Combined optical tomographic and magnetic ... - SPIE Digital Library

Combined optical tomographic and magnetic resonance imaging of tumor bearing mice J. Masciotti1*, G. Abdoulaev1, J. Hur1, J. Papa3, J. Bae3, J. Huang3, D. Yamashiro3, J. Kandel3, A. H. Hielscher1,2* 1

Dept. Biomedical Engineering, Columbia University, 500 West 120th Street, New York, NY 10027 2 Dept. Radiology, Columbia University, 660 West 168th Street, New York, NY 10032 3 Dept. Pediatrics & Surgery, Columbia University, 3959 Broadway, BHN 214, New York, NY 10032

ABSTRACT With the advent of small animal imaging systems, it has become possible to non-invasively monitor the progression of diseases in living small animals and study the efficacy of drugs and treatment protocols. Magnetic resonance imaging (MRI) is an established imaging modality capable of obtaining high resolution anatomical images as well as studying cerebral blood volume (CBV), cerebral blood flow (CBF), and cerebral metabolic rate of oxygen (CMRO2). Optical tomography, on the other hand, is an emerging imaging modality, which, while much lower in spatial resolution and insensitive to CBF, can separate the effects of oxyhemoglobin, deoxyhemoglobin, and CBV with high temporal resolution. In this study we present our first results concerning coregistration of MRI and optical data. By applying both modalities to imaging of kidney tumors in mice that undergo VEGF treatment, we illustrate how these imaging modalities can supplement each other and cross validation can be performed. Keywords: Optical tomography, Small animal imaging, MRI, Multimodality imaging

1. INTRODUCTION Small animal imaging systems have been receiving increasing attention over the past decade.1,2,3 The interest is motivated by advances in animal models of human diseases and the progress in their transgenic manipulation. By studying small animals it is possible to monitor the progression and treatment of these disease models and to link specific genes to normal and abnormal function at the molecular, cellular, and organ level. Historically, biochemical and physiological studies of organs and systems of interest have been through necropsy, which requires sacrificing animals at different stages of disease progression. This makes it impossible to study the temporal and spatial development of disease within a single animal and complicates the experimental procedures and statistical analysis. With the advent of novel small animal imaging systems, it is now possible to perform noninvasive assays for the monitoring of both the temporal and spatial progression of disease and other biological processes. Two such imaging modalities that have drawn considerable attention in recent years are magnetic resonance imaging (MRI) and diffuse optical tomography (DOT). MRI is a well-established imaging modality characterized by its ability to obtain anatomical images with high spatial resolution.4,5 Its temporal resolution is in the range of 0.01 Hz to 1 Hz. Small animal MRI systems are normally characterized by having small sample space and high magnetic field, and therefore better spatial resolution compared to human systems. Optical tomography, though offering poorer spatial resolution, has shown great promise in measuring physiologically important chromophore concentrations such as oxyhemoglobin (HbO2) and deoxyhemoglobin (Hb).6,7,8,9,10 Optical methods are also sensitive to cytochrome, blood volume, scattering properties, and can be used in combination with molecular markers.11,12,13 The temporal resolution is the range of 3 Hz to 50 Hz. Oximetry systems have been commercially available from Hamamatsu, Hitachi, Somanetics, ISS, NIRx, and Techen, but so far, have primarily been used to study brain imaging. Commercial optical imaging systems tend to cost considerably less than MRI systems. Another advantage is that optical tomography systems require less space and several prototypes exists that are portable. By combining the two modalities, one will be able to capture anatomical and physiological *

74

Corresponding authors’ e-mails: James Masciotti: ([email protected]), Andreas H. Hielscher: ([email protected]).

Optical Tomography and Spectroscopy of Tissue VI, edited by Britton Chance, Robert R. Alfano, Bruce J. Tromberg, Mamoru Tamura, Eva M. Sevick-Muraca, Proc. of SPIE Vol. 5693 (SPIE, Bellingham, WA, 2005) · 1605-7422/05/$15 · doi: 10.1117/12.590844

information at high spatial and temporal resolution. We are currently in the process of designing such a hybrid system. In the study at hand, we investigate the use of MRI and DOT to image the effects of vascular endothelial growth factor (VEGF) antagonists on the development of kidney tumors in mice. After a description of the tumor model, we present details on the MR as well as optical imaging system. We show initial results in which we compare optical tomographic images with MR images of treated and untreated tumors.

2. METHODS 2.1. Mouse Tumor Model To compare the potential and advantage of MRI and DOT in small animal imaging we studied tumor growth and regression in NCR athymic nude mice that carried orthotopically implanted kidney tumors. Tumors were allowed to grow for 42 days. (An example of a tumor is compared to a normal kidney and is shown in Figure 1) In order to assess the effect of perturbing established vasculature, one group of animals was treated with vascular endothelial growth factor (VEGF) antagonists, while a second control group was injected with control peptide. VEGF is requisite for blood vessel growth during embryonic development and tumorigenesis. VEGF blockade has recently been validated as a therapeutic strategy in clinical trials, leading to the approval of the anti-VEGF antibody bevacizumab by the FDA in February 2004. VEGF antagonists are currently studied to better understand the molecular consequences of perturbing tumor vasculature.14,15,16 It is has been shown in several animal studies that disruption of VEGF signaling can attenuate or even abolish tumor vasculature, producing marked tumor regression.17,18,19 In this study we imaged animals 1 day after the start of the treatment, and compared the MR images with optical tomographic images. 2.2. Magnetic Resonance Imaging (MRI) Our MRI system consists of a 9.4T magnet and a Bruker AvanceTM 400 Spectrometer (Figure 2a). The system was fitted with Bruker’s micro2.5 imaging gradient set, which allowed the use of a linearly polarized RF birdcage coil (diameter = 3.5 cm), which is shown in Figure 2b. The birdcage was accompanied by a mouse bedding insert which allowed the mouse to be slid into proper position. The magnetic field had good homogeneity in slices through the center of the coil. Therefore 3 slices, axial, sagittal, and coronal, were taken through the center of the imaging space. Before the mouse was inserted into the magnetic field, it was anesthetized with ketamine. Throughout the experiment the mouse was ventilated with a Harvard apparatus animal ventilator, which pumped air through an isoflourane vaporizer set to 1.5% for continuous anesthesia. The setup procedure for the MRI consisted of tuning and matching the RF probe to the Lamour frequency (which is 400 Mhz at 9.4T), calibration of the spectrometer frequencies, calibration of the transmitted pulse power, and calibration of the receiver gain. A T1 weighted multi-spin multi-echo (MSME) imaging sequence was used. The pulse repetition time was set to TR = 300 ms. The echo time was set to TE = 2.2ms. Image matrix dimensions were 128 x 128 over a field of view of 3cm x 3cm. The slice thickness was chosen to be 1mm in order to get higher signal. The number of acquisitions was set to 4, which produced a total scan time of just over 5 minutes. The tumor was identifiable in the coronal and axial planes.

Figure 1: Mouse kidney without (left) and with (right) tumor.

Figure 2: Bruker 9.4T magnet (left) Birdcage coil with mouse and ventilation tubing (right).

2.3. Optical Tomography For the optical tomographic measurements we employed a dynamic near-infrared optical tomographic (DYNOT) instrument.20 This instrument operates in continuous-wave mode. A beam from two laser diodes (wavelengths λ = 760 and λ = 830 nm) was sequentially coupled into different multimode fiber bundles, which deliver the laser light to

Proc. of SPIE Vol. 5693

75

various positions on the mouse abdomen. The laser diodes have a maximum optical output power of 400 mW at the distal end of the fiber but are typically operated at a mean optical output power of 100 mW; the optical power incident on the target is about 10 - 30 mW. The lasers are driven by a Newport model 8000 laser controller mainframe housing up to four modules, each serving one laser. Each module provides sinusoidal modulation of the laser diode current with individually selectable frequency and amplitude. The detectors are silicon photodiodes, which provide the required sensitivity, linear response over several orders of magnitude, and ease of operation. Programmable gain amplifiers are used to provide variable gain settings for different source-detector pairs. Fast detection over a large dynamic range, coupled with fast source switching is achieved by synchronizing adjustment of the sensitivity for all channels with source movement; thereby providing on-the-fly adaptive gain control. Both source and detector fibers are 1-mm multimode fiber bundles. A more detailed description of the DYNOT instrument can be found in reference [20]. The optical imaging probe consisted of a hollow Delrin cylinder (diameter = 3cm, height = 5 cm) and two fiber-holding rings, which were machined from acetate. When placed around the cylinder, these rings allow the ends of the fiber bundles to be in contact with the surface of the cylinder and can be slid up and down the cylinder to adjust the vertical position of the fibers. Each of the 2 rings had 24 holes drilled, spaced 15° apart allowing 12 source and 12 detector fibers arranged in an alternating pattern (see Figure 3). For the measurements presented in this work, we used 24 sources and 24 detectors, resulting in 576 source-detector combinations. Just under three full tomographic datasets, involving all 576 source-detector pairs, were acquired per second. A single time point consisted of illuminating each of the 24 light sources in turn and simultaneously (in parallel) detecting the transmitted light at all of the corresponding 24 detectors. Before placement in the optical probe the mice were anesthetized with ketamine. The cylinder was partially filled with 1% Intralipid which was used as a matching fluid in order to reduced edge effects during image Figure 3: Optical imaging head with reconstruction. The 1% intralipid was obtained by diluting 20% Intralipid mouse inserted. Consists of 24 sources (Sigma-Aldrich Corp. St. Louis, MO). The mice fit snugly in the cylinder. and 24 detectors. Before a mouse was place in the probe, the tumor were palpated. The rings were adjusted so that the tumor was in between the two rings. The rings were separated by 1.3 cm. After the mouse was placed in the probe, the optimal gain settings were found for each source detector pair. Subsequently the optical scan was performed which consisted of 1000 full tomographic scans and took just over 5 minutes. In order to remove motion artifacts, a 200 point subsection of the data, where motion was minimal, was taken and then averaged over in order to remove other noise. After mouse scans were completed a calibration scan was performed on a homogeneous medium consisting of 1% Intralipid suspension. 2.4. Reconstruction Algorithm In this work the 3D reconstruction of optical properties in the mouse was achieved using a PDE-constrained optimization approach.21,22 In this approach we seek to minimize an objective function Φ that quantifies the difference between measured and predicted signals. Minimization is subject to a set of constraints on optical properties and radiance that are given by the steady-state equation of radiative transfer23: Ω • ∇Ψ (x, Ω ) + (µ a (x ) + µ s (x ))Ψ (x, Ω ) = µ s (x ) ∫ k (Ω, Ω')Ψ (x, Ω')dΩ', 4π

(1)

where Ψ is the fluence (W/cm2), µa is the absorption coefficient (cm-1), µs is the scattering coefficient (cm-1), Ω is the directional vector, and k describes the probability that photons traveling in direction Ω ’ are scattered in direction Ω and is given by the Heyney-Greenstein scattering model: k (cos θ = Ω • Ω') =

76


(1 + g

1− g 2 2

− 2 g cos θ

)

3/ 2

,

(2)

where g is the anisotropy factor. Equation (1) is often referred to as a forward model, that predicts the detector readings given some approximation of the optical properties µa and µs. The source term is incorporated in to the boundary conditions: Ψ (x, Ω ) = S (x, Ω ) on Ω • n(x ) < 0 (3) Our formulation of the model requires the definition of an objective function that determines the discrepancy between measured data M and predicted detector data P generated by the forward model. Because the DYNOT instrument does not provide absolute measurements, due to unknown coupling losses in measurement heads, a calibration scheme is used which is similar to that suggested by Y. Pei et al.24 The calibration scheme requires that measurements of a homogeneous phantom of known optical properties are taken with the same experimental setup as when the target (the mouse) was imaged. As the phantom we used 1% Intralipid. Intuitively, the calibration scheme can be thought of as using the phantom to find the scaling factors needed to account for unknown losses in each of the source detector pair measurements. According to the calibration scheme, the standard least square norm objective function is modified to yield: m Psph ,d M s ,d

1 Φ (µ a , µ s , Ψ ) = ∑ ∑ 2 s d

M sph ,d

2

−

Psm,d

m Psph ,d M s , d M sph ,d

(Ψ ) +

2

β 2

R(µ a , µ s ),

(4)

where s and d are used for indexing the sources and detectors, Ψ is vector of radiances for all source Mph and Mm are the actual λ (nm) µ a (cm-1) µ s (cm-1) g ph measurements for the phantom and the mouse respectively, P and 760 .023 27 0.9 Pm are the forward model predicted measurements for the phantom 830 .027 22 0.9 25 and target respectively, R is the Tikhonov regularization term and Table 1: Optical properties for initialization β is a constant, in our case set to 0.01. The reconstruction was initialized with the optical properties shown in Table 1. The reconstruction code uses an iterative augmented Lagrangian method26 to solve the constrained optimization problem, in which the objective function (4) is being minimized, subject to constraints (1) that are considered for each source s. The finite volume discrete ordinates method27 was used for the ERT with 24 ordinates and a mesh with 6137 volumes. This mesh, generated with GiD mesh generator, is shown in Figure 4. Reconstruction maps of µa were produced while µs was assumed constant. Since all measurements were performed at two wavelengths, there were 2 sets of reconstructed absorption properties µaλ1 and µaλ2. It was assumed that, at each wavelength, the primary determinant of the absorption coefficient was a linear combination of oxyhemoglobin and deoxyhemoglobin: λ λ µ aλ = ε HbO [HbO2 ] + ε Hb [Hb], 2

(5)

where ελHbO2, ελHb, are the known extinction coefficients for deoxyhemoglobin and oxyhemoglobin at the given wavelengths, respectively. By simultaneously solving the set of algebraic equations at the two wavelengths, we calculated the concentrations of oxyhemoglobin [HbO2] and deoxyhemoglobin [Hb] as: λ

[Hb] =

Figure 4: Reconstruction mesh consisting of 6137 volumes.

λ

λ

λ

1 2 ε HbO µ a1 − ε HbO µa2 2 2

λ

λ

λ

λ

1 ε 2 2ε 1 ε Hb − ε Hb HbO 2 HbO 2

,

(6)

λ1 λ2 λ2 λ1 ε Hb µ a − ε Hb µa , [HbO2 ] = λ λ λ2 λ1 1 ε 2 ε Hb − ε Hb ε HbO HbO 2

(7)

2

The total hemoglobin concentration [THb] is simply the sum on [Hb] and [HbO2].


77

Figure 5: MR images (a: coronal view, b: axial view) and optical tomographic images (c: coronal view, d: axial view) of a tumorbearing mouse. The margins of the unaffected kidney and tumor have been traced and labeled in red and yellow, respectively. The optical images show the blood volume measured by total hemoglobin content. The dashed lines labeled (I) and (II) in Fig. 5c indicate the lines where optical source and detectors fibers where placed (see also Fig. 3). Fig. 5d shows an axial cross-section at the level of plane (I).

3. RESULTS 3.1. Comparison of MRI and optical tomographic images As first example we show MRI and optical images (Figure 5) obtained from tumor bearing mice. Figures 5a and 5b show coronal and axial cross-sections obtained with our MRI system. In both images one can clearly identify the kidney without the tumor (circled in red) and the tumor-bearing kidney (circled in yellow). The kidney with the tumor is enlarged. While these images contain good anatomical information and can be used to determine the size of the tumor, no information about blood volume, blood-oxygenation levels, or other physiological information can be gleaned. The corresponding optical images are shown in Figures 5c and 5d. They lack anatomical detail, but provide physiological important information not contained in the MR image. The dominating feature in the optical images is an area of increased blood volume (as indicated by increased total hemoglobin concentration [THb]). Comparing the optical images with the MR image it is apparent that the area of increased blood volume corresponds to the region occupied by the kidney-bearing tumor. An increase in blood volume is indeed expected in the tumor, since this tissue is highly vascularized as determined in previous studies.18 This example illustrates how optical and MRI images can provide complementary information, and can be used to cross-validate each other. 78


Figure 6: Optical Tomographic coronal cross-sections of deoxyhemoglobin (a and b) and total hemoglobin (c and d) for untreated (a and c) and treated (b and d) tumors.

Figure 7: Optical Tomographic axial cross-sections of deoxyhemoglobin (a and b) and total hemoglobin (c and d) for untreated (a and c) and treated (b and d) tumors.


79

3.2. Optical Images of VEGF treated and untreated mice Next we compared optical tomographic images of various hemoglobin-dependent parameters obtained from treated and untreated mice. Figure 6 and Figure 7 show coronal as well as axial cross sections of deoxy-hemoglobin concentrations [Hb] and total hemoglobin concentrations [THb]. The images were taken 43 days after tumor inoculation and 1 day after the start of the treatment regiment. Differences between the images of the treated and untreated mouse can already be seen. The untreated animals show a higher level of [Hb] and well as [THb] in the regions of the tumor. This result is in agreement with expected effects. As mentioned earlier, treatment with VEGF blockages should reduce the number of viable blood vessels in the tumor. With less vasculature the [THb] values should be smaller. Higher [Hb] values in the untreated tumors reflect the overall higher blood volume, but could also be caused by a higher oxygen extraction rate in the tumor. There is some indication in the images of untreated animals that the peak in the center of the tumors shows relatively higher [Hb] values, which would be in accordance with higher metabolic activity. Further studies will be necessary to confirm and elucidate this point.

4. SUMMARY AND OUTLOOK The presented work constitutes this first step towards a hybrid optical-tomographic/magnetic-resonance imaging system for small animals. We have imaged tumor-bearing mice with a Bruker 9.4T MR imaging system and a dual-wavelength, steady-state optical tomographic imaging instrument. The MR images can be used to obtain high-resolution anatomical information, such as tumor size, while the optical tomographic images provide information about physiologically relevant hemoglobin-dependent parameters; however at a lower spatial resolution. Employing a PDE-constrained image reconstruction scheme that uses the equation of radiative transport as model of light propagation in tissue, we have demonstrated the ability to determine oxy, deoxy, and total hemoglobin distribution inside the animals. Comparing optical images obtained from VEGF-treated and untreated mice, we were able to determine that tumors in untreated mice exhibit higher concentrations of deoxy and total hemoglobin that tumors in VEGF treated animals. These observations are in agreement with results from histopathological studies and fluorescence microscopy on frozen sections. Studies on larger animal populations will be necessary to confirm the positive findings and establish optical tomographic imaging as a viable small animal imaging modality. Furthermore, future work will focus on simultaneous MRI-optical image registration. In the current study the positions and orientations of the mice during both the MRI and optical scans were recorded. But because the two scans were taken at separate times in separate measurement heads, there is some uncertainty as to the position and orientation of the mouse and therefore the exact location of the tumor. To identify objects in both MRI and optical images more accurately, there is a need for an imaging probe that allows simultaneous imaging of both modalities. To this end we are currently developing an RF birdcage probe that has optical fiber probes embedded in its outer walls to allow fixation onto an animal’s surface.

5. ACKNOWLEDGEMENTS This work was supported in part by the National Institute of Biomedical Imaging and Bioengineering (NIBIB grant number 5R01-EB001900, A. H. Hielscher) and the National Cancer Institute (NCI grant numbers 5R01CA100451, J. Kandel, and 5R01CA088951, D. Yamashiro), which both are divisions of the National Institutes of Health (NIH).

6. REFERENCES 1. 2. 3. 4. 5.

80

S.R. Cherry, "In vivo molecular imaging and genomic imaging: new challenges for imaging physics," Phys. Med. Biol. 49, R13 (2004). M. G. Pomper, "Can small animal imaging accelerate drug development?," Journal Cellular Biochemistry Suppl. 39, pp. 211-220. (2002). P. D. Acton, H. F. Kung, "Small animal imaging with high resolution single photon emission tomography," Nuclear Medicine and Biology 30(8), pp. 889-895 (2003). G. A. Johnson, G. P. Cofer, B. Fubara, S. L. Gewalt, L. W. Hedlund, R. R. Maronpot, “Magnetic resonance histology for morphologic phenotyping,” Journal of Magnetic Resonance Imaging 16(4), pp. 423-429 (2002). E. X. Wu, K. K. Wong, M. Andrassy, H. Tang, “High-resolution in vivo CBV mapping with MRI in wild-type


6.

7.

8.

9. 10. 11. 12.

13. 14. 15.

16.

17. 18.

19.

20. 21. 22. 23. 24. 25. 26. 27.

mice,” Magnetic Resonance in Medicine 49(4), pp. 765-770 (2003). A. Y. Bluestone, M. Stewart, B. Lei, I.S. Kass, J. Lasker, G.S. Abdoulaev, A. H. Hielscher, “Three-dimensional optical tomographic brain imaging in small animals, Part I: Hypercapnia,” Journal of Biomedical Optics 9(5), pp. 1046-1062 (2004). A. Y. Bluestone, M. Stewart, J. Lasker, G.S. Abdoulaev, A. H. Hielscher, “Three-dimensional optical tomographic brain imaging in small animals, Part II: Unilateral Carotid Occlusion,” Journal of Biomedical Optics 9(5), pp. 1063-1073 (2004). J. P. Culver, T. Durduran, D. Furuya, C. Cheung, J. H. Greenberg, A. G. Yodh, “Diffuse optical tomography of cerebral blood flow, oxygenation, and metabolism in rat during focal ischemia,” Journal of Cerebral Blood Flow & Metabolism 23 pp. 911-924 (2003). G. Yu, T. Durduran, D. Furuya, J. H. Greenberg, A. G. Yodh, “Frequency-domain multiplexing system for in vivo diffuse light measurements of rapid cerebral hemodynamics,” Applied Optics 42(16), pp. 2931-2939 (2003). A.H, Hielscher, “Optical tomographic imaging of small animals,” Current Opinion in Biotechnology 16(1), pp. 7988 (2005). E. E. Graves, R. Weissleder, V. Ntziachristos, “Fluorescence Molecular Imaging of Small Animal Tumor Models,” Current Molecular Medicine 4, pp. 419-430 (2004). V. Ntziachristos, E. A. Schellenberger, J. Ripoll, D. Yessayan, E. Graves, A. Bodganov, L. Josephson, R. Weissleder, “Visualization of antitumor treatment by means of fluorescence molecular tomography with an annexin V-Cy5.5 conjugate,” Proceedings of the National Academy of Science of the United States of America 101(33), pp. 12294-12299 (2004). M. Doubrovin, I. Serganova, P. Mayer-Kuckuk, V. Ponomarev, R. G. Blasberg, "Multimodality in Vivo MolecularGenetic Imaging,” Bioconjugate Chemistry 15, pp.1376-1388 (2004). K. J. Kim, B. Li, J. Winer, M. Armanini, N. Gillett, H. S. Phillips, N. Ferrara, “Inhibition of vascular endothelial growth factor-induced angiogenesis suppresses tumor growth in vivo” Nature 362, pp. 841–844 (1993). H. P. Gerber, J. Kowalski, D. Sherman, D. A. Eberhard, N. Ferrara, “Complete Inhibition of Rhabdomyosarcoma Xenograft Growth and Neovascularization Requires Blockade of Both Tumor and Host Vascular Endothelial Growth Factor,” Cancer Research 60, pp. 6253–6258 (2000). M. Prewett, J. Huber, Y. Li, A. Santiago, W. O’Connor, K. King, J. Overholser, A. Hooper, B. Pytowski, L. Witte, et al, “Antivascular Endothelial Growth Factor Receptor (Fetal Liver Kinase 1) Monoclonal Antibody Inhibits Tumor Angiogenesis and Growth of Several Mouse and Human Tumors,” Cancer Res. 59, pp. 5209–5218 (1999). J. Glade-Bender, J. J. Kandel, D. J. Yamashiro, “VEGF blocking therapy in the treatment of cancer,” Expert Opinion on Biological Therapy 3(2), pp. 263-276 (2003). J. Z. Huang, J. S. Frischer, A. Serur, A. Kadenhe, A. Yokoi, K. W. McCrudden, T. New, K. O'Toole, S. Zabski, J. S. Rudge, J. Holash, G. D. Yancopoulos, D. J. Yamashiro, J. J. Kandel, “Regression of established tumors and metastases by potent vascular endothelial growth factor blockade,” Proceedings of the National Academy of Science of the United States of America 100(13), pp. 7785-7790 (2003). J. S. Frischer, J. Z. Huang, A. Serur, A. Kadenhe-Chiweshe, K. W. McCrudden, K. O'Toole, J. Holash, G. D. Yancopoulos, D. J. Yamashiro, J. J. Kandel, “Effects of potent VEGF blockade on experimental Wilms tumor and its persisting vasculature,” International Journal of Oncology 25(3), pp. 549-553 (2004). C.H. Schmitz, M. Löcker, J. M. Lasker, A. H. Hielscher, R. L. Barbour, "Instrumentation for fast functional optical tomography," Review of Scientific Instrumentation 73(2), pp. 429-439 (2002). L. Biegler, O. Ghattas, M. Heikenschloss, and B. van Bloemen Wanders (eds): Large-Scale PDE-Constrained Optimization, Lecture Notes in Computational Science and Engineering, 30, Springer Verlag, Berlin, (2003). G. Abdoulaev, K. Ren, A.H. Hielscher, "Optical tomography as a constrained optimization problem,” submitted to Inverse Problems. K. M. Case and P. F. Zweifel: Linear Transport Theory, Addison Wesley, Reading, Massachusetts (1967). Y. Pei, H. L. Graber, and R. L. Barbour: Influence of systematic errors in reference states on image quality and on stability of derived information for DC optical imaging, Applied Optics 40, 5755–5769 (2001). H. W. Engl, M. Hanke, and A. Neubauer: Regularization of Inverse Problems, Kluwer Academic Publishers, Dordrecht (1996). J. Nocedal and S.J. Wright: Numerical Optimization, Springer Verlag, New York, (1999). K. Ren, G. S. Abdoulaev, G. Bal, and A. H. Hielscher, “Algorithm for solving the equation of radiative transfer in the frequency domain,” Optics Letters 29(6), pp. 578-579 (2004).


81