to have both high surface area and high volume, which may obfuscate vessel-induced changes in volume (especially given the filtration effect noted above, whereby large cells are more likely to find their way into larger vessels).

To help disentangle these issues we modelled cell volume as a function of surface area, finding a high correlation between these parameters as expected (R² = 0.96, cf. 0.94 in the original report of Canham and Burton [1]). Surface area was scaled such that its units matched those of volume according to the expected geometric relationship between these quantities (i.e. surface area values were raised to the power 3/2). This model was then subtracted from the raw volume data to produce a “residual volume”, i.e. a volume estimate that is corrected for differences in the surface area of cells (inferred to be a proxy for their original size). Residual cell volume was only weakly associated with lumen diameter (R² = 0.13, p < 10⁻⁴).

As above, we next employed a bivariate model, combining lumen diameter and cell surface area, to attempt to explain residual cell volume. This provided mild predictive power when all cells were considered (R² = 0.23, p < 10⁻⁸), with the sign of the correlation suggesting that larger cells in narrower vessels tend to have reduced volume. Notably, this is opposite in sign to the typical association between cell volume and surface area, and is consistent with an interpretation that larger cells in narrower vessels may have some amount of their cellular fluid (water) squeezed out as they pass through narrow capillaries.

The “tipping point” described above suggests that cells may experience differing constraints depending on whether they are large relative to the vascular lumen. We therefore re-assessed the bivariate model of residual volume, including only those cells within the tipping point (i.e. those closer to the shallow line plotted in Fig. 6(B); 81 of 121 cells included). The data is plotted against the model in Fig. 7(A) (excluded cells in red), with significantly improved explanatory power shown (R² = 0.75, p < 10⁻²³). The improved model validity suggests that if cellular volume (water) is indeed lost, this is a phenomenon which occurs only for a subset of capillary vessels according to whether the average size of cells in those vessels is too large relative to the size of the lumen. The bivariate regression coefficient for this model suggests that, for the average-sized cell, approx. 11.3 fL is lost per 1 µm change in lumen diameter (i.e. ∼11% of the typical cell volume). For the bivariate model of the preceding paragraph, with all cells included, this value was instead 4.8 fL / µm (∼5% of typical cell volume).

Minimum cylindrical diameter: The concept of minimum cylindrical diameter (MCD) was described above, describing the narrowest tube through which a cell could fit without changing its volume or surface area. Measured in vitro, this parameter has been reported to be tightly conserved (e.g. 3.0 to 3.5 µm) across all red cells [1]. Our data showed a much wider variation, ranging from approx. 3 to 6 µm (Fig. 7(B)). More importantly, MCD in our data was strongly dependent on the size of the vascular lumen (R² = 0.60, p < 10⁻²⁴) which indicates that MCD of a cell is variable depending on the vessel it is confronted with. If cells beyond the tipping point are ignored, as described in the previous section, the model prediction improved substantially as shown by the solid line plotted in Fig. 7(B) (R² = 0.90, p < 10⁻⁴⁰).

A certain degree of correspondence between MCD and lumen size should be expected due to the network filtration effect described above, i.e. cells with larger MCD should be observed more commonly in wider tubes. However, it would be an incredible coincidence if almost every red cell happened to find a vessel only just wide enough to accommodate it; a far more likely interpretation is that cell shape is adjusted to match the vessel. As MCD is defined from only two parameters, volume and surface area, one of these must change in order for MCD to change. As discussed above, the weight of evidence from the literature suggests that surface area is constant which indicates that the variation in MCD must result from a change in cell volume. It is not possible to determine which is the case from our present dataset; in future, studying connected bifurcations and comparing the surface area and volume of the same cells in the parent and daughter vessels could answer this question.

Volume vs. sphericity: Canham and Burton reported a strong correlation between cell volume and the sphericity parameter, which relates volume to surface area to describe how round a cell is (a sphere encloses the most volume possible, for a given surface area) [1]. They attributed the correlation to a largely fixed relationship between volume and surface area that applies to all healthy cells drawn from an individual. We did not find any such correlation, as was mentioned above (p = 0.15). This suggests that volume and surface area are not so predictably related within narrow capillary tubes in vivo; the other findings above suggest that knowledge of the size of the vascular lumen is needed to accurately predict cell volume.

Geometric scaling: Finally, in their pioneering work Canham and Burton [1] argued that the shape of red cells “cannot be varying independently of the volume” on the basis that linear relationships provided better fits to their data than the expected power laws (e.g. surface area being proportional to volume raised to the 2/3 power). Our data from retinal capillaries in vivo do not accord with this; a simple brute optimization gave an optimal exponent of 0.70 in order for surface area to best predict volume, very close to the expected geometric value of 2/3. Again, contrary to prior work we conclude that the key determinants of cell shape (volume and/or surface area) must in fact be altered on passage through narrow capillary vessels. As argued above, evidence against a significant change in cell surface area is strong, suggesting that it is cell volume that must change.

4. Discussion

We have presented data describing variations in the shape of red cells observed in transit through capillaries of the living human retina. The data is consistent with the general thesis that red cells can encounter considerable resistance when attempting to navigate through the healthy retinal capillary network. This was evidenced by a network filtration effect whereby wider vessels were more likely to contain larger cells; by reductions in volume for large cells within narrow vessels consistent with the ejection of cellular water; and by pronounced, non-linear changes in cell separation and orientation resulting from modest changes in the size of cells relative to the diameter of capillary lumens.

Although the statistical correlations reported here were strong, it is important to emphasize the limited capacity of our method to accurately characterize 3D cell shape: we considered only the average cell due to signal-to-noise considerations, did not incorporate information in the axial dimension where resolution is much lower, and must also contend with limited transverse resolution due to the finite size of the eye’s pupil. These limitations are detailed further below, at the end of this section.

It has been proposed that narrower capillaries might facilitate gas exchange due to a reduced diffusion path between hemoglobin and tissue, as well as an increased cell surface area apposed to the vascular lumen due to the elongation of cells [5–7]. Our results did not show any direct elongation of cells as a result of smaller lumen diameters, due to the network filtration effect, and also showed a significant increase in cell separation in narrow vessels (spanning a range from 10 to 40 µm). Hence, our findings are not consistent with the idea that gas exchange should be improved in narrow vessels. Recent experimental studies also cast doubt on the idea that exchange is improved, showing that increased hemoglobin concentration actually suppresses gas exchange due to a more tortuous diffusion path [6]. Another major obstacle to gas exchange is permeability of the red cell membrane, which may be facilitated by specialized gas channels [45] and is enhanced upon osmotic swelling of cells [6]. This implies that the loss of cell volume suggested in our study may be accompanied by reduced membrane permeability. The above factors could of course be offset if flow velocity is commensurately higher in narrower vessels, permitting a greater volume flow rate of hemoglobin which is also a key driver of gas exchange [46]. However, we found no association between measures of flow velocity and lumen diameter (nor with any of our other outcome measures). Overall then, the net effect for narrow capillary lumens with their reduced density of red cells, more tortuous diffusion paths and reduced membrane permeability is anticipated to be a significant reduction in gas exchange.

Our observations are also not consistent with the view that red cells may freely adopt any efficient, geometrically possible shape of constant volume and surface area irrespective of their biomechanical properties [1,9,14]. Therefore, of the three categories of explanation outlined in the Introduction, regarding the reason for red cells to generally be wider than capillary lumens, the third category is most consistent with our results. Accordingly, we propose that capillaries which are narrow compared with the size of red cells allow tissue to minimize the potential for excessive oxygen exchange under baseline conditions.

Extending beyond baseline levels of metabolic activity, it is known that high demand for visual processing triggers targeted dilations and constrictions within the retinal microvascular network, improving the flow of red cells towards active pockets of neurons [12,47]. This capacity is underscored by strong heterogeneity in flow and in cellular distribution between capillaries observed under baseline conditions, with increased neural activity triggering reconfiguration across the network to achieve greater homogeneity in red cell distribution. This shift towards homogeneity is thought to be an important contributor to improvements in gas exchange [46]. Our results suggest pronounced differences in resistance to cellular flow between vessels as a key driver of network heterogeneity; in order to improve both flow and homogeneity under metabolic load, the network may preferentially dilate the initially higher-resistance “reserve” vessels. Our results further suggest that the healthy capillary network is efficiently tuned in this regard, with the diameter of the average capillary coinciding with a “tipping point” regarding the ease with which red blood cells may enter and traverse retinal capillaries. Such a scheme would allow the density and packing of cells to be varied dramatically following relatively minor changes in lumen diameter. However, this scheme may also suggest a vulnerability to disease processes wherein reduced ability to regulate diameter in response to varying conditions could produce ischemia (where vessels are too narrow) or support oxidative damage (where vessels are too wide).

Despite the strong evidence presented above that cellular resistance to flow encounters a “tipping point” at narrow lumen diameters and large cell sizes, we did not observe any significant correlation between measures of flow velocity and these parameters. Such a correlation would be expected for the idealized case of a single vessel in isolation, with a constant hematocrit, and a given pressure drop applied across its length. However in a real network, the narrower vessel at a bifurcation is expected to simply receive a far greater share of the plasma, which does not need to deform to enter a vessel and hence encounters lower resistance to flow [17]. This could allow the overall flow velocity in that vessel to be maintained. On the other hand the cells, especially those largest in size, would tend to travel down the fellow branch where they would encounter the least resistance to flow. Therefore in general, the lack of a correlation between flow velocity and lumen diameter in the capillary network is consistent with the interpretation that red cells may face significant resistance to flow in normal capillary transit.

The network filtration effect identified here demonstrates that the ease with which a red cell enters a given capillary tube depends on the size of the cell relative to the vascular lumen [17], and represents a simple mechanism by which the natural variation in red cell size (surface area) may be passively handled within a capillary network. This was an important effect to account for since it otherwise masks various associations (for example, cell length and width were not associated with one another unless cell size was also accounted for).

Our finding of a reduced cellular volume within narrow capillary tubes, beyond a network filtration effect, may be more controversial. However, we presented multiple lines of evidence supporting this conclusion. Firstly, a simple bivariate model of surface area and lumen diameter explained the estimated loss in volume which is consistent with the proposed mechanistic explanation of cells being too large relative to the capillary lumen. This model applied well only in a subset of vessels, though this may be a point in its favor as these vessels did correspond to the “tipping point” derived from considering cell orientation and separation. Secondly, the minimum tube diameter through which each cell could theoretically pass, given its volume and surface area, closely followed the size of the lumen in which each cell was observed; this would represent incredible fortune on the basis of network filtration alone. Compared with in vitro studies where minimum tube diameter is smaller than most capillaries and tightly conserved across the red cell population [1,9,11,48], our results strongly suggest that marked differences in the volume and/or surface area of cells are imposed by the limited extent of the capillary lumen. Finally, we appealed to geometric and statistical associations relating volume and surface area, which were used by earlier authors to present a model of constant volume and surface area in vitro [1]. Our findings in vivo are at odds with that earlier work. We note that each of the above points supporting a loss in cellular volume could equally well be explained by changes in cell surface area, however, volume changes are the most likely explanation given that red cells are observed to conserve their surface area across a variety of experimental conditions [1,4,10,41–44,49]. It is possible that by ejecting volume, resistance to cellular flow is reduced such that less power, supplied ultimately by the heart, is required to propel the same quantity of hemoglobin through the microvasculature. It will be necessary to track the same cells as they traverse capillary junctions to confirm these observations (i.e. namely, that volume of a given cell changes whilst surface area remains constant).

Some authors [40] have reported minimum cylindrical diameter with higher mean and spread, more comparable to that reported here than those from earlier work [1]. This suggests that differences in cellular shape between in vitro and in vivo studies may depend on the particular details of the preparation. This can include extraction, washing and spinning which may have unknown effects on the biomechanical properties of cells [50] (the above cited study [40] did not engage any such process). Other notable differences in some preparations include the lack of compression of the cell within a vessel at all [50], or otherwise the use of synthetic channels [9] which lack the endothelial glycocalyx that is known to be important in the lubrication of capillary flow [3]; the lack of cell-cell interactions at physiological hematocrits (for example, inserting one cell at a time through a flow channel); and the lack of a rhythmic, pulsatile variation in pressure driving the flow. Cardiac pulsatility is now a well-established feature of the retinal capillary network, despite persisting textbook accounts of steady or purely stochastic variability in flow at the capillary level, with velocity varying by around a factor of two across the cardiac cycle in most retinal capillaries [22–25,39,51].

Is there a plausible mechanism for reductions in the volume of red cells through narrow capillary tubes? Such a change should almost certainly be associated with a loss of cellular water, which could arise due to some combination of change in osmotic pressure, mechanical pressure, and/or membrane permeability. Red cell volume is highly sensitive to osmotic gradients, with red cells acting as “perfect osmometers” under certain conditions and within the bounds of a relatively fixed cell membrane surface area [42]. Whether an osmotic gradient exists within the capillaries is unclear, however, the relatively stagnant plasma boundary layer which occupies perhaps 30% of the capillary lumen [3] could support the formation of such a gradient. Gradients of various kinds must, of course, exist in order for metabolites to diffuse out of the capillary lumen and into tissue, and pH gradients secondary to CO₂ gradients exist within the microvasculature and are known to alter cell volume [52]. Osmotic shrinking of cells within the capillaries may have a functional benefit; experiments measuring the resistance to red cell passage through narrow pores (e.g. 2.6 µm) show that flow of cells relative to that of plasma is significantly facilitated upon hypertonic shrinkage of cells [49].

In regards to mechanical pressure, micropipette aspiration studies show that red cells drawn through narrow synthetic tubes do indeed lose volume. Total cell membrane surface area hardly changed in such experiments [10], as the red cell membrane is highly resistant to two-dimensional strain (though is highly amenable to one dimensional strain, i.e. shear). An estimated 12% of the cell volume is rapidly exchangeable with plasma, in a reversible fashion such that the original cell shape is recovered following the cessation of pressure [44]. Beyond this, shape changes may not be reversible; our figure of an average 11% loss in volume, presented above, may therefore prove to be an over-estimate.

As mentioned above, our results should be interpreted with caution in light of several limitations which may impact the ability of our method to accurately characterize cell shape in 3D. The first is the inherent difficulty in estimating a 3D shape from a 2D image profile; our method does not incorporate any information beyond the en face 2D image plane. To infer 3D cell shape from this 2D information we presumed a circular cross-section to the vascular lumen (or more precisely, the channel within the plasma layer through which cells pass). An example of the ambiguity faced is the interpretation of two cells with the same 2D image profile, but one aligned with the flow axis and the other perpendicular to it. The former was interpreted as a “sausage” shaped cell in a thin vessel, whereas the latter would be interpreted as a disciform cell seen edge-on in a wide vessel. Confirmation of our assumptions was not possible as the vessel wall and plasma layer are poorly discriminated in most of our images due to a lack of contrast; thus we can see only the boundary between cell and plasma, not that between plasma and lumen. Accordingly our approach is also blind to differences in the thickness of the plasma layer, required to confirm “slipper” shapes formed when cells congregate towards one side of a vessel. Such configurations are thought to be adopted as shear rates increase [53], maximizing the flow which can be produced by a given pressure gradient [54]. Similarly, our method is blind to the fine structure of folds in the cell membrane which have been documented by use of interferometric methods to accurately measure 3D shape [37] but are also visible directly under light microscopy [4]. Although our bright-field contrast mechanism is presumably similar to the latter, the transverse resolution inherent in imaging through the finite pupil of the eye limits the fine variations in shape which can be discerned. This is compounded by our consideration of the average cell, necessary due to limited signal-to-noise secondary to safety considerations in human retinal imaging, which will tend to average away differences in the location of folds, dimples etc in the cell membrane. Some of the above limitations may be circumvented by adaptive optics scanning laser ophthalmoscopy approaches able to visualize the vessel wall fine structure [55,56], however, interferometric methods are probably required to provide sufficient resolution in the axial dimension [36,37].

Another potential limitation implicit in our statistical findings is that variations in image focus (blur), or in the thresholding procedure to separate cells from plasma and from the surrounding retinal tissue, could increase the apparent size of cells which would increase co-variance between our measures of volume, surface area and lumen diameter. However, we found no direct correlation between the length of cells and the diameter of lumens (p = 0.19), which might be expected were these factors to explain a meaningful degree of the variance in our data. Outcome measures such as cell orientation and separation are also comparatively unaffected by such errors. The consistency of our estimated cell volumes and surface areas with established values further suggests that any effects of optical spreading of light do not impart systematic bias (e.g. it does not appear that measures are generally inflated beyond their true values).

A final potential limitation of the present work is that vessels may be inclined out-of-the-plane by a certain degree. This will tend to shorten cells whilst leaving width unaffected (not withstanding blur, addressed above), proportionally reducing our estimates of volume and surface area. It is also worth mentioning that cells may be oriented obliquely in the out-of-plane dimension, to which our method is completely blind. However, we do not expect the likelihood of cells being oriented out of the plane of our imaging device to be associated with their lumen diameter, and hence these limitations are not anticipated to impose systematic bias that would affect our conclusions. Finally, our imaging was conducted in the region adjacent to the foveal avascular zone, which features the least stratified portion of the retinal capillary network.

5. Conclusions

Measurement of the shape of individual red blood cells traversing the retinal capillary network suggests that resistance to cellular flow is heavily dependent on the size of cells relative to the capillary lumen, leading to significant and predictable variations in the shape, volume, orientation and packing of cells. In particular, water may be squeezed out of red cells to facilitate passage through narrow capillaries. Narrow capillaries in general are predicted to significantly reduce gas exchange, which suggests mitigation of oxidative damage as a key factor driving the evolution of narrow vessels. Further work is needed to observe these changes unfold by tracking the same cells across connected vascular junctions to explore the impact of experimental modulation of capillary diameter, and to more accurately characterize in vivo cell shape in 3D.