effective phase function for light scattered by blood

Effective phase function for light scattered by blood

Ioan Turcu

The scattering process induced in blood by a collimated laser beam is theoretically investigated. Anindividual red blood cell (RBC) has a scattering phase function strongly peaked in the forward direction.For far-field experiments, the small scattering volumes can be considered as “macroscopic particles”characterized by an effective scattering phase function. Using the single-cell phase function as “inputdata” the angular distribution of light scattered at small angles by the whole scattering volume, con-taining RBCs in suspension, is calculated analytically. The angular dispersion of the light scattered byblood can be approximately described by the same formula used to characterize the light scattered by asingle cell but with an effective, hematocrit-dependent anisotropy parameter. © 2006 Optical Society ofAmerica

OCIS codes: 290.0290, 290.4210, 290.5850.

1. Introduction

Light propagation through disperse systems, such asbiological cells in suspensions or organized in tissues,has been studied extensively. However, in most prac-tical cases, the scattering process modeling is an ex-tremely difficult task. Most of the biological cells havecharacteristic dimensions at least ten times higherthan the incident light wavelength � (radii a �� ) andrelative refraction indexes m close to 1 �m � 1 �� 1�.For all of them the scattering phase functions arestrongly peaked in the forward direction or, equiva-lently, the asymmetry parameter g (the mean cosineof the scattering angle) is close to 1: 1 � g �� 1.

One of the simplest biological systems used to testthe theoretical models is the red blood cell (RBC). Ifthe light-scattering properties of an individual RBCare relatively well understood,1–4 the spatial and an-gular distribution of the light intensity scattered by acollection of RBCs in suspension is extremely com-plex and only partly understood.5–8

There are many experiments using collimated la-ser beams in which one detects the scattered field faraway as compared to the relatively small scatteringvolume.2,5,7,9 Measured angle-resolved intensity ofscattered light is usually compared with the predic-

tions of Mie theory, the Rayleigh–Gans approxima-tion, and the anomalous diffraction approximation.1,2

There are also several empirical phase functions thatare frequently fitted to the measurements. For RBCsthe Henyey–Greenstein phase function and the two-parameter Gegenbauer kernel phase function arethe most commonly used.2,5,10 Additionally, rele-vant published works are dedicated to solving nu-merically the problem of electromagnetic wavescattering by RBCs with methods such as the three-dimensional (3D) boundary-element method,4,8 thesemianalytical T-matrix approach,11 the finite-difference time-domain method (FDTD),2,13 or thediscrete dipole approximation (DDA)13.

The essential contributions of the angle-resolvedscattering Monte Carlo simulations5,14,15 must be alsoemphasized. Monte Carlo techniques are widely usedfor solving radiative transfer problems using as inputthe single-cell phase function, sample parameters,the geometry of the experimental setup, etc. The sim-ulated light flux versus scattering angle, obtained asoutput, is compared with experimental data and theagreement extent is taken as a measure of the ade-quacy of the hypothetical phase function introducedas input.

The present contribution develops a new theoreti-cal model that gives, in a direct analytic way, infor-mation that is usually obtained by complex numericalcomputations or by Monte Carlo simulations. Usingthe single-cell phase function as “input data” the an-gular distribution of light scattered at small anglesby RBCs in suspension is calculated analytically for aquasi-ballistic scattering process. The proposed for-mulas can be used to fit the measured data and to find

I. Turcu ([email protected]) is with the National Institute forResearch and Development of Isotopic and Molecular Technolo-gies, P.O. Box 700, RO-400293 Cluj-Napoca, Romania.

Received 28 January 2005; revised 16 May 2005; accepted 30May 2005.

0003-6935/06/040639-09$15.00/0© 2006 Optical Society of America

1 February 2006 � Vol. 45, No. 4 � APPLIED OPTICS 639

the primary parameters of the light scattered byRBCs: the single scattering anisotropy and the RBCmean scattering cross section, respectively.

At distances larger than 10 cm, for example, thesmall scattering volume (approximately 1 mm3) de-limited by a collimated laser beam incident on a 1 mmthick cuvette can be considered a “macroscopic par-ticle” characterized by an effective scattering phasefunction.16 The main goal of this paper is to findsimple approximate analytic solutions for the effec-tive scattering phase function of the RBCs in suspen-sion.

2. Quasi-Ballistic Scattering

The present contribution focuses on the quasi-ballistic scattering process induced in a diffusiveplane-parallel sample, containing RBCs in suspen-sion, by a collimated laser beam incident along thenormal direction. We will restrict in the followingthe frequency range of the laser in the red domain ofthe spectrum where the photons are mainly scat-tered, with the absorption being extremely small. Thestandard example is the red light �� 633 nm� of aHe–Ne laser for which the ratio of the absorp-tion and scattering cross sections is very small,2�a��s � 10�2, and in the first approximation the pho-ton absorption can be neglected, � � �a � �s � �s.

The main hypothesis used throughout the paper isthat the scattered light remains strongly peaked inthe forward direction. This means that either thesuspensions are relatively diluted or the cuvettes arethin enough or both. It is very important for the scat-tering process to remain in the quasi-ballistic regime,which allows us to neglect the backward scattering inthe first approximation.

With the simplifying hypotheses introduced, thescattering process can be approximated simply as aredistribution of the incident photons in differentclasses, each indexed by the scattering order. Insidethe sample the number of incident photons decreasesby one at each scattering event, increasing by one thenumber of photons from the first-order scattered flux.It is very easy to observe that the rule is the same forany scattering order: the nth-order flux decreases ateach new scattering event (the scattered photon goesinto the flux n-1) and increases for each scatteringevent that occurs in the flux n-1. The photon scatter-ing is a random process, so the natural description isin probabilistic terms. In a stationary steady statethe total photon flux is independent of the depth in-side the sample and is always equal to the incidentflux.

The normalized photon flux � can be split in suc-cessive order scattering fluxes6,16:

�� 0�� s��, �s�� n�1

�n��, (1)

where �0 is the reduced incident flux, �s is thescattered flux, and �1, �2, · · ·, �n, · · · are the first-,second-, · · ·, nth-order scattered fluxes, respectively.

We have introduced the optical depth � � sz,

where s � ��s is the scattering coefficient �s�1 is the

photon mean free path), � is the cell density, �s is thescattering cross section, and z is the coordinate alongthe direction of the incident laser beam. In practicethe hematocrit H of a sample is frequently used,namely the volume fraction of the suspended RBCs.This parameter is directly related to the cell density� � H�v, where v is the mean volume of the RBC. Forconcentrated suspensions the hematocrit dependenceof the scattering coefficient must be amended totake into account the second-order term5,17 s �H�1 � H��v.

A simple “birth and death” model can describe thespatial variation of photons in the defined fluxes:

d�0��d��

� ��0��, �0�0� � 1, (2)

d�n��d��

� ��n�� n�1��,

�n�0� � 0, n � 1, 2, 3, . . . , (3)

�n�1

�n�� 1. (4)

Equation (2) describes the decrease of the incidentphoton flux due to the first scattering event. Thevariation of the nth-order scattered flux is given bythe imbalance between the number of photons scat-tered n times (which enrich �n) and the number ofphotons scattered n � 1 times (which leave �n andincrease �n�1), respectively.

We must emphasize that the simple form of theequations is an acceptable approximation in higherscattering orders only if the angular spreading of themultiply scattered photons remain reasonably small.

The reduced incident flux of the nonscattered pho-tons decays, as expected, exponentially:

�0�� e��, (5)

and the hierarchy of interconnected equations forhigher scattering orders can be solved iteratively:

�n�� n

n! e��, (6)

giving Poisson-type photon distributions9 inside thesample. Equation (6) shows us also that for a givenscattering order n, the maximum of the flux �n isattained at a value of optical depth equal to n:�n�� n�n�. In other words, if the depth inside thesample is n times larger than the photon mean freepath, the probability to find photons scattered n timesattains its maximum. Several scattering orders belowand above n are also present but with decreasingprobabilities, while the photons with scattering or-

640 APPLIED OPTICS � Vol. 45, No. 4 � 1 February 2006

ders that are not in the neighborhood of n are almostabsent.

Since the total flux remains unchanged at anydepth, the reduction of the incident flux must be com-pensated by the corresponding increase of the scat-tered flux:

�s�� n�1

�n

n! e�� e� � 1�e�� 1 � e�� 1 � �0��.

(7)

Usually it is not very easy to measure the total scat-tered flux. It is much more convenient to measure theintensity of light scattered in a solid angle � �sin � � � � A�D2 �� and � are the polar and azi-muthal angles of the detection direction) given by thecapturing area A of the photodetector and the samplephotodetector distance D.

To describe the angular dependence of scatteredlight, a new ingredient must be introduced: the an-gular scattering probability or, in other terms, theso-called scattering phase function:

f�, �; �, ��,

��1

1

d��0

2�

d��f�, �; �, �� 1, (8)

where the scattered and incident directions are givenby and � (the cosines of the polar angles) and bythe azimuthal angles � and ��, respectively.

If this angular distribution depends only on theangle between the scattered and incident directions,given by the unit vectors e and e�, the scattering pro-cess has an axial symmetry and is described conve-niently by the following expansion in terms ofLegendre polynomials17:

f�, �; �, �� f�cos �� 1

2� �l�0

�l � 1�2�flPl�cos ��,

g ��1

1

d��0

2�

d��cos � f�, �; �, ��,

cos � � e · e� (9)

The anisotropy parameter g is defined as the meancosine of the scattering angle.

One must emphasize that the light scattered by oneRBC is axially symmetric only in very particularcases when the incident light travels along the sym-metry axis of the cell. For an ensemble of RBCs thatare usually randomly oriented, taking the meanphase function as axially symmetric is a very goodapproximation. There are also some cases with non-random orientation (flowing blood, for example)where the blood cells align. The axisymmetric ap-

proximation is then appropriate only for special di-rections normal to the anisotropy plane.

The probability that an incoming photon travelingalong the direction � will be scattered by a single redblood cell along the direction is obtained by integratingthe phase function with respect to the azimuthal an-gle16,18:

p�, �� 0

2�

f�, �; �, ��d��

� �l�0

�l � 1�2�flPl��Pl��, (10)

and the anisotropy parameter g is obtained simply byintegration:

g ��1

1

p�, 1�d � f1. (11)

We can now introduce the steady-state angular de-pendent fluxes,

��, � � �0��, � � �s��, � � �n�0

�n��, �, (12)

solve the corresponding radiative transfer equa-tions,16,17 (see Appendix A), and find the analytic ex-pressions of the reduced incident flux �0 and thescattered flux �s respectively:

�0��, � � e�� 1�, (13)

�s��, � � �1 � e��feff ��, �. (14)

The scattered angular flux is given in terms of aneffective phase function defined by the followingformula:

feff��, � � �l�0

�l � 1�2�f̃l��Pl��, f̃l�� e�fl � 1

e� � 1.

(15)

Equation (15) gives explicitly the optical depth de-pendence of the scattered light angular distributionin an axisymmetric geometry.

Even more, each nth-order scattered flux can beexplicitly obtained:

�n��, � � fn��n

n! e��. (16)

Its nth-order scattering phase function is given by

fn�� l�0

�l � 1�2�flnPl��, (17)


which describes the angular spreading of the photonsscattered n times.

To simplify further the obtained results, one needsthe explicit form of the angular scattering probabilitycoefficients f1 for individual RBC.

3. Effective Henyey–Greenstein Phase Function

Basically there are two analytical phase functionsproposed in the literature, which approximate thescattering properties of RBCs. For individual cellsthe most appropriate seems to be the two-parameterGegenbauer kernel phase function.2,10 The scatteringis sharply peaked in the forward direction, with theanisotropy parameter being very close to 1 � g �0.995�. In practical cases the suspended RBCs areexposed also to the wave scattered by the neighboringcells, and the most appropriate phase function wasproved to be the Henyey–Greenstein phase function.5Additionally, due to the effect of neighboring cells, thesingle-cell scattering anisotropy at a physiological he-matocrit is significantly reduced.5

The proposed approach refers to multicellular sus-pensions with different hematocrits, and consequentlywe will use in the following the Henyey–Greensteinmodel � fl � gl � for the microscopic single-cell phasefunction:

fHG�� l�0

�l � 1�2�glPl�� 12

1 � g2

�1 � 2g � g2�3�2.

(18)

The effective phase function for any scattering ordercan be simply obtained by the substitution fl

n → gnl:

fn�� l�0

�l � 1�2�gnlPl�� 12

1 � g2n

�1 � 2gn � g2n�3�2.

(19)

The nth-order anisotropy parameter gn measuringthe angular spreading of the nth-order scattered flux,

gn ��1

1

fn��d � gn, (20)

is simply given as the nth power of the original single-cell anisotropy parameter.16

The effective phase function coefficients become

f̃l�� e�gl

� 1

e� � 1. (21)

Unfortunately, with the above exact formula, the ex-pansion in Legendre polynomials cannot be com-pressed in a general analytic form. But, as we willshow in the following, the effective phase function canbe given by a compact analytic formula introducing

approximate expressions of the expanding coeffi-cients.

It is not difficult to observe that the coefficients f̃l��take values in the restricted domain �0, 1� and (be-cause g � 1) are decreasing functions on both l and �variables. To keep the physics of the problem as sim-ple as possible we will introduce a �-dependent an-isotropy function g̃�� gG��, proposing the followingapproximate expression for the effective phase func-tion coefficients:

f̃l�� glG��, G�� 1. (22)

The main advantage of this power-law dependence isthat the expansion in Legendre polynomials can becompressed in a Henyey–Greenstein-type effectivephase function:

feff ��; � �12

1 � g2G(�)

�1 � 2gG(�) � g2G(�)�3�2. (23)

The physical meaning of the formula is extremelysimple. The whole scattering volume exposed to thelight beam can be modeled as a “macroscopic particle”having the same phase function as a single RBC butwith a �-dependent effective anisotropy parameter:g̃�� gG��. The power-law dependence describes thedecrease of the light scattering anisotropy. Just thisanisotropy decrease is the main reason for the failingof the quasi-ballistic light propagation for thickerblood samples and�or with higher hematocrits.

The major challenge is to find the explicit expres-sion for G�� that ensures the best approximation.Our proposal is the following formula:

G�� 1�e� � 1

e� � � � 1, (24)

which is a monotonous increasing function and veri-fies a global optimization criterion for the entire do-main of variation for the index l (see Appendix B fordetails). Nevertheless the suitability of the proposedapproximations should be finally proved or improvedby experiments.

Formulas (14), (15), (23), and (24) are the mainresults of the paper and are approximately validas long as the anisotropy remains sufficiently large�g̃�� 0.9, for example]. For the RBC case the an-isotropy parameter is relatively high �g 0.99�, butthis value seems to decrease with the sample hemat-ocrit.5 Accordingly we can say that the obtained for-mulas are good approximations in experiments forwhich � � 10.

An interesting property of the obtained formulas isthat despite the fact that they have been deduced insmall-angle approximations. They give the correctbehavior in the diffusion limit g̃�� → � → 0. Conse-quently, our expectation is that the validity of theformulas can be extended to samples with higher


hematocrits and�or thicker cuvettes. We emphasizethat the given limits are only estimated and the va-lidity of the approximations used, together with suit-able limitations, must come from experiments.

G�� can be replaced by simpler functions adequatein the following asymptotic cases:

G�� →1 � ��3, � � 1� � 1, � � 5, (25)

which describe approximately the scattering processin the two well-known limits: predominantly singlescattering and multiple scattering, respectively.Figure 1 illustrates a comparison between the func-tion G�� and each of its asymptotes.

The limits in Eq. (25) are a rough estimate obtainedby the condition that the discrepancy between thefunction G�� and the corresponding asymptotic ap-proximants be smaller than 5%.

To make the theoretical approach more accessiblewe will present several curves that illustrate themain dependencies predicted by the proposed model.In all cases the single-scattering anisotropy is takenas g � 0.98. In Fig. 2 the angular dependence of theeffective phase function [Eq. (23)] is given for severalvalues of the optical depth covering both single-scattering and multiple-scattering domains. We focusonly on small-angle scattering because the proposedmodel is asymptotically accurate only in this limit.One can observe the expected decrease of the scatter-ing anisotropy (the effective phase function becomesmore flat) for higher values of the optical depth. Wehighlight the fact that the optical depth increase canbe achieved in two ways: by increasing the samplethickness or by increasing the sample hematocrit. Inboth cases the small-angle scattering anisotropy di-

minishes with a predictable extent characterized byEq. (23).

In Fig. 3 the detection angle is kept fixed and thescattering efficiency is depicted as a function of opti-cal depth. For an optical depth of higher than 10 thepredictions remain qualitatively correct, but thequantitative accuracy of the proposed dependency isnot ensured by the approximations introduced.Therefore the upper limit of our model cannot beprecisely defined and must be established by experi-ments.

As expected, the amount of light scattered at afixed (small) angle increases with � in the single-scattering domain, attains a maximum due to the

Fig. 1. Asymptotic linear approximations for G��.Fig. 2. Angular dispersion of the scattered photons. The effectivephase function angular dependence shown for several values of theoptical depth �.

Fig. 3. Effective phase function and the scattered flux depen-dency on optical depth, given at three different scattering angles.


balance between single and multiple scattering, re-spectively, and after that decreases with increasingoptical depth. The shape of the curve modifies withthe scattering angle. The maximum decreases andshifts to the right by increasing the angles, and thecurve becomes flatter.

Concerning this behavior, it is significant to eval-uate the hematocrit Hm that assures maximum lightscattering efficiency for a suspension with thicknessd. Our predictions suggest that the optical depth forwhich the angular flux attains its maximum �m isangle dependent, but for scattering at small angles itis in the neighborhood of 2 ��m � 2�. The correspond-ing hematocrit can be simply evaluated from the re-lations � � ��sd � H�sd�v and can be approximatedas Hm � 2v��sd�. The ratio between the mean vol-ume and mean scattering cross section for humanRBC, can be also estimated from the published data,2and is roughly given by v��s � 1 m. For a cuvettewith a thickness d � 1 mm, for example, one obtainsHm � 2 � 10�3. Our predictions suggest that theangular flux of photons scattered at small angles at-tains its maximum at hematocrits that are more thantwo orders of magnitude smaller than the physiologicvalues. Even more, if one employs 1 cm thick cuvettes(commonly used in uv�vis spectrophotometry), thehematocrit corresponding to maximum scattering ef-ficiency is ten times smaller than compared to 1 mmcase.

4. Conclusions

There are many practical situations in light scatter-ing experiments on suspended RBCs when the inci-dent light is a narrow collimated laser beam ofapproximately 1 mm diameter. The thickness ofplane-parallel samples containing the RBCs is usu-ally also of the same order of magnitude, and theinformation concerning the scattering process is ob-tained by capturing the scattered far field by a pho-todetector or by a CCD camera. The main conclusionof the paper is that in a far-field experimental setupthe whole scattering volume exposed to the lightbeam can be modeled as a “macroscopic particle,”having its own effective scattering phase function.

Using a hierarchy of successive-order equations,one can find approximate formulas for the scatteredangular fluxes of any scattering order and also for thetotal scattered flux. It was deduced that, if one de-scribes the single RBC scattering by the Henyey–Greenstein phase function, the effective phase functiondescribing the suspension has the same analytic ex-pression. The property is also true for any scatteringorder. Moreover, the nth-order anisotropy parametermeasuring the angular spreading of the nth-orderscattered flux is simply given as the nth-order of theoriginal single-particle anisotropy parameter.

We have also found that one can define an effectiveanisotropy parameter that has a strong dependenceon optical depth and is a good approximation of themean cosine of the scattered light at any depth in-side the sample.

The results obtained have a general relevance andmay be used in many practical situations in whichone measures the light scattered by blood in thinlayers or diluted systems. In this context it may beuseful to compare our theoretical predictions to sev-eral reported experiments.

Steenbergen et al.7 have measured the 633 nmlight scattered by very thin layers �20–100 m� ofundiluted human blood subjected to simple shear.The measured angular distribution of the scatteredlight was very well fitted with Henyey–Greensteinfunctions. The measured anisotropy values decreasewith an increase in the layer thicknesses, in agree-ment with our model. As the measurements cannotbe performed with layers thinner than 20 m due tothe accuracy of the mechanical system, the main dif-ficulty that must be exceeded is to extrapolate themeasured values to zero layer thickness and to findthe single-scattering anisotropy g. The present contri-bution offers just the theoretical formula, g̃�sd� �gG�s d�, which may be used to perform such extrapo-lations.

In the same paper7 the authors follow an alternativestrategy, namely to measure the scattered light at aproper layer thickness and subsequently to find the valueof g with an inverse Monte Carlo technique. The resultsexhibit an increase of the multiple-scattering anisotropy�g̃ in our notation) with increasing values of g. MonteCarlo simulations are consistent with the power-typefunction g̃�� gG�� (with � fixed by the layer thick-ness) deduced with our theoretical model.

The Monte Carlo technique was used also byHammer et al.5 to simulate the angular distributionof light power scattered by a layer of whole blood,0.1 mm thick. A Gegenbauer kernel two-parameterphase function was used as a single RBC phase func-tion. The measured angle-resolved intensity of thescattered light was compared with results from theangle-resolved Monte Carlo simulation. The bestagreement between the measured and the calculateddata was achieved for a set of parameters that provethat the Henyey–Greenstein function (which can beobtained as a particular case of the two-parameterGegenbauer kernel phase function) is suitable as theeffective phase function for the description of lightscattering in blood with a physiological hematocrit.

Finally, we can say that the developed model seemsto describe quite well and in simple analytic termsthe angular distribution of the light scattered at asmall angle by RBCs in suspension. The model standsfor an alternative approach complementary to theMonte Carlo simulations or to other methods dedi-cated to numerically solving the problem of electro-magnetic wave scattering by RBCs.

The strength of the model consists in its ability todirectly fit the measured data and find the primaryparameters of the light scattering by RBCs. Theweaknesses are related to the fact that the approxi-mations used restrict the validity domain to scatter-ing at a small angle and also relatively small valuesof the optical depth. Additionally, the validity limits


cannot be rigorously established so that the experi-ment must be the final evaluation in validating thetheory and its limits.

Appendix A

The quasi-ballistic scattering can be described by thefollowing radiative transfer equation16,17:

d��, �

d�� , � � w�

�1

1

d�p�, ��, ��,

(A1)

where w � �s�� 1 is the RBC albedo. In the domainof small angles the cosine of the scattering angle isclose to unity �1 � �� 1�, an approximation thatwill be used to simplify the left-hand side of Eq. (A1),where the � dependence will be ignored: � 1. Put-ting all that together, one obtains the following ap-proximate radiative transfer equation:

d��, �d�

� ��, � ��1

1

d�p�, ��, ��.

(A2)

The angular-dependent fluxes �n��, � are solutionsof a hierarchy of interconnected integro-differentialequations16 succeeding each other according to thescattering order:

d��, �d�

� ��0��, �, �0�0, � � �� 1�, (A3)

d�n��, �d�

� ��n��, �

��1

1

p�, ��n�1��, ��d�,

�n�0, � � 0, n � 1, (A4)

�n�0

��1

1

�n��, �d � 1, (A5)

which are obtained introducing the detailed expres-sion of the angular flux ��, � � �n�0

�n��, � intoEq. (A2).

The solution of Eq. (A3) is simply given as a productof the � angular distribution and the exponential de-cay along the sample depth:

�0��, � � e�� 1�. (A6)

We will assume that this structure with the twovariables separated in two different functions is keptin any scattering order:

�n��, � � fn��Fn��. (A7)

The angular-dependent functions fn�� can be takenas the effective phase functions describing the angu-lar spreading of the nth-order scattered flux.

It is easy to verify that in this hypothesis the solu-tions are given by

Fn�� n

n! e��, Fn�0� � 0,

fn�� 1

1

p�, ��fn�1��d�, f0�� 1�.

(A8)

The integral can be easily performed:

fn�� l�0

�l � 1�2�flPl��k�0

�k � 1�2�fn�1,k

��1

1

d�Pl��Pk��

��l�0

�l � 1�2�flfn�1, lPl�� (A9)

using the standard expansion in Legendre polyno-mials,

fn�� l�0

�l � 1�2�fn,lPl��, (A10)

and the orthogonality property,

��1

1

dPl��Pk�� l,k

l � 1�2. (A11)

One obtains the following simple and importantrecurrence formula:

fn,l � fl fn�1,l � fln. (A12)

The phase function of the multiple scattered photonsbecomes

fn�� l�0

�l � 1�2�flnPl��. (A13)

In other words, the Legendre coefficients for the an-gular distribution of the photons scattered n timesare given by the nth power of the correspondingsingle-scattering coefficient. A somewhat surprisingfact is that the formula is correct also for the non-scattered photons. Indeed, for the “function” �, theexpansion coefficients are all equal to 1:


�� l�0

�l � 1�2�Pl��Pl�� → �� 1�

� �l�0

�l � 1�2�Pl��, (A14)

and accordingly

f0�� l�0

�l � 1�2�Pl�� 1�. (A15)

The scattered flux can be rewritten in the followingrelatively compact form:

�s��, � � e��l�0

�l � 1�2��n�1

1n! ��f1�n�Pl��

� e��l�0

�l � 1�2��e�f1 � 1�Pl��, (A16)

but the variable separability was lost.The total scattered flux at any depth inside the

sample can be obtained by integration and must begiven by Eq. (7):

�s�� 1

1

d�s��, � � �1 � e��. (A17)

The scattered angular flux can be expressed in termsof an effective phase function feff ��, �,

�s��, � � �1 � e��feff��, �, (A18)

defined by the following formula:

feff��, � �1

e� � 1 �l�0

�l � 1�2��e�fl � 1�Pl��,

(A19)

and also satisfying the normalization condition

��1

1

feff��, �d � 1. (A20)

Appendix B

The applicability of our results depends largely on theaccuracy of the following approximation:

e�gl� 1

e� � 1� g lG(�), (B1)

which will be analyzed to some extent. The majortask is to find an appropriate function G(�) that isable to minimize the discrepancy between the twosides of Eq. (B1). The proposed undertaking is rela-tively difficult to achieve because the two expressionsmust stay close to one another as much as possible

and index l must take integer values in the range�0, �.

We can use the fact that index l comes out on bothsides of Eq. (B1) in a power-type dependence gl withg � 1. This particularity allows us to introduce a newvariable, x � gl, so that the analysis can be reduced toa comparison of the functions ��x; �� and �� x; ��:

��x; �� e�x � 1

e� � 1, �� x; �� xG(�), 0 � x � 1, (B2)

both depending parametrically on �. Although by def-inition x is a discrete variable, for simplicity, theanalysis will be extended in continuum.

There are two reasons that ensure that �� x; �� is agood approximant: (i) both functions increase monot-onously with the variable x and (ii) take same valuesat the domain boundary. Until now G�� has been anarbitrary function, and an additional similarity cri-terion must be imposed to optimize the approxima-tions. One of the simplest requirements is theequality of definite integrals:

�0

1

��x; ��dx ��0

1

�� x; ��dx → G�� 1�e� � 1

e� � � � 1.

(B3)

The imposed condition is strong enough to find ex-plicitly the �-dependent function G(�).

In Fig. 4 the two functions ��x; �� and �� x; �� areshown for several values of the optical depth �, thedifferences being reasonably small in both single-scattering and multiple-scattering domains. It is cu-rious to some extent the fact that the discrepancy,

Fig. 4. Comparison between the functions ��x; �� and �� x; ��. Theoptical depth � is considered as parameter.


even relatively small, is most important at interme-diate � values. It decreases as expected at small op-tical depths but also decreases in the range of largervalues, where the approximations fail for other rea-sons: the quasi-ballistic character of light scatteringis gradually lost.

This research was supported by the Romanian Ed-ucation and Research Ministry, BIOTECH Program,research project 01-8-CPD-042. The author is grate-ful to Lorelai Ciortea for a thorough reading of themanuscript.

References1. G. J. Streekstra, A. G. Hoekstra, E. J. Nijhof, and R. M.

Heethaar, “Light-scattering by red blood cells in ektacytom-etry: Fraunhofer versus anomalous diffraction,” Appl. Opt. 32,2266– 2272 (1993).

2. M. Hammer, D. Schweitzer, B. Michel, E. Thamm, and A. Kolb,“Single scattering by red blood cells,” Appl. Opt. 37, 7410–7418(1998).

3. A. N. Shvalov, J. T. Soini, A. V. Chenyshev, P. A. Tarasov, E.Soini, and V. P. Maltsev, “Light-scattering properties of indi-vidual erythrocytes,” Appl. Opt. 38, 230–235 (1999).

4. S. T. Tsinopoulos and D. Polyzos, “Scattering of He-Ne laserlight by an average-sized red blood cell,” Appl. Opt. 38, 5499–5510 (1999).

5. M. Hammer, A. N. Yaroslavsky, and D. Schweitzer, “A scat-tering phase function for blood with physiological haemat-ocrit,” Phys. Med. Biol. 46, N65–69 (2001).

6. J. Kim and J. C. Lin, “Successive order scattering transportapproximation for laser light propagation in whole blood me-dium,” IEEE Trans. Biomed. Eng. 45, 505–510 (1998).

7. W. Steenbergen, R. Kolkman, and F. De Mul, “Light-scattering

properties of undiluted human blood subjected to simpleshear,” J. Opt. Soc. Am. A 16, 2959–2967 (1999).

8. S. T. Tsinopoulos, E. J. Sellountos, and D. Polyzos, “Lightscattering by aggregated red blood cells,” Appl. Opt. 41, 1408–1417 (2002).

9. L.-H. Wang and S. L. Jacques, “Error estimation of measuringtotal interaction coefficient of turbid media using collimatedlight transmission,” Phys. Med. Biol. 39, 2349–2354 (1994).

10. L. O. Reynolds and N. J. McCormick, “Approximate two pa-rameter phase function for light Scattering,” J. Opt. Soc. Am.70, 1206–1212 (1980).

11. A. M. K. Nilsson, P. Asholm, A. Karlsson, and A. Anderson-Engels, “T-matrix computations of light scattered by red bloodcells,” Appl. Opt. 37, 2735–2748 (1998).

12. A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 1995).

13. A. Karlsson, J. He, J. Swartling, and S. Andersson-Engels,“Numerical simulations of light scattering by red blood cells,”IEEE Trans. Biomed. Eng. 52, 13–18 (2005).

14. S. L. Jacques and L. Wang, “Monte Carlo modeling of lighttransport in tissue,” in Optical Thermal Response of Laser-Irradiated Tissue, A. Welch and M. J. C. van Gemert, eds.(Plenum, 1995), pp. 73–100.

15. A. N. Yaroslavsky, I. V. Yaroslavsky, T Goldbach, and H-J.Schwarzmaier, “Influence of scattering phase function approx-imation on the optical properties of blood determined from theintegrating sphere measurements,” J. Biomed. Opt. 4, 47–53(1999).

16. I. Turcu, “Effective phase function for light scattered by dis-perse systems—the small-angle approximation,” J. Opt. A:Pure Appl. Opt. 6, 537–543 (2004).

17. A. Ishimaru, Wave Propagation and Scattering in RandomMedia (Academic, 1978).

18. A. A. Kokhanovsky, “Analytical solutions of multiple light scat-tering problems: a review,” Meas. Sci. Technol. 13, 233–240(2002).


effective phase function for light scattered by blood

Documents