JT3A.25.pdf
Optics in the Life Sciences 2015 © OSA 2015
Molecular Fluorescence Tomography with Structured Light and Compressive Sensing Ruoyang Yao, Qi Pian and Xavier Intes* Department of Biomedical Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180, USA *Corresponding author:
[email protected]
Abstract: We apply compressive sensing based preconditioning techniques to improve timeresolved wide-field FMT reconstructions. By designing masks to illumination and detection experimental basis, the coherence of the sensitivity matrix is reduced and optical reconstructions are improved. OCIS codes: (070.6120) Spatial light modulators, (110.4234) Multispectral and hyperspectral imaging, (170.3010) Image reconstruction techniques, (170.3650) Lifetime-based sensing, (170.6920) Time-resolved imaging; (170.6960) Tomography;
1. Introduction Fluorescence Molecular Tomography has been an intense area of research focus over the last decade [1]. The increased availability of molecular fluorescent probes in conjunction with the high sensitivity of optical systems has positioned FMT has a powerful molecular imaging technique at the preclinical stage. However, the main applications of FMT are still limited to retrieve the bio-distribution of probes with low resolution. To increase information content beyond bio-distribution, and to improve tomographic resolution, time-resolved systems are required. Indeed, lifetime is well known contrast mechanism that is routinely employed in microscopy to probe the micro-environment, to increase multiplexing power and/or monitor protein-protein interactions when using Forster Resonance Imaging Transfer. However, the implementation of time-resolved tomographic instrumentation in preclinical settings is typically limited to cross-sectional imaging with long acquisition times [2,3], reducing considerably its appeal for preclinical investigations. To enable whole-body imaging and retain the benefit of time-resolved imaging system, our group has proposed a novel instrumental approach based on structured light to perform FMT [4]. We have demonstrated the potential of wide-field structured light illumination and detection strategies to perform almost real time DOT [5] and fast hyperspectral time-resolved wide-field FMT [6]. In both cases, we have employed bar-like illumination and detection patterns to acquire data with optimal Signal-to-Noise Ratio [7]. The inverse problem is then cast within a classical inverse problem framework in which the extended illumination and detection patterns are simulated within a Monte Carlo forward model [8]. However, such formulation leads still to a highly ill-posed/ill-conditioned problem due to the very small number of measurements compared to unknowns. Hence, reconstructions are typically sensitive to noise and model mismatch. To improve the reconstruction performances, we have proposed recently to take advantage of the inherent sparsity of the image space encountered in typical FMT applications. Indeed, FMT relies on “targeted probes” that accumulate preferentially in certain tissue, and hence create a sparse imaging space. Hence, Compressive Sensing based approaches are good candidates to address the inverse problem. For instance, we have proposed a Lp-norm approach coupled with early gates to yield high-resolution reconstructions [9]. Here we investigate one other potential use of CS based techniques for FMT. More precisely we investigate the potential of a CS-based preconditioning technique [10] on our wide-field illumination/detection patterns strategy. 2. Forward Model and Inverse Problem Combining coupled diffusion equation and Robin type boundary condition and assuming that absorption coefficient at excitation and emission wavelength are equal, the excitation light and emission light field are given by:
xi (r ) g x (r, r ') s i (r ')dr ', i 1,...N s
(1) ; mi (r ) g m (r, r ')xi (r ') axf (r ') dr ', i 1,... N s (2)
where subscripts x and m stand for the excitation and emission wavelength, r denotes location and s i denotes the ith source. a is the absorption coefficient, axf is fluorophore yield and g x / g m are Green’s functions. In case of weak fluorophore, the contribution of fluorophore absorption can be ignored, and then the measurement at the jth detector due to the ith source can be expresses as: i , j g emj (r )exi (r ) af (r ) dr
i 1,...N s , j 1,...N d
(3)
JT3A.25.pdf
Optics in the Life Sciences 2015 © OSA 2015
A linear relationship between the measured fluorescence b at each detector position for each illumination source and the unknown effective quantum yield x of the fluorophore can be formulated as Ax b , where A is the Jacobian matrix that establishes the relationship between the measurements and the unknowns. A is expressed as:
g 1em ,1ex1 ,1 g Nd,1ex1 ,1 A em 1 2 g em ,1ex ,1 N N g emd,1ex ,1s 4. Bounds on the Coherence
1 1 g em , N ex , N Nd 1 g em , N ex , N R M N 1 2 g em , N ex , N Nd Ns g em , N ex , N
(4)
Compressive sensing (CS) is an emerging field based on the discovery that sparse signals and images can be reconstructed from highly incomplete information. Besides sparsity, CS requires incoherence. Mathematically, the coherence of a matrix is defined as: a p , aq ( A) max (5) p,q, p q a p 2 aq 2 where ap and aq are two different columns of A. A less strict alternative measurement is cumulative coherence defined as: a p , aq 1 (k, A) max max (6) p Q k , pQ aq qQ a p 2 2 which is monotonically non-decreasing function in k. It has been shown that the decrease of cumulative coherence will lead to better recovery of sparse signals. Following [9], we can note that as A is actually the columnwise Knocker product of the excitation light field Φ and emission light field G: ex1 ,1 ex1 , N g 1em ,1 g 1em , N Φ R N s N G R Nd N (7) Nd Nd exN,1s exN,sN g em ,1 g em , N Nd 1 T Let ak be the kth column of sensitivity matrix A, k [ex1 ,k , , exN,sk ]T and gk [ gem , k , , gem, k ] , then we have:
A Φ G [1 g1 ,2 g 2 , ,N g N ]
(8)
Based on further derivations, the bounds on coherence can be obtained as [10]:
( A) 1 (k, A) K ( ΦT Φ I N
2 F
GT G I N
2 F
)
(9)
where E F denotes the Frobenius norm and K is a constant relevant to N, Ns, Nd and k, Φ and G are columnnormalized excitation and emission light field matrices. As a result, reducing the coherence of sensitivity matrix can be accomplished by reducing ΦT Φ I N
2 F
and GT G I N
2 F
.
5. Preconditioning strategy Preconditioners can be applied in the goal to bring ΦT Φ and G T G close to the identity matrix. This can be expressed as: Φ pre M s Φ G pre M d G (10) To compute the preconditioner, a Singular Value Decomposition is then preformed, Φ = Us Σs Vs , and the preconditioner of the excitation field is expressed as: M s ( Λ s I ) 1/ 2 U sT Λ s Σ s Σ sT (11)
JT3A.25.pdf
Optics in the Life Sciences 2015 © OSA 2015
where ε is a stabilizing parameter. This preconditioner is equivalent to an optical mask applied to the illumination basis. Note that this optical mask may have negative values, impossible to implement in experiments directly. In this case he mask is decomposed two matrices with non-negative entries: M s M (s ) (M (s ) ) (12) The preconditioner of the emission field, also known as the measurement mask can be computed with the same approach to yield an optical mask applied to the detection basis. Then the new sensitivity matrix (preconditioned) can be calculated using (8). 6. Simulation Results We performed a set of in silico experiments to test the improvements on incoherence as well as reconstruction fidelity when using this preconditioning scheme. We designed a synthetic phantom of size 40mm×40mm×20mm with murine optical properties. The whole imaging domain is discretized into 14,791 nodes (80,095elements), in our mesh based Monte Carlo code, with two spherical inclusions (radius 2.5mm) simulated in the center of the phantom, 18.85mm apart. 40 bar-shape patterns [2] are simulated for illumination and detection to yield time-resolved fluorescence data that replicates our typical experiments settings (Figure 2a). Then, precondition of the Jacobian is performed by computing the illumination and detection optical masks as described above. Results are compared under 3 conditions: 1) no masks are applied; 2) precondition the sensitivity matrix directly; 3) apply separate masks to illumination and detection. First, we report on the decrease of the coherence of the sensing matrix when applying preconditioning (Figure 1 c). Using area under curve as an indicator, separate masks Ms and Md reduce the AUC to 13.31% while the direct mask MA reduce the AUC to 11.02%. This result indicates that the CS-based preconditioning technique outperforms classical preconditioning to reduce coherence. Example of the preconditioned illumination and detection basis after the CS preconditioning are provided in Figure 2 b. a) c) b)
Fig. 1. A) Example of experimental illumination and detection basis example; b) Examples of preconditioned illumination and detection basis; c) Normalized coherence of the sensing matrix before and after conditioning..
Applying several CS-based algorithms, both greedy type (such as StOMP) and convex relation (such as ISTA), in conjunction with the preconditioner led to significant improvements in localization and volume in the optical reconstructions (not shown here due to lack of space). 7. References [1] C. Darne, Y. Lu, and E. M. Sevick-Muraca, "Small animal fluorescence and bioluminescence tomography: a review of approaches, algorithms and technology update," Phys. Med. Biol. 59, R1–R64 (2014). [2] M. J. Niedre, R. H. de Kleine, E. Aikawa, D. G. Kirsch, R. Weissleder, and V. Ntziachristos, "Early photon tomography allows fluorescence detection of lung carcinomas and disease progression in mice in vivo," Proc. Natl. Acad. Sci. USA 105, 19126–31 (2008). [3] K. M. Tichauer, R. W. Holt, F. El-Ghussein, Q. Zhu, H. Dehghani, F. Leblond, and B. W. Pogue, "Imaging workflow and calibration for CTguided time-domain fluorescence tomography," Biomed. Opt. Express 2, 3021–36 (2011) [4] J Chen, V Venugopal, F Lesage and X Intes, “Time Resolved Diffuse Optical Tomography with patterned light illumination and detection,” Optics Letters 35, 2121-2123 (2010). [5] S. Bélanger, M. Abran, X. Intes, C. Casanova, and F. Lesage, “Real-time diffuse optical tomography based on structured illumination,” J. Biomed. Opt. 15(1), 016006 (2010) [doi:10.1117/1.3290818]. [6] Q Pian, R Yao, L Zhao and X Intes, “Hyperspectral Time-Resolved Wide-Field Fluorescence Molecular Tomography based on Structured Light and Single Pixel-Detection,” Optics Letters, in press (2015). [7] V Venugopal, J Chen, F Lesage and X Intes, “Full-field time-resolved fluorescence tomography of small animals,” Optics Letters 35, 31893191 (2010). [8] J Chen and X Intes, “Comparison of Monte Carlo Methods for Fluorescence Molecular Tomography - Computational Efficiency,” Medical Physics 38 (10), 5788-5798 (2011). [9] L Zhao, H Yang, W Cong, G Wang and X Intes, “Lp Regularization for Early Time-Gate Fluorescence Molecular Tomography,” Optics Letters 39(14), 4156-4159 (2014). [10] Jin, A., B. Yazici, and V. Ntziachristos. "Light illumination and detection patterns for fluorescence diffuse optical tomography based on compressive sensing." (2014): 1-1
