2.1.

Theoretical Model for Capillary-Level Optical Oximetry

The RBC, or discocyte, is a biconcave disc-shaped cell containing hemoglobin. The complex refractive index of hemoglobin, $n=n^{\prime }+i{n}^{\prime \prime }$, is ${\mathrm{sO}}_2$ dependent, as is the light scattering created by RBC geometry.¹⁵ To quantitatively investigate RBCs’ backscattering spectra, we consider a light wave with unit amplitude propagating in direction ${\mathbf{s}}_0$ that encounters an RBC in plasma. For any far-field point $\mathbf{r}$, the scattering process generates a spherical wave $E_{\mathbf{s}}(\mathbf{r})$ propagating in direction $\mathbf{s}=\mathbf{r}/r$, where $r=|\mathbf{r}|$. This can be written as follows:

The differential scattering cross section is defined as

We use ${\sigma }_{\mathrm{s}}(-{\mathbf{s}}_0,{\mathbf{s}}_0)$ to present RBC backscattering spectrum, and ${|\mathbf{s}\times (\mathbf{s}\times {\mathbf{e}}^{(i)})|}_{\mathbf{s}=-{\mathbf{s}}_0}=1$. Additionally, we assume RBCs as homogenous within their boundaries, thus

Equation (4) gives the backscattering spectrum of one single cell with a fixed orientation. We assume RBCs have random orientations in blood, thus the backscattering spectrum of one randomly oriented cell is given by averaging ${\sigma }_{\mathrm{s}}(-{\mathbf{s}}_0,{\mathbf{s}}_0)$ over uniformly distributed incident directions

Fig. 1

Calculation of $\mathcal{F}\{R(\mathbf{r})\}$ for a single RBC.

To characterize the intrinsic size variation of RBCs, we measured RBC sizes in vitro from anticoagulated bovine blood. The RBCs were sphered isovolumetrically with 0.03% sodium dodecyl sulfate (SDS) in phosphate buffered saline (PBS) and pipetted onto a glass slide.²⁰ The radius of spherical RBCs was quantified from two-dimensional (2-D) bright-field microscopic images (Leica, $40\times$). The detailed sample preparation is described in Sec. 3.2. A microscopic image of spherical RBCs is shown as the inset in Fig. 2. The radius of spherical bovine RBCs roughly followed a Gaussian distribution, with a mean of $2.63\mu \mathrm{m}$, and a coefficient of variation (CV) of 5.54%. The CV is defined as the standard deviation divided by the mean value. The radius of spherical human RBCs measured by the same method also roughly followed a Gaussian distribution with a mean of $3.03\mu \mathrm{m}$ and a CV of 5.26%, shown in Appendix Fig. 9. As the sphering of RBCs was isovolumetric, we approximate the size (or volume) distribution of normal RBCs as that of the spherical RBCs characterized by microscope.

Fig. 2

The radius distribution of spherical bovine RBCs. Blue bars: the probability mass function of cell radius. The curve: Gaussian distribution (mean: $2.63\mu \mathrm{m}$ and CV: 5.54%). Spherical RBCs were made by adding 0.03% SDS to blood (ratio: $3:1$). Inset panel: A microscopic image of spherical RBCs. Scale bar: $10\mu \mathrm{m}$.

Considering the cell size, the RBC backscattering spectrum is given by averaging ${\sigma }_{\mathrm{b}}$ over size

2.2.

Geometric Model of Red Blood Cells

The geometric function for normal disk-shaped RBCs, or discocytes, is based on the Jacobi elliptic functions.²¹ To account for deformations of RBCs in circulation and particularly in capillaries, parameters in the geometric function of a discocytes were modified to obtain six types of deformed RBCs, including stomatocyte I, stomatocyte II, stomatocyte III, and sphero-stomatocyte in the bloodstream, and bullet type and parachute type in capillaries.²¹ The geometries of RBCs are inputs for simulations of the theoretical model.

2.3.

Refractive Indices of Red Blood Cells and Plasma

We used complex refractive indices of RBCs, oxygenated and deoxygenated, in our simulations. We assume RBCs are internally homogeneous filled with hemoglobin solution. Using Kramers–Kronig relations, we generated the complex refractive index of RBCs according to the absorption coefficient of hemoglobin, which is calculated from the tabulated molar extinction coefficient for hemoglobin in water.²² Since RBC hemoglobin content is usually between 300 and $360\mathrm{g}/\mathrm{L}$, we used $330\mathrm{g}/\mathrm{L}$ in calculation of RBC refractive indices.²³ The refractive index of plasma we used is 1.35.²⁴

3.1.

Visible Light Optical Coherence Tomography System

We used spectral domain vis-OCT, an optical technique that probes tissue structure with backscattered light, to experimentally validate our model for discocytes. Spectroscopic analysis with vis-OCT can provide spatially resolved spectral information, allowing extraction of backscattering spectra from individual RBCs. The homebuilt vis-OCT used in this study incorporates a super-continuum laser for broadband illumination and a spectrometer covering a wavelength range from 520 to 630 nm. The transverse and axial resolutions of the vis-OCT system are 15 and $1.7\mu \mathrm{m}$, respectively. A detailed description of the vis-OCT system is given in Ref. 3.

For RBC measurements, the following scanning protocol was used. In the fast scanning axis, each B-scan images consisted of 512 A-lines, covering a 0.55-mm length. In the slow scanning axis, 512 B-scans were required to cover a 0.55-mm range. The laser spot is translated along the fast scanning axis, whereas A-lines are acquired at a rate of 50 kHz, scanning back and forth between adjacent B-scans, so the total time to finish one vis-OCT scan is approximately 5.24 s. Although the transverse resolution is not high enough to resolve fine features of a single RBC with a typical long axis length of around $8\mu \mathrm{m}$, we were able to identify and extract spectrum from individual RBCs. This is possibly for two reasons. First, in our experiments, the RBCs are sparsely suspended. According to the detection volume ($0.55\times 0.55\times 3.02{\mathrm{mm}}^3$), sample concentration (1%), hematocrit (0.4), and the size of RBCs characterized by microscope, the average distance between two RBCs in the sample is approximately $26.7\mu \mathrm{m}$, which is much larger than the imaging resolution. Second, our scanning protocol oversampled in the transverse direction, making the distance between adjacent scan planes $1.07\mu \mathrm{m}$ to ensure sampling density was sufficient to distinguish the individual RBCs. Thus, we can still differentiate individual RBCs by vis-OCT in Fig. 4(b). Furthermore, for in vivo cases, where RBCs are closer to each other, the backscattering spectra measured by vis-OCT are averaged over many individual RBCs.

3.2.

Sample Preparation

A sample of spherical RBC was prepared for measurements of cell size distribution. The sample was made by first adding 0.03% SDS PBS solution to bovine blood (Quad Five) at a volume ratio of $3:1$. Then this mixture was diluted to a final concentration of 1% (bovine blood accounting for 1% of the whole sample volume) by adding more PBS. A drop of the sample was put on a glass slide and then imaged with a bright-field microscope (Leica, $40\times$). The radius of spherical RBCs was quantified from 2-D microscopic images.

Oxygenated and deoxygenated RBC samples were contained in glass petri dishes for vis-OCT measurements of backscattering spectra. Each glass petri dish contained approximately 10 mL of sample solution. For oxygenated samples, bovine blood was exposed to air for 1 h before dilution to 1% in PBS. For deoxygenated samples, 10% sodium dithionite (${\mathrm{Na}}_2{\mathrm{S}}_2{\mathrm{O}}_4$) in PBS was added to bovine blood at a ratio of $3:1$ before the blood was diluted to 1% in PBS. To ensure samples were fully oxygenated or deoxygenated, a commercial oxygen probe (MI-730, Microelectrodes, Inc.) was used to measure the ${\mathrm{pO}}_2$ independently.²⁵ To disperse RBCs in suspension, each sample was fully shaken before measurements were taken. For deoxygenated samples, each measurement was taken within 10 min of preparation with ${\mathrm{Na}}_2{\mathrm{S}}_2{\mathrm{O}}_4$ to avoid oxygenation from air exposure.

3.3.

Data Processing

RBC backscattering spectra were obtained by an inverse Fourier transform-short time Fourier transform method.²⁶ For each blood sample, seven independent vis-OCT scans ($0.55\times 0.55\times 3.02{\mathrm{mm}}^3$) were acquired. In each vis-OCT scan, a threshold-based algorithm was used to segment regions containing RBCs, and spectra from all these regions were averaged to extract a mean spectrum. Afterward, each mean spectrum was normalized by its maximum value and seven independent mean spectra from the same sample were further averaged to produce a final spectrum with a standard deviation.

4.1.

Model Validation by Mie Theory

We first tested the validity of the first-order Born approximation used in this model derivation. We compared the backscattering spectra calculated by our model and Mie theory of a single homogenous sphere with weak refractive index contrast to the background. The radius of the sphere was $2.25\mu \mathrm{m}$, comparable to an RBC in size. The refractive indices of the sphere and background were 1.391 and 1.35, respectively, close to those of RBCs and plasma. In Fig. 3(a), the geometry-dependent backscattering spectra calculated by both methods exhibit similar oscillatory characteristics, caused by the interference of light reflected from the top and bottom boundaries of a single sphere. This similarity validates the first-order Born approximation for calculation of backscattering spectra of large soft spheres similar to RBCs. We then applied the complex refractive index of ${\mathrm{HbO}}_2$ and Hb to spheres with the same radius to calculate their backscattering spectra by our model and Mie theory. The similar oscillatory character shown in Appendix Fig. 10 still dominates the backscattering spectra, making hemoglobin absorption contrast difficult to discern. However, when we averaged the spectra over size following a Gaussian distribution (mean: $2.25\mu \mathrm{m}$; CV: 6%), the geometry-dependent spectra were smoothed out and the absorption contrast clearly emerged. To confirm this contrast, we also calculated the spectra by Mie theory for comparison. Results by both methods display similar hemoglobin absorption contrast in Fig. 3(b), confirming the feasibility of our model and indicating the possibility for deriving absorption-dependent ${\mathrm{sO}}_2$ from backscattering at the cellular level.

Fig. 3

Normalized backscattering spectra by Mie theory and our model for spheres: (a) Uniform radius: $2.25\mu \mathrm{m}$; refractive index: 1.391; (b) mean radius: $2.25\mu \mathrm{m}$; CV: 6%; refractive index: complex refractive index of ${\mathrm{HbO}}_2$ and Hb. Refractive index of background: 1.35.

4.2.

Model Verification by Visible Light Optical Coherence Tomography

Our model was used to simulate the backscattering spectra of discocytes with different oxygenations. We then used vis-OCT to experimentally validate our model for discocytes by measuring the backscattering spectra of the prepared oxygenated and deoxygenated blood samples. The 2-D cross section and three-dimensional (3-D) configuration of a discocyte are shown in Fig. 4(a). The discocytes were assumed to have the same mean volume as the RBCs sphered by SDS, so their sizes also roughly followed a Gaussian distribution (mean long axis length: $7.5\mu \mathrm{m}$; CV: 6%). The simulation results of oxygenated and deoxygenated RBCs are shown in Figs. 4(c)–4(d), indicating similar contrast to that of hemoglobin absorption. To verify sample oxygenation, a commercial oxygen probe was used to measure the ${\mathrm{pO}}_2$ of the oxygenated and deoxygenated blood samples, which were $147.16\pm 5.04\mathrm{mm}\mathrm{Hg}$ and $0.342\pm 0.129\mathrm{mm}\mathrm{Hg}$, corresponding to ${\mathrm{sO}}_2$ values of $98.87\pm 0.35\%$ and approximately 0%, respectively. A vis-OCT B-scan image of suspended RBCs is shown in Fig. 4(b). The spectra of oxygenated and deoxygenated RBCs with standard deviations are shown in Figs. 4(c)–4(d). In each vis-OCT scan, between 1000 and 2000 RBCs were segmented for averaging backscattering spectra. The backscattering spectra calculated by Mie theory for equal-volumed spherical RBCs are shown for comparison. All results are normalized by their maximum values, respectively.

The results of our model, Mie theory, and experiments show similar absorption contrast in Figs. 4(c)–4(d) without oscillations. Compared with Mie theory, results of our model are closer to experiments in magnitude, especially for the deoxygenated samples. Although results of Mie theory show greater contrast, especially for the oxygenated samples, they are not well within the margin of error of experiments. This suggests that a cell’s geometry affects its backscattering spectrum, and our model can serve as a more accurate method to calculate backscattering spectra of nonspherical particles.

Fig. 4

(a) The geometry of a discocyte. (b) A pseudocolor B-scan image of RBCs in PBS by vis-OCT. Horizontal and vertical scale bars: $100\mu \mathrm{m}$. Normalized backscattering spectra of (c) oxygenated and (d) deoxygenated discocytes from vis-OCT and our model (mean long axis length: $7.5\mu \mathrm{m}$, CV: 6%), and equal-volumed spheres by Mie theory (mean radius: $2.625\mu \mathrm{m}$, CV: 6%).

4.3.

Model Simulations of Red Blood Cells with Different Mean Cellular Hemoglobin Contents

We calculated refractive indices of RBCs for simulations based on constant MCHC of $330\mathrm{g}/\mathrm{L}$, but practically this MCHC can vary between 300 and $360\mathrm{g}/\mathrm{L}$, resulting in variations of RBC refractive indices.²³ Thus, we explored how the variation in MCHC affects RBC backscattering spectra by setting the MCHC as 300, 320, 340, and $360\mathrm{g}/\mathrm{L}$, respectively. Under different MCHCs, the backscattering spectra of RBCs were calculated and averaged over orientation and size. The results are shown in Fig. 5. The oxygenation contrast in backscattering spectra of RBCs persists with different MCHCs, but higher MCHC tends to generate backscattering spectra with more intensive oxygenation contrast. This demonstrates the reasonability of using $330\mathrm{g}/\mathrm{L}$ as MCHC to do all our simulations.

Fig. 5

The influence of MCHC on backscattering spectra of RBCs.

4.4.

Model Simulations of Deformed Red Blood Cells

RBCs in circulation can deform by elastic and electrical forces, surface tension, and osmotic or hydrostatic pressures.²¹ In particular, RBCs can deform as bullet-like or parachute-like shapes as they pass through capillaries.^27,28 To account for this, we explored the effect of cell deformations in capillaries on backscattering spectra. For comparison, four general RBC deformations in the bloodstream are also explored. Six RBC deformations, including bullet type ($l=5.35\mu \mathrm{m}$, $h_{+}=4.125\mu \mathrm{m}$, $h_{-}=1.375\mu \mathrm{m}$, $m_{+}=0.12$, and $m_{-}=0.35$) and parachute type ($l=5.525\mu \mathrm{m}$, $h_{+}=4.55\mu \mathrm{m}$, $h_{-}=0.15\mu \mathrm{m}$, $m_{+}=0$, and $m_{-}=0.99$) in capillaries, and stomatocyte type I ($l=7.8\mu \mathrm{m}$), stomatocyte II ($l=6.525\mu \mathrm{m}$), stomatocyte III ($l=6.025\mu \mathrm{m}$), and sphero-stomatocyte ($l=5.825\mu \mathrm{m}$) in the bloodstream,²¹ which have the same mean volume, were inputs for model simulation. Their 2-D cross sections and 3-D configurations are on the top left of Figs. 6(a)–6(f). Sizes of all deformations followed a Gaussian distribution (CV: 6%). We compare their backscattering spectra in Figs. 6(a)–6(f), with all curves normalized by maximum values of the oxygenated RBCs, respectively. The contrasts are all similar, indicating that the backscattering spectra showing hemoglobin absorption contrast do not appreciably vary for differently deformed RBCs.

Fig. 6

The geometries of deformed RBCs and their backscattering spectra calculated by our model: (a) bullet type, (b) parachute type, (c) stomatocyte type I, (d) stomatocyte type II, (e) stomatocyte type III, and (f) sphero-stomatocyte. In each case, only the curves for the geometry on the top left are highlighted, whereas the others are semitransparent.