## Reconstruction of localized fluorescent target from multi-view continuous-wave surface images of small animal with
_{p} |

Biomedical Optics Express, Vol. 5, Issue 6, pp. 1839-1860 (2014)

http://dx.doi.org/10.1364/BOE.5.001839

Acrobat PDF (3425 KB)

### Abstract

Fluorescence diffuse optical tomography using a multi-view continuous-wave and non-contact measurement system and an algorithm incorporating the *l _{p}* (0

*< p*≤ 1) sparsity regularization reconstructs a localized fluorescent target in a small animal. The measurement system provides a total of 25 fluorescence surface 2D-images of an object, which are acquired by a CCD camera from five different angles of view with excitation from five different angles. Fluorescence surface emissions from five different angles of view are simultaneously imaged on the CCD sensor, thus leading to fast acquisition of the 25 images within three minutes. The distributions of the fluorophore are reconstructed by solving the inverse problem based on the photon diffusion equations. In the reconstruction process incorporating the

*l*sparsity regularization, the regularization term is reformulated as a differentiable function for gradient-based non-linear optimization. Numerical simulations and phantom experiments show that the use of the

_{p}*l*sparsity regularization improves the localization of the target and quantitativeness of the fluorophore concentration. A mouse experiment demonstrates that a localized fluorescent target in a mouse is successfully reconstructed.

_{p}© 2014 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]

4. A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. **50**, R1–R43 (2005). [CrossRef] [PubMed]

5. R. Weissleder, “Molecular imaging in cancer,” Science **321**, 1168–1171 (2006). [CrossRef]

6. D. J. Hawrysz and E. M. Sevick-Muraca, “Developments toward diagnostic breast cancer imaging using near-infrared optical measurements and fluorescent contrast agents,” Neoplasia **2**(5), 388–417 (2000). [CrossRef]

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

8. D. Y. Paithankar, U. A. Chen, B. W. Pogue, M. S. Patterson, and E. M. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light reemitted from random medium,” Appl. Opt. **36**(10), 2260–2272 (1997). [CrossRef] [PubMed]

9. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, “A submillimeter resolution fluorescence molecular imaging system for small animal imaging,” Med. Phys. **30**, 901–911 (2003). [CrossRef] [PubMed]

10. N. C. Deliolanis, T. Lasser, D. Hyde, A. Soubret, J. Ripoll, and V. Ntziachristos, “Free-space fluorescence molecular tomography utilizing 360° geometry projections,” Opt. Lett. **32**(4), 382–384 (2007). [CrossRef] [PubMed]

*L*

_{1}norm minimization by use of an expectation maximization algorithm for a linearized DOT inverse problem, and showed that reconstructed regions with abnormal optical properties were localized more sharply than other methods they used [20

20. N. Cao, A. Nehorai, and M. Jacob, “Image reconstruction for diffuse optical tomography using sparsity regularization and expectation-maximization algorithm,” Opt. Express , **15**(21), 13695–13708 (2007). [CrossRef] [PubMed]

21. T. Shimokawa, T. Kosaka, O. Yamashita, N. Hiroe, T. Amita, Y. Inoue, and M. Sato, “Hierarchical Bayesian estimation improves depth accuracy and spatial resolution of diffuse optical tomography,” Opt. Express **20**(18), 20427–20446 (2012). [CrossRef] [PubMed]

22. P. Xu, Y. Tian, H. Chen, and D. Yao, “L_{p} Norm Iterative Sparse Solution for EEG Source Localization,” IEEE Trans. Biomed. Eng. **54**(3), 400–409 (2007). [CrossRef] [PubMed]

23. P. M. Shankar and M. A. Neifeld, “Sparsity constrained regularization for multiframe image restoration,” J. Opt. Soc. Am. A **25**(5), 1199–1214 (2008). [CrossRef]

24. M. Freiberger, C. Clason, and H. Scharfetter, “Total variation regularization for nonlinear fluorescence tomography with an augmented Lagrangian splitting approach,” Appl. Opt. **49**(19), 3741–3747 (2010). [CrossRef] [PubMed]

*L*

_{1}norm minimization was numerically investigated by Han et al. [25

25. D. Han, X. Yang, K. Liu, C. Qin, B. Zhang, X. Ma, and J. Tian, “Efficient reconstruction method for L1 regularization in fluorescence molecular tomography,” Appl. Opt **49**(36), 6930–6937 (2010). [CrossRef] [PubMed]

26. D. Han, J. Tian, S. Zhu, J. Feng, C. Qin, B. Zhang, and X. Yang, “A fast reconstruction algorithm for fluorescence molecular tomography with sparsity regularization,” Opt. Express **18**(8), 8630–8646 (2010). [CrossRef] [PubMed]

*L*

_{1}norm minimization and Landweber iterative method with adaptive meshing of the finite element method (FEM) [27

27. H. Yi, D. Chen, X. Qu, K. Peng, X. Chen, Y. Zhou, J. Tian, and J. Liang, “Multilevel, hybrid regularization method for reconstruction of florescent molecular tomography,” Appl. Opt. **51**(7), 975–986 (2012). [CrossRef] [PubMed]

*L*

_{1}sparsity constraint was also applied to fluorescence/bioluminescence diffuse optical tomography (F/BDOT) [28

28. P. Mohajerani, A. A. Eftekhar, J. Huang, and A. Adibi, “Optimal sparse solution for fluorescent diffuse optical tomography: theory and phantom experimental results,” Appl. Opt. **46**(10), 1679–1685 (2007). [CrossRef] [PubMed]

29. Y. Lu, X. Zhang, A. Douraghy, D. Stout, J. Tian, T. F. Chan, and A. F. Chatziioannou, “Source reconstruction for spectrally-resolved bioluminescence tomography with aparse A priori information,” Opt. Express **17**(10), 8062–8088 (2009). [CrossRef] [PubMed]

30. S. Okawa and Y. Yamada, “Reconstruction of fluorescence/bioluminescence sources in biological medium with spatial filter,” Opt. Express **18**(12), 13151–13172 (2010). [CrossRef] [PubMed]

*l*(0 <

_{p}*p*≤ 1) sparsity regularization in the FDOT reconstruction algorithm by partial use of the focal underdetermined system solver (FOCUSS) algorithm [31

31. Z. He, A. Cichocki, R. Zdunek, and S. Xie, “Improved FOCCUS method with conjugate gradient iterations,” IEEE Trans. Signal Process. **57**(1), 399–404 (2009). [CrossRef]

32. S. Okawa, Y. Hoshi, and Y. Yamada, “Improvement of image quality of time-domain diffuse optical tomography with l_{p} sparsity regularization,” Biomed. Opt. Express **2**(12), 3334–3348 (2011). [CrossRef] [PubMed]

*p*-norm of the solution which is strictly constrained by the linear forward model.

*l*norm of the concentration of the fluorophore simultaneously. Numerical simulations, phantom and mouse experiments demonstrate that the

_{p}*l*(0 <

_{p}*p*≤ 1) sparsity regularization improves the localization of the fluorophore in the reconstructed images.

## 2. Method

### 2.1. Measurement system

*in vivo*optical imaging system, Clairvivo OPT (Shimadzu Co., Kyoto, Japan), was used for acquisition of the fluorescence surface images of objects in this study. Figure 1 shows the schematics for acquiring the fluorescence surface images. The system consisted of a horizontal animal table, excitation light sources, a CCD camera, reflecting mirrors, a camera lens, an optical filter and eight white LED light sources illuminating the animal from four different angles. The table was made of plexiglass and was transparent to visible and NIR light. A small animal either anesthetized or sacrificed was laid down and fixed by adhesive tapes at a right position on the table without using an animal holder. The maximum sizes of an object were 40 mm, 30 mm and 120 mm in width, height and length, respectively, which were limited by the view of the CCD camera. A gas port for anesthesia was installed at an end of the table, and would be connected to the head of a small animal when

*in vivo*measurements would be made.

^{2}illuminated the whole object from five different polar angles of 52, 120, 180, 240 and 308 degrees, i.e., Excitation 1 to Excitation 5 as shown in Fig. 1(b). Each laser light illuminated the object with a width of 40 mm and a length of more than 120 mm.

### 2.2. Forward modeling

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

*D*= 1/(3

*μ′*) is the diffusion coefficient with the reduced scattering coefficient

_{s}*μ′*,

_{s}*μ*the absorption coefficient of the background medium, and Φ the fluence rate at the position

_{a}*r*with the subscripts

*x*and

*m*indicating excitation and emission light, respectively.

*ε*and

*γ*are the molar extinction coefficient and the quantum yield, respectively, and

*N*is the fluorophore concentration. The light source

*q*

_{0}is distributed uniformly in the area of the surface illuminated by the excitation light. The boundary conditions are given as −

*n*·

*D*∇Φ

_{y}*= Φ*

_{y}*/(2*

_{y}*A*) where

*n*is the vector outward normal to the surface of the medium, subscript

*y*refers to

*x*or

*m*, and

*A*is the parameter depending on the ratio of the refractive indices,

*n*. In this study,

_{r}*n*is give as 1.36, then

_{r}*A*= 3. The forward solutions of light propagation are obtained by solving Eqs. (1) and (2) using the finite element method (FEM) [33

33. 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]

*r*) = −

*n*·

*D*∇Φ

_{m}*= Φ*

_{m}*/(2*

_{m}*A*), and are calculated from the solutions of Eqs. (1) and (2).

*, is given for the*

_{i,j,k,l}*i*-th excitation,

*j*-th angle of view,

*k*-th pixel in an surface image, and the

*l*-th voxel in a medium containing a unit fluorophore concentration where

*i*and

*j*= 1, 2,···,5,

*k*= 1, 2,···,

*K*,

*l*= 1, 2,···,

*L*. Here,

*K*is the total number of the pixels in the surface image, and

*L*is the total number of the voxels in the medium. Because Γ

*is the fluorescence intensity when the fluorophore of unit concentration is placed at the*

_{i,j,k,l}*l*-th voxel, it corresponds to the sensitivity of the

*l*-th voxel to the measured fluorescence intensity. Then the measured fluorescence intensities represented by a vector,

*F*, are calculated by the product of the sensitivity matrix,

*H*, and the fluorophore concentration vector,

*f*, consisting of the fluorophore concentrations at

*L*voxels as Eq. (3), where

*F*and

*f*are vectors of 5 × 5 ×

*K*and

*L*components, respectively, and

*H*is a matrix of 5 × 5 ×

*K*rows and

*L*columns consisting of Γ

*, as Eq. (4),*

_{i,j,k,l}### 2.3. Image reconstruction with l_{p} sparsity regularization

*f*, is carried out by minimizing the residual error between the measured data represented by a vector

*M*and the calculated data of the vector

*F*=

*Hf*, and is formulated by the following minimization problem with application of the

*l*sparsity regularization, where

_{p}*λ*is a regularization parameter, and

*f*is a component of the vector

_{l}*f*. The first and second terms in the bracket of Eq. (5) are the residual error term and the sparsity regularization term, respectively, and the second term is introduced to minimize the

*l*norm (0 <

_{p}*p*≤ 1) of the reconstructed

*f*. By this regularization

*f*becomes sparse as the exponential factor,

*p*, approaches zero. Changing the magnitude of

*p*adjusts the sparseness of

*f*so that images can be modified by taking some prior information about the fluorescent target into account.

*p*≤ 1, there exists a difficulty in calculating gradients of the function in the bracket of Eq. (5) when a gradient-based optimization is used to solve Eq. (5), because |

*f*|

_{l}^{p−1}goes to infinity at singular points where the values of

*f*are close to zero. To overcome this difficulty,

_{l}*f*is replaced by the following formulation with another variable

_{l}*z*used in FOCUSS algorithm [31

_{l}31. Z. He, A. Cichocki, R. Zdunek, and S. Xie, “Improved FOCCUS method with conjugate gradient iterations,” IEEE Trans. Signal Process. **57**(1), 399–404 (2009). [CrossRef]

*z*represents a vector consisting of

*z*. Now the gradient of the function in the bracket of Eq. (7) can be calculated without difficulty. A nonlinear conjugate gradient method [34

_{l}34. C. R. Vogel, *Computational Methods for Inverse Problems (Frontiers in Applied Mathematics)* (SIAM, Philadelphia, 2002). [CrossRef]

35. S. R. Arridge, “A gradient-based optimization scheme for optical tomography,” Opt. Express **12**(6), 213–226 (1998). [CrossRef]

*f*given by the solution using the Tikhonov regularization which is the case of

*p*= 2 in Eq. (5) [21

21. T. Shimokawa, T. Kosaka, O. Yamashita, N. Hiroe, T. Amita, Y. Inoue, and M. Sato, “Hierarchical Bayesian estimation improves depth accuracy and spatial resolution of diffuse optical tomography,” Opt. Express **20**(18), 20427–20446 (2012). [CrossRef] [PubMed]

*l*sparsity regularization.

_{p}^{®}Xenon processor W5580 (8M Cache, 3.20 GHz, 6.40 GT/s Intel

^{®}QPI), RAM with 24 GB and Matlab with Parallel Computing Toolbox were used. Preprocessing including FEM meshing, generation of the sensitivity matrix,

*H*, of Eq. (4) and singular value decomposition of

*H*necessary for reconstruction using the Tikhonov regularization required about 24 hours. Once preprocessing was carried out image reconstruction using the Tikhonov regularization took about 40 s and fifty iterations for reconstruction using the

*l*sparsity regularization took about 114 s, resulting in the total time of about 154 s.

_{p}## 3. Simulations and experiments

### 3.1. Numerical simulations

#### 3.1.1. Single target

*M*, are generated by solving the fundamental equations (1) and (2) with 3D-FEM by use of COMSOL Multiphysics ver. 3.5a (COMSOL Inc., MA, USA). The object is a cylinder having a diameter of 25 mm and a height of 50 mm with the

*x*-

*y*-

*z*coordinate as shown in Fig. 2. The cylinder is discretized into 24,485 tetrahedral elements.

*μ*and

_{a}*μ′*of the cylinder are given as 0.022 mm

_{s}^{−1}and 0.6 mm

^{−1}, respectively. The optical properties are given by referring those of the rat liver [36

36. W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. **26**, 2166–2185 (1990). [CrossRef]

*μ*is made smaller than that of rat liver by considering the existence of other tissues and organs. A single fluorescent target of indocyanine green (ICG) is assumed to be placed in the plane (

_{a}*x*–

*y*plane) perpendicular to the cylinder axis (

*z*-axis) at a height of 25 mm (

*z*= 0). ICG is a popular fluorescent diagnostic dye approved by Food and Drug Administration (FDA). The peak excitation and emission wavelengths of ICG are 780 and 830 nm, respectively [37

37. A. B. Milstein, S. Oh, K. Webb, C. A. Bouman, Q. Zhang, D. A. Boas, and R. P. Millane, “Fluorescence optical diffusion tomography,” Appl. Opt. **42**(16), 3081–3094 (2003). [CrossRef] [PubMed]

39. B. Toczylowska, E. Zieminska, G. Goch, D. Milej, A. Gerega, and A. Liebert, “Neurotoxic effects of indocyanine green-cerebellar granule cell culture viability study,” Biomed. Opt. Express , **5**(3), 800–816 (2014). [CrossRef] [PubMed]

*K*= 25 × 51 = 1275 data, and the total number of the measurement data is 5 × 5 × 25 × 51 = 31875. In the image reconstruction, the cylinder is discretized into

*L*= 25347 voxels with volumes of 1 mm

^{3}. Therefore, the matrix

*H*has 31875 rows and 25347 columns. In this case, the number of the measurement data, 31875, is larger than that of the unknowns, 25347, and the ill-posedness of the inversion process might be relaxed.

**Case (i):**The target has a volume of 1 mm^{3}and 100 pmol of ICG, and the depth, which means the distance from the center of the target to the nearest surface point in a radial direction, is varied as 4, 6, and 9 mm. When the coordinate of the target center is expressed by (*x*,_{c}*y*,_{c}*z*), these varied depths are described by (_{c}*x*,_{c}*y*,_{c}*z*) = (0.0, 8.5, 0.0), (0.0, 6.5, 0.0) and (0.0, 3.5, 0.0), respectively._{c}**Case (ii):**The target has a volume of 1.0 mm^{3}, and the quantity of ICG is changed as 100, 10, and 1 pmol with its depth fixed as 6 mm.**Case (iii):**The volume of the target is changed as 1, 8, and 27 mm^{3}with the quantity of ICG and depth of the target fixed as 100 pmol and 6 mm, respectively.

*c*represents photon counts detected by a CCD camera. Image reconstruction is carried out using the Tikhonov regularization or the

*l*sparsity regularization with

_{p}*p*= 0.5 or 1. The size of the voxel for reconstruction is 1 mm × 1 mm × 1 mm. The reconstructed quantity of ICG in the target is evaluated with a volume-of-interest (VOI) analysis. The VOI which surrounds the true target position is a cube with a side of 5 mm, and the quantity of ICG in the VOI,

*R*

_{VOI}, is expressed by the following equation,

*y*-position,

*Q*(

*y*), which is calculated by integrating the ICG concentrations (quantity in a unit volume of 1.0 mm

^{3}),

*N*(

*x*,

*y*,

*z*), over a rectangular parallelepiped volume of 1 mm × 1 mm × 50 mm along the

*z*-axis from

*z*= −25 mm to

*z*= 25 mm as expressed by the following,

#### 3.1.2. Multiple targets

*y*-axis with a distance of

*d*at the fixed

*y*-position in the

*x*–

*y*plane at

*z*= 0. Both targets have the same volume of 1 mm

^{3}. The distance,

*d*, and the ICG quantities in the targets are varied in the following cases (iv) and (v).

**Case (iv):***d*is varied as 4.0, 6.0, and 8.0 mm with*y*= 6.5 mm, resulting in the coordinates of the target centers of (_{c}*x*,_{c}*y*,_{c}*z*) = (±2.0, 6.5, 0.0), (±3.0, 6.5, 0.0), and (±4.0, 6.5, 0.0). The quantities of the ICG in the both targets are 100 pmol equally._{c}**Case (v):***d*is fixed as 6.0 mm with*y*= 6.5 mm, but the quantities of ICG in the targets are different. One target on the left hand side located at (_{c}*x*,_{c}*y*,_{c}*z*) = (−3.0, 6.5, 0.0) contains 100 pmol, and the other on the right hand side located at (_{c}*x*,_{c}*y*,_{c}*z*) = (3.0, 6.5, 0.0) contains 75, 50, or 25 pmol._{c}

*Q*(

*x*), is calculated in the manner similar to Eq. (9),

### 3.2. Phantom experiments

*μ*= 0.022 mm

_{a}^{−1}and

*μ′*= 0.6 mm

_{s}^{−1}, which were achieved by adjusting the concentrations of the pigment ink (0.31 wt%) and kaolin particle (1.95 wt%). A single fluorescent target with a diameter of 1 mm and a height of 3 mm was located in the plane at the height of 110 mm from the bottom of the phantom. The target contained 13 pmol ICG, and the depth of the target was changed as 4, 6, and 9 mm. The fluorescence surface images of the phantom were acquired using Clairvivo OPT. Image reconstruction was carried out for a partial region of the phantom with a diameter of 25 mm and a height of 50 mm, which included the target. The reconstructed target images were evaluated by a VOI analysis with the size of the VOI of 5 mm × 5 mm × 10 mm.

### 3.3. Mouse experiment

^{®}) was embedded in the abdomen of the mouse. Magnevist

^{®}(gadopentetate dimeglumine) is an FDA-approved agent and used clinically for MRI to visualize lesions with abnormal vascularity [40

40. J. Barkhausen, W. Ebert, J. F. Debatin, and H.-J. Weinmann, “Imaging of myocardial infarction: comparison of magnevist and gadophrin-3 in rabbits,” J. Am. Coll. Cardiol. **39**(8), 1392–1398 (2002). [CrossRef] [PubMed]

*K*= 532 data (about half that for the phantom experiments), and the total number of the measurement data used for image reconstruction was 5×5×532=13,300. Image reconstruction was carried out for the abdominal region of the mouse which was modeled with a total of

*L*= 6979 cubic voxels with each voxel having a side of 1.0 mm. For forward calculation to provide the matrix

*H*, an FEM mash having 125,613 tetrahedral elements was used to solve the photon diffusion equations (1) and (2). The FEM model of the mouse abdomen was generated from the MR image using a software, ZedView (LEXI Co., Ltd., Tokyo, Japan). The number of the measurement data of 13,300 was again larger than that of the unknowns of 6979. The optical properties of the liver of the mouse,

*μ′*= 0.56 mm

_{s}^{−1}and

*μ*= 0.072 mm

_{a}^{−1}[36

36. W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. **26**, 2166–2185 (1990). [CrossRef]

## 4. Results and discussions

### 4.1. Numerical simulations

#### 4.1.1. Single target

*Q*(

*y*) profiles (not shown) ranging from 4 mm to 6 mm. The deeper the target position is, the broader the target is reconstructed.

*l*sparsity regularization is used, the reconstructed targets are more sharply localized than using the Tikhonov regularization. The targets were localized at the correct positions with FWHMs of 3 mm and 1 mm when

_{p}*p*= 1 and 0.5, respectively. A smaller value of

*p*localizes the target more sharply, and the target reconstructed with

*p*= 0.5 is as sharp as the true image.

*R*

_{VOI}for Case (i).

*R*

_{VOI}using the Tikhonov regularization (blue dashed line) are about 30 % to 45 % of the true value of 100 pmol while they are 88 % to 99 % of the true value when the

*l*sparsity regularization (red solid and green chained lines) is used.

_{p}*R*

_{VOI}decreases as the target depth increases, probably due to diffusive light propagation and small signal-to-noise ratios of the detected fluorescence emission light. This effect of the target depth is often observed in FDOT and DOT. The results for Case (i) indicate that the

*l*sparsity regularization reconstructs the quantity of the fluorophore robustly even when the target is embedded deeply inside the medium.

_{p}*l*sparsity regularization even when the true quantity of ICG reduces from 100 pmol to 1 pmol. The reconstructed target is sharper as

_{p}*p*becomes smaller, and is slightly broader as the true quantity becomes smaller. FWHMs are approximately from 5 mm to 7 mm when the Tikhonov regularization is used, while they are approximately 1 mm to 2 mm when

*p*=0.5, and approximately 3 mm when

*p*=1.

*R*

_{VOI}, shown in Fig. 5 indicate that the reconstructed ICG quantity changes almost linearly to the change in the true ICG quantity. When using the

*l*sparsity regularization, the value of

_{p}*R*

_{VOI}ranges from 84 % to 95 % of the true values which changes from 100 pmol to 1 pmol while it ranges from 31 % to 36 % of the true values when using the Tikhonov regularization. As the true value decreases, the percentage decreases slightly due to lower signal-to-noise ratios of the measurement noise.

*l*sparsity regularization with successful recovery of the ICG quantity. The reconstructed values of

_{p}*R*

_{VOI}are 90 % to 95 % and 83 % to 94 % of the true value for

*p*=1 and 0.5, respectively. On the other hand, the values of

*R*

_{VOI}using the Tikhonov regularization are 36 % to 39 % of the true value. When the volume of the true target is 27 mm

^{3}, the reconstructed target image using the Tikhonov regularization looks the closest to the correct image, while those using the sparsity regularization with

*p*= 1 and 0.5 look closest to the true images when the volumes of the true targets are 8 mm

^{3}and 1 mm

^{3}, respectively. From these results, it seems that the appropriate

*p*value depends on the target size, and prior information obtained by other imaging modalities will be useful for selecting an appropriate

*p*value to reconstruct the target size faithfully.

#### 4.1.2. Multiple targets

*Q*(

*x*) along the line connecting the two target centers. When the distance between the targets is 4 mm, the targets are reconstructed separately only when

*p*= 0.5. The maxima of

*Q*(

*x*) are 3.5, 16.0, and 28.4 pmol for the Tikhonov regularization,

*p*= 1, and

*p*= 0.5, respectively. The targets are reconstructed larger in volume and smaller in concentration than the true ones. The reconstructed quantities of the two targets are almost the same. When the distance between the two targets is 6 mm, the two targets are recognized in all of the reconstructed images. Especially, the two targets are clearly separated when

*p*= 1 and 0.5. The maxima of

*Q*(

*x*) are 3.1, 12.0, and 25.3 pmol for the Tikhonov regularization,

*p*= 1, and 0.5, respectively. The two targets are separated well regardless of the

*p*value when the distance is 8 mm. The maxima of

*Q*(

*x*) are 3.7, 16.6, and 22.3 pmol for the Tikhonov regularization,

*p*= 1, and 0.5, respectively. In all cases, the reconstructed quantities are almost the same for the two targets.

*Q*(

*x*) revealing two targets at the fixed positions with different quantities of ICG. The reconstructed ICG quantities of the targets on the right hand side are smaller than those on the left hand side corresponding to the true quantities listed in Table 2. When the Tikhonov regularization is used, the reconstructed targets are not clearly separated similarly to the result for Case (iv). As

*p*decreases, the spatial resolution of the reconstructed targets is improved, and the difference in the reconstructed quantities of ICG between the two targets increases. When the true ICG quantity of the target on the right hand side is 25 pmol, it is difficult to find it in the reconstructed images when

*p*= 1 and 0.5 while it is observable when the Tikhonov regularization is used. This is because the ICG concentrations in the reconstructed images are normalized by the maximum in each image. When the Tikhonov regularization is used, the maximum in the target on the left hand side is very small and close to that on the right hand side as seen in Fig. 10(c).

*l*sparsity regularization. The

_{p}*l*sparsity regularization term in Eq. (5) is minimized when the reconstructed fluorophore distribution of the target is localized in a small number of voxels (especially a single voxel) and/or when the reconstructed fluorophore concentrations in the target are close to zero. In the cases of single target, both the residual error and sparsity regularization terms in Eq. (5) can be simultaneously minimized by localizing the fluorophore distribution in a single voxel.

_{p}*l*sparsity regularization terms is not minimized. In order to minimize the sum, it is necessary to allow the fluorophores to exist at multiple voxels with lower fluorophore concentrations than those in the case of single target. So, the

_{p}*l*sparsity regularization term becomes smaller by taking smaller fluorophore concentrations. Appropriate adjustment of the FEM mesh by use of sophisticated meshing such as dual-meshing or adaptive-meshing may solve this problem. [41

_{p}41. A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express , **12**(22), 5402–5417 (2004). [CrossRef] [PubMed]

42. M. Huang and Q. Zhu, “Dual-mesh optical tomography reconstruction method with a depth correction that uses a priori ultrasound information,” Appl. Opt. **43**(8), 1654–1662 (2006). [CrossRef]

### 4.2. Phantom experiments

*c*was the maximum photon counts in the fluorescence surface image.

_{max}*R*

_{VOI}as a function of the target depth. Similarly to the numerical simulations, the

*l*sparsity regularization localized the targets more sharply than the Tikhonov regularization. Smaller

_{p}*p*localized targets more sharply, and the deeper targets were reconstructed more broadly. The VOI analysis revealed that the recovered ICG quantities using the

*l*sparsity regularization were approximately 18 pmol, 11 pmol and 5 pmol against the true value of 13 pmol for the target depths of 4 mm, 6 mm and 9 mm, respectively, while those using the Thikonov regularization were approximately 3.5 pmol, 4 pmol and 1.5 pmol, respectively. The reasons why the recovered ICG concentrations using the

_{p}*l*sparsity regularization for the target depth of 4 mm were larger than the true value are not clear, but unintentional heterogeneity of the optical properties in the phantom and possible failure of the diffusion approximation for the case of the target depth of 4 mm may be the reasons.

_{p}### 4.3. Mouse experiment

*l*sparsity regularization while it was reconstructed in a broader region by use of the Tikhonov regularization. Figure 16 shows the profiles of

_{p}*Q*(

*y*) along the

*y*-axis passing through the reconstructed targets. The maximum of

*Q*(

*y*), 10 pmol, reconstructed by use of the

*l*sparsity regularization with

_{p}*p*= 0.5 was three times that with

*p*= 1 and ten times that by use of the Tikhonov regularization.

*l*sparsity regularization, although the position of the reconstructed target was about 2 mm away from the true position. The error in the reconstructed target position may have been caused by the difference between the background optical properties used in the reconstruction process and those of the actual mouse, i.e., the background optical properties were assumed homogeneous in the reconstruction process while they were heterogeneous in the actual mouse. Various factors such as movement, temperature and metabolism of mouse will affect measurements when a mouse is alive. In this study, a sacrificed mouse with an embedded glass tube containing ICG was used for the purpose of focusing discussions on the validation of the algorithm and the performances of the imaging system. The present imaging system required 3 minutes to obtain the fluorescent surface images for image reconstruction. Real-time tomographic imaging with shorter acquisition and reconstruction times than in the present study will be a next challenge.

_{p}### 4.4. Selection of regularization parameter

*l*sparsity regularization in this study adjusts the degree of sparsity of fluorophore distributions by varying the value of

_{p}*p*appearing in Eq. (6). As

*p*becomes smaller, the power of

*z*, 2/

*p*, becomes larger. Then the variations in the values of

*z*at different positions are enlarged more by smaller

_{l}*p*when

*z*are transformed to

_{l}*f*. When

_{l}*z*= 0.1 and 1.0 for example,

_{l}*p*= 0.5 provides

*f*= 0.0001 and 1.0, respectively, while

_{l}*p*= 1 provides

*f*= 0.01 and 1.0, respectively. Therefore, localization of the reconstructed target is improved with smaller

_{l}*p*through the transformation of

*z*to

_{l}*f*.

_{l}*z*by the manner of the Tikhonov regularization. Throughout this paper,

_{l}*λ*in Eq. (7) is given as 5 × 10

^{−7}which was empirically determined prior to this study by numerical simulations where noisier measurement data were used. Reconstructed images using

*λ*smaller than 5 × 10

^{−7}(not shown) were not so much different from the images obtained here.

*λ*was very small, the reconstructed images were found to be less affected by the noises contained in the measurement data and by the mismatches between the forward model and the actual geometry. This fact indicated that the noises and mismatches in this study were small enough to reconstruct good images without regularization. However, if noises in the data measured by other measuring systems are larger than those by Clairvivo Opt used in this study, larger

*λ*must be used to suppress the effect of the noises, and resultantly the reconstructed target images will be blurred more. When the

*l*sparsity regularization is used for noisier measurement data,

_{p}*λ*should be selected by a certain method such as the L-curve method.

### 4.5. Comparison with other sparsity regularization methods

*p*-norm (0 <

*p*< 1) as discussed in this paper. Usually, the sparse reconstruction is carried out with minimizing the 1-norm of the solution in the inverse problems for FDOT or DOT, and the non-differentiability of the 1-norm at its singular points requires various techniques proposed in the recent literatures as the following.

*L*1 sparsity constraint can be achieved [43

43. V. C. Kavuri, Z.-J. Lin, F. Tian, and H. Liu, “Sparsity enhanced spatial resolution and depth localization n diffuse optical tomography,” Biomed. Opt. Express , **3**(5), 943–957 (2012). [CrossRef] [PubMed]

*t*which gradually changed from zero to infinity to keep the changes in

*μ*within a certain range. Although the rate of the change in the parameter

_{a}*t*may affect the convergence process and quality of the reconstructed image, the effect of the sparsity constraint explained in the method was well demonstrated by phantom experiments and

*in vivo*functional brain imaging.

*L*

^{1}sparsity constraint and total variation method simultaneously, was proposed [44

44. J. Dutta, S. Ahn, C. Li, S. R. Cherry, and R. M. Leahy, “Joint L^{1} and total variation regularization for fluorescence molecular tomography,” Phys. Med. Biol. **57**, 1459–1476 (2012). [CrossRef] [PubMed]

*L*

^{1}norm of the fluorophore distribution was equal to the vector which had unities as the components when the reconstructed variables were larger than zero. Practically, the non-differentiability of the

*L*

^{1}norm penalty near zero was overcome by the method. The method must need a threshold value for the positivity constraint, although the method to constrain the solution was not mentioned in the literature. The

*L*

^{1}sparsity constraint worked well and improved the sparseness of the reconstructed fluorophore distribution.

*L*

_{1}norm. In the literature [20

20. N. Cao, A. Nehorai, and M. Jacob, “Image reconstruction for diffuse optical tomography using sparsity regularization and expectation-maximization algorithm,” Opt. Express , **15**(21), 13695–13708 (2007). [CrossRef] [PubMed]

*L*

_{1}norm minimization with the EM algorithm reconstructed more sharply localized distributions of the absorption coefficient than other methods such as the Tikhonov regularization and simultaneous iterative reconstruction technique.

*L*

_{1}(

*L*1,

*L*

^{1}or 1-) norm by use of the various methods including the

*l*sparsity regularization must be equivalent each other if they work appropriately under the same conditions of the measurement system. Every literature has reported similarly that the

_{p}*L*

_{1}norm minimization has localized the target more sharply than the Tikhonov regularization. And this is consistent with the results in this paper. The major feature of the method proposed in this paper is that the method can be applied to the

*p*-norm with 0 <

*p*< 1 even at singular points where other methods fail to work. The merit of the proposed method is that the method can obtain the fluorophore distribution localized more sharply than other methods which minimize the

*L*

_{1}norm. This can be a breakthrough to improve the spatial resolution of the reconstructed optical images in biomedicine, although the selection of an appropriate

*p*value is a problem remained for future work.

## 5. Conclusions

*l*sparsity regularization was developed for efficient reconstruction of the fluorescence targets. Regularization was done by minimizing the

_{p}*l*norm of the reconstructed distribution of the fluorophore concentration. The reconstructed distribution was transformed by introducing another variable for calculation of the gradient of the

_{p}*p*-norm (0 <

*p*≤ 1) to overcome the non-differentiability at the singular points.

*l*sparsity regularization reconstructs the targets with improved localization and with much narrower distributions of the fluorophore than the Tikhonov regularization. Also the

_{p}*l*sparsity regularization improves the correctness of the reconstructed fluorophore quantities.

_{p}*l*sparsity regularization alleviated the mismatches between the forward model and the real object. Through the numerical simulations, phantom and mouse experiments, the reconstructed distributions of the fluorophore with

_{p}*p*< 1 localized the target more sharply than those with

*p*= 1 as reported in some literatures previously.

*l*sparsity regularization will be a useful tool for FDOT in preclinical animal tests for developments of novel drugs.

_{p}## Acknowledgments

## References and links

1. | V. Ntziachristos, C.-H. Yung, C. Bremerand, and R. Weissleder, “Fluorescence molecular tomography resolves protease activity in vivo,” Nat. Med. |

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. | R. Weissleder, “Molecular imaging in cancer,” Science |

6. | D. J. Hawrysz and E. M. Sevick-Muraca, “Developments toward diagnostic breast cancer imaging using near-infrared optical measurements and fluorescent contrast agents,” Neoplasia |

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

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

9. | E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, “A submillimeter resolution fluorescence molecular imaging system for small animal imaging,” Med. Phys. |

10. | N. C. Deliolanis, T. Lasser, D. Hyde, A. Soubret, J. Ripoll, and V. Ntziachristos, “Free-space fluorescence molecular tomography utilizing 360° geometry projections,” Opt. Lett. |

11. | 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. |

12. | A. P. Gibson, T. Austin, N. L. Everdell, M. Schweiger, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, “Three-dimensional whole-head optical tomography for passive motor evoked responses in the neonate,” NueroImage |

13. | J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. |

14. | B. W. Pogue, T. O. McBride, J. Prewitt, U. Lösterberg, and K. D. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Appl. Opt. |

15. | 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. |

16. | P. K. Yalavarthy, B. W. Pogue, H. Dehghani, C. M. Carpenter, S. Jiang, and K. D. Paulsen, “Structural information within regularization matrices improves near infrared diffuse optical tomography,” Opt. Express |

17. | A. Douiri, M. Schweiger, J. Riley, and S. R. Arridge, “Anisotropic diffusion regularization methods for diffuse optical tomography using edge prior information,” Meas. Sci. Tech. |

18. | P. Hiltunen, D. Calvetti, and E. Somersalo, “An adaptive smoothness regularization algorithm for optical tomography,” Opt. Express |

19. | C. Panagiotou, S. Somayajula, A. P. Gibson, M. Schweiger, R. M. Leahy, and S. R. Arridge, “Information theoretic regularization in diffuse optical tomography,” J. Opt. Soc. Am. A |

20. | N. Cao, A. Nehorai, and M. Jacob, “Image reconstruction for diffuse optical tomography using sparsity regularization and expectation-maximization algorithm,” Opt. Express , |

21. | T. Shimokawa, T. Kosaka, O. Yamashita, N. Hiroe, T. Amita, Y. Inoue, and M. Sato, “Hierarchical Bayesian estimation improves depth accuracy and spatial resolution of diffuse optical tomography,” Opt. Express |

22. | P. Xu, Y. Tian, H. Chen, and D. Yao, “L |

23. | P. M. Shankar and M. A. Neifeld, “Sparsity constrained regularization for multiframe image restoration,” J. Opt. Soc. Am. A |

24. | M. Freiberger, C. Clason, and H. Scharfetter, “Total variation regularization for nonlinear fluorescence tomography with an augmented Lagrangian splitting approach,” Appl. Opt. |

25. | D. Han, X. Yang, K. Liu, C. Qin, B. Zhang, X. Ma, and J. Tian, “Efficient reconstruction method for L1 regularization in fluorescence molecular tomography,” Appl. Opt |

26. | D. Han, J. Tian, S. Zhu, J. Feng, C. Qin, B. Zhang, and X. Yang, “A fast reconstruction algorithm for fluorescence molecular tomography with sparsity regularization,” Opt. Express |

27. | H. Yi, D. Chen, X. Qu, K. Peng, X. Chen, Y. Zhou, J. Tian, and J. Liang, “Multilevel, hybrid regularization method for reconstruction of florescent molecular tomography,” Appl. Opt. |

28. | P. Mohajerani, A. A. Eftekhar, J. Huang, and A. Adibi, “Optimal sparse solution for fluorescent diffuse optical tomography: theory and phantom experimental results,” Appl. Opt. |

29. | Y. Lu, X. Zhang, A. Douraghy, D. Stout, J. Tian, T. F. Chan, and A. F. Chatziioannou, “Source reconstruction for spectrally-resolved bioluminescence tomography with aparse A priori information,” Opt. Express |

30. | S. Okawa and Y. Yamada, “Reconstruction of fluorescence/bioluminescence sources in biological medium with spatial filter,” Opt. Express |

31. | Z. He, A. Cichocki, R. Zdunek, and S. Xie, “Improved FOCCUS method with conjugate gradient iterations,” IEEE Trans. Signal Process. |

32. | S. Okawa, Y. Hoshi, and Y. Yamada, “Improvement of image quality of time-domain diffuse optical tomography with l |

33. | 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. |

34. | C. R. Vogel, |

35. | S. R. Arridge, “A gradient-based optimization scheme for optical tomography,” Opt. Express |

36. | W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. |

37. | A. B. Milstein, S. Oh, K. Webb, C. A. Bouman, Q. Zhang, D. A. Boas, and R. P. Millane, “Fluorescence optical diffusion tomography,” Appl. Opt. |

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

39. | B. Toczylowska, E. Zieminska, G. Goch, D. Milej, A. Gerega, and A. Liebert, “Neurotoxic effects of indocyanine green-cerebellar granule cell culture viability study,” Biomed. Opt. Express , |

40. | J. Barkhausen, W. Ebert, J. F. Debatin, and H.-J. Weinmann, “Imaging of myocardial infarction: comparison of magnevist and gadophrin-3 in rabbits,” J. Am. Coll. Cardiol. |

41. | A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express , |

42. | M. Huang and Q. Zhu, “Dual-mesh optical tomography reconstruction method with a depth correction that uses a priori ultrasound information,” Appl. Opt. |

43. | V. C. Kavuri, Z.-J. Lin, F. Tian, and H. Liu, “Sparsity enhanced spatial resolution and depth localization n diffuse optical tomography,” Biomed. Opt. Express , |

44. | J. Dutta, S. Ahn, C. Li, S. R. Cherry, and R. M. Leahy, “Joint L |

**OCIS Codes**

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

(170.6280) Medical optics and biotechnology : Spectroscopy, fluorescence and luminescence

**ToC Category:**

Image Reconstruction and Inverse Problems

**History**

Original Manuscript: March 17, 2014

Revised Manuscript: May 5, 2014

Manuscript Accepted: May 6, 2014

Published: May 19, 2014

**Citation**

Shinpei Okawa, Tatsuya Ikehara, Ichiro Oda, and Yukio Yamada, "Reconstruction of localized fluorescent target from multi-view continuous-wave surface images of small animal with lp sparsity regularization," Biomed. Opt. Express **5**, 1839-1860 (2014)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-5-6-1839

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### References

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