Two-dimensional bioluminescence tomography:numerical simulations and phantom experiments

Senhu Li, Qizhi Zhang, and Huabei Jiang

The reconstruction of internal light sources in bioluminescence tomography (BLT) is a challenginginverse problem because of the limited amount of information available compared with that for otherkinds of tomography such as fluorescence tomography in which external illumination sources are used.We demonstrated previously, using phantom experiments, that a target containing luciferases could bedetected tomographically when the target was located relatively close to the imaging boundary. Here wedescribe an improved BLT reconstruction method that can detect luciferase-containing targets locatedanywhere within an imaging domain. The method is tested with numerical simulations and furtherconfirmed with several phantom experiments. © 2006 Optical Society of America

OCIS codes: 170.3010, 110.6960.

1. Introduction

Bioluminescence imaging is able to image cellular andmolecular markers in intact living animals and isgreatly facilitating the noninvasive study of geneticregulatory elements, cell trafficking, and proteinfunctions.1–4 Whole-body bioluminescence imagingthat helps visualize gene expression and protein func-tion has been reported.5,6 The internal light producedby luciferase catalysts in a light-generating reactionthrough the oxidation of an enzyme-specific substrate(luciferin) can be measured with a charge-coupled de-vice (CCD) camera. However, bioluminescence imag-ing is limited to relatively superficial detection ofluciferases, and no depth information is available.

Methods of bioluminescence tomography (BLT) havebeen reported recently.7,8 Gu et al.8 developeda finite-element-based reconstruction method anddemonstrated that tomographic bioluminescence im-ages can be obtained experimentally. Wang et al.7 es-tablished a mathematical foundation for uniquereconstruction of a bioluminescence source distribu-tion in BLT. BLT can actually be based on the sameframework as fluorescence tomography, but here lightis collected from the object in the absence of externalillumination sources. The major advantage of BLT isthat there is no inherent background bioluminescence

in most tissues, and thus BLT yields high imagingcontrast.4 The disadvantage of the technique is that itmakes fewer source–detector pairs (projections) avail-able than does fluorescence tomography. This makesBLT a highly ill-posed inverse source problem.

However, one can make BLT well defined by intro-ducing a priori knowledge of the bioluminescent sourcestructure, as was shown in Ref. 7. In fact, ill-posedproblems constitute a fertile ground for the use of op-timization methods. When a problem presents an in-finite number of solutions, optimization methods makeit possible to reformulate the problem such that aunique solution can be obtained. The general principleis to define an objective function such that an optimalsolution can be chosen from a set of possible solutions.In such ill-posed inverse problems, we do not pursueonly one solution to the problem; actually we pursuethe best solution, which is unique. Based on this prin-ciple, in this paper we improve our finite-element re-construction algorithm, which was able to detecttargets at limited depths.8 Our goal is to eliminate thedepth limitation and to find the best solution for BLT.As an initial step, we focus in this paper on two-dimensional BLT. By introducing weighting factorscoupled with filtering functions at each iteration of ourreconstruction algorithm, we demonstrate, using bothsimulations and phantom experiments, that the im-proved algorithm is able to detect bioluminescent lighttargets at any depth within the imaging domain.

2. Methods and Materials

A. Reconstruction Algorithm

Details of our reconstruction algorithm were de-scribed elsewhere.8 Briefly, it iteratively updates

The authors are with the Department of Biomedical Engineer-ing, University of Florida, Gainesville, Florida 32611-6131. S. Li’se-mail address is sli@bme.ufl.edu.

Received 22 August 2005; revised 22 November 2005; accepted28 November 2005; posted 1 December 2005 (Doc. ID 64285).

0003-6935/06/143390-05$15.00/0© 2006 Optical Society of America

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the bioluminescent light source distribution, S�x, y�,based on the two-dimensional finite-element solutionto the bioluminescence light-propagation equation.The iterative procedure can be described as follows:

1. Initialize S�x, y� � 0;2. Compute photon density � based on the follow-

ing photon diffusion equation:

� · D���x, y� � �a��x, y� � �S�x, y� (1)

�D�� · in � ��, (2)

where D and �a are the diffusion and absorption co-

efficients, respectively, in is the unit normal vector forthe boundary, and � is a coefficient related to theinternal reflection at the boundary.

3. Compute the light-source profile update,�S � ��S1, �S2, . . . , �SN�T, from the following equa-tion (see Ref. 8 for more details):

�JTJ � I��S � JT���m� � ��c��. (3)

J is the Jacobian matrix that should be formed by��S at the boundary measurement sites; I isthe identity matrix; � is the regularization parameterthat is often determined by trial and error( � 0.1 was used in this study); �i

�m� and �i�c�, re-

spectively, are the measured and calculated data fori � 1, 2, . . . , M boundary locations.

4. Update S�x, y� until �S is less than a criterionvalue (usually 1%).

To improve the reconstruction algorithm for recov-ering targets embedded deep in scattering media, wemodify the updating vector at each iteration, coupledwith a constraint or filter on the source distribution.A modification of the updating vector is achieved byadding a depth-dependent weighting factor, as fol-lows:

Snew � Sold � W�S, (4)

where W is the weighting factor. For a circularimaging domain, W is empirically determined asd2�50 � 0.1, where d is the depth or distance of a nodefrom the boundary. For a square or rectangular do-main, W � �d1

1.5 � d21.5��40, where d1 and d2 are the

shortest two distances between a node and the foursides of the imaging domain. This weighting factor isintended to give the sources at deeper locations morechances to show up. However, the choice of W is em-pirical and was determined by trial and error throughnumerical simulations. The true mathematics behindthe modification of the updating vector warrants fur-ther investigation so the best formation of W can be

Fig. 1. Meshes used for the circular (top) and square (bottom)boundary models. Open circles, locations of given light sources forthe simulation study.

Fig. 2. (Color online) The experimental setup used in thisstudy.

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found. As a priori information, we also add a filter ateach iteration to eliminate nonphysical negativesources or low-intensity sources that often are shownas imaging artifacts.

B. Numerical Simulations

First we tested our algorithm by using a series ofsimulated data for both circular �5 cm in diameter)and square �3 cm � 3 cm� imaging domains, whereone or two targets (1 cm in diameter) were embedded.The finite-element meshes used are shown in Fig. 1.For the circular mesh, 96 detectors around the bound-ary were used, and the total number of nodes was406. For the square mesh, 120 detectors around theboundary were assumed (the distance between twodetectors is less than 1 mm), and the total number ofnodes was 402. The diffusion and absorption coeffi-cients of the background and target media for allcases tested were set to be 0.33 mm and 0.005 mm�1,respectively. One or two targets were placed at dif-ferent locations (open circles in Fig. 1). Sourcecoefficient Sk � 1.0 in the target region, whereasSk � 0 in the background region.

C. Phantom Experiments

In our phantom experiments, Quantilum recombi-nant luciferase (Promega, Milwaukee, Wisc.), a stan-

dard firefly luciferase with a peak emission at560 nm, was used as the source of enzymes, while aluciferase assay system (Promega) provided luciferin,magnesium, ATP, and other cofactors for generatinghigh total light output with a half-life of 10 min.Quantilum recombinant luciferase was diluted106-fold in 1� cell culture lysis reagent (supplied bythe luciferase assay system) and 1 mg�ml acetylatedbovine serum albumin (Sigma-Aldrich, St Louis,Mo.). The luciferase assay substrate was dissolvedinto luciferase assay buffer to produce the luciferaseassay reagent. Light was generated when the dilutedluciferase �350 �l� was combined with the luciferaseassay reagent �350 �l�.

We conducted measurements only with cubic phan-toms (square imaging domain) limited by availableexperimental apparatus. The luciferase–luciferin re-agent prepared was then placed in 1 cm cylindricalholes embedded at several depths from the surface ina 30 mm � 30 mm � 30 mm solid cubic phantom (1%Intralipid solution � 1% agar powder). The cubic

Fig. 3. (Color online) Simulated results for the circular boundarymodel: (a), (c), (e) reconstructed images; (b), (d), (f) plots of lightintensity across the center of the target (thinner lines, exact value;thicker curves, reconstructed results). Refer to Fig. 1 for the targetlocations indicated there by open circles.

Fig. 4. (Color online) Simu-lated results for the squareboundary model: (a), (c), (e),(g) reconstructed images; (b),(d), (f) plots of light intensityacross the center of the target(thinner lines, exact value;thicker curves, reconstructedresults). Refer to Fig. 1 for thetarget locations indicatedthere by open circles.

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phantom had D � 0.33 mm and �a � 0.0023 mm�1.In the experiments the luciferase–luciferin contain-ing the cubic phantom was mounted on a 360°rotation stage, as shown in Fig. 2. By rotating thephantom three times by 90°, we collected lumines-cence light emitted from each side of the phantomwith a thermoelectrically cooled ��30 °C�, front-illuminated CCD array (Roper Scientific Instrumen-tation, Trenton, N.J.) coupled with an optical lenssubsystem (Navitar Zoom 7000, Rochester, N.Y.).

3. Results and Discussion

The mesh used for the simulations with circular ge-ometry is shown in Fig. 1(a), in which a single lightsource at three different locations (indicated by open

circles in Fig. 1) was simulated. For convenience, thelight intensity on the open circle node and its neigh-boring nodes was set to 1 to represent the target (lightsource intensity and size).

Figure 3 shows the corresponding reconstructedresults for three target locations. As can be seen, thereconstructed images prove a very good detection ofthe location and size of the target. For the value of thelight-source intensity, the reconstructed images alsogive better than 97% accuracy compared to the exactvalue when the target was located at a depth of10 mm or less [see the one-dimensional profile plotsshown in Figs. 3(b) and 3(d)]. However, when thetarget was located almost at the center of the medium[Figs. 3(e) and 3(f)], the reconstruction accuracy wasreduced to 77% relative to the exact light-source in-tensity.

The simulations for the square model have givensimilar results to those for the circular model. Forthe square model, one or two targets were placed atseveral locations, as shown in Fig. 1(b). Threesingle-target cases and one two-target case werestudied, and Fig. 4 gives the reconstructed imagesand associated one-dimensional source intensityplots. As was done for the circular model cases, thelocation and size of the target(s) were accuratelydetected. We can see that, when the target was closeto the boundary [Figs. 4(a)–4(d)], the peak value ofthe light intensity was recovered with an accuracyof better than 98%. However, when the target was atthe center of the medium [Figs. 4(e) and 4(f)], thisquantitative prediction had an error of 32% relativeto the exact light-intensity value. Figure 4(g) pre-sents the reconstructed image for the two-target case,which shows that our algorithm is also effective forthe reconstruction of multiple targets.

Four phantom experiments were performed in thisstudy. Figures 5(a)–5(d) illustrate the target loca-tion(s) for three single-source cases and one two-sourcecase and Figs. 6(a)–6(d) give the corresponding recon-structed images. We immediately note that the phan-tom results provide quite accurate detection of thelocation and size of the target(s), which is consistentwith the simulation studies. Similarly, the light inten-sity of the source close to the center of the phantomwas underestimated, but those of the sources close tothe boundary were fairly acceptable. For both simula-tions and experiments the meshing scheme had a re-markable effect on the reconstructed images. As wecan see, we were not able to reconstruct a smoothboundary of the target sources with the relativelycoarse meshes used. A finer mesh might improve theimage reconstruction if more boundary measurementswere available.

In summary, we have shown that our improved two-dimensional BLT reconstruction algorithm with theweighting and threshold methods is able to detect tar-gets embedded at any depth in a centimeter-sized me-dium. Both the simulated and the experimentalresults obtained suggest that the shape, size, andintensity of bioluminescent light sources can be accu-

Fig. 5. Test geometry for the phantom experiments.

Fig. 6. (Color online) Images reconstructed from experimentaldata. Refer to Fig. 5 for the target locations indicated there by opencircles.

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rately predicted. We plan to test the improved algo-rithm, using animal experiments, in the near future.

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