J. Ripoll and V. Ntziachristos Vol. 20, No. 6 /June 2003/J. Opt. Soc. Am. A 1103

Iterative boundary method for diffuse opticaltomography

Jorge Ripoll

Institute for Electronic Structure and Laser, Foundation for Research and Technology–Hellas, P.O. Box 1527,71110 Heraklion, Greece

Vasilis Ntziachristos

Laboratory for Bio-Optics and Molecular Imaging, Center for Molecular Imaging Research, Massachusetts GeneralHospital and Harvard Medical School, Building 149, 13th Street 5406, Charlestown,

Massachusetts 02129-2060

Received November 14, 2002; revised manuscript received January 28, 2003; accepted January 30, 2003

The recent application of tomographic methods to three-dimensional imaging through tissue by use of lightoften requires modeling of geometrically complex diffuse–nondiffuse boundaries at the tissue–air interface.We have recently investigated analytical methods to model complex boundaries by means of the Kirchhoff ap-proximation. We generalize this approach using an analytical approximation, the N-order diffuse-reflectionboundary method, which considers higher orders of interaction between surface elements in an iterative man-ner. We present the general performance of the method and demonstrate that it can improve the accuracy inmodeling complex boundaries compared with the Kirchhoff approximation in the cases of small diffuse volumesor low absorption. Our observations are also contrasted with exact solutions. We furthermore investigateoptimal implementation parameters and show that a second-order approximation is appropriate for mostin vivo investigations. © 2003 Optical Society of America

OCIS codes: 110.6960, 170.3010, 170.5280, 170.3660, 170.5270.

1. INTRODUCTIONThe study of light transport through highly scatteringmedia such as living tissue has been the focus of recentresearch, mainly owing to its application in medicaldiagnosis1–8 and recently for its ability to probe molecularfunction in vivo.9 Light penetration of several centime-ters in tissue is possible in the 650–850-nm range owingto low tissue absorption in this spectral region.10 Lately,rigorous mathematical modeling of light propagation intissue (see Ref. 11 for a review), combined with techno-logical advancements in photon sources and detectiontechniques, has allowed the development of diffuse opticaltomography,12 (DOT), an imaging technique that three-dimensionally resolves and quantifies optical propertiesin dense media such as tissues. DOT has been recentlyused to elucidate hemoglobin variations for imaging bloodpooling,13 the accumulation of contrast agents in breasttumors,14 arthritic joint imaging,15,16 and functional ac-tivity in the brain17 and muscle.18 More recently similartomographic principles combined with appropriate fluoro-phores of molecular specificity, have facilitated the devel-opment of fluorescence molecular tomography, an imagingmethod that allows elucidation of molecular signatures orpathways in intact tissues.9,19

A typical tomographic problem for optical imaging ap-plication requires modeling of photon propagation in thediffuse medium under investigation. Tissue applicationsrequire such calculations in the presence of complexboundaries. Two major computational strategies for gen-erating such solutions are generally available: The first

1084-7529/2003/061103-08$15.00 ©

employs numerical methods such as finite differences orelements20 that are based on volume discretization, andthe second uses the boundary-element method21–23

(BEM), which solves numerically the surface integralequations for diffusive waves by discretizing the surface23

(in Ref. 23 the BEM method was implemented with anequivalent method termed the extinction theoremmethod). More recently we investigated the Kirchhoffapproximation24 (KA) as an alternative to modeling arbi-trary boundaries. This method, also known in physicaloptics as the tangent-plane method,25–28 has been exten-sively applied for modeling scattering of electromagneticwaves from random rough surfaces, and its limits of va-lidity have been previously established.29 The KAmethod uses the angular spectrum representation of thepropagating wave and employs the reflection coefficientsto calculate the scattered wave inside any arbitrary geom-etry. In imaging applications the KA method has beenshown to be highly efficient in the computation times re-quired for calculating the forward model.30 However, ithas also been demonstrated that the accuracy of themethod deteriorates as the boundary becomes more com-plex, especially for smaller volumes or in the presence oflow absorption.24 A major issue associated with the KAmethod’s limitations is an effect known as the shadowingeffect, which is evident at areas that, owing to the com-plex boundary, do not have direct visibility with the lightsource (shadow regions). The shadowing effect origi-nates from the fact that the KA is not capable of takinginto consideration high-order reflections at the interface,

2003 Optical Society of America

1104 J. Opt. Soc. Am. A/Vol. 20, No. 6 /June 2003 J. Ripoll and V. Ntziachristos

and therefore the calculation of the scattered wave at theshadow regions yields extremely high errors (.100%).Furthermore, for highly reflective boundaries or for situ-ations in which high spatial-frequency components of thescattered diffuse wave are generally detectable (as in thecase of small volumes, low absorption, and short source–detector distances from a nonplanar interface) the accu-racy of the KA may diminish. We have developed an it-erative analytical (explicit) method that overcomes theproblems of the KA while maintaining fast computationalcharacteristics. The N-order diffuse-reflection boundarymethod (DRBM) is a calculation approach that balancesthe computation performance and accuracy achieved be-tween the iterative solution to the BEM31,32 and the KA.The DRBM is here applied to the two-dimensional case indiffuse optical tomography of complex geometries small insize (,3 cm) with both low and high absorption coeffi-cients. A method equivalent to the BEM with constantsurface elements22 that yields accurate numerical solu-tions to the diffusion approximation is used for generat-ing forward solutions. This method was presented in de-tail elsewhere in the context of diffusive waves23 and willnot be introduced here.

This paper is organized as follows. In Section 2 we re-call the basic integral equations to be solved for the for-ward problem, and the standard KA method is presentedas in Ref. 24. The DRBM is then introduced in Section 3.Section 4 compares reconstruction results in two dimen-sions obtained by use of the DRBM and the BEM. Fi-nally, the findings are discussed in the last section.

2. INTEGRAL EQUATIONS AND THEKIRCHHOFF APPROXIMATIONThe integral equations related to tomographic optical im-aging of dense media are explained in detail in Ref. 24.The most relevant expressions are briefly presented here.Let us consider a general problem in diffuse tomographyin which we have a diffusive volume V as depicted inFig. 1, delimited by surface S, which contains the abnor-mality we want to image. This diffusive medium is char-

Fig. 1. Scattering geometry.

acterized by its absorption coefficient ma , the diffusioncoefficient D, and the refractive index n in and is sur-rounded by a nondiffusive medium of refractive indexnout . If in such a medium the light source is modulatedat a frequency v, the average intensity U(r, t)5 U(r)exp(2ivt) represents a diffuse photon densitywave33 and obeys the Helmholtz equation with a wavenumber k 5 (2ma /D 1 iv/cD)1/2, where c is the speed oflight in the medium. By use of the expression for theGreen function in a infinite homogeneous medium,

g~kur 2 r8u! 5exp~ikur 2 r8u!

Dur 2 r8u, (1)

the solution for arbitrary geometries can be expressed interms of a surface integral by means of Green’s theorem(see Ref. 24 for a detailed derivation) as

G~rs , rd! 5 g~kurs 2 rdu! 21

4pE

SF ]g~kur8 2 rdu!

]n8

11

CndDg~kur8 2 rdu!GG~rs , r8!dS8, (2)

where n8 is the surface normal pointing into the nondif-fusive medium (see Fig. 1) and G is the complete Greenfunction that takes into account the presence of theboundary. In Eq. (2) we have made use of the boundarycondition between the diffusive and the nondiffusivemedium,28,34,35 G(rs , r8)uS 5 2CndDn • ¹G(rs , r8)uS ,r8 P S, where Cnd takes into account the refractive-indexmismatch between both media.34,35 The total intensitymeasured at a detector point rd inside volume V that isdue to a source density distribution F is given in terms ofthe complete Green function G(rs , rd) as

U~rd! 51

4pE

VF~r8!G~r8, rd!dr8, rd P V. (3)

A rigorous solution to Eq. (2) can be found by means ofnumerical methods such as the BEM21,22 by discretizingthe surface and inverting the resulting matrix (see Ref. 23and references therein). This method gives a rigorous so-lution to the forward problem, but it requires large com-putation times, since it involves solving a large system ofequations with a nonsparse matrix. Therefore it is ap-propriate for surfaces with limited number of discretiza-tion points (typically, ,104).

When many forward solutions need to be generated,such as in most inverse schemes, an approximation to Eq.(2) for arbitrary three-dimensional boundaries is pre-ferred for manageable computation times and memory re-quirements. When the curvature of the surface understudy is of the order of the diffuse wavelength the KA canbe applied as a linear approximation to the BEM problemin Fourier space (i.e., a convolution in real space). TheKA method, also known in physical optics as the tangent-plane method, assumes that the surface is replaced ateach point by its tangent plane. Therefore the total av-erage intensity U at any point rp of the surface S is givenby the sum of the homogeneous incident intensity U inc

and that of the wave reflected from the local plane of nor-mal n(rp), Usc. The detailed derivation of the expres-

J. Ripoll and V. Ntziachristos Vol. 20, No. 6 /June 2003/J. Opt. Soc. Am. A 1105

sions is presented in Ref. 24 and will not be repeatedhere. By means of the reflection coefficient for diffusivewaves36 at a diffusive–nondiffusive interface,

Rnd~K! 5iCndDAk2 2 K2 1 1

iCndDAk2 2 K2 2 1, (4)

one finally reaches the KA approximation for the completeGreen function at each local plane rp that defines surfaceS as

GKA~rs , rp! 5 E2`

1`

@1 1 Rnd~K!#g̃~K, Z̄ !exp~iK • R̄!dK,

(5)

where g̃ is the homogeneous Green function in Fourierspace and (R̄, Z̄) are the coordinates of urs 2 rpu with re-spect to the plane defined by n̂(rp): Z̄ 5 (rs 2 rp)• @2n̂(rp)#, R̄ 5 Z̄ 2 (rs 2 rp). Introducing Eq. (5) intoEq. (2) and discretizing surface S into N differential areasDS, we obtain the expression for the KA in diffusive me-dia:

GKA~rs , rd! 5 g~kurs 2 rdu! 21

4p (p51

N F ]g~kurp 2 rdu!

]np

11

CndDg~kurp 2 rdu!G

3 GKA~rs , rp!DS~rp!. (6)

As discussed in Ref. 24, the KA method is a time- andmemory-efficient approximation that is valid at (1) diffusemedia bounded by convex surfaces, (2) geometries thatare larger than the decay length of the diffuse intensity(typically, R . 2.0 cm), and (3) at absorption values thatyield significant photon field attenuation (typically,ma . 0.05 cm21).

3. DIFFUSE-REFLECTION BOUNDARYMETHODA. First-Order Diffuse-Reflection Boundary MethodTo improve the regions of validity of analytical methodswhile maintaining favorable computation characteristics,we propose the use of image sources to model reflectionversus fast Fourier transforms (FFT) at the interface andan iterative approach that, in contrast to the KA approxi-mation, assumes high-order reflections at the boundary.This approach, the N-order DRBM can be envisaged as ahybrid between the KA and the iterative solution to thesurface integral equation (2). The method, in contrast tothe BEM solution, does not involve matrix inversion of anonsparse matrix and maintains the computational andimplementation simplicity of the KA method.

If we assume an approximation to the reflected wave[see Eq. (5)] by means of the method of image sources,37

the boundary values for the first-order DRBM can befound as

GDRBM~1 ! ~rs , rp! 5 @ g~R̄, Z̄ ! 2 g~R̄, Z̄ 1 CndD !#. (7)

The use of image sources avoids time-consuming FFTsthat would be necessary if Eq. (4) were directly consideredand, for the cases so far studied, gives equivalent perfor-mance characteristics.

By use of this expression, the first-order DRBM can becalculated similarly to Eq. (6) as a summation over N lo-cally planar discrete areas DS:

GDRBM~1 ! ~rs , rd! 5 g~kurs 2 rdu! 2

1

4p (p51

N F ]g~kurp 2 rdu!

]np

11

CndDg~kurp 2 rdu!GDS~rp!

3 @ g~R̄, Z̄ ! 2 g~R̄, Z̄ 1 CndD !#. (8)

Equation (8) yields computation times that are signifi-cantly shorter compared with the exact solution, since itis strictly an analytical approach involving no Fouriertransforms or matrix inversion. Computation times ofthis new expression for the first-order DRBM are shownin Fig. 2, compared with the results by use of the KA withFFTs and the rigorous solution with matrix inversion(BEM). From Fig. 2 we infer that the computation timefor the first-order DRBM and the KA grows linearly withthe number of surface elements, as compared with apower-law dependence for the BEM. Owing to the differ-ence in slope of their linearity dependence, the first-orderDRBM is 3 orders of magnitude faster than the KA.When compared with rigorous numerical solutions suchas the BEM, the first-order DRBM proves to be 5–6 or-ders of magnitude faster.

B. N-Order Diffuse-Reflection Boundary MethodTo improve the computation accuracy of the forward cal-culation, we introduce the N-order DRBM. This is amethod based on the scheme presented in Ref. 31 thatsolves iteratively a system of equations G – U 5 U(inc),where the matrix G defines the interaction of one surfacepoint with every other, U(inc) is the vector containing theincident average intensity at each surface point, and U is

Fig. 2. Computation times of the BEM (in minutes) versus theKA with FFTs (in seconds) and the DRBM (0.01 s). Results wereobtained from a Pentium III running at 650 MHz, with 256 MbRAM.

1106 J. Opt. Soc. Am. A/Vol. 20, No. 6 /June 2003 J. Ripoll and V. Ntziachristos

the vector of unknowns defining the surface values of theaverage intensity, i.e., a superposition of U(inc) with pho-ton contributions reflected from the boundary. This it-erative scheme consists of finding the steady-state solu-tion of

d

dtU 1 G – U 5 U~ inc!, (9)

which is approximated by using the Euler method withtime step t as

U~N ! 5 U~N21 ! 2 tG – U~N21 ! 1 tU~ inc!. (10)

Details of the derivation and implications of this ap-proach can be found in Ref. 31, its main idea residing inminimizing the change in the surface values between it-erations. In Eq. (10), if we use as the initial value U(0)

5 U(inc), we obtain the iterative BEM solution31 (conver-gence of the iterative BEM will be shown in Sec. 4 to-gether with that for the DRBM). However, to facilitatetime-efficient computations, we can approximate the ini-tial value by using the first-order DRBM from Eq. (7), i.e.,U(0) 5 GDRBM

(1) . To improve the accuracy of the calcula-tion, we can then iteratively apply the DRBM by rewrit-ing Eq. (2) and assuming that each of the surface points isa detector, similarly to solutions obtained with theBEM.21–23 The surface value at a surface point rk that isdue to a source at rs would be

GDRBM~N ! ~rs , rk! 5 GDRBM

~N21 !~rs , rk! 2 t • g~kurs 2 rku!

1 t1

4p (p51

N F ]g~kurp 2 rku!

]np

11

CndDg~kurp 2 rku!G

3 GDRBM~N21 !~rs , rk!DS~rp!. (11)

Equation (11) is the main expression for the N-orderDRBM and must be applied to each surface point. In thisexpression we must take the principal value of the inte-gral, i.e., exclude those points where the source anddetector coincide, (rk 5 rp), and substitute those withthe corresponding self-induction values. These are wellknown and depend on the medium and surface discretiza-tion area. Their expressions in two and three-dimensions can be found in Refs. 31 and 38, for example.As can be seen from Eq. (11), the factor t plays an impor-tant role and can be considered a relaxation parameter.The rate of convergence of the iteration procedure will de-pend on its value, as will be shown in Section 4.

Once all the surface values for the N order are found bymeans of Eq. (11), we can find the intensity anywhereinside the diffusive volume by using an equivalent ofEq. (2):

GDRBM~N ! ~rs , rd! 5 g~kurs 2 rdu! 2

1

4p (p51

N F ]g~kurp 2 rdu!

]np

11

CndDg~kurp 2 rdu!G

3 GDRMB~N ! ~rs , rp!DS~rp!. (12)

Computation times of Eq. (12) for the first four orders ofthe DRBM are shown in Fig. 3, which shows the rigorousBEM solution from Fig. 2 as a reference. From Fig. 3 wesee that there is a significant increase in computationtime the second-order DRBM is applied, as expected.However, we see that these computation times are still2–3 orders of magnitude faster than a matrix inversionmethod, while maintaining accuracy. Also, a main fea-ture of this figure is that as the number of surface pointsincreases, so does the validity of the first-order DRBM,the higher DRBM orders being necessary only for smallvolumes (i.e., a small number of surface discretizationpoints).

4. METHODSA. SimulationsWe performed two types of two-dimensional study. Thefirst study assumed a diffuse medium with a complexboundary and studied the DRBM performance in recon-structing the absorption coefficient. The second studytested the hypothesis that the use of approximate, easy-to-model boundaries could be used as alternatives to rig-orously calculating solutions in the presence of complexboundaries to image media with arbitrary surfaces.

The simulations considered the two-dimensional casepresented in Fig. 4 and employed 27 continuous intensitysources (i.e., modulation frequency v 5 0) and 27 detec-tors placed with a spacing of 0.5 mm, giving a field of viewof 13 mm. The outer geometry simulated was 2.5 cm inwidth and 1.5 cm in depth, typical in mouse imaging in aplanar imaging configuration. The forward field was cal-culated numerically by using the BEM as a gold standardfor the homogeneous medium and adding two absorbing

Fig. 3. Computation times (in minutes) of the BEM versus fourdifferent orders of the DRBM. Results for the BEM for a largenumber of surface points were found by extrapolation, owing tothe high computational cost. Results were obtained from a Pen-tium III running at 650 MHz, with 256 Mb RAM.

J. Ripoll and V. Ntziachristos Vol. 20, No. 6 /June 2003/J. Opt. Soc. Am. A 1107

objects with Gaussian profile of 1-mm FWHM located at(x, y) 5 (20.5 cm, 0.35 cm) and (x, y) 5 (0, 20.35 cm).Owing to the small sizes of these objects, their contribu-tion to the total wave was well described by the Born ap-proximation. Two diffusive media were considered, at-taining background absorption coefficients of ma 5 0.02cm21 and ma 5 0.2 cm21, respectively. The absorptioncoefficient maximum of the embedded objects was twicethe value of the background in both situations. In allcases, the background and object’s reduced scattering co-efficient was ms8 5 10 cm21, and the index of refractionwas that of water. To obtain the reconstructions, aweight matrix (or sensitivity matrix) was built, and theproduct of this matrix with local perturbations was usedto generate data perturbations, i.e., the scattered inten-sity. Reconstructions were obtained by use of 3 3 104 al-gebraic reconstruction iterations.12

B. Optimization of ConvergenceTo find an expression for the optimum value of the relax-ation parameter t, we performed several reconstructionsfor different background values and different t values andstudied the convergence of the root-mean-square error de-fined as

error~N ! 51

NdH(

i51

Nd F1 2UDRBM

~N !

UBEMG 2J 1/2

, (13)

where Nd is the number of detectors (27 in our case), N isthe order, and UDRBM and UBEM represent the average in-tensity obtained by with the DRBM and the BEM, respec-tively, that is due to a point source that was taken to beclosest to the complex interface in order to include the ef-fect of multiple reflections (see Fig. 4, the source to the farleft). The convergence rates for different background op-tical properties are presented in Fig. 5. This study al-lowed for two main observations. First, reconstruction fi-delity rapidly deteriorated when the maximum error persource was larger than 5%. This was also observed inRef. 30. Second, we found an empirical expression for tthat optimized the convergence for all realistic optical pa-rameters, i.e.,

Fig. 4. Geometry used for the simulation of diffuse optical to-mography. Circles represent sources (27 with 0.05-mm spac-ing); crosses represent detectors (27 with 0.05-mm spacing).

t 5 2imag$k%

AW1 1, (14)

where W is the minimum distance between surfacepoints, 1.5 cm in our case. When there is no wave damp-ing, Eq. (14) yields a relaxation parameter equal to unity.This expression was also tested for different system sizes,yielding in all cases good convergence rates. The depen-dence of the convergence rate on t and the DRBM order isshown in Fig. 6 for the case ms 5 10 cm21, ma 5 0.02cm21, which by means of Eq. (14) yields a value t 5 5.

C. ResultsFigures 7(a) and 7(b) depict the reconstructions obtainedwith the rigorous Green function by means of the BEMand are given as reference images to the DRBM results.In Figs. 7(c) and 7(d) we show the reconstructions for thefourth-order DRBM in the ma 5 0.02 cm21 case and thesecond-order DRBM in the ma 5 0.2 cm21 case, respec-tively. In Figs. 7(e) and 7(f ) we show the results obtainedfor the two media simulated by use of the KA method

Fig. 6. Error convergence versus order and relaxation param-eter t for the case ma 5 0.2 cm21, ms8 5 10 cm21. Modulationfrequency v 5 0.

Fig. 5. Convergence of the DRBM for different orders for thecases: ma 5 0.2 cm21, ms8 5 10 cm21, t 5 1 (solid curve);ma 5 0.02 cm21, ms8 5 10 cm21, t 5 1 (dotted–dashed curve);ma 5 0.02 cm21, ms8 5 10 cm21, t 5 2 (dashed curve); ma

5 0.02 cm21, ms8 5 5 cm21, t 5 2 (circles).

1108 J. Opt. Soc. Am. A/Vol. 20, No. 6 /June 2003 J. Ripoll and V. Ntziachristos

(equivalent to the first-order DRBM). Convergence ratesfor the cases presented in Figs. 7(c) and 7(d) are depictedin Fig. 8.

These convergence rates are compared with those ob-tained by using the iterative BEM method [i.e., theDRBM, but using the incident average intensity as theinitial value—see Eq. (10)], where we see that by usingthe first-order DRBM as an initial value we obtain morethan an order of magnitude improvement. Generally, fororders lower than those presented in Figs. 7(c) and 7(d),the object that is nearest the curved boundary is recon-structed with higher errors, whereas the object near theflat interface is fairly reconstructed in all cases. This isdue to the fact that the first-order DRBM (or the KAmethod) is exact in the presence of a plane but becomesless accurate as the curvature of the surface increases.For small geometries, such as the one under study, thefirst-order approximation is not capable of reconstructing

Fig. 7. Reconstruction of simulated geometry in Fig. 4 with useof the optimized relaxation parameter t [see Eq. (14)] and the ex-act Green function: (a) ma 5 0.02 cm21, ms8 5 10 cm21 and (b)ma 5 0.2 cm21, ms8 5 10 cm21; the DRBM for (c) fourth order,ma 5 0.02 cm21, ms8 5 10 cm21 and (d) second order, ma

5 0.2 cm21, ms8 5 10 cm21; and the KA (first-order DRBM) for(e) ma 5 0.02 cm21, ms8 5 10 cm21 and (f ) ma 5 0.2 cm21, ms85 10 cm21. See Fig. 8 for the errors corresponding to (c) and

(d), where we see that, to obtain good reconstructions, the errormust be under 5%. Modulation frequency v 5 0.

Fig. 8. Convergence of the DRBM and the iterative BEM for thecases presented in Figs. 7(c) and 7(d). The actual reconstructionvalues are marked with circles. Modulation frequency v 5 0.

the objects, since in both cases the error for N 5 1 isgreater than 5% [see Figs. 8(a) and 8(b)]. As we haveseen when comparing Figs. 7(c) and 7(d), higher ordersare needed when one deals with small absorption values,i.e., when the diffusive wave can travel deeper into themedium. This is expected, since, in that case, multiplereflections have a higher contribution. However, themain advantage of the DRBM is that for experimentalvalues of tissue the absorption coefficient is usually highowing to the presence of blood, and therefore the second-order DRBM will deliver accurate reconstructions [seeFig. 7(d)]. Also, as expected,39 high absorption values in-crease the resolution of DOT images.

Figure 9 depicts the results that we obtained when us-ing the rigorous expression for the Green function but ap-proximating the complex boundary of Fig. 4. Figures9(a) and 9(c) assumed an elliptical outline, representedwith a thick curve, around the geometry simulated. Thisapproximate boundary was chosen so that it encloses avolume similar to the one bounded by the exact interface.Finally, Figs. 9(b) and 9(d) depict the reconstructions ob-tained assuming infinite slab geometry and the method ofimage sources. Comparison between Figs. 7(a) and 7(b)and the results of Fig. 9 evinces the incapacity of using anapproximate boundary to accurately reconstruct objectsbounded by the complex boundary of Fig. 7. Thereforeapplication of higher-order methods such as the DRBM orthe BEM and a priori knowledge of the boundary shapewould be required in such small and complex geometries.As mentioned above regarding the lower-order solutionsof Fig. 7, the object that is nearest the curved boundaryfails to be reconstructed, whereas the one near the planeinterface is approximately retrieved.

5. DISCUSSIONAs novel tomographic systems emerge with a muchhigher data set, it becomes important to develop fast com-putation schemes for modeling diffuse media with arbi-trary boundaries. Current optical tomographic systemstypically operate with arrays of tens of sources and detec-tors, typically leading to 102 –103 available measure-ments. However, as the spatial sampling increases, for-

Fig. 9. Effect of using the exact Green function (BEM) but ap-proximating the geometry of Fig. 4. For optical properties ma

5 0.02 cm21, ms8 5 10 cm21: (a) Ellipse, (b) slab. For opticalproperties ma 5 0.2 cm21, ms8 5 10 cm21: (c) an ellipse, (d) aslab. Modulation frequency v 5 0.

J. Ripoll and V. Ntziachristos Vol. 20, No. 6 /June 2003/J. Opt. Soc. Am. A 1109

ward problems will require solutions for source–detectorpairs of the order of 103 –105, or higher. For example, theuse of CCD cameras as detector arrays, combined with anarray of sources in the tomographic sense, can readilyyield detector sets of the order of 105 or higher.

The DRBM offers an attractive solution for the calcula-tion of corresponding forward problems as it scales lin-early with the number of sources and not exponentially aswith matrix inversion techniques. The method solvesrigorously the surface integral equation by iteratively de-composing each scattering event into an order of reflec-tion with the interface. Thus the first-order approxima-tion to the DRBM would be essentially equivalent to theKA, but, for the following orders, it solves the surface in-tegral equation iteratively and thus deviates from the KA.In this manner, one can obtain solutions to any order ofaccuracy by adding orders of reflection. This methodol-ogy can account for shadow regions (areas that do nothave direct visibility with the light source) owing to thehigher reflection orders employed and can model high re-flectivity at the interfaces for media with arbitrary diffuseoptical properties. Since the set of N iterations does notemploy matrix inversion, this method retains computa-tional efficiency compared with more rigorous implicit so-lution methods based on the solution of a system of equa-tions such as the BEM. The DRBM obtains the maincontribution to the system of equations in a quick and ef-ficient manner, and the KA lacks high-order contribu-tions, whereas more rigorous methods such as the BEMor finite elements or differences needlessly take into ac-count all the multiple-scattering interactions at the inter-face. It is therefore clear that the KA will prove usefulfor imaging large volumes, whereas the N-order DRBMwill find greater application for imaging small diffusivevolumes.

The most important consideration is that the numberof DRBM iterations needed rarely exceeds two in the caseof imaging in tissue, owing to the high absorption ofblood, thus yielding fast iteration rates [see Fig. 6(d)].The fact that diffusive waves are highly damped reflectson very small contributions from multiply scattered diffu-sive waves. This means that by using the DRBM we ob-tain accurate solutions comparable with the BEM whileretaining time efficiency closer to that of the KA, asshown in Fig. 3. The DRBM can be used for several ap-plications, including generating the sensitivity functions(or weights) of the system, so that inversion schemes suchas algebraic reconstruction techniques, singular-value de-composition, or conjugate gradient-based methods may beapplied. Also, since it generates the complete Greenfunction for any three-dimensional geometry, it is possibleto apply it to improve already-existing reconstructionmethods that use the Born or Rytov approximations.These overall characteristics render the DRBM as an ap-pealing alternative to modeling diffusion problems in thepresence of complex boundaries.

ACKNOWLEDGMENTSJ. Ripoll acknowledges support from European Unionproject FMRX-CT96-0042. V. Ntziachristos gratefully ac-

knowledges National Institutes of Health grants 1 ROIEB000750-1 and R21-CA91807.

J. Ripoll, the corresponding author, can be reached bye-mail at jripoll@iesl.forth.gr.

REFERENCES1. A. G. Yodh and B. Chance, ‘‘Spectroscopy and imaging with

diffusing light,’’ Phys. Today 48, 34–40 (1995).2. S. K. Gayen and R. R. Alfano, ‘‘Emerging optical biomedical

imaging techniques,’’ Opt. Photon. News, March 1996, pp.16–22 .

3. E. B. Haller, ‘‘Time-resolved transillumination and opticaltomography,’’ J. Biomed. Opt. 1, 7–17 (1996).

4. M. A. Franceschini, K. T. Moesta, S. Fantini, G. Gaida, E.Gratton, H. Jess, W. W. Mantulin, M. Seeber, P. M. Schlag,and M. Kaschke, ‘‘Frequency-domain techniques enhanceoptical mammography: initial clinical results,’’ Proc. Natl.Acad. Sci. USA 94, 6468–6473 (1997).

5. V. Ntziachristos, X. H. Ma, and B. Chance, ‘‘Time-correlatedsingle photon counting imager for simultaneous magneticresonance and near-infrared mammography,’’ Rev. Sci. In-strum. 69, 4221–4233 (1998).

6. D. Grosenick, H. Wabnitz, H. Rinneberg, K. Moesta, and P.Schlag, ‘‘Development of a time-domain optical mammo-graph and first in vivo applications,’’ Appl. Opt. 38, 2927–2943 (1999).

7. V. Ntziachristos, A. G. Yodh, M. Schnall, and B. Chance,‘‘Concurrent MRI and diffuse optical tomography of breastafter indocyanine green enhancement,’’ Proc. Natl. Acad.Sci. USA 97, 2767–2772 (2000).

8. V. Ntziachristos and B. Chance, ‘‘Probing physiology andmolecular function using optical imaging: applications tobreast cancer,’’ Breast Cancer Res. 3, 41–46 (2001).

9. V. Ntziachristos, C. Tung, C. Bremer, and R. Weissleder,‘‘Fluorescence molecular tomography resolves protease ac-tivity in vivo,’’ Nature Med. 8, 757–760 (2002).

10. V. Ntziachristos, J. Ripoll, and R. Weissleder, ‘‘Would near-infrared fluorescence signals propagate through large hu-man organs for clinical studies?’’ Opt. Lett. 27, 333–335(2002).

11. S. R. Arridge, ‘‘Optical tomography in medical imaging,’’ In-verse Probl. 15, R41–R93 (1999).

12. A. Kak and M. Slaney, Principles of Computerized Tomog-raphic Imaging (IEEE Press, New York, 1988).

13. B. W. Pogue, S. P. Poplack, T. O. McBride, W. A. Wells, K. S.Osterman, U. L. Osterberg, and K. D. Paulsen, ‘‘Quantita-tive hemoglobin tomography with diffuse near-infraredspectroscopy: pilot results in the breast,’’ Radiography218, 261–266 (2001).

14. V. Ntziachristos, ‘‘Concurrent diffuse optical tomography,spectroscopy and magnetic resonance of breast cancer,’’Ph.D. dissertation (University of Philadelphia, Philadel-phia, Pa., 2000).

15. A. Klose, A. H. Hielscher, K. M. Hanson, and J. Beuthan,‘‘Two- and three-dimensional optical tomography of fingerjoints for diagnostics of rheumatoid arthritis,’’ in PhotonPropagation in Tissues IV, D. A. Benaron, B. Chance, M.Ferrari, and M. Kohl, eds., Proc. SPIE 3566, 151–160(1998).

16. Y. Xu, N. Iftimia, H. Jiang, L. L. Key, and M. B. Bolster,‘‘Imaging of in vitro and in vivo bones and joints withcontinuous-wave diffuse optical tomography,’’ Optics Ex-press 8, 447–451 (2001).

17. D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Klein-schmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E.L. Kermit, W. F. Cheong, and D. K. Stevenson, ‘‘Noninva-sive functional imaging of human brain using light,’’ J.Cereb. Blood Flow Metab. 20, 469–477 (2000).

18. E. M. C. Hillman, J. C. Hebden, M. Schweiger, H. Deh-ghani, F. E. W. Schmidt, D. T. Delpy, and S. R. Arridge,‘‘Time resolved optical tomography of the human forearm,’’Phys. Med. Biol. 46, 1117–1130 (2001).

1110 J. Opt. Soc. Am. A/Vol. 20, No. 6 /June 2003 J. Ripoll and V. Ntziachristos

19. V. Ntziachristos, C. Bremer, E. E. Graves, J. Ripoll, and R.Weissleder, ‘‘In vivo tomographic imaging of near-infraredfluorescent probes,’’ Mol. Imag. 1, 82–88 (2002).

20. S. R. Arridge, ‘‘Photon measurement density functions.Part I: Analytical forms,’’ Appl. Opt. 34, 7395–7409(1995).

21. G. Beer and J. O. Watson, Introduction to Finite andBoundary Element Methods for Engineers (Wiley, Chiches-ter, UK, 1992).

22. C. A. Brebbia and J. Dominguez, Boundary Elements, AnIntroductory Course (MacGraw-Hill, New York, 1989).

23. J. Ripoll and M. Nieto-Vesperinas, ‘‘Scattering integralequations for diffusive waves: detection of objects buriedin diffusive media in the presence of rough interfaces,’’ J.Opt. Soc. Am. A 16, 1453–1465 (1999).

24. J. Ripoll, V. Ntziachristos, R. Carminati, and M. Nieto-Vesperinas, ‘‘The Kirchhoff approximation for diffusivewaves,’’ Phys. Rev. E 64, 051917 (2001).

25. P. Beckmann, in Progress in Optics VI, E. Wolf, ed. (North-Holland, Amsterdam, 1961), pp. 55–69.

26. J. A. Ogilvy, Theory of Wave Scattering from Random RoughSurfaces (Adam Hilger, Bristol, 1991).

27. M. Nieto-Vesperinas, Scattering and Diffraction in PhysicalOptics (Pergamon, New York, 1996).

28. A. Ishimaru, Wave Propagation and Scattering in RandomMedia (Academic, New York, 1978).

29. J. A. Sanchez-Gil and M. Nieto-Vesperinas, ‘‘Light scatter-ing from random rough dielectric surfaces,’’ J. Opt. Soc. Am.A 8, 1270–1286 (1991).

30. J. Ripoll, M. Nieto-Vesperinas, R. Weissleder, and V. Ntzi-achristos, ‘‘Fast analytical approximation for arbitrary ge-

ometries in diffuse optical tomography,’’ Opt. Lett. 27, 527–529 (2002).

31. C. Macaskill and B. J. Kachoyan, ‘‘Iterative approach forthe numerical simulation of scattering from one- and two-dimensional rough surfaces,’’ Appl. Opt. 32, 2839–2847(1993).

32. K. Davey and I. Rosindale, ‘‘An iterative solution schemefor systems of boundary element equations,’’ Int. J. Numer.Methods Eng. 37, 1399–1411 (1994).

33. M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, ‘‘Re-fraction of diffuse photon density waves,’’ Phys. Rev. Lett.69, 2658–2661 (1992).

34. R. Aronson, ‘‘Boundary conditions for diffusion of light,’’ J.Opt. Soc. Am. A 12, 2532–2539 (1995).

35. R. Haskell, B. Tromberg, L. Svaasand, T. Tsay, T. Feng, andM. Mcadams, ‘‘Boundary conditions for the diffusion equa-tion in radiative transfer,’’ J. Opt. Soc. Am. A 11, 2727–2741(1994).

36. J. Ripoll and M. Nieto-Vesperinas, ‘‘Reflection and trans-mission coefficients for diffuse photon density waves,’’ Opt.Lett. 24, 796–798 (1999).

37. M. S. Patterson, B. Chance, and B. C. Wilson, ‘‘Time re-solved reflectance and transmittance for the noninvasivemeasurement of tissue optical properties,’’ Appl. Opt. 28,2331–2336 (1989).

38. T. Lu and D. O. Yevick, ‘‘Boundary element analysis of di-electric waveguides,’’ J. Opt. Soc. Am. A 19, 1197–1206(2002).

39. J. Ripoll, M. Nieto-Vesperinas, and R. Carminati, ‘‘Spatialresolution of diffuse photon density waves,’’ J. Opt. Soc. Am.A 16, 1466–1476 (1999).