High temporal resolution OCT using image-based retrospective gating

Madhusudhana Gargesha; Michael W. Jenkins; David L. Wilson; Andrew M. Rollins

doi:10.1364/OE.17.010786

High temporal resolution OCT using image-based retrospective gating

Madhusudhana Gargesha, Michael W. Jenkins, David L. Wilson, and Andrew M. Rollins

Optics Express
Vol. 17,
Issue 13,
pp. 10786-10799
(2009)
•https://doi.org/10.1364/OE.17.010786

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Madhusudhana Gargesha, Michael W. Jenkins, David L. Wilson, and Andrew M. Rollins, "High temporal resolution OCT using image-based retrospective gating," Opt. Express 17, 10786-10799 (2009)

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Related Topics
Table of Contents Category
- Medical Optics and Biotechnology
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Image analysis (100.2960)
- Multiframe image processing (110.4155)
- Medical and biological imaging (170.3880)
- Optical coherence tomography (170.4500)

About this Article
History
- Original Manuscript: March 11, 2009
- Revised Manuscript: April 27, 2009
- Manuscript Accepted: May 3, 2009
- Published: June 12, 2009
Virtual Issues
Virtual Journal for Biomedical Optics Vol. 4, Iss. 8

Accessible

Open Access

Abstract

High temporal resolution OCT imaging is very advantageous for analyzing cardiac mechanics in the developing embryonic heart of small animals. An image-based retrospective gating technique is presented to increase the effective temporal resolution of an OCT system and to allow visualization of systolic dynamics in 3D. The gating technique employs image similarity measures for rearranging asynchronously acquired input data consisting of a time series of 2D images at each z position along the heart volume, to produce a time sequence of 3D volumes of the beating heart. The study includes a novel robust validation technique, which quantitatively evaluates the accuracy of the gating technique, in addition to visual evaluations by 2D multiplanar reformatting (MPR) and 3D volume rendering. The retrospective gating and validation is demonstrated on a stage 14 embryonic quail heart data set. Using the validation scheme, it is shown that the gating is accurate within a standard deviation of 4.7 ms, which is an order of magnitude shorter than the time interval during which systolic contraction (~50 ms) occurs in the developing embryo. This gating method has allowed, for the first time, clear visualization of systolic dynamics of the looping embryonic heart in 3D.

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M. Gargesha, M. W. Jenkins, A. M. Rollins, and D. L. Wilson, “Denoising and 4D visualization of OCT images,” Opt. Express 16(16), 12313–12333 (2008).
[Crossref] [PubMed]
M. W. Jenkins, F. Rothenberg, D. Roy, V. P. Nikolski, Z. Hu, M. Watanabe, D. L. Wilson, I. R. Efimov, and A. M. Rollins, “4D embryonic cardiography using gated optical coherence tomography,” Opt. Express 14(2), 736–748 (2006).
[Crossref] [PubMed]
M. W. Jenkins, D. C. Adler, M. Gargesha, R. Huber, F. Rothenberg, J. Belding, M. Watanabe, D. L. Wilson, J. G. Fujimoto, and A. M. Rollins, “Ultrahigh-speed optical coherence tomography imaging and visualization of the embryonic avian heart using a buffered Fourier Domain Mode Locked laser,” Opt. Express 15(10), 6251–6267 (2007).
[Crossref] [PubMed]
M. W. Jenkins, O. Q. Chughtai, A. N. Basavanhally, M. Watanabe, and A. M. Rollins, “In vivo gated 4D imaging of the embryonic heart using optical coherence tomography,” J. Biomed. Opt. 12(3), 030505 (2007).
[Crossref] [PubMed]
M. W. Jenkins, P. Patel, H. Deng, M. M. Montano, M. Watanabe, and A. M. Rollins, “Phenotyping transgenic embryonic murine hearts using optical coherence tomography,” Appl. Opt. 46(10), 1776–1781 (2007).
[Crossref] [PubMed]
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[Crossref]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
J. Männer, L. Thrane, K. Norozi, and T. M. Yelbuz, “High-resolution in vivo imaging of the cross-sectional deformations of contracting embryonic heart loops using optical coherence tomography,” Dev. Dyn. 237(4), 953–961 (2008).
[Crossref] [PubMed]
W. Luo, D. L. Marks, T. S. Ralston, and S. A. Boppart, “Three-dimensional optical coherence tomography of the embryonic murine cardiovascular system,” J. Biomed. Opt. 11(2), 021014 (2006).
[Crossref] [PubMed]
I. V. Larina, N. Sudheendran, M. Ghosn, J. Jiang, A. Cable, K. V. Larin, and M. E. Dickinson, “Live imaging of blood flow in mammalian embryos using Doppler swept-source optical coherence tomography,” J. Biomed. Opt. 13(6), 060506 (2008).
[Crossref]
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[Crossref]
A. M. Davis, F. G. Rothenberg, N. Shepherd, and J. A. Izatt, “In vivo spectral domain optical coherence tomography volumetric imaging and spectral Doppler velocimetry of early stage embryonic chicken heart development,” J. Opt. Soc. Am. A 25(12), 3134–3143 (2008).
[Crossref]
A. Davis, J. Izatt, and F. Rothenberg, “Quantitative measurement of blood flow dynamics in embryonic vasculature using spectral Doppler velocimetry,” Anat. Rec. (Hoboken) 292(3), 311–319 (2009).
[Crossref]
M. A. Choma, S. D. Izatt, R. J. Wessells, R. Bodmer, and J. A. Izatt, “Images in cardiovascular medicine: in vivo imaging of the adult Drosophila melanogaster heart with real-time optical coherence tomography,” Circulation 114(2), e35–e36 (2006).
[Crossref] [PubMed]
S. A. Boppart, G. J. Tearney, B. E. Bouma, J. F. Southern, M. E. Brezinski, and J. G. Fujimoto, “Noninvasive assessment of the developing Xenopus cardiovascular system using optical coherence tomography,” Proc. Natl. Acad. Sci. U.S.A. 94(9), 4256–4261 (1997).
[Crossref] [PubMed]
U. J. Schoepf, C. R. Becker, R. D. Bruening, T. Helmberger, A. Staebler, P. Leimeister, and M. F. Reiser, “Electrocardiographically gated thin-section CT of the lung,” Radiology 212(3), 649–654 (1999).
[PubMed]
Y. P. P. Du, E. R. McVeigh, D. A. Bluemke, H. A. Silber, and T. K. F. Foo, “A comparison of prospective and retrospective respiratory navigator gating in 3D MR coronary angiography,” Int. J. Cardiovasc. Imaging 17(4), 287–294, discussion 295–296 (2001).
[Crossref] [PubMed]
J. Albers, J. M. Boese, C. F. Vahl, and S. Hagl, “In vivo validation of cardiac spiral computed tomography using retrospective gating,” Ann. Thorac. Surg. 75(3), 885–889 (2003).
[Crossref] [PubMed]
C. E. Woodhouse, W. R. Janowitz, and M. Viamonte., “Coronary arteries: retrospective cardiac gating technique to reduce cardiac motion artifact at spiral CT,” Radiology 204(2), 566–569 (1997).
[PubMed]
A. F. Kopp, S. Schroeder, A. Kuettner, M. Heuschmid, C. Georg, B. Ohnesorge, R. Kuzo, and C. D. Claussen, “Coronary arteries: retrospectively ECG-gated multi-detector row CT angiography with selective optimization of the image reconstruction window,” Radiology 221(3), 683–688 (2001).
[Crossref] [PubMed]
A. F. Kopp, S. Schroeder, A. Kuettner, A. Baumbach, C. Georg, R. Kuzo, M. Heuschmid, B. Ohnesorge, K. R. Karsch, and C. D. Claussen, “Non-invasive coronary angiography with high resolution multidetector-row computed tomography. Results in 102 patients,” Eur. Heart J. 23(21), 1714–1725 (2002).
[PubMed]
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[Crossref]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
R. Huber, D. C. Adler, and J. G. Fujimoto, “Buffered Fourier domain mode locking: Unidirectional swept laser sources for optical coherence tomography imaging at 370,000 lines/s,” Opt. Lett. 31(20), 2975–2977 (2006).
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2009 (1)

A. Davis, J. Izatt, and F. Rothenberg, “Quantitative measurement of blood flow dynamics in embryonic vasculature using spectral Doppler velocimetry,” Anat. Rec. (Hoboken) 292(3), 311–319 (2009).
[Crossref]

2008 (6)

I. V. Larina, N. Sudheendran, M. Ghosn, J. Jiang, A. Cable, K. V. Larin, and M. E. Dickinson, “Live imaging of blood flow in mammalian embryos using Doppler swept-source optical coherence tomography,” J. Biomed. Opt. 13(6), 060506 (2008).
[Crossref]

J. Männer, L. Thrane, K. Norozi, and T. M. Yelbuz, “High-resolution in vivo imaging of the cross-sectional deformations of contracting embryonic heart loops using optical coherence tomography,” Dev. Dyn. 237(4), 953–961 (2008).
[Crossref] [PubMed]

S. Rugonyi, C. Shaut, A. P. Liu, K. Thornburg, and R. K. K. Wang, “Changes in wall motion and blood flow in the outflow tract of chick embryonic hearts observed with optical coherence tomography after outflow tract banding and vitelline-vein ligation,” Phys. Med. Biol. 53(18), 5077–5091 (2008).
[Crossref] [PubMed]

K. Norozi, L. Thrane, J. Männer, F. Pedersen, I. Wolf, S. Mottl-Link, H. P. Meinzer, A. Wessel, and T. M. Yelbuz, “In vivo visualisation of coronary artery development by high-resolution optical coherence tomography,” Heart 94(2), 130 (2008).
[Crossref] [PubMed]

M. Gargesha, M. W. Jenkins, A. M. Rollins, and D. L. Wilson, “Denoising and 4D visualization of OCT images,” Opt. Express 16(16), 12313–12333 (2008).
[Crossref] [PubMed]

A. M. Davis, F. G. Rothenberg, N. Shepherd, and J. A. Izatt, “In vivo spectral domain optical coherence tomography volumetric imaging and spectral Doppler velocimetry of early stage embryonic chicken heart development,” J. Opt. Soc. Am. A 25(12), 3134–3143 (2008).
[Crossref]

2007 (6)

A. Mariampillai, B. A. Standish, N. R. Munce, C. Randall, G. Liu, J. Y. Jiang, A. E. Cable, I. A. Vitkin, and V. X. D. Yang, “Doppler optical cardiogram gated 2D color flow imaging at 1000 fps and 4D in vivo visualization of embryonic heart at 45 fps on a swept source OCT system,” Opt. Express 15(4), 1627–1638 (2007).
[Crossref] [PubMed]

M. W. Jenkins, P. Patel, H. Deng, M. M. Montano, M. Watanabe, and A. M. Rollins, “Phenotyping transgenic embryonic murine hearts using optical coherence tomography,” Appl. Opt. 46(10), 1776–1781 (2007).
[Crossref] [PubMed]

M. W. Jenkins, D. C. Adler, M. Gargesha, R. Huber, F. Rothenberg, J. Belding, M. Watanabe, D. L. Wilson, J. G. Fujimoto, and A. M. Rollins, “Ultrahigh-speed optical coherence tomography imaging and visualization of the embryonic avian heart using a buffered Fourier Domain Mode Locked laser,” Opt. Express 15(10), 6251–6267 (2007).
[Crossref] [PubMed]

M. W. Jenkins, O. Q. Chughtai, A. N. Basavanhally, M. Watanabe, and A. M. Rollins, “In vivo gated 4D imaging of the embryonic heart using optical coherence tomography,” J. Biomed. Opt. 12(3), 030505 (2007).
[Crossref] [PubMed]

R. Yelin, D. Yelin, W. Y. Oh, S. H. Yun, C. Boudoux, B. J. Vakoc, B. E. Bouma, and G. J. Tearney, “Multimodality optical imaging of embryonic heart microstructure,” J. Biomed. Opt. 12(6), 064021 (2007).
[Crossref]

B. A. Filas, I. R. Efimov, and L. A. Taber, “Optical coherence tomography as a tool for measuring morphogenetic deformation of the looping heart,” Anatomical Record-Advances in Integrative Anatomy and Evolutionary Biology 290(9), 1057–1068 (2007).
[Crossref]

2006 (5)

M. A. Choma, S. D. Izatt, R. J. Wessells, R. Bodmer, and J. A. Izatt, “Images in cardiovascular medicine: in vivo imaging of the adult Drosophila melanogaster heart with real-time optical coherence tomography,” Circulation 114(2), e35–e36 (2006).
[Crossref] [PubMed]

W. Luo, D. L. Marks, T. S. Ralston, and S. A. Boppart, “Three-dimensional optical coherence tomography of the embryonic murine cardiovascular system,” J. Biomed. Opt. 11(2), 021014 (2006).
[Crossref] [PubMed]

M. W. Jenkins, F. Rothenberg, D. Roy, V. P. Nikolski, Z. Hu, M. Watanabe, D. L. Wilson, I. R. Efimov, and A. M. Rollins, “4D embryonic cardiography using gated optical coherence tomography,” Opt. Express 14(2), 736–748 (2006).
[Crossref] [PubMed]

R. Huber, M. Wojtkowski, and J. G. Fujimoto, “Fourier Domain Mode Locking (FDML): A new laser operating regime and applications for optical coherence tomography,” Opt. Express 14(8), 3225–3237 (2006).
[Crossref] [PubMed]

R. Huber, D. C. Adler, and J. G. Fujimoto, “Buffered Fourier domain mode locking: Unidirectional swept laser sources for optical coherence tomography imaging at 370,000 lines/s,” Opt. Lett. 31(20), 2975–2977 (2006).
[Crossref] [PubMed]

2005 (2)

M. Liebling, A. S. Forouhar, M. Gharib, S. E. Fraser, and M. E. Dickinson, “Four-dimensional cardiac imaging in living embryos via postacquisition synchronization of nongated slice sequences,” J. Biomed. Opt. 10(5), 054001 (2005).
[Crossref] [PubMed]

J. L. Lucitti, K. Tobita, and B. B. Keller, “Arterial hemodynamics and mechanical properties after circulatory intervention in the chick embryo,” J. Exp. Biol. 208(10), 1877–1885 (2005).
[Crossref] [PubMed]

2004 (1)

S. A. de Winter, R. Hamers, M. Degertekin, K. Tanabe, P. A. Lemos, P. W. Serruys, J. R. T. C. Roelandt, and N. Bruining, “Retrospective image-based gating of intracoronary ultrasound images for improved quantitative analysis: the intelligate method,” Catheter. Cardiovasc. Interv. 61(1), 84–94 (2004).
[Crossref]

2003 (3)

V. X. D. Yang, M. L. Gordon, E. Seng-Yue, S. Lo, B. Qi, J. Pekar, A. Mok, B. C. Wilson, and I. A. Vitkin, “High speed, wide velocity dynamic range Doppler optical coherence tomography (Part II): Imaging in vivo cardiac dynamics of Xenopus laevis,” Opt. Express 11, 1650–1658 (2003).
[Crossref] [PubMed]

J. R. Hove, R. W. Köster, A. S. Forouhar, G. Acevedo-Bolton, S. E. Fraser, and M. Gharib, “Intracardiac fluid forces are an essential epigenetic factor for embryonic cardiogenesis,” Nature 421(6919), 172–177 (2003).
[Crossref] [PubMed]

J. Albers, J. M. Boese, C. F. Vahl, and S. Hagl, “In vivo validation of cardiac spiral computed tomography using retrospective gating,” Ann. Thorac. Surg. 75(3), 885–889 (2003).
[Crossref] [PubMed]

2002 (3)

T. M. Yelbuz, M. A. Choma, L. Thrane, M. L. Kirby, and J. A. Izatt, “Optical coherence tomography: a new high-resolution imaging technology to study cardiac development in chick embryos,” Circulation 106(22), 2771–2774 (2002).
[Crossref] [PubMed]

M. Kachelrieß, D.-A. Sennst, W. Maxlmoser, and W. A. Kalender, “Kymogram detection and kymogram-correlated image reconstruction from subsecond spiral computed tomography scans of the heart,” Med. Phys. 29(7), 1489–1503 (2002).
[Crossref] [PubMed]

A. F. Kopp, S. Schroeder, A. Kuettner, A. Baumbach, C. Georg, R. Kuzo, M. Heuschmid, B. Ohnesorge, K. R. Karsch, and C. D. Claussen, “Non-invasive coronary angiography with high resolution multidetector-row computed tomography. Results in 102 patients,” Eur. Heart J. 23(21), 1714–1725 (2002).
[PubMed]

2001 (2)

A. F. Kopp, S. Schroeder, A. Kuettner, M. Heuschmid, C. Georg, B. Ohnesorge, R. Kuzo, and C. D. Claussen, “Coronary arteries: retrospectively ECG-gated multi-detector row CT angiography with selective optimization of the image reconstruction window,” Radiology 221(3), 683–688 (2001).
[Crossref] [PubMed]

Y. P. P. Du, E. R. McVeigh, D. A. Bluemke, H. A. Silber, and T. K. F. Foo, “A comparison of prospective and retrospective respiratory navigator gating in 3D MR coronary angiography,” Int. J. Cardiovasc. Imaging 17(4), 287–294, discussion 295–296 (2001).
[Crossref] [PubMed]

1999 (1)

U. J. Schoepf, C. R. Becker, R. D. Bruening, T. Helmberger, A. Staebler, P. Leimeister, and M. F. Reiser, “Electrocardiographically gated thin-section CT of the lung,” Radiology 212(3), 649–654 (1999).
[PubMed]

1997 (3)

C. E. Woodhouse, W. R. Janowitz, and M. Viamonte., “Coronary arteries: retrospective cardiac gating technique to reduce cardiac motion artifact at spiral CT,” Radiology 204(2), 566–569 (1997).
[PubMed]

S. A. Boppart, G. J. Tearney, B. E. Bouma, J. F. Southern, M. E. Brezinski, and J. G. Fujimoto, “Noninvasive assessment of the developing Xenopus cardiovascular system using optical coherence tomography,” Proc. Natl. Acad. Sci. U.S.A. 94(9), 4256–4261 (1997).
[Crossref] [PubMed]

S. Yazdanfar, M. Kulkarni, and J. Izatt, “High resolution imaging of in vivo cardiac dynamics using color Doppler optical coherence tomography,” Opt. Express 1(13), 424–431 (1997).
[Crossref] [PubMed]

1996 (1)

Y. H. Gui, K. K. Linask, P. Khowsathit, and J. C. Huhta, “Doppler echocardiography of normal and abnormal embryonic mouse heart,” Pediatr. Res. 40(4), 633–642 (1996).
[Crossref] [PubMed]

1995 (1)

L. A. Taber, I. E. Lin, and E. B. Clark, “Mechanics of cardiac looping,” Dev. Dyn. 203(1), 42–50 (1995).
[Crossref] [PubMed]

Acevedo-Bolton, G.

Adler, D. C.

Albers, J.

Basavanhally, A. N.

Baumbach, A.

Becker, C. R.

Belding, J.

Bluemke, D. A.

Bodmer, R.

Boese, J. M.

Boppart, S. A.

Boudoux, C.

Bouma, B. E.

Brezinski, M. E.

Bruening, R. D.

Bruining, N.

Cable, A.

Cable, A. E.

Choma, M. A.

Chughtai, O. Q.

Clark, E. B.

L. A. Taber, I. E. Lin, and E. B. Clark, “Mechanics of cardiac looping,” Dev. Dyn. 203(1), 42–50 (1995).
[Crossref] [PubMed]

Claussen, C. D.

Davis, A.

Davis, A. M.

de Winter, S. A.

Degertekin, M.

Deng, H.

Dickinson, M. E.

Du, Y. P. P.

Efimov, I. R.

Filas, B. A.

Foo, T. K. F.

Forouhar, A. S.

Fraser, S. E.

Fujimoto, J. G.

Gargesha, M.

M. Gargesha, M. W. Jenkins, A. M. Rollins, and D. L. Wilson, “Denoising and 4D visualization of OCT images,” Opt. Express 16(16), 12313–12333 (2008).
[Crossref] [PubMed]

Georg, C.

Gharib, M.

Ghosn, M.

Gordon, M. L.

Gui, Y. H.

Y. H. Gui, K. K. Linask, P. Khowsathit, and J. C. Huhta, “Doppler echocardiography of normal and abnormal embryonic mouse heart,” Pediatr. Res. 40(4), 633–642 (1996).
[Crossref] [PubMed]

Hagl, S.

Hamers, R.

Helmberger, T.

Heuschmid, M.

Hove, J. R.

Hu, Z.

Huber, R.

Huhta, J. C.

Y. H. Gui, K. K. Linask, P. Khowsathit, and J. C. Huhta, “Doppler echocardiography of normal and abnormal embryonic mouse heart,” Pediatr. Res. 40(4), 633–642 (1996).
[Crossref] [PubMed]

Izatt, J.

Izatt, J. A.

Izatt, S. D.

Janowitz, W. R.

Jenkins, M. W.

M. Gargesha, M. W. Jenkins, A. M. Rollins, and D. L. Wilson, “Denoising and 4D visualization of OCT images,” Opt. Express 16(16), 12313–12333 (2008).
[Crossref] [PubMed]

Jiang, J.

Jiang, J. Y.

Kachelrieß, M.

Kalender, W. A.

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Supplementary Material (5)

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Fig. 1 A schematic of our image-based gating algorithm. As shown, the major steps are: (i) decimate images, (ii) determine cardiac period at each slice position, (iii) reassemble or wrap-around data to produce one correctly ordered, unevenly spaced set of images from a cardiac cycle at each slice, (iv) interpolate linearly to create even temporal spacing, (v) determine relative shift from one slice to the next (vi) determine absolute time shift at each slice, and (vii) reorder image data to create volumes over the cardiac cycle.

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Fig. 2 (A) Schematic showing proposed validation scheme. The real-time volume is captured in 103 ms, while image-based retrospective gating data represents a 4.9 ms time interval. The left part of Fig. 2(a) shows an en face real-time OCT image with the B-scan direction superimposed on the image. The right part of Fig. 2(a) shows various volumes with yellow arrows pointing to the B-scan that represents the identical phase in the cardiac cycle. (B, C) slice 31 from a real-time volume (volume scan A) and an image-based retrospective gated volume (volume 24). (D, E) slice 45 from a real-time volume (volume scan A) and an image-based retrospective gated volume (volume 24). In slice 31 the real-time volume and gated volume are not synchronized. The images are taken at different phases in the cardiac cycle as clearly seen within the yellow oval. Slice 45 in volume 24 is synchronized with slice 45 in the real-time volume as can be seen by the similarity in the 2 images.

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Fig. 3 (Media 1, 3.16 MB) A (2D + time) sagittal (en face) movie created from 4D retrospective gated OCT data of the stage 14 embryonic quail heart. The slice was chosen 313 μm deep into the tissue. With retrospective gating, cardiac systolic dynamics can be clearly visualized with a high temporal resolution in planes different from the OCT imaging plane.

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Fig. 4 4D Visualization of retrospective gated OCT data from stage 14 embryonic quail heart. Movies of 4D gated data (90 volumes in a single cardiac cycle, or ~270 volumes/s) are shown in sagittal or en face (a) (Media 2, 2.7 MB), transverse (b) (Media 3, 2.9 MB), and coronal (c) (Media 4, 3.2 MB) views. In each case, a cutaway view of the beating heart with a moving orthogonal 2D slicer in the plane of interest is shown.

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Fig. 5 (Media 5, 3.4 MB) Digitally reconstructed radiograph (DRR) volume rendering movie created using gated data from stage 14 embryonic quail heart. A summing of voxel intensities along rays cast through the volume perpendicular to the viewing plane has been used to create a 3D effect with the rotating and beating heart volume.

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Fig. 6 Validation of retrospective gating algorithm using two different reference volume scans A (left) and B (right). A linear fit is obtained through matching volume numbers in gated data as per the method described in Section 2.3. The slope of the fit obtained on gated data is 1.364 ms/slice for scan A and 1.120 ms/slice for scan B. The slope of reference scan is 1.275 ms/slice.

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Fig. 7 Validation of image-based retrospective gating using input data with different integer decimation factors. In each case, the B-scan frame was decimated by an integer number n in both x and y directions. Following decimation, the gating algorithm (Section 2.2) and the proposed validation scheme (Section 2.3) were applied. We observe that, up to a decimation factor of 8, the slope of the validation plot (slice number versus matching volume number in gated data) closely matches the ideal slope from the reference volume.

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Tables (1)

Table 1 Computation times for different steps of the image-based retrospective gating algorithm.

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Equations (6)

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τ_{i}^{'} = τ_{i} - ⌊ τ_{i} / T^{'} ⌋ T^{'}

D (z_{k}, T^{'}) = \sum_{j = 1}^{N_{τ} - 1} \sum_{m \in Z^{2}} {{| I_{m} [x_{m}, y_{m}, z_{k}, τ_{i (j + 1)}] - I_{m} [x_{m}, y_{m}, z_{k}, τ_{i (j)}] |}^{2} + {| {τ^{'}}_{j + 1} - {τ^{'}}_{j} |}^{2} / {T^{'}}^{2}}^{1 / 2}

Q_{k, k^{'}} (s) = \sum_{t = 0}^{L} R [I_{m} (x, y, z_{k}, t), I_{m} (x, y, z_{k^{'}}, t - s)],

Q_{k, k^{'}} (s) = \sum_{i = 1}^{P} \sum_{t = 0}^{L} R [I_{m} (x, y, z_{k}, t), I_{m} (x, y, z_{k^{(i)}}, t - s)],

\begin{array}{l} S_{m} = λ \\ S_{k} - S_{k^{'}} = s_{k, k^{'}}, f o r k, k^{'} = 1, 2, ..., m, ..., N_{z} a n d k < k^{'} \end{array}

\begin{array}{l} Q_{z_{m}} (t_{k}) = R [I_{r} (x, y, z_{m}, τ_{m}), I_{g} (x, y, z_{m}, t_{k})] \\ t_{z_{m}}^{'} = \arg \begin{array}{c} \max_{k = 1, 2, ... L} & Q_{z_{m}} (t_{k}) \end{array} . \end{array}

Task	Full spatial resolution (seconds)	Spatial decimation = 4 (seconds)
Estimate cardiac cycle time and “wrap around” sequence	56.767	5.6720
Interpolate data	578.688	32.655
Determine relative and absolute shifts	296.109	32.046
Rearrange full-resolution data	2648.2	2650.4