Image reconstruction in diffuse optical tomography based on simplified spherical harmonics approximation

Transcription

1 Imge reconstruction in diffuse opticl tomogrphy bsed on simplified sphericl hrmonics pproximtion Michel Chu, 1 Hmid Dehghni, 2,* 1 School of Physics, University of Exeter, EX4 1EZ, UK 2 School of Computer Science, University of Birminghm, B15 2TT, UK *h.dehghni@cs.bhm.c.uk Abstrct: The use of higher order pproximtions to the Rditive trnsport eqution, through simplified sphericl hrmonics expnsion (SP N ) in opticl tomogrphy re presented. It is shown tht, lthough the nisotropy fctor cn be modeled in the forwrd problem, its sensitivity to the mesured boundry dt is limited to superficil regions nd more importntly, due to uniqueness of the inverse problem it cnnot be determined using frequency domin dt. Imge reconstruction through the use of higher ordered models is presented. It is demonstrted tht t higher orders (for exmple SP 7 ) the imge reconstruction becomes highly under-determined due to the lrge increse in the number of unknowns which cnnot be dequtely recovered. However, reconstruction of diffuse prmeters, nmely opticl bsorption nd reduced sctter hve shown to be more ccurte where only the sensitivity mtrix used in the inverse problem is bsed on SP N method nd imge reconstruction is limited to these two diffuse prmeters Opticl Society of Americ OCIS codes: ( ) Inverse problems; ( ) Light propgtion in tissues References nd links 1. A. P. Gibson, J. C. Hebden, nd S. R. Arridge, Recent dvnces in diffuse opticl imging, Phys. Med. Biol. 50(4), R1 R43 (2005). 2. D. R. Leff, O. J. Wrren, L. C. Enfield, A. Gibson, T. Athnsiou, D. K. Ptten, J. Hebden, G. Z. Yng, nd A. Drzi, Diffuse opticl imging of the helthy nd disesed brest: systemtic review, Brest Cncer Res. Tret. 108(1), 9 22 (2008). 3. D. A. Bos, D. H. Brooks, E. L. Miller, C. A. DiMrzio, M. Kilmer, R. J. Gudette, nd Q. Zhng, Imging the body with diffuse opticl tomogrphy, IEEE Signl Process. Mg. 18(6), (2001). 4. R. Choe, A. Corlu, K. Lee, T. Durdurn, S. D. Konecky, M. Grosick-Koptyr, S. R. Arridge, B. J. Czerniecki, D. L. Frker, A. DeMichele, B. Chnce, M. A. Rosen, nd A. G. Yodh, Diffuse opticl tomogrphy of brest cncer during neodjuvnt chemotherpy: cse study with comprison to MRI, Med. Phys. 32(4), (2005). 5. H. Dehghni, B. W. Pogue, S. P. Poplck, nd K. D. Pulsen, Multiwvelength three-dimensionl ner-infrred tomogrphy of the brest: initil simultion, phntom, nd clinicl results, Appl. Opt. 42(1), (2003). 6. H. Jing, Y. Xu, N. Iftimi, J. Eggert, K. Klove, L. Bron, nd L. Fjrdo, Three-dimensionl opticl tomogrphic imging of brest in humn subject, IEEE Trns. Med. Img. 20(12), (2001). 7. L. C. Enfield, A. P. Gibson, N. L. Everdell, D. T. Delpy, M. Schweiger, S. R. Arridge, C. Richrdson, M. Keshtgr, M. Douek, nd J. C. Hebden, Three-dimensionl time-resolved opticl mmmogrphy of the uncompressed brest, Appl. Opt. 46(17), (2007). 8. B. W. Zeff, B. R. White, H. Dehghni, B. L. Schlggr, nd J. P. Culver, Retinotopic mpping of dult humn visul cortex with high-density diffuse opticl tomogrphy, Proc. Ntl. Acd. Sci. U.S.A. 104(29), (2007). 9. D. A. Bos, K. Chen, D. Grebert, nd M. A. Frnceschini, Improving the diffuse opticl imging sptil resolution of the cerebrl hemodynmic response to brin ctivtion in humns, Opt. Lett. 29(13), (2004). 10. J. P. Culver, A. M. Siegel, J. J. Stott, nd D. A. Bos, Volumetric diffuse opticl tomogrphy of brin ctivity, Opt. Lett. 28(21), (2003). 11. J. C. Hebden, A. Gibson, T. Austin, R. M. Yusof, N. Everdell, D. T. Delpy, S. R. Arridge, J. H. Meek, nd J. S. Wytt, Imging chnges in blood volume nd oxygention in the newborn infnt brin using three-dimensionl opticl tomogrphy, Phys. Med. Biol. 49(7), (2004). (C) 2009 OSA 21 December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24208

2 12. A. Y. Bluestone, G. Abdoulev, C. Schmitz, R. L. Brbour, nd A. H. Hielscher, Three-dimensionl opticl tomogrphy of hemodynmics in the humn hed, Opt. Express 9(6), (2001). 13. A. D. Klose, nd A. H. Hielscher, Fluorescence tomogrphy with simulted dt bsed on the eqution of rditive trnsfer, Opt. Lett. 28(12), (2003). 14. V. Ntzichristos, Fluorescence moleculr imging, Annu. Rev. Biomed. Eng. 8(1), 1 33 (2006). 15. E. M. Sevick-Murc, G. Lopez, J. S. Reynolds, T. L. Troy, nd C. L. Hutchinson, Fluorescence nd bsorption contrst mechnisms for biomedicl opticl imging using frequency-domin techniques, Photochem. Photobiol. 66(1), (1997). 16. G. Alexndrkis, F. R. Rnnou, nd A. F. Chtziionnou, Tomogrphic bioluminescence imging by use of combined opticl-pet (OPET) system: computer simultion fesibility study, Phys. Med. Biol. 50(4225), 4241 (2005). 17. C. H. Contg, nd M. H. Bchmnn, Advnces in in vivo bioluminescence imging of gene expression, Annu. Rev. Biomed. Eng. 4(1), (2002). 18. C. Kuo, O. 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Imging 18(3), (1999). 24. J. Riley, H. Dehghni, M. Schweiger, S. Arridge, J. Ripoll, nd M. Nieto-Vesperins, 3D opticl tomogrphy in the presence of void regions, Opt. Express 7(13), (2000). 25. S. R. Arridge, H. Dehghni, M. Schweiger, nd E. Okd, The finite element model for the propgtion of light in scttering medi: direct method for domins with nonscttering regions, Med. Phys. 27(1), (2000). 26. H. Dehghni, S. R. Arridge, M. Schweiger, nd D. T. Delpy, Opticl tomogrphy in the presence of void regions, J. Opt. Soc. Am. 17(9), (2000). 27. A. H. Hielscher, R. E. Alcouffe, nd R. L. Brbour, Comprison of finite-difference trnsport nd diffusion clcultions for photon migrtion in homogeneous nd heterogeneous tissues, Phys. Med. Biol. 43(5), (1998). 28. J. Ripoll, M. Nieto-Vesperins, S. R. Arridge, nd H. Dehghni, Boundry conditions for light propgtion in diffusive medi with nonscttering regions, J. Opt. Soc. Am. A 17(9), (2000). 29. K. M. Cse, nd P. F. Zweifel, Liner Trnsport Theory (Addidon-Wesley, Reding, MA, 1967). 30. H. B. Jing, Opticl imge reconstruction bsed on the third-order diffusion equtions, Opt. Express 4(8), (1999). 31. S. Wright, M. Schweiger, nd S. R. Arridge, Reconstruction in opticl tomogrphy using the P-N pproximtions, Mes. Sci. Technol. 18(1), (2007). 32. E. D. Aydin, C. R. E. de Oliveir, nd A. J. H. Goddrd, A comprison between trnsport nd diffusion clcultions using finite element-sphericl hrmonics rdition trnsport method, Med. Phys. 29(9), (2002). 33. A. D. Klose, U. Netz, J. Beuthn, nd A. H. Hielscher, Opticl tomogrphy using the time-independent eqution of rditive trnsfer Prt 1: forwrd model, J. Qunt. Spectrosc. Rdit. Trnsf. 72(5), (2002). 34. S. Chndrsekhr, Rditive Trnsfer (Clrendon Press, London, 1950). 35. A. D. Klose, nd E. W. Lrsen, Light trnsport in biologicl tissue bsed on the simplified sphericl hrmonics equtions, J. Comput. Phys. 220(1), (2006). 36. M. Chu, K. Vishwnth, A. D. Klose, nd H. Dehghni, Light trnsport in biologicl tissue using threedimensionl frequency-domin simplified sphericl hrmonics equtions, Phys. Med. Biol. 54(8), (2009). 37. G. Mrquez, L. V. Wng, S. P. Lin, J. A. Schwrtz, nd S. L. Thomsen, Anisotropy in the bsorption nd scttering spectr of chicken brest tissue, Appl. Opt. 37(4), (1998). 38. S. Nickell, M. Hermnn, M. Essenpreis, T. J. Frrell, U. Krämer, nd M. S. Ptterson, Anisotropy of light propgtion in humn skin, Phys. Med. Biol. 45(10), (2000). 39. S. R. Arridge, nd M. Schweiger, Photon-mesurement density functions. Prt2: Finite-element-method clcultions, Appl. Opt. 34(34), (1995). 40. S. R. Arridge, nd W. R. B. Lionhert, Nonuniqueness in diffusion-bsed opticl tomogrphy, Opt. Lett. 23(11), (1998). 41. H. Dehghni, B. W. Pogue, J. Shudong, B. Brooksby, nd K. D. Pulsen, Three-dimensionl opticl tomogrphy: resolution in smll-object imging, Appl. Opt. 42(16), (2003). 42. M. E. Emes, nd H. Dehghni, Wvelength dependence of sensitivity in spectrl diffuse opticl imging: effect of normliztion on imge reconstruction, Opt. Express 16(22), (2008). (C) 2009 OSA 21 December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24209

3 1. Introduction Ner infrred (NIR) opticl tomogrphy is non-invsive imging modlity in which the opticl properties within volume of interest cn be reconstructed using mesured trnsmission nd reflectnce NIR dt. These clculted opticl mps cn then be used to derive functionl nd structurl informtion bout the tissue being imged [1 3]. In recent yers, severl NIR imging systems hve been developed with specific pplictions such s the detection nd chrcteriztion of brest cncer [1,4 7] nd functionl imging of the brin [8 12]. More recently the use of biologicl mrkers for function specific opticl imging hs led to the development of novel imging systems nd lgorithms tht im to recover imges of function specific ctivity within smll nimls using, for exmple, fluorescence [13 15] or bioluminescence mrkers [16 20]. Most NIR imging systems rely on the use of model bsed imge reconstruction lgorithm in which the light distribution within the imging domin is pproximted to provide mtch to mesured boundry dt [21 23] nd therefore providing n estimte to the opticl prmeters within the medium being imged. The most commonly pplied model for the estimtion of NIR light propgtion in tissue is chieved through the Diffusion Approximtion (DA) to the Rditive Trnsport Eqution (RTE) through its robustness in implementtion nd computtionl speed nd flexibility. In order to ccurtely pply the DA, scttering effects within the medium must be dominnt over the bsorption, i.e. µ s >>µ, where µ s is the reduced scttering coefficient nd µ is the bsorption coefficient. Whilst these conditions re generlly pplicble in most soft tissue, the presence of non-scttering regions, such s the cerebrospinl fluid found within the brin, or smll source-detector seprtion, s encountered in smll niml imging, cn men tht the DA is not vlid for ll cses [24 28]. To ccurtely model NIR light propgtion in smll geometries, or models with high bsorption (such s for bioluminescence imging), higher ordered forwrd models re needed whereby the problem is no longer limited by the diffuse pproximtion. Whilst Monte Crlo simultions which re bsed on probbilistic models cn produce ccurte results, they tend to be computtionlly slow to be of clinicl use. Alterntively, numericl pproximtions to the RTE cn be used whereby light propgtion is estimted through well estblished prticle trnsport models bsed on complex integrl-differentil equtions, which cn ccount for both sptil nd directionl component of light propgtion. The sphericl hrmonics (P N ) pproximtion of the RTE [29], for exmple, expnds the ngulr components of rdince into series of sphericl hrmonics. This method hs been successfully pplied in opticl imging but lthough more pproprite in conditions where the DA is not vlid, it is still less thn idel due to the computtionl cost incurred [29 32]. The discrete ordintes method (S N ) is lso common implementtion of the RTE often encountered in nucler physics which discretises the ngulr components into number of discrete directions. This method hs lso been used successfully in opticl imging ppliction but s in the ltter cse results in much higher computtionl cost s compred to the DA [27,33,34]. More recently, the ppliction of Simplified Sphericl Hrmonics (SP N ) pproximtion to the RTE hs been pplied nd studied [35,36]. The use of SP N methods hve been demonstrted to give ccurte solutions for smll geometries nd in cses where the source/detector seprtion is smll. The SP N method hs the dvntge of requiring (N + 1)/2 equtions where N is the number of Legendre Polynomils. This is in contrst to (N + 1) 2 equtions for the P N pproximtion, nd N(N + 2) equtions for the S N method, where N is the number of llowed directions. Due to the computtionl complexities outlined bove, the vst mjority of existing model bsed reconstruction lgorithms mke use of the DA. The imge reconstructions therefore im to form imges of the bsorption coefficient, µ nd the diffusion coefficient κ = 1/3(µ s ) where µ s = (1-g)µ s nd µ s nd g re the scttering coefficient nd nisotropic fctor respectively. One drwbck of the ppliction of the DA is tht through the introduction of the reduced scttering coefficient, informtion bout the scttering phse function is lost [32] nd (C) 2009 OSA 21 December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24210

4 ssumed to be constnt t g = 0.9 which is ccepted s vlid for soft tissue. Little work hs been done to investigte the effect of vrying nisotropy fctor within imge reconstruction nd whether it is t ll possible to pproximte its vlue within the inverse problem. The effect of nisotropic structures tht ffect the propgtion of ner-infrred photons ccording to their direction is of current interest nd structures hve been experimentlly shown to cuse direction dependence of light propgtion in cses of chicken brest tissue nd humn skin [37, 38]. The theoreticl considertions hve so fr concentrted on developing wys to model the nisotropic propgtion of NIR light, nd less emphsis hs been put on exploring the implictions of nisotropic effects in imge reconstruction. In this study, SP N bsed models re used to study the effects of nisotropic scttering fctor on imge reconstruction. More specificlly the use of SP N methods hve been utilized to evlute the sensitivity of mesured boundry dt from frequency domin (FD) mesurement system (where mplitude nd phse of the fluence re used t typiclly 100 MHz modultion frequency) to not only opticl bsorption nd sctter but lso nisotropy fctor. Through clcultion of error-norm (residul) from theoreticl model, the uniqueness of this problem is demonstrted. Furthermore, frmework for the use of SP N methods in imge reconstruction is presented. 2. Theory 2.1 Forwrd model The forwrd problem is modeled using the SP N pproximtion to the RTE. The SP N equtions cn be derived by tking the plnr sphericl hrmonics pproximtion nd replcing the 1D derivtives with their 3D counterprts [35,36]. The SP 7 pproximtion results in set of four coupled equtions: 1. φ 1 φ1 = 3 µ Q φ 2 µ φ 3+ µ φ (1) φ 2+ µ 2 φ2 = 7 µ Q φ 1+ µ 2 φ µ 2 φ φ 3+ µ 2 4 φ3 = 11 µ Q µ φ 1+ µ 2 φ µ 2+ µ 4 φ (1b) (1c) (C) 2009 OSA 21 December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24211

5 φ 4+ µ φ4 = 15 µ Q φ 1 µ 2 φ µ 2+ µ 4 φ where Q is the isotropic source term, φn re composite moments of the fluence nd µ n is the n th order bsorption coefficient given by: ( ) ( ) ( ) µ µ µ (1d) n n x = t x s x g (2) where the totl ttenution coefficient µ t is given by the sum of the bsorption nd scttering coefficients (µ t = µ s ). In prctice, however, it is only possible to experimentlly record the totl fluence which is clculted from the composite moments s ϕ = φ 1 3 φ φ 3 35 φ (3) 4 If we ssume tht the composite moments re in fct mesureble, the SP N pproximtion introduces the possibility of reconstructing wider rnge of opticl prmeters. Wheres the DA llows the recovery of just µ nd µ s, the SP N pproximtion llows the recovery of µ, µ 1, µ 2, µ 3, µ n nd, from these vlues µ s nd g cn be clculted through Eq. (2). It is importnt to note tht lthough it is possible to mesure the ngulr dependence of the fluence t the boundry through the use of opticl fibers whose numericl perture cn be djusted, it is not cler how the composite moments s shown in Eq. (1) nd 3 my be mesured experimentlly. Nonetheless, for the purpose of the theoreticl study presented here, it will be ssumed tht these moments cn be mesured. The SP N equtions hve been implemented using the NIRFAST pckge nd previously vlidted using Monte Crlo models [36]. 2.2 Imge reconstruction The gol of the inverse problem is the recovery of the opticl properties, µ, by minimizing the difference between the mesured dt Φ M nd dt clculted by the forwrd model Φ C using modified Tinkhonov minimiztion pproch given by [22] min NM 2 NN M C 2 ( ) ( j 0) (4) 2 χ = Φ Φ + λ µ µ µ i= 1 j= 1 where NM is the number of mesurements, NN is the number of nodes in the FEM, µ 0 is n initil estimte of the opticl properties nd λ is the Tikhonov regulriztion prmeter. By considering only the first order terms nd pplying the Levenberg-Mrqurdt procedure, this leds to n updte vector where λ 2λ T ( ) 1 = J J+ I J Φ (5) T λ δ = nd J is the Jcobin mtrix ( δφ ), often referred to s the weight mtrix or sensitivity mtrix. The Jcobin defines the reltionship between chnges in clculted boundry dt due to smll chnges in the opticl properties. If both phse nd mplitude dt is vilble, the structure of the Jcobin becomes (C) 2009 OSA 21 December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24212

6 J δ ln I δ ln I δ ln I 1 2 NN δθ1 δθ1 δθ1 1 2 NN δ ln I δ ln I δ ln I NN = δθ2 δθ2 δθ δ ln I δ ln I δ ln I 1 2 δθ δθ δθ 1 2 NN NM NM NM NN NM NM NM where δlni i / j re sub-mtrices tht define chnges in the ith mesurement of log mplitude due to chnge in opticl properties t node j, nd δθ i / j define chnges in the ith mesurement of phse due to chnges in opticl properties t node j. In this study, the Jcobin s presented in sections is constructed using the perturbtion method nd for section 3.4 these re clculted using the Reciprocity pproch which ws vlidted using the perturbtion method [39]. As the opticl properties of the FEM mesh were defined by bsorption coefficient, scttering coefficient nd nisotropy fctor, the finl Jcobin ws mde up of three seprte kernels with the form NN (6) J = [ J ; J ; J ] (7) µ µ s g where J µ, J s µ nd J g re the Jcobins due to chnges in scttering coefficient, bsorption coefficient nd nisotropy respectively nd hve the form δ log I δθ J µ ; s = s s δ log I δθ J µ ; = (8) (8b) 3. Methods nd results 3.1 Sensitivity mpping J g δ log I δθ = ; δ g δ g The sensitivity of the SP 7 models to chnges in opticl properties were studied using circulr geometry of 43 mm rdius with 16 eqully spced nd collocted sources nd detectors s shown in Fig. 1. The Jcobin ws constructed using the perturbtion method before being mpped onto the mesh coordintes to produce mp of sensitivity to chnges in opticl properties. The reference dt ws clculted using homogeneous opticl properties of µ = 0.01 mm 1, µ s = 10 mm 1, g = 0.9 nd refrctive index (n) = 1.33 using modultion frequency of 100MHz. The sources were modeled s point sources locted t 1 mm inside the boundry, to correspond with one reduced scttering distnce s in DA. (8c) (C) 2009 OSA 21 December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24213

7 Fig. 1. FEM mesh, of 43mm rdius, contining 1785 nodes nd 3418 liner tringulr elements. Circles represent loction of sources, crosses represent loction of detectors. The totl sensitivity of log mplitude for ll sources nd detectors pirs to chnges in bsorption coefficient is mpped in Fig. 2(). Ares of high sensitivity exist in the regions surrounding the sources nd detectors with much lower bckground sensitivity decresing, s expected, towrds the centre of the domin. The negtive mgnitudes of sensitivity confirm tht increses in bsorption led to reductions in mesurements of intensity. Figure 2(b) shows the sensitivity of sme dt-type to scttering coefficient nd of similr trend to the previous cse. The sensitivity scle for scttering is gin negtive but hs much lower mgnitude thn tht of bsorption. Figure 2(c) shows tht the sensitivity of intensity mesurements to chnges in nisotropy fctor nd is gin highly loclized to the sources nd detectors. The mgnitude of sensitivity of g is severl orders lower thn tht of both bsorption nd scttering. Figures 2()-(b) show tht increses in both the bsorption nd scttering coefficients led to decreses in intensity mesurements. This suggests tht it is not possible to distinguish between the two effects. A decrese in intensity, for exmple, could be cused by either smll increse in bsorption or lrge increse in scttering. Fig. 2. Mps of sensitivity of log Amplitude dt to chnges in ) bsorption, b) scttering nd c) nisotropy Fig. 3. Mps of sensitivity of phse dt to chnges in ) bsorption, b) scttering nd c) nisotropy Figure 3() shows the sensitivity of phse mesurements to chnges in bsorption coefficient. It is evident tht the sensitivity is greter in the regions surrounding the sources nd detectors s compred to the regions in between them. Additionlly, the sensitivity (C) 2009 OSA 21 December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24214

8 lthough incresing to mximum t few scttering depths inside, it then decreses towrds the centre of the domin, which is consistent with previous findings [39]. The scttering sensitivity mp, Fig. 3(b), shows regions of very high sensitivity ner the sources nd detectors with much lower sensitivity elsewhere. The mgnitude of scttering sensitivity is gin much lower thn tht of bsorption but with positive scle. Sensitivity of phse mesurements to chnges in nisotropy fctor is gin limited to the regions surrounding the sources nd detectors with much lower mgnitude s compred to bsorption nd sctter. The results show tht using log mplitude dt lone, it is not possible to seprte chnges due to perturbtions in bsorption nd sctter properties since both result in the sme effect (i.e. increse in either bsorption or sctter results in decrese in log mplitude mesurements). With the introduction of phse dt, however, it cn be seen tht chnges in mplitude nd sctter hve opposite effects on such dt nd so it is possible to distinguish between perturbtions in bsorption or sctter. However, since scttering nd nisotropy fctor hve opposite sensitivities for both dt-types, n increse in one prmeter cn be compensted by decrese in the other nd vice-vers. Therefore the problem ppers nonunique for reconstruction of sctter nd nisotropy fctor. 3.2 Uniqueness of boundry dt Next the uniqueness of the imging problem is nlyzed. The problem of non-uniqueness rises when more thn one set of opticl property distributions led to n identicl set of boundry dt, in which cse, the simultneous recovery of multiple opticl properties, such s the bsorption nd scttering coefficients, is not possible [40]. The previously used circulr mesh, Fig. 1, ws used to test the uniqueness of the SP 7 equtions. A circulr nomly, with rdius 10 mm, ws inserted into the mesh s shown in Fig. 4. The opticl properties within the nomly were perturbed with either the bsorption coefficient rnging between mm 1 to mm 1, scttering coefficient rnging between 10mm 1 to 30mm 1 nd nisotropy fctor rnged between 0 nd 1 nd corresponding boundry dt for ll possible combintions were clculted. These wide rnge of vlues re chosen in order to llow comprehensive study of their effect on problem uniqueness. Reference dt ws lso clculted using the opticl properties of µ = mm 1, µ s = 20 mm 1 nd g = 0.9. The bsolute error (L1 norm) for ech dt-type between the reference nd ech set of perturbed dt ws the defined s ( ϕref ϕµ ) δ = (9) where ϕref is the reference homogeneous unperturbed boundry dt (either phse or log mplitude) nd ϕµ is the perturbed dt. Figure 5() shows the mp of error (Eq. (9) in mplitude dt for ech combintion of µ nd µ s. A lrge bnd of opticl property combintions exists for which the error between the reference dt nd perturbed dt flls to zero. These equivlent combintions led to identicl mplitude dt t the boundry nd, s such, mplitude dt lone is insufficient to recover both bsorption nd scttering properties simultneously. By introducing phse mesurements, however, the number of non-unique solutions cn be minimized. Figure 5(b) shows the error mp of phse mesurements for the sme rnge of opticl properties. In this cse, the opticl property combintions leding to identicl boundry dt lie within much more loclized region. By combining the two dt types, however, Fig. 5(c), the solution cn converge on smller rnge of opticl properties tht led to the perturbed boundry dt. The study ws then repeted with vrying bsorption coefficients nd nisotropy fctors. Figure 6() shows tht lrge bnd of non-unique opticl property combintions exists for mplitude mesurements. The errors in phse mesurements, Fig. 6(b), show n even lrger rnge of equivlent combintions. When combining the two dt types, however, it cn be seen tht the number of µ nd g combintions leding to the reference dt ctully flls within smller nrrow region, Fig. 6(c). (C) 2009 OSA 21 December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24215

9 The mp of error in mplitude mesurements with vrying µ s nd g, re lso shown in Fig. 7. Figure 7(), shows lrge bnd of equivlent combintions of the two prmeters for log mplitude dt. The error in phse dt ctully shows two lrge regions of non-unique opticl property combintions. The bnd of equivlent combintions in the mplitude error mp is prllel to one of the bnds in the phse error mp. As such, even when combining the two dt types, lrge bnd of equivlent combintions still exists, Fig. 7(c), nd therefore, even with both phse nd mplitude dt, it is not possible to distinguish between chnges due to µ s nd g. By direct visul comprison between Figs. 6(c) nd 7(c), it is evident tht there exists lrge rnge of bsorption, sctter nd nisotropy fctor tht would give rise to the sme boundry mesurements, even when considering both dt-types nd hence indicting the non-uniqueness of the problem to reconstruct ll three prmeters. Fig. 4. Circulr mesh with single inclusion of 10mm rdius used to test uniqueness of boundry mesurements. Fig. 5. Error mps of SP 7 dt with vrying opticl properties with rbitrry units of error. () mp of log(amplitude) with vrying µ (y xis) nd µ s (x xis), (b) sme s () but for phse nd (c) error mp of combined log(amplitude) nd Phse dt. (C) 2009 OSA 21 December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24216

10 Fig. 6. Sme s Fig. 5, but for () mp of log(amplitude) dt with vrying g nd µ, (b) sme s () but for phse, (c) sum of () nd (b) Fig. 7. Sme s Fig. 6, but for () mp of log(amplitude) dt with vrying g nd µ s, (b) sme s () but for phse, (c) sum of () nd (b) 3.3 Uniqueness with the introduction of higher order terms The use of higher ordered SP N pproximtions introduces number of new vribles, µ n, which re defined s in Eq. (2). These higher ordered bsorption coefficients re dependent on both µ s nd g nd s such, my mke it possible to seprte the effects due to chnges in the two prmeters, if it is ssumed tht the composite moments of the boundry dt cn be mesured. (C) 2009 OSA 21 December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24217

11 To test the uniqueness of the higher bsorption orders, the vlues of µ n for n = 1:4 (s vilble for SP 5 ) were clculted using opticl properties of µ = 0.01 mm 1, µ s = 20 mm 1 nd g = 0.5 s reference. The vlues of µ n were then re-clculted with vlues of the scttering coefficient rnging from 1mm 1 to 40mm 1 nd the nisotropy fctor rnging from 0.1 to 0.9. Figure 8() shows the difference between the reference vlue of µ 1 nd those clculted for ech combintion of scttering coefficient nd nisotropy fctor. There is lrge bnd of opticl property combintions for which the residul flls to zero nd therefore result in the sme vlue of µ 1. Figure 8(b) shows the sme differences between vlues of µ 2. There is still lrge number of scttering coefficient / nisotropy fctor combintions tht led to the sme vlue in this cse, lthough slightly smller thn for µ 1. Figures 8(c)-(d) show similr plots for µ 3 nd µ 4 respectively. It cn be seen tht s higher order moments of bsorption re considered, the number of scttering coefficient / nisotropy fctor combintions leding to the sme vlue reduces, but does not converge to prticulr combintion. As such, the scttering coefficient nd nisotropy fctors cnnot be simultneously identified. Fig. 8. Mps of residuls between () µ 1, (b) µ 2, (c) µ 3 nd (d) µ 4 for rnge of scttering coefficients nd nisotropy fctors s compred to set of reference vlues. The rnge of nisotropy fctors is listed on the x-xes whilst the y-xes represent the rnge of scttering coefficients. 3.4 Multi-prmeter imge reconstruction using the SP N pproximtion SP N bsed imge reconstruction lgorithms were developed for N = 1, 3, 5 nd 7 using the modified Tikhonov minimiztion method. Test boundry dt ws generted using the sme circulr geometry, Fig. 1, with µ = 0.01 mm 1, µ s = 10 mm 1, g = 0.9 nd n = A highly bsorbing nomly, with µ = 0.02 mm 1 nd highly scttering nomly with µ s = 20 mm 1, were inserted into the mesh s shown in Fig. 9. Both nomlies hve the sme nisotropic fctor nd refrctive index s the bckground. Ech of the SP N reconstruction lgorithms (for N = 1, 3, 5 nd 7) ws cpble of reconstructing different opticl properties, depending on the order N. In order to test the cpbility of ech of the SP N models, the opticl properties listed in Tble 1 were (C) 2009 OSA 21 December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24218

12 reconstructed, nd from these, vlues of µ nd µ s were extrcted. For the inverse model, the sme mesh s the forwrd model ws used without dded noise to ensure tht ny errors in the reconstructed opticl mps were only due to the complexity of the higher ordered pproximtions. Ech of the lgorithms were llowed to continue until there ws less thn 0.1% chnge in successive itertions. The initil regulriztion prmeter λ (Eq. (5) for ll lgorithms ws set to 10 times the mximum digonl of the mtrix J T J nd ws reduced by fctor of t ech itertion [22]. Figure 10() shows the reconstructed imges generted using n SP 1 bsed reconstruction lgorithm nd SP 1 forwrd dt. The loction of the two nomlies hs been ccurtely reconstructed. The bckground opticl properties hve been ccurtely recovered lthough the vlues of the two nomlies hve been over-estimted. The reconstructed imge shows mximum bsorption coefficient of mm 1 nd mximum scttering coefficient of 23.9 mm 1. The SP 1 reconstruction required 32 itertions t 3.1 second per itertion to recover the two relevnt unknowns. Fig. 9. FEM model with one highly bsorbing trget nd one highly scttering trget. Tble 1. Reconstructed vlues of SP N imge reconstruction lgorithms where the diffusion terms κ n = 1/(Aµ n) where A is constnt. Reconstruction model Forwrd dt used Unknown prmeters SP 1 SP 1 κ 1, µ SP 3 SP 3 κ 1, κ 3, µ 1, µ 2 SP 5 SP 5 κ 1, κ 3, κ 5, µ 1, µ 2, µ 4 SP 7 SP 7 κ 1, κ 3, κ 5, κ 7,µ 1, µ 2, µ 4, µ 6 The SP 3 reconstruction (using SP 3 dt) Fig. 10(b) hs lso ccurtely recovered the loction nd shpe of the two nomlies. The recovered bsorption nd scttering coefficients hve underestimted the trget vlues, returning mm 1 nd 17.3mm 1 respectively. Unlike the SP 1 imge, the SP 3 reconstruction shows signs of cross tlk between the bsorption nd scttering imges. The SP 3 reconstruction required 13 itertions t 7.3 second per itertion to recover the four relevnt unknowns. Reconstruction using the SP 5 model (using SP 5 dt) is shown in Fig. 10(c). In this cse, the opticl properties hve gin been overestimted with mximum bsorption coefficient of mm 1 nd mximum scttering coefficient of 21.4mm 1. The cross tlk between the bsorption nd scttering imges is still present. The SP 5 reconstruction required 34 itertions t 54 second per itertion to recover the six relevnt unknowns. The opticl mps generted by the SP 7 model (using SP 7 dt), Fig. 10(d) contins rtifct throughout the domin. The loction of the highly bsorbing trget hs been recovered with poor size nd contrst ccurcy. The reconstruction filed to recover the highly scttering trget. It is likely tht the filure of the SP 7 reconstruction is due to the highly underdetermined nture of the problem. The SP 7 model requires 8 unique vribles to be clculted t ech node, i.e unknowns, bsed on just 1920 boundry mesurements. The SP 7 reconstruction required 20 itertions t 52 second per itertion to recover the eight relevnt unknowns. (C) 2009 OSA 21 December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24219

13 Fig. 10. Reconstructed opticl mps using ) SP 1, b) SP 3, c) SP 5, d) SP 7 bsed reconstruction lgorithms 3.5 Multi-prmeter imge reconstruction nd hrd priori informtion The under-determined nture of the imge reconstruction problems cn be minimized with the use of prior informtion. MRI dt cn be used, for exmple, to determine regions of similr tissue types nd therefore, regions of homogeneous opticl properties [41]. This then reduces the problem of clculting opticl properties t every node to just smll number of known regions. The previous study ws repeted using region-bsed reconstruction for the SP N methods with N = 1, 3 nd 5. The SP 7 model hs been omitted from further studies s the increse in ccurcy over the SP 5 model hs been shown to be very smll nd neglectble [35]. It cn be seen (Fig. 11) tht the use of prior informtion, whereby insted of reconstruction the unknown prmeters t ech sptil vrible, homogeneous vlues re estimted for ech unknown region (in this cse 3) hs enbled the SP 5 reconstruction (when using SP 5 forwrd dt) to ccurtely recover both trgets. This is due to the vst reduction in the number of (C) 2009 OSA 21 December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24220

14 unknowns from 6 times the number of nodes (10710) to 6 times the number of regions (18). The sme findings (not shown) were found for SP 1 nd SP 3 cses. Fig. 11. Recovered opticl mp generted using SP 5 reconstruction with prior informtion. 3.6 Diffusion bsed imge reconstruction using the SP N pproximtion As discussed erlier, the SP N pproximtions, Eqs. (1)()-(d), re bsed on composite moments of fluence with the totl flux clculted using Eq. (2). The SP N reconstruction lgorithms introduced so fr were therefore bsed on vrious moments of the phse nd mplitude boundry dt which ttempted to recover ll moments of bsorption nd scttering coefficients. In prctice, however, there re no experimentl methods to esily mesure the ngulr components of mplitude nd phse nd so imge reconstructions must be bsed on totl vlues of fluence. A new reconstruction lgorithm ws therefore developed tht used the SP 5 model but only required mesurements of phse nd mplitude s vilble from the totl fluence nd not composite moments. The new reconstruction lgorithm uses the SP 5 forwrd model to clculte the composite moments of fluence. The totl fluence is then clculted using Eq. (3). The Jcobin is then constructed using Eqs. (8) nd (8b) where log of mplitude nd bsolute phse re extrcted from the clculted totl fluence. By eliminting the composite moments of fluence, however, it is only possible to reconstruct for µ. nd µ s (ssuming g = 0.9). This new reconstruction lgorithm is tested using SP 5 forwrd dt nd the resulting opticl mp is shown in Fig. 12(). The reconstruction hs performed well recovering the opticl prmeters within 10% of the trget vlues. For comprison, the sme forwrd dt is lso used to reconstruct opticl prmeters using the DA bsed lgorithm, Fig. 12(b). In this cse, the trget vlues hve been over-estimted nd boundry rtifct hs been introduced, indicting the mximum errors seen from high order model mismtch, is s expected ner the source / detector positions, which led to imge inccurcy nd rtifcts. 4. Discussions nd conclusions Forwrd models for imge reconstruction bsed on the simplified sphericl hrmonic pproximtion hve been presented. Sensitivity mps for mplitude nd phse dt re clculted, Figs. 2 nd 3, using SP 7 nd it is shown tht the sensitivity of mesured boundry dt for chnges in the nisotropy fctor re mny orders of mgnitude lower thn those of bsorption nd scttering coefficients. It is lso shown tht the sensitivity to both scttering coefficient nisotropy fctor is highly loclized to the regions djcent to the source nd detector (Figs. 2 nd 3). This suggests tht beyond few scttering lengths, the light distribution is fully diffuse nd s such, distnt chnges in scttering nd nisotropy fctors hve little effect on mesured dt. Although it is possible to normlize the sensitivity mps prior to use in imge reconstruction [42], it is lso evident tht n increse in scttering prmeter will hve n effect on boundry dt which is opposite of tht due to nisotropy fctor. This in turn indictes tht n increse in one prmeter cn be compensted by decrese on the other nd vice-vers. This effect is unlike tht seen for bsorption nd scttering, whereby the two prmeters cn be seprted using log mplitude nd phse dt, but it is uncler whether ll 3 prmeters cn be resolved using these 2 dt-types. (C) 2009 OSA 21 December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24221

15 Fig. 12. Diffusion bsed imges from SP 5 dt using () SP 5 bsed Jcobin nd (b) SP 1 bsed Jcobin. Through the use of error mps, whereby the difference in some reference dt with respect to dt in the presence of n nomly is clculted, it is shown tht using frequency domin dt, it is possible to distinguish, with some ccurcy, the bsorption nd scttering properties of the medium, Fig. 5, which hs lso been presented previously [21]. The sme principles hve been pplied to higher order models nd dditionlly the effect of the nisotropy fctor hs been studied s well s the opticl bsorption nd sctter. It is demonstrted, Fig. 6 nd 7, tht using frequency domin dt, lrge nd non-unique rnge of ll three opticl properties cn led to the sme mesured dt, indicting tht it is not possible to reconstruct ll three prmeters using only 2 dt-types. This finding is in line with erlier results, Fig. 3, nd demonstrte tht it is difficult to seprte for both scttering nd nisotropic fctors simultneously using frequency domin dt. Further work is needed to investigte whether other dt-types, for exmple using time resolved dt, nd / or spectrl dt, would llow seprtion of these vribles. Residul mps for different orders of bsorption coefficient (µ n ) s vilble using SP N models hve been clculted to investigte the possibility of resolving scttering nd nisotropy fctors, if µ n coefficients cn be mesured nd clculted. It is shown, Fig. 8, tht even if ll higher moments of the bsorption coefficients cn be clculted, there still exists lrge rnge of scttering nd nisotropy fctors tht would led to the sme µ n vlues, thus indicting tht these cnnot be esily resolved. Reconstruction lgorithms bsed on the SP N pproximtion hve been developed nd tested. For imge reconstruction, the sme FEM model s for the forwrd dt ws used nd no noise ws dded. The inverse crime hs been strictly committed, since we re concerned with the ccurcy of ech model in determining the unknown prmeters ssocited with ech model. Additionlly, noise free dt hve only used to only highlight the error seen due to model mismtch, rther thn the effects of noise within the dt, which cn show similr trends. It ws shown, Fig. 10()-(c), tht for N = 1, 3 nd 5 the reconstructions performed well, with the reconstructed vlues being within 24% of expected vlues with the worse results being obtined from SP 1 nd bsorption coefficient. This ccurcy could be further improved by the optimiztion of the regulriztion prmeter nd stopping criteri. The N = 7 model, however, filed to ccurtely recover the trget vlues, Fig. 10(d). The bsorption coefficient, lthough locted, is extremely over estimted, with some cross-tlk from the (C) 2009 OSA 21 December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24222

16 scttering vlue. The reconstructed sctter imge contins high vlued rtifcts nd cn be considered inccurte. By limiting the presented results to these lrger domin, it is lso worth noting tht the mjority of imge rtifcts re seen t the boundry, ner the source loctions, where SP 1 solution is known the be less ccurte. It is therefore expected tht the errors seen due to the lck of higher order pproximtions to be substntilly more significnt in smll geometry imging experiments. The poor performnce of the SP 7 model ws most likely due to the under-determined nture of the problem. As stted erlier, the SP 7 model contins 8 unique unknown vribles which need to be clculted t ech sptil loction (FEM node), i.e unknowns, bsed on just 1920 boundry mesurements (240 log mplitude nd 240 Phse, bsed on 16 co-locted source nd detectors nd 4 composite moments of fluence). In order to eliminte the lrge degree of freedoms, the ccurcy of the reconstruction lgorithms ws further improved by creting better determined problem. This ws chieved for the SP 1, SP 3 nd SP 5 pproximtions with the use of prior informtion. The prior informtion ws used to identify number of homogeneous regions, reducing the number of properties to be recovered. This simplifiction of the problem led to improvements in ll three of the reconstructions (for N = 1, 3 nd 5). Both the bsorbing nd scttering trgets were ccurtely recovered by ll models to within 2%, Fig. 11. The SP N equtions re bsed on composite moments of fluence, Eq. (2). In relity, however, experimentl systems cn only mesure the totl fluence t the boundry of the domin nd so the full SP N reconstruction lgorithms re of limited use in imge reconstruction where ll composite moments re needed. An lterntive method hs been proposed in which the forwrd models re bsed on bsolute fluence but the Jcobin mtrix ws clculted using the SP 5 model to llow the clcultion of bsorption (µ ) nd scttering (µ s ) coefficient only. The diffusion prmeter bsed imges reconstructed from simulted SP 5 dt whereby the Jcobin is bsed on either SP 1 or SP 5, Fig. 12, indicte tht lthough both models cn be used, the higher order model is more ccurte both in terms of quntittive nd qulittive nlysis. The trget vlues of the test problem clculted using SP 5 re recovered with more ccurcy s compred to the SP 1 bsed model with much less rtifct. Additionlly, eliminting the use of complex moments lso results in decresed computtion time nd memory requirements. The results presented in this work indicte tht for imge reconstruction whereby the DA is less vlid, in for exmple, smll niml imging nd / or where the bsorption coefficient is more dominnt, the higher order models bsed on simplified sphericl hrmonics cn be used to generte the sensitivity mtrix for diffusion bsed imge reconstruction, without the dditionl computtion complexity in terms of the number of unknown prmeters. The incorportion of these more ccurte models cn however llow for better ccurcy in terms of light propgtion models. Acknowledgements This work hs been sponsored by the Engineering nd Physicl Sciences Reserch Council, UK. (C) 2009 OSA 21 December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24223