Abstract
Optical molecular imaging is an important technique of studies at molecular level and provides promising tools to non-invasively delineate in vivo physiological and pathological activities at cellular and molecular levels, and it has been widely used for diagnosing, managing diseases, metastasis detection and drug development. From a mathematical perspective, this paper mainly focuses on the forward problem and inverse problem in biological tissues based on the radiative transfer equation (RTE). The forward problem is accustomed to describing photon propagation in biological tissues and the inverse problem is used to reconstruct internal source distribution from the signal detected on the external surface. We also introduce the detailed derivation of the RTE and Robin boundary condition and discretization of the forward problem, along with the reconstruction methods and iterative solution algorithms summarized for the inverse problem. Finally, the current and future challenges of optical molecular imaging are discussed. This survey aims to construct a mathematical method, a state-of-the-art framework for optical molecular imaging, from which future research may benefit.
Similar content being viewed by others
References
Weissleder R, Pittet M J. Imaging in the era of molecular oncology. Nature, 2008, 452: 580–589
Weissleder R, Ntziachristos V. Shedding light onto live molecular targets. Nat Med, 2003, 9: 123–128
Tian J. Molecular Imaging: Fundamentals and Application. Hangzhou: Zhejiang University Press, 2012
Weissleder R, Mahmood U. Molecular imaging. Radiology, 2001, 219: 316–333
Mssound T F, Gambhir S S. Molecular imaging in living subjects: seeing fundamental biological processes in a new light. Gene Dev, 2003, 17: 545–580
Cherry S R. In vivo molecular and genomic imaging: new challenges for imaging physics. Phys Med Biol, 2004, 49: R13–R48
Ntziachristos V, Ripoll J, Wang L V, et al. Looking and listening to light: the evolution of whole-body photonic imaging. Nat Biotechnol, 2005, 23: 313–320
Weissleder R. Molecular imaging in cancer. Science, 2006, 312: 1168–1171
Willmann J K, van Bruggen N, Dinkelborg L M, et al. Molecular imaging in drug development. Nat Rev Drug Discov, 2008, 7: 591–607
Lv Y J. Research of inverse problems in bioluminescence tomography (in Chinese). Dissertation for the Doctoral Degree. Beijing: Institute of Automation, Chinese Academy of Sciences, 2007
Arridge S R, Hebden J C. Optical imaging in medicine: II. Modelling and reconstruction. Phys Med Biol, 1997, 42: 841–853
Gibson A P, Hebden J C, Arridge S R. Recent advances in diffuse optical imaging. Phys Med Biol, 2005, 50: R1–R43
Harrach B. On uniqueness in diffuse optical tomography. Inverse Probl, 2009, 25: 055010
Boas D A, Brooks D H, Miller E L, et al. Imaging the body with diffuse optical tomography. IEEE Signal Proc Mag, 2001, 18: 57–75
Arridge S R, Schotland J C. Optical tomography: forward and inverse problems. Inverse Probl, 2009, 25: 123010
Contag C H, Bachmann M H. Advances in in vivo bioluminescence imaging of gene expression. Annu Rev Biomed Eng, 2002, 4: 235–260
Wang G, Hoffman E A, McLennan G, et al. Development of the first bioluminescent CT scanner. Radiology, 2003, 229: 566
Wang G, Shen H O, Cong W X, et al. Temperature-modulated bioluminescence tomography. Opt Express, 2006, 14: 7852–7871
Wang G, Cong W X, Durairaj K, et al. In vivo mouse studies with bioluminescence tomography. Opt Express, 2006, 14: 7801–7809
Cong W X, Wang G, Kumar D, et al. Practical reconstruction method for bioluminescence tomography. Opt Express, 2005, 13: 6756–6771
Wang G, Li Y, Jiang M. Uniqueness theorems in bioluminescence tomography. Med Phys, 2004, 31: 2289–2299
Jiang M, Zhou T, Cheng J T, et al. Image reconstruction for bioluminescence tomography from partial measurement. Opt Express, 2007, 15: 11095–11116
Ntziachristos V, Bremer C, Weissleder R. Fluorescence imaging with near-infrared light: new technological advances that enable in vivo molecular imaging. Eur Radiol, 2003, 13: 195–208
Ntziachristos V, Tung C H, Bremer C, et al. Fluorescence molecular tomography resolves protease activity in vivo. Nat Med, 2002, 8: 757–760
Ntziachristos V, Schellenberger E A, Ripoll J, et al. Visualization of antitumor treatment by means of fluorescence molecular tomography with an annexin V-Cy5.5 conjugate. Proc Nat Acad Sci, 2004, 101: 12294–12299
Deliolanis N, Lasser T, Hyde D, et al. Free-space fluorescence molecular tomography utilizing 360° geometry projections. Opt Lett, 2007, 32: 382–384
Tan Y, Jiang H. DOT guided fluorescence molecular tomography of arbitrarily shaped objects. Med Phys, 2008, 35: 5703–5707
Zhang B, Liu S Q, Cao X, et al. Fluorescence tomography reconstruction with simultaneous positron emission tomography priors. IEEE Trans Multimedia, 2013, 15: 1031–1038
Robertson R, Germanos M S, Li C, et al. Optical imaging of Cerenkov light generation from positron-emitting radiotracers. Phys Med Biol, 2009, 54: N355–N365
Li C Q, Mitchell G S, Cherry S R. Cerenkov luminescence tomography for small-animal imaging. Opt Lett, 2010, 35: 1109–1111
Hu Z H, Liang J M, Yang W D, et al. Experimental Cerenkov luminescence tomography of the mouse model with SPECT imaging validation. Opt Express, 2010, 18: 24441–24450
Spinelli A E, D’Ambrosio D, Calderan L, et al. Cerenkov radiation allows in vivo optical imaging of positron emitting radiotracers. Phys Med Biol, 2010, 55: 483–495
Ruggiero A, Holland J P, Lewis J S, et al. Cerenkov luminescence imaging of medical isotopes. J Nucl Med, 2010, 51: 1123–1130
Xu Y D, Chang E, Liu H G, et al. Proof-of-concept study of monitoring cancer drug therapy with Cerenkov luminescence imaging. J Nucl Med, 2012, 53: 312–317
Klose A D, Ntziachristos V, Hielscher A. The inverse source problem based on the radiative transfer equation in optical molecular imaging. J Comput Phys, 2005, 202: 323–345
Wang L V, Wu H I. Biomedical Optics: Principles and Imaging. Wiley-Interscience, 2007
Rice B W, Cable M D, Nelson M B. In vivo imaging of light-emitting probes. J Biomed Opt, 2001, 6: 432–440
Chandrasekhar S. Radiative Transfer. Oxford: Clarendon Press, 1950
Born M, Wolf E. Principals of Optics. 7th ed. Cambridge: Cambridge University Press, 1999
Qin C H. Research of bioluminescence tomography based on meshless method and construction of BLT prototype system (in Chinese). Dissertation for the Doctoral Degree. Beijing: Institute of Automation, Chinese Academy of Sciences, 2009
Ishimaru A. Wave Propagation and Scattering in Random Media. New York: IEEE Press, 1977
Toublanc D. Henyey-Greenstein and Mie phase functions in Monte Carlo radiative transfer computations. Appl Optics, 1996, 35: 3270–3274
Schweiger M, Arridge S R, Hiraoka M, et al. The finite element method for the propagation of light in scattering media: boundary and source conditions. Med Phys, 1995, 22: 1779–1792
Wang L V, Jacques S L, Zheng L Q. MCML-Monte Carlo modeling of light transport in multi-layered tissues. Comput Meth Prog Bio, 1995, 47: 131–146
Boas D, Culver J, Stott J, et al. Three dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head. Opt Express, 2002, 10: 159–170
Li H, Tian J, Zhu F, et al. A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo method. Acad Radiol, 2004, 11: 1029–1038
Ren N N, Liang J M, Qu X C, et al. GPU-based Monte Carlo simulation for light propagation in complex heterogeneous tissues. Opt Express, 2010, 18: 6811–6823
Gu X J, Xu Y, Jiang H B. Mesh-based enhancement schemes in diffuse optical tomography. Med Phys, 2003, 30: 861–869
Culver J P, Choe R, Holboke M J, et al. Three-dimensional diffuse optical tomography in the parallel plane transmission geometry: evaluation of a hybrid frequency domain/continuous wave clinical system for breast imaging. Med Phys, 2003, 30: 235–247
Qin C H, Tian J, Yang X, et al. Galerkin-based meshless methods for photon transport in the biological tissue. Opt Express, 2008, 16: 20317–20333
Lv Y J, Tian J, Cong W X, et al. A multilevel adaptive finite element algorithm for bioluminescence tomography. Opt Express, 2006, 14: 8211–8223
Razansky D, Ntziachristos V. Hybrid photoacoustic fluorescence molecular tomography using finite-element-based inversion. Med Phys, 2007, 34: 4293–4301
Wang D F, Liu X, Chen Y P, et al. A novel finite-element-based algorithm for fluorescence molecular tomography of heterogeneous media. IEEE Trans Inf Technol Biomed, 2009, 13: 766–773
Zhong J H, Tian J, Yang X, et al. Whole-body Cerenkov luminescence tomography with the finite element SP3 method. Ann Biomed Eng, 2011, 39: 1728–1735
Qin C H, Tian J, Lv Y J, et al. Three-dimensional bioluminescent source reconstruction method based on nodes of adaptive FEM. In: Proceedings of SPIE 6916, Progress in Biomedical Optics and Imaging. San Diego: SPIE, 2008. 69161K
Tian J, Bai J, Yan X P, et al. Multimodality molecular imaging. IEEE Eng Med Biol, 2008, 27: 48–57
Cong A X, Wang G. Multispectral bioluminescence tomography: methodology and simulation. Int J Biomed Imag, 2006, 2006: 57614
Dehghani H, Davis S C, Pogue B W. Spectrally resolved bioluminescence tomography using the reciprocity approach. Med Phys, 2008, 35: 4863–4871
Gong R F, Wang G, Cheng X L, et al. A novel approach for studies of multispectral bioluminescence tomography. Numer Math, 2010, 115: 553–583
Lv Y J, Tian J, Cong W X, et al. Spectrally resolved bioluminescence tomography with adaptive finite element: methodology and simulation. Phys Med Biol, 2007, 52: 4497–4512
Svenmarker P, Xu C T, Liu H C, et al. Multispectral guided fluorescence diffuse optical tomography using upconverting nanoparticles. Appl Phys Lett, 2014, 104: 073703
Feng J C, Jia K B, Yan G R, et al. An optimal permissible source region strategy for multispectral bioluminescence tomography. Opt Express, 2008, 16: 15640–15654
Naser M A, Patterson M S. Algorithms for bioluminescence tomography incorporating anatomical information and reconstruction of tissue optical properties. Biomed Opt Express, 2010, 1: 512–526
Qin C H, Zhu S P, Feng J C, et al. Comparison of permissible source region and multispectral data using efficient bioluminescence tomography method. J Biophoton, 2011, 4: 824–839
Naser M A, Patterson M S. Improved bioluminescence and fluorescence reconstruction algorithms using diffuse optical tomography, normalized data, and optimized selection of the permissible source region. Biomed Opt Express, 2011, 2: 168–184
Lu Y, Machado H B, Douraghy A, et al. Experimental bioluminescence tomography with fully parallel radiativetransfer-based reconstruction framework. Opt Express, 2009, 17: 16681–16695
Guo W, Jia K B, Zhang Q, et al. Sparse reconstruction for bioluminescence tomography based on the semigreedy method. Comput Math Method Med, 2012, 2012: 494808
Xu Z, Bai J. Analysis of finite-element-based methods for reducing the ill-posedness in the reconstruction of fluorescence molecular tomography. Prog Nat Sci, 2009, 19: 501–509
Han R Q, Liang J M, Qu X C, et al. A source reconstruction algorithm based on adaptive hp-FEM for bioluminescence tomography. Opt Express, 2009, 17: 14481–14494
Guo H B, Hou Y Q, He XW. Adaptive hp finite element method for fluorescence molecular tomography with simplified spherical harmonics approximation. J Innov Opt Heal Sci, 2014, 7: 1350057
Lv Y J, Tian J, Cong W X, et al. A multilevel adaptive finite element algorithm for bioluminescence tomography. Opt Express, 2006, 14: 8211–8223
Yu J J, He X W, Geng G H, et al. Hybrid multilevel sparse reconstruction for a whole domain bioluminescence tomography using adaptive finite element. Comput Math Method Med, 2013, 2013: 548491
Zhang B, Yang X, Qin C H, et al. A trust region method in adaptive finite element framework for bioluminescence tomography. Opt Express, 2010, 18: 6477–6491
He X W, Hou Y B, Chen D F, et al. Sparse regularization-based reconstruction for bioluminescence tomography using a multilevel adaptive finite element method. Int J Biomed Imag, 2011, 2011: 203537
Tikhonov A N, Aresenin V Y. Solutions of ill-posed Problems. Washington DC: V. H. Winston and Sons, 1977
Cao N, Nehorai A, Jacobs M. Image reconstruction for diffuse optical tomography using sparsity regularization and expectation-maximization. Opt Express, 2007, 15: 13695–13708
Gao H, Zhao H K. Multilevel bioluminescence tomography based on radiative transfer equation Part 1: l1 regularization. Opt Express, 2010, 18: 1854–1871
Zeng J S, Fang J, Xu Z B. Sparse SAR imaging based on L 1/2 regularization. Sci China Inf Sci, 2012, 55: 1755–1775
Donoho D L. Compressed sensing. IEEE Trans Inf Theory, 2006, 52: 1289–1306
Yu J J, Liu F, Wu J, et al. Fast source reconstruction for bioluminescence tomography based on sparse regularization. IEEE Trans Biomed Eng, 2010, 57: 2583–2586
Zhong J H, Tian J, Yang X, et al. L 1-regularized Cerenkov luminescence tomography with a SP 3 method and CT fusion. In: Proceedings of the Annual International Conference of the Engineering in Medicine and Biology Society. Boston: IEEE, 2011. 6158–6161
Zhang Q T, Chen X L, Xu X C, et al. Comparative studies of l p-regularization-based reconstruction algorithms for bioluminescence tomography. Biomed Opt Express, 2012, 3: 2816–2836
Yi H J, Chen D F, Li W, et al. Reconstruction algorithms based on l 1-norm and l 2-norm for two imaging models of fluorescence molecular tomography: a comparative study. J Biomed Opt, 2013, 18: 56013
Cao X, Zhang B, Wang X, et al. An adaptive Tikhonov regularization method for fluorescence molecular tomography. Med Biol Eng Comput, 2013, 51: 849–858
Shi J W, Liu F, Zhang G L, et al. Enhanced spatial resolution in fluorescence molecular tomography using restarted L1-regularized nonlinear conjugate gradient algorithm. J Biomed Opt, 2014, 19: 046018
Ye J Z, Chi C W, Xue Z W, et al. Fast and robust reconstruction for fluorescence molecular tomography via a sparsity adaptive subspace pursuit method. Biomed Opt Express, 2014, 5: 387–406
Zhu D W, Li C Q. Nonconvex regularizations in fluorescence molecular tomography for sparsity enhancement. Phys Med Biol, 2014, 59: 2901–2912
Feng J C, Qin C H, Jia K B, et al. Total variation regularization for bioluminescence tomography with the split Bregman method. Appl Optics, 2012, 51: 4501–4512
Rudin L, Osher S, Fatemi E. Nonlinear total variation based noise removal algorithms. Phys D, 1992, 60: 259–268
Zhu Y G, Shi Y Y. A fast method for reconstruction of total-variation MR images with a periodic boundary condition. IEEE Signal Proc Lett, 2013, 20: 291–294
Yao L, Jiang H B. Enhancing finite element-based photoacoustic tomography using total variation minimization. Appl Optics, 2011, 50: 5031–5041
Gao H, Zhao H K. Multilevel bioluminescence tomography based on radiative transfer equation. Part 2: total variation and l1 data fidelity. Opt Express, 2010, 18: 2894–2912
Dutta J, Ahn S, Li C Q, et al. Joint L 1 and total variation regularization for fluorescence molecular tomography. Phys Med Biol, 2012, 57: 1459–1476
Jin W M, He Y H. Iterative reconstruction for bioluminescence tomography with total variation regularization. In: Proceedings of SPIE 8553, Optics in Heath Care and Biomedical Optics V. Beijing: SPIE, 2012. 855333
Cabello J, Torres-Espallardo I, Gillam J E, et al. PET reconstruction from truncated projections using total-variation regularization for hadron therapy monitoring. IEEE Trans Nucl Sci, 2013, 60: 3364–3372
Schweiger M, Arridge S R, Nissilä I. Gauss-Newton method for reconstruction in diffusion optical tomography.Phys Med Biol, 2005, 50: 2365–2386
He X W, Liang J M, Wang X R, et al. Sparse regularization for quantitative bioluminescence tomography based on the incomplete variables truncated conjugate gradient method. Opt Express, 2010, 18: 24825–24841
Freiberger M, Clason C, Scharfetter H. Total variation regularization for nonlinear fluorescence tomography with an augmented Lagrangian splitting approach. Appl Optics, 2010, 49: 3741–3747
Zhang Q T, Zhao H, Chen D F, et al. Source sparsity based primal-dual interior-point method for three-dimensional bioluminescence tomography. Opt Commun, 2011, 284: 5871–5876
Han D, Tian J, Zhu S P, et al. A fast reconstruction for fluorescence molecular tomography with sparsity regularization. Opt Express, 2010, 18: 8630–8646
Wu P, Liu K, Zhang Q, et al. Detection of mouse liver cancer via a parallel iterative shrinkage method in hybrid opitcal/microcomputed in tomography imaging. J Biomed Opt, 2012, 17: 126012
Goldstein T, Osher S. The Split Bregman method for L1 regularized problems. SIAM J Imag Sci, 2009, 2: 323–343
Abascal J F, Chamorro-Servent J, Aguirre J, et al. Fluorescence diffuse optical tomography using the split Bregman method. Med Phys, 2011, 38: 6275–6284
Zhang H, Cheng L Z, Li J P. Reweighted minimization model for MR image reconstruction with split Bregman method. Sci China Inf Sci, 2012, 55: 2109–2118
Nikazad T, Davidi R, Herman G T. Accelerated perturbation-resilient block-iterative projection methods with application to image reconstruction. Inverse Probl, 2012, 28: 035005
Ding X T, Wang K, Jie B, et al. Probability method for Cerenkov luminescence tomography based on conformance error minimization. Biomed Opt Express, 2014, 5: 2091–2112
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Leng, C., Tian, J. Mathematical method in optical molecular imaging. Sci. China Inf. Sci. 58, 1–13 (2015). https://doi.org/10.1007/s11432-014-5222-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11432-014-5222-5
Keywords
- radiative transfer equation (RTE)
- regularization method
- source reconstruction
- optical molecular imaging
- mathematical method