## Experimental Bioluminescence Tomography with Fully Parallel Radiative-transfer-based Reconstruction Framework

Optics Express, Vol. 17, Issue 19, pp. 16681-16695 (2009)

http://dx.doi.org/10.1364/OE.17.016681

Acrobat PDF (1556 KB)

### Abstract

Bioluminescence imaging is a very sensitive imaging modality, used in preclinical molecular imaging. However, in its planar projection form, it is non-quantitative and has poor spatial resolution. In contrast, bioluminescence tomography (BLT) promises to provide three dimensional quantitative source information. Currently, nearly all BLT reconstruction algorithms in use employ the diffusion approximation theory to determine light propagation in tissues. In this process, several approximations and assumptions that are made severely affect the reconstruction quality of BLT. It is therefore necessary to develop novel reconstruction methods using high-order approximation models to the radiative transfer equation (RTE) as well as more complex geometries for the whole-body of small animals. However, these methodologies introduce significant challenges not only in terms of reconstruction speed but also for the overall reconstruction strategy. In this paper, a novel fully-parallel reconstruction framework is proposed which uses a simplified spherical harmonics approximation (SPN). Using this framework, a simple linear relationship between the unknown source distribution and the surface measured photon density can be established. The distributed storage and parallel operations of the finite element-based matrix make *SP _{N}
*-based spectrally resolved reconstruction feasible at the small animal whole body level. Performance optimization of the major steps of the framework remarkably improves reconstruction speed. Experimental reconstructions with mouse-shaped phantoms and real mice show the effectiveness and potential of this framework. This work constitutes an important advance towards developing more precise BLT reconstruction algorithms that utilize high-order approximations, particularly second-order self-adjoint forms to the RTE for

*in vivo*small animal experiments.

© 2009 Optical Society of America

## 1. Introduction

*in vivo*preclinical research [1

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

2. R. Weissleder, “Scaling down imaging: Molecular mapping of cancer in mice,” Nat. Rev. Cancer **2**, 11–18 (2002).
[CrossRef] [PubMed]

*in vivo*biological tissues have high absorption and scattering characteristics, planar BLI only indirectly reflects the activity of the targeted biological object via the surface photon distribution [3

3. J. Virostko, A. C. Powers, and E. D. Jansen, “Validation of luminescent source reconstruction using single-view spectrally resolved bioluminescence images,” Appl. Opt. **46**, 2540–2547 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=ao-46-13-2540.
[CrossRef] [PubMed]

*in vivo*mouse experiments, especially when monitoring the initiation and progression of tumors over time within the same subject. The aim of BLT is to quantitatively acquire 3

*D*information of bioluminescence sources, significantly improving the information quality of bioluminescence imaging.

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]

*D*BLT reconstruction algorithms are based on the diffusion equation, which is a simple approximation to the RTE [6

6. 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**, 6756–6771 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-18-6756.
[CrossRef] [PubMed]

7. X. Gu, Q. Zhang, L. Larcom, and H. Jiang, “Three-dimensional bioluminescence tomography with model-based reconstruction,” Opt. Express **12**, 3996–4000 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-17-3996.
[CrossRef] [PubMed]

8. Y. Lv, J. Tian, W. Cong, G. Wang, J. Luo, W. Yang, and H. Li, “A multilevel adaptive finite element algorithm for bioluminescence tomography,” Opt. Express **14**, 8211–8223 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-18-8211.
[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**, 365–367 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=ol-31-3-365.
[CrossRef] [PubMed]

3. J. Virostko, A. C. Powers, and E. D. Jansen, “Validation of luminescent source reconstruction using single-view spectrally resolved bioluminescence images,” Appl. Opt. **46**, 2540–2547 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=ao-46-13-2540.
[CrossRef] [PubMed]

10. A. D. Klose, “Transport-theory-based stochastic image reconstruction of bioluminescent sources,” J. Opt. Soc. Am. A **24**, 1601–1608 (2007), http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-24-6-1601.
[CrossRef]

*S*) and spherical harmonics (

_{N}*P*) methods, as two usual numerical approximations, can yield simulation solutions based on two types of formulations. Compared with first-order formulations, the operators acting on the second-order forms such as the even-odd parity (EOP) equations are self-adjoint [5]. This distinct advantage provides a straightforward application of the finite element methods (FEM) easily executed on complex heterogeneous geometries [11

_{N}11. C. R. E. de Oliveira, “An arbitrary geometry finite element method for multigroup neutron transport with anisotropic scattering,” Progr. Nucl. Energ. **18**, 227–236 (1986).
[CrossRef]

12. S. Wright, M. Schweiger, and S. R. Arridge, “Reconstruction in optical tomography using the PN approximations,” Meas. Sci. Technol. **18**, 79–86 (2007).
[CrossRef]

*N*as large as possible and then

*N*(

*N*+2) and (

*N*+1)

^{2}coupled equations corresponding to

*S*and

_{N}*P*methods need to be solved. This computational complexity, especially for the whole-body of small animals, creates a substantial challenge in the development of novel BLT reconstruction algorithms. Recently, a novel type of second-order approximation form, the simplified spherical harmonics (

_{N}*SP*) method, has been developed for optical imaging [13

_{N}13. A. D. Klose and E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. **220**, 441–470 (2006).
[CrossRef]

14. Y. Lu and A. F. Chatziioannou, “A parallel adaptive finite element method for the simulation of photon migration with the radiative-transfer-based model,” Commun. Numer. Methods Eng. **25**, 751–770 (2009).
[CrossRef]

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

*A priori*information plays an indispensable role in BLT reconstruction. Among the various types of

*a priori*information, multispectral measurements are important for achieving BLT reconstructions [16

16. 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]

17. 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**, 5421–5441 (2005).
[CrossRef] [PubMed]

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

19. 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**, 4225–4241 (2005).
[CrossRef] [PubMed]

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]

17. 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**, 5421–5441 (2005).
[CrossRef] [PubMed]

*P*-matrix needs to be computed, which is a very time-consuming step, although it can be obtained prior to acquiring the measured data. The BLT reconstruction is sensitive to various factors. Precalculating the

*P*-matrix can affect the reconstruction quality due to the use of different heterogeneous geometries between the calculation and the experiment. Diffuse optical tomography (DOT) has been investigated for several decades and its reconstruction algorithms are easily applied to BLT. In this case, the Jacobian matrix needs to be calculated for each iteration, which is time-consuming [7

7. X. Gu, Q. Zhang, L. Larcom, and H. Jiang, “Three-dimensional bioluminescence tomography with model-based reconstruction,” Opt. Express **12**, 3996–4000 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-17-3996.
[CrossRef] [PubMed]

6. 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**, 6756–6771 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-18-6756.
[CrossRef] [PubMed]

*SP*) equations. This framework uses finite element methods to process complex reconstruction domain geometries. The linear relationship between the unknown source and the spectrally-resolved measured data using the

_{N}*SP*approximation is established to achieve BLT reconstruction. To improve the reconstruction speed and enable BLT reconstruction for the whole body of the mouse, the finite element-based matrices are stored and operated in a parallel distribution mode. Furthermore, for the time-consuming problems of the key steps in the reconstruction, corresponding improvements are also performed, which significantly accelerate the reconstruction. Timing analysis demonstrates the improved performance of the proposed framework. Experimental reconstructions using mouse-shaped phantoms and real mice show the potential of this framework for practical BLT applications. The next section introduces the proposed fully parallel framework using the

_{N}*SP*approximation. In the third section, the performance tests and analysis are described and experimental BLT reconstructions also are demonstrated. Finally, we discuss relevant issues.

_{N}## 2. Methods

### 2.1. Radiative transfer equation and SPN approximation

*ψ*(

**r**, ŝ,

*λ*) denotes photons in the unit volume traveling from point r in direction ŝ. Based on the principle of energy conservation, the RTE suggests that the radiance

*ψ*(

**r**, ŝ,,

*λ*) is equal to the sum of all factors affecting it (including absorption

*µ*(

_{a}**r**,

*λ*), scattering

*µ*(

_{s}**r**,

*λ*), and source energy

*S*(

**r**, ŝ,

*λ*)) when light photons cross a unit volume [20].

*p*(ŝ, ŝ′) is the scattering phase function and gives the probability of a photon scattering anisotropically from direction ŝ′ to direction ŝ. Generally, the Henyey-Greenstein (HG) phase function is used to characterize this probability [21]:

*g*is the anisotropy parameter; cos

*θ*denotes the scattering angle and is equal to ŝ·ŝ′ when we assume that the scattering probability only depends on the angle between the incoming and scattering directions. When photons reach the body surface of a mouse, that is

*r*∈

*∂*Ω, some of them are reflected and cannot escape from the mouse body Ω because of the mismatch between the refractive indices

*n*for Ω and

_{b}*n*for the external medium. When the incidence angle

_{m}*θ*from the mouse body is not larger than the critical angle

_{b}*θ*(

_{c}*θ*=arcsin(

_{c}*n*) based on Snell’s law), the reflectivity

_{m}/n_{b}*R*(cos

*θ*) is given by [22

_{b}22. R. C. Haskell, L. O. Svaasand, T. Tsay, T. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A **11**, 2727–2741 (1994), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-11-10-2727.
[CrossRef]

*θ*is the transmission angle. Furthermore, we can get the exiting partial current

_{m}*J*+(

**r**) at each boundary point

**r**[13

13. A. D. Klose and E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. **220**, 441–470 (2006).
[CrossRef]

*P*method, the

_{N}*SP*approximation is obtained [13

_{N}13. A. D. Klose and E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. **220**, 441–470 (2006).
[CrossRef]

*µ*(

_{a,n}*λ*)=

*µ*(

_{a}*λ*)+

*µ*(

_{s}*λ*)(1-

*g*); when

^{n}*ψ*(λ) is expanded by the

*P*approximation,

_{N}*ϕ*(λ) are the

_{n}*Legendre moments*of

*ψ*(

*λ*) (2≤

*n*≤

*N*,

*N*is an odd positive integer). Although the

*SP*solution is asymptotic and cannot converge to an exact radiative transfer solution with the increase of

_{N}*N*, the simulation results have shown good agreement between the

*SP*approximation and Monte Carlo methods [14

_{7}14. Y. Lu and A. F. Chatziioannou, “A parallel adaptive finite element method for the simulation of photon migration with the radiative-transfer-based model,” Commun. Numer. Methods Eng. **25**, 751–770 (2009).
[CrossRef]

*N*+1)/2 boundary conditions can be obtained corresponding to (

*N*+1)/2 Eqs. (5). These boundary conditions are mixed and consist of linear combinations of the even-order

*ϕ*and their first derivatives. With respect to the

_{n}*composite moments*

*φ*of

_{n}*ϕ*, which are

_{n}*SP*approximation and its boundary conditions when practical measurements are performed at the wavelength

_{N}*λ*using bandpass filter:

_{k}*SP*

_{1}to

*SP*approximations can be found in [13

_{7}**220**, 441–470 (2006).
[CrossRef]

### 2.2. Fully parallel reconstruction algorithm

*SP*approximation [23]:

_{N}*φ*in boundary integration, we assume v·

_{i}*φ*are unknown variables in the boundary equations (Eq. (7b)). We obtain

_{i}*N*mesh subdomains

_{c}*𝓣*(1≤

_{c}*𝓣*≤

_{c}*N*), where

_{c}*N*is the number of the utilized CPUs. Regarding the finite element implementation, the space of linear finite elements

_{c}*𝓥*is introduced on

*𝓣.φ*(

_{i}*λ*) and

_{k}*S*(

_{i}*λ*) are approximated:

_{k}*φ*(

_{i,p}*λ*) and

_{k}*s*(

_{i,p}*λ*) are the discretized values at a discretized point

_{k}*p*when using the basis function

*υ*(

_{p}**r**);

*N𝓟*is the total number of the discretized points over the entire domain. Considering Eqs. (8), (9a), and (9b), for a volumetric element

*τ*, we have

_{e}*∂τ*is the boundary element if

_{e}*τ*is on the boundary and belongs to the respective subdomain in the parallel implementation. After assembling the submatrices on all the elements, we get

_{e}*IM*(

*λ*) corresponding to

_{k}*M*(

*λ*) although

_{k}**220**, 441–470 (2006).
[CrossRef]

*β*(

_{j}*λ*) can be calculated with respect to Eq. (4) when

_{k}*ψ*(

**r**, ŝ,

*λ*) is expanded;

*𝓣*after removing the rows in Eqs. (12). When the surface optical data are collected at

_{c}*K*wavelengths, we get

*γ*is the percentage at the wavelength λk of the total energy. It is usually considered as an ill-conditioned matrix because of the ill-posedness of BLT. The surface measured data

_{k}*S*is the upper bound of the source density; δ the regularization parameter; and

^{sup}*η*(·) the penalty function. Since all the data in Eq. (16) are distributed on

*N*CPUs, the optimization algorithms should work in parallel mode.

_{c}### 2.3. Further demonstration

24. G. Karypis and V. Kumar, “Multilevel k-way partitioning scheme for irregular graphs,” J. Parallel Distrib. Comput. **48**, 96–129 (1998).
[CrossRef]

*𝓐*between the spectrally-resolved measured data and the unknown source variable. The distribution of the measurable boundary discretized points is not uniform on each CPU. In addition,

27. J. Nocedal and S. J. Wright, *Numerical Optimization*, (Springer, New York, 1999).
[CrossRef]

## 3. Results

*in vivo*imaging system (CRI, Woburn, Massachusetts). This system uses a cooled CCD camera with a custom lens as the detector. The distinct characteristic of this system is that a liquid crystal tunable filter (LCTF) is used to acquire multispectral data. Generally, the filter bandpass width was set to 20

*nm*and the optical data was collected from a single view. The exposure time for each wavelength was adjusted to obtain high signal-to-noise ratio (SNR). After completing each optical signal acquisition, the phantom or mouse were scanned using an X-ray microCAT system (Siemens Preclinical Solutions, Knoxville, TN) to obtain CT images. The software Amira (Mercury Computer Systems, Inc. Chelmsford, MA) used the CT images to generate volumetric meshes for BLT reconstructions.

29. B. Kirk, J. W. Peterson, R. H. Stogner, and G. F. Carey, “libMesh: A C++ Library for Parallel Adaptive Mesh Refinement/Coarsening Simulations,” Eng. Comput. **22**, 237–254 (2006).
[CrossRef]

30. S. Balay, K. Buschelman, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, B. F. Smith, and H. Zhang, PETSc Web page, 2001, http://www.mcs.anl.gov/petsc.

24. G. Karypis and V. Kumar, “Multilevel k-way partitioning scheme for irregular graphs,” J. Parallel Distrib. Comput. **48**, 96–129 (1998).
[CrossRef]

*δ*was set to zero in the reconstructions. In order to test the proposed framework, we selected the

*SP*

_{1}(

*DA*),

*SP*

_{3}and

*SP*approximations to perform BLT reconstructions. All the simulations were performed on a cluster of 27 nodes (2 CPUs of 3.2GHz and 4 GB RAM at each node).

_{7}### 3.1. Mouse-shaped phantom experiments

_{2}and Disperse Red. To imitate the bioluminescence source, an optical fiber coupled to a green LED was embedded within the phantom. The emission spectrum of the LED was similar to that of a bioluminescence source. Its wavelength range was from 500

*nm*to 700

*nm*with a peak at about 567

*nm*. The photon distribution data at two wavelengths (580

*nm*and 640

*nm*) were used in the BLT reconstruction. Table 1 shows the optical properties (

*µ*and

_{a}*µ*′

*) at two wavelengths measured with the inverse adding-doubling method [31*

_{s}31. S. A. Prahl, M. J. C. van Gemert, and A. J. Welch, “Determining the optical properties of turbid media by using the adding-doubling method,” Appl. Opt. **32**, 559–568 (1993), http://www.opticsinfobase.org/abstract.cfm?URI=ao-32-4-559.
[CrossRef] [PubMed]

*SP*approximations, we set the anisotropic parameter g to 0.9. More detailed information about this phantom can be obtained elsewhere [16

_{N}16. 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]

*nm*and 640

*nm*respectively. They were acquired using an exposure time of 5

*min*. There are distinct differences between them because of the different optical properties at different wavelengths, which benefit the 3

*D*source localization.

*SP*

_{1},

*SP*

_{3}and

*SP*approximations. Due to the absence of a regularization parameter, the

_{7}*SP*

_{1}-based reconstruction was very sensitive to the measured noise and we could not obtain good source localization, as shown in Fig. 4(a). Figures 4(b) and 4(c) show the reconstructed results when

*SP*

_{3}and

*SP*approximations were used. From the figures it is clear that the source positions are reconstructed well when using high-order approximations. The localization errors are less than 1

_{7}*mm*in two directions, which can clearly be observed from Figs. 4(b) and 4(c). These reconstructed results show the importance and performance of high-order

*SP*approximations for BLT reconstruction.

_{N}### 3.2. Reconstruction performance optimization

*SP*

_{1}). In order to save a dense inverse matrix

*IM*(

*λ*), the

_{k}*SP*

_{1}-based reconstruction requires only about 94 MB of space compared to about 1.5 GB in the

*SP*-based reconstructions. Although the proposed fully parallel framework has the ability to process matrices with large dimensions by distributing the storage, reconstruction time becomes impractical with the increase of the approximation order

_{7}*N*and the number of used total wavelengths

*K*. Performance optimization is indispensable to improve the efficiency of the proposed framework. The quasi-Newton optimization method (BLMVM) has been selected to significantly reduce reconstruction time when compared to general Newton-type methods. Additionally, matrix inversion and the number of utilized CPUs further optimizes the reconstruction framework.

#### 3.2.1. Direct vs. iterative inversions

*N*points, the

_{𝓟}*SP*-based BLT reconstruction must process a

_{N}*N**

*N*𝓟×

*N**

*N*

*matrix. The computational complexity of the matrix inversion is*

_{𝓟}*O*((

*N**

*N*

_{𝓟})

^{3}) if direct inversion methods are used. The computational burden is significantly increased with the increase of

*N*and

*N*

*. When 10 CPUs were used in the above BLT reconstructions, the*

_{𝓟}*SP*

_{1}-based reconstruction required only 1,163.1sec, as opposed to 3,086.1sec for

*SP*

_{3}and 10,754.5sec for

*SP7*(Table 2).

*LU*-factorization-based relationship forming (i.e. forming Eq. (14)) utilized most of the total reconstruction time. For the

*SP*

_{1}and

*SP*approximations, the percentage increased from 58.0% to 96.8%, making it critically important to improve the performance of the matrix inversion. For iterative matrix inversion methods, the parallel incomplete

_{7}*LU*(

*ILU*) conjugate gradient (CG) method was used to accelerate the inversion. This preconditioner was provided by the Hypre open source package [32], developed by Lawrence Livermore National Laboratory (LLNL). For the

*SP*

_{1}- and

*SP*-based reconstructions, the total reconstruction time sharply decreased from 644.5 to 3,367.5sec, as shown in Table 2. Although the percentage of the relationship forming part in the total time increased regardless of

_{7}*SP*

_{1}and

*SP*approximations, the reconstruction speed was improved by a factor of 1.80 and 3.19 corresponding to the SP1 and SP7 approximations.

_{7}#### 3.3. Real mouse experiments

*in vivo*spectrum of a firefly luciferase-based source was used. This bead uses tritium (the half life is about 12 years) to excite phosphor which generates photons, making it a very stable source. The bead dimensions are 0.9

*mm*in diameter and 2.5

*mm*in length. Prior to performing the experiments, the mouse was anesthetized and the bead was injected into the thigh using a syringe. The photon distribution data at 580

*nm*and 660

*nm*were collected from the ventral view. The exposure time was set to 1.5

*min*. The volumetric mesh used in the reconstruction was generated using CT images of the mouse and contained 5,932 points and 24,120 tetrahedral elements. Figure 6(a) shows the mapped photon distribution after co-registration between the photograph of the mouse and the volumetric mesh. From the CT images, the tritium source can be clearly identified, as shown in Fig. 7. Since the photon propagation path consists almost totally of muscle, the reconstruction domain was considered to be homogeneous muscle tissue. The corresponding mouse muscle optical properties were then used in the reconstruction (Table 1).

*SP*

_{1},

*SP*

_{3}, and

*SP*approximations are shown in Fig. 7. The actual center position of the tritium source was at (51.8,-0.1). The reconstructed center position obtained from

_{7}*SP*

_{1},

*SP*

_{3}, and

*SP*approximations was at about (51.1,0.2). The

_{7}*SP*-based reconstruction was similar with the

_{7}*SP*

_{3}-based one regarding the source center position. Although the

*SP*

_{1}-based reconstruction was somewhat different compared to the

*SP*

_{3}-and

*SP*-based results, the source localization errors were measured to be less than 1

_{7}*mm*in two directions. This result was similar to the

*SP*

_{3}- and

*SP*-based reconstruction in the phantom experiments. The difference between phantom- and real mouse-based BLT reconstructions was that the tritium source could be localized well in the

_{7}*SP*

_{1}-based reconstruction. This was likely because the tritium source was superficial compared to the LED source in the mouse-shaped phantom. In addition, the mouse surface was more irregular than the mouse-shaped phantom surface, it should have the effect in the reconstruction.

## 4. Discussions and Conclusion

*SP*

_{1}and

*SP*approximations.

_{7}## Acknowledgement

## References and links

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

2. | R. Weissleder, “Scaling down imaging: Molecular mapping of cancer in mice,” Nat. Rev. Cancer |

3. | J. Virostko, A. C. Powers, and E. D. Jansen, “Validation of luminescent source reconstruction using single-view spectrally resolved bioluminescence images,” Appl. Opt. |

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

5. | E. E. Lewis and W. F. Miller Jr., , |

6. | 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 |

7. | X. Gu, Q. Zhang, L. Larcom, and H. Jiang, “Three-dimensional bioluminescence tomography with model-based reconstruction,” Opt. Express |

8. | Y. Lv, J. Tian, W. Cong, G. Wang, J. Luo, W. Yang, and H. Li, “A multilevel adaptive finite element algorithm for bioluminescence tomography,” Opt. Express |

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. |

10. | A. D. Klose, “Transport-theory-based stochastic image reconstruction of bioluminescent sources,” J. Opt. Soc. Am. A |

11. | C. R. E. de Oliveira, “An arbitrary geometry finite element method for multigroup neutron transport with anisotropic scattering,” Progr. Nucl. Energ. |

12. | S. Wright, M. Schweiger, and S. R. Arridge, “Reconstruction in optical tomography using the PN approximations,” Meas. Sci. Technol. |

13. | A. D. Klose and E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. |

14. | Y. Lu and A. F. Chatziioannou, “A parallel adaptive finite element method for the simulation of photon migration with the radiative-transfer-based model,” Commun. Numer. Methods Eng. |

15. | G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys. |

16. | 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. |

17. | 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. |

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

19. | 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. |

20. | T. Vo-Dinh, |

21. | A. Ishimaru, |

22. | R. C. Haskell, L. O. Svaasand, T. Tsay, T. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A |

23. | S. S. Rao, |

24. | G. Karypis and V. Kumar, “Multilevel k-way partitioning scheme for irregular graphs,” J. Parallel Distrib. Comput. |

25. | G. H. Golub and C. F. Van Loan, |

26. | M. Benzi, “Preconditioning techniques for large linear systems: a survey,” J. Comput. Phys. |

27. | J. Nocedal and S. J. Wright, |

28. | S. J. Benson and J. Moré, “A limited-memory variable-metric algorithm for bound-constrained minimization,” Technical Report ANL/MCS-P909-0901, Mathematics and Computer Science Division, Argonne National Laboratory (2001). |

29. | B. Kirk, J. W. Peterson, R. H. Stogner, and G. F. Carey, “libMesh: A C++ Library for Parallel Adaptive Mesh Refinement/Coarsening Simulations,” Eng. Comput. |

30. | S. Balay, K. Buschelman, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, B. F. Smith, and H. Zhang, PETSc Web page, 2001, http://www.mcs.anl.gov/petsc. |

31. | S. A. Prahl, M. J. C. van Gemert, and A. J. Welch, “Determining the optical properties of turbid media by using the adding-doubling method,” Appl. Opt. |

32. | R. D. Falgout and U. M. Yang, “hypre: A library of high performance preconditioners,” In |

**OCIS Codes**

(110.6960) Imaging systems : Tomography

(170.3010) Medical optics and biotechnology : Image reconstruction techniques

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

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: June 10, 2009

Revised Manuscript: July 24, 2009

Manuscript Accepted: August 13, 2009

Published: September 3, 2009

**Virtual Issues**

Vol. 4, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

Yujie Lu, Hidevaldo B. Machado, Ali Douraghy, David Stout, Harvey Herschman, and Arion F. Chatziioannou, "Experimental bioluminescence tomography with fully parallel radiative-transfer-based reconstruction framework," Opt. Express **17**, 16681-16695 (2009)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-17-19-16681

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### References

- V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weisslder, "Looking and listening to light: the evolution of whole body photonic imaging," Nat. Biotechnol. 23,313-320 (2005). [CrossRef] [PubMed]
- R. Weissleder, "Scaling down imaging: Molecular mapping of cancer in mice," Nat. Rev. Cancer 2,11-18 (2002). [CrossRef] [PubMed]
- J. Virostko, A. C. Powers, and E. D. Jansen, "Validation of luminescent source reconstruction using single-view spectrally resolved bioluminescence images," Appl. Opt. 46,2540-2547 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=ao-46-13-2540. [CrossRef] [PubMed]
- 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]
- E. E. Lewis and W. F. Miller, Jr., Computational Methods of Neutron Transport, (JohnWiley & Sons, New York, 1984).
- 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 13,6756-6771 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-18-6756. [CrossRef] [PubMed]
- X. Gu, Q. Zhang, L. Larcom, and H. Jiang, "Three-dimensional bioluminescence tomography with model-based reconstruction," Opt. Express 12,3996-4000 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-17-3996. [CrossRef] [PubMed]
- Y. Lv, J. Tian, W. Cong, G. Wang, J. Luo, W. Yang, and H. Li, "A multilevel adaptive finite element algorithm for bioluminescence tomography," Opt. Express 14,8211-8223 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-18-8211. [CrossRef] [PubMed]
- H. Dehghani, S. C. Davis, S. Jiang, B. W. Pogue, K. D. Paulsen, and M. S. Patterson, "Spectrally resolved bioluminescence optical tomography," Opt. Lett. 31,365-367 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=ol-31-3-365. [CrossRef] [PubMed]
- A. D. Klose, "Transport-theory-based stochastic image reconstruction of bioluminescent sources," J. Opt. Soc. Am. A 24,1601-1608 (2007), http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-24-6-1601. [CrossRef]
- C. R. E. de Oliveira, "An arbitrary geometry finite element method for multigroup neutron transport with anisotropic scattering," Progr. Nucl. Energ. 18,227-236 (1986). [CrossRef]
- S. Wright, M. Schweiger, and S. R. Arridge, "Reconstruction in optical tomography using the PN approximations," Meas. Sci. Technol. 18,79-86 (2007). [CrossRef]
- A. D. Klose and E. W. Larsen, "Light transport in biological tissue based on the simplified spherical harmonics equations," J. Comput. Phys. 220,441-470 (2006). [CrossRef]
- Y. Lu and A. F. Chatziioannou, "A parallel adaptive finite element method for the simulation of photon migration with the radiative-transfer-based model," Commun. Numer. Methods Eng. 25,751-770 (2009). [CrossRef]
- G. Wang, Y. Li, and M. Jiang, "Uniqueness theorems in bioluminescence tomography," Med. Phys. 31,2289-2299 (2004). [CrossRef] [PubMed]
- 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]
- 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. 50,5421-5441 (2005). [CrossRef] [PubMed]
- Y. Lv, J. Tian, W. Cong, G. Wang, W. Yang, C. Qin, and M. Xu, "Spectrally resolved bioluminescence tomography with adaptive finite element analysis: methodology and simulation," Phys. Med. Biol. 52,4497-4512 (2007). [CrossRef] [PubMed]
- 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. 50,4225-4241 (2005). [CrossRef] [PubMed]
- T. Vo-Dinh, Biomedical Photonics Handbook, (CRC Press, 2002).
- A. Ishimaru, Wave propagation and scattering in random media, (IEEE Press, 1997).
- R. C. Haskell, L. O. Svaasand, T. Tsay, T. Feng, M. S. McAdams, and B. J. Tromberg, "Boundary conditions for the diffusion equation in radiative transfer," J. Opt. Soc. Am. A 11,2727-2741 (1994), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-11-10-2727. [CrossRef]
- S. S. Rao, The finite element method in engineering, (Butterworth-Heinemann, Boston, 1999).
- G. Karypis and V. Kumar, "Multilevel k-way partitioning scheme for irregular graphs," J. Parallel Distrib. Comput. 48,96-129 (1998). [CrossRef]
- G. H. Golub and C. F. Van Loan, Matrix computations (3rd ed.), (Johns Hopkins University Press, 1996).
- M. Benzi, "Preconditioning techniques for large linear systems: a survey," J. Comput. Phys. 182,418-477 (2002). [CrossRef]
- J. Nocedal and S. J. Wright, Numerical Optimization, (Springer, New York, 1999). [CrossRef]
- S. J. Benson and J. Moré, "A limited-memory variable-metric algorithm for bound-constrained minimization," Technical Report ANL/MCS-P909-0901, Mathematics and Computer Science Division, Argonne National Laboratory (2001).
- B. Kirk, J. W. Peterson, R. H. Stogner, and G. F. Carey, "libMesh: A C++ Library for Parallel Adaptive Mesh Refinement/Coarsening Simulations," Eng. Comput. 22,237-254 (2006). [CrossRef]
- S. Balay, K. Buschelman, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, B. F. Smith, and H. Zhang, PETSc Web page, 2001, http://www.mcs.anl.gov/petsc.
- S. A. Prahl, M. J. C. van Gemert, and A. J. Welch, "Determining the optical properties of turbid media by using the adding-doubling method," Appl. Opt. 32,559-568 (1993), http://www.opticsinfobase.org/abstract.cfm?URI=ao-32-4-559. [CrossRef] [PubMed]
- R. D. Falgout and U. M. Yang, "hypre: A library of high performance preconditioners," in Proceedings of the International Conference on Computational Science-Part III, p. 632-641 (2002).

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