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Review of convergent beam tomography in single photon emission computed tomography
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1992 Phys. Med. Biol. 37 507
(http://iopscience.iop.org/0031-9155/37/3/002)
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Phys. Med. B i d , 1992, Val. 37, NO 3, 507-534. Printed in the UK
Review of convergent beam tomography in single photon
emission computed tomography
G T Gullbergt, G L Z e n g t , F L Datzz, P E Christian$, C-H Tung? and
H T Morgan§
t Department of Radiology, Medical Imaging Research Laboratory, University of Utah,
Salt Lake City, UT 84132, USA
i Department of Radiology, Division o f Nuclear Medicine, University of Utah, Salt Lake
City, UT 84132, USA
P Nuclear Engineering, Ohio Imaging o f Picker International, Bedford Heights, O H 44128,
USA
Received 4 November 1991
Abstract. Investigation of convergent-beam single photon emission computed tomography
(SPECT) is actively being pursued to e v a l ~ a t eits clinical potential. Fan-beam, cane-beam,
pin-hole and astigmatic collimators are being used with rotating gamma cameras h a h g
large crystal areas, to increase the sensitivity for emission and transmission computed
tomography of small organs such as t h e thyroid, brain or heart. With new multi-detector
SPECr systems, convergent-beam geometry offers the ability to simultaneously obtain
emission and transmission data necessary to quantify uptake of radiopharmaceutical
distributions in the hean. The development of convergent-beam geometry in SPECI requires
the integration of hardware and software. In considering hardware, the optimum detector
system for cone-beam tomography is a system that satisfies the data sufficiency condition
for which the scanning trajectory intersects any plane passing through the reconstructed
region of interest. However, the major development of algorithms has been for the data
insufficient case of single planar orbit acquisitions. The development o f these algorithms
have made possible the preliminary evaluation of this technology and the imaging of brain
and heart are showing significant potential for the clinical application of cone-beam
tomography. Presently, significant research activity is pursuing the development of
algorithms for data acquisitions that satisfy the data sufficiency condition and that can be
implemented easily and inexpensively on clinical SPECT systems.
1. Introduction
I n industrial a n d medical applications convergent-beam geometry (figure I ) has become
a n important tomographic technique that is useful in obtaining solutions of inverses
problems with better resolution, sensitivity a n d faster data acquisition times than with
parallel geometry. Jaszczak er a / (1979) originally applied convergent-beam geometry
to increase the sensitivity for single photon emission computed tomography (SPECT)
of the brain by utilizing a multi-slice, short-bore fan-beam collimator that focuses to
a line oriented parallel to the axis-of-rotation of a rotating gamma camera. This was
followed by a specially designed, long-bore fan-beam collimator for imaging the brain
that would allow the face of the collimator to approach near the head and still allow
the camera to clear the patient's shoulders (Tsui er a/ 1986). Later, Jaszczak and
coworkers (Jaszczak er a / 1986, 1988a, b, Floyd et a / 1986a) implemented cone-beam
collimati,v to further optimize the sensitivity of rotating gamma cameras for brain
0031-9155/92/030507+28$04.50
0 1992 IOP Publishing Ltd
507
508
G T Gullberg et ai
.... ._._
0
... -.
..
Figure 1. The purpose o f using fan- and cone-beam collimators is to utilize more of the
available cry~talarea.
SPECT. This technology was applied by our group to obtain the first cone-beam
tomography images of the heart (Gullberg et al 1991a, b). More recent developments
of multi-detector SPECT systems offer an additional increase in sensitivity by combining
converging collimation with multiple detectors. Also, these new multi-detector SPECT
systems offer the ability to simultaneously obtain emission and transmission data
necessary to correct for the variable attenuation in the chest in reconstructing the
uptake of radiopharmaceutical distributions in the heart (Tung et a/ 1991a).
To fully develop convergent-beam tomography, as well as to identify its strengths
and potential practical applications, requires research which simultaneously develops
and optimizes hoth hardware a n d software. For certain applications, algorithms can
produce good image quality and quantitative information in spite of hardware limitations that are imposed by the physics of the imaging process or that are due to design
trade-offsinfluenced by cost considerations. In some cases new algorithms are needed
because of scanning geometries that are used to improve the physical properties of
the imaging detection process such as the use of body contouring orbits to minimize
the fall-off of the geometric point response with distance from the collimator face, or
‘short-scan’ acquisitions to minimize attenuation artifacts. In other cases algorithms
are able to minimize reconstruction artifacts for those SPECT systems which, because
of cost constraints, cannot acquire sufficient data for cone-beam tomography. Depending upon the SPECT system, certain potential exists to develop algorithms that can
produce good image quality and quantitative information in spite of hardware limitations and trade-off considerations.
This paper reviews recent developments in hardware and algorithms in the application of convergent-beam tomography to clinical nuclear medicine. Phantom and patient
Convergent beam tomography
509
results are presented showing the potential for the use of this technology in cardiac
and brain imaging.
2. Hardware considerations
2.1. Roraling gamma camera
Today, the rotating scintillation camera is used more widely than fixed-ring systems
because of its ability to perform both body (figure 2) and brain (figure 3) SPECT as
well as conventional planar imaging. Most of the SPECT systems are single detector
Figure 2. Cone-beam collimation increases sensilivity for heart imaging. Single Circular
orbit scans do not satisfy the dava sufficiency condition but the reconstructions are of good
image quality because the heart can be positioned near the central plane of the cone heam
geometry.
Figure 3. Tilted cone beam tomography of the brain can give high-resolution transaxial
images. However, elongation near the top of the brain i s seen in sagittal and coronal images
because the titled cone-beam geometry places these structures further from the central
plane of the cone-beam geometry.
510
G T Gullberg
et
a/
but recently systems with three (figure 4) and four rotating detectors have been
commercially made available. Several manufacturers are also selling large field-of-view
dual camera systems that offer whole-body imaging and computed tomography capability. With convergent-beam collimators, these systems offer an advantage of a significant
increase in sensitivity for imaging small organs because of the large field-of-view of
up to 60 cm compared with the standard 40 cm detectors.
The new three-detector SPECT systems (figure 4) offera unique detector arrangement,
that lends itself to easily acquiring simultaneous transmission and emission computed
tomography studies, and make it possible to obtain a measure of the distribution of
attenuation coefficients in a n acceptable clinical imaging time. A transmission source
with either a different energy than or the same energy as the emission source is
positioned at the focal line or focal point, depending upon the collimator, opposite
one of the detectors. This detector acquires both transmission and emission data while
the other two detectors acquire only emission data. The projection data are preprocessed to subtract out cross-talk between the transmission and emission data sets.
Fan-beam and cone-beam collimation improves the sensitivity for both emission and
transmission studies and significantly reduces the source strengths required for the
transmission scans. The combination of multiple detectors with converging collimators
offers for the first time a reasonable approach to correcting for the variable attenuation
in the thorax which ha3 coristaiitly plagued cardiac SPECT imaging.
2.2. Collimarion
Several different convergent-beam collimator geometries have been used in SPECT with
rotating gamma cameras. These include fan-beam (figure l), cone-beam (figures 2 and
3), pin-hole (Palmer and Wollmer 1990) and astigmatic (figure 5 ) geometries with the
envelope of rays arranged symmetrically or asymmetrically (Jaszczak et a / 1986) about
TRANSM~SSIONSOURCE
/
Figure 4. Multi-detector SPECT systems increase the sensitivity and also offer the potential
to simultaneously collect emission and transmission data. The detector opposite the line
source collects both transmission and emission data, whereas the other two heads simultaneously collect only emission data.
Figure 5. The astigmatic geometry has two lines o f focus. One has a focal length of F,
and the other has a focal length of F2.
a central axis that is oriented either orthogonal (figure 2) or oblique (figure 3 (see also
Manglos 1989, Manglos e t a / 1989b)) to the axis of rotation. I n particular, the astigmatic
collimator (Hawman and Hsieh 1986) is a generalization of the fan- and cone-beam
collimators with two orthogonal lines of focus at different focal lengths. One focal line
is aligned parallel and the other is aligned perpendicular to the axis of rotation. A
fan-beam collimator is equivalent to placing the perpendicular line of focus at a distance
far from the face of the collimator, whereas a cone-beam collimator is an astigmatic
collimator with both lines of focus at the same focal length. As both focal lengths
increase, the astigmatic collimator approaches that of a parallel collimator.
Simulations of the Defrise phantom (Defrise 1989) in figure 6 are shown in figure
7 comparing cone-beam, fan-beam, parallel and astigmatic geometries. A generalized
astigmatic reconstruction algorithm was applied to the simulated projection data
generated over one complete rotation about the axis of rotation in figure 5. The
astigmatic reconstruction algorithm uses the same ramp filtering in the transaxial
direction as the Feldkamp cone-beam reconstruction algorithm (Feldkamp el a1 1984)
but backprojects along an astigmatic geometry. In each simulation this algorithm was
applied by setting the focal lengths appropriate for the astigmatic, fan-beam ( F , = 112,
F2 = 9999), cone-beam (F,= 112, F2 = 112) and parallel geometries ( F , = 9999,
F2 = 9999), where all dimensions are in units of vonels equal tn 0.6 cm.
The simulations (figure 7) illustrate that the cone-beam reconstruction has more
slice-to-slice crosstalk in the sagittal reconstructions of the Defrise phantom than the
fan-beam and parallel reconstructions (see also Zeng and Gullberg 1990a). The parallel
and fan-beam geometries give the best results and are expected to be equivalent since
sufficient data are acquired in one circular planar orbit for these geometries. A
comparison of the astigmatic (F,= 112, Fz = 72) and cone-beam reconstructions incicates that an astigmatic geometry with the focal line Fl equal tn the cone-beam focal
512
G T Gullberg et al
Figure 6. Defrise phantom C O ~ S ~ of
S ~ seven
S
parallel ellipsoidal discs of identical uniform
intensity equally spaced at 5 voxels apart parallel to the axis of rotation.
Figure 7. Sagittal reconrtructions of the Defrise phantom for ( 0 ) cone-beam ( F , = 112,
F,=112), ( b ) astigmatic (F,=112, F,=72). ( c ) astigmatic ( F , = 1 1 2 , F,=1521, ( d l fanbeam ( F l = I n , F,=9999), (<) parallel (F,=9999, F,=9999), and ( f )rotated fan-beam
( F , = 9999, Fz = 112) geometries.
Convergent beam tomography
513
length and the focal line F2 less than the cone-beam focal length will give poor results.
However, an astigmatic geometry ( F , = 112, F2= 152) with Fz longer than the conebeam focal length gives better results because it more closely approximates a fan-beam
geometry. Also, we see that the rotated fan-beam geometry gives the worst results of
all the simulations.
2.3. Orbits
Cone-beam collimation has the hest sensitivity for large crystals but, as illustrated in
the simulations in figure I , does not produce artifact-free reconstructions using a
scanning trajectory whose focal point remains in a plane. Presently, all SPECT systems
acquire data by scanning only in single planar orbits where the focal point traverses
either circular or elliptical curves lying within a single plane (figure 8 ( a ) ) . These data
acquisitions have been shown to give insufficient projection data (Tuy 1983, Smith
1985). The reconstruction, using the Feldkamp algorithm (Feldkamp et a/ 19841, is an
approximation except for the central plane which is reconstructed accurately (figure
7). Preliminary results of cone-beam tomography of the heart (Gullberg et a / 1991a, b)
look promising when the heart is positioned near the central plane of the cone-heam
geometry and demonstrate improved lesion detection. These conclusions are based on
a limited number of clinical studies and subjective evaluation b y clinical investigators.
For cone-beam tomography, we know that a detector system will acquire sufficient
data (Tuy 1983, Smith 1985) if the scanning trajectory of the focal point has at least
Figure 8. Data sufficiency condition. ((I) One circular planar orbit does not satisfy the
data sufficiency condition since a plane can be bound that passes through the object but
does not intersect the orbit. However, ( b ) a circle-and-line orbit, ( c ) a dual orthogonal
orbit, and ( d ) a helical orbit do satisfy the data sufficiencycondition.
5 14
G T Gullberg et a1
Figure 9. The circle-and-line orbit illustrated in figure 8 ( b ) can be implemented by first
rotating the camera with the gantry fired (motion I ) and then fixing the detector with
respect to the gantry and performing a linear translation of the gantry (motion 2).
I
,’
,’
Figure 10. Illustration of the application of a dual anhogonal orhit in figure 8 ( c ) lo
cone-beam tomography of the brain.
Convergent beam tomography
515
one point of intersection for any plane passing through the reconstructed region of
interest. In figure 8(a) we see that a planar circular orbit does not satisfy this data
sufficiency condition since there is a plane that passes through the object hut does not
intersect the orbit. However, non-planar orbits such as a circle-and-line orbit (figure
8(b)), a dual orthogonal orbit (figure S(c)) and a helical orbit (figure 8(d)) d o satisfy
the data sufficiency requirement.
These orbits could he implemented to satisfy the data sufficiency condition with
present SPECT systems at little additional cost. Figure 9 illustrates the implementation
of the circle-and-line orbit using a rectangular detector that first rotates through a
planar orbit while the gantry remains stationary and then the gantry performs one
linear translation while the detector remains fixed with respect to the gantry. A helical
orbit can easily be implemented by translating the gantry while simultaneously rotating
the detector. The dual orthogonal orbit would be particularly applicable for brain
imaging and could be implemented using one transaxial orbit and one half orbit that
scans over the top of the head as illustrated in figure 10. In looking at ways to improve
cone-beam tomography, we need to consider the gain obtained in image quality with
sufficient angular sampling versus the cost of additional necessary hardware.
3. Algorithm considerations
The application of various converging collimator geometries in clinical SPECT has
required the development of new algorithms to solve several technical problems. Here
we review the development of these algorithms that have made possible the clinical
implementation of convergent-beam tomography using present clinical SPECT systems.
3.1. Nun-circular filtered backprojection cone-beam reconstruction algorithm
During a patient scan the camera rotates in a non-circular orbit, with the centre of the
camera always pointing to the centre of rotation, and closely follows the patient’s body
contour to minimize the fall-off in resolution with distance from the collimator. New
fan-beam (Weinstein 1980) and c0n.e-beam (Gullberg et a1 1991b, Gullberg and Zeng
1992) reconstruction algorithms were developed to reconstruct projections from noncircular detector orbits. A cone-beam reconstruction algorithm (Gullberg et a / 1991b,
Gullberg and Zeng 1992) was derived using the non-circular fan-beam algorithm
developed by Weinstein (1980) and extending it to cone-beam geometry using the
method of Feldkamp et a / (1984) to derive the cone-beam reconstruction algorithm
for nun-circular planar orbits.
3.2. ‘Short-scan’filtered backprojection reconsrruction algorithm
In the application of SPECT to cardiac imaging, it was found that the reconstruction
of the posterior projections of emissions from a radiopharmaceutical localized in the
heart resulted in reconstruction artifacts (Coleman et a/ 1982, Tamaki el a/ 1982, Go
et a/ 1985, Eisner et a1 1986, Tsui et a/ 1989b). Therefore, an angular sampling range
is usually chosen so as not to acquire the highly attenuated posterior projections. For
fan- and cone-beam geometries new ‘short-scan’ reconstruction algorithms (Zeng and
516
G T Gullberg et al
Gullberg 1990b, 1991, Gullberg and Zeng 1992) were developed for less than 360",
non-circular detector obit acquisitions. The algorithms require that data are sampled
over an angular range of 180" plus the fan or cone angle plus an additional small
angle. For these conditions the algorithms accurately reconstruct fan-beam geometry
but can only accurately reconstruct the central plane of a cone-beam projection
acquisition since the reconstruction of the midplane is equivalent to the reconstruction
of a fan-beam data set. Therefore, short-scan cone-beam reconstructions of the midplane should be equivalent to full-scan cone-beam and parallel reconstructions if for
the short-scan acquisitions the central slice is sampled over a sufficient angular range
for fan-beam projections.
The computer-generated phantom in figure 11 is used here to analyse the short-scan
cone-beam reconstruction algorithm (Gullberg and Zeng 1992). Simulations in figure
12 compare reconstructions of parallel projections (128 projections over 360") with
reconstructions of full-scan circular (128 projections), short-scan circular (83 projections) and short-scan non-circular (83 projections) cone-beam projection data sets.
The projections ofthe source distribution were formed assuming no photon attenuation,
thus focusing on the differences between short- and full-scan reconstructions. The
parallel reconstruction is used as a standard for evaluating the reconstruction artifacts
associated with single planar orbit cone-heam tomography.
Figure I I . The ellipsoidal heart-lung simulation phantom. The following source densities,
p, were used: 1 (ventricular wall). p = I; 2 (intraventricular blood pool), p =O.Z; 3 (lungs),
p = 0.2: 4 (tissue), p = 0.2. The unit vuxd equals 0.6 cm. A defect ( p = 0) is illustrated in
the wall of the ventricle.
From the profiles, one sees that the reconstructions of the midplane (transverse
slice 32) are equivalent, as one would expect, for all reconstructions. The cone-beam
reconstructions of transaxial slices off the central slice are approximations due to the
nature of the incomplete data sampling. However, one observes from the profiles in
figure 12 that for a short distance off the midplane, the cone-beam full- and short-scan
Convergent beam tomography
517
Figure 12. Reconstructions of unattenuated projections of the simulated heart-lung phantom shown in fgure I I . Profiles compare the quantitative accuracy between parallel
geometry, cone-beam circular 'full-scan', circular 'shon-scan'. and non-circular 'short-scan'
reconstructions.
518
G T Gullberg et a1
reconstructions appear to be equivalent with the parallel reconstruction. Differences
become more apparent for slices further from the midplane.
These differences are better seen in coronal views such as the coronal slice shown
in figure 12 which is a distance of 10.5 voxels anterior to a central coronal slice. The
profile for the parallel reconstruction has a higher amplitude than the profiles for the
full- and short-scan cone-beam reconstructions. If we assume that the parallel reconstruction is accurate, one would infer that the differences in the profiles are due to the
insufficient sampling of the cone-beam projections. Differences between short- and
full-scan reconstructions are more apparent in background tissue where low-amplitude
artifacts are seen extending horizontally near the superior and inferior extensions of
the heart. Also, the cone-beam reconstructions show some additional low amplitude
artifacts extending vertically along the axis at the periphery ofthe simulated background
distribution. This is best appreciated in the short-scan reconstructions which exhibit
a broadening of the profile at the boundary as compared with the full-scan and parallel
reconstructions. It appears that the combination of the approximation of the cone-beam
reconstruction algorithm for non-central slice voxels and the short-scan data set together
result in artifacts that are seen in short-scan cone-beam reconstructions but are not
seen in full-scan reconstructions.
3.3. Truncated projecrion correction algorithms
. .
The magnification of a converging collimator can ircncatc both t:ensz?:ss:o~ 2nd
emission projection data. In some emission projections, the magnification can place
the radiopharmaceutical distribition in the tissue surrounding the organ of interest
outside the field-of-view. The application of the reconstruction filter to these truncated
projections produces spikes at the truncation edge in the filtered projections (figure
10.0
-10.0
1
0.0
20.0
40.0
60.0
Figure 13. A truncated projection profile of one sampled projection is shown I- - -) .
After filtering with a ramp filter the profile I---)
has a spike at the truncation edge.
To eliminate the ring artifact in the reconstruction caused by the spike, the filtered
projections are multiplied by a window function I--)
to give the result shown with the
full curve which is backprojected to form the reconstruction.
Convergent beam tomography
519
13), resulting in ring artifacts in the reconstruction (figure 14(a)). Clipping the truncation spikes after filtering and before backprojecting the filtered projections works well
to eliminate the reconstruction ring artifacts (figure 14( b)) if, such as in cardiac SPECT,
only the background distribution is truncated and is low compared to the organ of
interest (Zeng er a1 1990).
For transmission computed tomography (Malko er a1 1986, Tsui et a/ 1989a,
Manglos et nl 1990), which is used to obtain an attenuation map for iterative attenunation-correction reconstruction algorithms (see section 3.4), the limited field-of-view of
most SPECT detector systems makes it difficult to acquire transmission data of the
thorax with converging collimators without truncating the projections in most of the
projection views (Manglos er a/ 1990). Our approach (Tung et nl 1991b) has been to
reconstruct the attenuation distribution using an iterative E M reconstruction algorithm
(Lang and Carson 1984). The iterative algorithm solves an underdetermined system
of linear equations corresponding to only those projections that are measured. The
attenuation factors which are exponentials of partial line intergrals of the attenuation
distribution are calculated from the attenuation distribution reconstructed from the
truncated transmission projections.
Simulations in figure 15 show that even though the transmission image is distorted
at the boundary the attenuation correction method produces quantitative emission
reconstructions. Our hypothesis is that even though the transmission image is significantly distorted, the attenuation factors are measured accurately enough for those
attenuation factors that have the greatest influence upon the emission measurements
and, thus, are useful in producing quantitative emission reconstructions. The reconstruction of the truncated projections into an array larger than the size of the nontruncated portion of the projection, with the constraint that the support be equal to
the outer boundary of the object, significantly improves the image quality of the
transmission image and gives the best accuracy in the attenuation-correction reconstruction. However, we have seen from simulations that in general this is only somewhat
. .
Figure 14. Reconstmclions of truncated projections ( a ) before and ( b ) after applying the
window funclion illustrated in figure 13 to eliminate the truncation spikes. The transaxial
slices o f a 2"'n cardiac patient study were reconstructed from a 'short-scan' data set of 16
projections obtained using the 50cm focal length cone-beam collimator on the SX-300
SPECT system.
520
G T Gullberg et ~l
Figure 15. Simulation of attenuation correction using a truncated transmission scan.
Transmission images were reconstructed from a set of 120 fan-beam projections sampled
over 360' using 25 iterations ofthe EM algorithm. The focal length of the fan-beam geometry
was 70 pixels ( 1 pixel = 0.72 cm) and the elliptical phantom had an outer boundary with
major and minor axes of 50 and 30 pixels respectively. The transmission and emission
projections were truncated symmetrically by 20 pixels a n each side of each projection.
Firstly, the truncated transmission projections were reconstructed into a 24 x 24 array equal
to the dimensions of the non-truncated projections. Secondly, the truncated projections
were reconstructed into a 64 X 6 4 array. Thirdly, the truncated transmission projections
were reconstructed into a 6 4 x 64 array but the reconstruction was restricted to an elliptical
support equal to the outer boundary of the phantom. The emission images were reconstructed using 25 iterations of the EM algorithm. The attenuation factors for the emission
reconstruction were calculated from each of the three reconstructed attenuation distributions for the cases described above.
I
Convergent beam tomography
521
better than using the transmission reconstruction without support even if it is significantly distorted and in particular from the simulations in figure 15 it is difficult to
see any difference.
3.4. Iterative attenuation and geometric point response correction algorithms
I
Iterative reconstruction algorithms are computationally more intensive, especially for
cone-beam geometry (Manglos 1989, Manglos et al 1989a), than filtered backprojection
reconstruction algorithms but have the advantage of being able to accurately compensate for physical factors of attenuation (Gullberg et al 1989, Manglos et al 1991)
and a spatially varying geometric point response (Floyd et al 1986b, Tsui et al 1988,
Formiconi et al 1989, Zeng et al 1991, Zeng and Gullberg 1992b, Liang et a / 1992).
In particular, it is recognized that photon attenuation within the body is a major factor
contributing to the quantitative inaccuracy using SPECT to measure the in vivo distribution of radioactivity (Gullberg et al 1982, Lewis et al 1982, Webb et al 1983, Malko
et al 1986, Tsui et a/ 1989a). It has been shown that iterative attenuation correction
algorithms which model the attenuation process using a measured attenuation distribution of the thorax more accurately quantify the source distribution in the heart (Tsui
et al 1989a) than the more efficient filtered backprojection algorithms which use
pre-processing or post-processing attenuation correction techniques based upon either
an assumed constant attenuator or a measurement of the variable attenuator. The
disadvantage with single-detector SPECT systems has been the necessity to either acquire
simultaneous transmission and emision data (Bailey et a / 1987, Frey et al 1992), thus
contaminating either transmission or emission data because of cross-talk between
transmission and emission energy windows, or acquire separate emission and transmission scans thus doubling the patient imaging time.
The new multi-detector systems (see 5 2.1) offer the ability to simultaneously acquire
transmission and emission data (Tung ef al 1991a), thus making it feasible to acquire
the data necessary for variable attenuation correction in an acceptable clinical imaging
time. A transmission line source is positioned opposite one of the detectors (figure 4,
detector 3) of a multi-detector, fan-beam collimated SPECT system. Transmission and
emission data are acquired in one detector ( 3 ) while the other two detectors (1 and
2) simultaneously acquire emission data. Cross-talk from one source into the energy
window of the other source for the detector acquiring simultaneous emission and
transmission data is removed based upon separate measurements of scatter; one using
only the transmission source and one using only the emission source. For the data
obtained from the detector acquiring simultaneous transmission and emission data,
the fraction of cross-talk is removed from the projections before reconstruction. For
the detectors acquiring only emission data the cross-talk is less but is also measured
and subtracted out. If "Tc" is used as the transmission source and *'-TI as the emission
source, an estimate of the high-energy 20'TI photons that will appear in the 99Tc"
window of the detector for the simultaneous transmission and emission data is obtained
from measurements of the 99Tc" window of the other two detectors during the scan.
If "Tcm is used for both emission and transmission sources, the transmission data are
obtained from the detector obtaining the simultaneous transmission and emission data
by subtracting estimates of the emisssion data obtained from the other two detectors.
A simultaneous transmission and emission experiment was performed using the
Jaszczak elliptical phantom with cardiac insert shown in figure 16(a). The transmission
522
G T Gullberg et a/
Figure 16. ((I) Experimental set-up of the simultaneous emission and transmission phantom
experiment on the PRISM 3000. The transmission line source filled withqQTcc"is attached
to the detector so that it remains at the focal line as the detector follows a non-circular
orbit. The hean insert and background of the elliptical Iaszcrak phantom were filled with
201
TI. ( b ) Results of short axis reconstructions without and with attenuation correction
using 25 iterations of EM algorithm and attenuation factors calculated from the reconstructed attenuation distribution (not shown). T h e emission data from detectors I and 2 in
figure 4 were used in the reconstruction.
Convergent beam tomography
523
line source was filled with 7 . 4 lo8
~ Bq (20 mCi) of 99Tc", the wall and chamber of the
cardiac insert were filled with 4 . 4 4 lo6
~ Bq (120 pCi) and 1.14 x lo7 Bq (309 pCi) of
201
TI respectively and the background in the outer elliptical chamber was filled with
1.14xtO'Bq (309 pCi) of 201Tl,
Both the transmission (detector 3) and emission
(detectors 1 and,2) projection data were reconstructed from 120 projections sampled
over 360"using 30 iterations of the EM algorithm. The results in figure 16(b) demonstrate
an improvement in the uniform distribution of the in vivo radionuclide concentration
for the attenuation-corrected reconstruction.
In addition to attenuation it is also recognized that the spatially varying point
response of a gamma camera due to the collimator geometric response and scatter
results in loss of contrast and reconstruction artifacts that are seen as shape distortions
and density non-uniformity. Models for photon attenuation (Gullberg et a / 1989,
Manglos el a / 1991), the three-dimensional spatially varying geometric point response
(Tsui and Gullberg 1990) and scatter (Liang et al 1992) are incorporated into projection
and backprojection operations that are used in iterative reconstruction algorithms to
correct for these physical effects. In one approach, the space varying point response
can be modelled as a three-dimensional 'inverse cone' of rays emanating from each
detector bin (Zeng et a / 1991), which tramforms even parallel tomography into a
cone-beam reconstruction problem. The projection and backprojection operations sum
along each ray weighting the contribution to the sum by the geometric response and
attenuation factors that are calculated during the projection and backprojection summation along each ray. Another approach (Zeng and Gullberg 1992b, Liang el al 1992)
which is computation'ally more efficient is to use the shift invariant property of the
collimator geometric response at a fixed distance from the detector. In this model the
attenuation factor is calculated along the central ray of the point response for a sampled
bin as a function of the attenuating material between the point of interest and the
detector, and the attenuation is assumed to be constant over the spread of the point
response for that bin at a fixed distance from the collimator face.
The result of correcting for the spatially varying geometric point response in a
cone-heam tomography study of a patient heart is shown in figure 17. The corrected
images show a slight narrowing in the wall thickness of the left ventricle. In the
projection and backprojection operations the three-dimensional 'inverse-cone' structure
was used to model only the geometric response of the collimator and no attenuation
or scatter correction was implemented. Therefore, it is difficult to interpret the results
considering the complication of potential reconstructed attenuation artifacts being
present. To evaluate only the geometric response correction, a phantom study was
performed using the Jaszczak heart phantom without filling the outer cylindrical portion
with water. This minimized the attenuation in the projections of the source distributed
in the cardiac insert, thus allowing us to focus upon and to evaluate the point response
correction. The projections were obtained using a fan-heam collimator (50 cm focal
length) on the PRISM 3000 SPECT system. The results in figure 18 show significant
narrowing of the simulated heart wall with the geometric response correction.
3.5. Non-planar cone-beam reconstrucfion algorithms
As discussed in section 2.3, in cone-beam SPECT it is common practice to acquire scans
tranversing a single circular or non-circular planar orbit (figure 8 ( a ) )and reconstruct
the projection data using a Feldkamp-type reconstruction algorithm. These orbits do
524
G T Gullberg et al
Figure 17. Correction for the spatially varying geometric point response in a cone-beam
tomography study or a patient heart. An iterative EM algorithm was used to r e ~ o n s t r ~ ~ t
16 projections sampled over an angular range of approximately 214". The projections were
acquired on the SX-300 SPECT system using the 50 Cm racal length cane-beam collimator.
The images were reconstructed without (an the left) and with (on the right) modelling far
the geometric response in the projection and backprojeclion operations. Modelling of
photon attenuation and scatter was not included.
not satisfy the data sufficiency condition since one can find planes that pass through
the reconstructed region-of-interest, parallel to, but not intersecting, the plane of the
orbit. Therefore, reconstruction artifacts can result and these artifacts due to asymmetric
reconstructed point responses have to be weighed against the advantage gained with
increased sensitivity of cone-beam over parallel and fan-beam collimation.
Recognizing the potential problems with insufficient sampling of cone-beam projections, research is presently pursuing the use of non-planar orbit trajectories such as
those shown in figures 8, 9 and 10 that would give sufficient angular sampling and
could he implemented with minor modifications to present SPECT systems. The conebeam reconstruction algorithm developed by Grangeat (1987) can be used to reconstruct
cone-beam projections acquired from almost all orbit trajectories. However, the one
drawback of the approach is that it involves the sorting (or rebinning) of the cone-beam
data. We emphasize that even if the data sufficiency condition is satisfied, an efficient
convolution hackprojection reconstruction algorithm is not guaranteed. Two orbits for
which efficient reconstruction algorithms have been derived are dual orthogonal (Clack
et a/ 1991) and circle-and-line (figure 8 ( b ) ) orbits. Note that the orbit in figure S ( c )
has one complete orbit and one orthogonal half orbit which in practice is necessary
because one cannot pass the detector through the body. An algorithm for this orbit
has not been developed. The algorithm developed by Clack et al(l991) was developed
for two complete orthogonal orbits and approximates the cone-beam projections as
Radon planar integrals. The circle-and-line algorithm (Zeng and Gullberg 1992a) uses
a modification of Smith's algorithm (Smith 1990). The development of efficient filtered
backprojection reconstruction algorithms for other useful detector trajectories, such
Convergent beam tomography
525
~
~~~
Figure 1X. A phantom study was performed lo evaluate the ability to correct for a \pace
varying geometric response of a fan-beam collimator (50cm focal length) on PRISM.
Jasrczak hean phantom without water attenuator was used. Results of filtered backprojection algorithm (FBP), E M without and with geometric response corrrection. The results
were reconstructed from 120 projections over an angular range of360'. Modelling of photon
attenuation and scatter was not included.
as the helical orbit (figure 8 ( d ) ) ,is presently an area of research which has the potential
of significant future benefits.
Results in figure 19 compare the Feldkamp and iterative EM reconstruction of
cone-beam projections acquired using a planar circular orbit, with the reconstruction
of cone-beam projections acquired using dual orthogonal and circle-and-line orbits.
The sagittal reconstructions of the Defrise phantom are superior for the orbits which
satisfy the data sufficiency condition. However, notice that the iterative EM reconstruction does show an improvement over the Feldkamp reconstruction for a planar circular
orbit.
3.6. Geometrical parameter estimation algorithms
In addition to the need for special reconstruction algorithms for converging geometries,
it is important for good tomographic image quality to be able to measure precisely the
geometric parameters of the physical imaging system so as to accurately orient in space
collimator, camera and rotating gantry relative to the centre of rotation. A method was
developed to estimate the geometrical parameters for fan-beam (Gullberg et al 1987)
and cone-beam (Gullberg et al 1990) detector geometries from measured coordinates
of the centroid of a projected point source sampled over 360". Non-linear expressions
526
G T Gullberg et al
Figure 19. Comparisons of c o n e ~ h e a mreconstructions of the Defrise phantom for planar
and non-planar orhit acquisitions: ( a ) ideal sagittal slice: ( b ) Feldkamp reconstmetion of
a simulated single planar orbit; ( c ) iterative EM reconstruction o f a simulated single planar
orbit; ( d ) dual orthogonal orbit reconstruction (Clack el a1 1991); ( e ) circle-and-line orbit
reconstmaion (Zeng and Gullberg 1992a).
are derived for the coordinates of the centroids in terms of the geometrical parameters.
The Marquardt algorithm is used to determine the geometric parameters by fitting the
measured centroid data to the non-linear expressions of the geometric parameters. The
method is sensitive to systematic errors and is also useful to evaluate the accuracy of
the collimator construction.
3.7. Variance reconstnrction algorithm
An important aspect in evaluating the clinical efficacy of cone-beam tomography is to
determine the specificity and sensitivity of the technique which is most accurately
ascertained through very time consuming observer studies. As an initial attempt to at
least evaluate contrast resolution, an algorithm (Hahn et al 1988) was developed to
estimate the variance of the estimated intensity for each pixel reconstructed using the
Feldkamp algorithm. The estimated variance measures precession and thus is useful
to evaluate the minimal detectable contrast resolution. The approach developed by
Hahn et al (1988) can also be used to obtain covariance estimates which are useful
to evaluate the texture in the cone-beam reconstructions. We have also found that the
spatial distribution of variance estimates is useful to evaluate the asymmetry of the
reconstructed point response for single planar orbit cone-beam tomography.
Convergent beam tomography
521
4. Results
The development of hardware and algorithms has made possible the clinical evaluation
of converging-beam tomography. Preliminary results of phantom and patient studies
are presented here. These studies were performed using the collimators listed in table
1. The data were acquired from single planar orbits on the SX-300 and PRISM 3000
SPECT systems (Ohio Imaging of Picker International, Bedford Heights, Ohio).
Table 1. Collimators used in the phantom and patient studies. The hole length is for the
central hale of the converging collimators, the hole diameter is the measurement at the
centre of the tapered hole, and the focal length is measured from the front face of the
collimator. The tabulated resolution ( F W H M ) is the total system resolution (intrinsic plus
collimator).
Resolution
at IO cm
(mm1
Cone beam (SX-300)
Cone beam (SX-300)
Fan beam (SX-300)
Fan beam (PRISM 3000)
Parallel (SX-300)
Parallel (PRISM 3000)
6.0
6.7
11.4
7.8
9.1
7.8
Sensitivity
(counts mi"-'
+Ci-')"
Hale
Hole
diameter
length
Focal
length
(mm1
(cm)
(cm)
270
252
243
300
350
245
1.89
1.55
6.0
1.4
1.6
I.4
4.0
3.3
13.0
2.7
2.54
2.7
50
60
58
50
m
m
1 Ci = 3.7x IO" Bq.
4.1. Phantom study
A study was performed on the SX-300 SPECT system using the 20 cm circular Jaszczak
heart phantom. Experimental results in figure 20 show parallel and cone-beam reconstructions of transaxial and short axis slices through the cardiac insert. The LEGP
parallel hole collimator was used for the parallel study and the 60 cm focal length
cone-heam collimator was used for the cone-beam study which differsfrom the studies
by Gullherg et al (1991a) where a 50 cm focal length cone-beam collimator was used.
The cone-beam results were obtained using the full-scan cone-beam reconstruction
algorithm (Feldkamp et a / 1984) to reconstruct 128 projections (-2.8" between projections) sampled over 360". There was no pre- or post-filtering applied for noise suppression. The central slice of the cone-beam geometry which is equivalent to a fan-beam
projection data set was summed over 128 projection angles to give 56 713 counts. The
parallel-hole collimator results were reconstructed from 128 projections sampled over
360' of which a typical slice through the heart has a total of 126 597 counts summed
over 128 projections. Also, for the parallel reconstructions no filtering for noise was
used.
In comparing the results, we attempted to select anatomically the same transaxial
and short axis slices for each study. The improved resolution for the cone-beam
collimator is evidenced by the thinner region of activity. The wall thickness was
measured for the short axis slice shown in figure 20 to be 1.46 cm for the cone-beam
528
G T Gullberg et a1
Figure 20. Experimental results or the 20 cm circular Jaezczak phantom showing parallel
and cone-beam reconstructions of transaxial and shan axis slices through the cardiac insert.
The parallel study used a parallel-hole collimator on the SX-300SPECT system to obtain
128 projections over 360" of approximately 126597 total counts for a central transaxial
slice through the cardiac insert. The cane-beam study used the 60 cm focal length cone-beam
collimator on the SX-300 SPECT system to obtain 128 projections of 56713 Counts for the
central cone-beam transaxial section.
results and 2.36 cm for the LEGP parallel-hole collimator results which compares to
a known thickness of 1 cm. For the LEGP parallel hole collimator, the %RMSerror
was 14.62% (u=28.19*2.81, p = 192.88i3.99) as compared to 18.12% (u=32.96*
2.85, p = 181.88i4.03) for the cone-beam results.
The difference in reconstructed resolution between the cone-beam and LEGP
parallel-hole collimators correlates with a system resolution for the cone-beam collimator of 6.1 m m FWHM at 10 cm and for the LEGP parallel-hole collimator of 9.1 mm
FWHM at 10 cm. The differencein the %RMSerror is attributed to differences in sensitivity
(sensitivity for the LEGP parallel-hole collimator of 1.39 times that of the cone-beam
collimator) and differences in the slice thickness (0.63 cm for parallel and 0.45 cm for
cone beam).
4.2. 'Patient studies
Converging collimation has been applied in several patient studies of heart (Gullberg
et a / 1991a) and brain. Here we show one heart study and one brain study.
4.2.1. Heart. Figure 21 shows a representative scan from a patient study demonstrating
improved resolution of cone-beam cardiac SPECT. The cone-beam study was performed
using the 60 cm focal length cone-beam collimator on the SX-300 and the parallel
study was performed using the LEHR parallel-hole collimator on the PRISM. The
patient was exercised and at the peak of stress was injected with 1.11 x loRBq ( 3 mCi)
Convergent beam tomography
529
Figure 21. Cardiac patient study. A comparison is shown between parallel and cone-beam
reconstructions of cardiac transaxial and short axis slices. The parallel study used the
LEHR parallel hole on the PRISM SPECT system to obtain 60 projections over 180' of
approximately 125 034 total counts for a central transaxial slice. The cone-beam study used
the 60cm focal length cone-beam collimator an the SX-300 SPECT system to obtain 64
projections over 180' of approximately 49 116 total counts for the central cone-beam
transaxial slice.
of '"TI. The cone-heam results were obtained using the short-scan cone-heam reconstruction algorithm (Gullherg and Zeng 1992) to reconstruct 64 projections (-2.8"
between projections) sampled over 180". (Because of technical problems the angular
sampling range was less than the minimum required for the short-scan algorithm.) The
central slice of the cone-heam geometry which is equivalent to a fan-heam projection
data set was summed over 64 projection angles to give 49 116 counts. The parallel-hole
collimator results were reconstructed from 60 projections of which a typical slice
through the heart had a total of 125 034 counts summed over 60 projections. Before
reconstruction, both the cone-beam and parallel projections were filtered with a
two-dimensional Metz filter.
The better resolution in the cone-heam studies is seen with the thinner ventricular
wall. The wall thickness is 1.64 cm for the cone-heam results and 2.27 cm for the LEHR
parallel-hole collimator results. The study was done with a 60 cm focal length collimator
which we designed especially for the heart. It is felt that a 60 cm or even longer focal
length cone-beam collimator would have fewer problems with truncating the heart
than the 50 cm focal length cone-beam collimator used by Gullherg et a / (1991a).
4.2.2. Brain. Figure 22 shows a comparison of fan-beam and cone-heam tomography
study of the brain performed on the SX-300 SPECT system. The cone-heam study was
performed using the 50 cm focal length cone-beam collimator and the fan-beam study
was performed using the 58 cm long-bore fan beam collimator (Tsui et a / 1986). The
patient was injected with 9.25 X 10' Bq (25 mCi) of 9yTC"-HMPAo and 128 projects were
acquired for both studies. The cone-beam study was performed using a tilted cone-heam
acquisition shown in figure 3 (Manglos 1989, Manglos et al1989h) allowing the detector
530
G T Gullberg et a1
Figure 22. Brain patient study. Reconstructions of transaxial and sagittal slices using a
50 cm focal length cone-beam collimator and a 58 cm focal length fan-beam collimator.
to stay close to the head but still clear the patient shoulders. The long-bore design of
the fan-beam collimator accomplished the same purpose without having to tilt the
detector.
The cone-beam results show better resolution of structures in the brain than obtained
with the fan beam collimator. However, we see image elongation along the axis of
rotation near the top of the head in coronal and sagittal views due to the insufficient
convergent beam tomography
53 1
cone-beam data acquisition. The tilted detector acquisition makes this problem even
worsebecause the tilt of the detector places the top of the head further from the central
plane of the cone-beam collimator. A non-stationary filter has been developed by Cao
and Tsui (1992) which improves the slice-to-slice cross talk and image elongation.
5. Summary
Convergent-beam geometry provides improved sensitivity and improved resolution for
both emission and transmission computed tomography. The development of collimators
and algorithms has made possible the evaluation of this technology and the preliminary
results of imaging brain and heart are showing significant potential for the clinical
application of SPECT with convergent-beam geometry. We have found that a singledetector SPECT system using a high-resolution cone-beam collimator with tilted detector
acquisition can obtain high-resolution transaxial images of the brain. However, with
single planar orbit acquisition, we see image elongation and slice-to-slice cross-talk
due to insufficient cone-beam data acquisition. This elongation and slice-to-slice
crosstalk does not seem to be as much a problem with cone tomography of the heart
since the heart does not extend axially as far as the brain from the central plane of
the cone-beam geometry. However, detector systems and algorithms for non-planar
orbit cone-beam tomography are being developed which eliminate these reconstruction
artifacts associated with insufficient acquisition of cone-beam projections.
The development of convergent-beam geometry in SPECT requires the integration
of hardware and software through the optimization of reconstruction algorithms,
collimator design, orbit specifications and calibration techniques to obtain the best
image quality and to minimize the effects of photon attenuation, insufficient projection
sampling and data truncation. The challenge in using convergent-beam geometry for
spEcTis to continue to improve computing hardware in order to process, in a reasonable
clinical time, images that are reconstructed using algorithms that better model the
physics of the imaging detection process and are useful for complicated detector
geometries such as cone beam which gives improved geometric sensitivity. This technology has the potential to improve clinical diagnosis.
Acknowledgments
The research work presented in this manuscript was partially supported by NIH Grant
R01 HL 39792 and Picker International. We would also like to thank Biodynamics
Research Unit, Mayo Foundation for use of the Analyze software package. A grant
of computer time from the Utah Supercomputing Institute, which is funded by the
State of Utah and the IBM Corporation, is gratefully acknowledged.
Resume
Mire
au
point sur la tomographie par faisceau convergent en tomagraphie d'tmisaion B photon unique.
L'appon clinique des gkometries cowergentes en tomographie d'hission B photon unique (SPECT) fait
a c t d e m e n t I'objet d'etudes approfandies. Les collimateurs son1 utilisis sur les cameras scintillations
tournantes 'grand champ' astigmale de type: en eventail, conique, st.4nop6, afin d'augmenter la sensiblit.4,
aussi bien en tomographie d'hission que de transmission, pour des petits organes lek quae la thyroide, le
532
G T Gullberg
et a1
CaNeau ou .le coeur. Avec les nouveaux systemes SPECT multi-t$tes, la gComCtrie convergeme offre la
possihilite d'ohtenir simultanement les donnees dimirsion et de lransmission nicessaires B la quantification
d e la fixation d'une substance radiopharmaceutique dans le coeur. En SPECT. le dCveloppement des faisceaux
i giomltrie canvergente neeessile !'integration de logicids et de circuits electroniques dans les systemes
existants. Du point d e vue du materiel, le systeme de detection optimum pour la tomographie B faisceau
conique est un syateme qui satisfait H la condition suffisante selon laquelle les trajectories balayCes interceptem
n'imporie que1 plan passant B t m v m la zone reconstruite. Cependant, la plupan des algorithmes ont Pte
divelappCs d a m le cas de donnCes insuffirantes, acquires suivanl des orhites un s e d plan. Le d&velappement
de ces algorithmes a rendu possible I'evsluatian prCliminaire de cette technologic. et les images d e cerveau
et de coeur ont montrC la potentialit6 d e l a tomographie B faisceau conique pour les applications cliniques.
Actuellement, desacrivites de recherche sepoursuivent de fa$on significative dam l e domrine desalgorithmes
adapt& au traitement der donnles acquises respectant 'la condition d e donnees suffisantes' et pauvant Etre
iacilement et Cconomiquement implant& sur I t s appareils SPECT utilises en clinique.
Zusammenfassung
Uberhlick iiber die Konvergenirtrahl-Tomographie in der Ein~el-Photon-Emissionscamputenomographi~.
Unterauchungender konvergenten Einzel-Photon-Emissionscomputenomographie
(SPECT) wurden zur Bes-
timmung ihrer klinischen Bedeurung durchgefiihn. Facherstrahl-, Kegelstrahl-, Nadellach- und astigmatische
Kollimatoren wurden verwendet hei rotierenden Gammakameras mi1 gro4en Kristallflichen, um so das
anaprechvermbgen E r Emissions-und Transmissionscomputertamographie kleiner Organe, wie z.B. Schilddriise, Him udci H a z . zu erh6hen. Mil ncucn Multikopf-SECT Systemen kann man mit der Konvergenzstrahlgeometrie simultan Emissions- und T'ranrmissionsdaten erhalten, die zur Quantifirierung der Verteilung YO" Radiopharmareutika im H e m n d i g sind. Die Entwicklung der Konvergenzstrahlgeoemt~i~
erforden die lntegrierung YO" Hard- und Sdtwuare.Bezogiich der Hardware is, &as uptimair Dricktol,ysteni
LUT Kegelrtrahltomographie ein System, das die Hinljnglichkeitc-Bedingung der Dalen erfiillt, for die die
Ahtasttrajektorie jedc Ehenc durch das rekonstruierende Gebiet rchneidet. Die hauptrachliche Entwicklung
des Algorithmus erfolgte jedoch ffir d e n Fall der Nicht-Hinl;inglichkeit der Daten bei spezieller A n der
Datengewinnung. Die Entwicklung dieser Algorithmen ermbglicht eine varliiufige Rewertung dieser Techd o g i e ; insbesondere die Bildgehung yon Herr und Him zeigt die Bedeutung for die klinische Anwendung
der Kegelrtrahllomographie. Die Entwicklung YO" Algorithmen, die die Hinlanglichkeils-Bedingung der
Daten erfiillen und die leicht und preiswen bei klinischen SPECT-Systemen eingesetzt werden k c m e n , wird
durch verstirkte Forschungraktivitaten unterstiitzt.
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