Iterative boundary method for diffuse optical ... - OSA Publishing

J. Ripoll and V. Ntziachristos

Vol. 20, No. 6 / June 2003 / J. Opt. Soc. Am. A

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Iterative boundary method for diffuse optical tomography 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 General Hospital 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 light often 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 approximation. We generalize this approach using an analytical approximation, the N-order diffuse-reflection boundary method, which considers higher orders of interaction between surface elements in an iterative manner. We present the general performance of the method and demonstrate that it can improve the accuracy in modeling complex boundaries compared with the Kirchhoff approximation in the cases of small diffuse volumes or low absorption. Our observations are also contrasted with exact solutions. We furthermore investigate optimal implementation parameters and show that a second-order approximation is appropriate for most in vivo investigations. © 2003 Optical Society of America OCIS codes: 110.6960, 170.3010, 170.5280, 170.3660, 170.5270.

1. INTRODUCTION The study of light transport through highly scattering media such as living tissue has been the focus of recent research, mainly owing to its application in medical diagnosis1–8 and recently for its ability to probe molecular function in vivo.9 Light penetration of several centimeters in tissue is possible in the 650–850-nm range owing to low tissue absorption in this spectral region.10 Lately, rigorous mathematical modeling of light propagation in tissue (see Ref. 11 for a review), combined with technological advancements in photon sources and detection techniques, has allowed the development of diffuse optical tomography,12 (DOT), an imaging technique that threedimensionally resolves and quantifies optical properties in dense media such as tissues. DOT has been recently used to elucidate hemoglobin variations for imaging blood pooling,13 the accumulation of contrast agents in breast tumors,14 arthritic joint imaging,15,16 and functional activity in the brain17 and muscle.18 More recently similar tomographic principles combined with appropriate fluorophores of molecular specificity, have facilitated the development of fluorescence molecular tomography, an imaging method that allows elucidation of molecular signatures or pathways in intact tissues.9,19 A typical tomographic problem for optical imaging application requires modeling of photon propagation in the diffuse medium under investigation. Tissue applications require such calculations in the presence of complex boundaries. Two major computational strategies for generating such solutions are generally available: The first 1084-7529/2003/061103-08$15.00

employs numerical methods such as finite differences or elements20 that are based on volume discretization, and the second uses the boundary-element method21–23 (BEM), which solves numerically the surface integral equations for diffusive waves by discretizing the surface23 (in Ref. 23 the BEM method was implemented with an equivalent method termed the extinction theorem method). More recently we investigated the Kirchhoff approximation24 (KA) as an alternative to modeling arbitrary boundaries. This method, also known in physical optics as the tangent-plane method,25–28 has been extensively applied for modeling scattering of electromagnetic waves from random rough surfaces, and its limits of validity have been previously established.29 The KA method uses the angular spectrum representation of the propagating wave and employs the reflection coefficients to calculate the scattered wave inside any arbitrary geometry. In imaging applications the KA method has been shown to be highly efficient in the computation times required for calculating the forward model.30 However, it has also been demonstrated that the accuracy of the method deteriorates as the boundary becomes more complex, especially for smaller volumes or in the presence of low absorption.24 A major issue associated with the KA method’s limitations is an effect known as the shadowing effect, which is evident at areas that, owing to the complex boundary, do not have direct visibility with the light source (shadow regions). The shadowing effect originates from the fact that the KA is not capable of taking into consideration high-order reflections at the interface, © 2003 Optical Society of America

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J. Opt. Soc. Am. A / Vol. 20, No. 6 / June 2003

and therefore the calculation of the scattered wave at the shadow regions yields extremely high errors (⬎100%). Furthermore, for highly reflective boundaries or for situations in which high spatial-frequency components of the scattered diffuse wave are generally detectable (as in the case of small volumes, low absorption, and short source– detector distances from a nonplanar interface) the accuracy of the KA may diminish. We have developed an iterative analytical (explicit) method that overcomes the problems of the KA while maintaining fast computational characteristics. The N-order diffuse-reflection boundary method (DRBM) is a calculation approach that balances the computation performance and accuracy achieved between the iterative solution to the BEM31,32 and the KA. The DRBM is here applied to the two-dimensional case in diffuse optical tomography of complex geometries small in size (⬍3 cm) with both low and high absorption coefficients. A method equivalent to the BEM with constant surface elements22 that yields accurate numerical solutions to the diffusion approximation is used for generating forward solutions. This method was presented in detail elsewhere in the context of diffusive waves23 and will not be introduced here. This paper is organized as follows. In Section 2 we recall the basic integral equations to be solved for the forward problem, and the standard KA method is presented as in Ref. 24. The DRBM is then introduced in Section 3. Section 4 compares reconstruction results in two dimensions obtained by use of the DRBM and the BEM. Finally, the findings are discussed in the last section.

2. INTEGRAL EQUATIONS AND THE KIRCHHOFF APPROXIMATION The integral equations related to tomographic optical imaging 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 tomography in which we have a diffusive volume V as depicted in Fig. 1, delimited by surface S, which contains the abnormality we want to image. This diffusive medium is char-

Fig. 1.

Scattering geometry.


acterized by its absorption coefficient ␮ a , the diffusion coefficient D, and the refractive index n in and is surrounded by a nondiffusive medium of refractive index n out . If in such a medium the light source is modulated at a frequency ␻, the average intensity U(r, t) ⫽ U(r)exp(⫺i␻t) represents a diffuse photon density wave33 and obeys the Helmholtz equation with a wave number ␬ ⫽ (⫺␮ a /D ⫹ i ␻ /cD) 1/2, where c is the speed of light in the medium. By use of the expression for the Green function in a infinite homogeneous medium, g 共 ␬ 兩 r ⫺ r⬘ 兩兲 ⫽

exp共 i ␬ 兩 r ⫺ r⬘ 兩兲 D 兩 r ⫺ r⬘ 兩

,

(1)

the solution for arbitrary geometries can be expressed in terms of a surface integral by means of Green’s theorem (see Ref. 24 for a detailed derivation) as G 共 rs , rd 兲 ⫽ g 共 ␬ 兩 rs ⫺ rd 兩兲 ⫺

⫹

1 C ndD

1 4␲

冕冋

⳵ g 共 ␬ 兩 r⬘ ⫺ rd 兩兲

S

册

⳵ n⬘

g 共 ␬ 兩 r⬘ ⫺ rd 兩兲 G 共 rs , r⬘ 兲 dS⬘ ,

(2)

where n⬘ is the surface normal pointing into the nondiffusive medium (see Fig. 1) and G is the complete Green function that takes into account the presence of the boundary. In Eq. (2) we have made use of the boundary condition between the diffusive and the nondiffusive medium,28,34,35 G(rs , r⬘ ) 兩 S ⫽ ⫺C ndDn • ⵜG(rs , r⬘ ) 兩 S , r⬘ 苸 S, where C nd takes into account the refractive-index mismatch between both media.34,35 The total intensity measured at a detector point rd inside volume V that is due to a source density distribution ⌽ is given in terms of the complete Green function G(rs , rd ) as U 共 rd 兲 ⫽

1 4␲

冕

V

⌽ 共 r⬘ 兲 G 共 r⬘ , rd 兲 dr⬘ ,

rd 苸 V.

(3)

A rigorous solution to Eq. (2) can be found by means of numerical methods such as the BEM21,22 by discretizing the surface and inverting the resulting matrix (see Ref. 23 and references therein). This method gives a rigorous solution to the forward problem, but it requires large computation times, since it involves solving a large system of equations with a nonsparse matrix. Therefore it is appropriate for surfaces with limited number of discretization 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 preferred for manageable computation times and memory requirements. When the curvature of the surface under study is of the order of the diffuse wavelength the KA can be applied as a linear approximation to the BEM problem in Fourier space (i.e., a convolution in real space). The KA method, also known in physical optics as the tangentplane method, assumes that the surface is replaced at each point by its tangent plane. Therefore the total average intensity U at any point rp of the surface S is given by the sum of the homogeneous incident intensity U inc and that of the wave reflected from the local plane of normal n(rp ), U sc. The detailed derivation of the expres-



sions is presented in Ref. 24 and will not be repeated here. By means of the reflection coefficient for diffusive waves36 at a diffusive–nondiffusive interface, R nd共 K兲 ⫽

iC ndD 冑␬ 2 ⫺ K2 ⫹ 1 iC ndD 冑␬ 2 ⫺ K2 ⫺ 1

,

(4)

one finally reaches the KA approximation for the complete Green function at each local plane rp that defines surface S as G KA共 rs , rp 兲 ⫽

冕

⫹⬁

⫺⬁

¯ 兲 exp共 iK • R ¯ 兲 dK, 关 1 ⫹ R nd共 K兲兴 ˜g 共 K, Z (5)

where ˜g is the homogeneous Green function in Fourier ¯, Z ¯ ) are the coordinates of 兩 rs ⫺ rp 兩 with respace and (R ¯ ⫽ (rs ⫺ rp ) spect to the plane defined by n ˆ (rp ): Z ¯ ¯ • 关 ⫺n ˆ (rp ) 兴 , R ⫽ Z ⫺ (rs ⫺ rp ). Introducing Eq. (5) into Eq. (2) and discretizing surface S into N differential areas ⌬S, we obtain the expression for the KA in diffusive media: G

共 rs , rd 兲 ⫽ g 共 ␬ 兩 rs ⫺ rd 兩兲 ⫺

KA

1

N

兺 4␲

p⫽1

⫹

1 C ndD

g 共 ␬ 兩 rp ⫺ rd 兩兲

册

冋

⳵ g 共 ␬ 兩 rp ⫺ rd 兩兲 ⳵ np

⫻ G KA共 rs , rp 兲 ⌬S共 rp 兲 .

(6)

As discussed in Ref. 24, the KA method is a time- and memory-efficient approximation that is valid at (1) diffuse media bounded by convex surfaces, (2) geometries that are larger than the decay length of the diffuse intensity (typically, R ⬎ 2.0 cm), and (3) at absorption values that yield significant photon field attenuation (typically, ␮ a ⬎ 0.05 cm⫺1 ).

3. DIFFUSE-REFLECTION BOUNDARY METHOD A. First-Order Diffuse-Reflection Boundary Method To improve the regions of validity of analytical methods while maintaining favorable computation characteristics, we propose the use of image sources to model reflection versus fast Fourier transforms (FFT) at the interface and an iterative approach that, in contrast to the KA approximation, assumes high-order reflections at the boundary. This approach, the N-order DRBM can be envisaged as a hybrid between the KA and the iterative solution to the surface integral equation (2). The method, in contrast to the BEM solution, does not involve matrix inversion of a nonsparse matrix and maintains the computational and implementation 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 be found as 1兲 ¯, Z ¯ 兲 ⫺ g共 R ¯, Z ¯ ⫹ C ndD 兲兴 . (7) G 共DRBM 共 rs , rp 兲 ⫽ 关 g 共 R

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The use of image sources avoids time-consuming FFTs that would be necessary if Eq. (4) were directly considered and, for the cases so far studied, gives equivalent performance characteristics. By use of this expression, the first-order DRBM can be calculated similarly to Eq. (6) as a summation over N locally planar discrete areas ⌬S: 1兲 G 共DRBM 共 rs

, rd 兲 ⫽ g 共 ␬ 兩 rs ⫺ rd 兩兲 ⫺

1

N

兺 4␲

p⫽1

⫹

1 C ndD

册

冋


g 共 ␬ 兩 rp ⫺ rd 兩兲 ⌬S共 rp 兲

¯, Z ¯ 兲 ⫺ g共 R ¯, Z ¯ ⫹ C ndD 兲兴 . ⫻ 关 g共 R

(8)

Equation (8) yields computation times that are significantly shorter compared with the exact solution, since it is strictly an analytical approach involving no Fourier transforms or matrix inversion. Computation times of this new expression for the first-order DRBM are shown in Fig. 2, compared with the results by use of the KA with FFTs and the rigorous solution with matrix inversion (BEM). From Fig. 2 we infer that the computation time for the first-order DRBM and the KA grows linearly with the number of surface elements, as compared with a power-law dependence for the BEM. Owing to the difference in slope of their linearity dependence, the first-order DRBM is 3 orders of magnitude faster than the KA. When compared with rigorous numerical solutions such as the BEM, the first-order DRBM proves to be 5–6 orders of magnitude faster. B. N-Order Diffuse-Reflection Boundary Method To improve the computation accuracy of the forward calculation, we introduce the N-order DRBM. This is a method based on the scheme presented in Ref. 31 that solves iteratively a system of equations G – U ⫽ U( inc) , where the matrix G defines the interaction of one surface point with every other, U( inc) is the vector containing the incident average intensity at each surface point, and U is

Fig. 2. Computation times of the BEM (in minutes) versus the KA with FFTs (in seconds) and the DRBM (0.01 s). Results were obtained from a Pentium III running at 650 MHz, with 256 Mb RAM.

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the vector of unknowns defining the surface values of the average intensity, i.e., a superposition of U( inc) with photon contributions reflected from the boundary. This iterative scheme consists of finding the steady-state solution of

d dt

U ⫹ G – U ⫽ U共 inc兲 ,

(9)

which is approximated by using the Euler method with time step ␶ as U共 N 兲 ⫽ U共 N⫺1 兲 ⫺ ␶ G – U共 N⫺1 兲 ⫹ ␶ U共 inc兲 .

(10)

Details of the derivation and implications of this approach can be found in Ref. 31, its main idea residing in minimizing the change in the surface values between iterations. In Eq. (10), if we use as the initial value U( 0 ) ⫽ U( inc) , we obtain the iterative BEM solution31 (convergence of the iterative BEM will be shown in Sec. 4 together with that for the DRBM). However, to facilitate time-efficient computations, we can approximate the initial value by using the first-order DRBM from Eq. (7), i.e., (1) U( 0 ) ⫽ GDRBM . To improve the accuracy of the calculation, we can then iteratively apply the DRBM by rewriting Eq. (2) and assuming that each of the surface points is a detector, similarly to solutions obtained with the BEM.21–23 The surface value at a surface point rk that is due to a source at rs would be N兲 N⫺1 兲 G 共DRBM 共 rs , rk 兲 ⫽ G 共DRBM 共 rs , rk 兲 ⫺ ␶ • g 共 ␬ 兩 rs ⫺ rk 兩兲

⫹␶

N

1

兺 4␲

p⫽1

⫹

1 C ndD

冋

⳵ g 共 ␬ 兩 rp ⫺ rk 兩兲 ⳵ np

g 共 ␬ 兩 rp ⫺ rk 兩兲

册

N⫺1 兲 ⫻ G 共DRBM 共 rs , rk 兲 ⌬S共 rp 兲 .

(11)

Equation (11) is the main expression for the N-order DRBM and must be applied to each surface point. In this expression we must take the principal value of the integral, i.e., exclude those points where the source and detector coincide, (rk ⫽ rp ), and substitute those with the corresponding self-induction values. These are well known and depend on the medium and surface discretization area. Their expressions in two and threedimensions can be found in Refs. 31 and 38, for example. As can be seen from Eq. (11), the factor ␶ plays an important role and can be considered a relaxation parameter. The rate of convergence of the iteration procedure will depend on its value, as will be shown in Section 4. Once all the surface values for the N order are found by means of Eq. (11), we can find the intensity anywhere inside the diffusive volume by using an equivalent of Eq. (2):

Fig. 3. Computation times (in minutes) of the BEM versus four different orders of the DRBM. Results for the BEM for a large number of surface points were found by extrapolation, owing to the high computational cost. Results were obtained from a Pentium III running at 650 MHz, with 256 Mb RAM.

N兲 G 共DRBM 共 rs

, rd 兲 ⫽ g 共 ␬ 兩 rs ⫺ rd 兩兲 ⫺

1

N

兺 4␲

p⫽1

⫹

1 C ndD

g 共 ␬ 兩 rp ⫺ rd 兩兲

册

冋

N兲 ⫻ G 共DRMB 共 rs , rp 兲 ⌬S共 rp 兲 .


(12)

Computation times of Eq. (12) for the first four orders of the DRBM are shown in Fig. 3, which shows the rigorous BEM solution from Fig. 2 as a reference. From Fig. 3 we see that there is a significant increase in computation time the second-order DRBM is applied, as expected. However, we see that these computation times are still 2–3 orders of magnitude faster than a matrix inversion method, while maintaining accuracy. Also, a main feature of this figure is that as the number of surface points increases, so does the validity of the first-order DRBM, the higher DRBM orders being necessary only for small volumes (i.e., a small number of surface discretization points).

4. METHODS A. Simulations We performed two types of two-dimensional study. The first study assumed a diffuse medium with a complex boundary and studied the DRBM performance in reconstructing the absorption coefficient. The second study tested the hypothesis that the use of approximate, easyto-model boundaries could be used as alternatives to rigorously calculating solutions in the presence of complex boundaries to image media with arbitrary surfaces. The simulations considered the two-dimensional case presented in Fig. 4 and employed 27 continuous intensity sources (i.e., modulation frequency ␻ ⫽ 0) and 27 detectors placed with a spacing of 0.5 mm, giving a field of view of 13 mm. The outer geometry simulated was 2.5 cm in width and 1.5 cm in depth, typical in mouse imaging in a planar imaging configuration. The forward field was calculated numerically by using the BEM as a gold standard for the homogeneous medium and adding two absorbing



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objects with Gaussian profile of 1-mm FWHM located at (x, y) ⫽ (⫺0.5 cm, 0.35 cm) and (x, y) ⫽ (0, ⫺0.35 cm). Owing to the small sizes of these objects, their contribution to the total wave was well described by the Born approximation. Two diffusive media were considered, attaining background absorption coefficients of ␮ a ⫽ 0.02 cm⫺1 and ␮ a ⫽ 0.2 cm⫺1 , respectively. The absorption coefficient maximum of the embedded objects was twice the value of the background in both situations. In all cases, the background and object’s reduced scattering coefficient was ␮ ⬘s ⫽ 10 cm⫺1 , and the index of refraction was that of water. To obtain the reconstructions, a weight matrix (or sensitivity matrix) was built, and the product of this matrix with local perturbations was used to generate data perturbations, i.e., the scattered intensity. Reconstructions were obtained by use of 3 ⫻ 104 algebraic reconstruction iterations.12 B. Optimization of Convergence To find an expression for the optimum value of the relaxation parameter ␶, we performed several reconstructions for different background values and different ␶ values and studied the convergence of the root-mean-square error defined as error共 N 兲 ⫽

1 Nd

再兺冋 Nd

i⫽1

1⫺

N兲 U 共DRBM

U BEM

册冎 2

Fig. 5. Convergence of the DRBM for different orders for the ␮ a ⫽ 0.2 cm⫺1 , ␮ s⬘ ⫽ 10 cm⫺1 , ␶ ⫽ 1 (solid curve); cases: ␮ a ⫽ 0.02 cm⫺1 , ␮ s⬘ ⫽ 10 cm⫺1 , ␶ ⫽ 1 (dotted–dashed curve); ␮ a ⫽ 0.02 cm⫺1 , ␮ s⬘ ⫽ 10 cm⫺1 , ␶ ⫽ 2 (dashed curve); ␮ a ⫽ 0.02 cm⫺1 , ␮ s⬘ ⫽ 5 cm⫺1 , ␶ ⫽ 2 (circles).

1/2

,

(13)

where N d is the number of detectors (27 in our case), N is the order, and U DRBM and U BEM represent the average intensity obtained by with the DRBM and the BEM, respectively, that is due to a point source that was taken to be closest to the complex interface in order to include the effect of multiple reflections (see Fig. 4, the source to the far left). The convergence rates for different background optical properties are presented in Fig. 5. This study allowed for two main observations. First, reconstruction fidelity rapidly deteriorated when the maximum error per source was larger than 5%. This was also observed in Ref. 30. Second, we found an empirical expression for ␶ that optimized the convergence for all realistic optical parameters, i.e.,

Fig. 6. Error convergence versus order and relaxation parameter ␶ for the case ␮ a ⫽ 0.2 cm⫺1 , ␮ s⬘ ⫽ 10 cm⫺1 . Modulation frequency ␻ ⫽ 0.

␶⫽2

imag兵 ␬ 其

冑W

⫹ 1,

(14)

where W is the minimum distance between surface points, 1.5 cm in our case. When there is no wave damping, 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 dependence of the convergence rate on ␶ and the DRBM order is shown in Fig. 6 for the case ␮ s ⫽ 10 cm⫺1 , ␮ a ⫽ 0.02 cm⫺1 , which by means of Eq. (14) yields a value ␶ ⫽ 5.

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

C. Results Figures 7(a) and 7(b) depict the reconstructions obtained with the rigorous Green function by means of the BEM and are given as reference images to the DRBM results. In Figs. 7(c) and 7(d) we show the reconstructions for the fourth-order DRBM in the ␮ a ⫽ 0.02 cm⫺1 case and the second-order DRBM in the ␮ a ⫽ 0.2 cm⫺1 case, respectively. In Figs. 7(e) and 7(f ) we show the results obtained for the two media simulated by use of the KA method

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(equivalent to the first-order DRBM). Convergence rates for the cases presented in Figs. 7(c) and 7(d) are depicted in Fig. 8. These convergence rates are compared with those obtained by using the iterative BEM method [i.e., the DRBM, but using the incident average intensity as the initial value—see Eq. (10)], where we see that by using the first-order DRBM as an initial value we obtain more than an order of magnitude improvement. Generally, for orders lower than those presented in Figs. 7(c) and 7(d), the object that is nearest the curved boundary is reconstructed with higher errors, whereas the object near the flat interface is fairly reconstructed in all cases. This is due to the fact that the first-order DRBM (or the KA method) is exact in the presence of a plane but becomes less accurate as the curvature of the surface increases. For small geometries, such as the one under study, the first-order approximation is not capable of reconstructing

Fig. 7. Reconstruction of simulated geometry in Fig. 4 with use of the optimized relaxation parameter ␶ [see Eq. (14)] and the exact Green function: (a) ␮ a ⫽ 0.02 cm⫺1 , ␮ s⬘ ⫽ 10 cm⫺1 and (b) ␮ a ⫽ 0.2 cm⫺1 , ␮ s⬘ ⫽ 10 cm⫺1 ; the DRBM for (c) fourth order, ␮ a ⫽ 0.02 cm⫺1 , ␮ s⬘ ⫽ 10 cm⫺1 and (d) second order, ␮ a ⫽ 0.2 cm⫺1 , ␮ s⬘ ⫽ 10 cm⫺1 ; and the KA (first-order DRBM) for (e) ␮ a ⫽ 0.02 cm⫺1 , ␮ s⬘ ⫽ 10 cm⫺1 and (f ) ␮ a ⫽ 0.2 cm⫺1 , ␮ s⬘ ⫽ 10 cm⫺1 . See Fig. 8 for the errors corresponding to (c) and (d), where we see that, to obtain good reconstructions, the error must be under 5%. Modulation frequency ␻ ⫽ 0.


Fig. 9. Effect of using the exact Green function (BEM) but approximating the geometry of Fig. 4. For optical properties ␮ a ⫽ 0.02 cm⫺1 , ␮ s⬘ ⫽ 10 cm⫺1 : (a) Ellipse, (b) slab. For optical properties ␮ a ⫽ 0.2 cm⫺1 , ␮ s⬘ ⫽ 10 cm⫺1 : (c) an ellipse, (d) a slab. Modulation frequency ␻ ⫽ 0.

the objects, since in both cases the error for N ⫽ 1 is greater than 5% [see Figs. 8(a) and 8(b)]. As we have seen when comparing Figs. 7(c) and 7(d), higher orders are needed when one deals with small absorption values, i.e., when the diffusive wave can travel deeper into the medium. This is expected, since, in that case, multiple reflections have a higher contribution. However, the main advantage of the DRBM is that for experimental values of tissue the absorption coefficient is usually high owing to the presence of blood, and therefore the secondorder DRBM will deliver accurate reconstructions [see Fig. 7(d)]. Also, as expected,39 high absorption values increase the resolution of DOT images. Figure 9 depicts the results that we obtained when using the rigorous expression for the Green function but approximating the complex boundary of Fig. 4. Figures 9(a) and 9(c) assumed an elliptical outline, represented with a thick curve, around the geometry simulated. This approximate boundary was chosen so that it encloses a volume similar to the one bounded by the exact interface. Finally, Figs. 9(b) and 9(d) depict the reconstructions obtained assuming infinite slab geometry and the method of image sources. Comparison between Figs. 7(a) and 7(b) and the results of Fig. 9 evinces the incapacity of using an approximate boundary to accurately reconstruct objects bounded by the complex boundary of Fig. 7. Therefore application of higher-order methods such as the DRBM or the BEM and a priori knowledge of the boundary shape would be required in such small and complex geometries. As mentioned above regarding the lower-order solutions of Fig. 7, the object that is nearest the curved boundary fails to be reconstructed, whereas the one near the plane interface is approximately retrieved.

5. DISCUSSION

Fig. 8. Convergence of the DRBM and the iterative BEM for the cases presented in Figs. 7(c) and 7(d). The actual reconstruction values are marked with circles. Modulation frequency ␻ ⫽ 0.

As novel tomographic systems emerge with a much higher data set, it becomes important to develop fast computation schemes for modeling diffuse media with arbitrary boundaries. Current optical tomographic systems typically operate with arrays of tens of sources and detectors, typically leading to 102 – 103 available measurements. However, as the spatial sampling increases, for-



ward problems will require solutions for source–detector pairs of the order of 103 – 105 , or higher. For example, the use of CCD cameras as detector arrays, combined with an array of sources in the tomographic sense, can readily yield detector sets of the order of 105 or higher. The DRBM offers an attractive solution for the calculation of corresponding forward problems as it scales linearly with the number of sources and not exponentially as with matrix inversion techniques. The method solves rigorously the surface integral equation by iteratively decomposing each scattering event into an order of reflection with the interface. Thus the first-order approximation to the DRBM would be essentially equivalent to the KA, but, for the following orders, it solves the surface integral equation iteratively and thus deviates from the KA. In this manner, one can obtain solutions to any order of accuracy by adding orders of reflection. This methodology can account for shadow regions (areas that do not have direct visibility with the light source) owing to the higher reflection orders employed and can model high reflectivity at the interfaces for media with arbitrary diffuse optical properties. Since the set of N iterations does not employ matrix inversion, this method retains computational efficiency compared with more rigorous implicit solution methods based on the solution of a system of equations such as the BEM. The DRBM obtains the main contribution to the system of equations in a quick and efficient manner, and the KA lacks high-order contributions, whereas more rigorous methods such as the BEM or finite elements or differences needlessly take into account all the multiple-scattering interactions at the interface. It is therefore clear that the KA will prove useful for imaging large volumes, whereas the N-order DRBM will find greater application for imaging small diffusive volumes. The most important consideration is that the number of DRBM iterations needed rarely exceeds two in the case of imaging in tissue, owing to the high absorption of blood, thus yielding fast iteration rates [see Fig. 6(d)]. The fact that diffusive waves are highly damped reflects on very small contributions from multiply scattered diffusive waves. This means that by using the DRBM we obtain accurate solutions comparable with the BEM while retaining time efficiency closer to that of the KA, as shown in Fig. 3. The DRBM can be used for several applications, including generating the sensitivity functions (or weights) of the system, so that inversion schemes such as algebraic reconstruction techniques, singular-value decomposition, or conjugate gradient-based methods may be applied. Also, since it generates the complete Green function for any three-dimensional geometry, it is possible to apply it to improve already-existing reconstruction methods that use the Born or Rytov approximations. These overall characteristics render the DRBM as an appealing alternative to modeling diffusion problems in the presence of complex boundaries.

17.

ACKNOWLEDGMENTS

18.

J. Ripoll acknowledges support from European Union project FMRX-CT96-0042. V. Ntziachristos gratefully ac-

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knowledges National Institutes of Health grants 1 ROI EB000750-1 and R21-CA91807. J. Ripoll, the corresponding author, can be reached by e-mail at [email protected].

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