Review Article Uniqueness, Born Approximation, and Numerical Methods for Diffuse Optical

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 824501 5 pageshttpdxdoiorg1011552013824501

Review ArticleUniqueness Born Approximation and Numerical Methods forDiffuse Optical Tomography

Kiwoon Kwon

Department of Mathematics Dongguk University Seoul 100-715 Republic of Korea

Correspondence should be addressed to Kiwoon Kwon kwkwondonggukedu

Received 8 April 2013 Accepted 12 May 2013

Academic Editor Chang-Hwan Im

Copyright copy 2013 Kiwoon KwonThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Diffuse optical tomogrpahy (DOT) is to find optical coefficients of tissue using near infrared light DOT as an inverse problem isdescribed and the studies about unique determination of optical coefficients are summarized If a priori information of the opticalcoefficient is known DOT is reformulated to find a perturbation of the optical coefficients inverting the Born expansion which is aninfinite series expansion with respect to the perturbation and the a priori information Numerical methods for DOT are explainedas methods inverting first- or second-order Born approximation or the Born expansion itself

1 Introduction

DOT is to findoptical coefficients of tissue using near infraredlight DOT is known to be of low cost portable nonionizedand nonmagnetized And DOT has higher temporal reso-lution and more functional information than conventionalstructural medical imaging modalities such as magneticresonance imaging (MRI) and computerized tomography(CT) For the comparison to other functional imagingmodalities such as functional MRI (fMRI) photon emissiontomography (PET) and electroencephalogram (EEG) see [1]DOT is used in the area of breast imaging [2ndash4] functionalneuroimaging [5 6] brain computer interface (BCI) [7 8]and the study about seizure [9 10] newborn infants [11 12]osteoarthritis [13] and rat brain [14 15]

In this paper DOT is explained as an inverse problemwith respect to a forward problem formulated as an ellipticpartial differential equation Propagation of light in biolog-ical tissues is usually described by diffusion approximationequation in the frequency domain the simplest but nontrivialapproximation of the Boltzmann equation as follows

minusnabla sdot (120581nablaΦ) + (120583

119886+

119894120596

119888

)Φ = 119902 in Ω(1a)

Φ + 2119886] sdot (120581nablaΦ) = 0 on 120597Ω (1b)

where ] is an outer unit normal vectorΦ is a photon densitydistribution 120581 = 13(120583

119886+ 120583

1015840

119904) is an diffusion coefficient

120583

119886is an absorption coefficient 1205831015840

119904is a reduced scattering

coefficient and 119886 is a reflection coefficientIf 120597120581120597] = 0 on 120597Ω by setting Ψ = radic120581Φ and 119896 =

radic

Δradic120581radic120581 + 120583

119886120581 + 119894(120596119888120581) with 119868119898(119896) ge 0 we have

minusΔΨ + 119896

2

Ψ =

119902

radic120581

in Ω (2a)

Ψ + 2119886] sdot (120581nablaΨ) = 0 on 120597Ω (2b)

If 119896 is constant and 119902(sdot)radic120581 = 120575(sdot 119903

119904) for some source

point 119903119904 we have the following solution of (2a)

Ψ (119903) = 119877 (119903 119903

119904) =

119890

119894119896|119903minus119903119904|

4120587

1003816

1003816

1003816

1003816

119903 minus 119903

119904

1003816

1003816

1003816

1003816

(3)

DOT is to find the optical coefficients 120583119886andor 1205831015840

119904from

the measurement information Φ

119894119895which is the value of the

solution of (1a) and (1b) at 119903119894isin 120597Ω when 119902(119903) = 120575(119903 119903

119895) 119903

119895isin

120597ΩThe 119903119894and 119903119895are usually called source and detector point

respectivelyIn Section 2 the unique determination of the optical

coefficients is discussed and many known results are sum-marized for the uniqueness questions In Section 3 DOT is

2 Journal of Applied Mathematics

reformulated as to find perturbation of the optical coefficientinverting the Born expansionThe errors of the Born approx-imation in the Lebesgue and Sobolev norms are given InSection 4 numerical methods of DOT are mainly describedas themethods inverting the first- second- and higher-orderBorn approximation and Born expansion itself

2 Uniqueness

The research about unique determination of the optical coef-ficients in DOT is rare except [16] but it is a very importantissue for DOT as an inverse problem The determinaiton ofoptical coefficients (120583

119886 120583

119904) in (1a) and (1b) is equivalent to the

determination of 119896 in (2a) and (2b) when 120596 = 0When 120583

119886 120581 nabla120581 have upper and lower bound and 119902 is

contained in 119867

minus1

(Ω) or a Dirac delta function (1a) and (1b)have a unique solution Φ isin 119867

1

(Ω) and (2a) and (2b) have aunique solution Ψ isin 119867

1

(Ω) [17 18]Boundary value problem (1a) and (1b) with 119902(119903) = 120575(119903 119903

119904)

is equivalent to boundary value problem with (1a) for 119902 = 0

and nonzero Robin boundary condition replacing (1b) Thisargument can be proved using the function119867 in [19] There-fore DOT is redescribed as to find the optical coefficientsfrom the Robin-to-Dirichlet map defined as a map from119867

minus12

(120597Ω) to11986712(120597Ω) Using unique solvability of (1a) withtheDirichlet orNeumann boundary condition replacing (1b)the Robin-to-Dirichletmap is equivalent to theNeumann-to-Dirichlet map and to the Dirichlet-to-Neumann map Since119868119898(119896) gt 0 for 120596 = 0 119896 is not a Dirichlet eigenvalue of(2a) Therefore knowing the Dirichlet-to-Neumann map isalso equivalent to knowing farfield map in inverse scatteringproblem [20 21]

Using above results and many known results for inversescattering problem we summarized the uniqueness andnonuniqueness results as follows

Case 1 (120581 = 1 120583

119886= 1 + (119898 minus 1)120594

119863 119898 = 1) The cases for

positive constant119898 can be understood as special cases of Case2 The limiting cases 119898 = infin and 119898 = 0 could be consideredas (1a) inΩ119863with the Robin boundary condition (1b) on 120597Ωand the boundary condition on 120597119863 asΦ = 0 (sound-soft case)and 120597Φ120597] = 0 (sound-hard case) on 120597119863 The uniquenessfor sound-soft and sound-hard obstacle 119863 is considered in[20 22]

Case 2 (120581 = 1 120583

119886= 1 + (119898(119909) minus 1)120594

119863 where 119898(119909) = 1 on

120597119863) This case is called ldquoinverse transmission problemrdquo andthe uniqueness is solved in [20]

Case 3 (120581 = 1 120583

119886= 120583

119886(119909)) In [16] nonuniqueness of 120583

119886(119909)

is shown by assuming that refractive index is not determinedor we use only continuous wave light source However ifrefractive index is known and 120596 = 0 unique determinationof the optical coefficient is possible This nonuniqueness isproved by using [23]

Case 4 (120581 = 119868

119899+ (119870(119909) minus 119868

119899)120594

119863 120583

119886= 1 + (119898(119909) minus 1)120594

119863)

Here 119870(119909) is a unknown positive-definite matrix functionsuch that 119870(119909) = 119868

119899on 120597119863 and 119898(119909) is a positive function

such that 119898(119909) = 1 on 120597119863 The uniqueness of 119863 is solved in[21 24ndash28] and the nonuniqueness of119870(119909) is reported in [2429 30] Therefore although the domain of nonhomogenity119863 can be uniquely determined by infinite measurementsthe nonhomogeneous anisotropic diffusion coefficient 119870(119909)cannot be determined uniquely Similar results are known forthe nonuniqueness and illusion for anisotropic nonhomoge-neous electric conductivity [24 31]

3 Born Approximation

Suppose that we know a priori information 119909

0

= (120581

0

120583

0

119886)

about the optical coefficients 119909 = (120581 120583

119886) to find Then DOT

is redescribed as to find the perturbation 120575119909 = 119909 minus 119909

0

=

(120575120581 120575120583

119886)

Let Φ and Φ

0 be the solutions of (1a) and (1b) for 119909 and119909

0 respectivelyWhen 119902(119903) = 120575(119903 119903

119904)Φ andΦ0 are the Robin

function 119877(sdot 119903119904) and 1198770(sdot 119903

119904) for the optical coefficients 119909 and

119909

0 respectively119877(sdot 119903119904) is expanded as the Born expansion an

infinite series with respect to 1198770(sdot 119903119904) and 120575119909 as follows

119877 (119903 119903

119904) = 119877

0

(119903 119903

119904) +R119877

0

(sdot 119903

119904) +R

2

119877

0

(sdot 119903

119904) + sdot sdot sdot

(4)

where

(RΨ) (119903) = R (120575119909)Ψ (119903) = (R1Ψ) (119903) + (R

2Ψ) (119903)

(R1Ψ) (119903) = R

1(120575120583

119886) Ψ (119903)

= minusint

Ω

120575120583

119886(119903

1015840

) 119877 (119903 119903

1015840

)Ψ (119903

1015840

) 119889119903

1015840

(R2Ψ) (119903) = R

2(120575120581)Ψ (119903)

= minusint

Ω

120575120581 (119903

1015840

) nabla119877 (119903 119903

1015840

) nablaΨ (119903

1015840

) 119889119903

1015840

(5)

It is proved that the 119899th order term in the above Bornexpansion (4) is the 119899th order Frechet derivative divided by119899 [18] The bounds for the operatorsR

1andR

2are given in

the Sobolev Lebesque and weighted Sobolev spaces norms[18]The estimate for the Lebesque space norms forR

1is also

given in [32] Therefore there exist positive constants 1198621and

119862

2such that

1003817

1003817

1003817

1003817

R1

1003817

1003817

1003817

10038171199081119901(Ω)rarr119908

1119901(Ω)

le 119862

1

1003817

1003817

1003817

1003817

120575120583

119886

1003817

1003817

1003817

1003817119871infin(Ω)

(6a)1003817

1003817

1003817

1003817

R2

1003817

1003817

1003817

10038171199081119901(Ω)rarr119908

1119901(Ω)

le 119862

2120575120581

119871infin(Ω)

(6b)

for 119901 ge 1 The detailed estimate for the coefficients 1198621 119862

2in

special cases is given in [18 32]If

119862

3= 119862

1

1003817

1003817

1003817

1003817

120575120583

119886

1003817

1003817

1003817

1003817infin+ 119862

2120575120581

infinlt 1 (7)

holds then the error of the 119898th order Born approximation119861

119898

119877

0

(sdot 119903

119904) is given by

1003817

1003817

1003817

1003817

1003817

119877 (sdot 119903

119904) minus 119861

119898

119877

0

(sdot 119903

119904)

1003817

1003817

1003817

1003817

10038171198821119901le 119862

119898

3

1003817

1003817

1003817

1003817

119877(sdot 119903

119904)

1003817

1003817

1003817

10038171198821119901(Ω)

(8)

using the estimates (6a) and (6b)

Journal of Applied Mathematics 3

4 Numerical Methods

Numerical methods for DOT are explained in terms of solv-ing the Born approximation given in Section 3 Other numer-ical methods not categorized as inverting Born approxima-tion are commented in the final section

41 Linearized Methods Linearized DOT to find 120575119909 bysolving the first-order Born approximation is studied bymany researchers in the initial stage of DOT Although thismethod lacks exact recovery in most cases it is used stillfrequently when faster real-time computing is needed andvery good a priori information is given In this methodthe discretized problem is an algebraic equation and itis essential to use efficient matrix solver In the Jacobianmatrix the number of rows is the number of measurementsand the number of columns is the number of elementsto be determined In most cases the number of elementsis larger than the number of measurements in order toobtain higher resolution imageTherefore the algebraic equa-tion is usually an underdetermined system Efficient matrixsolvers including arithmetic reconstruction technique (ART)simultaneous arithmetic reconstruction technique (SART)simultaneous iterative reconstruction technique (SIRT) andthe Krylov space methods are studied [33 34] Since thelinearized problems also have ill-posed property of DOTthere are many researches about regularization methodsincluding the Tikhonov regularization The dependence onthe discretization error due to ill posedness of the linearizedmethod is reported in [35]

42 Nonlinear Methods Nonlinear method is to find 120575119909

by solving the Born expansion [36] This method is usu-ally formulated as optimization problem and is solved bythe Newton-type method including Levenberg-Marquardtmethod [37 38] A few softwares based on nonlinear methodwith finite element forward solver with corresponding refer-ences are as follows

(i) TOAST (Time-Resolved Optical Absorption andScattering Tomography) [39]

(ii) NIRFAST (Near InfraRed Florescence and SpectralTomography) [40 41]

(iii) PMI (Photon Migration Imaging) Toolbox [42]

Nonlinear method needs heavy computation due to largeiteration numbers To reduce the heavy computation thereare studies about efficient numerical techniques such asmultigrid domain decomposition [43] and adaptive [44]method

43 Inverse Born Approximation of Order Higher Than TwoThismethod is to find 120575119909 by solving second- and higher-orderBorn approximation In fact solving the Born approximationof order higher than two is implicit but can be approximatedby explicit inverse Born approximation Formal inverse of theBorn approximation is called inverse Born approximationThe first-order inverse Born approximation corresponds to

linearized DOT and inverse Born expansion itself corre-sponds to (nonlinear) DOT Higher-order methods improvethe order of convergence for lower-order methods The errorof the inverse Born approximation is given and analyzed interms of the Lebesgue space norms when 120575120581 = 0 [45] and interms of the Sobolev space norms for the second order [46]

44 Other Methods The solution of (1a) and (1b) is usuallysolved by finite element method [36] Another approachfor the forward problem is to compute directly the Robinfunction using the Fourier-Laplace transform [47ndash50] Thedisadvantage of this method is that it depends on the specialgeometry of the region of interest and the inverse Fourier-Laplace transfrom is known to be severely ill posed

Equations (1a) and (1b) could be replaced by probabilisticapproach in the Monte Carlo method [51] The method takesmuch more time than finite element method and highlydepends on random number generator The comparison offinite element method and the Monte Carlo method is doneby many papers including [52]

The diffusion approximation (1a) and (1b) is the first-order approximation of radiative transfer equation Thereare studies about DOT based on radiative transfer equation[53 54] and its 119899th order approximation [55]

5 Conclusion

Unique determination of DOT is surveyed The study aboutnonuniqueness for anisotropic diffusion coefficients and forunknown refractive index is also surveyed The perturbationof photon density with respect to the perturbed opticalcoefficient is expanded using the Born expansion and theerror analysis removing higher-order terms is given Thenumerical methods for DOT are described by inverting first-second- and higher-order Born approximation and the Bornexpansion itself is reviewed

Acknowledgment

This work is supported by Basic Science Research Programthrough the National Research foundation of Korea (NRF)funded by the Ministry of Education (2010-0004047)

References

[1] B WhiteDeveloping high-density diffuse optical tomography forneuroimaging [PhD thesis] Washington University St LouisMo USA 2012

[2] Q Zhang T J Brukilacchio A Li et al ldquoCoregistered tomo-graphic X-ray and optical breast imaging initial resultsrdquo Journalof biomedical optics vol 10 no 2 Article ID 024033 2005

[3] T Durduran R Choe G Yu et al ldquoDiffuse optical measure-ment of blood flow in breast tumorsrdquoOptics Letters vol 30 no21 pp 2915ndash2917 2005

[4] V Ntziachristos A G Yodh M Schnall and B ChanceldquoConcurrentMRI and diffuse optical tomography of breast afterindocyanine green enhancementrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 97 no6 pp 2767ndash2772 2000


[5] J P Culver A M Siegel M A Franceschini J B Mandevilleand D A Boas ldquoEvidence that cerebral blood volume canprovide brain activation maps with better spatial resolutionthan deoxygenated hemoglobinrdquoNeuroImage vol 27 no 4 pp947ndash959 2005

[6] M A Franceschini and D A Boas ldquoNoninvasive measurementof neuronal activity with near-infrared optical imagingrdquo Neu-roImage vol 21 no 1 pp 372ndash386 2004

[7] R Sitaram H Zhang C Guan et al ldquoTemporal classificationof multichannel near-infrared spectroscopy signals of motorimagery for developing a brain-computer interfacerdquo NeuroIm-age vol 34 no 4 pp 1416ndash1427 2007

[8] S M Coyle T E Ward and C M Markham ldquoBrain-computer interface using a simplified functional near-infraredspectroscopy systemrdquo Journal of Neural Engineering vol 4 no3 pp 219ndash226 2007

[9] EWatanabe A Maki F Kawaguchi Y Yamashita H Koizumiand Y Mayanagi ldquoNoninvasive cerebral blood volume mea-surement during seizures using multichannel near infraredspectroscopic topographyrdquo Journal of Biomedical Optics vol 5no 3 pp 287ndash290 2000

[10] K Buchheim H Obrig W V Pannwitz et al ldquoDecrease inhaemoglobin oxygenation during absence seizures in adulthumansrdquoNeuroscience Letters vol 354 no 2 pp 119ndash122 2004

[11] G Taga Y Konishi A Maki T Tachibana M Fujiwaraand H Koizumi ldquoSpontaneous oscillation of oxy- and deoxy-hemoglobin changes with a phase difference throughout theoccipital cortex of newborn infants observed using non-invasive optical topographyrdquo Neuroscience Letters vol 282 no1-2 pp 101ndash104 2000

[12] T Austin A P Gibson G Branco et al ldquoThree dimensionaloptical imaging of blood volume and oxygenation in theneonatal brainrdquoNeuroImage vol 31 no 4 pp 1426ndash1433 2006

[13] Z Yuan Q Zhang E Sobel and H Jiang ldquoThree-dimensionaldiffuse optical tomography of osteoarthritis Initial results in thefinger jointsrdquo Journal of Biomedical Optics vol 12 no 3 ArticleID 034001 2007

[14] A M Siegel J P Culver J B Mandeville and D A BoasldquoTemporal comparison of functional brain imaging with diffuseoptical tomography and fMRI during rat forepaw stimulationrdquoPhysics in Medicine and Biology vol 48 no 10 pp 1391ndash14032003

[15] J P Culver TDurduranD Furuya CCheung JHGreenbergand A G Yodh ldquoDiffuse optical tomography of cerebralblood flow oxygenation and metabolism in rat during focalischemiardquo Journal of Cerebral Blood Flow and Metabolism vol23 no 8 pp 911ndash924 2003

[16] S R Arridge and W R B Lionheart ldquoNonuniqueness indiffusion-based optical tomographyrdquo Optics Letters vol 23 no11 pp 882ndash884 1998

[17] D Gilbarg and N S Trudinger Elliptic Partial DifferentialEquations of Second Order vol 224 of Grundlehren der Math-ematischen Wissenschaften Springer Berlin Germany 2ndedition 1983

[18] K Kwon and B Yazıcı ldquoBorn expansion and Frechet derivativesin nonlinear diffuse optical tomographyrdquo Computers amp Mathe-matics with Applications vol 59 no 11 pp 3377ndash3397 2010

[19] C Miranda Partial Differential Equations of Elliptic TypeErgebnisse der Mathematik und ihrer Grenzgebiete Band 2Springer New York NY USA 1970

[20] D Colton and R Kress Inverse Acoustic and ElectromagneticScattering Theory vol 93 of Applied Mathematical SciencesSpringer Berlin Germany 2nd edition 1998

[21] V Isakov ldquoOn uniqueness in the inverse transmission scatteringproblemrdquoCommunications in Partial Differential Equations vol15 no 11 pp 1565ndash1587 1990

[22] A Kirsch and R Kress ldquoUniqueness in inverse obstacle scatter-ingrdquo Inverse Problems vol 9 no 2 pp 285ndash299 1993

[23] J Sylvester and G Uhlmann ldquoA global uniqueness theorem foran inverse boundary value problemrdquo Annals of MathematicsSecond Series vol 125 no 1 pp 153ndash169 1987

[24] K Kwon ldquoIdentification of anisotropic anomalous region ininverse problemsrdquo Inverse Problems vol 20 no 4 pp 1117ndash11362004

[25] K Kwon and D Sheen ldquoAnisotropic inverse conductivity andscattering problemsrdquo Inverse Problems vol 18 no 3 pp 745ndash756 2002

[26] M Piana ldquoOn uniqueness for anisotropic inhomogeneousinverse scattering problemsrdquo Inverse Problems vol 14 no 6 pp1565ndash1579 1998

[27] R Potthast ldquoOn a concept of uniqueness in inverse scatteringfrom an orthotopic mediumrdquo Journal of Integral Equations andApplications vol 11 pp 197ndash215 1999

[28] D Sheen and D Shepelsky ldquoInverse scattering problem for astratified anisotropic slabrdquo Inverse Problems vol 15 no 2 pp499ndash514 1999

[29] G Eskin ldquoInverse scattering problem in anisotropic mediardquoCommunications in Mathematical Physics vol 199 no 2 pp471ndash491 1998

[30] F Gylys-Colwell ldquoAn inverse problem for the Helmholtz equa-tionrdquo Inverse Problems vol 12 no 2 pp 139ndash156 1996

[31] K Kwon ldquoThe uniquess cloaking and illusion for electricalimpedance tomographyrdquo Submitted

[32] S Moskow and J C Schotland ldquoConvergence and stability ofthe inverse scattering series for diffuse wavesrdquo Inverse Problemsvol 24 no 6 Article ID 065005 16 pages 2008

[33] S R Arridge and J C Schotland ldquoOptical tomography forwardand inverse problemsrdquo Inverse Problems vol 25 pp 1ndash59 2009

[34] J Kaipio andE Somersalo Statistical andComputational InverseProblems vol 160 of Applied Mathematical Sciences SpringerNew York NY USA 2005

[35] M Guven B Yazici K Kwon E Giladi and X Intes ldquoEffectof discretization error and adaptive mesh generation in diffuseoptical absorption imaging Irdquo Inverse Problems vol 23 no 3pp 1115ndash1133 2007

[36] S R Arridge ldquoOptical tomography inmedical imagingrdquo InverseProblems vol 15 no 2 pp R41ndashR93 1999

[37] K Levenberg ldquoA method for the solution of certain non-linearproblems in least squaresrdquo Quarterly of Applied Mathematicsvol 2 pp 164ndash168 1944

[38] D Marquadt ldquoAn algorithm for least-squares estimation ofnonlinear parametersrdquo IEEE Transactions of Circuits and Sys-tems vol 26 1979

[39] TOAST website April 2013 httpweb4csuclacukresearchvistoast

[40] NIRFAST webiste April 2013 httpwwwdartmouthedunirnirfastindexphp

[41] H DehghaniM E Eames P K Yalavarthy et al ldquoNear infraredoptical tomography using NIRFAST algorithm for numerical


model and image reconstructionrdquo Communications in Numeri-calMethods in Engineering with Biomedical Applications vol 25no 6 pp 711ndash732 2009

[42] PMI Toolbox website April 2013 httpwwwnmrmghhar-vardeduPMItoolbox

[43] K Kwon B Yazıcı and M Guven ldquoTwo-level domain decom-position methods for diffuse optical tomographyrdquo Inverse Prob-lems vol 22 no 5 pp 1533ndash1559 2006

[44] M Guven B Yazici K Kwon E Giladi and X Intes ldquoEffectof discretization error and adaptive mesh generation in diffuseoptical absorption imaging IIrdquo Inverse Problems vol 23 no 3pp 1135ndash1160 2007

[45] S Moskow and J C Schotland ldquoNumerical studies of theinverse Born series for diffuse wavesrdquo Inverse Problems vol 25no 9 Article ID 095007 18 pages 2009

[46] K Kwon ldquoThe second-order Born approximation in diffuseoptical tomographyrdquo Journal of Applied Mathematics vol 2012Article ID 637209 15 pages 2012

[47] V A Markel and J C Schotland ldquoInverse problem in opticaldiffusion tomography I Fourier-Laplace inversion formulasrdquoJournal of the Optical Society of America A Optics Image Sci-ence and Vision vol 18 no 6 pp 1336ndash1347 2001

[48] V A Markel and J C Schotland ldquoInverse problem in opticaldiffusion tomography II Role of boundary conditionsrdquo Journalof the Optical Society of America A Optics and Image Scienceand Vision vol 19 no 3 pp 558ndash566 2002

[49] V A Markel V Mital and J C Schotland ldquoInverse problemin optical diffusion tomography III Inversion formulas andsingular-value decompositionrdquo Journal of the Optical Society ofAmerica A Optics and Image Science and Vision vol 20 no 5pp 890ndash902 2003

[50] V AMarkel J AOrsquoSullivan and J C Schotland ldquoInverse prob-lem in optical diffusion tomography IV Nonlinear inversionformulasrdquo Journal of theOptical Society of AmericaAOptics andImage Science and Vision vol 20 no 5 pp 903ndash912 2003

[51] L Wang S L Jacques and L Zheng ldquoMCMLmdashMonte Carlomodeling of light transport in multi-layered tissuesrdquo ComputerMethods and Programs in Biomedicine vol 47 no 2 pp 131ndash1461995

[52] K Kwon T Son K J Lee and B Jung ldquoEnhancement of lightpropagation depth in skin Cross-validation of mathematicalmodeling methodsrdquo Lasers in Medical Science vol 24 no 4 pp605ndash615 2009

[53] G S Abdoulaev and A H Hielscher ldquoThree-dimensional opti-cal tomography with the equation of radiative transferrdquo Journalof Electronic Imaging vol 12 no 4 pp 594ndash601 2003

[54] T Tarvainen M Vauhkonen V Kolehmainen and J P Kai-pio ldquoFinite element model for the coupled radiative transferequation and diffusion approximationrdquo International Journalfor Numerical Methods in Engineering vol 65 no 3 pp 383ndash405 2006

[55] E D Aydin C R E de Oliveira and A J H GoddardldquoA compariosn between transport and diffusion calculationsusing a finite element-spherical harmonics ratiation tranportmethodrdquoMedical Physics vol 2 pp 2013ndash2023 2002

Submit your manuscripts athttpwwwhindawicom

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


reformulated as to find perturbation of the optical coefficientinverting the Born expansionThe errors of the Born approx-imation in the Lebesgue and Sobolev norms are given InSection 4 numerical methods of DOT are mainly describedas themethods inverting the first- second- and higher-orderBorn approximation and Born expansion itself

2 Uniqueness

The research about unique determination of the optical coef-ficients in DOT is rare except [16] but it is a very importantissue for DOT as an inverse problem The determinaiton ofoptical coefficients (120583

119886 120583

119904) in (1a) and (1b) is equivalent to the

determination of 119896 in (2a) and (2b) when 120596 = 0When 120583

119886 120581 nabla120581 have upper and lower bound and 119902 is

contained in 119867

minus1

(Ω) or a Dirac delta function (1a) and (1b)have a unique solution Φ isin 119867

1

(Ω) and (2a) and (2b) have aunique solution Ψ isin 119867

1

(Ω) [17 18]Boundary value problem (1a) and (1b) with 119902(119903) = 120575(119903 119903

119904)

is equivalent to boundary value problem with (1a) for 119902 = 0

and nonzero Robin boundary condition replacing (1b) Thisargument can be proved using the function119867 in [19] There-fore DOT is redescribed as to find the optical coefficientsfrom the Robin-to-Dirichlet map defined as a map from119867

minus12

(120597Ω) to11986712(120597Ω) Using unique solvability of (1a) withtheDirichlet orNeumann boundary condition replacing (1b)the Robin-to-Dirichletmap is equivalent to theNeumann-to-Dirichlet map and to the Dirichlet-to-Neumann map Since119868119898(119896) gt 0 for 120596 = 0 119896 is not a Dirichlet eigenvalue of(2a) Therefore knowing the Dirichlet-to-Neumann map isalso equivalent to knowing farfield map in inverse scatteringproblem [20 21]

Using above results and many known results for inversescattering problem we summarized the uniqueness andnonuniqueness results as follows

Case 1 (120581 = 1 120583

119886= 1 + (119898 minus 1)120594

119863 119898 = 1) The cases for

positive constant119898 can be understood as special cases of Case2 The limiting cases 119898 = infin and 119898 = 0 could be consideredas (1a) inΩ119863with the Robin boundary condition (1b) on 120597Ωand the boundary condition on 120597119863 asΦ = 0 (sound-soft case)and 120597Φ120597] = 0 (sound-hard case) on 120597119863 The uniquenessfor sound-soft and sound-hard obstacle 119863 is considered in[20 22]

Case 2 (120581 = 1 120583

119886= 1 + (119898(119909) minus 1)120594

119863 where 119898(119909) = 1 on

120597119863) This case is called ldquoinverse transmission problemrdquo andthe uniqueness is solved in [20]

Case 3 (120581 = 1 120583

119886= 120583

119886(119909)) In [16] nonuniqueness of 120583

119886(119909)

is shown by assuming that refractive index is not determinedor we use only continuous wave light source However ifrefractive index is known and 120596 = 0 unique determinationof the optical coefficient is possible This nonuniqueness isproved by using [23]

Case 4 (120581 = 119868

119899+ (119870(119909) minus 119868

119899)120594

119863 120583

119886= 1 + (119898(119909) minus 1)120594

119863)

Here 119870(119909) is a unknown positive-definite matrix functionsuch that 119870(119909) = 119868

119899on 120597119863 and 119898(119909) is a positive function

such that 119898(119909) = 1 on 120597119863 The uniqueness of 119863 is solved in[21 24ndash28] and the nonuniqueness of119870(119909) is reported in [2429 30] Therefore although the domain of nonhomogenity119863 can be uniquely determined by infinite measurementsthe nonhomogeneous anisotropic diffusion coefficient 119870(119909)cannot be determined uniquely Similar results are known forthe nonuniqueness and illusion for anisotropic nonhomoge-neous electric conductivity [24 31]

3 Born Approximation

Suppose that we know a priori information 119909

0

= (120581

0

120583

0

119886)

about the optical coefficients 119909 = (120581 120583

119886) to find Then DOT

is redescribed as to find the perturbation 120575119909 = 119909 minus 119909

0

=

(120575120581 120575120583

119886)

Let Φ and Φ

0 be the solutions of (1a) and (1b) for 119909 and119909

0 respectivelyWhen 119902(119903) = 120575(119903 119903

119904)Φ andΦ0 are the Robin

function 119877(sdot 119903119904) and 1198770(sdot 119903

119904) for the optical coefficients 119909 and

119909

0 respectively119877(sdot 119903119904) is expanded as the Born expansion an

infinite series with respect to 1198770(sdot 119903119904) and 120575119909 as follows

119877 (119903 119903

119904) = 119877

0

(119903 119903

119904) +R119877

0

(sdot 119903

119904) +R

2

119877

0

(sdot 119903

119904) + sdot sdot sdot

(4)

where

(RΨ) (119903) = R (120575119909)Ψ (119903) = (R1Ψ) (119903) + (R

2Ψ) (119903)

(R1Ψ) (119903) = R

1(120575120583

119886) Ψ (119903)

= minusint

Ω

120575120583

119886(119903

1015840

) 119877 (119903 119903

1015840

)Ψ (119903

1015840

) 119889119903

1015840

(R2Ψ) (119903) = R

2(120575120581)Ψ (119903)

= minusint

Ω

120575120581 (119903

1015840

) nabla119877 (119903 119903

1015840

) nablaΨ (119903

1015840

) 119889119903

1015840

(5)

It is proved that the 119899th order term in the above Bornexpansion (4) is the 119899th order Frechet derivative divided by119899 [18] The bounds for the operatorsR

1andR

2are given in

the Sobolev Lebesque and weighted Sobolev spaces norms[18]The estimate for the Lebesque space norms forR

1is also

given in [32] Therefore there exist positive constants 1198621and

119862

2such that

1003817

1003817

1003817

1003817

R1

1003817

1003817

1003817

10038171199081119901(Ω)rarr119908

1119901(Ω)

le 119862

1

1003817

1003817

1003817

1003817

120575120583

119886

1003817

1003817

1003817

1003817119871infin(Ω)

(6a)1003817

1003817

1003817

1003817

R2

1003817

1003817

1003817

10038171199081119901(Ω)rarr119908

1119901(Ω)

le 119862

2120575120581

119871infin(Ω)

(6b)

for 119901 ge 1 The detailed estimate for the coefficients 1198621 119862

2in

special cases is given in [18 32]If

119862

3= 119862

1

1003817

1003817

1003817

1003817

120575120583

119886

1003817

1003817

1003817

1003817infin+ 119862

2120575120581

infinlt 1 (7)

holds then the error of the 119898th order Born approximation119861

119898

119877

0

(sdot 119903

119904) is given by

1003817

1003817

1003817

1003817

1003817

119877 (sdot 119903

119904) minus 119861

119898

119877

0

(sdot 119903

119904)

1003817

1003817

1003817

1003817

10038171198821119901le 119862

119898

3

1003817

1003817

1003817

1003817

119877(sdot 119903

119904)

1003817

1003817

1003817

10038171198821119901(Ω)

(8)

using the estimates (6a) and (6b)


4 Numerical Methods














5 Conclusion


Acknowledgment


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










4 Numerical Methods














5 Conclusion


Acknowledgment


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Review Article Uniqueness, Born Approximation, and Numerical Methods for Diffuse Optical