Multispectral Bioluminescence Tomography: Methodology and Simulation

Bioluminescent imaging has proven to be a valuable tool for monitoring physiological and pathological activities at cellular and molecular levels in living small animals. Using biological techniques, target cells can be tagged with reporters encoding several kinds of luciferase enzymes, which generate characteristic photons in a wide spectrum covering the infrared range. Part of the diffused light can reach the body surface of the small animal, be separated into several spectral bands using appropriate filters, and collected by a sensitive CCD camera. Here we present a bioluminescence tomography (BLT) method for a bioluminescent source reconstruction from multispectral data measured on the external surface, and demonstrate the advantages of multispectral BLT in a numerical study using a heterogeneous mouse chest phantom. The results show that the multispectral approach significantly improves the accuracy and stability of the BLT reconstruction even if the data are highly noisy.


INTRODUCTION
Bioluminescent imaging (BLI) is an emerging technology to monitor molecular and cellular activities in vivo using various small animal models. This new modality is extremely sensitive, cost-effective, and nontoxic for investigating a wide variety of diseases such as cancers and facilitating drug development [1][2][3]. Bioluminescence tomography (BLT) is a major frontier of BLI.
In bioluminescent imaging, target cells are labeled with reporter genes encoding luciferase enzymes in a living small animal. Upon a chemical reaction with a substrate luciferin in the presence of ATP and oxygen, the luciferase releases photons to allow observation of molecular and cellular activities [2]. Luciferase enzymes from firefly (Fluc), click beetle (CBGr68, CBRed), and Renilla reniformis (hRluc) are often utilized as reporter genes. It was shown that these four kinds of luciferase enzymes have different emission spectra: at temperature 37 • C, Fluc, CBGr68, CBRed, and hRLuc exhibit their spectral peaks of 612 nm, 543 nm, 615 nm, and 480 nm, respectively [4]. Recently, a tricolor reporter was also developed, which emits green to red light [5]. These results enable multispectral BLT and its biomedical applications.
The bioluminescent photon propagation in the biological tissue is subject to both scattering and absorption. A significant number of bioluminescent photons escape from the body surface of the animal [6]. Using optical filters of different spectral bands, the photons in each spectral band can be captured by a highly sensitive CCD camera. In any band, the photon propagation in the tissue is typically described by the radiative transport equation [7]. However, a direct solution to the transport equation is not practically affordable due to the computational complexity. Since the scattering predominates over the absorption in this context, the diffusion model can be used as a good approximation to the physical process [8]. With filter techniques, the optical properties of the tissue can be determined for every spectral band using diffuse reflectance measurements [9,10]. Diffuse optical tomography (DOT) can also be applied to reconstruct the band-specific spatially variable optical parameters [11]. Then, based on the diffusion approximation, the BLT can be formulated as an inverse source problem. This problem is severely underdetermined and ill-posed. Recently, Wang et al. described the BLT principles [12,13] and reported the uniqueness of the solution dependent on a priori knowledge [14]. Several reconstruction methods for BLT were reported with numerical and phantom data [15][16][17][18][19][20]. The methods were found to perform well when a priori knowledge such as permissible source region was specified to overcome the ill-posedness of the problem.

International Journal of Biomedical Imaging
In this paper, we present a multispectral approach to enhance the accuracy and stability of the BLT reconstruction. To avoid any confusion, we would underline that there can be two meanings attached to the concept of multispectral BLT. The first interpretation is that if we have only a single bioluminescent source that is spatially and spectrally distributed we can decompose its spectrum into a number of bands or channels for bandwise measurement, and then perform the source reconstruction. The second interpretation is that when we have multiple bioluminescent sources that are spatially and spectrally distributed independently, we can decompose their spectra respectively for according measurement and reconstruct all these generally different source distributions in an integrated fashion. As an initial investigation to investigate multispectral BLT, we assume the first interpretation of the multispectral BLT approach. Alternatively, we may also refer to our method as a multiband BLT algorithm in this context.
In this project, multiple sets of surface data corresponding to wavelength bands are measured to help improve the BLT reconstruction significantly. A finite-element discretization of the diffusion approximation model is utilized to establish a linear relationship between the source photon density and the measured surface photon density in each band. A linear least square optimization is performed for multispectral BLT reconstruction. In the next section, we will present the multispectral BLT methodology. Then, we will demonstrate the feasibility of our method in numerical simulation. In the last section, we will discuss the relevant issues and draw a conclusion.

Image reconstruction
For each spectral band, the diffusion equation (1) and its boundary condition (2) can be formulated into a matrix equation using the finite-element method as follows [23]: where {Φ ν } and {S ν } are the collection of all the nodal values of the photon density Φ ν (r) and source density S ν (r), respectively.
is a positive-definite matrix. Then, the photon density {Φ ν } can be obtained from (4): By performing a spectral analysis, the energy contribution of a bioluminescent source can be determined over the entire spectral interval [4] as represented by S p ν = ω ν S t , where ω ν ≥ 0 and τ ν=1 ω ν ≈ 1, S t denotes the total photon density, and S p ν the photon density within [w ν , w ν+1 ], as shown in Figure 1. From the spectral distribution of a bioluminescence source and (6), the following linear system is obtained: A. X. Cong and G. Wang where Because the measured bioluminescent data are typically corrupted by noise, it is not optimal to solve for {S t } directly from (7). Hence, we propose to use the following optimization procedure to find a regularized solution [24]: where U ν stands for an upper bound, S t i the values in {S t }, Λ a weight matrix, V Λ = V T ΛV, η a stabilizing function, and α the regularization parameter. This is a standard linear least square problem with constrains. From (7), the source densities S t are constrained by the measured multispectral information, which helps improve the stability of the reconstruction.

Spectral distributions
Bioluminescent spectral analysis experiment [4] shows that the four kinds of luciferase enzymes hRLuc, CNGr68, Fluc+, and CBRed have different emission peaks at 480 nm, 543 nm, 612 nm, and 615 nm, respectively, as shown in Figure 1. Their fusion creates broad emission spectra. The spectral properties of the luciferase light source should have a significant impact on how much of the light is transmitted to the external surface. The detected polychromatic photons carry more information about the bioluminescence source than the corresponding integrated signals. We used the experimental data in Figure 1 to estimate the energy distribution for our numerical simulation. Let us assume that target cells be tagged with reporters encoded with the four kinds of luciferase enzymes and that the cells emit photons in the spec-

Single-band reconstruction
The numerical simulation was performed using a heterogeneous numerical phantom that contains regions resembling lungs, heart, muscle, and bone. This cylindrical phantom had a diameter of 30 mm and height of 26 mm, as shown in Figure 2. The phantom was discredited into 6576 vertex nodes and 11340 prism elements. The two bioluminescent sources were embedded in the left lung, as shown in Figure 3(a). The first source was located at (−8.66, 3.46, 13.1), while the second one at (−10.21, −3.17, 13.1). Both sources had photon density of 300 pW/mm 3 . The permissible region was selected based on a priori knowledge, as shown in Figure 3(b). This region contained 308 elements. The optical parameters averaged over the spectral range [400 nm, 750 nm] for each type of structure in the heterogeneous phantom are listed in Table 1 [16,18].
The simulated measurement data on the 1024 detector points on the phantom side surface were generated according to the finite-element forward model. Then, the measurement data were corrupted with 20% Gaussian noise to simulate measurement errors. We set the stabilizing function η(X) = X T X and the regularization parameter α = 3.0 × 10 −8 , and then performed the source reconstruction using our published single-band BLT algorithm [18]. The reconstructed locations of the sources are shown in Figure 4(a). The photon densities of the sources are shown in Figure 5(a). The quantitative data on the reconstruction are listed in Table 2.

Multiband reconstruction
We performed the proposed multispectral BLT reconstruction using the same numerical model described in Section 3.2. According to the literature [19,20], various optical parameters (absorption, scattering) were assigned to different regions of the numerical heterogeneous phantom, as listed in Table 3.
Three measurement data sets on the 1024 detector points on the phantom side surface were similarly generated for spectral ranges [400 nm, 530 nm], [530 nm, 630 nm], and [630 nm, 750 nm], respectively. The data sets were also corrupted with 20% Gaussian noise. With the same definition for the stabilizing function and the regularization parameter in Section 3.2, the three simulated data sets were taken into the multiband/multispectral reconstruction of the singlesource distribution using our method described in Section 2.
The reconstructed locations of the sources are shown in Figure 4(b). The photon densities of the sources are shown in Figure 5(b). The multiband reconstruction is quantitatively compared to the single-band reconstruction in Table 2. The reconstructed results of multispectral BLT are clearly superior to those of single-spectral BLT.

DISCUSSIONS AND CONCLUSION
It seems that the multispectral approach is both feasible and beneficial. Target cells can now be tagged with reporters encoding several kinds of luciferase enzymes, resulting in a much broader emission spectrum. Dichroic filters can be used to separate the mixed signal into several wavelength A. X. Cong and G. Wang  bands. As a result, a much more effective measurement can be made to improve the BLT reconstruction of such a source distribution. This multiband method significantly reduces the ill-posedness of the BLT problem. Our simulation results have showed that the multiband reconstruction has a better accuracy than the single-band reconstruction, especially in the case of high noise.
In conclusion, we have presented the multiband reconstruction method for BLT based on finite-element analysis. The multispectral approach greatly increases the measurement information, effectively reduces the ill-posedness of BLT, and significantly improves the accuracy and stability of the bioluminescent source reconstruction. Physical phantom and in vivo small animal experiments will be performed in the future. Also, we are investigating the multispectral BLT in the second sense discussed in the first section. 6 International Journal of Biomedical Imaging