1.

Introduction

Aggregation of red blood cells (RBCs) results in the formation of linear or branched structures called rouleaux. RBC aggregation is dependent both on plasma and cellular factors. The most important physiological consequence of the RBC aggregation is that the adhering of the cells significantly affects the rheological properties of blood.^1,2 Increased RBC aggregation has been observed in the pathophysiology of numerous diseases with circulatory disorders, such as cardiovascular diseases, inflammation, diabetes, hyperlipidemia, sickle cell disease, thalassemia, and trauma.³ Because of this, many techniques were utilized to fully elucidate the specific mechanisms involved in RBC aggregation. Analysis of the light scattering by RBC aggregates is one of the most popular.

Various theoretical approaches have been applied to describe light scattering by RBC aggregates.^4–7 The optical properties of individual RBCs that have been well recognized^8–15 are usually a starting point in the study. Tsinopoulos et al.⁴ calculated light scattering by rouleaux (up to 8 RBCs per rouleaux) giving the differential scattering cross section of the units. Shvartsman and Fine⁵ described an effect of RBC aggregation on the transparency of blood using anomalous diffraction theory, which was previously utilized by Twersky⁹ in the study of scattering by biological suspensions. He et al,⁷ using finite-difference time-domain method, calculated scattering probability for multiple RBCs that may be a model of rouleaux. Additionally, the data obtained in the study of the optical properties of blood¹¹ allowed interpretation of optical coherence tomography signals from sedimenting RBCs.¹⁶ Pop and Neamtu⁶ studied the diffuse component of light scattered in suspension of aggregating RBCs. Using Mie theory, they determined the number of RBCs per aggregate as functions of time and hematocrit.

RBC aggregation causes an increase in transmitted light intensity, whereas the intensity of the reflected light decreases in the course of aggregation.¹⁷ The extensive study in the area has been performed and on this basis, study instruments for the measurement of RBC aggregation have been developed e.g., Myrenne RBC aggregometer,¹⁸ laser-assisted optical rotational cell analyzer,¹⁹ RheoScan-A,²⁰ and Sefam erythroaggregometer.^21–23 The parameters obtained from the instruments reflect the kinetics of the RBC aggregation. However, those instruments do not allow estimation of the number of RBCs per aggregate.²⁴

To explain the RBC aggregation, two opposite models have been developed. Within the bridging model, bridging forces due to plasma proteins adsorbed on the RBC membrane cause the adhering of the cells. In contrast, according to the depletion model, depletion forces due to decreasing of the protein concentration in the vicinity of the cell membrane contribute to the aggregation. Recently, both the theoretical study²⁵ as well as the experimental investigations²⁶ seem to confirm the depletion model of the aggregation. The nature and magnitude of the interactive forces between the RBCs manifests in the kinetics of the RBC aggregation. Many techniques for the assessment of the kinetics of RBC aggregation have been developed. The quantitative methods of description of these kinetics are particularly needed in clinical diagnostics.

The kinetics of rouleaux formation has been studied both theoretically and experimentally. One of the first approaches to the study of RBC aggregation was done by Ponder.²⁷ Using a microscopic technique, he determined the number of cells in rouleaux that appear at a given time in a dilute suspension of the cells at various temperatures. Ponder has shown that the Smoluchowski theory of colloid aggregation describes the size distribution of the rouleaux very well. Furthermore, Ponder argued that for small aggregates, the mean rouleaux size increases linearly with time. The investigations developed by Kernick et al.²⁸ for various temperatures and hematocrits have shown that after a linear increase in the rouleaux length a nonlinear increase appears. They have confirmed Ponder’s finding that Smoluchowski theory is adequate for small rouleaux. Samsel and Perelson²⁹ formulated a theory of the kinetics of rouleaux formation taking into account both irreversible as well as reversible reactions. Making a comparison between the theory and results obtained by Kernick et al.²⁸ Samsel and Perelson²⁹ argued that the kinetics of rouleaux formation is a result of reversible reactions. A new insight in the problem was done by Barshtein et al.³⁰ Using microscopic techniques, they investigated the rouleaux formation in a narrow gap flow chamber at small hematocrits. Especially, they have shown detailed results related to the kinetics of spontaneous rouleaux formation. They confirmed that in the initial stage of the process, the average aggregate growth is a linear function of time. However, after the initial stage, a critical region of slow growth was observed and the aggregate size strongly increased with time. Barshtein et al.³⁰ have shown that in the initial stage, the irreversible reactions of polymerization and condensation take place. The study of the kinetics performed by Shiga et al.³¹ using a rheoscope and image analyzer has shown that at the shear rate $7.5{\mathrm{s}}^{-1}$, the rate of rouleaux formation is proportional to hematocrit within the range from 0.1% to 0.6%. Moreover, they have shown an increase in the rate of rouleaux formation with the depth of the sample.

In this paper, the kinetics of the rouleaux formation was investigated by light scattering. Diluted suspensions of RBCs in autologous plasma were obtained from the blood of healthy donors. The intensity of the coherent component of light was measured at various hematocrits as a function of time and sample thickness. From the experimental data, the scattering coefficient of RBC suspensions was determined. The anomalous diffraction theory was used to calculate the scattering cross section of the rouleaux and the average number of erythrocytes per rouleaux. Finally, the kinetics of the rouleaux formation is described.

2.

Materials and Methods

2.2.

Coherent Component of Light Measurements

The experimental setup for this study is schematically shown in Fig. 1. The He–Ne laser at a wavelength of 632.8 nm is used as a light source. The laser light is collimated by the lenses ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$, and selected by the pinhole ${\mathrm{P}}_1$, which plays the part of a spatial filter. The beam of light is passing through the aperture ${\mathrm{A}}_1$. Next, the light is scattered in the sample S in the wedge-shaped container, which is moved on its horizontal axis by motorized stage MS. The coherent component of light is selected using lens ${\mathrm{L}}_3$ and aperture ${\mathrm{A}}_2$. The photomultiplier Ph is used as detector of intensity of the coherent component of light, and the signal is amplified and stored in a computer. To reduce the noise, the chopper Ch is used in conjunction with the amplifier (Am). During each scanning cycle, the data are collected for sample thicknesses from 0.2 to 0.4 mm.

Fig. 1

Scheme of the experimental setup: HeNe—laser, Ch—chopper, F—filter, ${\mathrm{L}}_1\text{-}{\mathrm{L}}_3$—lenses, ${\mathrm{A}}_1\text{-}{\mathrm{A}}_3$—apertures, S—sample, Ph—photomultiplier, Am—amplifier, PC—computer, MS—motorized stage.

2.3.

Theoretical Background of Light Scattering by Red Blood Cell Suspension

In the suspension of RBCs illuminated by He–Ne laser light at the wavelength of 632.8 nm, the scattering is dominant over absorption.⁸ According to the theory of absorption and scattering given by Twersky, the total transmittance is a sum of the coherent component and incoherent component.⁹ The coherent part of the transmittance $C$, called the collimated transmittance, has the form

Eq. (1)

$$C(t)=\exp [-{\mu }_{\mathrm{s}}(t)d],$$

where ${\mu }_{\mathrm{s}}(t)$ is the macroscopic scattering coefficient of the suspension and $d$ is the thickness of the sample. Due to the aggregation process, the scattering coefficient ${\mu }_{\mathrm{s}}(t)$ is a function of time. This macroscopic coefficient is related with the cross section of the rouleaux ${\sigma }_{\mathrm{s}}(t)$ by the expression

Eq. (2)

$${\mu }_{\mathrm{s}}(t)=\frac{{\sigma }_{\mathrm{s}}(t)H(1-H)}{V_{\mathrm{u}}(t)},$$

where $V_{\mathrm{u}}(t)$ is the volume of the rouleaux and $H$ is the hematocrit.

The relative index of refraction (refractive index of RBC per refractive index of the plasma) at the wavelength of the incident radiation 632.8 nm is about 1.04. We assume that RBCs as well as rouleaux can be modeled as spherical particles. Furthermore, we assume that rouleaux have the same relative index of refraction as the index for RBC. Indeed, we assume that the rouleaux consist of tightly packed cells. Under these assumptions, the anomalous diffraction theory can be used to estimate the scattering cross section ${\sigma }_{\mathrm{s}}(t)$ of the units. The scattering cross section ${\sigma }_{\mathrm{s}}(t)$ of the units is equal to³²

Eq. (3)

$${\sigma }_{\mathrm{s}}(t)=\pi r_{\mathrm{u}}^2(t)\{2-\frac{4}{\rho (t)}\sin \rho (t)+\frac{4}{{\rho }^2(t)}[1-\cos \rho (t)]\},$$

where

Eq. (4)

$$\rho (t)=\frac{4\pi r_{\mathrm{u}}(t)}{\lambda }(n-1).$$

In the above equation, $r_{\mathrm{u}}(t)$ is the radius of the unit at the time $t$, $n$ is the relative index of refraction and $\lambda$ is the wavelength of the radiation in the medium. Figure 2(a) shows the scattering cross section as a function of the number of the cells per rouleau. The parameters shown in Fig. 2 were calculated at the wavelength $\lambda =632.8\mathrm{nm}$, relative index of refraction $n=1.04$, and the volume of single cell $V=90\mu {\mathrm{m}}^3$ which corresponds to the radius of the sphere $r=2.78\mu \mathrm{m}$. The scattering coefficient was calculated at the hematocrit of the suspension $H=0.01$. The functions were calculated in the interval $1\le N\le 25$. The anomalous diffraction theory gives the scattering cross section of the single cell ${\sigma }_{\mathrm{s}}=65.58\mu {\mathrm{m}}^2$, which corresponds to that obtained by Steinke and Shepherd from Mie theory ${\sigma }_{\mathrm{s}}=66.62\mu {\mathrm{m}}^2$.⁸ The cross section nonlinearly increases as the number of the cells per rouleaux increases. The scattering coefficient first decreases as the number of the cells per rouleaux increases and at $15>N$ takes almost constant values. This means that in this case the inverse scattering problem can be resolved at $N<15$. Another restriction for the inverse scattering problem is the combination of the number of the cells per rouleaux, hematocrit of the suspension, and the thickness of the sample. Especially at large values of ${\mu }_{\mathrm{s}}d$, the diffuse component of light increases and it may be difficult to separate the coherent component from the diffuse component.

Fig. 2

The scattering cross section ${\sigma }_{\mathrm{s}}$ (a) and scattering coefficient ${\mu }_{\mathrm{s}}$ (b) as a function of the number of cells per rouleaux.

3.

Results

The effect of rouleaux formation on the intensity of the coherent component for four hematocrits is depicted in Fig. 3. At every hematocrit, the data for five scanning cycles are presented, and the time for the beginning of each cycle is denoted. The intensity during each scanning cycle is a function of the thickness of the sample and the time. The increase of the intensity in time is a result of the aggregation process. The slope of the intensity in the logarithmic scale systematically decreases with time. The results show that the changes of the intensity depend on the hematocrit of the samples. At the hematocrit 0.01, the temporal increase of the intensity is relatively small as compared to the increase at hematocrit 0.04. Furthermore, at higher hematocrits after a fast increase of the intensity at the early stages of the aggregation, much slower increase of the intensity can be observed.

Fig. 3

The dependence of the intensity of the coherent component $I_{\mathrm{c}}$ on the thickness $d$ of the sample and time $t$, for hematocrit (a) 0.01, (b) 0.02, (c) 0.03 and (d) 0.04. Each scanning cycle is displayed with the time that the cycle began.

The slope of the intensity versus sample thickness is used to determine the scattering coefficient of the suspensions. To find the effect of the sample thickness on the aggregation process, the slope is analyzed in two regions of the sample thickness: $200\mu \mathrm{m}<d<300\mu \mathrm{m}$ and $300\mu \mathrm{m}<d<400\mu \mathrm{m}$. The intensity at various sample thicknesses at given scanning cycles was measured in different times. We have assumed that within the time necessary to performance one scanning cycle the state of the aggregation of the cells does not substantially change.

Figure 4 shows the effect of hematocrit and sample thickness on the macroscopic scattering coefficient ${\mu }_{\mathrm{s}}$ ($t$) of the RBC suspension. The decrease of the scattering coefficient with time manifests the aggregation of the cells. Figure 4 shows a little difference in the temporal behavior of the scattering coefficient at the two intervals of sample thickness. For the higher hematocrit, the scattering coefficient was not determined for initial times at the sample thickness $300\mu \mathrm{m}<d<400\mu \mathrm{m}$, because at this thickness, the intensity of the coherent component takes very small values.

Fig. 4

The dependence of the scattering coefficient ${\mu }_{\mathrm{s}}$ on time at the thickness interval $200\mu \mathrm{m}<d<300\mu \mathrm{m}$ (empty squares) and $300\mu \mathrm{m}<d<400\mu \mathrm{m}$ (filled squares) for hematocrit (a) 0.01, (b) 0.02, (c) 0.03 and (d) 0.04.

The number of cells per aggregate at a given instant of time was numerically determined from the experimental data of the scattering coefficient using Eqs. (2) and (3). The number of cells per aggregate is shown in Fig. 5. For every hematocrit, 10 samples were taken from different donors. Thus, the error bars showing standard deviations reflect the subject to subject variability in the number of cells per aggregate. The time course of the aggregate formation has a common feature independent on the hematocrit of the samples. In the first stage of the process, the linear increase in the number of cells per aggregate takes place. Next, a critical region appears when aggregates take a constant size of about three erythrocytes. Finally, a further increase in the number of the cells can be observed. Critical size is found as a median between the last point of initial linear increase and the first point of further growth.

Fig. 5

The number of red blood cells (RBCs) per aggregate $N$ as a function of time at hematocrit (a) 0.01, (b) 0.02, (c) 0.03, and (d) 0.04 obtained at the sample thickness $200\mu \mathrm{m}<d<300\mu \mathrm{m}$.

Figure 6 shows the aggregate size in the critical region separating two regions of aggregate growth for the investigated hematocrit. It is seen that, independent of the hematocrit, the above defined critical size is equal to $3\pm 0.5$ erythrocytes.

Fig. 6

The critical size $N_{\mathrm{cr}}$ as a function of hematocrit obtained at the sample thickness $200\mu \mathrm{m}<d<300\mu \mathrm{m}$.

Figure 7 shows an effect of the sample thickness on the aggregation process. In this figure, the means of the number of cells per aggregate are shown and the standard deviation is omitted for clarity. The results show that the aggregation rate is a little higher at the sample thickness $300\mu \mathrm{m}<d<400\mu \mathrm{m}$ as compared to the rate at the thickness $200\mu \mathrm{m}<d<300\mu \mathrm{m}$.

Fig. 7

An effect of sample thickness on the kinetics of the rouleaux formation. The number of RBCs per aggregate $N$ as a function of time at hematocrit (a) 0.01, (b) 0.015, (c) 0.02 obtained at the sample thickness $200\mu \mathrm{m}<d<300\mu \mathrm{m}$ (empty squares) and $300\mu \mathrm{m}<d<400\mu \mathrm{m}$ (filled squares).

To show the effect of hematocrit on the kinetics of the rouleaux formation, the aggregation rate was calculated for the first stage of the process. The aggregation rate is defined as $\mathrm{\Delta }N/\mathrm{\Delta }t$, where $N$ is the number of RBCs per rouleaux and $t$ denotes the time. Figure 8 shows the aggregation rate in the first stage of the process as a function of hematocrit of the samples at the two sample thickness intervals.

Fig. 8

The dependency of the rate of aggregation $\mathrm{\Delta }N/\mathrm{\Delta }t$ on hematocrit in intervals $200\mu \mathrm{m}<d<300\mu \mathrm{m}$ (empty squares) and $300\mu \mathrm{m}<d<400\mu \mathrm{m}$ (filled squares).

The results obtained in this paper can be compared with the results obtained with the use of the microscopic technique by Kernick et al.²⁸ In the comparison shown in Fig. 9, we have taken into account the number of RBCs per rouleaux that appears at 600 s at various hematocrits of the samples. A comparison of the results shows that the numbers of cells per rouleaux obtained from the scattering technique are in quite good agreement with ones obtained from the microscopic technique. Especially, it can be seen that the number of cells per rouleaux first increases with the increase of the hematocrit, then for a short hematocrit range takes a constant critical size and finally increases more.

Fig. 9

Dependence of the number of the cells per rouleaux $N$ at $t=600\mathrm{s}$ on hematocrit. Empty squares represent results from depth interval $200\mu \mathrm{m}<d<300\mu \mathrm{m}$, filled squares represent results from interval $300\mu \mathrm{m}<d<400\mu \mathrm{m}$ and stars represent the result obtained by Kernick et al.²⁸

Figure 10 schematically shows isolines of equal numbers of RBCs per rouleaux that appear at a given time at various hematocrits. This figure is, indeed, a phase diagram of the RBC suspension. The lines were found as an estimation of the experimental data. The low coherent part of the transmittance at the beginning of the process for hematocrit above 0.03 does not allow a good description of these lines in the right bottom corner of this figure. The line which represents the dependence of the critical size ($N=3$) on hematocrit divides the diagram into two parts. Above the critical line, the density of isolines is much larger than below. This shows that up to three RBC per rouleaux growth of rouleaux are slower than they are later. Below the critical line, linear growth of the rouleaux takes place. A critical region divides the plot into regions of slower and faster aggregate growth. Thus, any vertical line in this diagram represents the growth of the aggregates with time at a given hematocrit, allowing the observation of the kinetics of their formation. On the other hand, any horizontal line in the diagram represents observation of the RBC suspensions in a fixed time at various hematocrits. In this way, the results shown in Fig. 9 correspond to the line $t=600\mathrm{s}$ in Fig. 10. Both the results obtained from the scattering experiment as well as the ones obtained by Kernick et al. shown in Fig. 9 reflect the characteristic three regions of the phase diagram very well. In this case, the growth of aggregate size with increasing hematocrit for given time as well as with increasing time for given hematocrit is intermediated by critical region.

Fig. 10

Isolines of the number of RBCs per averaged rouleaux for the sample thickness $200\mu \mathrm{m}<d<300\mu \mathrm{m}$. The solid lines are incremented by 0.1 RBC. Four dashed lines represent the isolines for 2, 3, 5, and 10 erythrocytes.

4.

Discussion

Although the study of the kinetics of RBC aggregate formation has a long history, there is no commonly accepted theory that describes the process. The results shown in this paper, together with others, support the thesis that in the early stage of rouleaux formation, the number of RBCs per rouleaux grows linearly in time. Duration of the stage of linear growth can be observed up to the time when aggregates attain the critical region. The linear growth was first postulated by Ponder,²⁷ where he showed that Smoluchowski’s theory of irreversible coagulation of colloids permits one to predict the distribution of the number of the cells per relatively short rouleaux. Kernick et al.²⁸ have found that the kinetics of the aggregation becomes nonlinear with time, but confirmed Ponder’s conclusions about the size distribution for a small rouleaux length. The detailed experimental evidence of the linear growth was presented by Barshtein et al.³⁰ and they have shown that the polymer formation model developed by Samsel and Perelson²⁹ can be adopted to explain the early stage of the RBC aggregation. Furthermore, they have shown that the process is performed by polymerization and condensation at equal constants of the interactions. The results presented in this paper support the model of the early spontaneous aggregation of RBCs. Furthermore, our results confirm only the thesis that the early stage of linear growth involves aggregation of a few cells.

One of the most important features of the rouleaux formation kinetics is the presence of the critical region that appears when rouleaux attain a critical size of about three erythrocytes. The critical region is followed by the region of further growth in size. The phenomenon was observed for the first time by Barshtein et al.³⁰ and the dependence of the critical time on hematocrit was found. Thus, from one side, the results presented in this paper confirm the findings of Barshtein et al.³⁰ and, from the other side, explain data obtained by Kernick et al.²⁸ Among the studies of the kinetics performed by Samsel and Perelson, the critical region was not found but it was shown that reversible reactions may lead to a state of saturation. Taking into account the reversible reactions, Samsel and Perelson²⁹ argued that data obtained by Kernick et al.²⁸ can be explained within the theory. In opposition to the conclusions drawn from the reversible reaction theory, we have shown that after the critical region, further growth of the aggregate size appeared. Further growth of aggregates after deviation from linearity was also observed by Barshtein et al.³⁰ The conclusion of Samsel and Perelson²⁹ comes from the fact that due to the limited number of data points and the short times of observations (up to 1 h) performed by Kernick et al.,²⁸ the critical region was recognized as a steady state. An extrapolation of results from Fig. 10 shows that at the $H=0.005$, the critical region is reached after about 50 min. A tentative explanation of the problem seems to be as follows: the initial linear growth of the rouleaux can be described by the polymer formation model. This model works up until the mean rouleaux length does not exceed the critical size. Formation of the small rouleaux undergo via face-to-face interactions because the reactive area of the wall of the existing rouleaux is much less than the area of their face. When the mean rouleaux length exceeds the critical size, the reactive area of the wall is comparable to or higher than the area of the face which results in enhanced aggregation. Thus, the results shown in this paper seem to support the thesis that after a linear growth of the rouleaux, the critical region followed by further growth of the aggregates appears.

As was mentioned above, we have found that the critical time is strongly dependent on hematocrit. There is a great difference in the critical time measured by Barshtein et al.³⁰ and the time when the critical regions begins in this study. The values of the time obtained in this study are about 10 times larger those that obtained by Barshtein et al.³⁰ Note that Barshtein et al.³⁰ measured the aggregation in a flow chamber having a $40\text{-}\mu \mathrm{m}$ gap while our measurements were performed in a vertically oriented container having a thickness between 200 and $400\mu \mathrm{m}$.

Finally, let us consider the comparison of the result obtained in this study with the results presented by Kernick et al.²⁸ It was shown for the first time, to our knowledge, that results obtained with the light scattering technique are in very good agreement with results obtained from the microscopic technique. Thus, this unexpected result of the study shows that the proposed light scattering technique may be promising in the study of RBC aggregation.

5.

Conclusion

In this study, the light scattering technique was successfully used to investigate the RBC aggregate suspension, which was mainly studied using a microscopic technique up until now. It was shown that for these suspensions, the anomalous diffraction theory can be used. Using a theoretical expression for the scattering coefficient, the scattering cross section and the size of the aggregates was obtained. The evolution in time of this size was observed. Three phases of this evolution were found. Initially, the linear increase of their size can be observed. Next, when aggregates gain an average size of about three erythrocytes, the critical region appears. Finally, further growth of their size takes place. Furthermore, the investigations show the dependence of the observed phenomena on hematocrit and sample thickness. The phases of aggregate evolution were detailed and described with the use of a phase diagram. This new approach indicates the need for a more detailed model of RBC aggregation in the early stage.