2.1.

Blood Preparation

Fresh human erythrocytes from a healthy blood donor were centrifuged three times and washed with isotonic phosphate buffer ( $300\mathrm{mosmol}∕\mathrm{L}$ , pH 7.4) to remove the blood plasma and free hemoglobin. The hematocrit was varied by diluting the sample with buffer to values 0.84 and 42.1%, which correspond to hemoglobin solution values of 0.27 and $12.9\mathrm{g}∕\mathrm{dL}$ , respectively. The hematocrit under static conditions was determined using a red blood cell counter (Micros 60 OT 18, ABX Diagnostics, Montpellier, France), and the oxygen saturation was measured with a blood gas analyzer (OPTI Care, AVL Medizintechnik GmbH, Bad Homburg, Germany). A miniaturized blood circulation setup was adapted with a roller pump (Stöckert Instruments GmbH, München, Germany) and a blood reservoir, which was constantly aerated with a gas mixture of ${\mathrm{O}}_2$ , ${\mathrm{N}}_2$ , and $\mathrm{C}{\mathrm{O}}_2$ to maintain both the oxygen saturation $(>99\%)$ and a constant pH (7.4). The temperature was kept constant at $20°\mathrm{C}$ . The blood was gently stirred within the reservoir to avoid creating concentration inhomogenities caused by sedimentation or cell aggregation. The blood was kept flowing by a specially designed, turbulence-free cuvette with a laminar flow and a sample thickness of 1020 and $116\mu \mathrm{m}$ for Hct 0.84 and 42.1%, respectively, which imitates the flow conditions in arterioles of similar diameters. The flow was regulated for each sample thickness to a constant wall shear rate of $600{\mathrm{s}}^{-1}$ at the cuvette windows. The shear rate decreases to values approaching zero at the center of the cuvette. Thus, the direction of the incident light was parallel to the velocity gradient. For additional measurements in the wavelength region at $415\mathrm{nm}$ where the hemoglobin absorption is very high, a cuvette was used with a sample thickness of $40\mu \mathrm{m}$ .

2.2.

Experimental Setup

The macroscopic optical parameters, diffuse reflectance $R_d$ , the total transmission $T_t$ , and the diffuse transmission $T_d$ ( $=T_t$ —transmission within an aperture of $5.3\mathrm{deg}$ ) of flowing blood were measured every $5\mathrm{nm}$ in the spectral range from $250\text{to}1100\mathrm{nm}$ using an integrating sphere spectrometer (Lambda 900, PerkinElmer, Rodgau-Jügesheim, Germany), which is a two-beam spectrometer with double monochromator system (Fig. 1 ). The light source consists of a deuterium lamp for the UV range and a wolfram halogen lamp for the visible∕near infrared (NIR) range. The light intensity was adapted to the absorption behavior of the sample to maintain the intensity of the signal in the optimal range of the detectors. This optical arrangement of a sample and a reference beam compensates for light intensity shifts, and changes in the inner sphere reflectivity by positioning the glass cuvette onto the sphere. The cuvette can be fixed in defined position at a constant distance to the sphere aperture, in front of or behind the integrating sphere, to measure the transmittance or reflectance spectra. For the measurement of $T_t$ the reflectance port is closed with a diffuse reflecting Spectralon® standard. $T_d$ is measured after the standard is removed, so that nonscattered and scattered transmitted light leaves the sphere within an angle of $5.3\mathrm{deg}$ . $R_d$ is measured relative to the reflectance standard by replacing the special Spectralon® standard with a certified, known reflectance spectrum by the sample that is inclined at an angle of $8\mathrm{deg}$ to the incoming light. The Fresnel reflectance of the cuvette glass leaves the sphere through the open Fresnel port to avoid interference with the diffuse reflectance. This experimental setup allows the measurement of macroscopic radiation distribution with an extremely reduced error potential. The accuracy of repetitive measurements of the reflectance and transmission measurements of a sample with defined optical properties (e.g., polystyrene spheres) was smaller than 0.01%. Measurements on flowing blood resulted in an accuracy of 0.02%, probably induced by flow-dependent local inhomogenities in the Hct.

Fig. 1

Schematic diagram of the integrating sphere spectrometer. The reference beam goes via mirrors M3 to M5 into the integrating sphere. The sample beam reaches the sample via mirrors M1 and M2.

2.3.

Principle of Inverse Monte Carlo Simulation

The IMCS method, which was first presented by Roggan ²³ could be well adapted to the geometry of the measurement system. The IMCS, shown schematically in Fig. 2 , uses forward Monte Carlo simulations iteratively to calculate optical parameters ${\mu }_a$ , ${\mu }_s$ , and $g$ on the basis of a given phase function and measured values $R_d^M$ , $T_t^M$ , and $T_d^M$ for reflection and transmission. The forward simulation used in the present study considers all kinds of radiation losses within the experimental setup, including the simulation of the exact geometry of the sample and the cuvette, the integrating sphere, the illuminating light beam, and all the diaphragms and apertures. A precise consideration of the radiation losses reduces the error with respect to the determination of the intrinsic parameters to a minimum.

Fig. 2

Flow chart of the inverse Monte Carlo simulation.

The IMCS presented here applies the Newton-Raphson method²⁴ to find an approximate solution of the nonlinear system of Eq. 1,

As a starting value for the Newton-Raphson method, an estimation of the parameters ${\mu }_a$ , ${\mu }_s$ , and $g$ from the Kubelka-Munk theory is used. The calculated $R_d^S$ , $T_t^S$ , and $T_d^S$ are then compared to measured values $R_d^M$ , $T_t^M$ , and $T_d^M$ ,

2.4.

Possibility of Determination of the Intrinsic Parameters

The found intrinsic parameters can be seen as the optical properties of the sample, with the prerequisite that the transformation function $f({\mu }_a,{\mu }_s,g)=>(R_d,T_t,T_d)$ is in one-to-one correspondence. This underlying transformation function is a priori unknown and depends not only on the phase function but also on the geometric position of the detectors, the geometry of the illuminating light beam, the geometry and the optical properties of the sample, and as a consequence, also on the wavelength. Therefore the one-to-one correspondence must be evaluated over the whole spectral range. Also, the propagation of the errors in the measurement of $R_d$ , $T_t$ , and $T_d$ into the resulting errors of the determined ${\mu }_a$ , ${\mu }_s$ , and $g$ is not directly predictable. Spectral ranges can exist where the system is underdetermined, making it impossible, for example, to determine ${\mu }_s$ and $g$ independently. In special cases, the one-to-one correspondence could be investigated by systematically searching for other ${\mu }_s$ and $g$ pairs of identical ${\mu }_s^{\prime }$ values, with which the IMCS remains below the error threshold. In all simulations presented in this study, there is no indication that an invalid one-to-one correspondence occurred at any time.

The high precision of the intensity measurement combined with the high resolving simulation leads to the system registering small deviations in the specified boundary condition. Examples of significant boundary conditions are the refractive index of the cuvette glass or the space between cuvette and aperture of the integrating sphere. Under certain conditions, such as special wavelength ranges, sample properties, and geometries, the spatial distribution of the radiation losses can influence the measured $R_d$ , $T_t$ , and $T_d$ values significantly. This means additional information is registered. As a consequence, with this system it is possible, for example, to find a reasonable set of ${\mu }_a$ , ${\mu }_s$ , and $g$ values even when $T_d$ is equal to $T_t$ , i.e., there are only two independent measurement parameters. Furthermore, it is possible using all three measurement parameters to evaluate the most appropriate effective phase function that leads to the lowest error or gives the most successful simulations at a fixed error threshold.

This IMCS makes it possible to compare different phase functions such as Mie²⁵ and Henyey-Greenstein,²⁶ or to fit an adaptable phase function to the optimal form such as the Reynolds-McCormick phase function with the variation factor $\alpha$ , as shown in this work.

Equation 4 shows the Reynolds-McCormick phase function:

2.5.

Measurement Performance

Different phase functions and their dependence on wavelength and hematocrit were investigated to evaluate the most appropriate effective phase function. A very time-consuming fit procedure (two to four weeks for five wavelengths) was used with a possible total of 36 variation cycles and up to ${10}^8\text{photons}$ per fit step with no given error threshold. The lowest deviation that could be reached by the simulation within a defined number of fit and variation steps indicates the most appropriate effective phase function. The $R_d$ , $T_t$ , and $T_d$ values of five selected wavelengths for the two different hematocrits were used for the simulation.

In a second step, a procedure that uses less simulation time (up to one week for the 171 wavelengths) but offers higher precision was established to determine the effective phase function. The procedure results in one effective phase function for the regarded wavelength range, assuming that the phase function is independent of the wavelength. The IMCS was carried out using Reynolds-McCormick phase functions with different $\alpha$ values for the two hematocrits to find the best phase function for the entire spectral range between 250 and $1100\mathrm{nm}$ . $R_d$ , $T_t$ , and $T_d$ were measured every $5\mathrm{nm}$ resulting in 171 datasets that had to be simulated by IMCS. To reduce the calculation expense, the threshold for the error of each of the three fitted signals was set to 0.05%, i.e., ${10}^6$ to ${10}^7$ start photons were calculated in each fit step. During the fitting procedure, the number of photons was increased dynamically according to the precision required. If the simulation was unable to find a solution for the three intrinsic optical parameters of a measured $R_d$ , $T_t$ , and $T_d$ dataset at a single wavelength, the calculation was stopped, the dataset marked with “error,” and the next ${\mu }_a$ , ${\mu }_s$ , and $g$ set was simulated. The final number of failed simulations occurring for the whole spectrum (171 wavelengths) was counted for each simulation, dependent on the variation factor $\alpha$ .

As described in Sec. 2.3, an error threshold of 0.1% was used for the simulation of the intrinsic optical parameters of blood using the determined effective phase functions. For both hematocrits, a total of three independent measurement series were carried out and independently simulated.

3.1.

Investigation of Different Phase Functions Depending on Wavelength and Hematocrit

Different phase functions were tested such as the Reynolds-McCormick (RM) phase function, the Henyey-Greenstein (HG), and the modified HG phase function. The IMCS without error threshold was applied to averaged datasets of $R_d$ , $T_t$ , and $T_d$ at 300, 415, 575, 700, and $900\mathrm{nm}$ for blood with Hct 0.84 and 42.1%. The best results were obtained with the Reynolds-McCormick phase function. Figure 3 shows the sum of the minimal absolute percentage deviation of $R_d$ , $T_t$ , and $T_d$ for Hct 0.84% for the five wavelengths depending on $\alpha$ , ranging from 0.5 to 3. Values of $\alpha$ smaller than 0.5 and higher than 3 lead to much higher errors and are not shown. The data show no indication of significant wavelength dependence within this range. No wavelength dependence was obtained for Hct 42.1% either. With regard to the mean values, the optimal value of $\alpha$ appears to be in the range between 1 and 2 for diluted and undiluted blood.

Fig. 3

Sum of the minimal absolute percentage deviation of $R_d$ , $T_t$ , and $T_d$ for Hct 0.84% for selected wavelengths and its mean values dependent on variation factor $\alpha$ ranging from 0.5 to 3.

3.2.

Precise Evaluation of the Reynolds-McCormick Phase Function Depending on Hematocrit

Due to wavelength independence, the more precise procedure could be used to determine the effective phase function. Figure 4 shows the absolute number of datasets of the 171 in the wavelength range $250\text{to}1100\mathrm{nm}$ that could not be successfully simulated dependent on $\alpha$ for Hct 0.84 and 42.1%. From these curves it can be deduced that the Reynolds-McCormick (RM) phase function with $\alpha$ between 1.2 and 1.3 is the best effective phase function for describing the real scattering behavior of flowing blood with Hct 0.84%. For undiluted blood with Hct 41.2%, the best effective phase function can be obtained with $\alpha$ between 1.6 and 1.8.

Fig. 4

Absolute number of unsuccessfully simulated datasets for nine simulations with different $\alpha$ values.

For the diluted blood, the result is in agreement with data from Yaroslavsky ¹⁰ and Hammer ⁹ obtained by goniometric measurements, indicating that the optimal $\alpha$ is 1 and 1.5, respectively. Hammer ⁹ also found the RM-phase function to be wavelength independent in the range $489\text{to}690\mathrm{nm}$ . However, the $\alpha$ value for undiluted blood is in contrast to the results presented by Hammer, ²¹ who found the optimal effective phase function with $\alpha =0.49$ (HG phase function) at $514\mathrm{nm}$ . Using the HG phase function in the Monte Carlo simulation used in this work allows no successful simulation at $514\mathrm{nm}$ and leads to 77% error datasets for the wavelength range $250\text{to}1100\mathrm{nm}$ .

3.3.

Optical Parameters of Diluted Blood

Using the evaluated Reynolds-McCormick phase functions with $\alpha =1.25$ for blood with Hct 0.84% and 1.7 for blood with Hct 42.1%, the absorption coefficient ${\mu }_a$ , the scattering coefficient ${\mu }_s$ , and the anisotropy factor $g$ were determined in the wavelength range $250\text{to}1100\mathrm{nm}$ . Figure 5 shows the spectra of ${\mu }_a$ , ${\mu }_s$ , $g$ , and ${\mu }_s^{\prime }$ as mean values of the simulation of three different $R_d$ , $T_d$ , and $T_t$ measurements compared with literature data and data calculated using the Mie theory for spherical scatterers of the same mean volume and complex refractive index as the measured erythrocytes. Where necessary, the literature data were rescaled. The error bars are standard deviations (SD) for selected wavelengths. For ${\mu }_a$ , the mean relative SD averaged over all 171 wavelengths is 1.9%, ranging from 0.06 to 8%. The mean SD for ${\mu }_s$ and $g$ is 0.85% (0.06 to 4.8%) and 0.036% (0.001 to 0.26%), respectively, leading to a mean SD of 1.6% for ${\mu }_s^{\prime }$ (0.11 to 28.8%). For ${\mu }_a$ , the spectrum of a hemoglobin (Hb) solution of the same Hb content is also shown. The absorption spectrum shows the typical maxima of oxygenated Hb at $415\mathrm{nm}$ and the double peak at 540 and $575\mathrm{nm}$ . The Mie curve is comparable to the measured absorption in the range of low absorption above $600\mathrm{nm}$ . At higher absorptions, Mie values are lower with a maximal difference of about 30% in the range around $500\mathrm{nm}$ . The absorption of the Hb solution shows only distinctly higher values at the highest ${\mu }_a$ peaks. If absorption decreases, the difference between the ${\mu }_a$ values of the Hb solution and blood decreases. This can be explained by the Sieve effect. When light passes through a suspension of absorbing particles, such as blood, photons that do not encounter red blood cells pass unattenuated by absorption. As a consequence, the transmitted light intensity is higher than it would be if all the hemoglobin were uniformly dispersed in the solution. The magnitude of this effect increases with increasing ${\mu }_a$ and decreasing Hct.²⁷ This phenomenon is also known as “absorption flattening,” because the heights of peaks in an absorption spectrum for a suspension are depressed relative to those for a homogeneous solution of the same average concentration. This describes exactly the effect observed in Fig 5. The spectrum of the scattering coefficient shows, in principle, a similar shape to that calculated by the Mie theory but is on average 22% lower, dependent on the wavelength. The values of Hammer ²¹ using the anomalous diffraction theory do not appear to give better agreement.

Fig. 5

(a) Absorption coefficient ${\mu }_a$ , (b) scattering coefficient ${\mu }_s$ , (c) anisotropy factor $g$ , and (d) reduced scattering coefficient ${\mu }_s^{\prime }$ of flowing red blood cells (wall shear rate $600{\mathrm{s}}^{-1}$ ) with a hematocrit value of 0.84% in the wavelength range $250\text{to}1100\mathrm{nm}$ , compared to ${\mu }_a$ of a hemoglobin solution $(0.27\mathrm{g}∕\mathrm{dL})$ , to values calculated by the Mie theory and to data from the literature.^{9, 19}

The anisotropy factor $g$ between 600 and $1100\mathrm{nm}$ is in the range of 0.99 and is identical to that calculated with the Mie theory. The Mie curve is obtained using the complex refractive index of hemoglobin²⁸ and shows at $415\mathrm{nm}$ the typical shape corresponding to the refractive index and indicates anomalous dispersion. The $g$ value of 0.9818 determined by Steinke and Sheperd¹⁹ at $633\mathrm{nm}$ for blood of a similar concentration is distinctly lower. Below $600\mathrm{nm}$ , the measured values start to decrease compared to the Mie theory. The curve of $g$ seems to be inversely related to the absorption spectrum, leading to a distinct minimum of 0.805 at $415\mathrm{nm}$ . That means that the anisotropy factor at this Hct can be only described by the Mie theory if the hemoglobin absorption is very low.

The ${\mu }_s^{\prime }$ spectrum is in principle the mirror image of the $g$ spectrum. Analogous to ${\mu }_s$ , the effective scattering coefficient ${\mu }_s^{\prime }$ in the range $460\text{to}1100\mathrm{nm}$ is on average 24% lower than the Mie theoretical values. Below $460\mathrm{nm}$ , the influence of the strongly decreasing $g$ factor becomes dominant and ${\mu }_s^{\prime }$ increases similarly, far above the values given by the Mie theory. There is currently no other data available in this high absorption region.

3.4.

Optical Parameters of Blood in Physiological Concentration

Figure 6 shows the spectra of ${\mu }_a$ , ${\mu }_s$ , $g$ , and ${\mu }_s^{\prime }$ as mean values of the simulation of three different $R_d$ , $T_t$ , and $T_d$ measurements for Hct 42.1% compared to the literature data and the Mie theory. Where necessary, the literature data were rescaled to Hct 42.1%. Equivalent to Fig. 5, the error bars are standard deviations (SD) for selected wavelengths. For ${\mu }_a$ , the mean relative SD is 3.1%, ranging from 0.03 to 42.9%. The mean SD for ${\mu }_s$ and $g$ is 3.9% (0.012 to 39.5%) and 0.44% (0.001 to 8.7%), respectively, leading to a mean SD of 2.54% for ${\mu }_s^{\prime }$ (0.006 to 29.6%). For ${\mu }_a$ , the spectrum of an Hb solution at the same Hb concentration is also shown. The relative differences between the measured spectra and the Mie curve are very similar to the ones at Hct 0.84%, indicating that the dependence of ${\mu }_a$ on the Hct is linear and independent of the wavelength. Compared to the measurements of Hct 0.84%, the ${\mu }_a$ peaks of blood and the Hb solution show a similar difference, indicating an unchanged Sieve effect in contrast to the expected decrease at increased Hct. Only in the range below $400\mathrm{nm}$ is the difference smaller. The highest ${\mu }_a$ value is reached at $415\mathrm{nm}$ with $105.9{\mathrm{mm}}^{-1}$ . Note that in contrast to the Sieve effect, all ${\mu }_a$ data from the literature^{2, 7, 21} (all values determined with IMCS with exception of Reynolds¹⁸) are higher than the absorption spectrum of the Hb solution of the same Hb content. This could be explained by the previously discussed problem of determination of light losses in the IMCS. If some of the losses in radiation are neglected, the consequence would be an overestimation of ${\mu }_a$ .

Fig. 6

(a) Absorption coefficient ${\mu }_a$ , (b) scattering coefficient ${\mu }_s$ , (c) anisotropy factor $g$ , and (d) reduced scattering coefficient ${\mu }_s^{\prime }$ of flowing red blood cells (wall shear rate $600{\mathrm{s}}^{-1}$ ) with a hematocrit value of 42.1% in the wavelength range $250\text{to}1100\mathrm{nm}$ , compared to ${\mu }_a$ of a hemoglobin solution $(12.9\mathrm{g}∕\mathrm{dL})$ , to values calculated by the Mie theory and to various data from the literature.^{2, 4, 7, 18, 21}

As discussed for the diluted blood, the spectrum of the scattering coefficient of undiluted blood shows in principle a similar form to the curve given by the Mie theory. This was obtained from the Mie scattering cross section ${\sigma }_s$ using the approximation equation²⁹ ${\mu }_s=\mathrm{Hct}(1-\mathrm{Hct})({\sigma }_s∕V)$ , where $V$ is the volume of the scatterer. However, depending on wavelength, the measured values are on average 59% lower, indicating that the prerequisite of independent scattering without interference phenomena is gradually eliminated by the decrease in the mean distances between the cells as discussed by Faber ²⁰ The data of Yaroslavsky ⁷ Reynolds,¹⁸ and Roggan ² confirms the measurements presented here in contrast to the Mie theoretical values. An exception to this is Hammer ²¹ who presented data at $514\mathrm{nm}$ , even higher than the value given by the Mie theory.

In contrast to the diluted blood, the anisotropy factor $g$ is distinctly lower than the Mie theoretical $g$ values in the wavelength range $600\text{to}1100\mathrm{nm}$ , and no value is above 0.97. In a similar way to the spectrum of the diluted blood, the measured values start to decrease in contrast to the Mie theory below $600\mathrm{nm}$ , and the decrease is also related to the absorption spectrum but is more pronounced, leading to a minimum of 0.66 at $415\mathrm{nm}$ . The values of Hammer ²¹ at $514\mathrm{nm}$ and Roggan ² at $633\mathrm{nm}$ are in good agreement with the simulated data. However, the data of Reynolds¹⁸ and Yaroslavsky ⁷ at 810, 960, and $1064\mathrm{nm}$ is in agreement with the Mie theory.

In the range of $600\text{to}1100\mathrm{nm}$ , the effective scattering coefficient ${\mu }_s^{\prime }$ agrees well with the Mie theory with values between 1.9 and 2.27 ${\mathrm{mm}}^{-1}$ . Below $600\mathrm{nm}$ , ${\mu }_s^{\prime }$ shows distinct maxima corresponding to the minima of the anisotropy factor and is up to 12 times $\left(415\mathrm{nm}\right)$ higher than values calculated with the Mie theory. The data presented by Roggan ² (flowing blood, IMCS) and Enejder ⁴ (flowing bovine blood, IMCS) is in good agreement with the data shown here, whereas the data of Yaroslavsky ⁷ (IMCS) and Reynolds¹⁸ at 810, 960, and $1064\mathrm{nm}$ is much lower $\left(0.5\text{to}0.7{\mathrm{mm}}^{-1}\right)$ . Hammer ²¹ (angle-resolved MCS) determined ${\mu }_s^{\prime }$ with $6.18{\mathrm{mm}}^{-1}$ at $514\mathrm{nm}$ , which is 2.38 times higher than the values presented in this work.

3.5.

General Discussion