Small animal fluorescence and bioluminescence tomography: a review of approaches, algorithms and technology update

Emerging fluorescence and bioluminescence tomography approaches have several common, yet several distinct features from established emission tomographies of PET and SPECT. Although both nuclear and optical imaging modalities involve counting of photons, nuclear imaging techniques collect the emitted high energy (100–511 keV) photons after radioactive decay of radionuclides while optical techniques count low-energy (1.5–4.1 eV) photons that are scattered and absorbed by tissues requiring models of light transport for quantitative image reconstruction. Fluorescence imaging has been recently translated into clinic demonstrating high sensitivity, modest tissue penetration depth, and fast, millisecond image acquisition times. As a consequence, the promise of quantitative optical tomography as a complement of small animal PET and SPECT remains high. In this review, we summarize the different instrumentation, methodological approaches and schema for inverse image reconstructions for optical tomography, including luminescence and fluorescence modalities, and comment on limitations and key technological advances needed for further discovery research and translation.

[1]  S. Frick,et al.  Compressed Sensing , 2014, Computer Vision, A Reference Guide.

[2]  Banghe Zhu,et al.  In vivo imaging of orthotopic prostate cancer with far-red gene reporter fluorescence tomography and in vivo and ex vivo validation , 2013, Journal of biomedical optics.

[3]  S. Jacques Optical properties of biological tissues: a review , 2013, Physics in medicine and biology.

[4]  Baogang Xu,et al.  Multimodal Fluorescence-Mediated Tomography and SPECT/CT for Small-Animal Imaging , 2013, The Journal of Nuclear Medicine.

[5]  Yujie Lu,et al.  A compact frequency-domain photon migration system for integration into commercial hybrid small animal imaging scanners for fluorescence tomography , 2012, Physics in medicine and biology.

[6]  Gultekin Gulsen,et al.  In vivo validation of quantitative frequency domain fluorescence tomography , 2012, Journal of biomedical optics.

[7]  Jie Tian,et al.  Detection of mouse liver cancer via a parallel iterative shrinkage method in hybrid optical/microcomputed tomography imaging , 2012, Journal of biomedical optics.

[8]  I. Buvat,et al.  A review of partial volume correction techniques for emission tomography and their applications in neurology, cardiology and oncology , 2012, Physics in medicine and biology.

[9]  Jimin Liang,et al.  Comparative studies of lp-regularization-based reconstruction algorithms for bioluminescence tomography , 2012, Biomedical optics express.

[10]  V. Ntziachristos,et al.  FMT-XCT: in vivo animal studies with hybrid fluorescence molecular tomography–X-ray computed tomography , 2012, Nature Methods.

[11]  Jimin Liang,et al.  Three-dimensional Noninvasive Monitoring Iodine-131 Uptake in the Thyroid Using a Modified Cerenkov Luminescence Tomography Approach , 2012, PloS one.

[12]  R. Leahy,et al.  Joint L1 and total variation regularization for fluorescence molecular tomography , 2012, Physics in medicine and biology.

[13]  Tony F. Chan,et al.  A Novel Sparsity Reconstruction Method from Poisson Data for 3D Bioluminescence Tomography , 2012, J. Sci. Comput..

[14]  E M Sevick-Muraca,et al.  Validating the Sensitivity and Performance of Near-Infrared Fluorescence Imaging and Tomography Devices Using a Novel Solid Phantom and Measurement Approach , 2012, Technology in cancer research & treatment.

[15]  E M Sevick-Muraca,et al.  Translation of near-infrared fluorescence imaging technologies: emerging clinical applications. , 2012, Annual review of medicine.

[16]  Jie Tian,et al.  Fast-Specific Tomography Imaging via Cerenkov Emission , 2012, Molecular Imaging and Biology.

[17]  Banghe Zhu,et al.  Improvement of fluorescence-enhanced optical tomography with improved optical filtering and accurate model-based reconstruction algorithms. , 2011, Journal of biomedical optics.

[18]  Simon R. Cherry,et al.  In vivo Cerenkov luminescence imaging: a new tool for molecular imaging , 2011, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[19]  E. Sevick-Muraca,et al.  MINIMIZING EXCITATION LIGHT LEAKAGE AND MAXIMIZING MEASUREMENT SENSITIVITY FOR MOLECULAR IMAGING WITH NEAR-INFRARED FLUORESCENCE , 2011 .

[20]  Yukio Yamada,et al.  Improvement of image quality of time-domain diffuse optical tomography with lp sparsity regularization , 2011, Biomedical optics express.

[21]  Michael Hünerbein,et al.  Current trends and emerging future of indocyanine green usage in surgery and oncology , 2011, Cancer.

[22]  Brian W Pogue,et al.  Toward whole-body optical imaging of rats using single-photon counting fluorescence tomography. , 2011, Optics letters.

[23]  L. Furenlid,et al.  SPECT detectors: the Anger Camera and beyond , 2011, Physics in medicine and biology.

[24]  Merlijn Hutteman,et al.  The clinical use of indocyanine green as a near‐infrared fluorescent contrast agent for image‐guided oncologic surgery , 2011, Journal of surgical oncology.

[25]  O Nalcioglu,et al.  A photo-multiplier tube-based hybrid MRI and frequency domain fluorescence tomography system for small animal imaging , 2011, Physics in medicine and biology.

[26]  Changqing Li,et al.  Simultaneous PET and Multispectral 3-Dimensional Fluorescence Optical Tomography Imaging System , 2011, The Journal of Nuclear Medicine.

[27]  Kami Kim,et al.  Bright and stable near infra-red fluorescent protein for in vivo imaging , 2011, Nature Biotechnology.

[28]  Xing Zhang,et al.  Cerenkov Luminescence Tomography for In Vivo Radiopharmaceutical Imaging , 2011, Int. J. Biomed. Imaging.

[29]  Edoardo Charbon,et al.  Hybrid Small Animal Imaging System Combining Magnetic Resonance Imaging With Fluorescence Tomography Using Single Photon Avalanche Diode Detectors , 2011, IEEE Transactions on Medical Imaging.

[30]  Jie Tian,et al.  Whole-Body Cerenkov Luminescence Tomography with the Finite Element SP3 Method , 2011, Annals of Biomedical Engineering.

[31]  Baoci Shan,et al.  A Dual Modality System for Simultaneous Fluorescence and Positron Emission Tomography Imaging of Small Animals , 2011, IEEE Transactions on Nuclear Science.

[32]  Jie Tian,et al.  Experimental Cerenkov luminescence tomography of the mouse model with SPECT imaging validation. , 2010, Optics express.

[33]  Xavier Intes,et al.  Full-field time-resolved fluorescence tomography of small animals. , 2010, Optics letters.

[34]  Xin Liu,et al.  A Combined Fluorescence and Microcomputed Tomography System for Small Animal Imaging , 2010, IEEE Transactions on Biomedical Engineering.

[35]  Eva M. Sevick-Muraca,et al.  Near-Infrared Fluorescence Imaging in Humans with Indocyanine Green: A Review and Update~!2009-12-07~!2009-12-23~!2010-05-26~! , 2010 .

[36]  Feng Gao,et al.  Simultaneous fluorescence yield and lifetime tomography from time-resolved transmittances of small-animal-sized phantom. , 2010, Applied optics.

[37]  Qingming Luo,et al.  Combined system of fluorescence diffuse optical tomography and microcomputed tomography for small animal imaging. , 2010, The Review of scientific instruments.

[38]  Arion F. Chatziioannou,et al.  In Vivo Mouse Bioluminescence Tomography with Radionuclide-Based Imaging Validation , 2010, Molecular Imaging and Biology.

[39]  Liji Cao,et al.  Bayesian reconstruction strategy of fluorescence-mediated tomography using an integrated SPECT-CT-OT system , 2010, Physics in medicine and biology.

[40]  F. Pigge,et al.  Synthesis of a DOTA--biotin conjugate for radionuclide chelation via Cu-free click chemistry. , 2010, Organic letters.

[41]  Simon R Cherry,et al.  Cerenkov luminescence tomography for small-animal imaging. , 2010, Optics letters.

[42]  Yuting Lin,et al.  Quantitative fluorescence tomography using a combined tri-modality FT/DOT/XCT system , 2010, Optics express.

[43]  Ge Wang,et al.  Differential Evolution Approach for Regularized Bioluminescence Tomography , 2010, IEEE Transactions on Biomedical Engineering.

[44]  Xing Zhang,et al.  A fast bioluminescent source localization method based on generalized graph cuts with mouse model validations. , 2010, Optics express.

[45]  O Nalcioglu,et al.  Combined Fluorescence and X-Ray Tomography for Quantitative In Vivo Detection of Fluorophore , 2010, Technology in cancer research & treatment.

[46]  Hongkai Zhao,et al.  Multilevel bioluminescence tomography based on radiative transfer equation Part 1: l1 regularization. , 2010, Optics express.

[47]  Hongkai Zhao,et al.  Multilevel bioluminescence tomography based on radiative transfer equation part 2: total variation and l1 data fidelity. , 2010, Optics express.

[48]  W. Cong,et al.  Bioluminescence tomography based on the phase approximation model. , 2010, Journal of the Optical Society of America. A, Optics, image science, and vision.

[49]  Vasilis Ntziachristos,et al.  Hybrid System for Simultaneous Fluorescence and X-Ray Computed Tomography , 2010, IEEE Transactions on Medical Imaging.

[50]  Scott C Davis,et al.  Pre-clinical whole-body fluorescence imaging: Review of instruments, methods and applications. , 2010, Journal of photochemistry and photobiology. B, Biology.

[51]  Hamid Dehghani,et al.  In vivo bioluminescence tomography with a blocking-off finite-difference SP3 method and MRI/CT coregistration. , 2009, Medical physics.

[52]  Hyun Keol Kim,et al.  PDE-Constrained Fluorescence Tomography With the Frequency-Domain Equation of Radiative Transfer , 2010, IEEE Journal of Selected Topics in Quantum Electronics.

[53]  Steven L. Jacques,et al.  Monte Carlo Modeling of Light Transport in Tissue (Steady State and Time of Flight) , 2010 .

[54]  Anne Koenig,et al.  Fluorescence diffuse optical tomography for free-space and multifluorophore studies. , 2010, Journal of biomedical optics.

[55]  I-Chih Tan,et al.  Near-Infrared Fluorescence Imaging in Humans with Indocyanine Green: A Review and Update. , 2010, Open surgical oncology journal.

[56]  Jie Tian,et al.  Spectrally resolved bioluminescence tomography with the third-order simplified spherical harmonics approximation. , 2009, Physics in medicine and biology.

[57]  S. Arridge,et al.  Optical tomography: forward and inverse problems , 2009, 0907.2586.

[58]  T. Chan,et al.  Source reconstruction for spectrally-resolved bioluminescence tomography with sparse a priori information. , 2009, Optics express.

[59]  Michael Z. Lin,et al.  Mammalian Expression of Infrared Fluorescent Proteins Engineered from a Bacterial Phytochrome , 2009, Science.

[60]  R. Leahy,et al.  A three-dimensional multispectral fluorescence optical tomography imaging system for small animals based on a conical mirror design. , 2009, Optics express.

[61]  Jean-Marc Dinten,et al.  Optical calibration protocol for an x-ray and optical multimodality tomography system dedicated to small-animal examination. , 2009, Applied optics.

[62]  Hamid Dehghani,et al.  A microcomputed tomography guided fluorescence tomography system for small animal molecular imaging. , 2009, The Review of scientific instruments.

[63]  G Allan Johnson,et al.  Development of a noncontact 3-D fluorescence tomography system for small animal in vivo imaging , 2009, BiOS.

[64]  Yuri Musienko,et al.  Advances in multipixel Geiger-mode avalanche photodiodes (silicon photomultiplies) , 2009 .

[65]  Hamid Dehghani,et al.  Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction. , 2009, Communications in numerical methods in engineering.

[66]  M. Patterson,et al.  Quantitative fluorescence imaging of point-like sources in small animals , 2008, Physics in medicine and biology.

[67]  I. G. Meerovich,et al.  Fluorescence diffuse tomography for detection of red fluorescent protein expressed tumors in small animals. , 2008, Journal of biomedical optics.

[68]  Ulrich Dirnagl,et al.  Fluorescence tomography technique optimized for noninvasive imaging of the mouse brain. , 2008, Journal of biomedical optics.

[69]  John C Rasmussen,et al.  Molecular imaging with optics: primer and case for near-infrared fluorescence techniques in personalized medicine. , 2008, Journal of biomedical optics.

[70]  A. Godavarty,et al.  Three-dimensional fluorescence-enhanced optical tomography using a hand-held probe based imaging system. , 2008, Medical physics.

[71]  J.-M. Dinten,et al.  In vivo fluorescence molecular optical imaging: from small animal towards clinical applications , 2008, 2008 16th Mediterranean Conference on Control and Automation.

[72]  Stephen B. Tuttle,et al.  Magnetic resonance-coupled fluorescence tomography scanner for molecular imaging of tissue. , 2008, The Review of scientific instruments.

[73]  E. Candès The restricted isometry property and its implications for compressed sensing , 2008 .

[74]  Huabei Jiang,et al.  Diffuse optical tomography guided quantitative fluorescence molecular tomography. , 2008, Applied optics.

[75]  Milton V. Marshall,et al.  Imaging of lymph flow in breast cancer patients after microdose administration of a near-infrared fluorophore: feasibility study. , 2008, Radiology.

[76]  Wenxiang Cong,et al.  Integral equations of the photon fluence rate and flux based on a generalized Delta-Eddington phase function. , 2008, Journal of biomedical optics.

[77]  Andrew K. Dunn,et al.  A Time Domain Fluorescence Tomography System for Small Animal Imaging , 2008, IEEE Transactions on Medical Imaging.

[78]  Andreas H. Hielscher,et al.  A PDE-constrained SQP algorithm for optical tomography based on the frequency-domain equation of radiative transfer , 2008 .

[79]  Mark A A Neil,et al.  Fluorescence lifetime imaging by using time-gated data acquisition. , 2007, Applied optics.

[80]  M S Patterson,et al.  Bioluminescence imaging of point sources implanted in small animals post mortem: evaluation of a method for estimating source strength and depth , 2007, Physics in medicine and biology.

[81]  Jie Tian,et al.  Spectrally resolved bioluminescence tomography with adaptive finite element analysis: methodology and simulation , 2007, Physics in medicine and biology.

[82]  P. Peltié,et al.  Noncontact fluorescence diffuse optical tomography of heterogeneous media. , 2007, Applied optics.

[83]  Vasilis Ntziachristos,et al.  Noncontact optical imaging in mice with full angular coverage and automatic surface extraction. , 2007, Applied optics.

[84]  B. Pogue,et al.  Weight-matrix structured regularization provides optimal generalized least-squares estimate in diffuse optical tomography. , 2007, Medical physics.

[85]  A. Adibi,et al.  Optimal sparse solution for fluorescent diffuse optical tomography: theory and phantom experimental results. , 2007, Applied optics.

[86]  Vasilis Ntziachristos,et al.  Three-Dimensional in Vivo Imaging of Green Fluorescent Protein-Expressing T Cells in Mice with Noncontact Fluorescence Molecular Tomography , 2007, Molecular imaging.

[87]  Shi Ke,et al.  Comparison of visible and near-infrared wavelength-excitable fluorescent dyes for molecular imaging of cancer. , 2007, Journal of biomedical optics.

[88]  B. Rice,et al.  Three-dimensional reconstruction of in vivo bioluminescent sources based on multispectral imaging. , 2007, Journal of biomedical optics.

[89]  Vasilis Ntziachristos,et al.  Free-space fluorescence molecular tomography utilizing 360° geometry projections , 2007 .

[90]  Anne Koenig,et al.  Whole body small animal examination with a diffuse optical tomography instrument , 2007 .

[91]  Vasilis Ntziachristos,et al.  Free-space fluorescence molecular tomography utilizing 360 degrees geometry projections. , 2007, Optics letters.

[92]  Edward W. Larsen,et al.  Light transport in biological tissue based on the simplified spherical harmonics equations , 2006, J. Comput. Phys..

[93]  Jie Tian,et al.  A multilevel adaptive finite element algorithm for bioluminescence tomography. , 2006, Optics express.

[94]  E. Hoffman,et al.  In vivo mouse studies with bioluminescence tomography. , 2006, Optics express.

[95]  Wolfgang Bangerth,et al.  Non-contact fluorescence optical tomography with scanning patterned illumination. , 2006, Optics express.

[96]  Rinaldo Cubeddu,et al.  Comparison of noncontact and fiber-based fluorescence-mediated tomography. , 2006, Optics letters.

[97]  Ge Wang,et al.  Multispectral Bioluminescence Tomography: Methodology and Simulation , 2006, Int. J. Biomed. Imaging.

[98]  David L Donoho,et al.  Compressed sensing , 2006, IEEE Transactions on Information Theory.

[99]  David A Boas,et al.  Fluorescence-lifetime-based tomography for turbid media. , 2005, Optics letters.

[100]  R. Leahy,et al.  Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging , 2005, Physics in medicine and biology.

[101]  Ge Wang,et al.  A finite-element-based reconstruction method for 3D fluorescence tomography. , 2005, Optics express.

[102]  Geoffrey McLennan,et al.  Practical reconstruction method for bioluminescence tomography. , 2005, Optics express.

[103]  B. Pogue,et al.  Combining near-infrared tomography and magnetic resonance imaging to study in vivo breast tissue: implementation of a Laplacian-type regularization to incorporate magnetic resonance structure. , 2005, Journal of biomedical optics.

[104]  A. Hielscher,et al.  Optical tomography as a PDE-constrained optimization problem , 2005 .

[105]  Vasilis Ntziachristos,et al.  Fluorescent protein tomography scanner for small animal imaging , 2005, IEEE Transactions on Medical Imaging.

[106]  Eva M. Sevick-Muraca,et al.  Improved Excitation Light Rejection Enhances Small-Animal Fluorescent Optical Imaging , 2005, Molecular imaging.

[107]  J. Culver,et al.  Time-dependent whole-body fluorescence tomography of probe bio-distributions in mice. , 2005, Optics express.

[108]  M. Eppstein,et al.  Three-dimensional fluorescence lifetime tomography. , 2005, Medical physics.

[109]  S R Arridge,et al.  Recent advances in diffuse optical imaging , 2005, Physics in medicine and biology.

[110]  A. Joshi,et al.  Adaptive finite element based tomography for fluorescence optical imaging in tissue. , 2004, Optics express.

[111]  Yi Liu,et al.  A practical method to determine the light source distribution in bioluminescent imaging , 2004, SPIE Optics + Photonics.

[112]  David Dussault,et al.  Noise performance comparison of ICCD with CCD and EMCCD cameras , 2004, SPIE Optics + Photonics.

[113]  David A Boas,et al.  Fluorescence optical diffusion tomography using multiple-frequency data. , 2004, Journal of the Optical Society of America. A, Optics, image science, and vision.

[114]  Vasilis Ntziachristos,et al.  Experimental fluorescence tomography of tissues with noncontact measurements , 2004, IEEE Transactions on Medical Imaging.

[115]  B. Rice,et al.  Quantitative Comparison of the Sensitivity of Detection of Fluorescent and Bioluminescent Reporters in Animal Models , 2004, Molecular imaging.

[116]  B. Rice,et al.  Quantitative comparison of the sensitivity of detection of fluorescent and bioluminescent reporters in animal models. , 2004, Molecular imaging.

[117]  A. Chatziioannou,et al.  Detector concept for OPET-a combined PET and optical imaging system , 2003, IEEE Transactions on Nuclear Science.

[118]  M. Eppstein,et al.  Fluorescence-enhanced optical imaging in large tissue volumes using a gain-modulated ICCD camera. , 2003, Physics in medicine and biology.

[119]  B. Pogue,et al.  Three-dimensional optical tomography: resolution in small-object imaging. , 2003, Applied optics.

[120]  C. Bouman,et al.  Fluorescence optical diffusion tomography. , 2003, Applied optics.

[121]  Vasilis Ntziachristos,et al.  A submillimeter resolution fluorescence molecular imaging system for small animal imaging. , 2003, Medical physics.

[122]  Anuradha Godavarty,et al.  Near- Infrared Fluorescence Imaging and Spectroscopy in Random Media and Tissues , 2003 .

[123]  M. Eppstein,et al.  Fluorescence-enhanced optical imaging in large tissue volumes using a gain-modulated ICCD camera. , 2003, Physics in medicine and biology.

[124]  Eva M Sevick-Muraca,et al.  Fluorescence-enhanced, near infrared diagnostic imaging with contrast agents. , 2002, Current opinion in chemical biology.

[125]  Daniel J. Hawrysz,et al.  Three-dimensional, Bayesian image reconstruction from sparse and noisy data sets: Near-infrared fluorescence tomography , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[126]  C. Drumm,et al.  Discrete Ordinates Approximations to the First- and Second-Order Radiation Transport Equations , 2002 .

[127]  R. Weissleder,et al.  Charge-coupled-device based scanner for tomography of fluorescent near-infrared probes in turbid media. , 2002, Medical physics.

[128]  C. Contag,et al.  Advances in in vivo bioluminescence imaging of gene expression. , 2002, Annual review of biomedical engineering.

[129]  R. Weissleder,et al.  Fluorescence imaging with near-infrared light: new technological advances that enable in vivo molecular imaging , 2002, European Radiology.

[130]  V. Ntziachristos,et al.  Projection access order in algebraic reconstruction technique for diffuse optical tomography. , 2002, Physics in medicine and biology.

[131]  R. Weissleder,et al.  Experimental three-dimensional fluorescence reconstruction of diffuse media by use of a normalized Born approximation. , 2001, Optics letters.

[132]  E. Sevick-Muraca,et al.  Fluorescence-enhanced absorption imaging using frequency-domain photon migration: tolerance to measurement error. , 2001, Journal of biomedical optics.

[133]  Eva M. Sevick-Muraca,et al.  Pharmacokinetics of ICG and HPPH-car for the Detection of Normal and Tumor Tissue Using Fluorescence, Near-infrared Reflectance Imaging: A Case Study¶ , 2000, Photochemistry and photobiology.

[134]  Jim E. Morel,et al.  A Self-Adjoint Angular Flux Equation , 1999 .

[135]  S. Arridge Optical tomography in medical imaging , 1999 .

[136]  C. L. Hutchinson,et al.  Fluorescence and Absorption Contrast Mechanisms for Biomedical Optical Imaging Using Frequency‐Domain Techniques , 1997, Photochemistry and photobiology.

[137]  M S Patterson,et al.  Imaging of fluorescent yield and lifetime from multiply scattered light reemitted from random media. , 1997, Applied optics.

[138]  J. S. Reynolds,et al.  Multipixel Techniques for Frequency‐Domain Photon Migration Imaging , 1997, Biotechnology progress.

[139]  Akira Ishimaru,et al.  Wave propagation and scattering in random media , 1997 .

[140]  D. Boas,et al.  Fluorescence lifetime imaging in turbid media. , 1996, Optics letters.

[141]  E. Sevick-Muraca,et al.  Quantitative optical spectroscopy for tissue diagnosis. , 1996, Annual review of physical chemistry.

[142]  B. Hooper Optical-thermal response of laser-irradiated tissue , 1996 .

[143]  Andrew Pollard,et al.  The TN Quadrature Set for the Discrete Ordinates Method , 1995 .

[144]  Rainer Koch,et al.  DISCRETE ORDINATES QUADRATURE SCHEMES FOR MULTIDIMENSIONAL RADIATIVE TRANSFER , 1995 .

[145]  Guriĭ Ivanovich Marchuk,et al.  Adjoint Equations and Analysis of Complex Systems , 1995 .

[146]  Ashleyj . Welch,et al.  Optical-Thermal Response of Laser-Irradiated Tissue , 1995 .

[147]  Richard Barrett,et al.  Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods , 1994, Other Titles in Applied Mathematics.

[148]  S Nioka,et al.  Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxygenation. , 1991, Analytical biochemistry.

[149]  S. Suzuki,et al.  Newly Developed Photomultiplier Tubes with Position Sensitivity Capability , 1985, IEEE Transactions on Nuclear Science.

[150]  W. Wiscombe The Delta–M Method: Rapid Yet Accurate Radiative Flux Calculations for Strongly Asymmetric Phase Functions , 1977 .

[151]  V. I. Lebedev,et al.  Quadratures on a sphere , 1976 .

[152]  V. Lebedev Values of the nodes and weights of ninth to seventeenth order gauss-markov quadrature formulae invariant under the octahedron group with inversion☆ , 1975 .