Abstract

Non-invasive injectable cellular therapeutic strategies based on sustained delivery of physiological levels of BMP-2 for spinal fusion are emerging as promising alternatives, which could provide sufficient fusion without the associated surgical risks. However, these injectable therapies are dependent on bone formation occurring only at the specific target region. In this study, we developed and deployed fluorescence gene reporter tomography (FGRT) to provide information on in vivo cell localization and viability. This information is sought to confirm the ideal placement of the materials with respect to the area where early bone reaction is required, ultimately providing three dimensional data about the future fusion. However, because almost all conventional fluorescence gene reporters require visible excitation wavelengths, current in vivo imaging of fluorescent proteins is limited by high tissue absorption and confounding autofluorescence. We previously administered fibroblasts engineered to produce BMP-2, but is difficult to determine 3-D information of placement prior to bone formation. Herein we used the far-red fluorescence gene reporter, IFP1.4 to report the position and viability of fibroblasts and developed 3-D tomography to provide placement information. A custom small animal, far-red fluorescence tomography system integrated into a commercial CT scanner was used to assess IFP1.4 fluorescence and to demark 3-D placement of encapsulated fibroblasts with respect to the vertebrae and early bone formation as assessed from CT. The results from three experiments showed that the placement of the materials within the spine could be detected. This work shows that in vivo fluorescence gene reporter tomography of cell-based gene therapy is feasible and could help guide cell-based therapies in preclinical models.

© 2013 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  26. Y. Lu, H. B. Machado, A. Douraghy, D. Stout, H. Herschman, and A. F. Chatziioannou, “Experimental bioluminescence tomography with fully parallel radiative-transfer-based reconstruction framework,” Opt. Express17(19), 16681–16695 (2009).
    [CrossRef] [PubMed]

2012 (2)

A. Vaccaro, “Bone morphogenetic protein and their complications,” J. Bone Joint Surg. Br.94, 169 (2012).

C. D. Darne, Y. Lu, I. C. Tan, B. Zhu, J. C. Rasmussen, A. M. Smith, S. Yan, and E. M. Sevick-Muraca, “A compact frequency-domain photon migration system for integration into commercial hybrid small animal imaging scanners for fluorescence tomography,” Phys. Med. Biol.57(24), 8135–8152 (2012).
[CrossRef] [PubMed]

2011 (3)

G. S. Filonov, K. D. Piatkevich, L. M. Ting, J. Zhang, K. Kim, and V. V. Verkhusha, “Bright and stable near-infrared fluorescent protein for in vivo imaging,” Nat. Biotechnol.29(8), 757–761 (2011).
[CrossRef] [PubMed]

Y. Lu, B. Zhu, C. Darne, I. C. Tan, J. C. Rasmussen, and E. M. Sevick-Muraca, “Improvement of fluorescence-enhanced optical tomography with improved optical filtering and accurate model-based reconstruction algorithms,” J. Biomed. Opt.16(12), 126002 (2011).
[CrossRef] [PubMed]

R. M. Olabisi, Z. Lazard, M. H. Heggeness, K. M. Moran, J. A. Hipp, A. K. Dewan, A. R. Davis, J. L. West, and E. A. Olmsted-Davis, “An injectable method for noninvasive spine fusion,” Spine J.11(6), 545–556 (2011).
[CrossRef] [PubMed]

2010 (2)

R. M. Olabisi, Z. W. W. Lazard, C. L. Franco, M. A. Hall, S. K. Kwon, E. M. Sevick-Muraca, J. A. Hipp, A. R. Davis, E. A. Olmsted-Davis, and J. L. West, “Hydrogel microsphere encapsulation of a cell-based gene therapy system increases cell survival of injected cells, transgene expression, and bone volume in a model of heterotopic ossification,” Tissue Eng. Part A16(12), 3727–3736 (2010).
[CrossRef] [PubMed]

Y. Lu, B. Zhu, H. Shen, J. C. Rasmussen, G. Wang, and E. M. Sevick-Muraca, “A parallel adaptive finite element simplified spherical harmonics approximation solver for frequency domain fluorescence molecular imaging,” Phys. Med. Biol.55(16), 4625–4645 (2010).
[CrossRef] [PubMed]

2009 (6)

Y. Lu, H. B. Machado, A. Douraghy, D. Stout, H. Herschman, and A. F. Chatziioannou, “Experimental bioluminescence tomography with fully parallel radiative-transfer-based reconstruction framework,” Opt. Express17(19), 16681–16695 (2009).
[CrossRef] [PubMed]

X. Shu, A. Royant, M. Z. Lin, T. A. Aguilera, V. Lev-Ram, P. A. Steinbach, and R. Y. Tsien, “Mammalian expression of infrared fluorescent proteins engineered from a bacterial phytochrome,” Science324(5928), 804–807 (2009).
[CrossRef] [PubMed]

M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, “Light transport in biological tissue using three-dimensional frequency-domain simplified spherical harmonics equations,” Phys. Med. Biol.54(8), 2493–2509 (2009).
[CrossRef] [PubMed]

K. S. Cahill, J. H. Chi, A. Day, and E. B. Claus, “Prevalence, complications, and hospital charges associated with use of bone-morphogenetic proteins in spinal fusion procedures,” JAMA302(1), 58–66 (2009).
[CrossRef] [PubMed]

R. Y. Tsien, “Constructing and exploiting the fluorescent protein paintbox (Nobel Lecture),” Angew. Chem. Int. Ed. Engl.48(31), 5612–5626 (2009).
[CrossRef] [PubMed]

M. Miyazaki, H. Tsumura, J. C. Wang, and A. Alanay, “An update on bone substitutes for spinal fusion,” Eur. Spine J.18(6), 783–799 (2009).
[CrossRef] [PubMed]

2007 (3)

M. Bikram, C. Fouletier-Dilling, J. A. Hipp, F. Gannon, A. R. Davis, E. A. Olmsted-Davis, and J. L. West, “Endochondral bone formation from hydrogel carriers loaded with BMP2-transduced cells,” Ann. Biomed. Eng.35(5), 796–807 (2007).
[CrossRef] [PubMed]

K. E. Adams, S. Ke, S. Kwon, F. Liang, Z. Fan, Y. Lu, K. Hirschi, M. E. Mawad, M. A. Barry, and E. M. Sevick-Muraca, “Comparison of visible and near-infrared wavelength-excitable fluorescent dyes for molecular imaging of cancer,” J. Biomed. Opt.12(2), 024017 (2007).
[CrossRef] [PubMed]

G. B. Bishop and T. A. Einhorn, “Current and future clinical applications of bone morphogenetic proteins in orthopaedic trauma surgery,” Int. Orthop.31(6), 721–727 (2007).
[CrossRef] [PubMed]

2006 (2)

B. T. Feeley, A. H. Conduah, O. Sugiyama, L. Krenek, I. S. Y. Chen, and J. R. Lieberman, “In vivo molecular imaging of adenoviral versus lentiviral gene therapy in two bone formation models,” J. Orthop. Res.24(8), 1709–1721 (2006).
[CrossRef] [PubMed]

A. D. Klose and E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys.220(1), 441–470 (2006).
[CrossRef]

2005 (2)

A. Minamide, M. Yoshida, M. Kawakami, S. Yamasaki, H. Kojima, H. Hashizume, and S. D. Boden, “The use of cultured bone marrow cells in type I collagen gel and porous hydroxyapatite for posterolateral lumbar spine fusion,” Spine30(10), 1134–1138 (2005).
[CrossRef] [PubMed]

A. S. Mistry and A. G. Mikos, “Tissue engineering strategies for bone regeneration,” Adv. Biochem. Eng. Biotechnol.94, 1–22 (2005).
[CrossRef] [PubMed]

2003 (2)

M. P. Lutolf, F. E. Weber, H. G. Schmoekel, J. C. Schense, T. Kohler, R. Müller, and J. A. Hubbell, “Repair of bone defects using synthetic mimetics of collagenous extracellular matrices,” Nat. Biotechnol.21(5), 513–518 (2003).
[CrossRef] [PubMed]

Z. Gugala, E. A. Olmsted-Davis, F. H. Gannon, R. W. Lindsey, and A. R. Davis, “Osteoinduction by ex vivo adenovirus-mediated BMP2 delivery is independent of cell type,” Gene Ther.10(16), 1289–1296 (2003).
[CrossRef] [PubMed]

1995 (1)

H. S. An, K. Lynch, and J. Toth, “Prospective comparison of autograft vs. allograft for adult posterolateral lumbar spine fusion: differences among freeze-dried, frozen, and mixed grafts,” J. Spinal Disord.8(2), 131–135 (1995).
[CrossRef] [PubMed]

1992 (1)

N. Ferrara, K. Houck, L. Jakeman, and D. W. Leung, “Molecular and biological properties of the vascular endothelial growth factor family of proteins,” Endocr. Rev.13(1), 18–32 (1992).
[PubMed]

1990 (1)

U. Heise, J. F. Osborn, and F. Duwe, “Hydroxyapatite ceramic as a bone substitute,” Int. Orthop.14(3), 329–338 (1990).
[CrossRef] [PubMed]

1965 (1)

M. R. Urist, “Bone: formation by autoinduction,” Science150(3698), 893–899 (1965).
[CrossRef] [PubMed]

Adams, K. E.

K. E. Adams, S. Ke, S. Kwon, F. Liang, Z. Fan, Y. Lu, K. Hirschi, M. E. Mawad, M. A. Barry, and E. M. Sevick-Muraca, “Comparison of visible and near-infrared wavelength-excitable fluorescent dyes for molecular imaging of cancer,” J. Biomed. Opt.12(2), 024017 (2007).
[CrossRef] [PubMed]

Aguilera, T. A.

X. Shu, A. Royant, M. Z. Lin, T. A. Aguilera, V. Lev-Ram, P. A. Steinbach, and R. Y. Tsien, “Mammalian expression of infrared fluorescent proteins engineered from a bacterial phytochrome,” Science324(5928), 804–807 (2009).
[CrossRef] [PubMed]

Alanay, A.

M. Miyazaki, H. Tsumura, J. C. Wang, and A. Alanay, “An update on bone substitutes for spinal fusion,” Eur. Spine J.18(6), 783–799 (2009).
[CrossRef] [PubMed]

An, H. S.

H. S. An, K. Lynch, and J. Toth, “Prospective comparison of autograft vs. allograft for adult posterolateral lumbar spine fusion: differences among freeze-dried, frozen, and mixed grafts,” J. Spinal Disord.8(2), 131–135 (1995).
[CrossRef] [PubMed]

Barry, M. A.

K. E. Adams, S. Ke, S. Kwon, F. Liang, Z. Fan, Y. Lu, K. Hirschi, M. E. Mawad, M. A. Barry, and E. M. Sevick-Muraca, “Comparison of visible and near-infrared wavelength-excitable fluorescent dyes for molecular imaging of cancer,” J. Biomed. Opt.12(2), 024017 (2007).
[CrossRef] [PubMed]

Bikram, M.

M. Bikram, C. Fouletier-Dilling, J. A. Hipp, F. Gannon, A. R. Davis, E. A. Olmsted-Davis, and J. L. West, “Endochondral bone formation from hydrogel carriers loaded with BMP2-transduced cells,” Ann. Biomed. Eng.35(5), 796–807 (2007).
[CrossRef] [PubMed]

Bishop, G. B.

G. B. Bishop and T. A. Einhorn, “Current and future clinical applications of bone morphogenetic proteins in orthopaedic trauma surgery,” Int. Orthop.31(6), 721–727 (2007).
[CrossRef] [PubMed]

Boden, S. D.

A. Minamide, M. Yoshida, M. Kawakami, S. Yamasaki, H. Kojima, H. Hashizume, and S. D. Boden, “The use of cultured bone marrow cells in type I collagen gel and porous hydroxyapatite for posterolateral lumbar spine fusion,” Spine30(10), 1134–1138 (2005).
[CrossRef] [PubMed]

Cahill, K. S.

K. S. Cahill, J. H. Chi, A. Day, and E. B. Claus, “Prevalence, complications, and hospital charges associated with use of bone-morphogenetic proteins in spinal fusion procedures,” JAMA302(1), 58–66 (2009).
[CrossRef] [PubMed]

Chatziioannou, A. F.

Chen, I. S. Y.

B. T. Feeley, A. H. Conduah, O. Sugiyama, L. Krenek, I. S. Y. Chen, and J. R. Lieberman, “In vivo molecular imaging of adenoviral versus lentiviral gene therapy in two bone formation models,” J. Orthop. Res.24(8), 1709–1721 (2006).
[CrossRef] [PubMed]

Chi, J. H.

K. S. Cahill, J. H. Chi, A. Day, and E. B. Claus, “Prevalence, complications, and hospital charges associated with use of bone-morphogenetic proteins in spinal fusion procedures,” JAMA302(1), 58–66 (2009).
[CrossRef] [PubMed]

Chu, M.

M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, “Light transport in biological tissue using three-dimensional frequency-domain simplified spherical harmonics equations,” Phys. Med. Biol.54(8), 2493–2509 (2009).
[CrossRef] [PubMed]

Claus, E. B.

K. S. Cahill, J. H. Chi, A. Day, and E. B. Claus, “Prevalence, complications, and hospital charges associated with use of bone-morphogenetic proteins in spinal fusion procedures,” JAMA302(1), 58–66 (2009).
[CrossRef] [PubMed]

Conduah, A. H.

B. T. Feeley, A. H. Conduah, O. Sugiyama, L. Krenek, I. S. Y. Chen, and J. R. Lieberman, “In vivo molecular imaging of adenoviral versus lentiviral gene therapy in two bone formation models,” J. Orthop. Res.24(8), 1709–1721 (2006).
[CrossRef] [PubMed]

Darne, C.

Y. Lu, B. Zhu, C. Darne, I. C. Tan, J. C. Rasmussen, and E. M. Sevick-Muraca, “Improvement of fluorescence-enhanced optical tomography with improved optical filtering and accurate model-based reconstruction algorithms,” J. Biomed. Opt.16(12), 126002 (2011).
[CrossRef] [PubMed]

Darne, C. D.

C. D. Darne, Y. Lu, I. C. Tan, B. Zhu, J. C. Rasmussen, A. M. Smith, S. Yan, and E. M. Sevick-Muraca, “A compact frequency-domain photon migration system for integration into commercial hybrid small animal imaging scanners for fluorescence tomography,” Phys. Med. Biol.57(24), 8135–8152 (2012).
[CrossRef] [PubMed]

Davis, A. R.

R. M. Olabisi, Z. Lazard, M. H. Heggeness, K. M. Moran, J. A. Hipp, A. K. Dewan, A. R. Davis, J. L. West, and E. A. Olmsted-Davis, “An injectable method for noninvasive spine fusion,” Spine J.11(6), 545–556 (2011).
[CrossRef] [PubMed]

R. M. Olabisi, Z. W. W. Lazard, C. L. Franco, M. A. Hall, S. K. Kwon, E. M. Sevick-Muraca, J. A. Hipp, A. R. Davis, E. A. Olmsted-Davis, and J. L. West, “Hydrogel microsphere encapsulation of a cell-based gene therapy system increases cell survival of injected cells, transgene expression, and bone volume in a model of heterotopic ossification,” Tissue Eng. Part A16(12), 3727–3736 (2010).
[CrossRef] [PubMed]

M. Bikram, C. Fouletier-Dilling, J. A. Hipp, F. Gannon, A. R. Davis, E. A. Olmsted-Davis, and J. L. West, “Endochondral bone formation from hydrogel carriers loaded with BMP2-transduced cells,” Ann. Biomed. Eng.35(5), 796–807 (2007).
[CrossRef] [PubMed]

Z. Gugala, E. A. Olmsted-Davis, F. H. Gannon, R. W. Lindsey, and A. R. Davis, “Osteoinduction by ex vivo adenovirus-mediated BMP2 delivery is independent of cell type,” Gene Ther.10(16), 1289–1296 (2003).
[CrossRef] [PubMed]

Day, A.

K. S. Cahill, J. H. Chi, A. Day, and E. B. Claus, “Prevalence, complications, and hospital charges associated with use of bone-morphogenetic proteins in spinal fusion procedures,” JAMA302(1), 58–66 (2009).
[CrossRef] [PubMed]

Dehghani, H.

M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, “Light transport in biological tissue using three-dimensional frequency-domain simplified spherical harmonics equations,” Phys. Med. Biol.54(8), 2493–2509 (2009).
[CrossRef] [PubMed]

Dewan, A. K.

R. M. Olabisi, Z. Lazard, M. H. Heggeness, K. M. Moran, J. A. Hipp, A. K. Dewan, A. R. Davis, J. L. West, and E. A. Olmsted-Davis, “An injectable method for noninvasive spine fusion,” Spine J.11(6), 545–556 (2011).
[CrossRef] [PubMed]

Douraghy, A.

Duwe, F.

U. Heise, J. F. Osborn, and F. Duwe, “Hydroxyapatite ceramic as a bone substitute,” Int. Orthop.14(3), 329–338 (1990).
[CrossRef] [PubMed]

Einhorn, T. A.

G. B. Bishop and T. A. Einhorn, “Current and future clinical applications of bone morphogenetic proteins in orthopaedic trauma surgery,” Int. Orthop.31(6), 721–727 (2007).
[CrossRef] [PubMed]

Fan, Z.

K. E. Adams, S. Ke, S. Kwon, F. Liang, Z. Fan, Y. Lu, K. Hirschi, M. E. Mawad, M. A. Barry, and E. M. Sevick-Muraca, “Comparison of visible and near-infrared wavelength-excitable fluorescent dyes for molecular imaging of cancer,” J. Biomed. Opt.12(2), 024017 (2007).
[CrossRef] [PubMed]

Feeley, B. T.

B. T. Feeley, A. H. Conduah, O. Sugiyama, L. Krenek, I. S. Y. Chen, and J. R. Lieberman, “In vivo molecular imaging of adenoviral versus lentiviral gene therapy in two bone formation models,” J. Orthop. Res.24(8), 1709–1721 (2006).
[CrossRef] [PubMed]

Ferrara, N.

N. Ferrara, K. Houck, L. Jakeman, and D. W. Leung, “Molecular and biological properties of the vascular endothelial growth factor family of proteins,” Endocr. Rev.13(1), 18–32 (1992).
[PubMed]

Filonov, G. S.

G. S. Filonov, K. D. Piatkevich, L. M. Ting, J. Zhang, K. Kim, and V. V. Verkhusha, “Bright and stable near-infrared fluorescent protein for in vivo imaging,” Nat. Biotechnol.29(8), 757–761 (2011).
[CrossRef] [PubMed]

Fouletier-Dilling, C.

M. Bikram, C. Fouletier-Dilling, J. A. Hipp, F. Gannon, A. R. Davis, E. A. Olmsted-Davis, and J. L. West, “Endochondral bone formation from hydrogel carriers loaded with BMP2-transduced cells,” Ann. Biomed. Eng.35(5), 796–807 (2007).
[CrossRef] [PubMed]

Franco, C. L.

R. M. Olabisi, Z. W. W. Lazard, C. L. Franco, M. A. Hall, S. K. Kwon, E. M. Sevick-Muraca, J. A. Hipp, A. R. Davis, E. A. Olmsted-Davis, and J. L. West, “Hydrogel microsphere encapsulation of a cell-based gene therapy system increases cell survival of injected cells, transgene expression, and bone volume in a model of heterotopic ossification,” Tissue Eng. Part A16(12), 3727–3736 (2010).
[CrossRef] [PubMed]

Gannon, F.

M. Bikram, C. Fouletier-Dilling, J. A. Hipp, F. Gannon, A. R. Davis, E. A. Olmsted-Davis, and J. L. West, “Endochondral bone formation from hydrogel carriers loaded with BMP2-transduced cells,” Ann. Biomed. Eng.35(5), 796–807 (2007).
[CrossRef] [PubMed]

Gannon, F. H.

Z. Gugala, E. A. Olmsted-Davis, F. H. Gannon, R. W. Lindsey, and A. R. Davis, “Osteoinduction by ex vivo adenovirus-mediated BMP2 delivery is independent of cell type,” Gene Ther.10(16), 1289–1296 (2003).
[CrossRef] [PubMed]

Gugala, Z.

Z. Gugala, E. A. Olmsted-Davis, F. H. Gannon, R. W. Lindsey, and A. R. Davis, “Osteoinduction by ex vivo adenovirus-mediated BMP2 delivery is independent of cell type,” Gene Ther.10(16), 1289–1296 (2003).
[CrossRef] [PubMed]

Hall, M. A.

R. M. Olabisi, Z. W. W. Lazard, C. L. Franco, M. A. Hall, S. K. Kwon, E. M. Sevick-Muraca, J. A. Hipp, A. R. Davis, E. A. Olmsted-Davis, and J. L. West, “Hydrogel microsphere encapsulation of a cell-based gene therapy system increases cell survival of injected cells, transgene expression, and bone volume in a model of heterotopic ossification,” Tissue Eng. Part A16(12), 3727–3736 (2010).
[CrossRef] [PubMed]

Hashizume, H.

A. Minamide, M. Yoshida, M. Kawakami, S. Yamasaki, H. Kojima, H. Hashizume, and S. D. Boden, “The use of cultured bone marrow cells in type I collagen gel and porous hydroxyapatite for posterolateral lumbar spine fusion,” Spine30(10), 1134–1138 (2005).
[CrossRef] [PubMed]

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R. M. Olabisi, Z. Lazard, M. H. Heggeness, K. M. Moran, J. A. Hipp, A. K. Dewan, A. R. Davis, J. L. West, and E. A. Olmsted-Davis, “An injectable method for noninvasive spine fusion,” Spine J.11(6), 545–556 (2011).
[CrossRef] [PubMed]

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U. Heise, J. F. Osborn, and F. Duwe, “Hydroxyapatite ceramic as a bone substitute,” Int. Orthop.14(3), 329–338 (1990).
[CrossRef] [PubMed]

Herschman, H.

Hipp, J. A.

R. M. Olabisi, Z. Lazard, M. H. Heggeness, K. M. Moran, J. A. Hipp, A. K. Dewan, A. R. Davis, J. L. West, and E. A. Olmsted-Davis, “An injectable method for noninvasive spine fusion,” Spine J.11(6), 545–556 (2011).
[CrossRef] [PubMed]

R. M. Olabisi, Z. W. W. Lazard, C. L. Franco, M. A. Hall, S. K. Kwon, E. M. Sevick-Muraca, J. A. Hipp, A. R. Davis, E. A. Olmsted-Davis, and J. L. West, “Hydrogel microsphere encapsulation of a cell-based gene therapy system increases cell survival of injected cells, transgene expression, and bone volume in a model of heterotopic ossification,” Tissue Eng. Part A16(12), 3727–3736 (2010).
[CrossRef] [PubMed]

M. Bikram, C. Fouletier-Dilling, J. A. Hipp, F. Gannon, A. R. Davis, E. A. Olmsted-Davis, and J. L. West, “Endochondral bone formation from hydrogel carriers loaded with BMP2-transduced cells,” Ann. Biomed. Eng.35(5), 796–807 (2007).
[CrossRef] [PubMed]

Hirschi, K.

K. E. Adams, S. Ke, S. Kwon, F. Liang, Z. Fan, Y. Lu, K. Hirschi, M. E. Mawad, M. A. Barry, and E. M. Sevick-Muraca, “Comparison of visible and near-infrared wavelength-excitable fluorescent dyes for molecular imaging of cancer,” J. Biomed. Opt.12(2), 024017 (2007).
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N. Ferrara, K. Houck, L. Jakeman, and D. W. Leung, “Molecular and biological properties of the vascular endothelial growth factor family of proteins,” Endocr. Rev.13(1), 18–32 (1992).
[PubMed]

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M. P. Lutolf, F. E. Weber, H. G. Schmoekel, J. C. Schense, T. Kohler, R. Müller, and J. A. Hubbell, “Repair of bone defects using synthetic mimetics of collagenous extracellular matrices,” Nat. Biotechnol.21(5), 513–518 (2003).
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N. Ferrara, K. Houck, L. Jakeman, and D. W. Leung, “Molecular and biological properties of the vascular endothelial growth factor family of proteins,” Endocr. Rev.13(1), 18–32 (1992).
[PubMed]

Kawakami, M.

A. Minamide, M. Yoshida, M. Kawakami, S. Yamasaki, H. Kojima, H. Hashizume, and S. D. Boden, “The use of cultured bone marrow cells in type I collagen gel and porous hydroxyapatite for posterolateral lumbar spine fusion,” Spine30(10), 1134–1138 (2005).
[CrossRef] [PubMed]

Ke, S.

K. E. Adams, S. Ke, S. Kwon, F. Liang, Z. Fan, Y. Lu, K. Hirschi, M. E. Mawad, M. A. Barry, and E. M. Sevick-Muraca, “Comparison of visible and near-infrared wavelength-excitable fluorescent dyes for molecular imaging of cancer,” J. Biomed. Opt.12(2), 024017 (2007).
[CrossRef] [PubMed]

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G. S. Filonov, K. D. Piatkevich, L. M. Ting, J. Zhang, K. Kim, and V. V. Verkhusha, “Bright and stable near-infrared fluorescent protein for in vivo imaging,” Nat. Biotechnol.29(8), 757–761 (2011).
[CrossRef] [PubMed]

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M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, “Light transport in biological tissue using three-dimensional frequency-domain simplified spherical harmonics equations,” Phys. Med. Biol.54(8), 2493–2509 (2009).
[CrossRef] [PubMed]

A. D. Klose and E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys.220(1), 441–470 (2006).
[CrossRef]

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M. P. Lutolf, F. E. Weber, H. G. Schmoekel, J. C. Schense, T. Kohler, R. Müller, and J. A. Hubbell, “Repair of bone defects using synthetic mimetics of collagenous extracellular matrices,” Nat. Biotechnol.21(5), 513–518 (2003).
[CrossRef] [PubMed]

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A. Minamide, M. Yoshida, M. Kawakami, S. Yamasaki, H. Kojima, H. Hashizume, and S. D. Boden, “The use of cultured bone marrow cells in type I collagen gel and porous hydroxyapatite for posterolateral lumbar spine fusion,” Spine30(10), 1134–1138 (2005).
[CrossRef] [PubMed]

Krenek, L.

B. T. Feeley, A. H. Conduah, O. Sugiyama, L. Krenek, I. S. Y. Chen, and J. R. Lieberman, “In vivo molecular imaging of adenoviral versus lentiviral gene therapy in two bone formation models,” J. Orthop. Res.24(8), 1709–1721 (2006).
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K. E. Adams, S. Ke, S. Kwon, F. Liang, Z. Fan, Y. Lu, K. Hirschi, M. E. Mawad, M. A. Barry, and E. M. Sevick-Muraca, “Comparison of visible and near-infrared wavelength-excitable fluorescent dyes for molecular imaging of cancer,” J. Biomed. Opt.12(2), 024017 (2007).
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R. M. Olabisi, Z. W. W. Lazard, C. L. Franco, M. A. Hall, S. K. Kwon, E. M. Sevick-Muraca, J. A. Hipp, A. R. Davis, E. A. Olmsted-Davis, and J. L. West, “Hydrogel microsphere encapsulation of a cell-based gene therapy system increases cell survival of injected cells, transgene expression, and bone volume in a model of heterotopic ossification,” Tissue Eng. Part A16(12), 3727–3736 (2010).
[CrossRef] [PubMed]

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A. D. Klose and E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys.220(1), 441–470 (2006).
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R. M. Olabisi, Z. Lazard, M. H. Heggeness, K. M. Moran, J. A. Hipp, A. K. Dewan, A. R. Davis, J. L. West, and E. A. Olmsted-Davis, “An injectable method for noninvasive spine fusion,” Spine J.11(6), 545–556 (2011).
[CrossRef] [PubMed]

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R. M. Olabisi, Z. W. W. Lazard, C. L. Franco, M. A. Hall, S. K. Kwon, E. M. Sevick-Muraca, J. A. Hipp, A. R. Davis, E. A. Olmsted-Davis, and J. L. West, “Hydrogel microsphere encapsulation of a cell-based gene therapy system increases cell survival of injected cells, transgene expression, and bone volume in a model of heterotopic ossification,” Tissue Eng. Part A16(12), 3727–3736 (2010).
[CrossRef] [PubMed]

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N. Ferrara, K. Houck, L. Jakeman, and D. W. Leung, “Molecular and biological properties of the vascular endothelial growth factor family of proteins,” Endocr. Rev.13(1), 18–32 (1992).
[PubMed]

Lev-Ram, V.

X. Shu, A. Royant, M. Z. Lin, T. A. Aguilera, V. Lev-Ram, P. A. Steinbach, and R. Y. Tsien, “Mammalian expression of infrared fluorescent proteins engineered from a bacterial phytochrome,” Science324(5928), 804–807 (2009).
[CrossRef] [PubMed]

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K. E. Adams, S. Ke, S. Kwon, F. Liang, Z. Fan, Y. Lu, K. Hirschi, M. E. Mawad, M. A. Barry, and E. M. Sevick-Muraca, “Comparison of visible and near-infrared wavelength-excitable fluorescent dyes for molecular imaging of cancer,” J. Biomed. Opt.12(2), 024017 (2007).
[CrossRef] [PubMed]

Lieberman, J. R.

B. T. Feeley, A. H. Conduah, O. Sugiyama, L. Krenek, I. S. Y. Chen, and J. R. Lieberman, “In vivo molecular imaging of adenoviral versus lentiviral gene therapy in two bone formation models,” J. Orthop. Res.24(8), 1709–1721 (2006).
[CrossRef] [PubMed]

Lin, M. Z.

X. Shu, A. Royant, M. Z. Lin, T. A. Aguilera, V. Lev-Ram, P. A. Steinbach, and R. Y. Tsien, “Mammalian expression of infrared fluorescent proteins engineered from a bacterial phytochrome,” Science324(5928), 804–807 (2009).
[CrossRef] [PubMed]

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Z. Gugala, E. A. Olmsted-Davis, F. H. Gannon, R. W. Lindsey, and A. R. Davis, “Osteoinduction by ex vivo adenovirus-mediated BMP2 delivery is independent of cell type,” Gene Ther.10(16), 1289–1296 (2003).
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C. D. Darne, Y. Lu, I. C. Tan, B. Zhu, J. C. Rasmussen, A. M. Smith, S. Yan, and E. M. Sevick-Muraca, “A compact frequency-domain photon migration system for integration into commercial hybrid small animal imaging scanners for fluorescence tomography,” Phys. Med. Biol.57(24), 8135–8152 (2012).
[CrossRef] [PubMed]

Y. Lu, B. Zhu, C. Darne, I. C. Tan, J. C. Rasmussen, and E. M. Sevick-Muraca, “Improvement of fluorescence-enhanced optical tomography with improved optical filtering and accurate model-based reconstruction algorithms,” J. Biomed. Opt.16(12), 126002 (2011).
[CrossRef] [PubMed]

Y. Lu, B. Zhu, H. Shen, J. C. Rasmussen, G. Wang, and E. M. Sevick-Muraca, “A parallel adaptive finite element simplified spherical harmonics approximation solver for frequency domain fluorescence molecular imaging,” Phys. Med. Biol.55(16), 4625–4645 (2010).
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[CrossRef] [PubMed]

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M. P. Lutolf, F. E. Weber, H. G. Schmoekel, J. C. Schense, T. Kohler, R. Müller, and J. A. Hubbell, “Repair of bone defects using synthetic mimetics of collagenous extracellular matrices,” Nat. Biotechnol.21(5), 513–518 (2003).
[CrossRef] [PubMed]

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H. S. An, K. Lynch, and J. Toth, “Prospective comparison of autograft vs. allograft for adult posterolateral lumbar spine fusion: differences among freeze-dried, frozen, and mixed grafts,” J. Spinal Disord.8(2), 131–135 (1995).
[CrossRef] [PubMed]

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Mawad, M. E.

K. E. Adams, S. Ke, S. Kwon, F. Liang, Z. Fan, Y. Lu, K. Hirschi, M. E. Mawad, M. A. Barry, and E. M. Sevick-Muraca, “Comparison of visible and near-infrared wavelength-excitable fluorescent dyes for molecular imaging of cancer,” J. Biomed. Opt.12(2), 024017 (2007).
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A. Minamide, M. Yoshida, M. Kawakami, S. Yamasaki, H. Kojima, H. Hashizume, and S. D. Boden, “The use of cultured bone marrow cells in type I collagen gel and porous hydroxyapatite for posterolateral lumbar spine fusion,” Spine30(10), 1134–1138 (2005).
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A. S. Mistry and A. G. Mikos, “Tissue engineering strategies for bone regeneration,” Adv. Biochem. Eng. Biotechnol.94, 1–22 (2005).
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M. Miyazaki, H. Tsumura, J. C. Wang, and A. Alanay, “An update on bone substitutes for spinal fusion,” Eur. Spine J.18(6), 783–799 (2009).
[CrossRef] [PubMed]

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R. M. Olabisi, Z. Lazard, M. H. Heggeness, K. M. Moran, J. A. Hipp, A. K. Dewan, A. R. Davis, J. L. West, and E. A. Olmsted-Davis, “An injectable method for noninvasive spine fusion,” Spine J.11(6), 545–556 (2011).
[CrossRef] [PubMed]

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M. P. Lutolf, F. E. Weber, H. G. Schmoekel, J. C. Schense, T. Kohler, R. Müller, and J. A. Hubbell, “Repair of bone defects using synthetic mimetics of collagenous extracellular matrices,” Nat. Biotechnol.21(5), 513–518 (2003).
[CrossRef] [PubMed]

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R. M. Olabisi, Z. Lazard, M. H. Heggeness, K. M. Moran, J. A. Hipp, A. K. Dewan, A. R. Davis, J. L. West, and E. A. Olmsted-Davis, “An injectable method for noninvasive spine fusion,” Spine J.11(6), 545–556 (2011).
[CrossRef] [PubMed]

R. M. Olabisi, Z. W. W. Lazard, C. L. Franco, M. A. Hall, S. K. Kwon, E. M. Sevick-Muraca, J. A. Hipp, A. R. Davis, E. A. Olmsted-Davis, and J. L. West, “Hydrogel microsphere encapsulation of a cell-based gene therapy system increases cell survival of injected cells, transgene expression, and bone volume in a model of heterotopic ossification,” Tissue Eng. Part A16(12), 3727–3736 (2010).
[CrossRef] [PubMed]

Olmsted-Davis, E. A.

R. M. Olabisi, Z. Lazard, M. H. Heggeness, K. M. Moran, J. A. Hipp, A. K. Dewan, A. R. Davis, J. L. West, and E. A. Olmsted-Davis, “An injectable method for noninvasive spine fusion,” Spine J.11(6), 545–556 (2011).
[CrossRef] [PubMed]

R. M. Olabisi, Z. W. W. Lazard, C. L. Franco, M. A. Hall, S. K. Kwon, E. M. Sevick-Muraca, J. A. Hipp, A. R. Davis, E. A. Olmsted-Davis, and J. L. West, “Hydrogel microsphere encapsulation of a cell-based gene therapy system increases cell survival of injected cells, transgene expression, and bone volume in a model of heterotopic ossification,” Tissue Eng. Part A16(12), 3727–3736 (2010).
[CrossRef] [PubMed]

M. Bikram, C. Fouletier-Dilling, J. A. Hipp, F. Gannon, A. R. Davis, E. A. Olmsted-Davis, and J. L. West, “Endochondral bone formation from hydrogel carriers loaded with BMP2-transduced cells,” Ann. Biomed. Eng.35(5), 796–807 (2007).
[CrossRef] [PubMed]

Z. Gugala, E. A. Olmsted-Davis, F. H. Gannon, R. W. Lindsey, and A. R. Davis, “Osteoinduction by ex vivo adenovirus-mediated BMP2 delivery is independent of cell type,” Gene Ther.10(16), 1289–1296 (2003).
[CrossRef] [PubMed]

Osborn, J. F.

U. Heise, J. F. Osborn, and F. Duwe, “Hydroxyapatite ceramic as a bone substitute,” Int. Orthop.14(3), 329–338 (1990).
[CrossRef] [PubMed]

Piatkevich, K. D.

G. S. Filonov, K. D. Piatkevich, L. M. Ting, J. Zhang, K. Kim, and V. V. Verkhusha, “Bright and stable near-infrared fluorescent protein for in vivo imaging,” Nat. Biotechnol.29(8), 757–761 (2011).
[CrossRef] [PubMed]

Rasmussen, J. C.

C. D. Darne, Y. Lu, I. C. Tan, B. Zhu, J. C. Rasmussen, A. M. Smith, S. Yan, and E. M. Sevick-Muraca, “A compact frequency-domain photon migration system for integration into commercial hybrid small animal imaging scanners for fluorescence tomography,” Phys. Med. Biol.57(24), 8135–8152 (2012).
[CrossRef] [PubMed]

Y. Lu, B. Zhu, C. Darne, I. C. Tan, J. C. Rasmussen, and E. M. Sevick-Muraca, “Improvement of fluorescence-enhanced optical tomography with improved optical filtering and accurate model-based reconstruction algorithms,” J. Biomed. Opt.16(12), 126002 (2011).
[CrossRef] [PubMed]

Y. Lu, B. Zhu, H. Shen, J. C. Rasmussen, G. Wang, and E. M. Sevick-Muraca, “A parallel adaptive finite element simplified spherical harmonics approximation solver for frequency domain fluorescence molecular imaging,” Phys. Med. Biol.55(16), 4625–4645 (2010).
[CrossRef] [PubMed]

Royant, A.

X. Shu, A. Royant, M. Z. Lin, T. A. Aguilera, V. Lev-Ram, P. A. Steinbach, and R. Y. Tsien, “Mammalian expression of infrared fluorescent proteins engineered from a bacterial phytochrome,” Science324(5928), 804–807 (2009).
[CrossRef] [PubMed]

Schense, J. C.

M. P. Lutolf, F. E. Weber, H. G. Schmoekel, J. C. Schense, T. Kohler, R. Müller, and J. A. Hubbell, “Repair of bone defects using synthetic mimetics of collagenous extracellular matrices,” Nat. Biotechnol.21(5), 513–518 (2003).
[CrossRef] [PubMed]

Schmoekel, H. G.

M. P. Lutolf, F. E. Weber, H. G. Schmoekel, J. C. Schense, T. Kohler, R. Müller, and J. A. Hubbell, “Repair of bone defects using synthetic mimetics of collagenous extracellular matrices,” Nat. Biotechnol.21(5), 513–518 (2003).
[CrossRef] [PubMed]

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C. D. Darne, Y. Lu, I. C. Tan, B. Zhu, J. C. Rasmussen, A. M. Smith, S. Yan, and E. M. Sevick-Muraca, “A compact frequency-domain photon migration system for integration into commercial hybrid small animal imaging scanners for fluorescence tomography,” Phys. Med. Biol.57(24), 8135–8152 (2012).
[CrossRef] [PubMed]

Y. Lu, B. Zhu, C. Darne, I. C. Tan, J. C. Rasmussen, and E. M. Sevick-Muraca, “Improvement of fluorescence-enhanced optical tomography with improved optical filtering and accurate model-based reconstruction algorithms,” J. Biomed. Opt.16(12), 126002 (2011).
[CrossRef] [PubMed]

Y. Lu, B. Zhu, H. Shen, J. C. Rasmussen, G. Wang, and E. M. Sevick-Muraca, “A parallel adaptive finite element simplified spherical harmonics approximation solver for frequency domain fluorescence molecular imaging,” Phys. Med. Biol.55(16), 4625–4645 (2010).
[CrossRef] [PubMed]

R. M. Olabisi, Z. W. W. Lazard, C. L. Franco, M. A. Hall, S. K. Kwon, E. M. Sevick-Muraca, J. A. Hipp, A. R. Davis, E. A. Olmsted-Davis, and J. L. West, “Hydrogel microsphere encapsulation of a cell-based gene therapy system increases cell survival of injected cells, transgene expression, and bone volume in a model of heterotopic ossification,” Tissue Eng. Part A16(12), 3727–3736 (2010).
[CrossRef] [PubMed]

K. E. Adams, S. Ke, S. Kwon, F. Liang, Z. Fan, Y. Lu, K. Hirschi, M. E. Mawad, M. A. Barry, and E. M. Sevick-Muraca, “Comparison of visible and near-infrared wavelength-excitable fluorescent dyes for molecular imaging of cancer,” J. Biomed. Opt.12(2), 024017 (2007).
[CrossRef] [PubMed]

Shen, H.

Y. Lu, B. Zhu, H. Shen, J. C. Rasmussen, G. Wang, and E. M. Sevick-Muraca, “A parallel adaptive finite element simplified spherical harmonics approximation solver for frequency domain fluorescence molecular imaging,” Phys. Med. Biol.55(16), 4625–4645 (2010).
[CrossRef] [PubMed]

Shu, X.

X. Shu, A. Royant, M. Z. Lin, T. A. Aguilera, V. Lev-Ram, P. A. Steinbach, and R. Y. Tsien, “Mammalian expression of infrared fluorescent proteins engineered from a bacterial phytochrome,” Science324(5928), 804–807 (2009).
[CrossRef] [PubMed]

Smith, A. M.

C. D. Darne, Y. Lu, I. C. Tan, B. Zhu, J. C. Rasmussen, A. M. Smith, S. Yan, and E. M. Sevick-Muraca, “A compact frequency-domain photon migration system for integration into commercial hybrid small animal imaging scanners for fluorescence tomography,” Phys. Med. Biol.57(24), 8135–8152 (2012).
[CrossRef] [PubMed]

Steinbach, P. A.

X. Shu, A. Royant, M. Z. Lin, T. A. Aguilera, V. Lev-Ram, P. A. Steinbach, and R. Y. Tsien, “Mammalian expression of infrared fluorescent proteins engineered from a bacterial phytochrome,” Science324(5928), 804–807 (2009).
[CrossRef] [PubMed]

Stout, D.

Sugiyama, O.

B. T. Feeley, A. H. Conduah, O. Sugiyama, L. Krenek, I. S. Y. Chen, and J. R. Lieberman, “In vivo molecular imaging of adenoviral versus lentiviral gene therapy in two bone formation models,” J. Orthop. Res.24(8), 1709–1721 (2006).
[CrossRef] [PubMed]

Tan, I. C.

C. D. Darne, Y. Lu, I. C. Tan, B. Zhu, J. C. Rasmussen, A. M. Smith, S. Yan, and E. M. Sevick-Muraca, “A compact frequency-domain photon migration system for integration into commercial hybrid small animal imaging scanners for fluorescence tomography,” Phys. Med. Biol.57(24), 8135–8152 (2012).
[CrossRef] [PubMed]

Y. Lu, B. Zhu, C. Darne, I. C. Tan, J. C. Rasmussen, and E. M. Sevick-Muraca, “Improvement of fluorescence-enhanced optical tomography with improved optical filtering and accurate model-based reconstruction algorithms,” J. Biomed. Opt.16(12), 126002 (2011).
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G. S. Filonov, K. D. Piatkevich, L. M. Ting, J. Zhang, K. Kim, and V. V. Verkhusha, “Bright and stable near-infrared fluorescent protein for in vivo imaging,” Nat. Biotechnol.29(8), 757–761 (2011).
[CrossRef] [PubMed]

Toth, J.

H. S. An, K. Lynch, and J. Toth, “Prospective comparison of autograft vs. allograft for adult posterolateral lumbar spine fusion: differences among freeze-dried, frozen, and mixed grafts,” J. Spinal Disord.8(2), 131–135 (1995).
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X. Shu, A. Royant, M. Z. Lin, T. A. Aguilera, V. Lev-Ram, P. A. Steinbach, and R. Y. Tsien, “Mammalian expression of infrared fluorescent proteins engineered from a bacterial phytochrome,” Science324(5928), 804–807 (2009).
[CrossRef] [PubMed]

Tsumura, H.

M. Miyazaki, H. Tsumura, J. C. Wang, and A. Alanay, “An update on bone substitutes for spinal fusion,” Eur. Spine J.18(6), 783–799 (2009).
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M. R. Urist, “Bone: formation by autoinduction,” Science150(3698), 893–899 (1965).
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A. Vaccaro, “Bone morphogenetic protein and their complications,” J. Bone Joint Surg. Br.94, 169 (2012).

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G. S. Filonov, K. D. Piatkevich, L. M. Ting, J. Zhang, K. Kim, and V. V. Verkhusha, “Bright and stable near-infrared fluorescent protein for in vivo imaging,” Nat. Biotechnol.29(8), 757–761 (2011).
[CrossRef] [PubMed]

Vishwanath, K.

M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, “Light transport in biological tissue using three-dimensional frequency-domain simplified spherical harmonics equations,” Phys. Med. Biol.54(8), 2493–2509 (2009).
[CrossRef] [PubMed]

Wang, G.

Y. Lu, B. Zhu, H. Shen, J. C. Rasmussen, G. Wang, and E. M. Sevick-Muraca, “A parallel adaptive finite element simplified spherical harmonics approximation solver for frequency domain fluorescence molecular imaging,” Phys. Med. Biol.55(16), 4625–4645 (2010).
[CrossRef] [PubMed]

Wang, J. C.

M. Miyazaki, H. Tsumura, J. C. Wang, and A. Alanay, “An update on bone substitutes for spinal fusion,” Eur. Spine J.18(6), 783–799 (2009).
[CrossRef] [PubMed]

Weber, F. E.

M. P. Lutolf, F. E. Weber, H. G. Schmoekel, J. C. Schense, T. Kohler, R. Müller, and J. A. Hubbell, “Repair of bone defects using synthetic mimetics of collagenous extracellular matrices,” Nat. Biotechnol.21(5), 513–518 (2003).
[CrossRef] [PubMed]

West, J. L.

R. M. Olabisi, Z. Lazard, M. H. Heggeness, K. M. Moran, J. A. Hipp, A. K. Dewan, A. R. Davis, J. L. West, and E. A. Olmsted-Davis, “An injectable method for noninvasive spine fusion,” Spine J.11(6), 545–556 (2011).
[CrossRef] [PubMed]

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Opt. Express (1)

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M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, “Light transport in biological tissue using three-dimensional frequency-domain simplified spherical harmonics equations,” Phys. Med. Biol.54(8), 2493–2509 (2009).
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[CrossRef] [PubMed]

C. D. Darne, Y. Lu, I. C. Tan, B. Zhu, J. C. Rasmussen, A. M. Smith, S. Yan, and E. M. Sevick-Muraca, “A compact frequency-domain photon migration system for integration into commercial hybrid small animal imaging scanners for fluorescence tomography,” Phys. Med. Biol.57(24), 8135–8152 (2012).
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A. Minamide, M. Yoshida, M. Kawakami, S. Yamasaki, H. Kojima, H. Hashizume, and S. D. Boden, “The use of cultured bone marrow cells in type I collagen gel and porous hydroxyapatite for posterolateral lumbar spine fusion,” Spine30(10), 1134–1138 (2005).
[CrossRef] [PubMed]

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R. M. Olabisi, Z. Lazard, M. H. Heggeness, K. M. Moran, J. A. Hipp, A. K. Dewan, A. R. Davis, J. L. West, and E. A. Olmsted-Davis, “An injectable method for noninvasive spine fusion,” Spine J.11(6), 545–556 (2011).
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Tissue Eng. Part A (1)

R. M. Olabisi, Z. W. W. Lazard, C. L. Franco, M. A. Hall, S. K. Kwon, E. M. Sevick-Muraca, J. A. Hipp, A. R. Davis, E. A. Olmsted-Davis, and J. L. West, “Hydrogel microsphere encapsulation of a cell-based gene therapy system increases cell survival of injected cells, transgene expression, and bone volume in a model of heterotopic ossification,” Tissue Eng. Part A16(12), 3727–3736 (2010).
[CrossRef] [PubMed]

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Figures (2)

Fig. 1
Fig. 1

The reconstructed gene reporter distribution in the cross–sections with the maximal reconstructed values (the first, second and third rows). Top 80%, 90%, and 99% reconstructed values are shown, respectively. The fourth row shows the position of the cross-sections. “M1”, “M2”, and “M3” are Mouse 1, 2, and 3, respectively.

Fig. 2
Fig. 2

The first column shows that anatomical imaging information of new bone from BMP-2 using CT scanning with high spatial resolution mode. The imaging information of new bone for M3 cannot be acquired from the CT scanning. The second column is the reconstructed results from 3-D fluorescence gene reporter tomography. The arrows show the reconstructed new bone information. “M1”, “M2”, and “M3” are Mouse 1, 2, and 3, respectively.

Tables (1)

Tables Icon

Table 1 Quantitative analysis of the CT and optical imaging information based on Fig. 2. “N.A.” is due to the absence of new bone imaging information. The distance errors were calculated regarding the center positions of the new bone in the CT and optical reconstruction.

Equations (10)

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{ · 1 3 μ a1 x,m φ 1 x,m + μ a x,m φ 1 x,m ( 2 3 μ a x,m ) φ 2 x,m = [ Q μ a sf ϕ x ] m ( 2 3 μ a x,m ) φ 1 x,m · 1 7 μ a3 x,m φ 2 x,m +( 4 9 μ a x,m + 5 9 μ a2 x,m ) φ 2 x,m = [ 2 3 Q μ a sf ϕ x ] m
{ ( 1+ B 1 3 μ a1 x,m )ν· φ 1 x,m ( D 1 μ a3 x,m )ν· φ 2 x,m =( 1 2 + A 1 ) φ 1 x,m +( 1 8 + C 1 ) φ 2 x,m + [ s ^ ·ν<0 S( s ^ )2| s ^ ·ν |d s ^ ] x ( D 2 μ a1 x,m )ν· φ 1 x,m +( 1+ B 2 7 μ a3 x,m )ν· φ 2 x,m =( 1 8 + C 2 ) φ 1 x,m ( 7 24 + A 2 ) φ 2 x,m + [ s ^ ·ν<0 S( s ^ )( 5 | s ^ ·ν | 3 3| s ^ ·ν | )d s ^ ] x
J +,m,b =( 1 4 + J 0 )( φ 1 m 2 3 φ 2 m )( 0.5+ J 1 3 μ a1 m )ν· φ 1 m + 1 3 ( 5 16 + J 2 ) φ 2 m ( J 3 7 μ a3 m )ν· φ 2 m
[ M 1 φ 1 m M 1 φ 2 m M 2 φ 1 m M 2 φ 2 m ][ φ 1 m φ 2 m ]=[ B m 2 3 B m ][ μ a sf ]
b pq m = Ω Q ϕ x v p · v q dr
{ φ 1 m =( I M 1 φ 1 m 2 3 I M 1 φ 2 m ) B m μ a sf φ 2 m =( I M 2 φ 1 m 2 3 I M 2 φ 2 m ) B m μ a sf
J +,m,b = β 1 φ 1 m,b + β 2 φ 2 m,b =( β 1 G 1 + β 2 G 2 ) μ a sf =G μ a sf
J T +,m,b =A μ a sf
J T +,m,b =[ J 1 +,m,b J n v +,m,b J N v +,m,b ], A=[ G 1 G n v G N v ]
min 0< μ a sf < μ a sf,sup θ( μ a sf ): A μ a sf J T +,m,b 2

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