OSA's Digital Library

Biomedical Optics Express

Biomedical Optics Express

  • Editor: Joseph A. Izatt
  • Vol. 2, Iss. 11 — Nov. 1, 2011
  • pp: 3179–3193
« Show journal navigation

Bioluminescence tomography using eigenvectors expansion and iterative solution for the optimized permissible source region

Mohamed A. Naser and Michael S. Patterson  »View Author Affiliations


Biomedical Optics Express, Vol. 2, Issue 11, pp. 3179-3193 (2011)
http://dx.doi.org/10.1364/BOE.2.003179


View Full Text Article

Acrobat PDF (4211 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

A reconstruction algorithm for bioluminescence tomography (BLT) has been developed. The algorithm numerically calculates the Green’s function at different wavelengths using the diffusion equation and finite element method. The optical properties used in calculating the Green’s function are reconstructed using diffuse optical tomography (DOT) and assuming anatomical information is provided by x-ray computed tomography or other methods. A symmetric system of equations is formed using the Green’s function and the measured light fluence rate and the resulting eigenvalue problem is solved to get the eigenvectors of this symmetric system of equations. A space can be formed from the eigenvectors obtained and the reconstructed source is written as an expansion of the eigenvectors corresponding to non-zero eigenvalues. The coefficients of the expansion are found to obtain the reconstructed BL source distribution. The problem is solved iteratively by using a permissible source region that is shrunk by removing nodes with low probability to contribute to the source. Throughout this process the permissible region shrinks from the entire object to just a few nodes. The best estimate of the reconstructed source is chosen that which minimizes the difference between the calculated and measured light fluence rates. 3D simulations presented here show that the reconstructed source is in good agreement with the actual source in terms of locations, magnitudes, sizes, and total powers for both localized multiple sources and large inhomogeneous source distributions.

© 2011 OSA

1. Introduction

Bioluminescence tomography (BLT) has recently attracted much interest as a molecular imaging technique that can be used as a noninvasive visualization of disease progression and response to treatment [1

1. 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]

4

4. J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27(5), 48–57 (2008). [CrossRef] [PubMed]

]. Direct in vivo bioluminescence imaging does not give quantitative information about the distribution of the light source (corresponding to luciferase concentration) due to multiple light scattering and absorption in tissue. To correctly estimate the distribution of luciferase in tissue from the bioluminescence image, BLT algorithms need to be developed.

In the current work, a 3D BLT algorithm based on diffusion theory [19

19. S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20(2), 299–309 (1993). [CrossRef] [PubMed]

26

26. 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(2), 024007 (2007). [CrossRef] [PubMed]

] has been developed to reconstruct inhomogeneous bioluminescence source distributions with good accuracy in terms of location, power, magnitude, and size. The Green’s function for the object is found by numerically solving the diffusion equation using the method of finite elements at different wavelengths. The scattering and absorption coefficients of all tissues are reconstructed using the diffuse optical tomography algorithm developed in [17

17. M. A. Naser and M. S. Patterson, “Algorithms for bioluminescence tomography incorporating anatomical information and reconstruction of tissue optical properties,” Biomed. Opt. Express 1(2), 512–526 (2010). [CrossRef] [PubMed]

] assuming known anatomical information obtained by another imaging modality such as x-ray CT. The calculated Green’s function and the measured light fluence rate are normalized to equate the contributions from all wavelengths. A symmetric system of equations is formed using the Green’s function and the measured light fluence rate. An eigenvalue problem is then solved to get the eigenvectors of the formed symmetric system of equations. The system of equations is rank deficient because the number of unknowns (source magnitude at each location) exceeds the number of data values (light fluence rate). Therefore, many eigenvalues of the system will be zero. A set of eigenvectors corresponding to non-zero eigenvalues is used as a complete set of vectors where the reconstructed source can be written as an expansion of these vectors. The set of vectors is assumed to be complete not in the mathematical sense but in the physical sense that any physical source distribution can formed from a combination of these vectors. The coefficients of the expansion are solved to obtain the source distribution. The removal of the eigenvectors corresponding to zero eigenvalues reduces the ill-posedness of the problem and imposes constraint on the reconstruction of the source without a need of using a regularization penalty term which reduces the accuracy of the reconstruction. The solution is repeated several times. At every iteration, the permissible source region is shrunk by removing nodes with low probability to contribute to the source. Throughout this process the permissible region shrinks from the entire object to just a few nodes. The best estimate of the source is that which minimizes the difference between the calculated and measured light fluence rates.

2. BLT reconstruction

The diffusion equation and the Robin-type boundary condition at a specific wavelength at zero modulation frequency are given by [27

27. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(2), R41–R93 (1999). [CrossRef]

,28

28. A. D. Klose, “Transport-theory-based stochastic image reconstruction of bioluminescence sources,” J. Opt. Soc. Am. A 24(6), 1601–1608 (2007). [CrossRef]

]

[.κ(r,λ)+μa(r,λ)]φ(r,λ)=s(r,λ)      (rΩ)
(1)
[1+2κ(r,λ)An^(r)μa(r,λ)]φ(r,λ)=0         (rΩ)
(2)

where φ(r,λ) is the light fluence rate at position r and wavelength λ; Ω and Ω are the domain and its boundary respectively, and the source distribution is given by s(r,λ). The spatial distribution of the tissue optical properties at wavelength λ is given by the absorption coefficient μa(r,λ) and the diffusion coefficient κ(r,λ), where the diffusion coefficient is defined by

κ(r,λ)=13(μa(r,λ)+μs(r,λ)),
(3)

where μs(r,λ) is the reduced scattering coefficient μs=μs(1g), μs is the scattering coefficient and g is the anisotropy factor. n^(r)is a unit vector pointed outwardly normal to Ω, and A is derived from Fresnel’s law as [29

29. H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25(6), 711–732 (2009). [CrossRef] [PubMed]

]

A=(2/(1R0))1+|cosθc|31|cosθc|2,
(4)

where θc=sin1(1/η) is the critical angle, R0=(η1)2/(η+1)2, and η is the tissue refractive index at the boundary while the air refractive index is assumed to be 1.

Equation (1) can be solved numerically using the method of finite elements (FE), and the discrete version of Eq. (1) as shown in [18

18. M. A. Naser and M. S. Patterson, “Improved bioluminescence and fluorescence reconstruction algorithms using diffuse optical tomography, normalized data, and optimized selection of the permissible source region,” Biomed. Opt. Express 2(1), 169–184 (2011). [CrossRef] [PubMed]

] is given by

φ(λ)=G(λ)s(λ)
(5)

where G(λ)is the Green’s function matrix at wavelength λ obtained by discretizing the diffusion equation. Assuming that the source magnitude is not a function of wavelength or, equivalently, the source spectrum is known in the range of measurement, for multi-spectral measurements at the detectors on the object surfaceΩ, Eq. (5) for N wavelengths in a normalized form can be given by

(G(d,R;λ1)/max(φ(d;λ1))G(d,R;λ2)/max(φ(d;λ2))G(d,R;λN)/max(φ(d;λN)))s(R)=(φ(d;λ1)/max(φ(d;λ1))φ(d;λ2)/max(φ(d;λ2))φ(d;λN)/max(φ(d;λN)))
(6)

or concisely GdRsR=φd, where d is the detector index at the boundary Ω, and R is the source permissible region index which contains the index of nodes of the finite element mesh inside the domain Ω. This corresponds to the permissible source region which could be considered initially to be the whole domain Ω. The normalization of the Green’s function and light fluence rate by the maximum value of the light fluence rate at each wavelength improves the efficiency of the algorithm as shown before in [18

18. M. A. Naser and M. S. Patterson, “Improved bioluminescence and fluorescence reconstruction algorithms using diffuse optical tomography, normalized data, and optimized selection of the permissible source region,” Biomed. Opt. Express 2(1), 169–184 (2011). [CrossRef] [PubMed]

]. Usually the number of rows of the matrix GdR in Eq. (6) which equals N×length(d)is smaller than the number of columns, length(R). A symmetric system of equations can be obtained from Eq. (6):

(GdRTGdR)sR=(GdRT)*φd,
(7)

sR=1Mcivi
(8)

where viis an eigenvector of the matrix corresponding to non-zero eigenvalue and M is the number of chosen eigenvectors. The coefficients ci of the expansion in Eq. (8) can be obtained by substituting the source expansion in Eq. (8) into Eq. (7) and multiplying both sides of Eq. (7) by the eigenvectors as follows:

j=1MviT(GdRTGdR)vjcj=viT(GdRT)*φdGv*c=φvc=Gv1*φv
(9)

where [Gv]i,j=viT(GdRTGdR)vjand [φv]i=viT(GdRT)*φd.

The matrix Gv is a real symmetric matrix with dimensions M×M which is directly invertible and much smaller in size than the matrix(GdRTGdR). The system of equations obtained in Eq. (9) from Eq. (7) saves computation time by inverting a much smaller, directly invertible matrix and limits the source space domain by considering only the eigenvectors that contribute to the source distribution. To guarantee the robustness of the algorithm in the presence of noise, not all eigenvectors corresponding to non-zero eigenvalues should be used in expansion (8). Otherwise the matrix Gvwill be close to singular which will give inaccurate coefficients when using noisy data. Instead, for noise levels of 2%–10%, we found empirically that using eigenvectors such that the lowest corresponding eigenvalue is within 104of the highest eigenvalue ensures robustness in the presence of noisy data. Since the space of eigenvectors used in expansion (8) does not include all eigenvectors corresponding to non-zero eigenvalues, the accuracy of the reconstruction could be affected. Thus, increasing the number of eigenvectors used in the expansion within the same domain of eigenvalues will ensure the robustness and improve the accuracy. This can be done by rescaling the system of equations given in Eq. (7) such that every row is normalized by the maximum value of that row. This will re-scale the eigenvalues to give more corresponding eigenvectors in the domain of eigenvalues (the max value and 104 of the maximum value). Thus, the eigenvectors extraction is performed after normalizing the matrix(GdRTGdR). Physically, this normalization could be seen as giving more weight to source nodes which are far from detectors and have lower contribution to the light fluence rate compared to source nodes close to detectors.

4. Results and discussions

The 3D object (Fig. 1(a)
Fig. 1 (a) A 3D finite element mesh for the object used in the simulation. The mesh is a regular Delaunay tetrahedral with 2310 nodes and 12633 tetrahedrons. The object represents an idealized section of a mouse abdomen which illustrates 4 different tissue regions: adipose, bowel, kidneys, and bone. (b) A schematic of the object showing the detector set up and the coordinates.
) used in the simulations is a cylinder 25 mm in diameter and 16 mm in height and represents a finite element mesh of an idealized section of a mouse abdomen where adipose, bowel, kidney, and bone tissues are identified and assigned realistic wavelength-dependent optical properties. It is assumed that different regions are distinguished through CT x-ray imaging or other imaging technique and all elements within each region have the same optical properties. For the BLT, 64 uniformly distributed detectors are used around the circumference and 15 layers of these 64 detectors separated by 1 mm are used in the z-direction as shown in Fig. 1(b). The total number of detector readings at each wavelength isNd=15×64=960.

Figure 2
Fig. 2 2D sections illustrate the positions of the 3 spherical bioluminescence sources located in the bone and two kidneys. (a) A transverse cross section at the middle of the object shows the bowel, kidneys, and bone, (b) Sagittal section S1 passes through the bone and bowel, and (c) Coronal section S2 passes through the two kidneys.
shows three cross sections of the object that illustrate the positions of the bioluminescence sources to be reconstructed by the proposed algorithm. The circular transverse cross section Fig. 2(a) is taken at the middle of the object (z = 0) parallel to the upper and lower surfaces of the object. Figure 2(b) is a sagittal cross section that passes through the bowel and bone while Fig. 2(c) shows a coronal cross section passing through the two kidneys. Figure 2 shows 3 spherical bioluminescence sources of 5 mm and 3 mm diameter centered in the bone at (0, −7, 0) and two kidneys (−4.5, −2.5, 0) and (4.5, −2.5, 0), respectively.

In the first step, the optical properties of all tissues at the wavelengths 500, 550, 600, 650, and 700 nm were reconstructed using the DOT algorithm described in [17

17. M. A. Naser and M. S. Patterson, “Algorithms for bioluminescence tomography incorporating anatomical information and reconstruction of tissue optical properties,” Biomed. Opt. Express 1(2), 512–526 (2010). [CrossRef] [PubMed]

]. 16 external sources uniformly distributed around the middle of object have been used for the illumination. Associated with each source, 15 layers of detectors were uniformly distributed across the object height with a separation between layers of 1 mm. Each layer contains 49 detectors forming a field of view of 270 degree. The light fluence rate at the “detectors” could be derived from the corresponding pixels of the image captured by a CCD camera. It is assumed that an accurate segmented 3D image set is provided by x-ray CT, MR or by a deformable atlas of mouse anatomy. All elements within each region in the object are assumed to have the same optical properties. The light fluence rate at the boundary is written as a Taylor series expansion around an initial guess corresponding to a homogenous object with the same optical properties in all regions. The fluence rate is approximated by the first three terms, including the first and second order derivatives which are calculated by the direct method and give the change of light fluence rate at the boundary due to small changes in the tissue optical properties in each region [17

17. M. A. Naser and M. S. Patterson, “Algorithms for bioluminescence tomography incorporating anatomical information and reconstruction of tissue optical properties,” Biomed. Opt. Express 1(2), 512–526 (2010). [CrossRef] [PubMed]

]. The first and second order derivatives are used in an iterative algorithm to reconstruct the tissue optical properties. For this simulation, forward finite element calculations of the light fluence rate were performed using the true optical properties and Gaussian noise, which is an approximation to the Poisson measurement error distribution, of 2% of the signal was added to obtain the data for reconstruction, i.e. φ=φ×(1+2%randn), where randn is a Matlab function that generates random numbers drawn from a normal distribution with mean zero and standard deviation one. In Table 2

Table 2. True [30,31] and reconstructed scattering coefficients (mm−1)for each region in the heterogeneous object at five different wavelengths

table-icon
View This Table
| View All Tables
, the true and reconstructed scattering coefficients, μs andμsr respectively, are presented while Table 3

Table 3. True [30,31] and reconstructed absorption coefficients (mm−1) for each region in the heterogeneous object at five different wavelengths

table-icon
View This Table
| View All Tables
shows the true and reconstructed absorption coefficients, μaandμar respectively. The true scattering and absorption coefficients were calculated using the data reported in [30

30. G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. 50(17), 4225–4241 (2005). [CrossRef] [PubMed]

,31

31. S. A. Prahl, Oregon Medical Laser Clinic (2001), http://omlc.ogi.edu/spectra/index.html.

]. The maximum relative error obtained for scattering coefficients is 22% at 600 nm while the maximum relative error obtained for the absorption coefficients is 21% at 700 nm.

Figure 3
Fig. 3 BLT reconstruction of three uniform bioluminescence sources localized in the left and right kidneys (3 mm diameters), and the bone (5 mm diameter). The “anatomy” is shown in Figs. 1 and 2. The actual sources in (a), (b), and (c) have magnitude 1 nW/ mm3. The reconstructed sources are shown in (d), (e), and (f). The total powers of actual and reconstructed sources are in good agreement within 6% and the source magnitudes are reconstructed within 20% of the actual value.
shows the actual and reconstructed bioluminescence sources. The magnitudes of the three sources are 1 nW/mm3, and the total powers of the three sources are 5.65, 5.65, and 40.71 nW, respectively. The transverse, sagittal, and coronal cross sections of the reconstructed sources in Figs. 3(d), (e), and (f) show good agreement with the actual sources in terms of source magnitude, total power and size. The total power of the reconstructed sources are 5.31, 5.49, and 41.4 nW. The source magnitude varies between 0.8 to 1.2 nW/mm3 which is within 20% of the true value.

The evolution of the objective function with the number of source nodes in the permissible region is illustrated in Fig. 4
Fig. 4 The evolution of the BLT iterative solution to obtain the optimized reconstruction. (a) The objective function as a function of the number of finite element mesh source nodes in the permissible source region. (b) The reconstructed BL source distribution after the first iteration. (c) The reconstructed BL source distribution when the objective function is minimized.
. Figure 4(a) shows the objective function f=minλG˜(d,R;λ)s(R)φ˜(d;λ)1 as a function of the number of source nodes in the permissible source region. The permissible region has been shrunk in 60 iterations by removing nodes corresponding to low source magnitude, assuming that source nodes with low magnitude are less likely to be in the optimized source region. Figures 4(b) and 4(c) show the reconstructed source distribution in nW/mm3 after the first iteration which includes a permissible region of 1050 nodes and at the optimized permissible region which includes 126 nodes and minimizes the objective function. After the first iteration, the reconstructed source powers in the two kidneys and bone are 3.37, 3.8, and 51.93 nW, respectively which are within 40% of their actual values. The source magnitudes and sizes cannot be reconstructed perfectly after the first iteration due to the degeneracy of the solution. When the size of the source permissible region becomes comparable to the size of the actual source, the objective function has its lowest value and good accuracy in reconstructing the source power, magnitude, and size can be obtained, as shown in Fig. 4(c). When the permissible region is further reduced, the size of the source permissible region becomes smaller than the size of the actual source. Hence, the objective function increases again and the reconstruction of the source power, magnitude and size becomes less accurate.

Figure 6
Fig. 6 BLT reconstruction of three uniform bioluminescence sources localized in the left kidney (1 nW/ mm3), right kidney (2 nW/ mm3), and bone (3 nW/ mm3). The total powers of the three sources are 5.65, 11.31, and 122.14 nW, respectively. The actual sources are shown in (a), (b), and (c). The reconstructed sources are shown in (d), (e), and (f). The total powers of the reconstructed sources are 8.7, 10.94, and 120.14 nW respectively. The smallest source could not be properly reconstructed in this case because a much larger source is located nearby and dominates the emission of light measured at the boundary.
shows the BLT reconstruction for multiple sources with different magnitudes and sizes. In this case, one source is greater in both size and magnitude and hence has much larger total power than other sources and dominates the measured light fluence rate. The sources in the left and right kidneys are 1 nW/mm3 and 2 nW/mm3 respectively and the source in the bone is 3 nW/mm3. The three sources have total powers of 5.65, 11.31, and 122.14 nW respectively. The reconstructed sources are shown in Figs. 6(d), (e), and (f). The reconstructed sources powers are 8.7, 10.94, and 120.14 nW respectively. The size and magnitude of the smallest source in the left kidney could not be reconstructed in this case. When there are large differences between source powers or there is a large inhomogeneous source distribution, the shrinking of the permissible region cannot be done correctly. The assumption that nodes with low source magnitude are least likely to be in the optimized source region is no longer true.

Figure 8
Fig. 8 BLT reconstruction of a large non-uniform source distribution that fills the entire left kidney. The actual source in (a) and (b) has a hot spot of 3 mm diameter with magnitude 5 nW/mm3 which is five times the magnitude of the background. The reconstructed source in (c) and (d) shows good agreement with the actual source in terms of the total reconstructed power; however the details of the in homogeneity of the source could not be reconstructed correctly.
shows the BLT reconstruction for a non-uniform source distribution where the inhomogeneity of the source is high. In this case, the hot spot has a magnitude of 5 nW/mm3 which is 5 times that of the surrounding background. The total powers of the actual and reconstructed sources are 70.63 and 70.8 nW. However, the inhomogeneity of the source could not be reconstructed correctly in this case. The uniform background shows a smaller size and a higher magnitude, which varies from 1.5 to 2.5 nW/mm3, than the actual source. The objective function curve with respect to the number of source nodes for the reconstructed source shown in Fig. 8 does not show a deep valley with a clear global minimum as shown in Fig. 4(a). Instead, the curve is slowly increasing after the global minimum indicating that different source distributions can generate the same error and hence are indistinguishable. Therefore, shrinking the permissible region reduces the possible solutions, and if the permissible region equals the actual source size, a unique solution can be obtained for uniform and low inhomogeneity source distributions as shown in Figs. 3, 5, and 7. However, for high inhomogeneity source distributions as in Figs. 6 and 8, the solution is still not unique and different size and magnitude sources that have the same total power can generate the same light fluence rate at the object boundary. One way to improve the uniqueness of the solution is to increase the available data points by doing measurements at more wavelengths.

5. Conclusion

Acknowledgments

This work was supported in part by grants from the Ontario Ministry of Research and Innovation postdoctoral fellowship program.

References and links

1.

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]

2.

R. Weissleder and U. Mahmood, “Molecular imaging,” Radiology 219(2), 316–333 (2001). [PubMed]

3.

J. K. Willmann, N. van Bruggen, L. M. Dinkelborg, and S. S. Gambhir, “Molecular imaging in drug development,” Nat. Rev. Drug Discov. 7(7), 591–607 (2008). [CrossRef] [PubMed]

4.

J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27(5), 48–57 (2008). [CrossRef] [PubMed]

5.

X. Song, D. Wang, N. Chen, J. Bai, and H. Wang, “Reconstruction for free-space fluorescence tomography using a novel hybrid adaptive finite element algorithm,” Opt. Express 15(26), 18300–18317 (2007). [CrossRef] [PubMed]

6.

G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys. 31(8), 2289–2299 (2004). [CrossRef] [PubMed]

7.

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol. 50(23), 5421–5441 (2005). [CrossRef] [PubMed]

8.

S. Ahn, A. J. Chaudhari, F. Darvas, C. A. Bouman, and R. M. Leahy, “Fast iterative image reconstruction methods for fully 3D multispectral bioluminescence tomography,” Phys. Med. Biol. 53(14), 3921–3942 (2008). [CrossRef] [PubMed]

9.

H. Dehghani, S. C. Davis, S. Jiang, B. W. Pogue, K. D. Paulsen, and M. S. Patterson, “Spectrally resolved bioluminescence optical tomography,” Opt. Lett. 31(3), 365–367 (2006). [CrossRef] [PubMed]

10.

W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express 13(18), 6756–6771 (2005). [CrossRef] [PubMed]

11.

X. He, J. Liang, X. Qu, H. Huang, Y. Hou, and J. Tian, “Truncated total least squares method with a practical truncation parameter choice scheme for bioluminescence tomography inverse problem,” Int. J. Biomed. Imaging 2010, 291874 (2010). [CrossRef] [PubMed]

12.

H. Huang, X. Qu, J. Liang, X. He, X. Chen, D. Yang, and J. Tian, “A multi-phase level set framework for source reconstruction in bioluminescence tomography,” J. Comput. Phys. 229(13), 5246–5256 (2010). [CrossRef]

13.

J. Feng, K. Jia, G. Yan, S. Zhu, C. Qin, Y. Lv, and J. Tian, “An optimal permissible source region strategy for multispectral bioluminescence tomography,” Opt. Express 16(20), 15640–15654 (2008). [CrossRef] [PubMed]

14.

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 sparse a priori information,” Opt. Express 17(10), 8062–8080 (2009). [CrossRef] [PubMed]

15.

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]

16.

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

17.

M. A. Naser and M. S. Patterson, “Algorithms for bioluminescence tomography incorporating anatomical information and reconstruction of tissue optical properties,” Biomed. Opt. Express 1(2), 512–526 (2010). [CrossRef] [PubMed]

18.

M. A. Naser and M. S. Patterson, “Improved bioluminescence and fluorescence reconstruction algorithms using diffuse optical tomography, normalized data, and optimized selection of the permissible source region,” Biomed. Opt. Express 2(1), 169–184 (2011). [CrossRef] [PubMed]

19.

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20(2), 299–309 (1993). [CrossRef] [PubMed]

20.

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22(11), 1779–1792 (1995). [CrossRef] [PubMed]

21.

H. Jiang, “Frequency-domain fluorescent diffusion tomography: a finite-element-based algorithm and simulations,” Appl. Opt. 37(22), 5337–5343 (1998). [CrossRef] [PubMed]

22.

A. H. Hielscher, A. D. Klose, and K. M. Hanson, “Gradient-based iterative image reconstruction scheme for time-resolved optical tomography,” IEEE Trans. Med. Imaging 18(3), 262–271 (1999). [CrossRef] [PubMed]

23.

X. Gu, Q. Zhang, L. Larcom, and H. Jiang, “Three-dimensional bioluminescence tomography with model-based reconstruction,” Opt. Express 12(17), 3996–4000 (2004). [CrossRef] [PubMed]

24.

N. V. Slavine, M. A. Lewis, E. Richer, and P. P. Antich, “Iterative reconstruction method for light emitting sources based on the diffusion equation,” Med. Phys. 33(1), 61–68 (2006). [CrossRef] [PubMed]

25.

D. C. Comsa, T. J. Farrell, and M. S. Patterson, “Quantification of bioluminescence images of point source objects using diffusion theory models,” Phys. Med. Biol. 51(15), 3733–3746 (2006). [CrossRef] [PubMed]

26.

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(2), 024007 (2007). [CrossRef] [PubMed]

27.

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(2), R41–R93 (1999). [CrossRef]

28.

A. D. Klose, “Transport-theory-based stochastic image reconstruction of bioluminescence sources,” J. Opt. Soc. Am. A 24(6), 1601–1608 (2007). [CrossRef]

29.

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25(6), 711–732 (2009). [CrossRef] [PubMed]

30.

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. 50(17), 4225–4241 (2005). [CrossRef] [PubMed]

31.

S. A. Prahl, Oregon Medical Laser Clinic (2001), http://omlc.ogi.edu/spectra/index.html.

OCIS Codes
(170.3880) Medical optics and biotechnology : Medical and biological imaging
(170.6960) Medical optics and biotechnology : Tomography

ToC Category:
Image Reconstruction and Inverse Problems

History
Original Manuscript: September 27, 2011
Revised Manuscript: October 27, 2011
Manuscript Accepted: October 25, 2011
Published: October 26, 2011

Citation
Mohamed A. Naser and Michael S. Patterson, "Bioluminescence tomography using eigenvectors expansion and iterative solution for the optimized permissible source region," Biomed. Opt. Express 2, 3179-3193 (2011)
http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-2-11-3179


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. 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]
  2. R. Weissleder and U. Mahmood, “Molecular imaging,” Radiology219(2), 316–333 (2001). [PubMed]
  3. J. K. Willmann, N. van Bruggen, L. M. Dinkelborg, and S. S. Gambhir, “Molecular imaging in drug development,” Nat. Rev. Drug Discov.7(7), 591–607 (2008). [CrossRef] [PubMed]
  4. J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag.27(5), 48–57 (2008). [CrossRef] [PubMed]
  5. X. Song, D. Wang, N. Chen, J. Bai, and H. Wang, “Reconstruction for free-space fluorescence tomography using a novel hybrid adaptive finite element algorithm,” Opt. Express15(26), 18300–18317 (2007). [CrossRef] [PubMed]
  6. G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys.31(8), 2289–2299 (2004). [CrossRef] [PubMed]
  7. A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50(23), 5421–5441 (2005). [CrossRef] [PubMed]
  8. S. Ahn, A. J. Chaudhari, F. Darvas, C. A. Bouman, and R. M. Leahy, “Fast iterative image reconstruction methods for fully 3D multispectral bioluminescence tomography,” Phys. Med. Biol.53(14), 3921–3942 (2008). [CrossRef] [PubMed]
  9. H. Dehghani, S. C. Davis, S. Jiang, B. W. Pogue, K. D. Paulsen, and M. S. Patterson, “Spectrally resolved bioluminescence optical tomography,” Opt. Lett.31(3), 365–367 (2006). [CrossRef] [PubMed]
  10. W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express13(18), 6756–6771 (2005). [CrossRef] [PubMed]
  11. X. He, J. Liang, X. Qu, H. Huang, Y. Hou, and J. Tian, “Truncated total least squares method with a practical truncation parameter choice scheme for bioluminescence tomography inverse problem,” Int. J. Biomed. Imaging2010, 291874 (2010). [CrossRef] [PubMed]
  12. H. Huang, X. Qu, J. Liang, X. He, X. Chen, D. Yang, and J. Tian, “A multi-phase level set framework for source reconstruction in bioluminescence tomography,” J. Comput. Phys.229(13), 5246–5256 (2010). [CrossRef]
  13. J. Feng, K. Jia, G. Yan, S. Zhu, C. Qin, Y. Lv, and J. Tian, “An optimal permissible source region strategy for multispectral bioluminescence tomography,” Opt. Express16(20), 15640–15654 (2008). [CrossRef] [PubMed]
  14. 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 sparse a priori information,” Opt. Express17(10), 8062–8080 (2009). [CrossRef] [PubMed]
  15. 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]
  16. N. Cao, A. Nehorai, and M. Jacobs, “Image reconstruction for diffuse optical tomography using sparsity regularization and expectation-maximization algorithm,” Opt. Express15(21), 13695–13708 (2007). [CrossRef] [PubMed]
  17. M. A. Naser and M. S. Patterson, “Algorithms for bioluminescence tomography incorporating anatomical information and reconstruction of tissue optical properties,” Biomed. Opt. Express1(2), 512–526 (2010). [CrossRef] [PubMed]
  18. M. A. Naser and M. S. Patterson, “Improved bioluminescence and fluorescence reconstruction algorithms using diffuse optical tomography, normalized data, and optimized selection of the permissible source region,” Biomed. Opt. Express2(1), 169–184 (2011). [CrossRef] [PubMed]
  19. S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys.20(2), 299–309 (1993). [CrossRef] [PubMed]
  20. M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys.22(11), 1779–1792 (1995). [CrossRef] [PubMed]
  21. H. Jiang, “Frequency-domain fluorescent diffusion tomography: a finite-element-based algorithm and simulations,” Appl. Opt.37(22), 5337–5343 (1998). [CrossRef] [PubMed]
  22. A. H. Hielscher, A. D. Klose, and K. M. Hanson, “Gradient-based iterative image reconstruction scheme for time-resolved optical tomography,” IEEE Trans. Med. Imaging18(3), 262–271 (1999). [CrossRef] [PubMed]
  23. X. Gu, Q. Zhang, L. Larcom, and H. Jiang, “Three-dimensional bioluminescence tomography with model-based reconstruction,” Opt. Express12(17), 3996–4000 (2004). [CrossRef] [PubMed]
  24. N. V. Slavine, M. A. Lewis, E. Richer, and P. P. Antich, “Iterative reconstruction method for light emitting sources based on the diffusion equation,” Med. Phys.33(1), 61–68 (2006). [CrossRef] [PubMed]
  25. D. C. Comsa, T. J. Farrell, and M. S. Patterson, “Quantification of bioluminescence images of point source objects using diffusion theory models,” Phys. Med. Biol.51(15), 3733–3746 (2006). [CrossRef] [PubMed]
  26. 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(2), 024007 (2007). [CrossRef] [PubMed]
  27. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl.15(2), R41–R93 (1999). [CrossRef]
  28. A. D. Klose, “Transport-theory-based stochastic image reconstruction of bioluminescence sources,” J. Opt. Soc. Am. A24(6), 1601–1608 (2007). [CrossRef]
  29. H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng.25(6), 711–732 (2009). [CrossRef] [PubMed]
  30. G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol.50(17), 4225–4241 (2005). [CrossRef] [PubMed]
  31. S. A. Prahl, Oregon Medical Laser Clinic (2001), http://omlc.ogi.edu/spectra/index.html .

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited