Research Article Solution of Radiative Transfer Equation...

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Research ArticleSolution of Radiative Transfer Equationwith a Continuous and Stochastic Varying RefractiveIndex by Legendre Transform Method

M Gantri

Unit of Thermal Radiation Department of Physics Faculty of Sciences of Tunis Tunis El-Manar University 1060 Tunis Tunisia

Correspondence should be addressed to M Gantri gantrimohamedgmailcom

Received 6 February 2014 Revised 17 April 2014 Accepted 21 April 2014 Published 9 June 2014

Academic Editor Kumar Durai

Copyright copy 2014 M Gantri This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The present paper gives a new computational framework within which radiative transfer in a varying refractive index biologicaltissue can be studied In our previous works Legendre transform was used as an innovative view to handle the angular derivativeterms in the case of uniform refractive index spherical medium In biomedical optics our analysis can be considered as a forwardproblem solution in a diffuse optical tomography imaging scheme We consider a rectangular biological tissue-like domainwith spatially varying refractive index submitted to a near infrared continuous light source Interaction of radiation with thebiological material into the medium is handled by a radiative transfer model In the studied situation the model displays twoangular redistribution terms that are treated with Legendre integral transform The model is used to study a possible detectionof abnormalities in a general biological tissue The effect of the embedded nonhomogeneous objects on the transmitted signal isstudied Particularly detection of targets of localized heterogeneous inclusions within the tissue is discussed Results show thatmodels accounting for variation of refractive index can yield useful predictions about the target and the location of abnormalinclusions within the tissue

1 Introduction

A special attention in diffuse optical tomography is focusedon the development of methods for detection of photonsproviding the information concerning optical parametersof the explored medium This gives the targets of local-ized nonhomogeneous inclusion arising in tissues due tovarious pathologies like tumor formation local increase inblood volume and other abnormalities [1ndash4] In radiativetransfer theory the most used parameters in modeling laserradiation interaction with biological tissue are absorptionand scattering [5ndash7] However some other studies evokeda significant variation of refractive index of abnormal bio-logical tissues especially in the near infrared range Moreprecisely experimental results [8 9] showed that the tissueof malignant tumors could manifest an increase of therefractive index which can attain until 10 of that of anormal tissue which encircles them So medical imagingby diffuse optical tomography should take advantage from

the emergence of a third contrast parameter which is therefractive index This led to the appearance of a big numberof numerical and fundamental works in the field of radiativetransfer in a varying refractive index biological mediumWhile the conventional radiative transfer equation (RTE)has been widely used to study interaction of near infraredradiation with biological media there exist a number ofworks dealing with a modified radiative transfer equationin spatially varying refractive index media [10 11] Someof these papers are interested in varying refractive indexbiological tissues [12ndash14] In the present paper our first con-cern is to contribute to the usability of the radiative transfertheory in a potential optical tomography setting in medicalimaging At this level studying the effect of refractive indexon the transmitted light through a biological rectangularlayer should be crucial This could improve detectabilityof heterogeneous objects in a typical tomography schemeHowever it is important to note that in a varying refractiveindex medium the rays are not straight lines but curves So

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2014 Article ID 814929 7 pageshttpdxdoiorg1011552014814929

2 Computational and Mathematical Methods in Medicine

even in a rectangular geometry the varying index radiativetransfer equation displays the classical form of the angularderivative terms commonly appearing when dealing withspherical and cylindrical geometries with uniform refractiveindex [15ndash17] This finding gives rise to the use of Legendretransform as a manner for modeling these terms Althoughthis technique was used by Sghaier et al [17] in a uniformrefractive index spherical domain as an innovative view tohandle these terms it prevails as useful in this contemporaryproblem This fact is our second concern in this paperSo we present a computational RTE-based model suitablefor basic diffuse optical tomography forward problem withspatially varying refractive index biological medium Wetreat angular derivative terms by using the Legendre integraltransform techniqueWe investigate cases concerning opticaltomography applications Results concerning the effect of therefractive index variation on the detected signal are shown

2 Mathematical Model

In this work the radiative transfer equation in a humanbiological tissue is described by using a stationary varyingrefractive index RTE [18 19]

Ω sdot nablaΨ ( 119903

Ω) + (120583

119886( 119903) + 120583

119904( 119903)) Ψ ( 119903

Ω)

+

1

119899

nabla119899 sdot nabla ΩΨ( 119903

Ω) minus

2

119899

(Ω sdot nabla119899)Ψ ( 119903

Ω)

=

120583119904(119903)

4120587

int

4120587

0

119875 (Ω

Ω

1015840)Ψ (119903

Ω

1015840) 119889 (

Ω

1015840) + 119878 ( 119903

Ω)

(1)

where

(i) Ψ( 119903 Ω) is the directional energetic radiance at thespatial position vector 119903 = (119909 119910 119911)

(ii) 119899( 119903) is the refractive index distribution

(iii) 120583119886( 119903) and 120583

119904( 119903) are the absorption and scattering

coefficients respectively

(iv) 119888 = 119888vac119899119903 is the ratio of speed of light in a vacuum119888vac and the refractive index 119899

119903

(v) the source term 119878( 119903Ω) is an injected radiance at the

mediumrsquos boundary

(vi) the phase function 119875( Ω

Ω

1015840) describes the probability

that during a scattering event a photon with direc-tion

Ω

1015840 is scattered in the direction Ω

Equation (1) takes into account the fact that the rays arenot straight lines but curves It involves terms that illustratethe expansion or the contraction of the cross section of thetube of light rays in the medium

For a two-dimensional problem and in Cartesian coordi-nate system of the 119909-119910 plane the terms due to the refractiveindex variation can be expressed as

1

119899

nabla119899 sdot nabla ΩΨ minus

2

119899

(Ω sdot nabla119899)Ψ

= minus

1

119899

120597119899

120597119909

(sin120593120597Ψ120597120593

+ 2 cos120593Ψ)

+

1

119899

120597119899

120597119910

(cos120593120597Ψ120597120593

minus 2 sin120593Ψ)

(2)

where cos120593 and sin120593 are the Cartesian coordinates of theunit direction vector in the 119909-119910 plane In fact we assume thatthe radiance of out of plane directions is negligible By usingnotations 120585 = cos120593 and 120578 = sin120593 (2) displays the classicalform of the angular redistribution term commonly appearingwhen dealing with spherical and cylindrical geometries withuniform refractive index [20]1

119899


2

119899


=

1

2119899

2

120597119899

2

120597119909

120597

120597120585

[(1 minus 120585

2

)Ψ] +

120597119899

2

120597119910

120597

120597120578

[(1 minus 120578

2

)Ψ]

(3)

The angular redistribution terms will be noted

119863119909=

120597

120597120585

[(1 minus 120585

2

)Ψ]

119863119910=

120597

120597120578

[(1 minus 120578

2

)Ψ]

(4)

so (3) becomes

1

119899


2

119899

(Ω sdot nabla119899)Ψ =

1

2119899

2

120597119899

2

120597119909

119863119909+

120597119899

2

120597119910

119863119910

(5)

3 Numerical Method

In our numerical implementation we use a rectangulardomainwhich is divided into a set of 119868times119869 elementary uniformvolumes Δ119881 = Δ119909Δ119910Δ119911 with a uniform unitary depth (Δ119911 =1) The angular discretization is obtained through a discreteordinate technique This yields a set of119872 discrete directions120593119898 119898 = 1 sdot sdot sdot119872 giving a set of angular discrete direction

cosines (120585119898 120578119898) 119898 = 1 sdot sdot sdot119872 An orientation depending

on the incident ray direction is adopted for each cell [7]Calculations are done by using integration of (1) over anelementary volume Δ119881 for each discrete directionThis gives

Δ119910120585119898(Ψ119898119864

minus Ψ119898119882

) + Δ119909120578119898(Ψ119898119873

minus Ψ119898119878)

+ Δ119909Δ119910 (120583119886+ 120583119904) Ψ119898119875

+

1

119899

(

120597119899

120597119909

119863119898119909

+

120597119899

120597119910

119863119898119910)

= Δ119909Δ119910(119878119898119875

+ 120583119904

119872

sum

1198981015840=11198981015840= 119898

11990811989810158401199011198981198981015840Ψ1198981015840119875)

(6)

Computational and Mathematical Methods in Medicine 3

where119863119898119909

and119863119898119910

are the discrete angular derivative termsat the angular ordinates 120585

119898and 120578

119898 respectively and 119908

119898is

a weighting factor The discrete term of Henyey-Greensteinphase function is written as

1199011198981198981015840 =

1 minus 119892

2

4120587(1 + 119892

2minus 2119892 (120585

1198981205851198981015840 + 1205781198981205781198981015840))

32

(7)

where 119892 is the anisotropy factor If the direction cosines arepositive the directional radiance is known on the faces 119882and 119878 and they are unknown on the faces 119864 and 119873 of the(119894 119895)-cell and also in the centre 119875 Therefore we need twocomplementary relations to eliminateΨ

119898119864andΨ

119898119873 this can

be obtained by using interpolation formula

Ψ119898119875

= 120572Ψ119898119864

+ (1 minus 120572)Ψ119898119882

Ψ119898119875

= 120572Ψ119898119873

+ (1 minus 120572)Ψ119898119878

(8)

where 120572 is an interpolation parameter Using these relations(6) becomes

Δ119910120585119898

120572

(Ψ119898119875

minus Ψ119898119882

) +

Δ119909120578119898

120572

(Ψ119898119875

minus Ψ119898119878)

+ Δ119909Δ119910 (120583119886+ 120583119904) Ψ119898119875

+

1

2119899

2[

120597119899

2

120597119909

119863119898119909

+

120597119899

2

120597119910

119863119898119910]

= Δ119909Δ119910(119878119898119875

+ 120583119904

119872

sum

1198981015840=11198981015840= 119898

11990811989810158401199011198981198981015840Ψ1198981015840119875)

(9)

Theoretically if we know the solution in the (119894 119895)-cell wecan do calculus over the cells (119894 + 1 119895) and (119894 119895 + 1) using theboundary conditions and the following relations

Ψ119898119882

(119894 + 1 119895) = Ψ119898119864

(119894 119895) 119894 = 1 sdot sdot sdot 119868 minus 1

Ψ119898119864

(119894 119895 + 1) = Ψ119898119873


(10)

If the direction cosines are both positive we get thefollowing equation

Δ119910120585119898

120572

(Ψ119898119894119895

minus Ψ119898119894minus1119895

) +

Δ119909120578119898

120572

(Ψ119898119894119895

minus Ψ119898119894119895minus1

)

+ Δ119909Δ119910 (120583119886+ 120583119904) Ψ119898119894119895

+

1

2119899

2[

120597119899

2

120597119909

119863119898119909

+

120597119899

2

120597119910

119863119898119910]

= Δ119909Δ119910(119878119898119894119895

+ 120583119904

119872

sum

1198981015840=1

11990811989810158401199011198981198981015840Ψ1198981015840119894119895)

(11)

this gives

Ψ119898119894119895

= [Ψ119898119894119895

minus

119899119894119895

2119888Δ119905

Ψ119898119894119895

+

Δ119910120585119898

120572

Ψ119898119894minus1119895

+

Δ119909120578119898

120572

Ψ119898119894119895minus1

+

1

2119899

2

119894119895

[Δ119910 (119899

2

119894119895minus 119899

2

119894minus1119895)119863119898119909119894119895

+ Δ119910 (119899

2

119894119895minus 119899

2

119894119895minus1)119863119898119909119894119895

]

+Δ119909Δ119910(119878119898119894119895

+ 120583119904119894119895

119872

sum

1198981015840=1

11990811989810158401199011198981198981015840Ψ1198981015840119894119895)]

times [

Δ119910120585119898

120572

+

Δ119909120578119898

120572

+ Δ119909Δ119910 (120583119886119894119895

+ 120583119904119894119895)]

minus1

(12)

31 Numerical Treatment of Angular Derivative Terms withFinite Legendre Transform As is explained in [17] we con-sider the following Legendre transforms

L(

120597

120597120585

[(1 minus 120585

2

)Ψ] 119904) = int

1

minus1

120597

120597120585

[(1 minus 120585

2

)Ψ]119875119904(120585) 119889120585

L(

120597

120597120578

[(1 minus 120578

2

)Ψ] 119904) = int

1

minus1

120597

120597120578

[(1 minus 120578

2

)Ψ]119875119904(120578) 119889120578

(13)

where 119875119898is the119898 degree Legendre polynomial According to

Sturn-Liouville equation defining Legendre Polynomials wecan write

int

1

minus1

120597

120597120578

[(1 minus 120585

2

) 120595] 119875119904(120585) 119889120585

=

119904 (119904 + 1)

2119904 + 1

[int

1

minus1

Ψ119875119904+1(120585) 119889120585 minus int

1

minus1

Ψ119875119904minus1(120585) 119889120585]

int

1

minus1

120597

120597120578

[(1 minus 120585

2

) 120595] 119875119904(120578) 119889120578

=

119904 (119904 + 1)

2119904 + 1

[int

1

minus1

Ψ119875119904+1(120578) 119889120578 minus int

1

minus1

Ψ119875119904minus1(120578) 119889120578]

(14)

To obtain derivative terms we make use of numericalquadrature

119872

sum

119898=1

119908119898119863120585119898119875119904(120585119898)

=

119904 (119904 + 1)

2119904 + 1

[

119872

sum

119898=1

119908119898119875119904+1(120585119898) minus

119872

sum

119898=1

119908119898119875119904minus1(120585119898)]

119904 = 1 119872 minus 1


119872

sum

119898=1

119908119898119863120578119898119875119904(120578119898)

=

119904 (119904 + 1)

2119904 + 1

[

119872

sum

119898=1

119908119898119875119904+1(120578119898) minus

119872

sum

119898=1

119908119898119875119904minus1(120578119898)]

119904 = 1 119872 minus 1

(15)

The above systems of equations are closed by the obviousrelations

119872

sum

119898=1

119908119898119863120585119898

= int

1

minus1

120597

120597120578

[(1 minus 120585

2

) 120595] 119889 (120585)

119872

sum

119898=1

119908119898119863120578119898

= int

1

minus1

120597

120597120578

[(1 minus 120578

2

) 120595] 119889 (120578)

(16)

So discrete derivative terms in the (119894 119895)-cell into the mediumcan be obtained by solving the linear algebraic equations

119860120585

119863120585119894119895

= 119861120585119894119895 119860

120578

119863120578119894119895

= 119861120578119894119895 (17)

where

A120585

= (

1199081

1199082

119908119872

11990811198751(1205851) 119908

21198751(1205852) 119908

1198721198751(120585119872)

1199081119875119872minus1

(1205851) 1199082119875119872minus1

(1205852) 119908

119872119875119872minus1

(120585119872)

)

D120578119894119895

= (

1198631205851119894119895

1198631205852119894119895

119863120585119872119894119895

)

B120578119894119895

=

(

(

(

(

(

0

2

3

119872

sum

1198981015840=1

11990811989810158401205951198981015840119894119895(1198752(1205851198981015840)) minus (119875

0(1205851198981015840))

119872(119872 minus 1)

2119872 minus 1

119872

sum

1198981015840=1

11990811989810158401205951198981015840119894119895(119875119872(1205851198981015840) minus 119875

119872minus2(1205851198981015840))

)

)

)

)

)

(18)

and 119860120578119863120578119894119895 119861120578119894119895

are straightforward by replacing 120585 by 120578A set of weights are judiciously chosen with an equally

spaced distribution of angular ordinates (120585 120578) The matrices119860120585and 119860

120578must be regular and they are inversed each only

once in all the calculation process

To solve (12) we use successive iterations to actualise theimplicit internal source term in the right member So thisgives

Ψ

119896+1

119898119894119895= [

Δ119910120585119898

120572

Ψ

119896+1

119898119894minus1119895+

Δ119909120578119898

120572

Ψ

119896+1

119898119894119895minus1

+

1

2119899

2

119894119895

[Δ119910 (119899

2

119894119895minus 119899

2

119894minus1119895)119863119898119909119894119895

+ Δ119910 (119899

2

119894119895minus 119899

2

119894119895minus1)119863119898119909119894119895

]

+Δ119909Δ119910(119878

119896+1

119898119894119895+ 120583119904119894119895

119872

sum

1198981015840=1

11990811989810158401199011198981198981015840Ψ

119896

1198981015840119894119895)]

times [

Δ119910120585119898

120572

+

Δ119909120578119898

120572

+ Δ119909Δ119910 (120583119886119894119895

+ 120583119904119894119895)]

minus1

(19)

The iteration process is repeated until a convergencecriterion is attempted To improve convergence speed weuse a successive overrelaxation method So the updated valueΨ

119896+1

119898119894119895is a linear combination of the iterated value Ψ119896

119898119894119895and

the previously computed value

(Ψ

119896+1

119898119894119895)

updated= 120588Ψ

119896

119898119894119895+ (1 minus 120588)Ψ

119896+1

119898119894119895 (20)

where 120588 is a relaxation parameter whose value is usuallybetween 1 and 2 The solution is obtained when the relativediscrepancy value

120576 =

Ψ

119896+1

119898119894119895minus Ψ

119896

119898119894119895

Ψ

119896

119898119894119895

(21)

is smaller than a tolerance value In all our calculus we havetaken 10minus8 as a tolerance value As initial condition we takea field of null radiance Also all our calculations are done inthe case of interpolation diamond scheme (120572 = 05)

If the direction cosines are not both positive the prece-dent equations are valid provided that the orientationWESNof cells is done according to the direction of propagation [7]In all our investigations the injected power source is assumedto be equivalent to a forward collimated monochromaticintensity placed at a source point on themiddle of the bottomside of the boundary Results shown below are obtained byusing a continuous wave source with a uniform equivalentintensity value of 50mWsdotcmminus1

32 Boundary Conditions On the boundary the radiance isthe sum of the external source contribution and the partlyreflected radiance due to the refractive index mismatch at theboundary

(i) Ψ( 119903 Ω) = 119878( 119903119887Ω) + 119877 sdot Ψ( 119903

Ωref)

(ii) 119887sdotΩ lt 0 and

119887sdotΩref = minus119887 sdot

Ω

where 119903119887is a position on the boundary and

119887is an outer

normal unit vector The reflectivity 119877 can be calculated foreach direction using Fresnelrsquos relations


To present our results we use the detected fluence ratewhich is given in a (119894

119889 119895119889)-detector point on the boundary as

Φ119889=

119872

sum

119898=1

(1 minus 119877119898119894119889119895119889

)119908119898Ψ119898119894119889119895119889

(22)

with119877119898119894119889119895119889

=

1

if 120593119898gt arcsin(

119899air119899119894119889119895119889

)

1

2

[(

119899119894119889119895119889

cos120593119898minus 119899air cos120593119905

119899119894119889119895119889

cos120593119898+ 119899air cos120593119905

)

2

+(

119899119894119889119895119889

cos120593119905minus 119899air cos120593119898

119899119894119889119895119889

cos120593119905+ 119899air cos120593119898

)

2

]

else

120593119905= arcsin(

119899air sin120593119898119899119894119889119895119889

) 119899air asymp 1

(23)

Also we make use of a normalized detected fluence ratedefined as

Φ119873=

Φ119889

(1119863)sum

119863

119889=1119908

1015840

119889Φ119889

(24)

where 119863 is the number of the detector points on one sideof the boundary and 119908

1015840

119889is a weighting factor from the

generalized trapezoidal integration ruleIn all calculations we have used 28 detector points

on each side Also all calculus is carried out by using 16uniformly distributed discrete directions and a space grid of121 times 121 cells

4 Results and Discussion

41 Continuous Varying Refractive Index Medium In thisinvestigation we study near infrared radiation transport ina rectangular medium exposed to a continuous collimatedsource which is placed on the bottom side of the boundaryFigure 1 shows the considered medium it is assumed tobe 2 times 2 cm sized with varying refractive index Withinthe medium we consider a 119909-axis linear refractive indexvariation with different gradient values To show the effectof the refractive index on detected fluence rate we haveused a weakly absorbing and forward scattering backgroundmedium whose optical parameters are shown in Figure 1

Figures 2 and 3 represent the response of the mediumthrough the detected signal on the top and right side of theboundary in the case of linear variation of refractive indexWe note that the response of the medium varies linearlyaccording to the gradient of refractive index The maximumtransmission moves to regions of height index (Figure 2)Indeed according to boundary conditions of the medium(Descartes Laws) the transmission window increases propor-tionally of refractive index between the medium and outsidemedium (air)

Detector points

Top side

Right side

Source

y

x1 cm

120583s = 25 cmminus1

120583a = 05 cmminus1

g = 08

n = 033 + 120575 timesx

L

Figure 1 A test-medium with background properties and detectorpoints

16

14

12

10

08

06

04

02

0000 05 10 15 20

Top side (cm)

Nor

mal

ized

flue

nce r

ate (

au)

n = 156

n = 13 + 02 lowast xL




Figure 2 Response of a continuous varying refractive indexmedium detected fluence rate on the top side gradient effect

On the other hand the right side (Figure 3) shows theresponse of the medium detected on the right side of theboundary in the case of linear and parabolic variationrespectively The detected fluence rate curves presents distin-guished effect of refractive index gradient in linear case Thetransmission zone on the boundary increases with decreasinggradient values Even though most detected transmissionis obtained near the source a weak gradient can augmenttransmitted radiation relatively far from the sourcewhile highvalues of gradient can block transmitted radiation within themedium

42 Effect of Stochastic Varying Refractive Index In thisinvestigation we keep the same medium precedent but wewill study the stochastic variation refractive indexWe control


24

20

16

12

08

04

00

00 05 10 15 20

Right side (cm)

n = 156





Nor

mal

ized

flue

nce r

ate (

au)

Figure 3 Response of a continuous varying refractive indexmedium detected fluence rate on the right side gradient effect

16

14

12

10

08

06

04

0200 05 10 15 20

Top side (cm)

Nor

mal

ized

flue

nce r

ate (

au)

Stochastic varying RI in [12ndash17]Stochastic varying RI in [125ndash175]Stochastic varying RI in [13ndash18]

Figure 4 Response of stochastic varying refractive index mediumon the top side

this variation by uniform random fixing of a given intervalThe other optical properties are the same as in the precedentinvestigation In such cases there is an obvious effect ofthe heterogeneity on the detected signal especially when therefractive index is increased or decreased by 10 in a stochas-tic way The detected signal on different sides of the mediumis shown in Figures 4 and 5 There is a significant effect ofthis variation on the detected signal on the top side andright side because there is strong perturbation in mediumThese findings highlight the potential of refractive index asa possible detection parameter of a tumor in a surroundingsafe tissue These findings open prospects for further study

20

16

12

08

04

0000 05 10 15 20

Right side (cm)

Nor

mal

ized

flue

nce r

ate (

au)


Figure 5 Response of stochastic varying refractive index mediumon the right side

of the effect of local refractive variation in domain andfrequency-domain schemes in diffuse optical tomography

5 Conclusion

This study attempted to develop a computational way helpingin detection of abnormalities in a biological tissue Thisshould enable predictions of eventual tumor existence whenusing a diffuse optical tomography scheme The used modelis based on stationary radiative transfer equation includinga possible continuous and stochastic variation of refractiveindex In particular computational technique of Legendretransform is extended to handle angular derivative termsarising by the varying refractive index consideration Theobtained computational model is implemented to investi-gate some practical situations in diffuse optical tomography(DOT) setting Obtained results showed that variation ofrefractive index can yield useful predictions about the targetand the location of abnormal inclusions within the tissueThese findings open prospects for further study of the effectof local refractive variation in time-domain and frequency-domain schemes in diffuse optical tomography

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] S R Arridge and J C Hebden ldquoOptical imaging in medicinerdquoPhysics in Medicine and Biology vol 42 pp 841ndash853 1997

[2] S R Arridge andM Schweiger ldquoA gradient-based optimisationscheme for optical tomographyrdquoOptics Express vol 2 no 6 pp213ndash226 1998


[3] V Ntziachristos A H Hielscher A G Yodh and B ChanceldquoDiffuse optical tomography of highly heterogeneous mediardquoIEEE Transactions on Medical Imaging vol 20 no 6 pp 470ndash478 2001

[4] A H Hielscher A Y Bluestone G S Abdoulaev et al ldquoNear-infrared diffuse optical tomographyrdquo Optics Express vol 18 no5-6 pp 313ndash337 2002

[5] A D Klose ldquoThe forward and inverse problem in tissue opticsbased on the radiative transfer equation a brief reviewrdquo Journalof Quantitative Spectroscopy and Radiative Transfer vol 111 no11 pp 1852ndash1853 2010

[6] C D Klose A D Klose U Netz J Beuthan and A HHielscher ldquoMultiparameter classifications of optical tomo-graphic imagesrdquo Journal of Biomedical Optics vol 13 no 5Article ID 050503 2008

[7] H Trabelsi M Gantri E Sediki and R Ben Salah ldquoCom-putational laser spectroscopy in a biological tissuerdquo Journal ofBiophysics vol 2010 Article ID 253763 9 pages 2010

[8] B B Das F Liu and R Alfano ldquoTime-resolved fluorescenceand photonmigration studies in biomedical andmodel randommediardquo Reports on Progress in Physics vol 60 pp 227ndash292 1997

[9] M Ohmi Y Ohnishi K Yoden and M Haruna ldquoIn vitrosimultaneous measurement of refractive index and thicknessof biological tissue by the low coherence interferometryrdquo IEEETransactions on Biomedical Engineering vol 47 no 9 pp 1266ndash1270 2000

[10] D Lemonnier and V LeDez ldquoDiscrete ordinate solution ofradiative transfer across a slab with variable refractive indexrdquoJournal of Quantitative Spectroscopy and Radiative Transfer vol9 pp 195ndash204 2005

[11] L H Liu ldquoBenchmark numerical solutions for radiative heattransfer in two-dimensional medium with graded index dis-tributionrdquo Journal of Quantitative Spectroscopy and RadiativeTransfer vol 102 no 2 pp 293ndash303 2006

[12] J Beuthan O Minet J Helfmann M Herrig and G MullerldquoThe spatial variation of the refractive index of biological cellsrdquoPhysics in Medicine and Biology vol 41 pp 369ndash382 1996

[13] H Dehghani B Brooksby K Vishwanath B W Pogue and KD Paulsen ldquoThe effects of internal refractive index variation innear-infrared optical tomography a finite element modellingapproachrdquo Physics in Medicine and Biology vol 48 no 16 pp2713ndash2727 2003

[14] T Khan and H Jiang ldquoA new diffusion approximation to theradiative transfer equation for scattering media with spatiallyvarying refractive indicesrdquo Journal of Optics A Pure andAppliedOptics vol 5 no 2 pp 137ndash141 2003

[15] J R Tsai and M N Zisik ldquoRadiation in cylindrical symmetrywith anisotropic scattering and variable propertiesrdquo Journal ofHeat and Mass Transfer vol 43 pp 2651ndash2658 1990

[16] G Jia Y Yener and J W Cipolla ldquoRadiation between two con-centric spheres separated by a participatingmediumrdquo Journal ofQuantitative Spectroscopy and Radiative Transfer vol 46 no 1pp 11ndash19 1991

[17] T Sghaier M S Sifaoui and A Soufiani ldquoStudy of radiationin spherical media using discrete ordinates method associatedwith the finite Legendre transformrdquo Journal of QuantitativeSpectroscopy and Radiative Transfer vol 64 no 4 pp 339ndash3512000

[18] J Tualle and E Tinet ldquoDerivation of the radiative transferequation for scattering media with a spatially varying refractiveindexrdquo Optics Communications vol 228 pp 33ndash38 2003

[19] H A Ferwerda ldquoRadiative transfer equation for scatteringmedia with a spatially varying refractive indexrdquo Journal ofOptics A Pure and Applied Optics vol 1 no 3 pp L1ndashL2 1999

[20] M Modest Radiative Heat Transfer Academic Press 2003

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


MEDIATORSINFLAMMATION

of


Behavioural Neurology

EndocrinologyInternational Journal of



Disease Markers


BioMed Research International

OncologyJournal of



Oxidative Medicine and Cellular Longevity


PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of



Computational and Mathematical Methods in Medicine

OphthalmologyJournal of


Diabetes ResearchJournal of



Research and TreatmentAIDS


Gastroenterology Research and Practice


Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom


even in a rectangular geometry the varying index radiativetransfer equation displays the classical form of the angularderivative terms commonly appearing when dealing withspherical and cylindrical geometries with uniform refractiveindex [15ndash17] This finding gives rise to the use of Legendretransform as a manner for modeling these terms Althoughthis technique was used by Sghaier et al [17] in a uniformrefractive index spherical domain as an innovative view tohandle these terms it prevails as useful in this contemporaryproblem This fact is our second concern in this paperSo we present a computational RTE-based model suitablefor basic diffuse optical tomography forward problem withspatially varying refractive index biological medium Wetreat angular derivative terms by using the Legendre integraltransform techniqueWe investigate cases concerning opticaltomography applications Results concerning the effect of therefractive index variation on the detected signal are shown

2 Mathematical Model

In this work the radiative transfer equation in a humanbiological tissue is described by using a stationary varyingrefractive index RTE [18 19]

Ω sdot nablaΨ ( 119903

Ω) + (120583

119886( 119903) + 120583

119904( 119903)) Ψ ( 119903

Ω)

+

1

119899

nabla119899 sdot nabla ΩΨ( 119903

Ω) minus

2

119899

(Ω sdot nabla119899)Ψ ( 119903

Ω)

=

120583119904(119903)

4120587

int

4120587

0

119875 (Ω

Ω

1015840)Ψ (119903

Ω

1015840) 119889 (

Ω

1015840) + 119878 ( 119903

Ω)

(1)

where

(i) Ψ( 119903 Ω) is the directional energetic radiance at thespatial position vector 119903 = (119909 119910 119911)

(ii) 119899( 119903) is the refractive index distribution

(iii) 120583119886( 119903) and 120583

119904( 119903) are the absorption and scattering

coefficients respectively

(iv) 119888 = 119888vac119899119903 is the ratio of speed of light in a vacuum119888vac and the refractive index 119899

119903

(v) the source term 119878( 119903Ω) is an injected radiance at the

mediumrsquos boundary

(vi) the phase function 119875( Ω

Ω

1015840) describes the probability

that during a scattering event a photon with direc-tion

Ω

1015840 is scattered in the direction Ω

Equation (1) takes into account the fact that the rays arenot straight lines but curves It involves terms that illustratethe expansion or the contraction of the cross section of thetube of light rays in the medium

For a two-dimensional problem and in Cartesian coordi-nate system of the 119909-119910 plane the terms due to the refractiveindex variation can be expressed as

1

119899


2

119899


= minus

1

119899

120597119899

120597119909

(sin120593120597Ψ120597120593

+ 2 cos120593Ψ)

+

1

119899

120597119899

120597119910

(cos120593120597Ψ120597120593

minus 2 sin120593Ψ)

(2)

where cos120593 and sin120593 are the Cartesian coordinates of theunit direction vector in the 119909-119910 plane In fact we assume thatthe radiance of out of plane directions is negligible By usingnotations 120585 = cos120593 and 120578 = sin120593 (2) displays the classicalform of the angular redistribution term commonly appearingwhen dealing with spherical and cylindrical geometries withuniform refractive index [20]1

119899


2

119899


=

1

2119899

2

120597119899

2

120597119909

120597

120597120585

[(1 minus 120585

2

)Ψ] +

120597119899

2

120597119910

120597

120597120578

[(1 minus 120578

2

)Ψ]

(3)

The angular redistribution terms will be noted

119863119909=

120597

120597120585

[(1 minus 120585

2

)Ψ]

119863119910=

120597

120597120578

[(1 minus 120578

2

)Ψ]

(4)

so (3) becomes

1

119899


2

119899

(Ω sdot nabla119899)Ψ =

1

2119899

2

120597119899

2

120597119909

119863119909+

120597119899

2

120597119910

119863119910

(5)

3 Numerical Method

In our numerical implementation we use a rectangulardomainwhich is divided into a set of 119868times119869 elementary uniformvolumes Δ119881 = Δ119909Δ119910Δ119911 with a uniform unitary depth (Δ119911 =1) The angular discretization is obtained through a discreteordinate technique This yields a set of119872 discrete directions120593119898 119898 = 1 sdot sdot sdot119872 giving a set of angular discrete direction

cosines (120585119898 120578119898) 119898 = 1 sdot sdot sdot119872 An orientation depending

on the incident ray direction is adopted for each cell [7]Calculations are done by using integration of (1) over anelementary volume Δ119881 for each discrete directionThis gives

Δ119910120585119898(Ψ119898119864

minus Ψ119898119882

) + Δ119909120578119898(Ψ119898119873

minus Ψ119898119878)

+ Δ119909Δ119910 (120583119886+ 120583119904) Ψ119898119875

+

1

119899

(

120597119899

120597119909

119863119898119909

+

120597119899

120597119910

119863119898119910)

= Δ119909Δ119910(119878119898119875

+ 120583119904

119872

sum

1198981015840=11198981015840= 119898

11990811989810158401199011198981198981015840Ψ1198981015840119875)

(6)


where119863119898119909

and119863119898119910


119898and 120578


119898is


1199011198981198981015840 =

1 minus 119892

2

4120587(1 + 119892

2minus 2119892 (120585

1198981205851198981015840 + 1205781198981205781198981015840))

32

(7)


119898119864andΨ

119898119873 this can


Ψ119898119875

= 120572Ψ119898119864

+ (1 minus 120572)Ψ119898119882

Ψ119898119875

= 120572Ψ119898119873

+ (1 minus 120572)Ψ119898119878

(8)


Δ119910120585119898

120572

(Ψ119898119875

minus Ψ119898119882

) +

Δ119909120578119898

120572

(Ψ119898119875

minus Ψ119898119878)

+ Δ119909Δ119910 (120583119886+ 120583119904) Ψ119898119875

+

1

2119899

2[

120597119899

2

120597119909

119863119898119909

+

120597119899

2

120597119910

119863119898119910]

= Δ119909Δ119910(119878119898119875

+ 120583119904

119872

sum

1198981015840=11198981015840= 119898

11990811989810158401199011198981198981015840Ψ1198981015840119875)

(9)


Ψ119898119882

(119894 + 1 119895) = Ψ119898119864


Ψ119898119864

(119894 119895 + 1) = Ψ119898119873


(10)


Δ119910120585119898

120572

(Ψ119898119894119895

minus Ψ119898119894minus1119895

) +

Δ119909120578119898

120572

(Ψ119898119894119895

minus Ψ119898119894119895minus1

)

+ Δ119909Δ119910 (120583119886+ 120583119904) Ψ119898119894119895

+

1

2119899

2[

120597119899

2

120597119909

119863119898119909

+

120597119899

2

120597119910

119863119898119910]

= Δ119909Δ119910(119878119898119894119895

+ 120583119904

119872

sum

1198981015840=1

11990811989810158401199011198981198981015840Ψ1198981015840119894119895)

(11)

this gives

Ψ119898119894119895

= [Ψ119898119894119895

minus

119899119894119895

2119888Δ119905

Ψ119898119894119895

+

Δ119910120585119898

120572

Ψ119898119894minus1119895

+

Δ119909120578119898

120572

Ψ119898119894119895minus1

+

1

2119899

2

119894119895

[Δ119910 (119899

2

119894119895minus 119899

2

119894minus1119895)119863119898119909119894119895

+ Δ119910 (119899

2

119894119895minus 119899

2

119894119895minus1)119863119898119909119894119895

]

+Δ119909Δ119910(119878119898119894119895

+ 120583119904119894119895

119872

sum

1198981015840=1

11990811989810158401199011198981198981015840Ψ1198981015840119894119895)]

times [

Δ119910120585119898

120572

+

Δ119909120578119898

120572

+ Δ119909Δ119910 (120583119886119894119895

+ 120583119904119894119895)]

minus1

(12)


L(

120597

120597120585

[(1 minus 120585

2

)Ψ] 119904) = int

1

minus1

120597

120597120585

[(1 minus 120585

2

)Ψ]119875119904(120585) 119889120585

L(

120597

120597120578

[(1 minus 120578

2

)Ψ] 119904) = int

1

minus1

120597

120597120578

[(1 minus 120578

2

)Ψ]119875119904(120578) 119889120578

(13)



int

1

minus1

120597

120597120578

[(1 minus 120585

2

) 120595] 119875119904(120585) 119889120585

=

119904 (119904 + 1)

2119904 + 1

[int

1

minus1

Ψ119875119904+1(120585) 119889120585 minus int

1

minus1

Ψ119875119904minus1(120585) 119889120585]

int

1

minus1

120597

120597120578

[(1 minus 120585

2

) 120595] 119875119904(120578) 119889120578

=

119904 (119904 + 1)

2119904 + 1

[int

1

minus1

Ψ119875119904+1(120578) 119889120578 minus int

1

minus1

Ψ119875119904minus1(120578) 119889120578]

(14)


119872

sum

119898=1

119908119898119863120585119898119875119904(120585119898)

=

119904 (119904 + 1)

2119904 + 1

[

119872

sum

119898=1

119908119898119875119904+1(120585119898) minus

119872

sum

119898=1

119908119898119875119904minus1(120585119898)]

119904 = 1 119872 minus 1


119872

sum

119898=1

119908119898119863120578119898119875119904(120578119898)

=

119904 (119904 + 1)

2119904 + 1

[

119872

sum

119898=1

119908119898119875119904+1(120578119898) minus

119872

sum

119898=1

119908119898119875119904minus1(120578119898)]

119904 = 1 119872 minus 1

(15)


119872

sum

119898=1

119908119898119863120585119898

= int

1

minus1

120597

120597120578

[(1 minus 120585

2

) 120595] 119889 (120585)

119872

sum

119898=1

119908119898119863120578119898

= int

1

minus1

120597

120597120578

[(1 minus 120578

2

) 120595] 119889 (120578)

(16)


119860120585

119863120585119894119895

= 119861120585119894119895 119860

120578

119863120578119894119895

= 119861120578119894119895 (17)

where

A120585

= (

1199081

1199082

119908119872

11990811198751(1205851) 119908

21198751(1205852) 119908

1198721198751(120585119872)

1199081119875119872minus1

(1205851) 1199082119875119872minus1

(1205852) 119908

119872119875119872minus1

(120585119872)

)

D120578119894119895

= (

1198631205851119894119895

1198631205852119894119895

119863120585119872119894119895

)

B120578119894119895

=

(

(

(

(

(

0

2

3

119872

sum

1198981015840=1

11990811989810158401205951198981015840119894119895(1198752(1205851198981015840)) minus (119875

0(1205851198981015840))

119872(119872 minus 1)

2119872 minus 1

119872

sum

1198981015840=1

11990811989810158401205951198981015840119894119895(119875119872(1205851198981015840) minus 119875

119872minus2(1205851198981015840))

)

)

)

)

)

(18)

and 119860120578119863120578119894119895 119861120578119894119895






Ψ

119896+1

119898119894119895= [

Δ119910120585119898

120572

Ψ

119896+1

119898119894minus1119895+

Δ119909120578119898

120572

Ψ

119896+1

119898119894119895minus1

+

1

2119899

2

119894119895

[Δ119910 (119899

2

119894119895minus 119899

2

119894minus1119895)119863119898119909119894119895

+ Δ119910 (119899

2

119894119895minus 119899

2

119894119895minus1)119863119898119909119894119895

]

+Δ119909Δ119910(119878

119896+1

119898119894119895+ 120583119904119894119895

119872

sum

1198981015840=1

11990811989810158401199011198981198981015840Ψ

119896

1198981015840119894119895)]

times [

Δ119910120585119898

120572

+

Δ119909120578119898

120572

+ Δ119909Δ119910 (120583119886119894119895

+ 120583119904119894119895)]

minus1

(19)


119896+1


119898119894119895and


(Ψ

119896+1

119898119894119895)

updated= 120588Ψ

119896

119898119894119895+ (1 minus 120588)Ψ

119896+1

119898119894119895 (20)


120576 =

Ψ

119896+1

119898119894119895minus Ψ

119896

119898119894119895

Ψ

119896

119898119894119895

(21)




(i) Ψ( 119903 Ω) = 119878( 119903119887Ω) + 119877 sdot Ψ( 119903

Ωref)



Ω


119887is an outer





Φ119889=

119872

sum

119898=1

(1 minus 119877119898119894119889119895119889

)119908119898Ψ119898119894119889119895119889

(22)

with119877119898119894119889119895119889

=

1


119899air119899119894119889119895119889

)

1

2

[(

119899119894119889119895119889

cos120593119898minus 119899air cos120593119905

119899119894119889119895119889

cos120593119898+ 119899air cos120593119905

)

2

+(

119899119894119889119895119889

cos120593119905minus 119899air cos120593119898

119899119894119889119895119889

cos120593119905+ 119899air cos120593119898

)

2

]

else

120593119905= arcsin(

119899air sin120593119898119899119894119889119895119889

) 119899air asymp 1

(23)


Φ119873=

Φ119889

(1119863)sum

119863

119889=1119908

1015840

119889Φ119889

(24)


1015840







Detector points

Top side

Right side

Source

y

x1 cm

120583s = 25 cmminus1

120583a = 05 cmminus1

g = 08

n = 033 + 120575 timesx

L


16

14

12

10

08

06

04

02

0000 05 10 15 20

Top side (cm)

Nor

mal

ized

flue

nce r

ate (

au)

n = 156









24

20

16

12

08

04

00

00 05 10 15 20

Right side (cm)

n = 156





Nor

mal

ized

flue

nce r

ate (

au)


16

14

12

10

08

06

04

0200 05 10 15 20

Top side (cm)

Nor

mal

ized

flue

nce r

ate (

au)




20

16

12

08

04

0000 05 10 15 20

Right side (cm)

Nor

mal

ized

flue

nce r

ate (

au)




5 Conclusion




References



























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















where119863119898119909

and119863119898119910


119898and 120578


119898is


1199011198981198981015840 =

1 minus 119892

2

4120587(1 + 119892

2minus 2119892 (120585

1198981205851198981015840 + 1205781198981205781198981015840))

32

(7)


119898119864andΨ

119898119873 this can


Ψ119898119875

= 120572Ψ119898119864

+ (1 minus 120572)Ψ119898119882

Ψ119898119875

= 120572Ψ119898119873

+ (1 minus 120572)Ψ119898119878

(8)


Δ119910120585119898

120572

(Ψ119898119875

minus Ψ119898119882

) +

Δ119909120578119898

120572

(Ψ119898119875

minus Ψ119898119878)

+ Δ119909Δ119910 (120583119886+ 120583119904) Ψ119898119875

+

1

2119899

2[

120597119899

2

120597119909

119863119898119909

+

120597119899

2

120597119910

119863119898119910]

= Δ119909Δ119910(119878119898119875

+ 120583119904

119872

sum

1198981015840=11198981015840= 119898

11990811989810158401199011198981198981015840Ψ1198981015840119875)

(9)


Ψ119898119882

(119894 + 1 119895) = Ψ119898119864


Ψ119898119864

(119894 119895 + 1) = Ψ119898119873


(10)


Δ119910120585119898

120572

(Ψ119898119894119895

minus Ψ119898119894minus1119895

) +

Δ119909120578119898

120572

(Ψ119898119894119895

minus Ψ119898119894119895minus1

)

+ Δ119909Δ119910 (120583119886+ 120583119904) Ψ119898119894119895

+

1

2119899

2[

120597119899

2

120597119909

119863119898119909

+

120597119899

2

120597119910

119863119898119910]

= Δ119909Δ119910(119878119898119894119895

+ 120583119904

119872

sum

1198981015840=1

11990811989810158401199011198981198981015840Ψ1198981015840119894119895)

(11)

this gives

Ψ119898119894119895

= [Ψ119898119894119895

minus

119899119894119895

2119888Δ119905

Ψ119898119894119895

+

Δ119910120585119898

120572

Ψ119898119894minus1119895

+

Δ119909120578119898

120572

Ψ119898119894119895minus1

+

1

2119899

2

119894119895

[Δ119910 (119899

2

119894119895minus 119899

2

119894minus1119895)119863119898119909119894119895

+ Δ119910 (119899

2

119894119895minus 119899

2

119894119895minus1)119863119898119909119894119895

]

+Δ119909Δ119910(119878119898119894119895

+ 120583119904119894119895

119872

sum

1198981015840=1

11990811989810158401199011198981198981015840Ψ1198981015840119894119895)]

times [

Δ119910120585119898

120572

+

Δ119909120578119898

120572

+ Δ119909Δ119910 (120583119886119894119895

+ 120583119904119894119895)]

minus1

(12)


L(

120597

120597120585

[(1 minus 120585

2

)Ψ] 119904) = int

1

minus1

120597

120597120585

[(1 minus 120585

2

)Ψ]119875119904(120585) 119889120585

L(

120597

120597120578

[(1 minus 120578

2

)Ψ] 119904) = int

1

minus1

120597

120597120578

[(1 minus 120578

2

)Ψ]119875119904(120578) 119889120578

(13)



int

1

minus1

120597

120597120578

[(1 minus 120585

2

) 120595] 119875119904(120585) 119889120585

=

119904 (119904 + 1)

2119904 + 1

[int

1

minus1

Ψ119875119904+1(120585) 119889120585 minus int

1

minus1

Ψ119875119904minus1(120585) 119889120585]

int

1

minus1

120597

120597120578

[(1 minus 120585

2

) 120595] 119875119904(120578) 119889120578

=

119904 (119904 + 1)

2119904 + 1

[int

1

minus1

Ψ119875119904+1(120578) 119889120578 minus int

1

minus1

Ψ119875119904minus1(120578) 119889120578]

(14)


119872

sum

119898=1

119908119898119863120585119898119875119904(120585119898)

=

119904 (119904 + 1)

2119904 + 1

[

119872

sum

119898=1

119908119898119875119904+1(120585119898) minus

119872

sum

119898=1

119908119898119875119904minus1(120585119898)]

119904 = 1 119872 minus 1


119872

sum

119898=1

119908119898119863120578119898119875119904(120578119898)

=

119904 (119904 + 1)

2119904 + 1

[

119872

sum

119898=1

119908119898119875119904+1(120578119898) minus

119872

sum

119898=1

119908119898119875119904minus1(120578119898)]

119904 = 1 119872 minus 1

(15)


119872

sum

119898=1

119908119898119863120585119898

= int

1

minus1

120597

120597120578

[(1 minus 120585

2

) 120595] 119889 (120585)

119872

sum

119898=1

119908119898119863120578119898

= int

1

minus1

120597

120597120578

[(1 minus 120578

2

) 120595] 119889 (120578)

(16)


119860120585

119863120585119894119895

= 119861120585119894119895 119860

120578

119863120578119894119895

= 119861120578119894119895 (17)

where

A120585

= (

1199081

1199082

119908119872

11990811198751(1205851) 119908

21198751(1205852) 119908

1198721198751(120585119872)

1199081119875119872minus1

(1205851) 1199082119875119872minus1

(1205852) 119908

119872119875119872minus1

(120585119872)

)

D120578119894119895

= (

1198631205851119894119895

1198631205852119894119895

119863120585119872119894119895

)

B120578119894119895

=

(

(

(

(

(

0

2

3

119872

sum

1198981015840=1

11990811989810158401205951198981015840119894119895(1198752(1205851198981015840)) minus (119875

0(1205851198981015840))

119872(119872 minus 1)

2119872 minus 1

119872

sum

1198981015840=1

11990811989810158401205951198981015840119894119895(119875119872(1205851198981015840) minus 119875

119872minus2(1205851198981015840))

)

)

)

)

)

(18)

and 119860120578119863120578119894119895 119861120578119894119895






Ψ

119896+1

119898119894119895= [

Δ119910120585119898

120572

Ψ

119896+1

119898119894minus1119895+

Δ119909120578119898

120572

Ψ

119896+1

119898119894119895minus1

+

1

2119899

2

119894119895

[Δ119910 (119899

2

119894119895minus 119899

2

119894minus1119895)119863119898119909119894119895

+ Δ119910 (119899

2

119894119895minus 119899

2

119894119895minus1)119863119898119909119894119895

]

+Δ119909Δ119910(119878

119896+1

119898119894119895+ 120583119904119894119895

119872

sum

1198981015840=1

11990811989810158401199011198981198981015840Ψ

119896

1198981015840119894119895)]

times [

Δ119910120585119898

120572

+

Δ119909120578119898

120572

+ Δ119909Δ119910 (120583119886119894119895

+ 120583119904119894119895)]

minus1

(19)


119896+1


119898119894119895and


(Ψ

119896+1

119898119894119895)

updated= 120588Ψ

119896

119898119894119895+ (1 minus 120588)Ψ

119896+1

119898119894119895 (20)


120576 =

Ψ

119896+1

119898119894119895minus Ψ

119896

119898119894119895

Ψ

119896

119898119894119895

(21)




(i) Ψ( 119903 Ω) = 119878( 119903119887Ω) + 119877 sdot Ψ( 119903

Ωref)



Ω


119887is an outer





Φ119889=

119872

sum

119898=1

(1 minus 119877119898119894119889119895119889

)119908119898Ψ119898119894119889119895119889

(22)

with119877119898119894119889119895119889

=

1


119899air119899119894119889119895119889

)

1

2

[(

119899119894119889119895119889

cos120593119898minus 119899air cos120593119905

119899119894119889119895119889

cos120593119898+ 119899air cos120593119905

)

2

+(

119899119894119889119895119889

cos120593119905minus 119899air cos120593119898

119899119894119889119895119889

cos120593119905+ 119899air cos120593119898

)

2

]

else

120593119905= arcsin(

119899air sin120593119898119899119894119889119895119889

) 119899air asymp 1

(23)


Φ119873=

Φ119889

(1119863)sum

119863

119889=1119908

1015840

119889Φ119889

(24)


1015840







Detector points

Top side

Right side

Source

y

x1 cm

120583s = 25 cmminus1

120583a = 05 cmminus1

g = 08

n = 033 + 120575 timesx

L


16

14

12

10

08

06

04

02

0000 05 10 15 20

Top side (cm)

Nor

mal

ized

flue

nce r

ate (

au)

n = 156









24

20

16

12

08

04

00

00 05 10 15 20

Right side (cm)

n = 156





Nor

mal

ized

flue

nce r

ate (

au)


16

14

12

10

08

06

04

0200 05 10 15 20

Top side (cm)

Nor

mal

ized

flue

nce r

ate (

au)




20

16

12

08

04

0000 05 10 15 20

Right side (cm)

Nor

mal

ized

flue

nce r

ate (

au)




5 Conclusion




References



























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















119872

sum

119898=1

119908119898119863120578119898119875119904(120578119898)

=

119904 (119904 + 1)

2119904 + 1

[

119872

sum

119898=1

119908119898119875119904+1(120578119898) minus

119872

sum

119898=1

119908119898119875119904minus1(120578119898)]

119904 = 1 119872 minus 1

(15)


119872

sum

119898=1

119908119898119863120585119898

= int

1

minus1

120597

120597120578

[(1 minus 120585

2

) 120595] 119889 (120585)

119872

sum

119898=1

119908119898119863120578119898

= int

1

minus1

120597

120597120578

[(1 minus 120578

2

) 120595] 119889 (120578)

(16)


119860120585

119863120585119894119895

= 119861120585119894119895 119860

120578

119863120578119894119895

= 119861120578119894119895 (17)

where

A120585

= (

1199081

1199082

119908119872

11990811198751(1205851) 119908

21198751(1205852) 119908

1198721198751(120585119872)

1199081119875119872minus1

(1205851) 1199082119875119872minus1

(1205852) 119908

119872119875119872minus1

(120585119872)

)

D120578119894119895

= (

1198631205851119894119895

1198631205852119894119895

119863120585119872119894119895

)

B120578119894119895

=

(

(

(

(

(

0

2

3

119872

sum

1198981015840=1

11990811989810158401205951198981015840119894119895(1198752(1205851198981015840)) minus (119875

0(1205851198981015840))

119872(119872 minus 1)

2119872 minus 1

119872

sum

1198981015840=1

11990811989810158401205951198981015840119894119895(119875119872(1205851198981015840) minus 119875

119872minus2(1205851198981015840))

)

)

)

)

)

(18)

and 119860120578119863120578119894119895 119861120578119894119895






Ψ

119896+1

119898119894119895= [

Δ119910120585119898

120572

Ψ

119896+1

119898119894minus1119895+

Δ119909120578119898

120572

Ψ

119896+1

119898119894119895minus1

+

1

2119899

2

119894119895

[Δ119910 (119899

2

119894119895minus 119899

2

119894minus1119895)119863119898119909119894119895

+ Δ119910 (119899

2

119894119895minus 119899

2

119894119895minus1)119863119898119909119894119895

]

+Δ119909Δ119910(119878

119896+1

119898119894119895+ 120583119904119894119895

119872

sum

1198981015840=1

11990811989810158401199011198981198981015840Ψ

119896

1198981015840119894119895)]

times [

Δ119910120585119898

120572

+

Δ119909120578119898

120572

+ Δ119909Δ119910 (120583119886119894119895

+ 120583119904119894119895)]

minus1

(19)


119896+1


119898119894119895and


(Ψ

119896+1

119898119894119895)

updated= 120588Ψ

119896

119898119894119895+ (1 minus 120588)Ψ

119896+1

119898119894119895 (20)


120576 =

Ψ

119896+1

119898119894119895minus Ψ

119896

119898119894119895

Ψ

119896

119898119894119895

(21)




(i) Ψ( 119903 Ω) = 119878( 119903119887Ω) + 119877 sdot Ψ( 119903

Ωref)



Ω


119887is an outer





Φ119889=

119872

sum

119898=1

(1 minus 119877119898119894119889119895119889

)119908119898Ψ119898119894119889119895119889

(22)

with119877119898119894119889119895119889

=

1


119899air119899119894119889119895119889

)

1

2

[(

119899119894119889119895119889

cos120593119898minus 119899air cos120593119905

119899119894119889119895119889

cos120593119898+ 119899air cos120593119905

)

2

+(

119899119894119889119895119889

cos120593119905minus 119899air cos120593119898

119899119894119889119895119889

cos120593119905+ 119899air cos120593119898

)

2

]

else

120593119905= arcsin(

119899air sin120593119898119899119894119889119895119889

) 119899air asymp 1

(23)


Φ119873=

Φ119889

(1119863)sum

119863

119889=1119908

1015840

119889Φ119889

(24)


1015840







Detector points

Top side

Right side

Source

y

x1 cm

120583s = 25 cmminus1

120583a = 05 cmminus1

g = 08

n = 033 + 120575 timesx

L


16

14

12

10

08

06

04

02

0000 05 10 15 20

Top side (cm)

Nor

mal

ized

flue

nce r

ate (

au)

n = 156









24

20

16

12

08

04

00

00 05 10 15 20

Right side (cm)

n = 156





Nor

mal

ized

flue

nce r

ate (

au)


16

14

12

10

08

06

04

0200 05 10 15 20

Top side (cm)

Nor

mal

ized

flue

nce r

ate (

au)




20

16

12

08

04

0000 05 10 15 20

Right side (cm)

Nor

mal

ized

flue

nce r

ate (

au)




5 Conclusion




References



























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of



















Φ119889=

119872

sum

119898=1

(1 minus 119877119898119894119889119895119889

)119908119898Ψ119898119894119889119895119889

(22)

with119877119898119894119889119895119889

=

1


119899air119899119894119889119895119889

)

1

2

[(

119899119894119889119895119889

cos120593119898minus 119899air cos120593119905

119899119894119889119895119889

cos120593119898+ 119899air cos120593119905

)

2

+(

119899119894119889119895119889

cos120593119905minus 119899air cos120593119898

119899119894119889119895119889

cos120593119905+ 119899air cos120593119898

)

2

]

else

120593119905= arcsin(

119899air sin120593119898119899119894119889119895119889

) 119899air asymp 1

(23)


Φ119873=

Φ119889

(1119863)sum

119863

119889=1119908

1015840

119889Φ119889

(24)


1015840







Detector points

Top side

Right side

Source

y

x1 cm

120583s = 25 cmminus1

120583a = 05 cmminus1

g = 08

n = 033 + 120575 timesx

L


16

14

12

10

08

06

04

02

0000 05 10 15 20

Top side (cm)

Nor

mal

ized

flue

nce r

ate (

au)

n = 156









24

20

16

12

08

04

00

00 05 10 15 20

Right side (cm)

n = 156





Nor

mal

ized

flue

nce r

ate (

au)


16

14

12

10

08

06

04

0200 05 10 15 20

Top side (cm)

Nor

mal

ized

flue

nce r

ate (

au)




20

16

12

08

04

0000 05 10 15 20

Right side (cm)

Nor

mal

ized

flue

nce r

ate (

au)




5 Conclusion




References



























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















24

20

16

12

08

04

00

00 05 10 15 20

Right side (cm)

n = 156





Nor

mal

ized

flue

nce r

ate (

au)


16

14

12

10

08

06

04

0200 05 10 15 20

Top side (cm)

Nor

mal

ized

flue

nce r

ate (

au)




20

16

12

08

04

0000 05 10 15 20

Right side (cm)

Nor

mal

ized

flue

nce r

ate (

au)




5 Conclusion




References



























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of








































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of
















Research Article Solution of Radiative Transfer Equation...

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