Empirical model functions to calculate hematocrit-dependent optical properties of human blood

Empirical model functions to calculate hematocrit-dependentoptical properties of human blood

Martina Meinke, Gerhard Müller, Jürgen Helfmann, and Moritz Friebel

The absorption coefficient, scattering coefficient, and effective scattering phase function of human redblood cells (RBCs) in saline solution were determined for eight different hematocrits (Hcts) between0.84% and 42.1% in the wavelength range of 250–1100 nm using integrating sphere measurements andinverse Monte Carlo simulation. To allow for biological variability, averaged optical parameters weredetermined under flow conditions for ten different human blood samples. Based on this standard blood,empirical model functions are presented for the calculation of Hct-dependent optical properties for theRBCs. Changes in the optical properties when saline solution is replaced by blood plasma as thesuspension medium were also investigated. © 2007 Optical Society of America

OCIS codes: 170.1470, 290.5820, 300.1030, 290.7050.

1. Introduction

Detailed information about the light-scattering andabsorption properties of human blood plays an im-portant role in the many diagnostic and therapeuticapplications in laser medicine, hematology, and rou-tine medical diagnostics. Furthermore the opticalproperties of blood are required for a number ofoptical methods in order to calculate the light distri-bution in blood perfused tissues, e.g., optical tomog-raphy, photodynamic therapy, and laser-inducedthermotherapy.

According to the transport theory, the optical prop-erties of blood can be described by the intrinsic opticalparameters: absorption coefficient �a, scattering co-efficient �s, and anisotropy factor g. Several attemptshave been made to determine the optical propertiesof red blood cells (RBCs) or whole blood and hemo-globin (Hb) solutions.1–14 These parameters couldbe determined for highly diluted and undilutedflowing blood using the double integrating spheretechnique combined with inverse Monte Carlo sim-ulation (iMCS).1,5,7,9 The iMCS calculates the photon

trajectories on the basis of a given phase function,which describes the statistical angle distribution of ascattering event. Therefore it is essential to use anappropriate effective phase function to give a realisticdescription of the real radiation distribution withinthe investigated medium.9 Appropriate effective phasefunctions for flowing RBCs with a hematocrit (Hct) of0.84% and 42.1% could be evaluated in the wavelengthrange of 250–1100 nm using the double integratingsphere technique combined with a high-precisioniMCS.7

The aim of this study is to derive a set of empiricalformulas to predict the optical properties of a fullyoxygenated blood sample under defined flow condi-tions and a random Hct. Therefore the dependenceof optical parameters on the Hct was investigatedwithin a continuous Hct range including detailedevaluation of Hct-dependent effective phase func-tions.

It is known that the optical behavior of blood de-pends on various physiological parameters such asoxygen saturation, osmolarity, flow conditions, andaggregation.15–22 In addition to the hemoglobin con-centration (Hbc) and the oxygen saturation, whichcan change the absorption spectrum of Hb massively,Hct is the most important physiological parameterwith regard to its influence on absorption and scat-tering properties and is widely used as a character-istic parameter for blood concentration.

However, Hct is the product of the red blood cellconcentration (RBCc) and the mean-corpuscular vol-ume (MCV), and Hbc is the product of the RBCc andthe mean-corpuscular hemoglobin content (MCH). Ahigh Hb concentration is responsible for the higher

M. Meinke ([email protected]) and G. Müller are withthe Institut für Medizinische Physik and Lasermedizin, CampusBenjamin Franklin, Charité-Universitätsmedizin Berlin, Fabeck-strasse 60-62, 14195 Berlin, Germany. J. Helfmann and M. Friebelare with the Laser- und Medizin-Technologie GmbH, Berlin, Fa-beckstrasse 60-62, 14195 Berlin, Germany.

Received 15 June 2006; revised 28 September 2006; accepted 16October 2006; posted 20 October 2006 (Doc. ID 72038); published13 March 2007.

0003-6935/07/101742-12$15.00/0© 2007 Optical Society of America

1742 APPLIED OPTICS � Vol. 46, No. 10 � 1 April 2007

refractive index of the RBC compared with bloodplasma. Even at a constant Hct, the blood parametersRBCc, Hbc, MCV, and MCH show an individual vari-ability, which can influence the absorption and scat-tering properties. Because it is not possible to preparethese parameters for experiments, the biological vari-ability always leads to an uncertainty in the deter-mination of the optical parameters at a defined Hct.Investigation of the optical influence of this biologicalvariability is another aim of this study.

Finally, the influence of the medium in which theRBCs are suspended was investigated. Human eryth-rocyte concentrates are often used instead of wholeblood due to the availability and less complicatedhandling, and blood plasma is often replaced by sa-line solution. Furthermore blood products such aserythrocyte concentrates are available only withaqueous buffer solutions as a medium. It is knownthat plasma can show different optical properties5

and influence the optical parameters of blood signif-icantly. Thus a further task of this paper is quanti-fication of the influence of plasma on the opticalparameters of blood compared with saline solution.

To reach these ends the optical parameters �a, �s,and g were determined on one sample dependent onthe Hct for eight different values from 0.84% to 42.1%using the Reynolds–McCormick phase function23

with an evaluated optimal parameter � for eachHct.

The influence of the biological variability of the sizeand shape of the RBCs and their Hb content on theoptical parameters was estimated by investigation often blood samples from ten different donors diluted tothe same Hct of 8.6%. These ten measurements re-sulted in averaged data for the optical parameters ofa standard blood sample. The Hct-dependent opticalparameters determined on one blood sample wererelated to the averaged standard data of a Hct of8.6% to obtain a statistical correction and, as a con-sequence, improve the validity of the measured opti-cal parameters for a blood sample of a known Hct.

In another step model, functions were determinedfor the calculation of optical properties for selectedHcts and wavelength regions using a data table ofbiologically averaged values of �a, �s, and g. Further-more, Mie calculations were made to see if changes inthe optical properties of the RBCs could be estimatedwhen blood plasma was used instead of saline solu-tion.

2. Material and Methods

A. Blood Preparation

Fresh human erythrocytes from healthy blood donorswere centrifuged three times and washed with anisotonic phosphate buffer (300 mosmol�L, pH 7.4) toremove the blood plasma and free hemoglobin. Toexamine the biological variability, erythrocytes fromten different blood donors were diluted with a bufferto a Hct of 8.6%. Erythrocytes from one blood donorwere used to investigate the Hct dependence. The Hctwas varied by diluting the sample with a buffer to

eight different values: 0.84%, 4.0%, 5.9%, 8.6%, 17.1%,25.6%, 33.2%, and 42.1%.

The influence of the medium on the optical prop-erties of the RBCs was investigated using one bloodsample of a Hct of 41.2%, washed with an isotonicphosphate buffer. One half of the sample was centri-fuged, and the supernatant was replaced by cell-freeblood plasma.

The Hct was determined using a RBC counter(Micros 60 OT 18, ABX Diagnostics, Montpellier,France). All the samples were oxygenated in excess of98%. The oxygen saturation was determined with ablood gas analyzer (OPTI Care, AVL MedizintechnikGmbH, Bad Homburg). These systems also provideinformation on the total Hb, MCV of erythrocytes,mean-Hbc and content within the cell, and the dis-tribution of the cell size. A miniaturized blood circu-lation setup was used with a roller pump (SorinGroup, Germany) and a blood reservoir, which wasconstantly aerated with a gas mixture of O2, N2, andCO2. The temperature was kept constant at 20 °C. Toavoid sedimentation or cell aggregation, the bloodwas gently stirred within the reservoir. The bloodwas kept flowing by a specially designed, turbulence-free cuvette with a laminar flow and a sample thick-ness of 1020 �m for a Hct of 0.84% and 116 �m for aHct of 4.0%–42.1%. The flow was regulated for eachsample thickness to a constant wall shear rate of600 s�1 at the cuvette windows. For the RBCs sus-pended in plasma, the shear rate was reduced rela-tive to the higher viscosity, as compared with water,to give equal shear forces and flow conditions.

B. Methods

The macroscopic optical parameters, diffuse reflec-tance Rd, total transmission Tt, and diffuse transmis-sion Td of all the blood samples were measured in thespectral range from 250 to 1100 nm using an integrat-ing sphere spectrometer (PerkinElmer, Lambda 900).A measurement was carried out every 5 nm leading to171 data sets. The experimental setup used, as de-scribed elsewhere,7,24 allows the measurement of themacroscopic radiation distribution with an error ofless than 0.1%. A special iMCS program was used,which considers the geometry of the optical setup andall radiation losses. Simulations with 106–107 pho-tons lead to statistical errors of less than 0.1%. Theerror of the optical parameters cannot be directlycalculated because the transformation function be-tween the macroscopic and the intrinsic parametersis a priori unknown. Typical standard deviations ofthree determinations for �a, �s, and �1 � g� are in therange of 3%–5%. The iMCS is a suitable method togive valid results for evaluation of the optimal effec-tive phase function and the calculated intrinsic opti-cal parameters. The iMCS uses an estimated set ofstart parameters �a, �s, and g from the Kubelka–Munk theory to calculate the resulting values Rd, Tt,and Td. These are then compared with the Rd, Tt, andTd values, which have been measured experimen-tally. By systematic variation of �a, �s, and g, the

1 April 2007 � Vol. 46, No. 10 � APPLIED OPTICS 1743

deviation of the simulated Rd, Tt, and Td values fromthe measured ones are minimized until a set of opti-cal parameters is found where the deviations arewithin the error threshold. Due to the high sensitivityof the simulation, it is possible to fit an adaptablephase function to the optimal form, e.g., the Reynolds–McCormick phase function by variation of the � factoras shown before.7

An error threshold of 0.1% for the deviation be-tween measured and simulated Rd, Tt, and Td wasused for calculation of the intrinsic optical parame-ters of blood using the determined effective phasefunctions. For all blood samples, a total of three in-dependent measurement series were carried out andindependently simulated. For calculations accordingto Mie theory, the refractive index of the used plasmawas measured with an Abbe refractometer at fourwavelengths (400, 500, 600, 700 nm). A whole refrac-tive index spectrum was calculated according to theSellmeier approximation:

n�� 1 �c1�

2

�2 � �12, (1)

with the fit parameters c1 � 0.7909 and �1 � 100.00.

3. Results and Discussion

A. Biological Variability

The influence of the biological variability in the statis-tical distribution of the RBCs size and shape and the

Hb content of the RBCs were estimated by investi-gation of blood samples from ten different donors withan identical Hct of 8.6%. The mean values and rangesin the RBCc, MCV, mean corpuscular hemoglobinconcentration (MCHC), and a Hbc of the ten sampleswere 1.008 � 106��l �0.96–1.05 � 106�, 86.5 �m3

(82–92.7), 32.9 g�L (30.4–35.3), and 2.87 g�dL (2.72–3.22 g�dL). Each sample was measured and sim-ulated a total of three times. The iMCS was carriedout using the Reynolds–McCormick phase functionwith � � 1.6. Figure 1 shows the averaged values of�a, �s, and g in the wavelength range of 250–1100 nmincluding the standard deviations. Furthermore theeffective scattering coefficient ��s � �s �1g� is pre-sented. The relative standard deviation for �a rangesfrom 1.9% to 8.5% with an average value of 4.9%. �s

shows relative standard deviations from 3.7% to13.7% with an average value of 8.4%. g and ��s showrelative standard deviations, which nearly correlatewith the Hb absorption. The relative standard devi-ation of �1 � g� ranges from 6.5% to 25.1% with anaverage value of 12.8%. The respective range of ��swas 5.3% to 23.3%, and the average value was 16.0%.The values of the standard optical parameters are pre-sented in Table 1.

Thus the spectrally averaged deviations of the scat-tering parameters induced by biological variabilitiesare approximately two to four times higher thanthose of preparation, measurement, and simulation,which normally range from 3% to 5%. Only the ab-sorption coefficient shows averaged standard devia-tions in a comparable range. The uncertainty in the

Fig. 1. Mean values of �a, �s, g, and ��s of ten blood samples of the RBCs in saline solution with a Hct of 8.6% from ten different donorsincluding the standard deviations in the wavelength range of 250–1100 nm.


determination of the optical parameters is induced bythe biological variabilities in the blood parametersRBCc, MCV, MCHC, and Hb concentration, whichcannot be prepared independently for experimentsbut are routinely determined by commercial clinicalblood analyzers. Therefore an attempt was made tocorrect the measured optical parameters theoreti-cally on the basis of the determined blood parame-ters. Based on the approximation that RBCs arespheres, Mie theory can be used to calculate changesin the scattering parameters induced by changes inthe sphere diameter given by the MCV and in thecomplex refractive index, which can be estimated

from the MCHC. The absorption coefficient was di-rectly normalized to the mean Hb concentration of allten samples of 2.87 g�L. However, the standard de-viations of the theoretically corrected parameterswere not significantly lower than those of the originalones. It has to be mentioned that even the meanvalues of MCHC, MCH, and MCV themselves showwide statistical distributions. The latter, for example,is normally between 80 and 100 �m3, but the distri-bution includes cells with 20 and 200 �m3. Thereforethe optical variability of blood of the defined Hct isdue to the variability in the mean values of otherblood parameters and their statistical distributions.

Table 1. Data of Standard Optical Parameters �aSt, �sSt, gSt, and �sSt� of the RBCs in Saline Solution with a Hct of 8.6% Dependent on Wavelengtha

Lambda(nm)

�aSt

(1�mm)�sSt

(1�mm) gSt

�sSt�(1�mm)

Lambda(nm)

�aSt

(1�mm)�sSt

(1�mm) gSt

�sSt�(1�mm)

250 7.52 27.9 0.877 3.43 470 3.13 31.3 0.9777 0.699255 7.93 27.2 0.880 3.27 475 2.79 31.5 0.9793 0.652260 8.60 26.6 0.883 3.10 480 2.52 31.8 0.9806 0.619265 9.41 25.7 0.883 3.01 490 2.20 32.0 0.9820 0.575270 9.82 25.4 0.882 2.99 500 2.04 32.2 0.9831 0.545275 9.93 25.2 0.884 2.93 510 1.98 32.2 0.9835 0.531280 9.74 25.3 0.889 2.80 520 2.51 31.5 0.9838 0.510285 9.19 25.5 0.899 2.59 530 3.97 29.5 0.9794 0.609290 8.24 25.9 0.9120 2.28 540 5.05 28.2 0.9755 0.691295 7.04 26.7 0.9268 1.96 550 4.40 29.0 0.9779 0.642300 6.00 27.4 0.9389 1.68 560 3.60 30.1 0.9804 0.590305 5.52 27.7 0.9449 1.53 570 4.56 28.8 0.9777 0.641310 5.56 27.5 0.9442 1.54 580 4.56 28.9 0.9771 0.662315 5.95 26.9 0.9398 1.62 590 1.90 32.1 0.9827 0.556320 6.50 26.0 0.9335 1.73 600 0.478 33.9 0.9854 0.496325 7.13 25.4 0.9280 1.83 610 0.170 34.2 0.9858 0.487330 7.69 25.2 0.9239 1.92 620 0.0812 34.3 0.9861 0.477335 8.10 24.9 0.9208 1.97 630 0.0496 34.2 0.9863 0.469340 8.38 24.7 0.9192 1.99 640 0.0348 34.3 0.9865 0.462345 8.48 24.6 0.9191 1.99 660 0.0251 34.1 0.9868 0.450350 8.37 24.6 0.9215 1.93 670 0.0239 33.9 0.9871 0.439355 8.07 24.7 0.9257 1.84 690 0.0243 33.6 0.9872 0.430360 7.69 24.7 0.9308 1.71 700 0.0246 33.4 0.9872 0.427365 7.40 24.7 0.9345 1.61 720 0.0284 32.9 0.9871 0.426370 7.37 24.5 0.9356 1.58 740 0.0360 32.4 0.9870 0.422375 7.72 24.0 0.9323 1.62 760 0.0461 31.8 0.9868 0.420380 8.58 23.2 0.9242 1.76 780 0.0558 31.2 0.9867 0.415385 9.98 22.2 0.9105 1.99 800 0.0641 30.8 0.9868 0.407390 11.86 21.1 0.892 2.29 820 0.0762 30.5 0.9867 0.407395 14.21 20.1 0.869 2.64 840 0.0853 30.5 0.9867 0.405400 16.86 19.1 0.845 2.96 860 0.0953 29.7 0.9864 0.403405 19.66 18.3 0.824 3.23 880 0.104 29.8 0.9861 0.413410 21.80 17.9 0.810 3.39 900 0.106 28.4 0.9857 0.405415 22.66 18.6 0.812 3.48 920 0.111 28.1 0.9854 0.411420 21.54 19.8 0.831 3.35 940 0.117 27.7 0.9847 0.423425 18.62 21.8 0.861 3.03 960 0.125 26.8 0.9840 0.429430 14.93 23.8 0.893 2.55 980 0.133 26.1 0.9836 0.428435 11.52 25.3 0.9200 2.02 1000 0.128 25.8 0.9837 0.421440 8.95 26.8 0.9405 1.60 1020 0.118 25.7 0.9839 0.412445 7.11 28.0 0.9543 1.28 1040 0.104 25.3 0.9837 0.412450 5.79 29.0 0.9629 1.08 1060 0.0879 25.0 0.9838 0.406455 4.83 30.0 0.9685 0.946 1080 0.0795 24.6 0.9841 0.392460 4.11 30.5 0.9724 0.841 1100 0.0740 24.5 0.9842 0.387465 3.56 31.0 0.9756 0.757

aThe standard deviations are discussed in Subsection 3.A.


To minimize the unavoidable statistical uncer-tainty when measuring one blood sample, the ob-tained biologically averaged parameters �a, �s, g, and��s are used in this paper as statistical standard pa-rameters to which the Hct-dependent optical param-eters determined on one blood sample will be related.

B. Evaluation of Appropriate Phase Functions Dependenton the Hemocrit

As described by Friebel et al.,7 the Reynolds–McCormick phase function could be evaluated as anappropriate effective phase function for blood with aHct of 0.84% and 42.1%, having � values of 1.25 and1.7, respectively. The evaluation procedure is basedon the assumption that the scattering properties ofblood can be described by the Reynolds–McCormickphase function using one � value for the whole spec-tral range of 250–1100 nm. Using a special time-extensive iMCS without error threshold, as describedby Friebel et al.,7 it has been shown that at least one� value could be found for the whole spectral rangefor the description of the scattering properties of theexamined blood. The same could be found for all otherblood concentrations between a Hct of 0.84% and42.1%. The best � for the whole spectrum between250 and 1100 nm was found by simulating all 171data sets of Rd, Tt, and Td with a fixed error thresholdof 0.05%. The best � was indicated by the fit with thelowest number of simulations exceeding the errorthreshold. Figure 2 shows the best � values for eachHct measured. For highly diluted blood of a Hct of0.84%, � is 1.23. The optimal � value increases withblood concentration up to a Hct of approximately 10%where � is in the range of 1.6–1.7. At higher Hctvalues, � shows no significant dependence on a Hctwith values between 1.65 and 1.7. The mean error ofthe determination of the best � is approximately�0.1. The increase in Hct reduces the mean distancebetween erythrocytes. This might lead to the phe-nomenon that the scattering in one cell is no longerindependent of the surrounding cells. This interfer-ence of scattered waves (collective scattering) by

many cells with small intercellular distances willchange the values of g and � compared with the sit-uation of large intercellular distances at a Hct of0.84%.

C. Optical Parameters Dependent on the Hematocrit

The macroscopic optical parameters Rd, Tt, and Td ofa RBC sample from one donor suspended in salinesolution and diluted to eight different concentrationswere measured, and the iMCS was carried out usingthe evaluated Hct-dependent effective phase func-tions. To minimize the influence of the biological vari-ability, the determined �a, �s, g, and ��s spectra of allthe blood concentrations were related to the averagedstandard values at a Hct of 8.6%. Biologically aver-aged optical parameters, dependent on a Hct, can beobtained by replacing the measured spectrum of aHct of 8.6% by the averaged standard spectrum of aHct of 8.6% and then recalculating all the other Hct-dependent spectra related to a Hct of 8.6%. Figure 3shows the biologically averaged �a, �s, g, and ��s de-pendent on a wavelength for the eight different bloodconcentrations between a Hct of 0.84% and 42.1%.

It is obvious that �a, �s, and ��s increase continu-ously with a Hct over the whole wavelength rangewhereas g decreases. The fundamental wavelengthdependence of the optical parameters was discussedpreviously.7 To quantify the dependence on the Hct,the relative intrinsic parameters calculated as a pa-rameter of a given Hct divided by the parameter at aHct of 8.6% were investigated in more detail. Figure4 shows the relative intrinsic parameters �a_rel,�s_rel, g_rel, and ��s_rel dependent on wavelength.

Up to a Hct of 25.6%, �a_rel shows no indication ofwavelength dependence in the whole wavelengthrange. At a Hct of 33.2% and more distinctly at a Hctof 42.1%, �a_rel is in the region below 600 nm,slightly higher than in the range of 600–1100 nm.This increase correlates with the Hb absorption ex-cept in the area of highest absorption at 415 nm.Instead of a positive peak, �a_rel shows a negativepeak at a Hct of 33.2%, which increases at a Hct of42.1%. This is probably a saturation effect due to themaximal optical density at this wavelength leadingto a nonsignificant transmission signal. �s_rel and��s_rel show no wavelength dependence below a Hct of17.1%. Above a Hct of 17.1%, �s_rel and ��s_rel areonly independent of a wavelength between 600 and1100 nm. The wavelength dependence above a Hct of17.1% in the range of 250–600 nm increases with theHct but seems to be inversely related to the Hb ab-sorption. The relative anisotropy factor g_rel is onlywavelength independent above 600 nm for all Hcts.Below 600 nm, g_rel shows a strong dependence onHb absorption, which increases with the Hct. Increas-ing absorption leads to a decrease of g_rel. The g_relspectrum at higher Hcts is practically the mirror im-age of the absorption coefficient.

Within the wavelength and Hct ranges, where therelative parameters were approximately independentof the wavelength, the intrinsic parameters were av-eraged over the whole wavelength range and related

Fig. 2. Most appropriate � values for eight different blood con-centrations between a Hct of 0.84% and 42.1%.


to the Hct. Figure 5 shows the averaged relative in-trinsic parameter �a_rel dependent on a Hct for thewavelength ranges of 250–600 nm and 600–1100 nmwith approximation curves and correlation coeffi-cients. The averaged �a_rel in the wavelength rangeof 250–600 nm increases linearly and slightly steeperthan in the range of 600–1100 nm. The linear in-crease of �a_rel with a Hct indicates that the absorp-tion cross section a can be seen as constant over thewhole Hct range.

The higher steepness of �a_rel in the spectral re-gion of high Hb absorption, compared with the lowabsorption area above 600 nm, is probably due to thesieve effect. The sieve effect describes the attenuatedabsorption when the absorbing particles are not ho-mogeneously distributed. It increases markedly withthe absorption within the cell but decreases with theincreasing Hct. The influence of the Hct increaseswith the absorption.7,25 Therefore the attenuation ofthe absorption induced by the sieve effect occursmainly in the wavelength range of 250–600 nm,where the Hb absorption is high and decreases withincreasing Hct.

Figure 6 shows the averaged relative intrinsic pa-rameter �s_rel in the wavelength range of 600–1100nm dependent on a Hct including approximationcurve and correlation coefficients. The figure alsoshows �s_rel for the single wavelengths of 250 and415 nm. �s_rel is seen to be linear up to approxi-mately a Hct of 10%. Above a Hct of 10%, �s_relshows a saturation effect. At 250 nm, where �a is �2orders of magnitude higher, the saturation effect is

more distinct. When �a is at a maximum at 415 nm,the saturation effect is maximal. The relative �s iscompared with the approximation by Twersky26 whodescribed the Hct dependence of �s as

�s �Hct�1 � Hct�

MCV s, (2)

where s is the scattering cross section. For compar-ison, �s_rel was calculated using the scattering crosssection, which was estimated from the simulated �s

and the RBCc at a Hct of 0.84% where independentscattering is probably dominant.7 �s_rel shows a sim-ilar saturation effect with increasing Hct, but all thevalues remain below the ones calculated according toEq. (1). At a Hct of 42.1%, the measured �s_rel is 29%lower. A similar saturation effect above a Hct of 10%was observed by Roggan et al.1 at 633 nm and Faberet al.14 at 800 nm.

At 250 and 415 nm where the saturation effectstarts earlier, �s_rel at a Hct of 42.1% is 35% and43%, respectively, lower than predicted by Twersky.The saturation effect seems to increase with increas-ing absorption as depicted in Fig. 4.

The saturation effect of �s is probably induced bythe decrease of the mean distance between cells.Even in highly diluted blood of a Hct of 0.86%, themean distance is not large enough to ensure the ex-istence of independent single scattering. Under flowconditions, the RBCs tend to have a higher concen-tration in the central part of the flow leading to fur-ther reduced distances or contacts between cells.

Fig. 3. (Color online) Biologically normalized values of �a, �s, g, and ��s of RBCs in saline solution of eight different blood concentrationsbetween a Hct of 0.84% and 42.1% in the wavelength range of 250–1100 nm. Only the curves of a Hct of 0.84% and 42.1% are speciallymarked because all the curves change continuously with the Hct.


Therefore interfering waves from neighboring cellscannot be neglected, and collective scattering be-comes increasingly important.

The absorption-dependent increase of the saturationeffect of �s is in agreement with Mie theory. An in-crease in the imaginary part of the refractive index ofspheres of comparable size leads to a decrease in thescattering cross section and therefore to a reduced �s.

The averaged relative anisotropy factor g_rel forthe spectral range of 600–1100 nm dependent on theHct is depicted in Fig. 7. As presented for �s_rel, the

Hct dependence is also shown for the single wave-length of 250 and 415 nm.

Regarding the standard deviation of 3%–5% for�1 � g� including preparation, measurement, andsimulation, the error of g_rel results in approxi-mately 1%. Therefore the differences in g_rel from aHct of 0.84%–8.6% are not significant for 250 and415 nm. Above a Hct of 8.6%, g_rel decreases pro-gressively to 0.88 at a Hct of 42.1% at 250 nm and to0.82 at 415 nm. Comparable with �s_rel, g_rel shows

Fig. 4. (Color online) Relative intrinsic parameters �a_rel, �s_rel, g_rel, and ��s_rel of eight different concentrations of RBCs in salinesolution between a Hct of 0.84% and 42.1% related to a Hct of 8.6% in the wavelength range of 250–1100 nm. Only the curves of a Hctof 0.84% and 42.1% are specially marked because all the curves change continuously with the Hct.

Fig. 5. Averaged relative intrinsic parameter �a_rel dependenton the Hct for the wavelength ranges of 250–600 nm and 600–1100 nm with approximation lines.

Fig. 6. Averaged relative intrinsic parameter �s_rel dependent onthe Hct for the wavelength range of 600–1100 nm with approxi-mation curves and for the wavelengths of 250 and 415 nm and�s_rel calculated using Eq. (1).


a maximal decrease in Hct at 415 nm where the ab-sorption is at a maximum indicating the decrease of gwith an increasing Hct is dependent on Hb absorption.

A decrease of g with increasing Hct is also a conse-quence of the decrease of the mean distance betweencells.7 The increase in the complex refractive index inregions of high absorption results in an increase inreflection and a decrease in transmittance resultingin increased backscattering of the photons, which isidentical to a decrease of g.

Figure 8 shows the averaged relative intrinsic pa-rameter ��s_rel dependent on the Hct in the wave-length range of 600–1100 nm and for the singlewavelength of 250 and 415 nm. The averaged ��s_relincreases linearly with the Hct up to a Hct of 42.1%.At 250 nm and 415 nm, ��s_rel increases linearly onlyup to a Hct of 25.6% or 17.1%, respectively. At Hctsabove these values, ��s_rel increases continuously butwith decreasing steepness. At a Hct of 42.1% ��s_rel isat 250 nm 10% and at 415 nm 24% lower than theaveraged value for the wavelength range of 600–1100 nm. As shown for �s_rel, also ��s_rel shows sat-uration effects at 250 and 415 nm. At 415 nm, the

saturation starts at a lower Hct than at 250 nmindicating the absorption dependence of the effect.According to the results of Roggan et al.1 who deter-mined ��s of RBCs in saline solution in the Hct rangeof 2.5%–70% at 633 nm, ��s_rel will probably alsoshow a saturation effect above a Hct of 45% in thewavelength range from 600 to 1100 nm where the Hbabsorption is low.

It should be noted that the Hct-dependent satura-tion effect of �s_rel and the Hct-dependent decreaseof g_rel starts at much lower Hcts where the effectivescattering coefficient ��s_rel still increases linearlywith the Hct. These saturation effects of the scatter-ing parameters, in principle, correspond to the theo-retical considerations that the scattering of blood of aHct at 100% would be zero due to the disappearanceof the refractive index difference between cells andplasma.27 As a consequence, there must be a maxi-mum between a Hct of 0% and 100% including asaturation effect in a certain Hct range.

As discussed above, the Hct-dependent nonlineareffects of �s, ��s, and g may be explained by the de-crease in independent single scattering and an in-crease in collective scattering induced by the decreaseof the mean distance between cells. But the specialdistribution is not entirely homogeneous leading toincreasing occurrence of cells, which are very close orin contact with their neighbors. This double or triplecluster of cells may be interpreted as one large cell.This leads to a reduction in the scattering particleconcentration combined with an increase in the scat-tering cross section. The concentration is a three-dimensional parameter whereas the scattering crosssection is greatly influenced by the geometric crosssection, which is two dimensional. Therefore the in-crease in the scattering cross section does not equal-ize the reduction of cell concentration resulting in areduced �s. The reduction of the scattering efficiencyper cell induced by contact areas studied on RBCaggregates was described by Enejder et al.16 Accord-ing to Mie theory, an increase in the diameter of ascattering sphere in a comparable size range leads,on average, to a decrease in the anisotropy factor forthe respective wavelength range. At the beginningof these collective scattering phenomena, the Hct-dependent effects on �s and g seem to compensateeach other at low blood concentrations. With a fur-ther increase in blood concentration, ��s also shows asaturation effect because the increasing number ofcell-to-cell contacts results in increased cell concen-tration and reduces the total boundary area betweenvolumes of different refractive indices. This wouldlead in the end to a complete disappearance of thescattering at a Hct of 100%.

D. Influence of the Medium on the Optical Parameters ofRed Blood Cells

To investigate the optical influence of the suspensionmedium, RBCs from the same donor were suspendedin both saline solution and blood plasma. Figure 9shows �a, �s, g, and ��s spectra of RBCs suspended ina 0.9% saline solution versus RBCs suspended in

Fig. 7. Averaged relative intrinsic parameter g_rel dependent onthe Hct for the wavelength range of 600–1100 nm with approxi-mation curves and for the wavelengths of 250 and 415 nm.

Fig. 8. Averaged relative intrinsic parameter ��s_rel dependent onthe Hct for the wavelength range of 600–1100 nm with approxi-mation curves and for the wavelengths of 250 and 415 nm.


blood plasma. An Hct in the physiological range of41.2% was chosen. In addition, the optical parame-ters for the RBCs in plasma are depicted, which weretheoretically determined by Mie theory. Because ofthe restricted validity of Mie theory for RBCs,7 theoptical parameters were not absolutely, but relativelycalculated from the simulated parameters of theRBCs in saline solution according to the changed re-fractive index of the medium. Mie calculations werecarried out averaging over different sphere diameterscorresponding to the MCV of the blood sample and itsstatistical distribution. The complex refractive indexof the cells was calculated from the MCHC accordingto the model function of Friebel and Meinke.3

The absorption of RBCs in plasma in the spectralrange up to 600 nm is only 3.2% lower, on average,than in saline solution. Above 600 nm, the differenceincreases to 16.6%, on average. The absorption ofplasma itself, which shows normal bands in the rangeof 250–400 nm, does not seem to significantly influ-ence the RBC absorption in this case. Mie theory alsopredicts over the whole wavelength range a decreasein the absorption cross section of RBCs in plasma by2.3%, on average. This does not reach the extent ofthe measured absorption decrease in the range of600–1100 nm.

A differentiated discussion of the optical parame-ters between the ranges of 250–600 nm and 600–1100 nm is based on the different optical behavior of

RBCs in these ranges. Below 600 nm, high absorp-tion bands drastically influence the scattering prop-erties and therefore determine the optical behavior.Above 600 nm, the absorption is low and the scatter-ing properties dominate.

The scattering coefficient of RBCs in plasma, in therange of 250–600 nm, is on average 5.5% lower thanin saline solution. Above 600 nm, �s in plasma is onaverage 9.4% lower. Calculated scattering cross sec-tions according to Mie theory show a strong increaseup to 37% between 250 and 600 nm. A reduction inthe scattering cross section occurs above 640 nm andincreases with the wavelength up to 29%.

The anisotropy factor in plasma from 250 to600 nm is on average 0.22% lower than in salinesolution. Above 600 nm, the difference increases toan averaged value of 0.77%. The Mie theory shows ageneral increase of g but stronger values from 250 to600 nm (0.61%) than in the range of 600–1100 nm(0.14%).

As a consequence, the effective scattering ��s ofRBCs in plasma between 250 and 600 nm is 10%lower, on average, than in saline solution. In therange of 600–1100 nm, the reduction of ��s in plasmais 35%, on average, which can be almost exactly pre-dicted by Mie theory (36.5%). Below 600 nm, Mietheory gives ��s values, which are on average 32%below the respective values for saline solution.

Fig. 9. Biologically normalized values of �a, �s, g, and ��s dependent on the wavelength of RBCs suspended in 0.9% saline solution andsuspended in blood plasma (Hct of 41.2%). In addition, the optical parameters are depicted for spheres of the same volume together withthe refractive index in plasma, theoretically determined by Mie theory from the parameters of the RBCs in saline solution.


As expected, the substitution of blood plasma forsaline solution leads to a significant decrease in thescattering cross section. It also brings about an in-crease in the anisotropy factor due to the rise in therefractive index of the medium resulting in a de-crease of the refractive index difference between cellsand medium. With the exception of the scatteringcross section from 250 to 600 nm, the changes in alloptical parameters for undiluted blood suspended inplasma can be generally confirmed by Mie theory.However precise predictions of the optical parame-ters when replacing saline solution of a RBC suspen-sion by plasma or vice versa on the basis of thechanged refractive indices of the medium are onlypossible for ��s from 600 to 1100 nm. This is to beexpected according to the results of Friebel et al.7 whoshowed that Mie theory for undiluted blood is validonly for ��s in this spectral region.

The scattering and absorption coefficients of plasmaitself are negligible compared with the parameters ofRBC suspensions of physiological concentration. Incontrast, when investigating diluted blood, the scat-tering and absorption properties of plasma may in-fluence the composed optical properties. As shown byMeinke et al.5 at a blood sample of a Hct of 8.6%, �a

in the wavelength range below 300 nm of RBCs inplasma exceeds the one of RBCs in NaCl by up to 55%induced by absorption of the plasma itself. �s wasdetermined to be 15% lower in the plasma suspen-sion, which is a bigger difference than at a Hct of41.2%. This may have been caused by systematic er-rors in the blood preparation. The experimental ex-change of saline solution by plasma may be lesscomplete at the high Hct because the ratio of thevolume of the exchanged liquid to the irreplaceablerest volume between the centrifuged RBCs is smaller.Such uncontrolled dilution with saline solution leadsto a reduced refractive index of the plasma and con-sequently to a reduction in the refractive index-dependent effects. The anisotropy factor of RBCs inplasma of a Hct of 8.6% was shown to be made up ofthe anisotropy factor of the RBCs and the one ofplasma, leading to a decrease in g in wavelengthsbelow 500 nm and an increase above 500 nm com-pared with the saline solution suspension. Thus es-timation of the changes in optical parameters is morecomplicated at low Hcts because the optical proper-ties of the RBCs are significantly influenced by theoptical properties of plasma, which shows a widerbiological variability than the RBCs in certain spec-tral regions especially in the absorption. Thereforefor diluted blood, the intrinsic optical parameters ofplasma must be measured separately and put to-gether with the parameters of the RBCs in addition tothe effect of the changed refractive index.

E. Model Functions

The linear increase of �a_rel with a Hct over thewhole wavelength range, with the exception of thearea of high absorption between 400 and 430 nm,allows a simple estimation of the absorption coeffi-cient of a blood sample of RBCs in saline solution of

a known Hct or Hb content. In combination withTable 1, which gives the standard optical parameters�aSt, �sSt, and gSt of blood at a Hct of 8.6% and a Hb of2.87 g�L, �a of the RBCs in saline solution can beapproximately calculated at any Hct from 250 to400 nm and from 430 to 600 nm using the approxi-mation formulas in Fig. 5:

�a��, Hct� � 0.1233�aSt��Hctfor 250 nm � 400 nm, 430 nm � 600 nm,

0.84% Hct 42.1%. (3)

In the wavelength range of 600–1100 nm, the modelfunction is

�a��, Hct� � 0.1206�aSt��Hctfor 600 nm � 1100 nm, 0.84% Hct 42.1%.

(4)

For the area of 400–430 nm and Hcts over 25.6%,�a_rel has to be interpolated from nearby Hct valuesread from Fig. 3. The mean deviations of �a values arecalculated using Eqs. (3) and (4), and the originalsimulated values averaged over all defined wave-lengths range from 1.0% error at a Hct of 42.1% to11% at a Hct of 0.84%. The scattering coefficient �s ofa blood sample of the RBCs in saline solution of aknown Hct can be estimated using a model functionover the whole wavelength range of 250–1100 nm forHcts below 17.1%. In combination with the data �sStfrom Table 1, �s can be approximately calculated forthe RBCs in saline solution with any Hct by using

�s��, Hct� � ��0.0015 Hct2 � 0.1268 Hct��sSt��for 250 nm � 1100 nm, 0.84% Hct 17.1% for 600 nm �

1100 nm, 17.1 Hct 42.1%. (5)

Above a Hct of 17.1% and below 600 nm, �s_rel has tobe interpolated from nearby Hct values read fromFig. 3. The mean deviations of the calculated �s val-ues, using Eq. (5) and the simulated original ones, areaveraged over all defined wavelength ranges from2.3% at a Hct of 42.1% to 29% at a Hct of 0.84%. Therelative anisotropy factor g can be estimated only forall Hcts in the wavelength region of 600–1100 nm.Using the standard parameters gSt in Table 1, anapproximation formula (6) for g of the RBCs in salinesolution can be derived for any Hct:

g��, Hct� � ��2.684 � 10�6 Hct2 � 2.373 � 10�4 Hct

� 1.003�gSt�� for 600 nm �

1100 nm, 0.84% Hct 42.1%. (6)

Below 600 nm, g_rel has to be interpolated fromnearby Hct values read from Fig. 3 or for Hcts below25% backcalculated from �s and ��s. For �1-g�, themean deviations of the calculated values using Eq. (6)and the simulated original ones are averaged over all


defined wavelengths between 1.7% and 8.5% not cor-related with the Hct.

The effective scattering coefficient ��s can be cal-culated from �s and g or by a linear model function[Eq. (7)] over the whole range from 250 to 1100 nmfor Hcts below 17.1%. Above a Hct of 17.1%, the for-mula is valid only in the wavelength range of 600–1100 nm:

��s��, Hct� � 0.1167 � ��sSt��Hct for 250 nm �

1100 nm, 0.84% Hct 17.1%.for 600 nm � 1100 nm,17.1 Hct 42.1%. (7)

Above a Hct of 17.1% and below 600 nm, ��s_rel can beinterpolated from Fig. 3 as described for �s and g.Calculations using Eq. (7) show mean deviationsfrom the simulated original �s� values averaged overall defined wavelengths between 0.5% at a Hct of42.1% and 23% at a Hct of 0.84%. Because of the lowvalues of the optical parameters �a, �s, and ��s at a Hctlower than 4%, the relative errors for these parame-ters are disproportionally high. Excluding this lowHct, the relative mean deviations of the calculatedoptical parameters of all other Hcts and all definedwavelengths result in 3.9% for �a, 4.0% for �s, and2.5% for ��s, which are in agreement with the meandeviation for �1 � g� of 4.8%.

4. Conclusion

For this study, a high-precision integrating spheresetup for measuring Rd, Tt, and Td in combinationwith a high-resolution iMCS was used to determinethe intrinsic optical parameters �a, �s, and g of hu-man RBCs under flow conditions dependent on aHct in the wavelength range of 250–1100 nm. TheReynolds–McCormick phase function could be evalu-ated as an appropriate effective phase function foreach Hct within the investigated Hct range of 0.84%–42.1%. It has been shown for each investigated RBCconcentration that at least one � value can be foundfor the whole spectral range for the description of thescattering properties. The optimal value of the factor� at a Hct of 0.84% is 1.23. � increases to values in therange of 1.6–1.7 with a blood concentration up to aHct of approximately 10% and seems to be constantat higher Hct values up to 42.1%.

The influence of the biological variability of blood ofdefined concentration to the intrinsic optical param-eters could be shown by investigating ten blood sam-ples of the same Hct of 8.6% of ten different donors.The relative standard deviation averaged for �a overall wavelengths was 4.9% and for �s was 8.4%. Therelative standard deviation of �1 � g� shows an aver-age value of 12.8%. The corresponding value of ��s is16.0%. This uncertainty in the determination of theoptical parameters, which is higher than any othermeasurement error, is mainly induced by the variousstatistical distribution of the mean blood parameterscell volume (MCV) and Hb concentration within thecell (MCHC) within the blood samples. To minimize

this genuine biological error, the biologically aver-aged parameters for the RBCs in saline solution of aHct of 8.6% were used as statistical standard param-eters �aSt, �sSt, gSt, and ��sSt, which were related to theHct-dependent optical parameters determined on oneblood sample.

Based on these optical standard parameters forblood of a Hct of 8.6%, model functions could be de-rived to estimate the optical parameters �a, �s, and gof a given blood sample with a Hct between 0.86% and42.1% for 600–1100 nm where the Hb absorption islow. For �a, an additional model function could begiven for the wavelength range from 250 to 600 nm.The error of the calculation of the optical parametersusing these empirical model functions is lower thanthe error due to the biological variability. Influence ofthe medium on the optical parameters of the RBCswas investigated by comparing the optical parame-ters of the RBCs suspended in saline solution with aHct of 42.1% with those suspended in plasma.

The substitution of blood plasma for saline solutionleads to a decrease in the scattering cross section by5.5%, on average, between 250 and 600 nm and by9.4% between 600 and 1100 nm. The anisotropy fac-tor increases by 0.22%, on average, between 250 and600 nm and by 0.77% between 600 and 1100 nm dueto the rise in the refractive index of the medium,which results in a decrease of the refractive indexdifference between cells and medium. The absorptioncross section of the RBCs is also slightly decreased by3.2% and 16.6%, on average, within the respectivewavelength regions. With the exception of the scat-tering cross section in the wavelength range of 250–600 nm, the changes in all the optical parameters forundiluted blood suspended in plasma can be gener-ally confirmed by Mie theory. However precise pre-dictions of the optical parameters of undiluted bloodwhen replacing the medium are possible only for ��s inthe wavelength range of 600–1100 nm where ��s de-creases by 35%, on average.

This work was supported by the Federal Ministryof Education and Research (grant 13N7522). Theauthors also thank the Sorin Group DeutschlandGmbH, Munich, for placing the equipment at ourdisposal. The blood bags were kindly provided bythe Department of Transfusion Medicine, Charité-Universitätsmedizin Berlin, Germany.

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Empirical model functions to calculate hematocrit-dependent optical properties of human blood