1.

Introduction

The knowledge of the optical properties of human blood¹ (as a whole and also as components) plays an important role for many diagnostic and therapeutic applications in laser medicine and medical diagnostics. As a main component of blood, not only by their volume but also by their role, red blood cells (RBCs) can be easily collected, isolated, and handled. They have no internal macroscopic structure. Therefore, knowledge of the optical properties of RBCs in suspension, such as the scattering and absorption coefficients and the scattering phase function, is of crucial importance for the development of laser medicine. The dedicated literature, especially related to RBC optical properties, covers a diversity of aspects ranging from theoretical ^{2, 3, 4, 5, 6, 7, 8} and/or experimental ^{9, 10, 11, 12, 13, 14, 15} approaches to medical applications with diagnostic purposes.^{16, 17}

The morphology of RBC aggregation in whole blood and in RBC suspensions at rest and in shear flow is successfully studied by optical methods.^{18, 19, 20, 21, 22} Previous studies of RBC aggregation have suggested that the aggregation process involves three steps. It starts with the formation of pairs from the cells that associate with each other side by side. Primary aggregates comprising only two adjacent cells develop further into linear aggregates of different lengths known as rouleaux.^{23, 24, 25} These rouleaux are bound end-to-end and end-to-side. They form a three-dimensional (3-D) network that is tightened by cohesive forces into large dense aggregates (containing thousands of RBCs) separated by plasma layers. It is commonly accepted that such aggregation is mostly due to the nonspecific adsorption of the aggregating plasma proteins on the surfaces of adjacent RBCs. Nonetheless, there is evidence that protein adsorption is not the only mechanism for aggregation.^{26, 27}

Due to its importance in hemorheology, the RBC aggregation process has many approaches. Measurement of optical microscopy,²⁸ zeta sedimentation rate,²⁹ image analysis in a flow chamber,³⁰ ultrasound backscattering,^{31, 32} and photometry^{19, 20, 21, 22, 33} are among the methods that have been employed to quantitatively study the aggregation. Despite the fact that RBC aggregation can be assessed using different techniques, two main approaches have been developed in the aggregometry. The first is based on the registration of light transmission through the studied blood layer in a cone-plate^{21, 22} or plate-plate³⁴ viscometer. The second approach is based on the registration of the backscattered light from a blood layer in a Couette-type viscometer.^{19, 20, 35}

This paper reports the results of a quantitative investigation of the RBC aggregation process. The main goals of the present study are to measure, with a very high resolution, the angular spreading of light forward scattered at small angles by RBCs in suspensions, at low hematocrits, during the aggregation process; to determine the behavior of aggregated RBCs’ optical properties (scattering cross section and effective scattering anisotropy) during the aggregation process, comparing the measured data with the predictions of the new effective phase function model³ and Mie theory; and to estimate the potential of the proposed technique (light forward-scattering technique) as a new method for quantitative investigation of the cellular aggregation process. In a previous published paper,¹¹ we made an accurate investigation of the angular spreading of light forward scattered at small angles by RBCs in suspensions, when all RBCs are singly dispersed.

2.

Theoretical Aspects

2.1.

Forward-Scattered Light Distribution

The theoretical approach using a red light at $632.8\mathrm{nm}$ is facilitated by the fact that RBCs have a very small absorption cross section¹² ${\sigma }_a$ versus scattering cross section¹¹ ${\sigma }_s$ ( ${\sigma }_a\approx 0.2\mu {\mathrm{m}}^2$ , ${\sigma }_s\approx 68.4\mu {\mathrm{m}}^2$ , ${\sigma }_a∕{\sigma }_s<{10}^{-2}$ ), and consequently, almost all photons interacting with RBCs are scattered. Since the mean size of RBCs is roughly ten times larger than the wavelength of the incident radiation and the relative refractive index of the internal $(n_i\approx 1.40)$ and external $(n_e\approx 1.33)$ media is close to one $(n_r=n_i∕n_e\approx 1.05)$ , the light is scattered in the forward direction in a narrow angular range, with the scattering process being extremely anisotropic. For these reasons, we measured the intensity of light scattered forward at small angles. Another reason, which is in fact more important, is that the photon flux can be described analytically in the small angle approximation³ by an effective phase function $f_{\mathit{eff}}(\tau ,\mu )$ (Henyey-Greenstein type),^{3, 11} if the two RBC parameters are known: the mean value of the scattering cross section ${\sigma }_s$ and the scattering anisotropy $g$ (the mean cosine of the scattering angle). This new effective phase function model^{3, 11} can be briefly described by the following equations:

Eq. 1

$${\Phi }_s={\varphi }_o(1-e^{-\tau }),\frac{{\varphi }_s(\tau ,\mu )}{{\varphi }_o}\approx (1-e^{-\tau })f_{\mathit{eff}}(\tau ,\mu ),$$

$$\tau =\mu d\approx \frac{{\sigma }_s}{v}Hd,$$

Eq. 2

$$f_{\mathit{eff}}(\tau ,\mu )\approx \frac{1}{2}\frac{1-g_{\mathit{eff}}^2\left(\tau \right)}{{[1-2g_{\mathit{eff}}\left(\tau \right)\mu +g_{\mathit{eff}}^2\left(\tau \right)]}^{3∕2}},$$

$$g_{\mathit{eff}}\left(\tau \right)=g^{\frac{(\tau -1)e^{\tau }+1}{e^{\tau }-\tau -1}},$$

where ${\Phi }_s$ is the total scattered photon flux, ${\varphi }_o$ is the incident photon flux, $\tau$ is the optical depth, ${\varphi }_s$ is the angular flux of the photons scattered in the direction given by the angle between the incident and the observation directions (axial symmetry setup), $\mu$ is the extinction coefficients ( $\mu =\cos \theta$ , where $\theta$ is the scattering polar angle), $d$ is the sample thickness, $v$ is the single RBC mean volume, and $H$ is the hematocrit (volumic fraction of RBCs).

In Eq. 1, the attenuated flux of the unscattered photons is given by ${\varphi }_o\cdot e^{-\tau }$ , and the term $(1-e^{-\tau })$ shows in which way the total scattered light increases with optical depth, while the term $f_{\mathrm{eff}}$ $(\tau ,\mu )$ describes the spatial distribution of the scattered light. The physical meaning of Eq. 2 is that the whole scattered volume exposed to the light beam can be modeled as a “macroscopic“ particle having the same Henyey-Greenstein type phase function as for a single RBC but with the scattering anisotropy $g$ replaced by $\tau$ -dependent effective anisotropy function $g_{\mathit{eff}}\left(\tau \right)$ . The model described by Eqs. 1, 2 is based on some approximations.^{3, 11} Its domain of validity is related to the attained desired accuracy and depends on the set of values for the $g$ , $\theta$ , and $\tau$ parameters. The domain of validity depends strongly on $\tau$ and much less on $\theta$ and $g$ . Further, it will be shown that the scattering anisotropy can be considered constant during the RBC aggregation process. Therefore, for a value of the scattering anisotropy $g=0.979$ (experimental value for RBCs)¹¹ and a chosen angular domain $\theta =1\mathrm{deg}\text{to}4\mathrm{deg}$ , the domain of optical depth will be $\tau ⩽10$ to ensure an accuracy higher than 95%.

The behavior for angular spreading of light scattered at small angles by a diluted suspension is revealed by three parameters: scattering anisotropy, scattering cross section, and concentration (expressed by hematocrit or scattering centers density). Based on Mie theory for equivolumetric spherical particles, the dependences of scattering anisotropy and scattering cross section by the radius of scattering centers can be obtained, as it is shown in Fig. 1 . Two main observations arise from these dependences: in the first approximation, the scattering anisotropy can be considered independent of the scattering centers radius, and the scattering cross section has a quadratic dependence by radius of scattering centers.

Fig. 1

Theoretical dependences by radius of scattering centers obtained from Mie theory for scattering anisotropy and scattering cross section. The scattering cross section is well-fitted by a parabolic function. All dependences start from $r_o=2.78\mu \mathrm{m}$ (the radius of a single RBC volume-equivalent sphere); below this value, there is no physical meaning for RBCs.

The RBC aggregation process leads to a decrease of scattering centers per volume unit due to the fact that each aggregate becomes a new scattering center. Generally, in RBC suspensions, during the aggregation process, there is a heterogeneous distribution of aggregates having different number of cells and, consequently, different size. Moreover, the aggregation process being a dynamic one, the mean size of aggregates, respectively the mean number of cells per aggregate, will increase in time. An other important fact is that in an RBC suspension, the hematocrit does not change during aggregation. As a result, the two ways to expres RBC suspension concentration, by hematocrit or scattering centers density, are not equivalent anymore.

3.

Experimental Materials and Methods

3.2.

Experimental Setup

The light source used was a $632.8\text{-}\mathrm{nm}$ linearly polarized continuous-wave HeNe laser (JDS Uniphase, Model 1125P) with a power of $5\mathrm{mW}$ in ${\mathrm{TEM}}_{\mathrm{oo}}$ mode. The beam diameter at $1∕e^2$ was $0.81\mathrm{mm}$ , and the beam divergence $1\mathrm{mrad}$ . The laser beam was directed onto a $200\text{-}\mu \mathrm{m}$ -thick optical glass cuvette ( $\sim 20\text{-}\mu \mathrm{l}$ volume) containing the sample of RBCs in suspension at rest. A complementary metal oxide aemiconductor (CMOS)³⁸ sensor monochrome camera (PixeLINK, Model PL-A741) with a resolution of $1280\times 1024\text{pixels}$ at $27\mathrm{fps}$ was used to detect the small-angle scattered light. For monitoring the RBC aggregation process, the data were transferred in real time to a computer that captured an image (in bitmap file format) at each $10\mathrm{s}$ during $10\min$ , for each sample. The schematic diagram of the experimental setup it is shown in Fig. 2 .

Fig. 2

Schematic diagram of the experimental setup used for angle-resolved measurements of light intensity scattered by RBC suspension at small angles.

4.

Results and Discussion

The angular-resolved measurements of light intensity scattered by an RBC suspension from a $200\text{-}\mu \mathrm{m}$ -thick optical glass cuvette, during $10\min$ of the RBC aggregation process were performed at 1 to 4 off-axis deg, at hematocrits in the range of $3.5\cdot {10}^{-2}⩽H⩽{10}^{-1}$ . In rest condition, the RBC aggregation process involves formation of 3-D large dense aggregates (containing thousands of RBCs) separated by plasma layers. This stage of aggregation is impossible in blood or suspended RBC layers as thin as those on the order of $25\mu \mathrm{m}\text{to}40\mu \mathrm{m}$ ,^{1, 19, 21, 22, 23} typical for cone-plate or plate-pate aggregometers. For this reason, we selected a $200\text{-}\mu \mathrm{m}$ -thick optical glass cuvette that is five or fifteen times thinner than the sample thickness of $1\mathrm{mm}$ used for backscattering^{13, 19} measurements or $3\mathrm{mm}$ for ultrasound³² studies, respectively. The upper hematocrit limit $(H={10}^{-1})$ is about five times smaller than the physiological value. It was chosen in order to keep the scattered light according to CMOS sensor sensibility, without exceeding the limit of the theory validity domain (see Sec. 2.1). Meanwhile, the lower hematocrit limit $(H=3.5\cdot {10}^{-2})$ was established in order to have enough cells closer to a realistic aggregation process. For the angular domain, the upper value was limited by the validity domain of the theory, and the lower value was chosen to avoid the transmitted (unscattered) photons. Due to the position of the laser beam on the cuvette, the very small significant sedimentation observed for upper hematocrits during measurement time (selected no longer than $10\min$ ) does not influence the recorded data.

Figure 3 shows a few examples of the averaged speckle images $(960\times 256\text{pixels})$ captured during the RBC aggregation process and the optical microscopy images for the lower and upper hematocrit limit at three moments in time. Looking at each series $(H=\mathrm{const.})$ of averaged speckle images, there are obvious changes in the angular spreading of light intensity from a distribution almost isotropic at $t=0\min$ to a high anisotropic one at $t=8\min$ . This behavior is due to changes in the RBC suspension during the aggregation process, changes that can be observed in the optical microscopy images corresponding to the averaged speckle images. It is easy to observe in which way, starting from RBCs singly dispersed, the formatted rouleaux are bound end-to-end and end-to-side into 3-D, large, dense aggregates with a globular shape. Optical microscopy images were done for a supplementary confirmation of the RBC aggregate formation. They are somewhat unclear due to $200\text{-}\mu \mathrm{m}$ depth of the optical glass cuvette but are similar to those reported in another study.²⁸ Unfortunately, this kind of unclear optical image acquired at a low rate $1∕\min$ cannot be used to make a computerized analysis for the distribution of the RBC population into aggregate size, similar to the $40\text{-}\mu \mathrm{m}$ -depth samples reported in the literature.²³

Fig. 3

The angular distribution ( $1\mathrm{deg}\text{to}4\mathrm{deg}$ ) of scattered light by aggregated RBCs in averaged speckle images (up—for each group of two images) and the associated optical microscopy images (down) during aggregation process at $H=3.5\cdot {10}^{-2}$ for: (a) $0\min$ , (b) $4\min$ , and (c) $8\min$ , and at $H={10}^{-1}$ for same moments of time in (d), (e), and (f).

From the images captured at each $10\mathrm{s}$ and converted to a numerical matrix (256 gray levels), series of angular profiles (one for each hematocrit) giving the angular spreading of the scattered light intensity have been obtained during the RBC aggregation process. Considering for the RBC aggregation process the working hypothesis described in Sec. 2.2, the angular dependence of quasi-ballistic scattered photons from Eq. 2 was used as a fit function of the experimental data and also to determine the effective anisotropy parameter $g_{\mathit{eff}}$ . The effective scattering anisotropy summarizes the information concerning the angular distribution of the scattered photons. Due to their graphic superposition, Fig. 4 shows only a few angular profiles of the scattered light intensity from the $H={10}^{-1}$ series.

Fig. 4

Angle-resolved measurements for the averaged intensity of light scattered at small angles by RBCs suspended in plasma at different time moments during their aggregation process for $H={10}^{-1}$ series, relatively well-fitted by a Henyey-Greenstein-type phase function, as in Eq. 2.

The decrease of the scattered light intensity with the increase of scattering angle $\theta$ , which is relatively well-fitted by a Henyey-Greenstein type phase function as in Eq. 2, depends parametrically on the sample hematocrit. For each investigated hematocrit, the scattering effective anisotropy behavior during the RBC aggregation process was obtained as a fit parameter of the angular-resolved measurements of scattered light intensity at different moments in time (Fig. 5 ). The $t$ -dependent scattering effective anisotropy $g_{\mathit{eff}}\left(t\right)$ presents two regions: the first is characterized by a gradient that increases monotonically with the hematocrit, while for the second, the gradient is almost the same and has a very small value for all hematocrits. Each curve starts at different anisotropy of the scattering process, lower for higher hematocrit and vice versa, but all tend to have relatively the same value (around 0.972). In other words, the curves reveal the information concerning the angular redistribution of light intensity scattered by an aggregated RBC suspension. Moreover, based on our theoretical approach, the theoretical $r$ -dependence of scattering effective anisotropy $g_{\mathit{eff}}\left(r\right)$ can be obtained from Eqs. 2, 4, 5 for each studied hematocrit. The relation between the optical depth ${\tau }_o$ and hematocrit $H$ given by the linear dependence ${\tau }_o=152\cdot H$ , obtained from Eq. 5 for $d=200\mu \mathrm{m}$ , $v=90\mu {\mathrm{m}}^3$ , and ${\sigma }_{so}=68.4\mu {\mathrm{m}}^2$ was used. These theoretical dependences of scattering effective anisotropy $g_{\mathit{eff}}\left(r\right)$ are given in Fig. 6 , where the curves start from $r_o=2.78\mu \mathrm{m}$ (the radius of a single RBC volume-equivalent sphere); below this value, there is no physical meaning for RBC aggregates. The theoretical dependence of $g_{\mathit{eff}}\left(r\right)$ and the experimental dependence of $g_{\mathit{eff}}\left(t\right)$ were used to find the change of the aggregate mean radius in time during the RBC aggregation process for each investigated hematocrit (Fig. 7 ).

Fig. 5

The time dependences of the effective scattering anisotropy parameter during the first $10\min$ of the RBC aggregation process for each studied hematocrit. Each series of data $(H=\mathrm{const.})$ was obtained as a fit parameter for angular profiles taken from angle-resolved measurements at each $10\mathrm{s}$ .

Fig. 6

The theoretical dependence of the effective scattering anisotropy parameter on one aggregate mean radius during the first $10\min$ of the RBC aggregation process for each studied hematocrit.

Fig. 7

The aggregate mean radius in time during the RBC aggregation process for each studied hematocrit. The values were obtained by comparing experimental $t$ -dependent scattering effective anisotropy $g_{\mathrm{eff}}\left(t\right)$ with theoretical $r$ -dependent scattering effective anisotropy $g_{\mathrm{eff}}\left(r\right)$ .

Another important parameter, the scattering cross section ${\sigma }_s$ , which is a measure of the scattering efficiency, can be obtained from Eq. 3 if the time dependence of the aggregate mean radius is known. Figure 8 shows the time dependence of the scattering cross section ${\sigma }_s$ during the RBC aggregation process at each studied hematocrit. The $r$ -dependent scattering cross section during the RBC aggregation process from Eq. 3 was compared with the Mie theory prediction. This was done by using ${\sigma }_{s\mathit{Mie}}\left(r\right)={\sigma }_{so\mathit{Mie}}{(r∕r_o)}^2$ as a fitting function for the scattering cross section obtained from Mie theory. Here, ${\sigma }_{so\mathit{Mie}}$ is the scattering cross section at $r=r_o$ ; in other words, ${\sigma }_{\mathit{so}\mathit{Mie}}$ represents the scattering cross section for a single particle. Figure 9 gives the resulting data of the scattering cross section ${\sigma }_s$ from all investigated hematocrits together with the Mie theory prediction. The values of scattering cross section obtained from experimental data for any hematocrit follow very well a parabolic function. At small aggregate mean radius, up to a value of $9\mu \mathrm{m}$ , our results are in agreement with Mie theory, but after that, the differences become significant. In Mie theory, the particles are spherical and identical, while the RBC aggregate particles are not identical and are far from spherical shape (see microscopy images in Fig. 3). For these reasons, our results for a radius above $9\mu \mathrm{m}$ are not in agreement with Mie theory. Moreover, by fitting the scattering cross section obtained from Mie theory, we can obtain ${\sigma }_{so\mathit{Mie}}=49.5\mu {\mathrm{m}}^2$ at $r=r_o$ , which is different from the experimental value¹¹ ${\sigma }_{so}=68.4\mu {\mathrm{m}}^2$ used for calculating ${\sigma }_s\left(r\right)$ . This difference is also a reason for the discrepancy between our results and the Mie theory prediction.

Fig. 8

The time dependence of scattering cross section ${\sigma }_s$ during the RBC aggregation process for each studied hematocrit. The dependences were obtained based on the previously determined aggregate mean radius.

Fig. 9

The $r$ -dependent scattering cross section during the RBC aggregation process obtained for all studied hematocrits and their parabolic fitting function. Due to graph superposition, three sets of our calculated data were shifted with: (1) $+600\left[\mu {\mathrm{m}}^2\right]$ , (2) $+400\left[\mu {\mathrm{m}}^2\right]$ , and (3) $+200\left[\mu {\mathrm{m}}^2\right]$ . Comparatively, the Mie theory prediction is presented for the $r$ -dependent scattering cross section and its parabolic fit.

Based on our working hypothesis, which considers a spherical shape of RBC aggregates and the same number of cell per aggregate, $N_a$ can be determined. At each moment in time, from the previously determined aggregate mean radius, taking into consideration $v=90\mu {\mathrm{m}}^3$ , the number of cells per aggregate was calculated (Fig. 10 ). From the time dependence of the number of cells per aggregate $N_a$ (Fig. 10), we can define two different behavior regions. One of them, which starts with the first moment of the RBC aggregation process and lasts to about $3\min$ , is characterized by a slow increase of the $N_a$ parameter. This behavior can be assigned to the first phase of the RBC aggregation process, where the aggregate is represented especially by the formation of rouleaux with different lengths. For this first region, our working hypothesis uses rough approximations. The other region is characterized by a high increase of the $N_a$ parameter. This region reveals the second phase of the RBC aggregation process characterized by the formation of 3-D, large, dense aggregates with a globular shape, which are better approximated by our working hypothesis. There is a third region missing in which the $N_a$ parameter tends to a plateau, due to our $10\text{-}\min$ experimental limitation setup.

Fig. 10

The time dependence of the number of cells per aggregate $N_a$ obtained from the previously determined aggregate mean radius, taking into consideration $v=90\mu {\mathrm{m}}^3$ .

5.

Conclusions

Our work was focused on following the main aspects of a quantitative investigation of the RBC aggregation process by light a scattering techniques at small angles:

RBCs in suspension at four different hematocrits were investigated, and the $t$ -dependent scattering effective anisotropy $g_{\mathit{eff}}\left(t\right)$ was obtained from experimental measurement based on theoretical approach (new effective phase function model and our working hypothesis). This was compared with the theoretical prediction of $r$ -dependent scattering effective anisotropy $g_{\mathit{eff}}\left(r\right)$ to find out the aggregate mean radius that quantitatively characterizes the RBC aggregation process. Further, the dependences of the mean scattering cross section in time and by the aggregate mean radius were determined. The dependences of the mean scattering cross section by the aggregate mean radius were compared with the prediction of Mie theory for equivolumetric spherical particles. Last, the time dependence of the mean number of cells per aggregate was calculated. This parameter can be used for a quantitative characterization of the RBC aggregate formation process as a two-phase process.

The potential of the forward light-scattered techniques can be revealed even with our simple theoretical approach. Also, this can be achieved in the absence of a strong theoretical model for the RBC aggregation process and of a complementary one related to the scattered light intensity by RBCs during the aggregation process.

One can conclude that the light-scattering technique is a valuable alternative to classical aggregometric methods based on turbidimetric or backscattering measurements and can be used successfully in a quantitative investigation of the cellular aggregation process, due to its high sensitivity. The best results are obtained by detecting the off-axis scattered light at angles as small as possible. These results represent only the first several steps in a research field with a very high potential. Future work will be focused on the one hand, on the systematic investigation of several cases of relevant pathologies, and on the other hand, on the investigation of other cells with a relevant biomedical role, such as platelets. The final goal is to develop new, sensitive, and minimally invasive techniques for biological and medical diagnosis.