The inverse source problem based on the radiative transfer equation in optical molecular imaging

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1 Journl of Computtionl Physics 202 (2005) The inverse source problem bsed on the rditive trnsfer eqution in opticl moleculr imging Alexnder D. Klose, *, Vsilis Ntzichristos b, Andres H. Hielscher Deprtments of Biomedicl Engineering nd Rdiology, Columbi University, ET351 Mudd Building, MC 8904, 500 West 120th Street, New York, NY 10027, USA b Center for Moleculr Imging Reserch, Msschusetts Generl Hospitl & Hrvrd Medicl School, Chrlestown, MA 02129, USA Received 18 December 2003; received in revised form 14 June 2004; ccepted 10 July 2004 Avilble online 11 September 2004 Abstrct We present the first tomogrphic reconstruction lgorithm for opticl moleculr imging tht is bsed on the eqution of rditive trnsfer. The reconstruction code recovers the sptil distribution of fluorescent sources in highly scttering biologicl tissue. An objective function, which describes the discrepncy of mesured ner-infrred light with predicted numericl dt on the tissue surfce, is itertively minimized to find solution of the inverse source problem. At ech itertion step the predicted dt re clculted by forwrd model for light propgtion bsed on the eqution of rditive trnsfer. The unknown source distribution is updted long serch direction tht is provided by n djoint differentition technique. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Fluorescence tomogrphy; Fluorescence imging; Inverse source problem; Moleculr imging; Eqution of rditive trnsfer; Discrete ordintes method; Adjoint differentition; Algorithmic differentition; Scttering medi; Tissue optics 1. Introduction Trditionl biomedicl imging techniques differentite pthologicl from norml tissue by detecting mcroscopic chnges in tissue structures. Using moleculr imging, on the other hnd, one tries to monitor the development of disese-ssocited processes on moleculr level prior to the ppernce of mcroscopic tissue chnges. Specificlly designed moleculr probes re used s source of imge contrst. Tomogrphic imging techniques, such s positron emission tomogrphy (PET), single photon emission * Corresponding uthor. Tel.: ; fx: E-mil ddress: k2083@columbi.edu (A.D. Klose) /$ - see front mtter Ó 2004 Elsevier Inc. All rights reserved. doi: /j.jcp

2 324 A.D. Klose et l. / Journl of Computtionl Physics 202 (2005) computed tomogrphy (SPECT), nd mgnetic resonnce tomogrphy (MRI), hve lredy widely dopted the concept of this new imging modlity, nd hve been successful in imging of gene expressions, protein protein interctions, nd drug effects t cellulr nd moleculr level [1 8]. In ddition opticl moleculr imging hs emerged which uses ner-infrred fluorescent probes [1,9 14]. A fluorescent biochemicl mrker is injected into biologicl system nd will emit ner-infrred light upon excittion by n externl light source. From mesurements of the light intensity on the tissue surfce one seeks to determine the sptil concentrtion distribution of the mrker inside the tissue. Different biochemicl mechnisms within the tissue cn influence the fluorescent probe concentrtion distribution. For exmple, the biochemicl mrker ccumultes in tissue prts with specific trgets such s cell receptors [10,11,15 17], or the fluorophore is quenched in its ntive stte nd only emits light fter ctivtion, i.e. when specific enzyme is encountered [10,11,18 20]. In both cses the fluorescent light signl is proportionl to the ccumulted or ctivted fluorescent probe concentrtion. Most work in opticl moleculr imging hs been limited to direct imging of fluorescent light tht escpes the surfce of smll nimls [10,12,15,18,21,22]. In this instnce, n exct locliztion of light-emitting sources inside the tissue is not possible. However, fluorescent source distributions in smll nimls could be determined by recording tomogrphicl dt sets nd employing pproprite tomogrphic imge reconstruction schemes [19,20,23]. A mjor difficulty in determining the fluorescent source distribution is imposed by multiple scttering of photons tht propgte through biologicl tissue. The men free pth of scttered photons in biologicl tissue is typiclly in the rnge cm for ner-infrred light. Tht limits the ppliction of wellestblished imge reconstruction methods of trnsmission nd emission tomogrphy. For exmple, X- ry computed tomogrphy (CT) dels only with non-scttered or single-scttered photons nd the prticle trnsport within the tissue cn be described by solution of n integrl eqution [24]. An inversion formul is used to determine the tissue prmeters of interest such s the X-ry ttenution coefficient. In emission tomogrphy, such s PET or SPECT, similr inversion formuls cn be employed. In contrst to CT which requires the solution of n integrl eqution, in ner-infrred fluorescence tomogrphy n integro-differentil eqution needs to be solved tht tkes multiple scttering into ccount. This integro-differentil eqution is known s eqution of rditive trnsfer (ERT). Solving the ERT for given fluorescent source distribution is lso referred to s solving the forwrd problem. The problem of finding the fluorescence source distribution from mesured light intensities on the tissue surfce is clled the inverse problem. To solve the inverse source problem in highly scttering medi using the ERT one cn use explicit nd implicit methods [25]. Explicit methods provide nlyticl solutions to the inverse source problem directly from mesured dt. No forwrd problem for rditive trnsfer needs to be solved. Explicit methods bsed on the ERT re limited to simple medium geometries with sptilly non-vrying opticl prmeters [26 30]. For more complex geometries nd heterogeneous medi no explicit methods re vilble nd implicit methods need to be employed. Implicit methods for solving the inverse source problem itertively utilize solution of forwrd model to provide predicted mesurement dt. An updte of n initil source distribution is sought by minimizing functionl tht describes the goodness of fit between the predicted nd experimentl dt. Implicit methods re computtionlly expensive when the forwrd model is bsed on the ERT. Implicit methods bsed on the ERT hve been used before in vrious scientific fields such s in inverse het source problems or inverse hydrologic bioluminescence problems [25,31 36], but not yet in opticl moleculr imging. The inverse fluorescent source problem in opticl moleculr imging hs lredy been solved using the diffusion eqution s forwrd model for light propgtion [37 48]. The diffusion model is n pproximtion to the ERT nd hs limittions in opticlly thin medi, in medi with smll geometries where boundry effects re dominnt, nd in medi where sources nd detectors re not sufficiently fr prt [49 52]. This poses prticulr problems, for exmple, in the re of smll niml fluorescence imging, where fluorescent sources re potentilly very close to detectors on the tissue surfce.

3 A.D. Klose et l. / Journl of Computtionl Physics 202 (2005) An imge reconstruction method bsed on the ERT hs the potentil to overcome these limittions. Therefore, we hve developed tomogrphic reconstruction scheme bsed on the ERT tht solves the inverse fluorescent source problem by recovering the sptil distribution of fluorescent sources in scttering medi. This method is similr to previously developed reconstruction scheme tht ws pplied to different inverse problem in opticl tomogrphy, where the unknown sptil distribution of the intrinsic bsorption nd scttering properties of biologicl tissue ws sought [53 56]. Our fluorescence imge reconstruction technique is n implicit method nd cn be viewed s nonliner optimiztion pproch. A forwrd model for light trnsport bsed on the ERT predicts the detector redings on the tissue boundry for given initil source distribution. An objective function is defined tht is the v 2 -error norm of the predicted nd mesured detector redings. An updting scheme is used to itertively modify the initil distribution long serch direction nd determines new source distribution inside the medium. The serch direction is provided by the derivtive of the objective function with respect to the present source distribution. The optimiztion process is finished fter the mesured nd predicted dt mtch nd minimum of the objective function is found. The finl source distribution is displyed in n imge. Besides using the ERT for the first time in opticl moleculr imge reconstruction we lso introduce n djoint differentition method s novelty to fluorescence tomogrphy. An djoint differentition technique computes the grdient of the objective function by exploiting the numericl structure of the light propgtion model. The concept of the djoint differentition technique cn lso be pplied to similr inverse problems or sensitivity nlysis where the derivtive of n error function is sought. A distinct dvntge of using tht technique is its reltively simple numericl implementtion nd the resulting low computtionl costs. In previous Letter we lredy presented first numericl results for reconstructing the bsorption coefficient nd the quntum yield of two-dimensionl fluorophore distribution using synthetic mesurement dt [57]. The imge reconstruction method could be employed in two different modes, the bsorption-contrst mode nd fluorescent-contrst mode. The present rticle describes the physicl nd numericl detils of the bsorption contrst mode for reconstructing the fluorophore bsorption coefficient. In ddition to the previous work, we included prtly reflected boundry conditions (Fresnel reflection) tht tke the refrctive index mismtch t the ir tissue interfce into ccount. Furthermore, for the first time we tested the reconstruction lgorithm with experimentl dt obtined from three-dimensionl, nisotropiclly scttering, tissue-like fluorescent phntom. In the following sections we will first describe the ERT-bsed forwrd model for light propgtion in tissue using finite-difference discrete-ordintes method. Next we present the nonliner optimiztion technique tht is used to minimize the objective function. This prt includes specifics of the derivtive clcultion of the objective function by mens of n djoint differentition technique. Finlly, we illustrte the performnce of the reconstruction code using experimentl dt. 2. Forwrd model for light propgtion in tissue The forwrd model of fluorescent system in tissue consists of two stges: (1) excittion of fluorophores inside tissue nd (2) subsequent emission of fluorescent light. Externl light sources with wvelength k x illuminte the tissue surfce nd the light propgtes through the tissue. Fluorescent molecules in their ground stte with specific extinction coefficient in units of M 1 cm 1 bsorb the light nd re elevted into n excited stte. Some proportion, defined by the quntum yield g, of the excited fluorescent molecules emits gin light t different wvelength, k m > k x, nd returns to its ground stte. The fluorophore with concentrtion c in units of M constitutes n internl light source with the strength proportionl to g nd the fluorophore bsorption l x!m ¼ c in units of cm 1 t the excittion wvelength k x. The light originting from the fluorescent source distribution escpes the tissue nd is mesured t the tissue surfce.

4 326 A.D. Klose et l. / Journl of Computtionl Physics 202 (2005) Eqution of rditive trnsfer The forwrd model for light propgtion determines the fluorescent light distribution given n externl excittion light source t the tissue boundry nd sptil distribution of opticl properties. The light distribution for both the excittion nd emission wvelengths cn be described by hierrchl system of two time-independent ERTs for the rdince w(r,x) with units of W cm 2 sr 1. The first eqution of rditive trnsfer (ERT I) describes the propgtion of the excittion light t the wvelength k x originting from n externl light source S x (r,x) with units of W cm 3 sr 1. The ERT is n energy blnce eqution for the rdince w x (r,x) nd is given s Z X rw x þðl x! þ l x!m þ l x s Þwx ¼ l x s 4p pðx X 0 Þw x ðx 0 ÞdX 0 þ S x : Here, the sptil position r =(x, y, z) of the rdince is given in Crtesin coordintes. The direction X =(J, u) of the rdince is given in sphericl coordintes, but cn lso be expressed in Crtesin coordintes X =(n,g,l) with n ¼ sin # cos u; g ¼ sin # sin u; ð2þ l ¼ cos #: Both coordinte systems re shown in Fig. 1. The propgting light undergoes scttering nd bsorption processes tht re described by the scttering, l x s ðrþ, nd bsorption, lx ðrþ, coefficients t the wvelength k x in units of cm 1. The bsorption coefficient l x ðrþ consists of the intrinsic bsorption lx! ðrþ of the tissue nd the bsorption l x!m ðrþ due to the fluorochrome. The scttering phse function p(xæx 0 ) with units of sr 1 gives the probbility tht single photon coming from X 0 is deflected by n ngle h into X. The ngle h encloses the directions formed by X nd X 0 in the intervl h 2 [0,p] with XÆX 0 = cosh. A commonly pplied scttering phse function in tissue optics is the Henyey Greenstein function [58 60], which is given by 1 g 2 pðcos hþ ¼ ð3þ 4pð1 þ g 2 2g cos hþ 3=2 with the normliztion condition 2p Z 1 1 pðcos hþd cos h ¼ 1: ð1þ ð4þ Fig. 1. Coordinte systems for sptil position r nd direction X.

5 The rdition field described by the ERT I [Eq. (1)] excites fluorescent molecules t position r. The mount of excited fluorescent molecules depends linerly on the bsorbed fluence / x (r) if sturtion nd bleching effects re neglected. The fluence distribution / x (r) in units of W cm 2 is obtined by integrting the rdince w x (r,x) t position r over ll directions X: Z / x ðrþ ¼ w x ðr; XÞdX: ð5þ 4p The excited fluorophore with the quntum yield g nd bsorption coefficient l x!m ðrþ constitutes light source t the fluorescence wvelength k m with the strength S m ðr; XÞ ¼ 1 4p glx!m ðrþ/ x ðrþ ð6þ in units of W cm 3 sr 1. It emits isotropic light since ll directionl informtion is lost fter excittion. The second eqution of rditive trnsfer (ERT II) describes the light propgtion t the wvelength k m originting from sptilly distributed fluorescent light sources S m (r,x). We obtin the ERT II with the fluorescent source term (6) Z X rw m þðl m þ lm s Þwm ¼ l m s pðx X 0 Þw m ðx 0 ÞdX 0 þ 1 4p 4p glx!m / x : ð7þ The rdince distribution w m (r,x) depends on the opticl tissue prmeters, l m ðrþ nd lm s ðrþ, t wvelength k m nd on the fluorescent source distribution. The fluorescent light escpes through the tissue boundries nd is mesured by detector t position r d. The light intensity / m (r d ) is given in units of W cm 2 with Z / m ðr d Þ¼ w m ðr d ; XÞdX ð8þ nx>0 A.D. Klose et l. / Journl of Computtionl Physics 202 (2005) with n s the norml outwrd vector t the boundry Boundry condition Biologicl tissue hs n opticl refrctive index (n > 1) tht is different from the refrctive index of the surrounding medium such s ir (n 0 = 1). Light is reflected nd refrcted when escping the medium due to the refrctive index mismtch (n > n 0 ) t the tissue ir interfce. The escping rdince w(x) long the outwrd direction X with n ÆX > 0 is prtly reflected t the interfce nd contributes to the light propgtion inside the tissue. Furthermore, some frction of the light leving the tissue long X chnges its direction into X 00 due to refrction. The reltion between the outwrd direction X, the outwrd direction X 00 of the refrcted rdince, nd the inwrd direction X 0 of the reflected rdince is shown in Fig. 2. The rdince tht is reflected bck into the medium cn be obtined from the reltion wðx 0 Þ¼RwðXÞ; n X 0 < 0; ð9þ where R is the reflectivity, which determines the mount of reflected light. R is given by FresnelÕs lw [61]! R ¼ 1 sin 2 ðb Þ 2 sin 2 ðb þ Þ þ tn2 ðb Þ : ð10þ tn 2 ðb þ Þ The ngle is enclosed by the norml vector n nd the outwrd direction X of the escping rdince ¼ rccosðn XÞ: ð11þ The ngle b is enclosed by the norml vector n nd the outwrd direction X 00 of the refrcted light nd is determined by mens of SnellÕs lw

6 328 A.D. Klose et l. / Journl of Computtionl Physics 202 (2005) Fig. 2. Reltion between the outwrd directions X nd X 00, nd the inwrd direction X 0 t the ir tissue interfce. Light is reflected into the medium (refrctive index n) long X 0 nd refrcted long X 00 when leving the medium. b ¼ rcsin n sinðþ : ð12þ n 0 Totl reflection (R = 1) is tken into ccount for b > p/2. As cn be clerly seen, for refrctive index mtch (n = n 0 ) of both medi t the interfce we obtin the none-reentry boundry condition wðx 0 Þ¼0; n X 0 < 0: ð13þ 2.3. Finite-difference discrete-ordintes method Both trnsport equtions, Eqs. (1) nd (7), need to be solved numericlly becuse no nlyticl solutions of the ERT re vilble for sptilly heterogeneous medi with finite geometricl boundries [62]. Since the numericl solution method is the sme for both trnsport equtions, we will omit the superscript x nd m in both equtions for distinguishing the excittion nd fluorescence field. We employ finite-difference discrete-ordintes ðs N Þ method tht converts the integro-differentil eqution into system of lgebric equtions [62 71]. The rdince w(r, X), which is continuous function in spce, is replced by finite set of H discrete rdinces. The ERT is substituted by set of pproximte lgebric equtions. The size of the lgebric system with its H unknown rdince vlues depends on the sptil (finite-difference) nd ngulr (discrete-ordintes) discretiztion. First, the direction X is replced with set of discrete ordintes X k =(n k,g k,l k ) with full level symmetry [65,66,70,72]. The totl number of ordintes X k with k 2 {1..K} is given by K ¼ NðN þ 2Þ nd N the number of direction cosines of the S N method. The integrl in the ERT is pproximted with qudrture rule Z pðx X 0 ÞwðX 0 ; rþdx 0 XK w k 0pðX k X k 0ÞwðX k 0; rþ ¼ XK w k 0p kk 0w k 0ðrÞ ð14þ 4p k 0 ¼1 where w k 0 re weights determined by full level symmetry of the ordintes [66]. The ngulr discretiztion yields set of K coupled differentil equtions for the rdince w(r,x k )=w k (r) in the directions X k : k 0 ¼1 X k rw k ðrþþðl ðrþþl s ðrþþw k ðrþ ¼l s ðrþ XK k 0 ¼1 w k 0p kk 0w k 0ðrÞþS k ðrþ: ð15þ

7 The fluence /(r) is obtined with /ðrþ ¼ XK k¼1 w k w k ðrþ: Next, the continuous sptil vrible r is discretized on three-dimensionl Crtesin grid constituting prllelepiped. Hence, the rdince w k (r) is only defined on grid points r =(x m,y n,z l ) with m 2 {1..M}, n 2 {1..N}, nd l 2 {1..L}. The grid spcing between djcent points is given by Dx, Dy, nd Dz. We define for given direction X k nd grid point (mnl) the rdince w mnlk. The totl number of unknown w mnlk is H = MÆNÆLÆK. Furthermore, the sptil derivtives rw k ðrþ ¼ðow k =ox; ow k =oy; ow k =ozþ in Eq. (15) re substituted with first-order finite-difference pproximtions known s step method [65,66,70,73,74]. The difference formul of the step method depends on the direction X k of the ngulr-dependent rdince w mnlk. The set of ll ordintes X k for the unit sphere is subdivided into eight octnts nd we obtin eight different difference formuls for the rdince w mnlk. For exmple, for ll ordintes X k with positive direction cosine n k > 0 we hve the following finite difference term long the x-xis ow ox w mnl w m 1nl : ð17þ Dx The difference formul chnges for negtive direction cosine n k < 0 nd we obtin ow ox w mþ1nl w mnl : ð18þ Dx The sme finite-difference pproch is pplied long the y-xis nd z-xis for direction cosines g k nd l k. Hence, the discretized ERT for grid point (mnl) is, for exmple, for ll ordintes X k within the octnt n k >0,g k >0,ndl k >0: n k w mnlk w m 1nlk Dx ¼½l s Š mnl X K k 0 ¼1 A.D. Klose et l. / Journl of Computtionl Physics 202 (2005) þ g k w mnlk w mn 1lk Dy w k 0p kk 0w mnlk0 þ S mnlk : þ l k w mnlk w mnl 1k Dz þð½l s Š mnl þ½l Š mnl Þw mnlk Using the short-term nottion (d x,d y,d z ) for the discretized sptil derivtive of ðo=ox; o=oy; o=ozþ we obtin n lgebric system of equtions for the discretized ERT for ll grid points nd ordintes ð16þ ð19þ fn k d x þ g k d y þ l k d z þ½l Š mnl þ½l s Š mnl gw mnlk ¼½l s Š mnl X K k 0 ¼1 w k 0p kk 0w mnlk 0 þ S mnlk : ð20þ The elements w mnlk cn be cst into vector W nd we obtin in mtrix nottion AW ¼ BW þ S; ð21þ with A s the discretized streming nd collision opertor, B s the discretized integrl opertor or in-sctter term, nd S s source term [68]. Furthermore, we introduce in Eq. (19) the vector nottion [..] for ll opticl prmeters tht will be used in the remining rticle. The nth element of vector [..] is denoted by [..] n Source itertion We solve the system (21) for the rdince vector W by employing source itertion (SI) method [65,66,68,73]. This method is n itertive build-up of the rdince. By strting from n initil source term

8 330 A.D. Klose et l. / Journl of Computtionl Physics 202 (2005) (z = 0), i.e. n externl light source or n internl fluorescent source, n initil rdince vector W 0 is computed s shown in Eq. (22) AW 0 ¼ S: ð22þ The rdince W 0 is now used to determine the in-sctter source term BW 0 on the right hnd side of Eq. (23). Since the right hnd side for the next source itertion step (z + 1) is known from the previous step new rdince vector W z+1 cn be determined AW zþ1 ¼ BW z þ S: ð23þ A single itertion step for computing W z +1 is lso clled trnsport sweep. The itertion process is repeted until the reltive difference of the rdince of subsequent itertion steps is smller thn predefined vlue j given by the reltion W zþ1 W z < j: ð24þ W zþ1 The rdince vector W Z of the lst trnsport sweep Z is the solution of the discretized ERT. For exmple, t ech trnsport sweep single vector element w zþ1 mnlk for ll ordintes X k with n k >0,g k > 0, nd l k > 0 [see Eq. (19)] is clculted with P S mnlk þ½l s Š mnl w k 0p kk 0w z mnlk þðn k=mxþw zþ1 0 m 1nlk þðg k=myþw zþ1 mn 1lk þðl k=mzþw zþ1 mnl 1k w zþ1 mnlk ¼ k 0 : ð25þ ðn k =MxÞþðg k =MyÞþðl k =MzÞþ½l Š mnl þ½l s Š mnl According to the ERT t the excittion wvelength, see Eq. (1), the source term S mnlk is the externl light source S x mnlk. The opticl prmeters re ½l Š mnl ¼½l x! Š mnl þ½l x!m Š mnl, ½l s Š mnl ¼½l x s Š mnl. At the emission wvelength, see Eq. (7), we replce S mnlk with the fluorescent source term S m mnlk ¼ð1=4pÞg½lx!m Š mnl / x mnl, nd the opticl prmeters with ½l Š mnl ¼½l m Š mnl nd ½l sš mnl ¼½l m s Š mnl Delt Eddington method for highly nisotropiclly scttering medi A strongly nisotropic scttering phse function (g > 0.7) would require mny discrete ordintes to suffice the normliztion condition of the phse function in Eq. (4). Insted of using mny discrete ordintes (e.g. S 16 method with 288 ordintes) tht increses the computtionl cost we employ Delt Eddington (DE) method. The DE method requires only, for exmple, n S 6 pproximtion with 48 ordintes for medi with g 0.8. In doing so, the scttering coefficient l s in both ERTs is replced with l DE s ¼ð1 g 3 Þl s : ð26þ The Henyey Greenstein function is lso expnded into series of Legendre polynomils up to second order nd is replced in both ERTs [Eqs. (1) nd (7)]. More detils of the DE method cn be found in [75 77]. 3. Inverse source problem The inverse source problem derives the unknown sptil source distribution S m (r), which is proportionl to the fluorophore bsorption coefficient l x!m ðrþ in Eq. (6), from the mesured fluorescent light t wvelength k m escping the tissue surfce. Since the fluorophore bsorption coefficient is liner function of the fluorophore concentrtion inside the tissue reconstructed mp of the fluorophore bsorption cn directly be trnslted into mp of the concentrtion distribution of the biochemicl probe. All other intrinsic tissue properties, i.e. the opticl prmeters l x s ðrþ; lm s ðrþ; lx! ðrþ; nd l m ðrþ, re typiclly given, for

9 exmple, from previously performed reconstruction. Reconstruction lgorithms in opticl tomogrphy for bsorption nd scttering coefficients re widely vilble nd hve been documented [53,54,56,78]. Furthermore, the quntum yield g of the fluorophore is lso known nd usully provided by the mnufcturer Nonliner optimiztion The sptil distribution of the fluorophore bsorption, l x!m ðrþ, is reconstructed by pplying nonliner optimiztion technique to n objective function U tht is n explicit function of l x!m ðrþ: Ul x!m ðrþ :¼ uð/ m l x!m ðrþþ : ð27þ The function u(/ m (r d )) describes the difference between the mesured, M m (r d ), nd predicted dt, / m (r d ) [see Eq. (8)], t detector positions r d for ll D source detector pirs uð/ m ðr d ÞÞ ¼ 1 2 X D d¼1 / m ðr d Þ M m 2 ðr d Þ : ð28þ r m ðr d Þ The quntity r m exhibits the confidence we hve in the ccurcy of our mesurement dt. It is lrgely influenced by the system noise of the experimentl set-up tht mesures the light intensities t the tissue boundry. The min contribution to r m for n individul source detector pir is the shot noise of lser diode (source) nd chrged-coupled-devices (CCD) cmer (detector). Shot noise is fundmentl property of the quntum nture of light nd vries ccording p to Poisson distribution. Its power is proportionl to the squre root of the mesured signl r m ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M m ðr d Þ. We minimize the objective function by using limited-memory Broyden Fletcher Goldfrb Shnno (BFGS) technique tht belongs to the clss of qusi-newton methods [56]. This technique updtes itertively n initil guess of the fluorophore bsorption long serch direction. Once the minimum is found, the finl result is the unknown distribution of the fluorophore bsorption coefficients. The updting procedure cn be formulted s [79,80] ½l x!m Š iþ1 ¼½l x!m Š i þ i u i ; ð29þ where ½l x!m Š i is vector t itertion step i contining set of fluorophore bsorption coefficients ½l x!m Š mnl Š iþ1 is obtined. The vector u i is serch for ll sptil grid points (mnl) from which the new vector ½l x!m direction. The prmeter i is the step length in the direction u i. After the updte in Eq. (29) ws performed new serch direction u i+1 is determined with [81,82] u iþ1 ¼ du iþ1 þ cs þ ky ð30þ dl x!m with the vectors s nd y: s ¼ l x!m iþ1 l x!m i y ¼ du iþ1 du i : ð31þ dl x!m dl x!m The sclrs c nd k re defined by c ¼ 1 þ yt y s T ½dU=dl x!m s T y s T y A.D. Klose et l. / Journl of Computtionl Physics 202 (2005) Š iþ1 þ yt ½dU=dl x!m s T y Š iþ1 ; k ¼ st ½dU=dl x!m s T y Š iþ1 : ð32þ

10 332 A.D. Klose et l. / Journl of Computtionl Physics 202 (2005) At ech itertion step i new derivtive ½dU=dl x!m Š i needs to be clculted in order to determine the serch direction u i. The derivtive clcultion is n essentil prt of the imge reconstruction lgorithm becuse it is computtionlly expensive. We hve developed n djoint differentition method to compute the grdient. The detils of this technique for the ppliction in fluorescence tomogrphy re described in the next section Adjoint differentition The optimiztion method requires clculting the derivtive ½dU=dl x!m Š of the objective function tht is function of vribles ½l x!m Š mnl for ll grid points (mnl). To pproximte the derivtive for single vrible ½l x!m Š mnl t grid point (mnl) we could pply the divided difference for sufficiently smll perturbtion Dl x!m, du U ½lx!m Š mnl þ Dl x!m U ½l x!m Š mnl : ð33þ dl x!m mnl Dl x!m This method, however, is computtionlly too expensive, since for ech perturbtion of ½l x!m Š mnl t grid point (mnl) seprte forwrd clcultion hs to be performed. Tht leds to totl of (M ÆNÆL +1) forwrd clcultions when ll unknown fluorophore bsorption coefficients re considered. Insted of perturbing ech component of ½l x!m Š we employ n djoint method [83,84]. A prticulr implementtion of tht method is the djoint differentition technique [85 94]. We hve lredy pplied this technique to the imge reconstruction problem of intrinsic tissue properties in opticl tomogrphy bsed on the ERT [55]. The djoint differentition method, lso termed s computtionl or lgorithmic differentition in the reverse direction, is directly pplied to the existing numericl code of the forwrd model. The min dvntge of this pproch is tht the grdient cn be clculted ccording to simple rules t level of single steps in the forwrd code insted of solving n djoint eqution of rditive trnsfer tht constitutes n entire new numericl problem [83,93,95,96]. Furthermore, the djoint differentition technique computes the derivtive in period of time equivlent to only one to three forwrd clcultions. The forwrd model of light propgtion t wvelength k m, tht provides solution to the ERT [Eq. (7)] nd vlue of the objective function for given fluorophore bsorption distribution, is decomposed into sequence of single differentible functions. This sequence of functions is built up in the forwrd direction s the solution of the forwrd model is computed. Applying systemticlly the chin rule of differentition to ech single function in the reverse direction numericl vlue of the derivtive of the objective function with respect to the fluorophore bsorption distribution is obtined Decomposition of the forwrd model The djoint differentition pproch exploits the lgorithmic structure of the numericl forwrd model given by the source itertion scheme, see Eq. (23). The itertive build-up of the forwrd model clcultes sequence (W 0,W 1,...,W z,...,w Z ) of rdince distributions tht converges towrds solution w m of the ERT in Eq. (7). The rdince vector W z+1 t the trnsport sweep z + 1 is function of the rdince vector W z t the previous trnsport sweep z nd of given fluorophore bsorption ½l x!m Š. This function W zþ1 ðw z ; ½l x!m ŠÞ is represented by the itertion rule in Eq. (25) for n individul vector element [W z +1 ] mnlk. The finl vlue W Z of the lst trnsport sweep is used for the determintion of the function P 2 uðw Z Þ¼ 1 X D w k w Z ðdþk M m ðr d Þ k 2 M m ; ð34þ ðr d Þ d¼1

11 where the subscript d pertins to detector position r d on the Crtesin mesh with grid point (d) =(mnl). Thus, the objective function U, s defined in Eq. (27), is composition of the function u nd Z functions W z of ll trnsport sweeps: U ¼ u W Z W Z 1 W zþ1 W z W 1 W 0 : ð35þ The opertion is defined s composite function W zþ1 W z ¼ W zþ1 ðw z ; ½l x!m ŠÞ W z ð½l x!m ŠÞ :¼ W zþ1 ðw z ð½l x!m ŠÞ; ½l x!m ŠÞ: ð36þ Strting with n initil input vector ½l x!m Š vlue of the objective function U cn be obtined. Tht lso defines the forwrd direction of the forwrd lgorithm Algorithmic differentition of the forwrd model The derivtive of the objective function U with respect to the input prmeter of the forwrd lgorithm, i.e. the fluorophore bsorption vector ½l x!m Š, is given by the vector du ¼ XZ oðu W Z W Z 1 W z Þ : ð37þ dl x!m ol x!m z¼0 Ech component of the bove sum cn be obtined by pplying the chin rule of differentition long the forwrd direction of the forwrd lgorithm oðu W Z W Z 1 W z Þ ¼ ou ol x!m ow Z ow Z ow Z 1 owzþ1 ow Z 1 Z 2 ow ow z ow z ol x!m : ð38þ Eq. (38) consists minly of repeted mtrix mtrix multiplictions with ow zþ1 =ow z nd ow z =ol x!m s mtrices. The derivtive ½ou=ol x!m Š is not included in the sum since u is not n explicit function of ½l x!m Š nd thus its derivtive vnishes. We will now provide n lgorithmic procedure tht derives the grdient of U by pplying systemticlly the chin rule of differentition long the reverse direction of the forwrd lgorithm in order to void repeted mtrix mtrix multiplictions. The lgorithmic differentition in the reverse direction requires only mtrix vector multiplictions tht leds to less computtionl effort. This procedure cn be chieved by tking the trnspose, or djoint, of ech component in (38) oðu W Z W Z 1 W z T Þ ¼ owz ol x!m T ow zþ1 T owz 1 T ow Z ow Z 2 ow Z 1 T T ou : ð39þ ow Z ol x!m ow z Thus, we obtin with Eqs. (37) nd (39) T du ¼ XZ ow z T T ou dl x!m ol x!m ow z : ð40þ z¼0 A.D. Klose et l. / Journl of Computtionl Physics 202 (2005) Additionlly, the reltion between subsequent steps, ½oU=oW z Š T nd ½oU=oW zþ1 Š T, in Eq. (40) is for ll z < Z, T ou ow z ¼ owzþ1 T T ou ow z ; ð41þ ow zþ1 nd for z = Z, T ou ¼ ou T : ð42þ ow Z ow Z

12 334 A.D. Klose et l. / Journl of Computtionl Physics 202 (2005) By providing the derivtives ½oU=oW z Š; ow z =ol x!m ; nd ow zþ1 =ow z for ech trnsport sweep we re ble to clculte the djoint derivtive vector ½dU=dl x!m Š T in Eq. (40) by stepping through the forwrd lgorithm in the reverse direction. Strting with the lst trnsport sweep we compute the prtil derivtive in Eq. (42) by differentiting Eq. (34) nd obtin for ech component t detector position r d : P ou ¼ ou w k w Z ðdþk M m ðr d Þ k ¼ M m w k : ð43þ ðr d Þ ow Z ðdþk ow Z ðdþk Next, the derivtive of w z mnlk ow z mnlk o½l x!m with respect to ½lx!m Š mnl in Eq. (40) is given by ð1=4pþg/ x mnl ðn k =MxÞþðg k =MyÞþðl k =MzÞþ½l m Š mnl þ½lm s Š : ð44þ mnl ¼ Š mnl Finlly, the mtrix components in Eq. (41) re given by the prtil derivtives ow zþ1 mnlk ow z ¼ ½l m s Š mnl w k 0p kk 0d mm 0 nn 0 ll 0 þ n k ow zþ1 m 1nlk m 0 n 0 l 0 k Mx ow z þ g k ow zþ1 mn 1lk 0 m 0 n 0 l 0 k My ow z þ l k ow zþ1 mnl 1k 0 m 0 n 0 l 0 k Mz ow z 0 m 0 n 0 l 0 k 0 n k Mx þ g k My þ l k Mz þ½lm s Š mnl þ½lm Š mnl with d mm 0 nn 0 ll 0 ¼ d mm 0d nn 0d ll 0 nd d 0 ¼ 1 if 0 ¼ 0 if 0 6¼ : We lso pproximte the prtil derivtives on the right-hnd-side in Eq. (45) with ow zþ1 m 1nlk ow z m 0 n 0 l 0 k 0 :¼ d ðm 1Þm 0 nn 0 ll 0 kk 0 ow zþ1 mn 1lk ow z m 0 n 0 l 0 k 0 :¼ d mm 0 ðn 1Þn 0 ll 0 kk 0 ow zþ1 mnl 1 ow z m 0 n 0 l 0 k 0 :¼ d mm 0 nn 0 ðl 1Þl 0 kk 0 since w zþ1 m 1nlk ; wzþ1 mn 1lk ; nd wzþ1 mnl 1k re slowly vrying functions of wz m 0 n 0 l 0 k 0 ð45þ ð46þ for sufficiently smll Dx, Dy, nd Dz. We found tht in generl the error introduced by this pproximtion is negligible. The differentition of the objective function long the reverse direction of the forwrd lgorithm constitutes computtionl dvntge over the differentition long the forwrd direction. In the reverse direction mtrix lwys opertes on vector [see Eq. (39)]. For exmple, multipliction of squred mtrix with size of q 2 elements nd vector with q elements involves totl of 2q 2 q opertions (q multiplictions nd q 1 summtions for ech new vector element). However, by clculting the derivtive in the forwrd direction [see Eq. (38)] mtrix opertes on mtrix. A mtrix mtrix multipliction requires totl of 2q 3 q 2 opertions (q multiplictions nd q 1 summtions for ech new mtrix element). Since mtrix vector multiplictions consist of q times fewer opertions thn mtrix mtrix multiplictions computing the derivtive long the reverse direction involves less computtionl effort nd is q times fster. 4. Inverse problem of totl bsorption Besides l x!m ðrþ of the fluorophore the totl bsorption coefficient l x ðrþ ¼lx! ðrþþl x!m ðrþ t the excittion wvelength k x cn be reconstructed to provide informtion bout the intrinsic tissue nd fluorophore bsorption. In this cse we define n objective function, Uðl x ðrþþ ¼ uð/x ðl x ðrþþþ, similr to Eq. (28) s

13 uð/ x ðr d ÞÞ ¼ 1 2 X D d¼1 ð/ x ðr d Þ M x ðr d ÞÞ 2 M x : ð47þ ðr d Þ The predicted detector redings / x (r d ) re clculted by solving the ERT in Eq. (1). The mesurement dt M x (r d ) re given t the excittion wvelength k x. The objective function Uðl x ðrþþ is itertively minimized by using the nonliner optimiztion method to find the unknown distribution of bsorption coefficients l x ðrþ. Agin, we employ the djoint differentition technique pplied to the forwrd lgorithm of Eq. (1) to provide the derivtive vector of U with respect to ½l x Š. The derivtive vector ½dU=dl x Š is computed in the sme mnner s explined in Section 3.2. Eq. (40) is modified nd we obtin T du ¼ XZ ow zt T ou dl x ol x z¼0 ow z : ð48þ Agin, Eq. (41) s pplied to ERT I is used for clculting the subsequent step ½oU=oW z Š T from ½oU=oW zþ1 Š T in the reverse direction. A mtrix element of ow z =ol x in Eq. (48) is S x ow z mnlk þ½lx s Š P mnl mnlk k o½l x Š ¼ 0 mnl A.D. Klose et l. / Journl of Computtionl Physics 202 (2005) w k 0p kk 0w z 1 mnlk 0 n k Mx þ g k My þ l k Mz þ½lx Š mnl þ½lx s Š mnl n k Mx wz m 1nlk þ g k My wz mn 1lk þ l k Mz wz mnl 1k 2 2 : ð49þ n k k k Mx My Mz þ½lx Š mnl þ½lx s Š mnl 5. Experimentl results In the following section we show the first experimentl results of recovering the sptil distribution of fluorescent sources in tissue-like phntom using the trnsport-theory bsed reconstruction code. Experimentl dt were recorded t both the excittion nd emission wvelengths in order to reconstruct the totl bsorption, l x ¼ lx! þ l x!m, nd fluorophore bsorption, l x!m. The instrumentl design we used for cquiring the experimentl dt re described in more detil by Grves et l. [97]. The given tissue-like phntom hd size of 4 cm 4 cm 1.3 cm with the opticl prmeters l x s ¼ 30 cm 1 nd l x! ¼ 0:4 cm 1 t the excittion wvelength, nd l m s ¼ 30 cm 1 nd l m ¼ 0:4 cm 1 t the emission wvelength, see Fig. 3. The refrctive index of the phntom ws n = The nisotropy fctor ws ssumed to be g = 0.8. The highly forwrd-peked scttering phse function nd the use of n S 6 method with only 48 ordintes required the DE method to suffice the normliztion condition [Eq. (4)]. Hence, the scttering coefficient for both wvelengths ws rescled with reltion (26) tht yielded Fig. 3. Scttering phntom with three embedded tubes. Tubes I nd II contined fluorochrome with l x!m medium. Tube III is purely bsorbing nd hd higher bsorption thn the bckground medium. < l x of bckground

14 336 A.D. Klose et l. / Journl of Computtionl Physics 202 (2005) l DE s ¼ 14:64 cm 1. All forwrd clcultions nd reconstructions were performed on grid with sptil seprtion Dx = Dy = Dz = 0.1 cm. Two fluorescent tubes (I nd II) with dimeter of 0.2 cm were embedded inside the medium t depths d I = 0.55 cm nd d II = 0.55 cm, mesured from the top plne of the phntom (d = 0 cm). Both tubes contined fluorochrome (Cy5.5 Amershm Phrmci Biotech, NJ, USA) tht pertins to the group of crbocynine dyes [12]. This fluorescent dye hd n extinction coefficient = 250,000 M 1 cm 1 nd quntum yield g = The concentrtion inside the tubes ws c = M yielding fluorophore bsorption l x!m ¼ 0:05 cm 1. The intrinsic bsorption ws l x! ¼ 0cm 1 nd sme scttering properties s bckground medium were present. Additionlly, n bsorbing tube (III) with dimeter of 0.3 cm nd no fluorescent properties ws included t depth d III = 0.3 cm. This tube hd n incresed bsorption coefficient with respect to the bckground medium. Thirty-two source fibers were plced on the bottom plne of the phntom (d = 1.3 cm) on n re of cm 2, nd 150 eqully distributed detector points were on the top plne (d = 0 cm) on n re of cm 2. The source fibers illuminted the phntom t the excittion wvelength k x = 675 nm. The light propgtion model took the prtilly collimted light source with n perture p/3 into ccount by ssigning non-zero source power density S(X k ) > 0 to only four ordintes tht pointed inside the medium. The escping light t the detection plne ws recorded t the excittion nd fluorescent wvelength (k m = 694 nm) using wvelength selective filters nd CCD cmer. We obtined tomogrphicl dt set with D = source detector pirs. In the next sections we show the reconstructed fluorophore bsorption distribution nd the totl bsorption distribution by employing mesurement dt sets t either the fluorescence wvelength, the excittion wvelength, or t both wvelengths Fluorophore bsorption reconstruction using mesurement dt t fluorescent wvelength k m Given the experimentl set-up we independently mesured for 32 sources the fluorescent light tht escped the top plne of the medium. The mesured dt becme input to the imge reconstruction lgorithm. The optimiztion process strted from n initil guess ½l x!m Š 0 ¼ 0cm 1. The optimiztion ws stopped when the reltive difference j(u i +1 U i )/U i j of the objective function between two consecutive itertion steps becme smller thn It took 28 itertions with 28 djoint derivtive nd 33 forwrd clcultions for ech source position. One complete forwrd clcultion comprised solving the ERT for ll 32 source positions. Hence, one complete djoint derivtive clcultion involved the solution of the ERT for ll source positions of the forwrd code. Therefore, the reconstruction time ws pproximtely 25 h on Linux Beowulf Cluster consisting of 10 Intel Pentium III Xeon processors with 2.4 GHz clock rte. The reconstructed fluorophore distribution l x!m ðrþ is shown in Fig. 4. In this cse it ws not possible to derive quntittive informtion bout the fluorophore distribution due to the unknown source strength t the excittion wvelength k x. Therefore, we disply in Fig. 4 reltive fluorophore bsorption coefficient distribution with rbitrry units. The reltive fluorophore bsorption distribution is normlized to its mximum vlue. Thus, the imge scle is with mximum fluorophore bsorption t 1 nd no fluorophore bsorption t Fluorophore bsorption reconstruction using mesurement dt t fluorescent wvelength k m nd excittion wvelength k x Absolute vlues of the mesured light intensity t the fluorescent wvelength k m need to be ccessible for given excittion source strength t k x to reconstruct the bsolute fluorophore bsorption distribu-

15 A.D. Klose et l. / Journl of Computtionl Physics 202 (2005) Fig. 4. Reconstructed reltive fluorophore bsorption l x!m in different depths d mesured from top plne (d = 0 cm). The fluorophore distribution is normlized to its mximum vlue. The excittion sources were locted t d = 1.3 cm nd the detector plne for mesuring the fluorescent light ws locted t d = 0 cm. Both fluorescent tubes (Tube I nd II) cn be clerly seen. tion. However, the experimentl sitution does often not llow to determine the bsolute fluorescent light intensity, e.g. due to unknown excittion source strengths, unknown filter ttenution nd fiber coupling coefficients, nd unknown losses t the tissue ir interfces [44,45,48,98,99]. These quntities cn be described by re-scling fctor tht depends on the prticulr experimentl set-up. This re-scling fctor cn be obtined by relting mesured light intensities ~M x 0 t the excittion wvelength kx for the homogeneous phntom with known opticl properties without ny tubes present to numericlly predicted light intensities / x 0 for the sme medium. The mesured fluence ~M m t the emission wvelength k m for the phntom with the unknown fluorophore distribution is re-scled by tht fctor which is defined s the rtio / x 0 = ~M x 0. Hence, the re-scled mesurement dt Mm t detector positions r d re given by the reltion M m M ðr d Þ¼ ~ m ðr d Þ/ x 0 ðr dþ ~M x 0 ðr ; ð50þ dþ

16 338 A.D. Klose et l. / Journl of Computtionl Physics 202 (2005) nd re independent of ll experimentl design-dependent light intensity losses. Subsequently, the re-scled mesurement dt M m (r d ) becme input to the objective function (28) of the imge reconstruction lgorithm. A similr re-scling pproch is described elsewhere [44,45,48]. The reconstructed fluorophore bsorption l x!m is shown in Fig. 5. The imges show the positions of the fluorescent tubes I nd II. The mximum vlue of the reconstructed fluorophore bsorption distribution is l x!m ¼ 0:032 cm 1. The fluorophore bsorption l x!m ws reconstructed strting from homogeneous initil guess ½l x!m Š 0 ¼ 0cm 1. The reconstruction process ws terminted when the reltive difference j(u i +1 U i )/U i j of subsequent itertion steps of the optimiztion process ws smller thn The optimiztion technique needed 23 itertions tht included totl of 35 forwrd clcultions nd 23 djoint derivtive clcultions. The computtion time ws pproximtely 21 h Totl bsorption reconstruction using mesurement dt t excittion wvelength k x The sptil distribution of the totl bsorption coefficient l x ¼ lx! þ l x!m ws reconstructed given the experimentl mesurement dt t excittion wvelength k x. Agin, we needed to re-scle the mesurement dt ~M x ðr d Þ due to the unknown light intensity losses of the experimentl design: Fig. 5. Reconstructed fluorophore bsorption l x!m in different depths d mesured from top plne (d = 0 cm). The excittion sources were locted t d = 1.3 cm. The excittion nd fluorescent light intensities were mesured t the detector positions in the top plne t d = 0 cm. Both fluorescent tubes (Tube I nd II) cn be clerly seen.

17 A.D. Klose et l. / Journl of Computtionl Physics 202 (2005) M x M ðr d Þ¼ ~ x ðr d Þ/ x 0 ðr dþ ~M x 0 ðr : ð51þ dþ The re-scled mesurement dt M x (r d ) becme input to the objective function [see lso Eq. (47)] of the reconstruction lgorithm. The imge reconstruction strted from n initil guess ½l x Š0 ¼ 0:4 cm 1. The stop criterion ws set to j(u i +1 U i )/U i j <10 2. The reconstruction process ws finished fter 58 itertions with totl of 58 forwrd nd 58 djoint derivtive clcultions. The computtion time ws pproximtely 40 h. All reconstructed imges for different depths mesured from the top plne of the phntom re shown in Fig. 6. All three tubes (I, II, nd III) cn be seen in the imges. The centrl tube (III) is the highly bsorbing tube tht contins no fluorophore nd hs higher bsorption coefficient thn the bckground medium. Its mximum vlue of bsorption is l x ¼ 1:29 cm 1 t depth d = 0.4 cm. Both fluorescent tubes, I nd II, hve n bsorption coefficient l x < 0:1 cm 1 smller thn the bckground medium in tomogrphic plne d = 0.6 cm. Fig. 6. Reconstructed totl bsorption l x in different depths d mesured from top plne (d = 0 cm). The bckground medium hs n bsorption of l x ¼ 0:4 cm 1. The excittion sources were locted t d = 1.3 cm nd the detectors were locted t d = 0 cm. Both fluorescent tubes (I nd II) cn be seen s drk res with low bsorption l x < 0:1 cm 1 due to the fluorophore. The purely bsorbing tube (III) with l x > 1cm 1 is present in the center s white elongted re.

18 340 A.D. Klose et l. / Journl of Computtionl Physics 202 (2005) Discussion nd summry We hve developed reconstruction lgorithm for recovering the sptil distribution of fluorescent sources in highly scttering medi. The code provides tomogrphic imges of the fluorophore bsorption coefficient tht cn be used in moleculr imging of biologicl tissue. The presented reconstruction lgorithm includes two originl spects tht hve not been used before in fluorescence tomogrphy. First, the forwrd model for light propgtion is bsed on the ERT nd not the widely employed diffusion pproximtion. By not relying on the diffusion pproximtion, this lgorithm hs the potentil to provide more ccurte solutions in cses where the diffusion pproximtion fils, such s in medi with smll geometries, medi with high bsorption coefficients, or smll source detector seprtions. Hving provided n ERT-bsed reconstruction code we re now in the position to quntify the dvntges in future studies. Second, n djoint differentition method for fluorescence tomogrphy hs been developed. This method is used to compute the derivtive of n objective function to provide serch directions for the nonliner optimiztion technique. By pplying the chin rule of differentition to ll trnsport sweeps of the forwrd lgorithm the derivtive cn be built up in n itertive mnner. The chin rule cn be executed in two different wys, either in the forwrd direction or reverse direction of the numericl code of the light propgtion model. By pplying the chin rule of differentition in the forwrd direction mtrix mtrix multiplictions need to be performed for ll trnsport sweeps s cn be seen in Eq. (38). On the other hnd, in the reverse or djoint direction, s shown by Eqs. (40) nd (41), only mtrix vector multiplictions re necessry tht considerbly decreses the mount of computtionl opertions. The djoint differentition technique hs lso the dvntge tht the existing code of the forwrd model cn be utilized to compute the derivtive. A numericl implementtion of n djoint ERT s shown, for exmple, by Ustinov et l. [95] is not required. In generl, the tomogrphic imges clculted from experimentl mesurement dt show qulittively ccurte distribution of the fluorophore bsorption coefficient [Figs. 4 nd 5] nd the totl bsorption coefficient [Fig. 6]. The fluorescent tubes I nd II cn clerly be resolved with the originl dimeters of 0.2 cm in tomogrphic plne for specific depth d. The depth resolution is not s good since detectors were only plced on the top plne opposite to the sources. In order to obtin better depth resolution the medium needs to be illuminted nd the escping light needs to be mesured from different views. We further observe tht the resolution of the fluorescent imges is higher thn the imges of the totl bsorption. We still notice imge rtifcts close to the detector plne (d 0.1 cm) nd to the source plne (d 1.3 cm) s cn be seen, for exmple, in Fig. 5. These rtifcts re most likely cused by the high sensitivity of the objective function with respect to chnges in the fluorophore bsorption coefficient in the vicinity of source nd detector points. We used two different types of mesurement dt for reconstructing the fluorophore bsorption distribution. First, only mesurement dt t the emission wvelength were employed, nd second, mesurement dt t the emission nd excittion wvelengths were used. In the first cse, the imge reconstruction method ws ble to reconstruct qulittively the fluorophore bsorption distribution. The tomogrphic imges in Fig. 4 show both fluorescent tubes with significntly higher light emission thn the bckground medium. However, no quntittive informtion, i.e. the bsolute fluorophore bsorption coefficient, could be retrieved becuse of unknown excittion source strengths nd unknown intensity losses t the tissue ir interfce. Quntittive informtion of l x!m ðrþ my still be obtined by using set of reference solutions with well-known concentrtions of Cy5.5. Furthermore, our reconstruction results show tht ll numericl pproximtions nd ssumptions mde within the light propgtion model (step method, S 6 pproximtion, DE method) nd within the optimiztion process (using pproximte prtil derivtives for the djoint differentition, uncertinty of intrinsic opticl prmeters) hve reltively little impct on the qulittive imge reconstruction. On the other hnd, the reconstructed imges re noisier when compred to imges in Fig. 5 tht uses re-scled mesurement dt. Tht is

19 mostly due to the unknown intensity losses which is dependent on ech individul source nd detector point position. In the second cse, where mesurements t the emission nd excittion wvelengths re used, the uncertinties of unknown intensity losses cn be overcome. This is chieved by re-scling the mesurements t k m prior to reconstruction with mesurements t k x tken from the homogeneous medium. In this wy we could reconstruct imges with little noise s shown in Fig. 5. Besides tht, we were lso ble to determine the bsolute fluorophore bsorption coefficient. Tht enbles us to derive the fluorophore nd moleculr probe concentrtion for future smll niml imging system without employing reference solutions of Cy5.5. Another importnt chrcteristic of using re-scled mesurement dt re tht it my correct shortcomings of the used light propgtion model. For exmple, the reconstruction method my be less prone to errors introduced by corse sptil Crtesin grid or errors cused by ry-effects nd flse scttering [69,100]. In contrst, reconstruction code tht uses re-scled mesurement dt my not be ble to model correctly light propgtion in physicl domins where the sme code might work when unprocessed dt re used. Besides the distribution of the fluorophore bsorption l x!m, we cn lso reconstruct the totl bsorption distribution l x ¼ lx! þ l x!m s illustrted in Fig. 6 tht shows ll three tubes. This exmple is of prticulr interest, becuse it dels with non-diffusive regime. In Tube III the bsorption coefficient ðl x > 1cm 1 Þ is pproximtely six times smller thn the reduced scttering coefficient l 0x s ¼ð1 gþl x s ¼ 6cm 1 of the bckground medium. Hielscher et l. [101] hve shown tht the diffusion model inccurtely describes light propgtion in this domin. Despite the ill-posedness of the inverse source problem we still chieve quite ccurte reconstruction results. In our exmple we hve n underdetermined inverse problem with 23,534 unknowns but only 4800 mesurement dt. Tht we nevertheless get resonble results my be explined by two mjor spects. First, only 1.21% of the phntomõs volume contined fluorophores with fluorophore bsorption coefficient l x!m l x!m lrger thn zero. The remining phntom volume (98.79%) hd no fluorescent properties with ¼ 0cm 1. Since we strted the optimiztion process from n initil guess ½l x!m Š 0 ¼ 0cm 1 we ssigned to most of the unknown fluorophore bsorption coefficients the correct vlue. This is resonble pproch, s this mimics the prcticl sitution. In prctice, n dministered fluorescent moleculr probe will minly ccumulte in the side of interest, while the rest of the tissue will not show incresed levels of fluorophore concentrtion. Therefore, n initil guess of ½l x!m Š 0 ¼ 0cm 1 for the entire medium will provide n initil guess close to the true solution, which will often result in stisfctory convergence of the optimiztion process. Second, due to the source detector configurtion of the experimentl set-up we hve limited field of view (FOV). Consequently, the objective function is most sensitive to fluorophore bsorption distribution within phntom volume tht is enclosed by the source re (2.16 cm 2 ) nd detector re (4.75 cm 2 ) on the bottom nd top plne of the phntom. The enclosed volume with pproximtely 7000 unknown fluorophore bsorption coefficients only ccounts for 30% of the entire phntom volume. Fluorophore bsorption coefficients l x!m A.D. Klose et l. / Journl of Computtionl Physics 202 (2005) outside of the trgeted volume do not or only little contribute to the overll sensitivity du=dl x!m of the objective function U. Hence, the inverse source problem becomes less underdetermined. The presence of mesurement noise nd model error prohibit the uniqueness of the inverse problem. A comprehensive uncertinty nlysis, tht describes the impct of the mesurement error on the imge error, hs not been performed yet, neither for trnsport-theory-bsed reconstruction methods nor for the widely pplied diffusion-theory-bsed reconstruction methods. In the future, other pproches could be considered to decrese the ill-posedness. For exmple, some reserchers employ different regulriztion methods such s Tikhonov regulriztion; or the inverse source problem cn lso be formulted within Byesin frmework tht llows the incorportion of prior knowledge nd uses ll vilble informtion bout the problem model (forwrd nd inverse model) within the objective function [48].