_{a}, anisotropy factor g, and effective scattering coefficient μ

^{′}

_{s}—are determined for human red blood cells of hematocrit 42.1% dependent on the shear rate in the wavelength range 250 to 1100 nm. Integrating sphere measurements of light transmittance and reflectance in combination with inverse Monte-Carlo simulation are carried out for different wall shear rates between 0 and 1000 s

^{-1}. Randomly oriented cells show maximal μ

_{a}, μ

_{s}, and μ

^{′}

_{s}values. Cell alignment and elongation, as well as the Fahraeus effect at increasing shear rates, lead to an asymptotical decrease of these values. The anisotropy factor shows this behavior only below 600 nm, dependent on absorption; above 600 nm, g is almost independent of shear rate. The decrease of μ

^{′}

_{s}is inversely correlated with the hemoglobin absorption. Compared to randomly oriented cells, aggregation reduces all parameters by a different degree, depending on the hemoglobin absorption. It is possible to evaluate the influence of collective scattering phenomena, the absorption within the cell, and the cell shape.

## 1.

## Introduction

The development of spectroscopic methods for blood analysis requires knowledge of the light-scattering and absorption properties of human blood. It is also important for many diagnostic and therapeutic applications in laser medicine, hematology, and medical routine diagnosis.

According to the radiation transport theory, the optical properties of blood can be described by the intrinsic optical parameters: absorption coefficient
${\mu}_{a}$
, scattering coefficient
${\mu}_{s}$
, and anisotropy factor
$g$
, together with an appropriate phase function. Various approaches have been taken to determine the optical properties of blood.
^{1, 2, 3, 4, 5, 6, 7, 8, 9, 10} Due to the high optical density of blood, especially at physiological concentrations, it has not been possible in most cases to determine the parameters
${\mu}_{s}$
and
$g$
separately. Only the effective scattering coefficient
${\mu}_{s}^{\prime}={\mu}_{s}$
$(1-g)$
could be determined for single wavelengths or small spectral ranges where the absorption of hemoglobin is low. Using the double integrating sphere technique combined with an inverse Monte Carlo simulation (iMCS),
^{1, 4, 5, 6} all three optical parameters of diluted, as well as undiluted, flowing blood could be determined independently in a spectral range of 250 to
$1100\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
, including the spectral areas of high hemoglobin absorption. It is known that the optical behavior of blood depends on various physiological parameters such as oxygen saturation, pH, aggregation, and hematocrit.
^{11, 12, 13, 14, 15} One important parameter is the flow-induced shear force that can cause characteristic changes in cell morphology and organization of red blood cells.
^{3, 16, 17, 18, 19, 20, 21, 22, 23, 24} The position-dependent shear force
$\tau \left(x\right)$
can be described by the shear rate
$\gamma $
, which is defined as the velocity gradient in the direction normal to the flow and independent of the blood viscosity
$\eta $
.

In blood vessels under conditions of constant laminar flow, there is a radially dependent shear rate with a minimum at the center and a maximum at the wall of the vessel.^{25} Flow-dependent changes in the optical properties have to be considered for numerous applications, such as blood oxymetry, determination of the Hct in cardiopulmonary systems, laser Doppler blood flowmetry, and diagnostic measurements of the erythrocyte aggregation/disaggregation behavior.
^{26, 27, 28, 29, 30}

The characteristic changes in cell morphology and organization of red blood cells (RBCs) can be linked to various ranges of shear rate.^{23, 24} At shear rates below
$60\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$
or at flow stop, RBCs show aggregation-forming rouleaux-shaped aggregates, which disintegrate at higher shear rates. This aggregation is derived from plasma proteins. Washed RBCs form no aggregates and are randomly oriented at flow stop. RBCs in plasma show random orientation at shear rates between 100 to
$200\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$
when the aggregates are disintegrated. At shear rates exceeding
$400\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$
, RBCs are aligned, and finally, in the range above
$1000\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$
, they are deformed into ellipsoid with the long axis running parallel to the flow.^{18, 19}

In vessels of diameters below
$300\phantom{\rule{0.3em}{0ex}}\mathrm{\mu}\mathrm{m}$
, the RBCs tend to move near the center of the tube (transverse migration). This leads to a radial RBC concentration profile with a thin cell-free plasma layer close to the wall of the tube. Due to the velocity profile, a difference occurs between the mean velocities of RBCs and the suspending fluid, leading to a dynamic hematocrit
${\mathrm{Hct}}_{\mathrm{D}}$
within the tube, which is lower than the static hematocrit
${\mathrm{Hct}}_{\mathrm{S}}$
(Fahraeus effect).^{25, 31}

Gaehtgens, Albrechts, and Kreutz^{32} found that the Fahraeus effect decreases with decreasing flow. In experiments using flow-through cuvettes, the effects of orientation, elongation, and transversal migration occur simultaneously. To exclude the influence of cell deformation in experiments, the RBCs can be fixed with glutaraldehyde, and substituting the blood plasma for saline solution can prevent cell aggregation.

The aim of the present study is to investigate the influence of shear rate on the optical properties of human blood at physiological Hct in the wavelength range 250 to $1100\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ . To do this, the optical parameters ${\mu}_{a}$ , ${\mu}_{s}$ , and $g$ were determined for human RBCs dependent on the shear rate for six different values from 0 to $1000\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$ . To facilitate comparison with data from the literature, ${\mu}_{s}^{\prime}$ was also calculated. In the main investigations, native RBCs were suspended in saline solution to avoid aggregation. Cell aggregation at flow stop was investigated in additional experiments where the RBCs were suspended in plasma. Fixed cells were used to investigate the influence of cell elongation, and the influence of the Hct was investigated in selected experiments using diluted RBC suspensions of Hct 8.6%.

All shear rates given in this work are the wall shear rates at the cuvette window. Given a cuvette with a rectangular cross section (width
$b\gg $
height
$d$
), the shear rate can be estimated for Newtonian fluids according to Eq. 2,^{20}

The fact that blood is a non-Newtonian fluid and therefore exhibits higher wall shear rates has to be considered when comparing absolute values.^{33} In the present experiments, the cuvette thickness is constant and therefore the mean flow velocity can be recalculated and used as a variable parameter instead of the shear rate. The wall shear rate was calculated to compare the data in this study with data from the literature where the same formula was used.^{20}

The mean velocity ${v}_{m}$ can be easily determined from the volume flow $\stackrel{\u2022}{V}=dV\u2215dt$ according to ${v}_{\mathrm{m}}=\stackrel{\u2022}{V}\u2215\mathrm{bd}$ .

## 2.

## Materials and Methods

## 2.1.

### Blood Preparation

To measure native RBCs in saline solution, fresh human erythrocytes from healthy blood donors were centrifuged three times $\left(1000\phantom{\rule{0.3em}{0ex}}\mathrm{g}\right)$ and washed with isotonic phosphate buffer ( $300\phantom{\rule{0.3em}{0ex}}\mathrm{mosmol}\u2215\mathrm{L}$ , pH 7.4) to remove the blood plasma and free hemoglobin. This procedure does not influence the biological function of the RBCs, but inhibits the formation of aggregates. The blood concentration was varied by diluting the sample with buffer. Two different hematocrit values, 8.6 and 42.1%, were investigated. The suspensions with Hct 42.1% of native intact RBCs are referred to as “RBC in $\mathrm{NaCl}$ ” in the diagrams shown in the chapter results. To investigate the influence of flow-induced changes on the cell shape, the RBCs of one blood sample were fixed by treatment with glutaraldehyde (0.15%). The treatment of the cell membrane with glutaraldehyde fixed the red blood cell in the normal biconcave shape. This suspension of non-native RBCs is called “RBC fixed.” To study the aggregation phenomena, RBCs of another sample were suspended in human plasma by replacing the supernatant with cell-free blood plasma after the last washing procedure. These suspensions contained native intact RBCs and plasma proteins, giving the possibility of forming aggregates at low shear rates and are called “RBC plasma.”

The hematocrit was determined using a red blood cell counter (Micros 60 OT 18, ABX Diagnostics, Montpellier, France). All samples were oxygenated in excess of 98%. Lower oxygen saturations may lead to drastically different results in the optical parameters.^{11} The oxygen saturation was determined with a blood gas analyzer (OPTI Care, AVL Medizintechnik GmbH, Bad Homburg). A miniaturized blood circulation setup was used with a roller pump (Sorin Group, Germany) and a blood reservoir, which was constantly aerated with a gas mixture of
${\mathrm{O}}_{2}$
,
${\mathrm{N}}_{2}$
, and
${\mathrm{CO}}_{2}$
. The temperature was kept constant at
$20\xb0\mathrm{C}$
. The blood was gently stirred to avoid uncontrolled sedimentation or cell aggregation within the reservoir. The blood was kept flowing by a customized turbulence-free cuvette with laminar flow and a sample thickness of
$116\phantom{\rule{0.3em}{0ex}}\mathrm{\mu}\mathrm{m}$
. The flow was adjusted for each sample to keep constant wall shear rates of 0, 50, 100, 200, 600, and
$1000\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$
at the cuvette windows.

## 2.2.

### Spectral Measurements

The diffuse reflectance
${R}_{d}$
, the total transmission
${T}_{t}$
, and the diffuse transmission
${T}_{d}$
of all blood samples were measured using an integrating sphere spectrometer (Perkin Elmer, Lambda 900, Rodgau-Jügesheim, Germany) in the spectral range of 250 to
$1100\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
at data intervals of
$5\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
. The experimental setup was described by Friebel, ^{6} and allowed the measurement of macroscopic radiation distribution with an error of less than 0.1%. A special iMCS program was used which, taking into consideration various radiation losses, gives valid results for the calculated intrinsic optical parameters.^{1} Hct-dependent effective phase functions for RBCs flowing with a shear rate of
$600\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$
was evaluated previously using the double integrating sphere technique.^{6} For shear rate changes, no significant dependence on the phase function could be observed. This is due to the relatively small shear-rate dependent changes in the optical parameters compared to Hct changes. Therefore, effective phase functions determined for
$600\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$
were used for all shear rates in the range 0 to
$1000\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$
. The iMCS was carried out using the Reynolds-McCormick phase^{34} function with
$\alpha =1.6$
for Hct 8.6% and
$\alpha =1.7$
for Hct 42.1%. The error in this method for selecting phase functions can lead to deviations in the determination of the optical parameters of
$\pm 1\%$
for
${\mu}_{a}$
,
$\pm 2\%$
for
${\mu}_{s}$
,
$\pm 2\%$
for
$(1-g)$
, and
$\pm 1.5\%$
for
${\mu}_{s}^{\prime}$
. An error threshold of 0.1%, i.e., the difference between measured and simulated macroscopic radiation distribution, was used for the simulation of the intrinsic optical parameters of blood. For all blood samples, a total of three independent measurement series were carried out and independently simulated using about
${10}^{7}$
photons for each simulation. The standard deviation of the three measurements was 2 to 5% for
${\mu}_{a}$
,
${\mu}_{s}$
,
$(1-g)$
, and
${\mu}_{s}^{\prime}$
, as is shown in the diagrams.

## 3.

## Results

To exclude the optical influence of biological variability of RBC samples with identical Hct taken from different donors, all intrinsic optical parameters given in this work are relative values related to those determined at a standard shear rate of
$600\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$
. The absolute values can be obtained by multiplication with the averaged parameters of standard blood of the same Hct, which was measured under standard conditions (wall shear rate
$600\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$
, oxygen saturation 98 to 100%, osmolarity
$300\phantom{\rule{0.3em}{0ex}}\mathrm{mosmol}\u2215\mathrm{l}$
, pH 7.4,
$20\xb0\mathrm{C}$
) and was presented in a previous paper.^{15}

To reduce the number of diagrams, only the optical parameters dependent on wavelength for RBCs in $\mathrm{NaCl}$ have been selected to be shown in Figs. 1, 2, 3, 4 . In Fig. 5 , the wavelength dependency of ${\mu}_{a}$ and ${\mu}_{s}^{\prime}$ for RBCs in plasma is exemplarily shown for diluted blood.

The fixed cells were investigated at the shear rates used for native RBCs in $\mathrm{NaCl}$ , but only the mean values of the relative optical properties at $1000\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$ have been shown, because at lower shear rates, no significant difference could be found. RBCs in plasma are only measured and shown at flow stop and $600\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$ to investigate the formation of aggregates.

The measurements on native RBCs in $\mathrm{NaCl}$ and in plasma were carried out using Hct 42.1% and 8.4%, but because of similar results, only the data from the investigations on Hct 42.1% were shown in Figs. 1, 2, 3, 4. In Sec. 3.5 and Fig. 5, selected specific characteristics are presented from diluted blood.

## 3.1.

### Absorption

Figure 1a shows the relative values of ${\mu}_{a}$ of RBC suspended in saline solution with Hct 42.1% dependent on wavelength for six different wall shear rates. With regard to the error tolerance, the relative absorption coefficient ${\mu}_{a}\_\mathrm{rel}$ for all adjusted shear rates is constant throughout the investigated wavelength range, so that a mean relative absorption coefficient can be calculated for $250\phantom{\rule{0.3em}{0ex}}\mathrm{nm}<\lambda <1100\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ . The mean relative values of RBCs in saline solution (RBC in $\mathrm{NaCl}$ ), of fixed RBCs (RBC fixed), and RBC suspended in plasma (RBC plasma), are presented in Fig. 1b dependent on shear rate.

The mean relative absorption of RBCs in saline solution is at a maximum at $1.064\pm 0.01$ at flow stop. With increasing shear rates, the mean ${\mu}_{a}\_\mathrm{rel}$ decreases continuously to a value of $1.0\pm 0.009$ at $200\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$ . Above $200\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$ , ${\mu}_{a}\_\mathrm{rel}$ shows no significant change up to $1000\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$ with a value of $0.993\pm 0.015$ . The fixed RBCs in saline solution show a comparable shear rate value within the error tolerance. RBCs suspended in plasma show wavelength-dependent effects at flow stop. For the RBCs in plasma, the absorption in the wavelength range 250 to $600\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ is lower than in the range 600 to $1100\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ . For this reason, two mean values are presented for RBCs in plasma in Fig. 1b, one mean value for ${\mu}_{a}\_\mathrm{rel}$ with $\lambda <600\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ and one for ${\mu}_{a}\_\mathrm{rel}$ with $\lambda >600\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ . Mean ${\mu}_{a}\_\mathrm{rel}$ values decreased to $0.99\pm 0.04$ in the wavelength range below $600\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ , whereas above $600\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ , ${\mu}_{a}\_\mathrm{rel}$ had a mean value of $1.043\pm 0.04$ .

## 3.2.

### Scattering

Figure 2a shows the relative values of ${\mu}_{s}$ dependent on wavelength for different shear rates of RBCs in saline solution. As observed for the absorption, ${\mu}_{s}\_\mathrm{rel}$ of RBCs suspended in saline solution shows no significant dependence on wavelength but is significantly influenced by changes in the shear rate. The mean ${\mu}_{s}\_\mathrm{rel}$ is with $1.2\pm 0.03$ at a maximum at flow stop, and decreases substantially with increasing shear rate up to $1.05\pm 0.02$ at $100\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$ [Fig. 2b]. Above $100\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$ , ${\mu}_{s}\_\mathrm{rel}$ decreases slightly to a value of $1.037\pm 0.02$ at $200\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$ , and finally to $0.974\pm 0.016$ at $1000\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$ [Fig. 2b]. For fixed RBC, ${\mu}_{s}\_\mathrm{rel}$ is equivalent to the values of native RBC up to $600\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$ . At $1000\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$ , ${\mu}_{s}\_\mathrm{rel}$ is slightly higher with a value of 0.995.

Corresponding to ${\mu}_{a}\_\mathrm{rel}$ , also for ${\mu}_{s}\_\mathrm{rel}$ of RBCs suspended in plasma, two averaged values at flow stop for the wavelength ranges 250 to $600\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ $(0.78\pm 0.04)$ and above $600\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ $(0.9\pm 0.08)$ can be differentiated, which are considerably smaller than the values for RBCs in saline solution.

## 3.3.

### Anisotropy

Figure 3a shows the relative values of the anisotropy factor $g$ of RBCs suspended in saline solution for different wall shear rates dependent on wavelength. In contrast to ${\mu}_{a}\_\mathrm{rel}$ and ${\mu}_{s}\_\mathrm{rel}$ , $g\_\mathrm{rel}$ shows a wavelength dependence below $600\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ . Regarding the statistical uncertainty of the simulation, $g\_\mathrm{rel}$ shows no significant changes above $600\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ with regard to shear rate and wavelength. Nevertheless, a mean $g\_\mathrm{rel}$ averaged over this spectral range was calculated for each shear rate and is depicted versus the shear rate in Fig. 3b, including $g\_\mathrm{rel}$ at $415\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ , to characterize the shear rate dependence in the highly absorbing spectral range. Analogous values were determined for fixed RBCs at $1000\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$ . Two average values for the wavelength ranges were calculated above and below $600\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ for RBCs suspended in plasma at flow stop.

In the wavelength range 600 to $1100\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ , $g\_\mathrm{rel}$ of RBCs in saline solution is only at $1.002\pm 0.0008$ , significantly higher than at $600\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$ at flow stop. At $1000\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$ , $g\_\mathrm{rel}$ of the fixed RBC is not affected, resulting in values of about 1.

By contrast, at $415\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ , $g\_\mathrm{rel}$ of untreated RBCs in saline solution has a value of $1.052\pm 0.01$ at a maximum at flow stop. With increasing shear rates, $g\_\mathrm{rel}$ decreases asymptotically to values of $1.01\pm 0.01$ at $200\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$ and $0.997\pm 0.005$ at $1000\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$ .

The $g\_\mathrm{rel}$ values of fixed RBCs at $415\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ at a shear rate of $1000\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$ show a slight increase to $1.002\pm 0.005$ , and for the spectral range above $600\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ , the $g\_\mathrm{rel}$ is comparable to the native RBCs in saline solution. At flow stop, $g\_\mathrm{rel}$ of RBCs suspended in plasma remains at 1 in the wavelength range above $600\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ . However, in the range 250 to $600\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ , the mean $g\_\mathrm{rel}$ decreases to $0.984\pm 0.01$ .

## 3.4.

### Effective Scattering Coefficient

Figure 4a gives the calculated relative values of the effective scattering coefficient ${\mu}_{s}^{\prime}\_\mathrm{rel}$ versus wavelength for different shear rates. Corresponding to the anisotropy factor, ${\mu}_{s}^{\prime}\_\mathrm{rel}$ shows a different behavior below and above $600\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ . In the range above $600\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ , the mean ${\mu}_{s}^{\prime}\_\mathrm{rel}$ of RBCs in saline solution is at a maximum with $1.11\pm 0.03$ [Fig. 4b] at flow stop, and decreases significantly with increasing shear rate to $1.02\pm 0.01$ at $100\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$ . Above $100\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$ , there is a slight decrease in ${\mu}_{s}^{\prime}\_\mathrm{rel}$ to $1.01\pm 0.01$ at $200\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$ , and finally to $0.97\pm 0.015$ at $1000\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$ . In the wavelength range 250 to $600\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ , ${\mu}_{s}^{\prime}\_\mathrm{rel}$ shows with $1.056\pm 0.02$ at flow stop, and $1.03\pm 0.02$ at $50\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$ , significantly lower ${\mu}_{s}^{\prime}\_\mathrm{rel}$ values than above $600\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ . At a shear rate of $1000\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$ , ${\mu}_{s}^{\prime}\_\mathrm{rel}$ is, in the range 250 to $600\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ , with a value of $0.97\pm 0.015$ , not significantly higher than above $600\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ . The fixed RBCs show slightly higher ${\mu}_{s}^{\prime}\_\mathrm{rel}$ values of 1 at $1000\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$ for both spectral ranges.

For RBCs suspended in plasma at flow stop, ${\mu}_{s}^{\prime}\_\mathrm{rel}$ results in a mean value of $0.92\pm 0.04$ in the wavelength range 250 to $600\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ and $1.03\pm 0.03$ in the range 600 to $1100\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ .

## 3.5.

### Diluted Blood

With regard to statistical uncertainties, the experiments on blood (Hct of 8.6%) lead to comparable results for the relative values, which indicate that there is no significant Hct dependence of shear-rate induced changes on the optical properties of blood in this Hct range. However, the determination of the optical parameters below
$600\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
, where the absorption of hemoglobin is high, showed reduced statistical errors due to the low optical density of *diluted* blood. The shear-rate induced changes in particular in the optical parameters of RBCs suspended in plasma could be analyzed more precisely. Figure 5 shows the spectrum from 250 to
$1100\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
of the relative values of
${\mu}_{a}$
and
${\mu}_{s}^{\prime}$
of RBCs suspended in plasma with an Hct of 8.6% at flow stop, compared to a shear rate of
$600\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$
. The spectrum of
${\mu}_{a}\_\mathrm{rel}$
is the mirror image of the absorption spectrum of hemoglobin. The mean values of
${\mu}_{a}\_\mathrm{rel}$
, averaged over the wavelength ranges 250 to
$600\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
$(0.98\pm 0.03)$
, and 600 to
$1100\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
$(1.027\pm 0.03)$
, are comparable to those of Hct 42.1% [see Fig. 1b]. In the same way, the spectrum of
${\mu}_{s}^{\prime}\_\mathrm{rel}$
represents the inverse absorption spectrum. The mean values for the wavelength range 250 to
$600\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
are
$0.89\pm 0.05$
and
$1.02\pm 0.03$
for the range 600 to
$1100\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
, and also correspond to those of Hct 42.1% [see Fig. 4b]. Analogous results were obtained for the parameters
${\mu}_{s}\_\mathrm{rel}$
and
$g\_\mathrm{rel}$
. This indicates that the wavelength dependence of the decrease in absorption and scattering of aggregated RBC is correlated with the hemoglobin absorption. This phenomenon should be also assumed for physiological Hct levels such as 42.1%.

## 4.

## Discussion

## 4.1.

### Red Blood Cell Aggregation at Flow Stop

Cell aggregation of RBCs suspended in plasma at flow stop compared to RBCs in saline solution leads to a decrease in optical parameters that is correlated with the absorption of hemoglobin. The decrease of
${\mu}_{a}$
and the enhanced effect at high absorption can be explained by the sieve effect. Photons that do not hit red blood cells pass unattenuated by absorption. This leads to a reduced absorption effect comparable to homogeneously dispersed hemoglobin. Aggregation of RBCs leads to a more heterogeneous dispersion of the absorbing hemoglobin and therefore to greater absorption reduction. The sieve effect increases considerably with absorption^{6, 35} and aggregation. This phenomenon, also called absorption flattening, is in agreement with the results of the measurements at Hct 8.6% presented in Fig. 5, which shows the correlation between the absorption and the decrease of
${\mu}_{a}\_\mathrm{rel}$
at flow stop. The same result can be obtained by Mie calculations assuming spherical aggregates of 2, 3, and 4 times the single RBC volume. In this case, the increase in the absorption cross section is lower than the evoked decrease in concentration of the growing aggregates, leading to a decrease of its product, the absorption coefficient. This phenomenon increases with the size of the aggregates and with hemoglobin absorption. An aggregate of three-fold volume, for example, leads to a reduction in
${\mu}_{a}$
to 99% in the wavelength range 600 to
$1100\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
and to 77% at
$415\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
.

The decrease of
${\mu}_{s}\_\mathrm{rel}$
,
$g\_\mathrm{rel}$
, and
${\mu}_{s}^{\prime}\_\mathrm{rel}$
due to RBC aggregation is analogous to the saturation effect of the linear increase of
${\mu}_{s}$
and
${\mu}_{s}^{\prime}$
with Hct and the Hct-dependent decrease of
$g$
, demonstrated by Meinke
^{15} Induced by the high cell concentration with very small cell-to-cell distances, aggregated or loosely associated double or triple cells occur under flow conditions. This effect leads to collective scattering with a decrease in the scattering cross section per cell, and consequently to an attenuation of the linear increase of
${\mu}_{s}$
and
${\mu}_{s}^{\prime}$
, with Hct. Furthermore, the collective scattering induces a reduction of the forward scattering, thus reducing
$g$
. These effects were shown to increase with absorption, i.e., with the imaginary part of the complex refractive index. As a consequence, cell aggregation at constant Hct leads to an absolute decrease of
${\mu}_{s}$
,
${\mu}_{s}^{\prime}$
, and
$g$
, which is enhanced by
${\mu}_{a}$
.

The results of
${\mu}_{s}$
and
$g$
in the wavelength range 600 to
$1100\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
are in qualitative agreement with the results of T-matrix computations of spheroids by Eneijder
^{20} This work shows cell-aggregation induced reduction of the scattering cross section and relative changes in
$g$
of about 0.003, which are below the measurement errors of the respective values presented here. Furthermore, results of
${\mu}_{a}$
and
${\mu}_{s}^{\prime}$
in the wavelength range 600 to
$1100\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
correspond to the absorption- and effective scattering coefficients of undiluted bovine blood at flow stop. These were experimentally determined by Eneijder, who found
${\mu}_{a}$
and
${\mu}_{s}^{\prime}$
to be approximately 1.03, relative to a shear rate of
$800\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$
at
$800\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
. In this investigation, the parameters were found to be independent of wavelength in the range 650 to
$900\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
, where the absorption is low.

The aggregation induced decrease of
${\mu}_{a}$
and
${\mu}_{s}$
are consistent with the findings of Shvartsman and Fine,^{16} who suggested that the natural pulsatile signal is driven by shear-force originated fluctuations of aggregation length.

## 4.2.

### Shear Rate Range 0 to $200\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$

Untreated RBCs suspended in saline solution are randomly oriented at flow stop and have a maximum absorption and scattering coefficient, and therefore a maximum absorption and scattering cross section per cell, because cell concentration does not change. Cell morphology associated with higher degrees of organization, such as aligned RBCs, exhibited lower scattering coefficients as explained by T-matrix computations.^{20} At increasing shear rates, a decrease in the random cell orientation reduces this effect, and as a consequence, the cross sections of
${\mu}_{a}$
and
${\mu}_{s}$
decrease asymptotically. Above a shear rate of
$200\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$
, only minimal changes can be observed, indicating that most of the effect has vanished. The same effect shows
${\mu}_{s}^{\prime}$
and
$g$
in the wavelength range below
$600\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
.

The Fahraeus effect, i.e., the dynamic decrease of the Hct, should also cause a decrease in the cell concentration dependent parameters
${\mu}_{a}\_\mathrm{rel}$
and
${\mu}_{s}\_\mathrm{rel}$
with increasing shear rate. At flow stop, the hematocrit is identical with the static hematocrit determined during blood preparation. Under flow conditions, a reduced dynamic Hct appears, which decreases with increasing shear rates. According to Barbee and Cokelet,^{31} the relative Hct decrease can be estimated to be 9.5% for a vessel diameter of
$116\phantom{\rule{0.3em}{0ex}}\mathrm{\mu}\mathrm{m}$
. In flow cuvettes with a flat cross section, this effect should be smaller than in tubes, due to the mainly 2-D parabolic flowing profile, but it could also be responsible for a significant part of the measured relative
${\mu}_{a}$
and
${\mu}_{s}$
changes of 6.4 and 17%. Since the anisotropy factor
$g$
is not directly dependent on cell concentration,
$g\_\mathrm{rel}$
should not be influenced by the Fahraeus effect.

The anisotropy factor is proportional to the degree of absorption. At
$415\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
where the absorption is at a maximum,
$g\_\mathrm{rel}$
has the highest value for randomly oriented cells. In spectral regions where
${\mu}_{a}$
is very low
$(<0.1\phantom{\rule{0.3em}{0ex}}{\mathrm{mm}}^{-1})$
,
$g$
is almost invariant to changes in shear rate. The principle shape of the anisotropy spectra has been discussed in Friebel
^{6} The spectrum of the absolute anisotropy factor at a wall shear rate of
$600\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$
is inversely related to the hemoglobin absorption, i.e.,
$415\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
is at a minimum. The absorption-dependent decrease in
$g$
of undiluted RBC suspensions shows the highest deviation to the predictions of the Mie theory^{36} compared with other optical parameters. The Mie theory shows a similar influence of the absorption on
$g$
, but only at absorption values that are 10 times higher than those of blood. The fact that reducing cell alignment and deformation will cause
$g$
to increase, a disproportionally high value in the spectral regions of high absorption indicates an approximation to the Mie theory. The same effect could be observed in ongoing investigations when the RBCs were made to swell to more spheroid forms by reducing the osmolarity.

${\mu}_{s}^{\prime}$
also shows a maximum at flow stop, but the maximum decreases with increasing absorption, which is the expected inverse behavior of the anisotropy factor. The values of
${\mu}_{s}^{\prime}$
in the wavelength range 600 to
$1100\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
are in accordance with the measurements on whole bovine blood at
$800\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
of Eneijder, ^{20} who found
${\mu}_{s}^{\prime}\_\mathrm{rel}$
with over 1.05 at a shear rate of
$100\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$
.

Roggan
^{1} found relative
${\mu}_{a}$
and
${\mu}_{s}^{\prime}$
values of 1.09 and 1.11 at flow stop for RBCs in saline solution of Hct 41% at
$633\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
, which are comparable to the values presented in these investigations.

## 4.3.

### Shear Rate Range 200 to $600\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$

In this shear rate range, the red blood cells start to align with the flow, and the aggregates of RBCs in plasma are completely dissociated.^{37} The absorption cross section does not change within this shear rate range, indicating that cell alignment has no significant influence on
${\mu}_{a}$
, and the dynamic Hct decrease is only effective in the shear rate range from 0 to
$200\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$
. As the Fahraeus effect has the same influence on
${\mu}_{a}$
and
${\mu}_{s}$
, it can be concluded that the small additional decreasing effect on
${\mu}_{s}\_\mathrm{rel}$
of 3.7% is only evoked by a decrease in the cross section induced by increasing cell alignment and decreasing random cell orientation. The latter also causes the slight decrease in the relative anisotropy factor, in the spectral range below
$600\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
, because
$g$
is not directly dependent on cell concentration. As a consequence, a decrease of
${\mu}_{s}^{\prime}\_\mathrm{rel}$
of about 1% is also induced by cell alignment and decreasing random cell orientation. There is partial agreement with the results of Steenbergen, Kolkman, and de Mul,^{38} who found that by using inverse MCS, the total attenuation coefficient
${\mu}_{t}={\mu}_{a}+{\mu}_{s}$
of undiluted blood in plasma measured by within a
$100\text{-}\mathrm{\mu}\mathrm{m}$
cuvette at
$633\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
, is approximately constant in the shear rate range 150 to
$500\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$
. However, below
$150\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$
, there is a strong variation of
${\mu}_{t}$
possibly induced by clustering effects. In contrast,
$g$
was found to increase by 10% with the shear rate in the range 50 to
$500\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$
based on the assumption that the Henyey-Greenstein phase function^{39} renders a satisfactory description for single scattering by whole blood, which is in contrast to the results of prior studies^{6} and consequently to the method used in the present work.

## 4.4.

### Shear Rates at $1000\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$

At these high shear rates, RBCs start to elongate. The decrease in ${\mu}_{s}\_\mathrm{rel}$ when the shear rate is increased from 600 to $1000\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$ does not occur when the cells are fixed, indicating that cell elongation has an additional decreasing effect on the scattering cross section. Cell elongation also reduces $g\_\mathrm{rel}$ ; the effect vanishes after cell fixation, which suggests that elongation causes a further higher degree of organization. At a shear rate of $1000\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$ , the absorption coefficient of untreated and fixed cells shows no significant change compared to $600\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$ , indicating that cell elongation as well as alignment has no influence on the absorption cross section.

The
${\mu}_{a}\_\mathrm{rel}$
and
${\mu}_{s}^{\prime}\_\mathrm{rel}$
values of RBCs in saline solution, dependent on shear rate in the wavelength range 600 to
$1100\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
, are in agreement with previous measurements of RBCs in saline solution of Hct 41% at
$633\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
made by Roggan, ^{1} who determined relative
${\mu}_{a}$
and
${\mu}_{s}^{\prime}$
values of 0.96 and 0.97 (values interpolated) at a shear rate of
$1000\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$
. Furthermore, the T-matrix computations^{20} of aligned and elongated spheroids in this spectral range resulted in a reduction of the scattering cross section and an almost unchanged
$g$
factor.

## 5.

## Conclusion

In the work presented here, it could be shown that the wall shear rate has an influence on the optical parameters ${\mu}_{a}$ , ${\mu}_{s}$ , $g$ , and ${\mu}_{s}^{\prime}$ of human blood induced by effects on the cell shape and cell organization of the RBCs. Moreover, a dependence was found to exist between the shear rate dependent effects and the hemoglobin absorption within the RBCs. The maximal absorption and scattering cross sections were found for randomly oriented RBCs. Increasing the shear rate leads to an asymptotic decrease in the absorption and scattering cross sections and of $g$ at spectral ranges with high hemoglobin absorption. Up to a shear rate of $200\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{-1}$ , a dynamic hematocrit decrease (Fahraeus effect) leads to reduction in the cell concentration and therefore to a further decrease in ${\mu}_{a}$ and ${\mu}_{s}$ . At low absorption, $g$ is almost independent of shear rate. ${\mu}_{s}^{\prime}$ values decrease with the shear rate, but this behavior is attenuated to increasing hemoglobin absorption. Cell aggregation at flow stop leads to an absorption-dependent decrease in ${\mu}_{a}$ , ${\mu}_{s}$ , $g$ , and ${\mu}_{s}^{\prime}$ . The decrease in ${\mu}_{a}$ can be explained by the sieve effect and by calculations for accreted spheres according to the Mie theory. An increase in collective scattering leads to a reduction of the scattering cross section per cell and the forward scattering. Measurements on diluted blood with Hct 8.6% showed no significant differences in the shear rate dependence of the optical parameters.

From these results, it follows that to estimate the optical properties of blood, the flow conditions have to be taken into account. Moreover, not only the collective scattering phenomena but also the absorption within the cell, together with the cell shape, are important parameters for the scattering properties of blood.

## Acknowledgments

This work was supported by the Federal Ministry of Education and Research (grant number 13N7522). The authors also wish to thank the Sorin Group Deutschland GmbH, Munich, for placing the equipment at our disposal. The blood bags were kindly provided by the Departments of Transfusion Medicine, Charité-Universitätsmedizin Berlin, Germany.