BSu3A.89.pdf Biomedical Optics and 3D Imaging OSA 2012

Bioluminescence Tomography With A PDE-Constrained Algorithm Based On The Equation Of Radiative Transfer

Hyun Keol Kim1, Andreas H. Hielscher1,2

1Department of Biomedical Engineering, Columbia University, 500 West, 120th St., New York, NY 10027, USA 2Department of Radiology, Columbia University, 660 West, 168th St., New York, NY 103277, USA

Author e-mail address: hkk2107@columbia.edu

Abstract: We present the first bioluminescence tomography algorithm that makes use of the PDE-constrained concept, which has shown to lead to significant savings in computation times in similar applications. Implementing a sequential quadratic programming (SQP) method, we solve the forward and inverse problems simultaneously. Using numerical results we show that the PDE-constrained SQP approach leads to ~10 fold increase in convergence when compared to a standard unconstrained method. ©2011 Optical Society of America OCIS codes: (170.3010) Image reconstruction techniques; (170.6280) Spectroscopy, fluorescence and luminescence

1. Introduction Bioluminescence tomography (BLT) is an imaging modality that makes use of bioluminescent markers that emits light when certain biochemical environments are encountered, which can thus detect molecular processes associated with the development of diseases [1]. BLT recovers the spatial distribution of bioluminescence inside the medium from measured intensities on the tissue surface. The most commonly used bioluminescence marker is luciferases, which has a broad light emission spectrum with a peak between 538 nm and 570 nm. In this wavelength range, the intrinsic tissue absorption is relatively high [2]. Therefore the equation of radiative transfer (ERT) is more appropriate as a model of light propagation caused by luciferase, rather than its diffusion approximation (DA) that limits its use to scattering-dominant, low-absorbing media [3]. However, ERT-based BLT codes require a large amount of computation times when they are used with unconstrained approaches [4-8] that require a large number of forward runs to complete convergence. Thus it is highly desirable to develop computationally efficient ERT-based BLT codes. To this end, we present the first PDE-constrained bioluminescence image reconstruction algorithm. We evaluate the performance of the PDE-constrained BLT scheme using numerical results by comparing the new algorithm with an unconstrained code that makes use of the limited-memory Broyden–Fletcher–Goldfarb–Shanno (lm-BFGS) method [4,6].

2. Method The optical bioluminescence tomographic problem can be formulated in more general terms as follows:

min f (!,q) = 12

Qd! ! zd( )2 + R(q)d" subject to C = A ! ! b(q) = 0 (1)

where f (!,q) is the objective function that quantifies the difference between predictions Qd! and measurements

zd of emitted light made on the tissue surface; R(q) is the regularization term used to stabilize the optimization behavior; C is the discretized version of the radiative transfer equation. In this work, we introduce a PDE-approach method that uses a SQP method [9] to solve the forward and inverse problems simultaneously. The SQP method solves the following quadratic problem for the step !x = (!q,!!) , then one will set xk+1 = xk +!xk .

min gkT!xk +12!xxTLxx (xk,!x )!xx

subject to(Ck,q )!qk + (Ck," )!"k +Ck = 0,

(2)

BSu3A.89.pdf Biomedical Optics and 3D Imaging OSA 2012

where Lxx (xk,!k ) is the Hessian of the Lagrangian function L(x,!) = f (x)+!T (A" ! b) and !k is the estimate for the Lagrange multiplier at the current iterate. The inverse and forward iterates, !qk and !!k , can be obtained respectively as

!qk = "(Hkr )"1gkr (3)

!!k = "(Ck,! )"1 (Ck,q )!qk +C[ ], (4)

where Hrk and gkr are the reduced Hessian and the reduced gradient respectively. The detailed description of this algorithm can be found in reference [9].

3. Results In the following examples, we demonstrate the performance of the proposed rSQP scheme as applied to the reconstruction of bioluminescent sources inside the medium. The numerical phantom has a circular shape with a diameter of 2cm, and one single bioluminescent source with a diameter of 0.15 cm is embedded inside the medium. The bioluminescent source has a power density of 10 W/cm3, which is to be found by the rSQP algorithm. For numerical experiments, we considered spectrally resolved data: 600, 610, 620 nm. The optical properties of the background medium are µa = 0.28 cm-1, !µs = 16.7 cm-1 at 600 nm, and µa = 0.16 cm-1, !µs = 16.4 cm-1 at 610 nm and µa = 0.11 cm-1, !µs = 16.1 cm-1 at 620 nm. The 72 detector positions are spaced equally around the surface. We generated two sets of synthetic data corrupted by 15 dB noise: a single wavelength data and a three wavelengths data, both are obtained by solving the radiative transfer equation for a given set of optical properties.

We started the reconstruction from a homogeneous initial guess of q , which is set to zero over the entire medium. A regularization term is added to reduce instabilities due to random errors in the input data. To evaluate the performance of the PDE-constrained BLT code, we compared the CPU time and accuracy attained with this algorithm to results obtained with the unconstrained BLT code that employs a so-called lm-BFGS algorithm. To quantify our results we used the deviation ! ! [0,"] and correlation ! ! ["1,1] factors [9]: the smaller ! and the larger ! , the better image quality. For the single wavelength data, both methods are similar to each other with respect to accuracy: !SQP = 0.68, !SQP = 0.98 and ! lm-BFGS = 0.62, ! lm-BFGS= 0.98. For the CPU times, the PDE-constrained rSQP method converges faster than the unconstrained lm-BFGS method by a factor of about 6.7: the PDE-constrained method took only 5.8 min to converge, while the unconstrained lm-BFGS code required 41 min. This acceleration can be explained by the fact that since the PD-constrained code treats the forward and inverse problems independently, it does not require the complete solution of the forward problem until the final minimum is reached. In other words, the PDE-constrained algorithm solves the forward equation given by the ERT (1) with a loose tolerance in the 10-2-10-3 range during the optimization process, which takes only a fraction of the time required for the exact solution of the forward problem with a much tighter tolerance ( ! 10-8). Fig. 1 shows this advantage of the PDE-constrained method: the two methods show a similar decrease in the error function with respect to the iteration number but show large difference with respect to CPU times.

(a) (b)

Fig. 1 Objective function values of the two methods with respect to iteration numbers and CPU times.

BSu3A.89.pdf Biomedical Optics and 3D Imaging OSA 2012

Fig. 2 shows the results of the second data set with the multi-spectral data (600, 610, 620 nm). As before, ! and ! are measured for the two methods: !SQP = 0.74, !SQP = 0.94 and ! lm-BFGS = 0.73, ! lm-BFGS = 0.95. As expected, the multi-spectral data produced more accurate results than the single wavelength data, especially in the absolute strength of the target source. Furthermore, we observed again a substantial decrease in the time to convergence. The PDE-constrained code reaches convergence in 11.2 min while the unconstrained code required 98 min. In summary, we present here a PDE-constrained reduced Hessian sequential quadratic programming (rSQP) method that solves the bioluminescence inverse source problem in a computationally efficient manner. The proposed algorithm solves the forward and inverse problems simultaneously by treating the forward and inverse variables independently, which consequently leads to a significant time saving in the reconstruction process. Numerical experiments with simulated data show that our rSQP method increases the reconstruction speed by a factor of ~10 as compared to a standard lm-BFGS code, which is believed to be the most efficient unconstrained method. The algorithm presented here promises to accelerate the reconstruction process in combination with any forward model, and this advantage will be greatest for cases where small geometries typical for small-animal imaging are considered that requires the RTE as a forward model. This work was supported in part by a grant from the National Cancer Institute (NCI) at the National Institutes of Health (Grant # NCI-U54CA126513-039001).

4. References

[1] D. K. Welsh and S. A. Kay, “Bioluminescence imaging in living organisms,” Curr. Opin. Biotechnol. 16, 73–78 (2005). [2] W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissue,” IEEE J. Quantum Electron. 26, 2166–

2185 (1990). [3] A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration

in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998). [4] A. D. Klose, “Transport-theory-based stochastic image reconstruction of bioluminescent sources”, J. Opt. Soc. Am. A 24(6), 1601-1608

(2007). [5] H. Gao, H. Zhao, “Multilevel bioluminescence tomography based on radiative transfer equation Part 1: l1 regularization”, Optics Express 18,

1855 (2010). [6] 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”, Optics Express 17, 8062 (2009). [7] H. Dehghani, S. C. Davis, and B. W. Pogue, “Spectrally resolved bioluminescence tomography using the reciprocity approach,” Med. Phys.

35, 4863.4871(2008). [8] S. Ahn, A. J. Chaudhari, F. Darvas, Ch. A. Bouman, and R. M. Leahy, “Fast iterative image reconstruction methods for fully 3D multispectral

bioluminescence tomography,” Phys. Med. Biol. 53, 3921.3942(2008). [9] Hyun Keol Kim, Andreas H. Hielscher, “A PDE-constrained SQP algorithm for optical tomography based on the frequency-domain equation

of radiative transfer”, Inverse problems 25, 015010(20pp).

(a) PDE-constrained (rSQP): 600, 610, 620 nm (b) unconstrained (lm-BFGS): 600, 610, 620 nm

Fig. 2 Reconstruction results obtained with the two methods for the 15dB noisy data simulated with 600, 610 and 620 nm.