2.1.

Blood Preparation

Fresh erythrocyte concentrates from healthy human donors with high mean corpuscular hemoglobin concentration (MCHC) values were selected, washed with phosphate buffer, and concentrated by centrifugation to hematocrit values of 75 to 80%. The hemoglobin concentration was measured using a clinical blood analyzer (Micros 60 OT 18, ABX Diagnostics, Montpellier, France). The blood samples were completely hemolyzed by freezing $(-60°\mathrm{C})$ and thawing the sample for a total of three times. The cell membranes were removed using ultra-centrifugation (Allegra 64 R, Beckman Coulter GmbH, Krefeld, Germany) at $\mathrm{62,000}\mathrm{g}$ for $120\min$ . Finally, the highly concentrated hemoglobin solution was filtered through a $0.2\text{‐}\mu \mathrm{m}$ microfilter to ensure the removal of residual cell membranes. This technique enables the preparation of cell-free hemoglobin solutions with concentrations of 28.7 and $30.6\mathrm{g}∕\mathrm{dL}$ , which are close to the physiological hemoglobin concentration within the erythrocyte $(31\text{to}36\mathrm{g}∕\mathrm{dL})$ without adding chemicals. Lower concentrations (2.67 and $15.3\mathrm{g}∕\mathrm{dL}$ ) of hemoglobin were prepared by diluting the hemoglobin (Hb) solution of known total hemoglobin content with physiological saline solution. The prepared Hb solutions were adjusted to pH 7.4, and all measurements were carried out at $22°\mathrm{C}$ . 100% oxygen saturation of the Hb was achieved by equilibration with oxygen for $15\min$ before measurement. Deoxygenation (0% oxygen saturation) was effected by diluting the concentrated Hb solution with sodium dithionite in phosphate buffer (0.3% w∕w) maintaining the pH at 7.4.

2.2.

Experimental Setup

Transmittance of the diluted hemoglobin solution $(2.67\mathrm{g}∕\mathrm{dL})$ was measured for oxygenated and deoxygenated solutions using a two-beam VIS-NIR spectrometer (Lambda 900, Perkin Elmer, Rodgau-Jugesheim, Germany). The sample thickness was varied from $1\text{to}10\mathrm{mm}$ within the wavelength region, depending on the absorption. To prove the validity of Beer-Lambert’s law at physiological hemoglobin concentrations, solutions of 15.2 and $30.6\mathrm{g}∕\mathrm{dL}$ were measured using a sample thickness of $0.02\text{to}1.0\mathrm{mm}$ .

A modified integrating sphere spectrometer (Lambda 900, Perkin Elmer, Rodgau-Jugesheim, Germany) was used to determine the Fresnel reflectance (Fig. 1 ). The illuminating light beam is reflected on mirror 1 slanted at $45\mathrm{deg}$ and reaches the surface of the hemoglobin solution at an angle close to $90\mathrm{deg}$ . The Hb solution is filled into a $50\text{‐}\mathrm{mm}$ -deep cylindrical vessel with a black inner surface to avoid diffuse reflectance from the depth of the sample, and an iris diaphragm is mounted to block any possible backscattered light. The Fresnel reflectance goes back vertically through the diaphragm and is reflected on mirror 1, then reaches the inner surface of the integrating sphere for detection. For the calculation of the Fresnel reflectance, the measurement was repeated with a precision mirror 2 of exact known reflectance, which is interchangeable with the vessel of Hb solution. To evaluate the method, pure water was measured, for which the refractive index is known. No deoxygenated Hb was prepared for the Fresnel reflectance measurements, because the Hb in the highly concentrated solution is at a level of near saturation. The addition of solid sodium dithionite to 0.3 % w∕w would lead to precipitation, and as a consequence causes light backscattering, which would interfere with the Fresnel reflectance.

Fig. 1

Principle setup of the measurement.

2.3.

Theoretical Background

For transmittance measurements on diluted absorbing solutions, the transmittance is given by Beer-Lambert’s law:

If a light beam is reflected rectangularly on the surface of an absorbing material, the relationship between the amplitude of the relected beam ${\tilde{E}}_r$ and the incident beam ${\tilde{E}}_i$ is given by the Fresnel formula:

Since the energy is proportional to the square of the amplitude, the reflectance $R$ can be defined as the relationship ${\tilde{E}}_r^2∕{\tilde{E}}_i^2$ . Then $R$ is given by:

3.1.

Transmittance Measurements

The absorption coefficients of an Hb solution calculated using Eq. 1 at a concentration of $2.67\mathrm{g}∕\mathrm{dL}$ from $250\text{to}1100\mathrm{nm}$ are shown in Fig. 2 for oxygenated and deoxygenated hemoglobin compared with reference data.^{7, 8} The Soret absorption band appears at $415\mathrm{nm}$ with an absorption coefficient of about $48{\mathrm{mm}}^{-1}$ . The other typical absorption maxima are at 274, 344, 542, 577, and $915\mathrm{nm}$ for oxygenated hemoglobin. At $1130\mathrm{nm}$ , the values approach those of the water absorption, and above $1250\mathrm{nm}$ the absorption curves are equal. The absorption of deoxygenated hemoglobin shows the expected shift of the Soret peak, the fusion of the two maxima at 542 and $577\mathrm{nm}$ to one peak at $555\mathrm{nm}$ , and an enhanced absorption in the red light. The absorption of the deoxygenated hemoglobin already approximates that of the water absorption at about $1000\mathrm{nm}$ and is identical above $1130\mathrm{nm}$ . The isosbestic point, where absorption is independent of the oxygenation level, is clearly visible at $805\mathrm{nm}$ .

Fig. 2

Absorption coefficient of oxygenated and deoxygenated hemoglobin solution at a concentration of $2.67\mathrm{g}∕\mathrm{dL}$ (measured with different sample thicknesses and calculated using Beer-Lambert’s law) compared with some reference data and the absorption spectrum of water.^{7, 8}

The results of the investigations using highly concentrated hemoglobin solutions, close to the concentration inside the red blood cell, are shown in Fig. 3 . The absorption spectra of Hb solutions of 15.3 and $30.6\mathrm{g}∕\mathrm{dL}$ in the wavelength range of $250\text{to}1100\mathrm{nm}$ are compared with absorption values measured at a concentration of $2.67\mathrm{g}∕\mathrm{dL}$ (shown in Fig. 2) linearly extrapolated to concentrations of 15.3 and $30.6\mathrm{g}∕\mathrm{dL}$ . The curves show a high correlation with almost no differences within the range of the error limits, indicating that Beer-Lambert’s law is still valid, even at such high concentrations.

Fig. 3

Measured absorption coefficients of highly concentrated hemoglobin solutions of 30.6 and $15.3\mathrm{g}∕\mathrm{dL}$ (sample thickness $0.02\mathrm{mm}$ ) compared to the extrapolated data calculated from the $2.67\mathrm{g}∕\mathrm{dL}$ concentration.

3.2.

Reflectance Measurements

The measured Fresnel reflectance of Hb solutions at $28.7\mathrm{g}∕\mathrm{dL}$ and pure water for control are shown in Fig. 4 . The reflectance generally decreases with increasing wavelength. The shape of the Hb curves at the point of highest absorption (approximately $415\mathrm{nm}$ ) is notable, indicating anomalous refraction.

Fig. 4

Measured Fresnel reflections of Hb solutions with a concentration of $28.7\mathrm{g}∕\mathrm{dL}$ and of pure water, as illustrated in Fig. 1.

In Fig. 5 , the upper diagram shows the absorption constant $k$ in the wavelength range of $250\text{to}1100\mathrm{nm}$ calculated using Eq. 4, and the absorption values extrapolated to the Hb concentration $28.7\mathrm{g}∕\mathrm{dL}$ from the spectra of the hemoglobin concentration level $2.67\mathrm{g}∕\mathrm{dL}$ . Using this data, the real part of refractive index $n$ for hemoglobin $(28.7\mathrm{g}∕\mathrm{dL})$ is calculated from the Fresnel reflectance according to Eq. 5, and is shown in the lower diagram of Fig. 5. The measured reference values of the refractive index of water are also shown.^{9, 10} Similar to the Fresnel reflectance, $n$ generally decreases with increasing wavelength with a maximum at the UV region (normal diffraction). The wavelength dependence of the real part of the refractive index of the hemoglobin solution is similar to the one of water, but parallel shifted by about 0.08 units to higher values. In contrast to the water curve, the slope of the Hb curve in the UV region from $250\text{to}310\mathrm{nm}$ is steeper. In the range of $360\text{to}510\mathrm{nm}$ , the real part of the refractive index shows the typical S-shape of an anomalous diffraction with a point of inflexion at $415\mathrm{nm}$ , corresponding to the absorption maximum of hemoglobin, where $k$ is at a maximum. Furthermore, at 545 and $580\mathrm{nm}$ , small relative maxima are visible, which correspond to the relative absorption maxima. These small maxima are within the error limits and are therefore not significant.

Fig. 5

Top: Absorption constant $k$ depends on the wavelength calculated using Eq. 4 and the absorption values extrapolated to the Hb concentration $28.7\mathrm{g}∕\mathrm{dL}$ . Bottom: Real part of refractive index $n$ calculated from the Fresnel reflectance, following Eq. 5 compared to the refractive index of water^{9, 10} and $n$ values obtained by subtractive Kramers-Kronig analysis.⁴

4.1.

Imaginary Part of the Refractive Index

The imaginary part of the refractive index, characterized by the absorption constant $k$ , is proportional to the absorption coefficient. The absorption coefficients determined in the present study are in agreement with most of the values found in the literature. Figure 2 shows a comparison of the measured absorption spectra of hemoglobin at a concentration of $2.67\mathrm{g}∕\mathrm{dL}$ , as well as some values from other authors and the absorption spectrum of water.

The results depicted in Fig. 3 show that the absorption coefficient is linearly correlated to the hemoglobin concentration up to $30.6\mathrm{g}∕\mathrm{dL}$ . This means that the absorption cross section is constant in this range. For the first time, these direct measurements show definitely that Beer-Lambert’s law is valid for Hb concentrations that are close to the physiological values within the red blood cells (about $31\text{to}36\mathrm{g}∕\mathrm{dL}$ ). This verifies the current extrapolations.

4.2.

Real Part of the Refractive Index

For pure water, the determined refractive index is equal to the literature values of water within the error limit, as shown in Fig. 5, lower diagram, indicating that the method using reflectance measurements to determine the refractive index is valid.

This curve of the refractive index is different in shape and height compared to other investigations. It has to be noted that, to our knowledge, no studies exist where the refractive index of hemoglobin in physiological high concentration were directly measured, whereas some authors have extrapolated or calculated this data. Based on the work of Stoddard and Adair,¹¹ who did measurements with a refractometer using white light, Barer’s formula $n_{\mathrm{Hb}}=n_{{\mathrm{H}}_2\mathrm{O}}+0.001942{\mathrm{c}}_{\mathrm{Hb}}$ , a Hb concentration of $28.7\mathrm{g}∕\mathrm{dL}$ would result, for example, in a $n_{\mathrm{Hb}}$ of 1.388 at $589\mathrm{nm}$ . Faber ⁴ presented a refractive index at $800\mathrm{nm}$ of $1.392\pm 0.001$ for a $27.9\mathrm{g}∕\mathrm{dL}$ Hb concentration using optical coherence tomography. Using transmittance measurements of erythrocyte monolayer, Khairullina¹² gives $n$ values for intact erythrocytes with physiological Hb concentration $(31\text{to}35\mathrm{g}∕\mathrm{dL})$ of 1.406 to 1.409 in the wavelength range $380\text{to}800\mathrm{nm}$ and without any distinct wavelength dependence.

The refractive index for the hemoglobin solution of $28.7\mathrm{g}∕\mathrm{dL}$ in this investigation is $1.408\pm 0.003$ at $589\mathrm{nm}$ and $1.403\pm 0.003$ at $800\mathrm{nm}$ , where the concentration of hemoglobin is a little lower than a normal physiological concentration. Therefore, our value is higher than values from previous investigations. This could be due to the practical problems involved in preparing such high hemoglobin concentrations and its correct determination. Stoddard and Adair used highly purified hemoglobin, whereas in our investigations, the refractive index of a hemoglobin solution was measured as it naturally occurs in human red blood cells, i.e., including salts $(\sim 0.7\mathrm{g}∕\mathrm{dL})$ , and other organic compounds $(\sim 0.2\mathrm{g}∕\mathrm{dL})$ .¹³ This is essential, as methods used to calculate the light distribution in blood or blood perfused tissues require the refractive index data that relate to the situation in situ, and it is known that the refractive index of multicomponent solutions can be influenced by nonabsorbing components.¹⁴ Therefore, the additional compounds will lead to an increase in the refractive index compared to pure hemoglobin solutions of identical concentration. Another problem is the tendency of highly concentrated biomolecules to adsorb on nonphysiological surfaces,¹⁵ such as the optical glass of a refractometer, which can lead to falsified results.

The wavelength dependence of the real part, obtained from subtractive Kramers-Kronig analysis of the absorption data of Prahl,⁷ as presented by Faber ⁴ (see Fig. 5), exhibits the same maximum amplitude in the anomalous dispersion $(\Delta n=0.018\text{units})$ . The two maxima at 545 and $580\mathrm{nm}$ also have the same range of amplitude $(\Delta n=0.002)$ , even though they are near the error limit or within it. The increase of 0.008 from $800\mathrm{nm}\text{to}430\mathrm{nm}$ is low compared to our studies with an increase of 0.024, and the curve in the UV region has lower values than at higher wavelengths, which also differ from our results. When calculating the refractive index for a defined wavelength range, using the Kramers-Kronig relation, it is necessary to know, if not the whole spectrum, at least the adjacent spectral areas, which is not the case for the spectral range below $250\mathrm{nm}$ . However, a high increase in absorption of hemoglobin in the UV range below $250\mathrm{nm}$ is probable and would, if neglected, lead to incorrect results in the Kramers-Kronig calculation. This would also explain the generally low increase with decreasing wavelength in these calculations.

Similarly, Hammer ³ calculated the refractive index of a Hb solution with a concentration of $34\mathrm{g}∕\mathrm{dL}$ in the wavelength range of $400\text{to}700\mathrm{nm}$ using Tycko’s real part for some wavelengths,¹⁶ Latimer’s approximate formulas⁶ for dispersion of liquids, and absorption spectra from van Assendelft.¹⁷ The general form of the curve is qualitatively similar to Faber’s and to our results. The refractive index is about 1.402 at $700\mathrm{nm}$ ; the two maxima at 545 and $580\mathrm{nm}$ show only half the range of amplitude (0.001). The increase from $700\text{to}430\mathrm{nm}$ of 0.002 is less than 10% of the observed increase in our study. The maximal amplitude for the anomalous dispersion between 430 and $400\mathrm{nm}$ has only half the value compared with our results. These deviations can also be explained by the previously mentioned problems found with calculating wavelength-dependent refractive indices.