## Reconstruction of fluorescence/bioluminescence sources in biological medium with spatial filter

Optics Express, Vol. 18, Issue 12, pp. 13151-13172 (2010)

http://dx.doi.org/10.1364/OE.18.013151

Acrobat PDF (9789 KB)

### Abstract

We propose a new method for reconstruction of emitting source distributions by use of a spatial filte and a successive updating process of the forward model for fluorescence/bioluminescenc diffuse optical tomography. The spatial filte transforms a set of the measurement data to a single source strength at a position of interest, and the forward model is updated by use of the estimated source strengths. This updating process ignores the dispensable source positions from reconstruction according to the reconstructed source distribution, and the spatial resolution of the reconstructed image is improved. The estimated sources are also used for the reduction of artifacts induced by noises based on the singular value decomposition. Some numerical experiments show the advantages of the proposed method by comparing the present results with those obtained by the conventional methods of the least squares method and Algebraic Reconstruction Technique. Finally the criteria for practical use of the method are quantitatively presented by the simulations for 2D and 3D geometries.

© 2010 Optical Society of America

## 1. Introduction

1. V. Ntziachristos, C-H. Yung, C. Bremerand, and R. Weissleder, “Fluorescence molecular tomography resolves protease activity *in vivo*,” Nat. Med. **8** (7), 757–760 (2002). [CrossRef] [PubMed]

2. V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. **23**(3), 313–320 (2005). [CrossRef] [PubMed]

3. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Prob. **15**, R41–R93 (1999). [CrossRef]

*in-vivo*molecular imaging of the drug delivery process can be realized. F/BDOT also allows us to study disease and to evaluate the efficien y of a treatment over time for an identical animal [1

1. V. Ntziachristos, C-H. Yung, C. Bremerand, and R. Weissleder, “Fluorescence molecular tomography resolves protease activity *in vivo*,” Nat. Med. **8** (7), 757–760 (2002). [CrossRef] [PubMed]

6. D. Grosenick, H. Wabnitz, H. H. Rinneberg, T. Moesta, and P. M. Schlag, “Development of a time-domain optical mammography and fir t *in vivo* applications,” Appl. Opt. **38**(13), 2927–2943 (1999). [CrossRef]

12. R. Weissleder, “Molecular Imaging in Cancer,” SCIENCE **321**, 1168–1171 (2006). [CrossRef]

15. K. Vishwanath, B. Pogue, and M-A. Mycek, “Quantitative fluore cence lifetime spectroscopy in turbid media: comparison of theoretical, experimental and computational method,” Phys. Med. Biol. **47**, 3387–3405 (2002). [CrossRef] [PubMed]

16. J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Time-resolved imaging of fluore cent objects embedded in turbid media,” Opt. Lett. **20**(5), 489–491 (1995). [CrossRef] [PubMed]

22. L. Zhang, F. Gao, H. He, and H Zhao, “Three-dimensional scheme for time-domain fluore cence molecular tomography based on Laplace transforms with noise-robust factors,” Opt. Express **16**(10), 7214–7223 (2008). [CrossRef] [PubMed]

20. F. Gao, H. Zhao, L. Zhang, Y. Tanikawa, A. Marjono, and Y. Yamada, “A self-normalized, full time-resolved method for fluore cence diffuse optical tomography,” Opt. Express **16**(17), 13104–13121 (2008). [CrossRef] [PubMed]

25. M. A. O’Leary, D. A. Boas, X. D. Li, B. Chance, and A. G. Yodh, “Fluorescence lifetime imaging in turbid media,” Opt. Lett. **21**(2), 158–160 (1996). [CrossRef] [PubMed]

29. B. D. van Veen and K. M. Buckly, “Beamforming: A versatile approach to spatial filtering” IEEE ASSP Mag. **15**, 4–23 (1988). [CrossRef]

30. S. Baillet, J. C. Mosher, and R. M. Leahy, “Electromagnetic brain mapping,” IEEE Signal Process Mag. **18**, 14–30 (2001). [CrossRef]

35. C. Kuo, O. Coquoz, T. L. Troy, H. Xu, and B. W. Rice, “Three-dimensional reconstruction of in vivo bioluminescent sources based on multispectral imaging,” J. Biomed. Opt. **12**, 024007 (2007). [CrossRef] [PubMed]

## 2. Methods

### 2.1. Forward model of light propagation

3. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Prob. **15**, R41–R93 (1999). [CrossRef]

*r*is the position vector,

*D*= 1/(3

*µ*′

*) is the diffusion coefficien with the reduced scattering coefficien*

_{s}*µ*′

*,*

_{s}*µ*is the absorption coefficient Φ is the fluenc rate which is the integral of the photon radiance over all directions, and

_{a}*q*

_{0}is the fluorescenc light source. Equation (1) can be solved under the boundary condition expressed by Eq. (2),

*A*= (1 +

*R*)/(1 −

_{f}*R*),

_{f}*R*is the internal reflectio ratio, and

_{f}*n*is the outward normal to the surface ∂Ω of the medium. The light intensities measured at the surface, Γ(

*r*), are the flu es of the fluenc rates at the surface given by the left hand side of Eq. (2).

*q*

_{0}(

*r*), is described as

*q*

_{0}=

*ηµ*Φ

_{af}*, where*

_{x}*η*is the quantum yield of the fluorophore

*µ*is the absorption coefficien of the fluorophor and Φ

_{af}*is the fluenc rate of the excitation light. The mechanisms of the emissions are different between bioluminescence and fluorescence However, propagation of light emitted by either bioluminescence or fluorescenc probes is described by Eq. (1) commonly. It is possible to reconstruct the fluorescenc properties such as*

_{x}*η*and

*µ*after the reconstruction of

_{af}*q*

_{0}when Φ

*is given. Therefore, this study is focused on the reconstruction of*

_{x}*q*

_{0}which can be converted to either fluorescenc or bioluminescence light sources.

*Q*are the vectors of the discretized fluenc rates and the source strengths at

*N*nodes of FEM, and the

*N*×

*N*matrices

*K*and

*C*are calculated with the basis functions and the optical properties [36

36. M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the Finite-Element Method for the Forward and Inverse Model in Optical Tomography,” J. Math. Imaging Vis. **3**, 263–283 (1993). [CrossRef]

*B*is for the boundary condition. The entries of the matrices are,

*K*= ∫

_{ij}_{Ω}

*D*∇

*ψ*·∇

_{i}*ψ*,

_{j}dr*C*= ∫

_{ij}_{Ω}

*µ*and

_{a}ψ_{i}ψ_{j}dr*B*= 1/(2

_{ij}*A*) ∫

_{∂Ω}

*ψ*, where

_{i}ψ_{j}dγ*i*and

*j*denote the indexes assigned to the nodes,

*B*is the integral over the boundary surface, and

_{ij}*ψ*is a basis function.

_{i}*Q*is given, the vector-matrix equation (3) can be solved for Φ,

*G*= {

*K*(

*D*) +

*C*(

*µ*) +

_{a}*B*}

^{−1}.

*D*∇

*Φ, and they can be readily calculated as 1/(2*

_{n}*A*)Φ due to the boundary condition, Eq. (2). Since the flu es can be measured only at a limited number of positions,

*M*, at the surface of the medium, the forward model in F/BDOT is formulated as follows,

*m*is a vector composed of the measurable flu es at

*M*positions, and the

*M*×

*N*matrix

*L*consists of the

*M*row vectors corresponding to the measurement positions among the rows of the matrix

*G*.

### 2.2. Inverse problem to reconstruct the light source distribution

#### 2.2.1. Spatial filte

*Q*. The components of

*Q*are linearly mixed into

*m*by

*L*. To reconstruct the source term, we design a spatial filte which extracts a single component in the source vector from the measurement data. The spatial filte exploits the differences in the contributions of spatially different sources to the measurement data. The spatial filte is constructed for every node with the source strength to reconstruct the whole distribution of the source strengths. The spatial filte

*w*we employ in this study transforms the set of the measurements

_{k}*m*to the estimation of a source strength

*q̂*at a specifi location. The estimated source distribution

_{k}*Q̂*= [

*q̂*

_{1},

*q̂*

_{2}, ⋯,

*q̂*]

_{N}*is obtained with the*

^{T}*M*×

*N*matrix

*W*which consists of the spatial filter

*w*,

_{k}*q̂*is the

_{k}*k*-th source estimated by

*w*, which is the

_{k}*k*-th column vector of

*W*, represents the spatial filte targeting on the

*k*-th source strength

*q*, and

_{k}*l*is the

_{i}*i*-th column vector of

*L*.

*l*represents the contribution of the single source with a unit strength in the

_{i}*i*-th position to the measurement data. The spatial filter are constructed for all nodes of FEM model where the sources can exist.

*k*-th position agrees with the true one when the spatial filte and the column vectors

*l*hold

_{j}*w*, where

^{T}_{k}l_{j}*δ*is the Kronecker’s delta. However, in the ill-posed inverse problem such as the source reconstruction for F/BDOT in this paper, it is not possible to obtain such a spatial filte for every

_{kj}*l*since the matrix

_{j}*L*is singular. For successful estimation,

*w*(

^{T}_{k}l_{j}*k*≠

*j*) appearing in Eq. (7) should be close to zero, and

*w*should be as close to unity as possible. In other words, the spatial filte is the solution of the following maximization problem,

^{T}_{k}l_{k}*LL*) is singular, its inverse matrix in Eq. (9) can be substituted by a pseudo-inverse matrix. The coefficien in Eq. (9), {

^{T}*l*(

^{T}*LL*)

^{T}^{−1}

*l*}

_{i}^{−1/2}, differentiates the spatial filter from the Moore-Penrose pseudo inversion.

*w*) = 1,

^{T}_{k}l_{k}*w*= (

_{k}*LL*)

^{T}^{−1}

*l*is obtained as a spatial filte. In this case, the reconstructed distribution of the sources coincides with the minimum norm solution by the Moore-Penrose matrix inverse, which is difficul to reconstruct the emitting sources deeply embedded inside the medium in the underdetermined inverse problem. On the other hand, the proposed spatial filte has an advantage that the maximum of the estimated source strengths is located at the position of the source expected to make the largest contribution to the measurement data, regardless of the distance from the detectors. This comes from Eq. (8), which means that the spatial filte is the vector which maximizes

_{k}*q̂*estimated from the measurement data generated by the single source with the unit strength at the targeted

_{k}*k*-th position.

*w*)

^{T}_{k}l_{k}^{2}≥ (

*w*)

^{T}_{j}l_{k}^{2}, since the spatial filte

*w*maximizes the squared product of (

_{k}*w*)

^{T}l_{k}^{2}, where

*w*is the arbitrary vector satisfying ∥

*w*∥

^{T}L^{2}= 1. If the source

*q*contributes to the measurement data to a great extent, i.e.

_{k}*m*≃

*l*, the estimated source strength

_{k}q_{k}*q̂*=

_{k}*w*at the

^{T}_{k}m*k*-th position takes a larger value than those at the other positions. Thus we can expect that the spatial filter identify the correct locations of strong sources, even if the spatial filte can not obtain

*w*sufficientl close to unity and as the result the estimated source strength cannot agree with the true one.

^{T}_{k}l_{k}*Q*such as the smoothness, the sparseness, etc. The regularization technique is employed to obtain the solution with the characteristics expected for the true solution. On the other hand, the method proposed in this paper pays attention to the relation among the components of

*Q*. The spatial filte investigates which source is large and which is not.

*L*depends on the shape of the medium, the optical properties and the positions of the detectors. The spatial filte depends on

*L*.

#### 2.2.2. Updating of the forward model

_{∞}denotes the maximum norm. Then the forward model is updated by use of the weight,

*p*, as the following,

_{i}*m*=

*LPQ*, where

*P*is the diagonal matrix with

*p*. The weight,

_{i}*p*, takes the value in the range of [0, 1].

_{i}*p*= 1 means that the

_{i}*i*-th source position is necessary for the forward model, while

*p*= 0 means that the source position can be ignored. The spatial filte is recalculated with the updated matrix

_{i}*L*←

*LP*.

*Q̂*of

*m*=

*LQ̂*. However, the solution can relatively compares the source strengths among the source positions. Now we introduce a coefficien for the adjustment,

*α*, by solving the optimization problem as follows,

*Q̄*, is estimated by the following equation,

#### 2.2.3. Reduction of artifacts based on SVD

*ε*represents the sum of the noises and the errors.

*L*which consists of the products,

*l*, first

_{i}q̄_{i}*L*is decomposed as

*L*=

*UDV*where

^{T}*U*and

*V*are the

*M*×

*M*and

*N*×

*N*matrices respectively, whose columns form the orthonormal basis, and

*D*is the

*M*×

*N*matrix which has the singular values

*s*

_{1},

*s*

_{2}, ⋯,

*s*, ⋯,

_{j}*s*and (

_{M}*N*−

*M*) components with zero in the diagonal. The

*M*× 1 basis vectors,

*u*(

_{j}*j*= 1, 2, ⋯

*M*) form the matrix

*U*= [

*u*

_{1},

*u*

_{2},⋯,

*u*]. The basis vector and the singular value with the identical index

_{M}*j*correspond with each other. Note that the measurement data are represented by a linear combination of the basis vectors

*u*, i.e.

_{j}*m*=

*U*, where y is a

_{y}*M*× 1 vector determined uniquely by

*U*. Therefore, by removing the basis vectors corresponding to the small singular values from the measurement data, i.e. by setting the corresponding components of

*y*as zeros as the following procedure,

*ε*in Eq. (14) can be reduced,

*H*is the

*M*×

*M*diagonal matrix with the

*j*-th diagonal components

*H*having 1 or 0 with depending on whether the

_{jj}*j*-th basis vector is the signal or the noise space. The number of the basis vectors selected as the signals depends on the situations. The number of the basis vectors for the signals increase as the SNR is high.

## 3. Results and discussions of numerical experiments

### 3.1. Conditions of numerical experiments

^{−1}, respectively, by assuming the application to breast cancer screening using fluorescen dye working in the near-infrared wavelength range [9

9. J. C. Hebden, H. Veenstra, H. Dehghani, E. M. C. Hillman, M. Schweiger, S. R. Arridge, and D. T. Delpy, “Three-dimensional time-resolved optical tomography of a conical breast phantom,” Appl. Opt. **40**(19), 3278–32887 (2001). [CrossRef]

*A*in Eq. (2) is set as unity assuming no internal reflection Some regions with unit source strength are included in the medium.

### 3.2. Improvement in localization by updating the forward model

*x,y*) =(20 mm, 0 mm) with the source strength of unity. From Fig. 1, it is understood that updating the forward model improves the localization of the reconstructed emitting source regions.

*m*−

*LQ̄*∥

^{2}/∥

*m*∥

^{2}×100 (%), decreases with the increase of the number of iterations in successful cases. This feature can help users choose a reconstructed image which is recognized as an appropriate one from a physiological point of view.

### 3.3. Comparison with the conventional methods

37. C. R. Vogel,*Computational Methods for Inverse Problems (Frontiers in Applied Mathematics)* (SIAM, Philadelphia, 2002). [CrossRef]

*Q*=

*L*(

^{T}*LL*)

^{T}^{−1}

*m*. To minimize the norm of the solution, the locations of the reconstructed emitting sources are shifted to the surface of the medium as shown in Fig. 3(b). It is obvious that the solution with weakly emitting sources provide small norm.

*m*−

*LQ*∥

^{2}[38

38. C. C. Paige and M. A. Saunders, “LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares,” ACM trans. on Math. Software **8**(1), 43–71 (1982). [CrossRef]

20. F. Gao, H. Zhao, L. Zhang, Y. Tanikawa, A. Marjono, and Y. Yamada, “A self-normalized, full time-resolved method for fluore cence diffuse optical tomography,” Opt. Express **16**(17), 13104–13121 (2008). [CrossRef] [PubMed]

### 3.4. Reduction of the artifacts induced by various noises and errors

#### 3.4.1. Reduction of the effects of the random noises in the measurement data

*SD*= 0.01). In Fig. 4 (a-1), the averages of the reconstructed source strengths clearly show the single emitting source region and most of SDs are less than 0.02, which are sufficientl small compared with the average values. In this case, after SVD of the updated matrix

_{noise}*L*, we select the larger singular values that yield more than 98 percent (

*λ*= 0.98) of the total, and the corresponding basis vectors

_{SVD}*u*are chosen as the signal space. The noise space is removed by Eq.(15) in every iteration. By using the proposed method of removing the noise space the emitting source region can be well identifie in most of the reconstructions with the noise of

_{i}*SD*=0.01. The artifacts tend to appear near the detectors far from the emitting sources. The light intensities measured by the detectors far from the emitting sources are weak with low SNR. Without removing the noise space, it is difficul to identify the emitting sources in most cases. The strong artifacts are reconstructed widely and randomly.

_{noise}*SD*=0.001, the proposed reconstruction scheme is quite robust, and the emitting sources can always be found at the correct positions. When the noise is significant

_{noise}*SD*= 0.1, as shown in Fig. 4 (c), the reconstructed images can not provide the information of the fluorophore although removing the noise space is carried out.

_{noise}#### 3.4.2. Reduction of the artifacts induced by the background emissions

*q*, at each node is assigned as (a) 0.0001, (b) 0.001 or (c) 0.01. Figure 5 plots the measured intensities from the target and background emissions. The number of the updating iterations is six.

_{BG}*λ*= 0.995, the noise reduction is found to be insufficient The uniformly distributed background emissions are measured equally by all the detectors and appear as the artifact emitting source localized at the center of the circular medium. When the background emission is not uniform, the reconstructed artifact sources will be distributed corresponding to the nonuniformity. The image shown in Fig. 6 (a-2) for the case with

_{SVD}*λ*= 0.97 demonstrates that the influenc of the background is completely removed.

_{SVD}*λ*= 0.97 is shown in Fig. 6 (b-1) and is accompanied by the artifact. When

_{SVD}*λ*= 0.8, the single emitting source region is well localized as shown in Fig. 6 (b-2). It is difficul to reconstruct the single target source region at the correct position for the case (c) as shown in Figs. 6 (c-1) and (c-2). Particularly, in Fig. 6(c-2) with

_{SVD}*λ*= 0.8 the target source region disappears due to the low SNR.

_{SVD}#### 3.4.3. Influenc of the mismatch in the optical properties

*µ*′

*and*

_{s}*µ*.

_{a}*µ*′

*= 0.8 mm*

_{s}^{−1}. And the inhomogeneous one has a particular sector regions in the range of (3/4)

*π*≤

*θ*≤ (5/4)

*π*with different

*µ*′

*, (a) 0.72, (b) 0.4 or (c) 0.08 mm*

_{s}^{−1}. The measured light intensities in the particular region increase with the decrease in

*µ*′

*.*

_{s}*µ*′

*=0.8 mm*

_{s}^{−1}. The number of the updating iterations is six. The true source distribution is shown in Fig. 1 (h).

*λ*are successful in the cases (a) and (b), respectively. The artifact reduction alleviates the mismatches. On the other hand, it is not possible to reconstruct the images without the artifacts as shown in Fig. 8 (c-1) and (c-2) in the case (c).

_{SVD}*µ*′

*of the particular region is so small that the measured light intensities increase to around 10% of the maximum of the measurement data as shown in Fig. 7. The artifacts appear to compensate the mismatch in the optical properties. The artifact is readily removed when the correct distribution of*

_{s}*µ*′

*is used for the reconstruction as shown in Fig. 8 (c-3). In the case of*

_{s}*µ*′

*= 8.0 mm*

_{s}^{−1}in the particular region which is 10 times larger than

*µ*′

*in the rest of the medium, the emitting source region is reconstructed well (Fig. 8(d)). The light intensities measured by the detectors far from the emitting source region are essentially so small that the strong attenuation by the particular region hardly affects the measurement data.*

_{s}*µ*is studied by the same manner as that in

_{a}*µ*′

*. Figure 9 shows the measured intensities for the homogeneous medium with*

_{s}*µ*= 0.007 mm

_{a}^{−1}and the inhomogeneous media with

*µ*of (a) 0.0035 or (b) 0.0007 mm

_{a}^{−1}in the particular region.

*µ*′

*is given as 0.8 mm*

_{s}^{−1}uniformly. The smaller

*µ*is, the larger intensities measured in the particular region are.

_{a}*µ*= 0.007 mm

_{a}^{−1}in the cases (a) and (b), respectively. The appropriate

*λ*of 0.99 and 0.97 recovers the emitting source regions. Insufficien removing of the noise space with

_{SVD}*λ*= 0.99, leads to the artifacts as shown in Fig. 10 (b-2). Figure 10 (b-3) shows that the reconstruction with the true distribution of

_{SVD}*µ*does not produce any artifacts. The large

_{a}*µ*in the particular region hardly affects the image as shown in Fig. 10(c).

_{a}### 3.5. Spatial resolution of the reconstructed images

*SD*= 0.01 are added to the simulated measurement data.

_{noise}*x,y*) = (−10, 0) and (10, 0), the reconstruction with

*λ*= 0.98 and seven updating iterations separates two emitting sources clearly as shown in Fig. 11 (a). The number of the updating iterations to obtain well localized source regions increases when the true sources are embedded deeply. When the centers of the two sources are at (

_{SVD}*x,y*) = (−7.5, 0) and (7.5, 0), the reconstruction with

*λ*of 0.97 and seven updating iterations localizes two emitting sources separately (Fig. 11(b)).

_{SVD}*x,y*) = (−5.0, 0) and (5.0, 0), the two emitting sources are not clearly separated although two peaks are recognizable as shown in Fig. 11 (c). The number of the iteration is seven, and the separation is not improved by the updating iteration anymore. Figure 11 (d) shows the reconstructed image with

*λ*= 0.99 and six iterations when the centers of the emitting regions are (

_{SVD}*x,y*) = (−7.5, 20) and (7.5, 20). Two emitting source regions in Fig. 11(d) are separated as well as in Fig. 11 (b). The proposed method has a potential to localize two emitting source regions separately when the two source regions are separated more than about 10 mm.

## 4. Advanced simulations of FDOT

### 4.1. 2D reconstruction

#### 4.1.1. Generation of the measurement data

*π*≤

*θ*≤ (5/4)

*π*with

*µ*=0.0070 mm

_{a}^{−1}and

*µ*′

*= 0.72 mm*

_{s}^{−1}at the excitation wavelength. In the second sector region in −(1/4)

*π*≤

*θ*≤ (1/4)

*π*,

*µ*=0.0056 mm

_{a}^{−1}and

*µ*′

*=0.80 mm*

_{s}^{−1}, and in the third section

*µ*=0.0070 mm

_{a}^{−1}and

*µ*′

*=0.80 mm*

_{s}^{−1}. Figures 12(a) and (b) show the true distributions of

*µ*and

_{a}*D*= 1/(3

*µ*′

*), respectively.*

_{s}*ε*, is 0.013

*µ*M

^{−1}mm

^{−1}and the quantum yield,

*η*, is 0.016. We assume two regions with high concentrations of ICG. One is a circular region having a radius of 2.5 mm with its center at (

*x,y*) = (0, 20) and with the concentration of ICG,

*N*, of 5

*µ*M. Therefore,

*µ*of this region increases by

_{a}*µ*(=

_{af}*εN*) = 0.065 mm

^{−1}to 0.072 mm

^{−1}due to ICG. The other is a circular region having a radius of 5 mm with its center at (

*x,y*) = (0,−20) and with N of 1

*µ*M leading to

*µ*in this circular region of 0.020 mm

_{a}^{−1}. Besides, ICG is distributed uniformly in the medium with the concentration of 0.001

*µ*M which causes the background emission. The true distribution of

*µ*+

_{a}*µ*is shown in Fig. 12(c).

_{af}*µ*and

_{a}*µ*′

*at the emission wavelength are assumed to be equal to those at the excitation wavelength.*

_{s}*, is shown in Fig. 12(d). The fluenc rate of the fluorescenc light is calculated with the true fluorescenc light sources,*

_{x}*q*

_{0}=

*ηµ*Φ

_{af}*which is shown in Fig. 12(e). There exist strong emission from the two regions containing the concentrated ICG and the background emissions observed clearly near the excitation light sources. The detectors for the fluorescenc light are placed at 16 positions in the middle of the neighboring two excitation light sources. The excitation light is filtere out and does not contaminate the measurement data. Gaussian noises with*

_{x}*SD*= 0.01 are added to the measurement data.

_{noise}#### 4.1.2. Reconstruction of the fluorescenc light sources

*µ*and

_{a}*µ*′

*are assumed in the medium with the values of 0.007 mm*

_{s}^{−1}and 0.8 mm

^{−1}, respectively. Figures 13(a) to (e) show the reconstructed images when

*λ*are (a) 0.995, (b) 0.9, and (c), (d) and (e) 0.8, respectively. The numbers of the updating iterations are six for cases (a) to (c), f ve for case (d), and seven for case (e).

_{SVD}*λ*. Less strong artifacts appear in the left half of the circular medium due to the mismatch in

_{SVD}*µ*′

*. The influenc of the mismatch in*

_{s}*µ*′

*in the left half is more significan than that in*

_{s}*µ*in the right half. Although the two emitting source regions are reconstructed, the effect of removing the noise space is not sufficient

_{a}*λ*smaller than that in Fig. 13(a). Comparing with Fig. 13 (a), the artifact at the center of the medium is eliminated, and the localization of two emitting sources is improved. From these results,

_{SVD}*λ*= 0.80 seems more preferable.

_{SVD}*λ*= 0.8 and six iterations shown in Fig. 13(c) are the closest to the true ones among the three cases (c), (d) and (e).

_{SVD}*λ*and the number of iterations are somewhat influential

_{SVD}### 4.2. 3D reconstruction

*µ*′

*and*

_{s}*µ*listed in Table 1 [42,43

_{a}43. Y. Lv, J. Yian, W. Cong, G. Wang, W. Yang, C. Qin, and M. Xu, “Spectrally resolved bioluminescence tomography with adaptive finit element analysis: methodology and simulation,” Phys. Med. Biol. **52**, 4497–4512 (2007). [CrossRef] [PubMed]

*x,y*) = (−6, 0, 12) and (6, 0, 18) with the radius of 1.5 mm. The ICG concentrations in the regions are given as 1

*µ*M. The illuminating and the detecting positions are placed in the

*x*−

*y*planes of

*z*= 9 mm and 21 mm. Each plane has eight illuminating and eight detecting positions with an equal spacing. The detectors are placed in the middle of the two neighboring illuminating positions. The measurement data are generated by the same manner as in the 2D reconstruction in the previous section. The reconstructions are conducted from the measurement data of the 16 measured fluorescenc light intensities. The FEM calculation employs 5631 nodes and 24000 elements for the generation of the measurement data.

^{−6}.

*µ*= 0.0018 mm

_{a}^{−1}and

*µ*′

*=0.996 mm*

_{s}^{−1}, are used in the reconstruction. In addition to the two emitting regions, ICG is uniformly distributed with the concentration of 0.0001

*µ*M in the whole medium. The measurement data contain the Gaussian noise with

*SD*= 0.01. Figure 15(b) shows the images of the source strength reconstructed from the noisy measurement data with seven iterations and

_{noise}*λ*= 0.995. Strong artifacts are making the emitting regions unclear.

_{SVD}*λ*=0.9 and 0.8, the emitting sources are successfully reconstructed from the noisy data as shown in Figs. 15 (c) and 15(d), respectively, although some artifacts are observed. Also the positions of the emitting sources are slightly shifted from the true positions partly due to the coarser mesh than that for Fig. 15(a).

_{SVD}## 5. Conclusions

## References and links

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2. | V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. |

3. | S. R. Arridge, “Optical tomography in medical imaging,” Inverse Prob. |

4. | A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. |

5. | M. S. Patterson and B. W. Pogue, “Mathematical model for time-resolved and frequency-domain fluore cence spectroscopy in biological tissues,” Appl. Opt. |

6. | D. Grosenick, H. Wabnitz, H. H. Rinneberg, T. Moesta, and P. M. Schlag, “Development of a time-domain optical mammography and fir t |

7. | D. Grosenick, K. T. Moesta, M. Möller, J. Mucke, H. Wabnitz, B. Gebauer, C. Stroszczynski, B. Wassermann, P. M. Schlag, and H. Rinneberg,“Time-domain scanning optical mammography: I. Recording and assessment of mammograms of 154 patients,” Phys. Med. Biol. |

8. | D. Grosenick, H. Wabnitz, K. T. Moesta, J. Mucke, P. M. Schlag, and H. Rinneberg, “Time-domain scanning optical mammography: II. Optical properties and tissue parameters of 87 carcinomas,” Phys. Med. Biol. |

9. | J. C. Hebden, H. Veenstra, H. Dehghani, E. M. C. Hillman, M. Schweiger, S. R. Arridge, and D. T. Delpy, “Three-dimensional time-resolved optical tomography of a conical breast phantom,” Appl. Opt. |

10. | T. Yates, C. Hebdan, A. Gibson, N. Everdell, S. R. Arridge, and M. Douek, “Optical tomography of the breast using a multi-channel time-resolved imager,” Phys. Med. Biol. |

11. | G. Boverman, E. L. Miller, A. Li, Q. Zhang, T. Chaves, D. H. Brooks, and D. A. Boas “Quantitative spectroscopic optical tomography of the breast guided by imperfect a priori structural information,” Phys. Med. Biol. |

12. | R. Weissleder, “Molecular Imaging in Cancer,” SCIENCE |

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15. | K. Vishwanath, B. Pogue, and M-A. Mycek, “Quantitative fluore cence lifetime spectroscopy in turbid media: comparison of theoretical, experimental and computational method,” Phys. Med. Biol. |

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18. | A. B. Milstein, J. J. Scott, S. Oh, D. A. Boas, R. P. Millane, C. A. Bouman, and K. J. Webb, “Fluorescence optical diffuse tomography using multiple-frequency data,” J. Opt. Soc. Am. A |

19. | D. Y. Paithankar, U. A. Chen, B. W. Pogue, M. S. Patterson, and E. M. Sevick-Muraca, “Imaging of fluore cent yield and lifetime from multiply scattered light reemitted from random medium,” Appl. Opt. |

20. | F. Gao, H. Zhao, L. Zhang, Y. Tanikawa, A. Marjono, and Y. Yamada, “A self-normalized, full time-resolved method for fluore cence diffuse optical tomography,” Opt. Express |

21. | A. Marjono, A. Yano, S. Okawa, F. Gao, and Y. Yamada, “Total light approach of time-domain fluore cence diffuse optical tomography,” Opt. Express |

22. | L. Zhang, F. Gao, H. He, and H Zhao, “Three-dimensional scheme for time-domain fluore cence molecular tomography based on Laplace transforms with noise-robust factors,” Opt. Express |

23. | F. Gao, H. Zhao, Y. Tanikawa, and Y. Yamada, “A linear, featured-data scheme for image reconstruction in time-domain fluore cence molecular tomography,” Opt. Express , |

24. | M. J. Eppstein, D. E. Doughety, D. J. Hawrysz, and E. M. Sevick-Muraka, “Three-Dimensional Baysian Optical Image Reconstruction with Domain Decomposition,” IEEE trans. Med. Imaging |

25. | M. A. O’Leary, D. A. Boas, X. D. Li, B. Chance, and A. G. Yodh, “Fluorescence lifetime imaging in turbid media,” Opt. Lett. |

26. | A. Soubret, J. Ripoll, D. Yessayan, and V. Ntziachristos, “Three-dimensional fluore cent tomography in presence of absorption: Study of the normalized Born approximation,” in |

27. | A. X. Cong and G. Wang, “A finite-elemen reconstruction method for 3D fluore cence tomography,” Opt. Express |

28. | G. Wang, W. Cong, K. Durairaj, X. Qian, H. Shen, P. Sinn, E. Hoffman, G. McLennan, and M. Henry, “ |

29. | B. D. van Veen and K. M. Buckly, “Beamforming: A versatile approach to spatial filtering” IEEE ASSP Mag. |

30. | S. Baillet, J. C. Mosher, and R. M. Leahy, “Electromagnetic brain mapping,” IEEE Signal Process Mag. |

31. | B. D. van Veen, W. van Drongelen, M. Yuchtman, and A. Suzuki, “Localization of brain electrical activity via linear constrained minimum variance spatial filte,” IEEE trans. Biomed. Eng. |

32. | S. Okawa and S. Honda, “MEG Analysis with Spatial Filtered Reconstruction,” IEICE Trans. on Fundam. Electron. Commun. Comput. Sci. |

33. | S. Okawa and S. Honda, “Dipole estimation with a combination of noise reduction and spatial filte,” International Congress Series |

34. | S. Okawa and Y. Yamada, “Source estimation with spatial filte for fluore cence diffuse optical tomography,” in |

35. | C. Kuo, O. Coquoz, T. L. Troy, H. Xu, and B. W. Rice, “Three-dimensional reconstruction of in vivo bioluminescent sources based on multispectral imaging,” J. Biomed. Opt. |

36. | M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the Finite-Element Method for the Forward and Inverse Model in Optical Tomography,” J. Math. Imaging Vis. |

37. | C. R. Vogel, |

38. | C. C. Paige and M. A. Saunders, “LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares,” ACM trans. on Math. Software |

39. | C. W. Groetsch, |

40. | C. W. Groetsch, |

41. | S. Holder, |

42. | S. Okawa and Y. Yamada, “3D Light Source Reconstruction with Spatial Filter for Fluorescence/ Bioluminescence Diffuse Optical Tomography,ffi |

43. | Y. Lv, J. Yian, W. Cong, G. Wang, W. Yang, C. Qin, and M. Xu, “Spectrally resolved bioluminescence tomography with adaptive finit element analysis: methodology and simulation,” Phys. Med. Biol. |

**OCIS Codes**

(100.3190) Image processing : Inverse problems

(170.3880) Medical optics and biotechnology : Medical and biological imaging

**ToC Category:**

Image Processing

**History**

Original Manuscript: March 25, 2010

Revised Manuscript: May 27, 2010

Manuscript Accepted: May 28, 2010

Published: June 3, 2010

**Virtual Issues**

Vol. 5, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

Shinpei Okawa and Yukio Yamada, "Reconstruction of fluorescence/bioluminescence sources in biological medium with spatial filter," Opt. Express **18**, 13151-13172 (2010)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-18-12-13151

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### References

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