Accuracies of the diffusion approximation and itssimilarity relations for laser irradiatedbiological media

Gilwon Yoon, Scott A. Prahl, and Ashley J. Welch

The accuracy of the diffusion approximation is compared with more accurate solutions for describing lightinteraction with biological tissues. Generally the diffusion approximation underestimates the light distribu-tion in the surface region, and, for high albedos, it significantly underestimates the fluence rate. Thisdifference is only a few percent for albedos of less than 0.5 due to the dominance of collimated light. As theanisotropy of scattering increases, deviations increase. In general, fluxes can be computed more accuratelywith the diffusion approximation than fluence rates. For anisotropic scattering, better results can beobtained by simple transforms of optical coefficients using the similarity relations. The similarity relationsimprove flux calculations, but computed fluence rates have substantial errors for high albedo and the largeindex of refraction differences at the surface.

1. Introduction

Mathematical descriptions of light absorption andscattering are important in understanding the interac-tion of laser light with biological media for treatmentand diagnostics. One mathematical technique for de-scribing light propagation in biological media is radia-tive transfer theory. Unfortunately, the general solu-tion is not known, and accurate solutions are limited tosimple conditions or slab geometries. Numerical solu-tions are more versatile but usually require extensivecomputer memory and time.

On the other hand, the diffusion approximation hasbeen widely used for describing light propagation inbiological media especially when scattering dominatesabsorption. 1 4 In the diffusion approximation, thephase function is represented by the first two terms ofthe Legendre polynomial expansion. This providesmathematical convenience, and the phase function ischaracterized by a single anisotropy factor.

In this paper, solutions of the diffusion approxima-tion are examined. However, it is difficult to quantifyerrors for general conditions since many parametersare involved. Nevertheless, the type and trend of

Gilwon Yoon is with National Institute for Health & MedicalResearch (INSERM, U279), France; S. A. Prahl is with The LaserCenter, Academic Medical Center, The Netherlands; and A. J.Welch is with University of Texas at Austin, Biomedical Engineer-ing Program, Austin, Texas 78712.

Received 28 November 1988.0003-6935/89/122250-06$02.00/0.( 1989 Optical Society of America.

errors can be studied. A 1-D slab geometry has beenselected to reduce complexities. Solutions from thediscrete ordinate method5 and from the tables of vande Hulst6 which are more accurate than solutions usingthe diffusion approximation are used as references. Inaddition, simple transforms of the optical coefficientsusing the similarity relations are examined for improv-ing the accuracy of computed fluxes and fluence ratesfor anisotropic scattering.

11. Diffusion Approximation

The radiative transfer equation is given as7

s * VL(r,s) + juL(r,s) = Asf p(s,s')L(r,s')dw', (1)

where L(r,s) is the radiance (W/cm2sr) at position r inthe s direction (s is the directional unit vector), it is theattenuation coefficient (1/cm) defined as the sum ofthe absorption coefficient ,ua (1/cm), and the scatteringcoefficient Iu (1/cm). The phase function p(s,s') rep-resents scattering contribution from s' to the s direc-tion and defined as

( ) J p(s,s')dw' = 1,

where co denotes the solid angle. It is often convenientto represent the total radiance in terms of the colli-mated radiance L(rs) and the diffuse radianceLd(r,s). The collimated radiance which is attenuatedby direct absorption and scattering is given by

dL,(r,s)/ds =-,uL,(rs). (2)

In the diffusion approximation, the diffuse radiance isapproximated as follows:

2250 APPLIED OPTICS / Vol. 28, No. 12 / 15 June 1989

Ld(r,s) Ud(r) + Fd(r) * s(3/47r),R(0) = (ni - n,)2 ifi =

where Ud(r) = 4,rLd(r,s)d/47r is the average radi-ance, and Fd(r) = 4,Ld(r,s)sdw is the diffuse fluxvector. Equation (3) may be considered as the firsttwo terms of a Taylor's expansion of Ld.

For a uniform collimated beam normally incident ona slab, the radiance varies only in the propagationdirection z. The collimated light expressed by Eq. (2)decays in proportion to exp(-,gtz). By inserting Eqs.(2) and (3) into Eq. (1), the following differential equa-tion in terms of Ud is obtained 7 :

a2Ud(z)/z 2 - 3/1altrUd(Z) = -(3LsItr + 3g9Ast)F,(Z)/47r, (4)

where Atr = Aa + (1 -g) 8,,g = average cosine angle of phase function (or

anisotropy factor) defined asf 4,rP(ss')Adw'1 4,rP(ss')dw',

/ = cosine angle of s with respect to z,F,(z) = Linc exp(-Atz); collimated flux, and Lin is

the incident irradiance (W/cm2).The collimated flux has a component in the z-direc-

tion only. The diffuse radiance can be obtained asfollows:

Ld(z,A) = Ud(z) + (3/47r)gitF,(z)A/1tr VUd(z)WA/tr. (5)

A. Boundary Conditions

For diffusion approximation, the following condi-tions at a surface for diffuse light have been imple-mented 7 :

fS2,,flLd(r,s)s - ndw = 0, (6)

where n is the normal unit vector directed into themedium and the integration is taken over the hemi-sphere. The diffuse flux entering the medium is set tozero. However, some portion of diffuse light in thetissue is reflected at boundaries where there is a mis-matched index of refraction. For biological tissues,values of the index of refraction of -1.4 have beenreported.8 For mismatched boundary conditions, thetotal diffuse flux reflected into the medium is part ofthe outward diffuse flux. This can be implemented asfollows:

fS2rsLd(rs)s -ndw = f 2 ,,. R(s --n)Ld(r,s)(s * -n)dw, (7)

where R is the reflectance whose value can be deter-mined from the index of refraction using the Fresnelequation. For unpolarized light, the reflected radi-ance with respect to the incident angle R(0i) is repre-sented by the reflected electric fields perpendicular tothe plane of incidence R 1 and parallel to the plane ofincidence R 11.

R(Oi) = 2 [R2 + R2]

R(O) = 1.0 if Oi >0, = critical angle.

Oi and ni represent the incident angle and index of theincident medium, respectively, and Ot and nt representthe transmitted angle and index of the transmittedmedium.

A similar technique used for Eq. (6) by Ishimaru7

has been employed to derive the boundary conditions:Fd in Eq. (3) is separated into normal and tangentialcomponents to the surface and placed in Eq. (7). Us-ing Eq. (8), the following expressions for a slab whosethickness is d are obtained:

Ud(z) - AhOUd(z)oz = - A(9,1tr)F,(z)/27r at z = 0,

Ud(z) + AhOUd(z)/az = A(g/ut/,%)F,(z)/2r at z = d, (9)

where A = [1 + R 2]/[1 - R1] and h = 2/3,4tr, assumingthe medium index is greater than or equal to the indexof the environment:

R = 2f' R(A)dA + Al;

R2 = 3fS cR( u) 2dt + ,3;

jc = coso,.

The boundary conditions in Eq. (7) have been appliedfor the diffusion approximation by Keijzer et al.9 andStar and Marijnissen. 4 For mathematical conve-nience, they simplified R(Oi): Keijzer et al. adaptedR(Oi) = R(O) for Oi S Oc and R(Oi) = 1.0 for 0i > 0,. Starand Marijnissen used exponential functions fitting theFresnel equation to improve accuracy. In this paper,exact values of R(0j) are computed numerically.

111. Similarity Relations

There can be different sets of Ma,1s and the averagecosine angle of the phase function g, which providesimilar estimates of radiance in diffusion approxima-tion calculations. These are the so-called similarityrelations. The following can be formulated by assum-ing the same behavior in a diffusion domain6:

Ma = Ma;

(1 -g)As = (1 -g)yA (10)

The primes indicate transformed parameters. In gen-eral, there is more than one choice. Each of them,however, distorts the results one way or the other.Notable differences exist among different similarityrelations in the surface region, but usually these differ-ences become negligible beyond several optical depths.A practical matter is how to select a specific set ofoptical coefficients and to study how the computation-al results are affected.

One motive for using the similarity relations is totransform anisotropic scattering into isotropic scatter-ing. In this case, we have

1 sin2 (0 -°)

2 sin2(0, + 0,)tan2(0i - 0,)

tan2 (0i + 0,) J (8) (11)M = a M= (1 -g)A g =0.

This eases the computational burden, and only theabsorption coefficient and effective scattering coeffi-

15 June 1989 / Vol. 28, No. 12 / APPLIED OPTICS 2251

where

(3)

for the highly forward peak since the diffusion func-tion is a poor approximation for anisotropic scattering.For the following Henyey-Greenstein phase function,

0 1 2 3 4

optical depth

Fig. 1. Forward and backward diffuse fluxes for a uniform colli-mated irradiance of 100 W/cm 2 are shown as a function of depth.Albedo = 0.99, g = 0, sample thickness = 4.0, and matched boundaryconditions are assumed. For high albedo and isotropic scattering,there are little differences between diffusion and discrete ordinate

solutions.

) ~ ~ ~ ~ a~lbedo=0.99. t . .. . ~ ~ ~~~ - discrete ordinate.1 - r- - - * diffusion)-I_ ) I I_ _

optical depth

Fig. 2. Total fluence rates computed with the diffusion approxima-tion and the discrete ordinate method with respect to differentalbedos. Differences are larger for high albedos. g = 0, samplethickness = 4.0, matched boundary conditions, and uniformly colli-

mated irradiance of 100 W/cm 2 are assumed.

cient M8, are needed to characterize the tissue proper-ties.

Another relation can be formulated with the addi-tion of scattering in the exact forward direction. Thisis of practical importance since highly forward scatter-ing may be approximated using the delta function.Adding the delta function to the phase functionchanges the scattering coefficient only. Suppose thatthe phase function is represented by the delta functionand another phase function pl(s,s'),

p(s,s') = f(s - s') + (1 - f)pl(ss'), (12)

where f represents the amount of the forward peak.Equation (1) can be formulated with p1(s,s') and (1 -f)MS instead of p(s,s') and gs, respectively.

If a phase function is approximated by the aboveform with pi(s,s') in a simple expression, computationis reduced. Accuracy depends on how well the approx-imate function represents a real function. A methodproposed by Joseph et al.,10 which is known as the 6-Eddington approximation, utilizes the above form ofphase function: P1 consists of the first two terms ofthe Legendre polynomials; i.e., the diffusion phasefunction and delta function are added to compensate

where the P, terms are Legendre polynomials, Josephet al. determined the value off by setting the 5-Edding-ton phase function equal to g and representing thesecond moment as a Henyey-Greenstein function.This yields

Ma Ma,

g' = g/(1 + g).

Equation (14) satisfies Eq. (10). The 5-Eddingtonapproximation can be considered as one special case ofthe similarity relations.

IV. Comparisons with More Accurate Solutions

Computations are for a slab irradiated by a normallyincident uniform laser beam. For anisotropic scatter-ing, the Henyey-Greenstein phase function is as-sumed. Solutions of the diffusion approximation us-ing the original coefficients and the coefficientscalculated from Eqs. (11) and (14) are compared withsolutions using the discrete ordinate method usingtwenty-four fluxes5 and van de Hulst's solutions.6Our primary interest is in the following quantities:

forward flux S2 ,zL(r,s)s * zdw;backward flux f 2 ,.._,L(r,s)s (-z)dw;fluence rate f4,L(r,s)dw.

However, radiance L(r,s) is also examined. In laserphotothermal reactions, the local rate of heat genera-tion given by (Ma X fluence rate) is required to calculatetemperatures within laser irradiated tissue.

A. Isotropic Scattering Case

For isotropic scattering (g = 0), the optical coeffi-cients given by Eqs. (11) and (14) are the same as theoriginal coefficients. First, we examine the case wherethe diffusion approximation is known to be accurate,i.e., high albedo, isotropic scattering, and matchedboundary conditions. The albedo is defined as Ms/Mt.Fairly accurate 1-D solutions are obtained with thediffusion approximation in terms of fluxes, i.e., for-ward, backward, and net fluxes. For example, diffuseforward and diffuse backward fluxes show little differ-ences compared with discrete ordinate solutions (Fig.1). In this illustration, optical depth is defined as r =Mt X geometrical depth, and the following parametersare used: albedo 0.99, collimated irradiance 100 W/cm2, an optical thickness of 4.0. For both models, thecollimated fluxes are the same and decrease accordingto exp(-optical depth).

Usually, larger errors are associated with fluencerate calculations (or the rate of heat generation) thanwith estimates of flux. This is illustrated in Fig. 2where total fluence rates computed with diffusion anddiscrete ordinate models are plotted as a function ofoptical depth. The fluence rates at the front surfacecomputed with both models are almost the same, but

2252 APPLIED OPTICS / Vol. 28, No. 12 / 15 June 1989

80

60

40eJ2

a20

Phg(SS') = (4) E (2n + 1)g`Pn(ss'), (13)

300

As = (1 -g2

)aU,

(14)

200

100

07

U

V

C

03

00 1

lower fluence rates are computed by the diffusion ap-proximation in the subsurface region. Also, differ-ences between diffusion and discrete ordinate solu-tions are greater for large albedos. For small albedos,the radiance is dominated by collimated light com-pared with diffuse light due to small scattering. Thedifference in the fluence rates estimated by the diffu-sion approximation (including the solution for colli-mated light) can be negligible when the albedo is small-er than 0.5.

B. Anisotropic Scattering

The diffusion approximation would be expected tobe less accurate for anisotropic scattering than forisotropic scattering since the diffusion phase funtion isnot suitable for representing highly anisotropic scat-tering. This is unfortunate since biological tissues areknown to be highly forward scattering."1 As g ap-proaches 1.0 (or -1.0), the accuracy of diffusion ap-proximation decreases.

For example, in Fig. 3, reflection is plotted withrespect to values of g for the Henyey-Greenstein phasefunction. The same parameters used in Fig. 1 areassumed, i.e., an albedo = 0.99, optical thickness = 4.0,and matched boundaries. Reflections reported by vande Hulst for these conditions are included as referencevalues. Reflection estimated by the diffusion approx-imation remains accurate up to values of g of -0.6;beyond that value, errors increase rapidly. Computedvalues for transmission show a similar trend: errorsincrease as g increases (from calculations not includedin this paper). Although it is not physically possible,negative reflections or transmissions greater than oneare computed with the diffusion approximation as gapproaches 1.0. It has been observed that for forwardscattering, the diffusion approximation for fluxes islow in the front surface region and high in the tissue.Thus low reflection and high transmission are comput-ed.

When the similarity relation of Eq. (11) is used,reflection calculations are slightly less accurate thanwith the original coefficients up to -g = 0.6, but betterresults are obtained for higher g terms (Fig. 3). Equa-tion (14) derived from the 5-Eddington approximationyields the most accurate fluxes for all conditions com-pared with the other two. Errors become smaller for gnear 1.0 due to its inclusion of a delta-function torepresent forward peak. In the 5-Eddington approxi-mation, the first three moments of the phase functionare exact with those of the Henyey-Greenstein formu-la.

Isotropic conditions achieved with Eq. (11), whichprovides better estimates of fluxes for high g values, donot provide better estimates of the fluence rate thanthe original coefficients. Total fluence rates comput-ed from three different sets of related optical coeffi-cients are compared with discrete ordinate solutions inFig. 4. For this comparison, high albedo and forwardscattering for the original coefficients and matchedboundary conditions are assumed (albedo = 0.99 and g= 0.8). Deeper in the tissue, higher fluence rates from

0.75

0.50 - I']A [3]- van de Hulst

.B 0.25

0.00

-0.25 0.0 0.2 0.4 0.6 0.8 1.0

average cosine angle of phase function, g

Fig. 3. Reflection computed using the diffusion approximationwith respect to the average cosine angle of phase functiong. Albedo= 0.99, sample thickness = 4.0, matched boundaries, and the Hen-yey-Greenstein phase function are assumed. [1] Original coeffi-cients (Ma = 0.99,g) are converted into, [2] u8 = (1 -g) ,u andg' = Ous-ingEq. (11), [3] , = (1-g 2)gsandg' =g/(1 +g) usingEq. (14). Ma =0.01 is the same for all cases. The values of van de Hulst are used as

references.

0::

cB

2

250

200

150

100

50

0 1 2 3 4

optical depth

Fig. 4. Total fluence rates computed using [1] original coefficients(Ma = 0.01,A, = 0.99,g = 0.8), [2] Ma, = 0.01,Mu4 = 0.198,andg' = 0 usingEq. (11), [3] A, = 0.01, A,, = 0.356, and g' = 0.444 using Eq. (14) arecompared with discrete ordinate solutions. Sample thickness = 4.0

and matched boundary are assumed.

the original coefficients, and lower fluence rates fromisotropic coefficients from Eq. (11) are observed in thefigure. On the other hand, the 6-Eddington of Eq. (14)yields accurate estimates at the front and rear surfacesand lower fluence rates in between.

The profiles of fluence rates shown in Fig. 4 aretypical for the case of forward scattering and matchedboundary conditions. As g increases,the slopes offluence rate curves from the original coefficients aresteeper at the near and front surface. Often negativefluence rates are calculated, whereas high fluence ratesare obtained in the tissue. The isotropic coefficientsobtained from Eq. (11) produce slightly higher fluencerates at the front surface but much lower fluence ratesdeep in the tissue. In general, for matched boundaryconditions, estimates using the 6-Eddington approxi-mation of Eq. (14) are accurate at the boundaries, butlower fluence rates are estimated in the tissue. Thecoefficients of Eq. (14) produce better results in thediffusion approximation than either the original coef-ficients or those given by Eq. (11).

15 June 1989 / Vol. 28, No. 12 / APPLIED OPTICS 2253

C. Mismatched Boundary Conditions

Biological tissues usually have higher indices of re-fraction than the air or the surrounding fluid. Light isinternally reflected and trapped, which causes higherintensity compared with the matched boundaries.The larger the tissue index is, the higher fluence ratesare generated.

In Fig. 5, the fluence rates for two different g's of 0.0and 0.8 are calculated for an index of refraction of 1.4to represent the biological tissue and an albedo of 0.99.The increase in the fluence rate due to internal reflec-tion is very significant especially for high albedos: foralbedo = 0.99 and g = 0, the fluence rate at the frontsurface increases by more than a factor of 2 (see thediscrete ordinate solutions in Figs. 2 and 5). However,when the bdiffusion approximation is used, thefluence rate increases by only 80%. It is interesting tonote that the largest underestimation by the diffusionapproximation is observed in the subsurface region forthe matched boundaries and at the front surface forthe mismatched boundaries.

Comparison of the isotropic scattering transformand forward scattering transform is also featured inFig. 5. For the diffusion approximation, the fluencerates for g = 0.8 are computed using Eq. (14). For g =0.8, light penetrates deeper in the tissue due to forwardscattering. Errors of the diffusion solutions are highfor all optical depths. For albedos smaller than 0.5,differences in the fluence rates between diffusion anddiscrete ordinate solutions are very small for eithermismatched boundary conditions or anisotropic scat-tering. The diffusion approximation as well as othermodels do not show large differences from Beer's lawwhen absorption is dominant."

D. Radiance

The diffusion approximation uses the first twoterms of Legendre polynomials to represent radiance.Therefore, estimates of radiance are accurate in thediffusion region, i.e., beyond several optical depths orat least a few optical depths. Estimates of radianceare not accurate at the boundaries or for thin sampleswhere radiances are highly anisotropic. For example,for matched boundary conditions and assuring that nodiffuse flux enters the tissue [see Eq. (6)], negativereflectance values are shown in Fig. 3 for high g. Theexact boundary conditions for the radiative transferequation are no radiance entering a medium for thematched indices. Each ray entering the tissue must bedetermined from the Fresnel equation for the mis-matched indices. In the diffusion approximation, theboundary conditions for each ray cannot be imple-mented.

For anisotropic scattering, it is more difficult topredict the radiance profiles. For example, angulardistributions of light intensity computed using theoriginal coefficients and Eqs. (11) and (14) are illus-trated in Fig. 6. Radiances at an optical depth of 3.0are compared with discrete ordinate solutions. Analbedo = 0.99, g = 0.8, optical thickness of sample =

600 -

2

500-

400-

300-

200-

100 -

0

g=0- - discrete ordinate* diffusion

g = 0.8.............

2 3 4

optical depth

Fig. 5. Total fluence rates for a tissue index of 1.4 are computedfrom diffusion and discrete ordinate models. Sample thickness =4.0, albedo = 0.99, and a uniform irradiance of 100 W/cm 2 areassumed. For g = 0.8 Eq. (14) is used to transform the coefficients.For diffuse light, the boundary conditions in Eq. (9) are implement-ed. For collimated light, the external reflection of 2.8% is consid-

ered only at the front surface.

i1

sFE

-.5

a

6

4

2

10-

80

discrete ordinate

0 2

0 It

0 [,, 3] A

[2 A O ,J O A .

0'~11a aa110 0

100-

80-

S7

60-

45 90 135 181

-g 40 - l8 20-

discreteordinate

[11II [21 [3)

angle [degree]

Fig. 6. Diffuse radiance (W/cm 2 sr) and collimated flux (W/cm 2 ) atan optical depth of 3.0. Albedo = 0.99, g = 0.8, sample thickness =4.0, and matched boundary conditions are assumed: [1] orginalcoefficients (a = 0.01, a = 0.99,g = 0.8); [2] Ma = 0.01, Ms = 0.198, andg'= 0 using Eq. (11); [3] MA = 0.01,u = 0.356,andg' = 0.444usingEq.

(14) are compared with discrete ordinate solutions.

4.0, and matched boundary conditions are assumed forcomputation. In the figure 00 indicates the directionof beam propagation, and 180° is the opposite direc-tion. For irradiation with a uniform collimated beammost diffuse light at an optical depth of 3.0 is confinedwithin the ±450 cone in the forward direction (discreteordinate solutions, solid line in Fig. 6). On the otherhand, the three diffusion solutions have flattened pro-files. The collimated flux has intensity only in thedirection of beam propagation. It is reduced from 100to 5.0 W/cm2, which is smaller than the diffuse radi-ance at this optical depth. The collimated intensity ofthe discrete ordinate method is much smaller than thediffuse intensity at 0°.

Among the three diffusion calculations described inthis paper, the original coefficients yield higher diffuseradiance but the smallest collimated radiance. Trans-

2254 APPLIED OPTICS / Vol. 28, No. 12 / 15 June 1989

11 , l

forms of the optical properties reduce diffuse light andincrease collimated light for the compensation of for-ward peaks; i.e., Eqs. (11) and (14) reduce effectivescattering. Using Eq. (11), anisotropic scattering isconverted into isotropic scattering and the effectivescattering coefficient becomes smaller by a factor of 1- g. Therefore, there is less diffuse light and lowerattenuation of collimated light, i.e., exp(-Ma - Ms).For Eq. (14), the effective scattering reduces by a fac-tor of 1 - g2. Equation (14) seems to be better thaneither of the other diffusion solutions when the totalradiance of both diffuse and collimated light is consid-ered. The original coefficients hardly represent a for-ward peak. In general, all three solutions based on thediffusion model are poor approximations of the angu-lar distribution of light except in the diffusion regionsaway from the source and boundaries.

V. Conclusion

Accuracies of the models based on the diffusion ap-proximation have been examined in a slab geometryirradiated by a uniform normal beam. Similar trends,even though the magnitudes may not be of the sameorder, are also observed in other geometries includingthe 2-D axial symmetry model. For high albedos andisotropic scattering, fluxes are estimated accurately bythe diffusion approximation, but lower fluence rates(or rates of heat generation) are computed in the sur-face region. For low albedos, fluxes are less accurate,but the total fluence rates are better estimated due tothe dominance of collimated light. In general, thediffusion approximation estimates low reflection andhigh transmission for forward scattering and is notrecommended when g is larger than -0.6. The analy-sis using the diffusion approximation for highly aniso-tropic scattering can be very misleading.

For anistropic scattering, simple parametric trans-forms using the similarity relations provide better re-sults. The similarity relations provide many improve-ments in terms of fluxes. The -Eddingtonapproximation of Eq. (14) yields the best flux esti-mates for all conditions. In predicting the fluencerate, the similarity transform of Eq. (11) is no betterthan the original coefficients. Equation (14) is betterthan the other two in estimating fluence rates, butsubstantial errors for high albedos and large indexdifferences still exist.

Some methods are available for improving the diffu-sion approximation. For example, the diffusion ap-

proximation for large absorption has been studied,'2

but its application is restricted to diffuse incidence.The PN approximation is an extension of the diffusionapproximation for the general function, but it encoun-ters computational complexities for highly anisotropicscattering and 2- or 3-D geometries. Geometrical flex-ibility and mathematical convenience are advantagesof using the diffusion approximation. These meritshave to be compromised with decreased accuracy.

This work was funded in part by the Office of NavalResearch under contract N14-86-0875.

The authors would like to thank Willem Star for hisvery useful remarks.

Gilwon Yoon is on leave from the Utah Laser Insti-tute of the University of Utah.

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0

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