A slice-by-slice blurring model and kernel evaluation using the Klein

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A slice-by-slice blurring model and kernel evaluation using the Klein-Nishina formula for 3D
scatter compensation in parallel and converging beam SPECT
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2000 Phys. Med. Biol. 45 1275
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Phys. Med. Biol. 45 (2000) 1275–1307. Printed in the UK
PII: S0031-9155(00)07327-9
A slice-by-slice blurring model and kernel evaluation using the
Klein–Nishina formula for 3D scatter compensation in parallel
and converging beam SPECT
Chuanyong Bai, Gengsheng L Zeng and Grant T Gullberg
Department of Radiology, University of Utah, 729 Arapeen Drive, Salt Lake City,
Utah 84108-1218, USA
E-mail: bai@pet.upmc.edu
Received 1 September 1999, in final form 21 February 2000
Abstract. Converging collimation increases the geometric efficiency for imaging small organs,
such as the heart, but also increases the difficulty of correcting for the physical effects of attenuation,
geometric response and scatter in SPECT. In this paper, 3D first-order Compton scatter in nonuniform scattering media is modelled by using an efficient slice-by-slice incremental blurring
technique in both parallel and converging beam SPECT. The scatter projections are generated
by first forming an effective scatter source image (ESSI), then forward-projecting the ESSI. The
Compton scatter cross section described by the Klein–Nishina formula is used to obtain spatial
scatter response functions (SSRFs) of scattering slices which are parallel to the detector surface.
Two SSRFs of neighbouring scattering slices are used to compute two small orthogonal 1D blurring
kernels used for the incremental blurring from the slice which is further from the detector surface to
the slice which is closer to the detector surface. First-order Compton scatter point response functions
(SPRFs) obtained using the proposed model agree well with those of Monte Carlo (MC) simulations
for both parallel and fan beam SPECT. Image reconstruction in fan beam SPECT MC simulation
studies shows increased left ventricle myocardium-to-chamber contrast (LV contrast) and slightly
improved image resolution when performing scatter compensation using the proposed model.
Physical torso phantom fan beam SPECT experiments show increased myocardial uniformity and
image resolution as well as increased LV contrast. The proposed method efficiently models the 3D
first-order Compton scatter effect in parallel and converging beam SPECT.
1. Introduction
In single photon emission computed tomography (SPECT), a large number of scattered
photons are detected because of the wide energy window used (typically 15–20%). Image
reconstruction without scatter compensation has degraded image quality (i.e. reduced contrast
and resolution) and biased quantitation (Jaszczak et al 1985). Intensive efforts have been made
to compensate for the scatter effect in SPECT in order to improve the quantitative and qualitative
accuracy of the reconstructed images. While these efforts have improved the reconstructed
images, none of these methods can perform fast and accurate scatter compensation in nonuniform scattering objects in converging beam SPECT.
A class of widely used and widely studied scatter compensation methods is based on
the estimation of the scatter component in the photo-peak projection data and subsequent
subtraction or deconvolution of the scatter contribution from the measured projection data.
Many of the subtraction-based scatter compensation methods use multiple energy window
0031-9155/00/051275+33$30.00
© 2000 IOP Publishing Ltd
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acquisition methods (Jaszczak et al 1984, 1985, Lowry and Cooper 1987, Koral et al 1988,
1990, Ogawa et al 1991, Gagnon et al 1989, Halama et al 1988, DeVito et al 1989, Hamill
and Devito 1989, King et al 1992, Frey et al 1992) to estimate the scatter contribution. The
estimated scatter component can be subtracted from the photo-peak projection data to obtain
the scatter-compensated projection data. Scatter compensation methods in this class are fast
and simple, but increase the noise in the reconstructed images. Alternatively, the deconvolution
methods use kernels that are obtained from physical measurements using a Gamma camera
(Axelsson et al 1984, Floyd et al 1985a, Msaki et al 1987, 1989, Meikle et al 1994). Usually,
the scatter component is deconvolved from the projection data before reconstruction.
Another class of scatter compensation methods is based on modelling scatter effects (the
scatter response function) in the projector–backprojector (transition matrix) used in iterative
reconstruction methods (Floyd et al 1985b, Frey et al 1993, Frey and Tsui 1993, Beekman
et al 1993, 1996, Welch et al 1995). The techniques for modelling the non-uniform attenuation
effect (Tsui et al 1989), the detector response effect (Tsui et al 1988, 1994, Formiconi et al
1989, Zeng et al 1991, Gilland et al 1994) and the scatter effect (Floyd et al 1985b, 1986, 1987,
Veklerov et al 1988, Frey and Tsui 1990, 1993, 1994, 1996, Bowsher and Floyd 1991, Frey
et al 1992, 1993, Beekman et al 1993, 1994, 1995, 1996, 1997, Riauka and Gortel 1994, Cao
et al 1994, Meikle et al 1994, Walrand et al 1994, Ju et al 1995, Welch et al 1995, Kadrmas
et al 1996, 1997, 1998, Hutton et al 1996, Riauka et al 1996, Wells et al 1997, Hutton 1997) in
a realistic projector–backprojector (transition matrix) may lead to accurate compensation for
these image-degrading effects in SPECT image reconstruction. We call this class of methods
reconstruction-based scatter compensation methods.
The accuracy of a reconstruction-based scatter compensation method is dependent upon
the accuracy of the scatter model used. A complicating factor in modelling scatter is that the
scatter response is generally different for every point in the object to be imaged.
Several techniques have been developed for calculating the transition matrix (scatter
model) in the projector–backprojector so that it is a function of the anatomy being imaged.
One technique computes the transition matrix using Monte Carlo (MC) simulations (Floyd
et al 1985b, 1986, 1987, Veklerov et al 1988, Frey and Tsui 1990, Bowsher and Floyd 1991).
This approach requires large data storage capacities and long computation times. A second
technique actually makes physical measurements (Beekman et al 1994, Walrand et al 1994). A
third technique is referred to as slab derived scatter estimation. This technique first calculates
and stores the scatter response tables for a point source behind slabs of a range of thicknesses,
and tunes the model to various object shapes including non-uniform attenuators (Frey et al
1993, Frey and Tsui 1993, Beekman et al 1993, 1996, 1997, Beekman and Viergever 1995).
A table occupying a few megabytes of memory is sufficient for representing this scatter model
for fully 3D reconstruction. A fourth method is based upon the integration of the Klein–
Nishina formula in non-uniform media (Riauka and Gortel 1994, Cao et al 1994, Riauka
et al 1996, Wells et al 1998). Like Monte Carlo techniques this technique requires large data
storage capacities and long computation times. A fifth method uses a variable attenuation
map to model first-order scatter at each image site by projecting and backprojecting along all
possible lines of scatter using a ray-driven projector–backprojector (Welch et al 1995, Welch
and Gullberg 1996, 1997, Laurette et al 1999).
Reconstruction-based scatter compensation methods (Floyd et al 1986, Frey et al 1993,
Beekman et al 1996, Kadrmas et al 1998) have been shown to result in images with
less variance when compared with subtraction-based scatter compensation methods (Frey
et al 1992, Beekman et al 1997). The main disadvantage of reconstruction-based scatter
compensation is that the scatter models tend to be very computationally intensive. Recent
advances in fast implementations of reconstruction-based methods such as using coarse-grid
Blurring model and kernel evaluation for SPECT
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modelling approaches (Kadrmas et al 1998), unmatched projector–backprojector pairs (Zeng
and Gullberg 1996, 1997, 2000, Welch and Gullberg 1996, 1997, Kamphuis et al 1998)
and accelerated iterative algorithms (Hudson and Larkin 1994, Byrne 1996), can make the
reconstruction time for reconstruction-based scatter correction techniques more reasonable.
The method presented in this paper is based on use of an incremental blurring technique
to model the scatter in the projector. The incremental blurring technique was first developed
to model the geometric detector response in the projector–backprojector pair (McCarthy and
Miller 1991, Wallis et al 1996, Di Bella et al 1996, Zeng et al 1998). In this work, the
incremental blurring approach is used to decrease the computational requirement for modelling
scatter.
Frey and Tsui (1996) proposed an effective scatter source estimation (ESSE) scatter
model to model 3D scatter response in SPECT. This technique produces an effective scatter
source which is used to generate the estimated scatter projection by forward-projecting the
effective scatter source, using a projector that models both attenuation and geometric detector
response effects. The projector used for forward-projecting the effective scatter source can
be implemented using the incremental blurring technique for modelling the non-uniform
attenuation and depth-dependent geometric point response. The activity image is convolved
with several kernels to obtain the effective scatter source. The effective scatter source kernel
and the relative scatter attenuation coefficient kernel used for the convolutions are generated
by Monte Carlo simulations and are assumed to be independent of the source position. The
technique models the non-uniform attenuation effect from the scattering point to the detector,
but does not model the non-uniform attenuation from the source to the scattering point. The
method does not have the capacity to provide an exact scatter estimate when extended to
converging beam collimation.
Zeng et al (1999) were the first to propose using a slice-by-slice blurring technique to
model 3D first-order Compton scatter in SPECT for parallel beam geometry, and Bai et al
(1998b) extended the technique to model 3D first-order Compton scatter in SPECT for fanbeam geometry. The scatter projection was generated by first forming an effective scatter
source image (similar to the effective scatter source in Frey and Tsui’s (1996) model) using
a slice-by-slice blurring model, then forward-projecting the effective scatter source image
using a slice-by-slice blurring model which modelled the non-uniform attenuation effect and
the depth-dependent geometric detector response effect. The small blurring kernels were
obtained by recursively changing a parameter which was used to calculate the kernels until
the generated first-order Compton scatter projection was best matched to the Monte Carlo
generated first-order Compton scatter projection. The kernels were stationary, and were used
to model first-order Compton scatter for Compton scatter compensation in SPECT imaging.
The incremental blurring method proposed in this paper to model first-order Compton
scatter is based on the Klein–Nishina formula. The key to this method is to transform the
Klein–Nishina formula so that the scatter angle is expressed in terms of the spatial coordinates,
i.e. the positions of the point source, the scattering point and the detection point, as well as
the focal length of the converging beam collimator. An effective scatter source image (ESSI)
can be obtained using a slice-by-slice blurring model, with the blurring kernels approximated
from the transformed Klein–Nishina formula. Scatter projections can be obtained by forwardprojecting the ESSI using a projector, which models non-uniform attenuation and geometric
detector response effects.
In section 2 the Klein–Nishina formula is introduced and the slice-by-slice blurring model
is derived. In sections 3 and 4 the model is evaluated by comparing the scatter point response
functions (SPRFs) and the scatter response of a distributed source generated using the proposed
model with those generated from Monte Carlo (MC) simulations. The model is also evaluated
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by comparing the measured total point response of a fan-beam SPECT system with that
generated using the proposed model. Fan-beam SPECT reconstructions are used to show
the effect of performing fully 3D scatter compensation using the proposed model. Section 5
includes some discussion and, finally, in section 6, conclusions are drawn from the previous
results and discussions.
2. Theory
Compton scatter is an important interaction mechanism when high-energy photons interact
with materials. This scatter is accompanied by energy transfer to the material. Within the
diagnostic energy range of nuclear medicine, the linear attenuation coefficients of body tissues
are essentially composed of contributions from Compton scatter and photoelectric absorption
(Alvarez and Macovski 1976, Phelps et al 1975). We propose in this paper the use of a sliceby-slice blurring model for first-order Compton scatter with the justification that 80% to 90%
of the scattered events detected in an 18% 99m Tc emission window are caused by first-order
Compton scatter (Floyd et al 1984).
2.1. Modelling first-order Compton scatter in SPECT
2.1.1. The Klein–Nishina formula. The photon energy and scatter-angle-dependent Compton
scatter differential cross section of an electron is expressed by the Klein–Nishina formula (Klein
and Nishina 1928, Johns and Cunningham 1974). In SPECT, we consider the scatter effect of
a voxel in the scattering object to be given by the differential cross-section dσ/d, defined by
the equation
dσ
(1 + cos2 θ )[1 + α(1 − cos θ)] + α 2 (1 − cos θ)2
= R0 F (θ, E)
(1)
= R0
d
[1 + α(1 − cos θ)]3
where R0 is a scatter factor dependent upon the composition of the scattering medium and the
volume of the voxel. That is to say, R0 depends upon the total number of free and loosely
bound electrons in the voxel, θ is the scatter angle, α is the ratio of the energy E of the incident
photon to the energy of an electron at rest (511 keV) and F (θ, E) is the expression in the large
brackets of equation (1). The equation for the differential cross section can be simplified as
the product of R0 and the factor F (θ, E) (which is only dependent upon the photon energy and
scatter angle). For non-uniform scattering objects, R0 can be different from voxel to voxel.
2.1.2. Scatter point response function (SPRF). The generation of a first-order Compton
scatter point response function (SPRF) at the detector, for a point source SS (figure 1), can be
decomposed into four consecutive steps:
(a) Step 1. Photons propagate from the point source to a scattering voxel SC. At the same
time the intensity is attenuated by a factor of ASS→SC , which is the total attenuation effect
from the point source to the scattering voxel.
(b) Step 2. Photons are scattered at SC with the scatter probability described by the Klein–
Nishina formula in equation (1): R0 F (θ, E). The effective scatter at the point SC for the
point source at SS is
E(SC, SS) = ISS ASS→SC R0 F (θ, E)
where ISS is the point source intensity.
(2)
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Figure 1. Fan beam geometry. The x-axis is the central ray of the fan beam geometry and the
y-axis is the detector surface.
(c) Step 3. Photons that scatter at SC propagate to the detector surface and are selectively
detected at detector position D according to the collimator geometry, energy window and
photopeak energy. If the total attenuation effect from SC to D is ASC→D , then the number
of photons that are scattered at point SC and detected at point D is
S(D, SC) = ISS ASS→SC R0 F (θ, E)ASC→D P (E)
(3)
where P (E) is the probability that a photon with energy E is detected for the given energy
window. Throughout this section, in order to simplify the derivation of the model, we
assumed that P (E) is a window function of energy E with value 1 when E is within
the energy window and 0 when E is outside the energy window. Appendix B provides a
discussion of this assumption.
(d) Step 4. The scatter point response function SPRF is the total number of scattered photons
detected by each point on the detector surface. This corresponds to the summation of
S(D, SC) over all the scattering points SC in the scattering medium:
SPRF(D) =
S(D, SC).
(4)
SC
2.1.3. Definitions of ESSI and ISSI. The results for the first two steps in section 2.1.2 (in
equation (2)) are stored in a scatter image array for all of the scattering voxels. This is defined
as the effective scatter source image (ESSI) for the point source SS.
Multiplication by R0 in equation (2) is a voxel-wise multiplication operation and is
dependent on the composition and size of the voxel. When computing the ESSI, this operation
can be performed in the final step. The intermediate result of the first two steps, before the
performance of the multiplication operation, is defined as the intermediate scatter source image
(ISSI). The ISSI is dependent upon the collimator geometry and positions of the point source
and scattering voxels.
Both the ESSI and ISSI have the dimension of the 3D image to be reconstructed. For each
projection angle, the ISSI is generated and stored in a 3D array. The ESSI is then generated
from the ISSI and stored in the same array. This same array is used for the generation of the
ISSI and ESSI for the next projection angle.
2.2. Slice-by-slice blurring model for first-order Compton scatter response
2.2.1. The spatial scatter response function (SSRF). In this section, we rewrite F (θ, E) in
equation (1) for a given collimator geometry in terms of the geometrical parameters and the
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positions of the point source, the scattering point and the detection point. Fan beam geometry
is used to keep our analysis general.
Figure 1 illustrates a fan beam geometry positioned in the fan direction. Photons emitted
from a point source SS(xSS , ySS ) are scattered by a scattering point SC(xSC , ySC ) on a scattering
slice which is parallel to the detector surface and closer to the detector surface than the point
source. Some of the scattered photons are detected by the camera at point D(0, yd ). The y-axis
is on the detector surface and the x-axis is along the central ray of the fan beam geometry. The
focal point is at FP(f, 0), where f is the focal length. The distance in the x-direction from the
scattering point to the point surface is x. A scattering slice at a distance x closer to the
detector surface than the point source will be denoted by ‘the scattering slice x’ throughout
the rest of this paper. The unit used for distance is ‘bin’, which is the projection bin size.
The scatter angle at SC(xSC , ySC ) can be expressed in terms of f , xSS , ySS , yd , and x as:
yd (f − xSS + x)
−1
− arctan(yd /f ).
− ySS (x)
(5)
θ = arctan
f
Note that the coordinates (xSC , ySC ) of the scattering point can be determined from f ,
xSS , ySS , yd and x. As a special case for the parallel geometry, the scatter angle is
θ = arctan[(yd − ySS )/(x)].
Substituting equation (5), as well as f and E, into F (θ, E) in equation (1), for a given
point source position, allows one to examine the behaviour of F as a function of yd and x
in both the fan direction and the parallel direction. To do this we will define a new function
Gfan (yd , x) = F (θ, E) for a given energy in the plane of the fan beam direction. This
essentially provides a 1D spatial scatter response function (SSRF) for first-order Compton
scatter for a scattering slice at a distance x from the point source. By the same token, we
define Gpar (zd , x) = F (θ, E) in the plane of the parallel beam direction, where zd is the
coordinate in the z direction (parallel beam direction).
Figure 2 plots the SSRF Gfan (yd , x) for x = 1, 2, . . . , 5, with photon energy
E = 140 keV, focal length f = 104 and the point source at (30, 0). The unit is bin size
which is equal to 0.625 cm. The physical focal length is 65 cm.
Equation (5) is a correct formula only for scatter events that occur in the plane containing
the source and perpendicular to the focal line. In the plane which is parallel to the focal
line, a similar equation is obtained from equation (5) by letting f go to infinity. These two
equations are used to obtain the two 1D functions Gfan (yd , x) and Gpar (zd , x). In this
paper, we approximate Gfan (yd , x) and Gpar (zd , x) using two 1D Gaussian functions.
The convolution of these two 1D Gaussian functions gives the approximation of the SSRF,
F (θ, E) = F (yd , zd , E), on the detector surface for an arbitrary scatter point in the 3D volume.
2.2.2. The ISSI for slice-by-slice blurring model. In order to generate the ISSI for the
scattering slice x = 1, the point source is convolved first with the SSRF of the slice in the
fan direction, then convolved with the SSRF in the parallel direction, and the result is finally
attenuated by the slice. This is given in the following equation:
ISSI|x=1 = A1 · ((ISS ∗ Gfan (yd , 1)) ∗ Gpar (zd , 1))
(6)
where A1 is an attenuation operator which gives the effective attenuation from the point source
to the scattering slice x = 1, SS is the location of the point source, Gfan (yd , 1) is the SSRF
in the fan direction (y direction) for the scattering slice x = 1, where yd is the coordinate in
the y direction, and Gpar (zd , 1) is the SSRF in the parallel direction (z direction), where zd is
the coordinate in the z direction; · stands for the effect of the attenuation and ∗ stands for 1D
convolution.
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Figure 2. Spatial scatter response function Gfan (yd , x) for x = 1, 2, . . . , 5.
In the same way, the ISSI for the scattering slice x = 2 can be generated as
ISSI|x=2 = A12 · A1 · ((ISS ∗ Gfan (yd , 2)) ∗ Gpar (zd , 2))
(7)
where A12 is an attenuation operator which gives the contribution of the effective attenuation
for the point source from the scattering slice x = 1 to the scattering slice x = 2, and
A12 · A1 gives the effective attenuation from the point source to the scattering slice x = 2.
If we compare the spatial scatter response functions for the scattering slices x = 2 and
x = 1 (for example, the function Gfan (yx , x) shown in figure 2 for the fan direction), we
see that we can approximate the SSRFs for x = 2 by convolving the SSRFs for x = 1 with
small scatter kernels that model the effect of scatter from slice x = 1 to x = 2, as shown
in equations (8) and (9):
12
(yd ) ∗ Gfan (yd , 1)
Gfan (yd , 2) = Kfan
12
Gpar (zd , 2) = Kpar (zd ) ∗ Gpar (zd , 1)
12
(yd )
Kfan
(8)
(9)
12
Kpar
(zd )
and
are the small 1D scatter kernels from x = 1 to x = 2 in the
where
fan and parallel directions respectively.
Thus, equation (7) can be rewritten as
12
12
(yd )} ∗ Kpar
(zd )).
ISSI|x=2 = A12 · ({[A1 · ((ISS ∗ Gfan (yd , 1)) ∗ Gpar (zd , 1))] ∗ Kfan
(10)
Noticing that the part of equation (10) in square brackets is the same as the right-hand
side of equation (6), one can rewrite equation (10) as
12
12
(yd )) ∗ Kpar
(zd )].
ISSI|x=2 = A12 · [(ISSI|x=1 ∗ Kfan
(11)
Equation (11) expresses the basic principle behind the proposed slice-by-slice blurring
model for first-order Compton scatter. Once the ISSI for the scattering slice x = 1 is
generated using equation (6), we can compute the ISSIs for each succeeding scattering slice
by performing slice-by-slice blurring using
n[n+1]
n[n+1]
(yd )) ∗ Kpar
(zd )]
ISSI|x=n+1 = An[n + 1] · [(ISSI|x=n ∗ Kfan
(12)
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where ISSI|x=n+1 is the ISSI for scattering slice x = n + 1, ISSI|x=n is the ISSI for
scattering slice x = n, An[n + 1] is the effective attenuation from scattering slice x = n to
n[n+1]
n[n+1]
x = n + 1, and Kfan
(yd ) and Kpar
(zd ) are the two small 1D kernels for the incremental
blurring from x = n to x = n + 1 in the fan and parallel directions respectively.
When the slice-by-slice blurring method is used for parallel beam geometry and cone
beam geometry one needs only to substitute the SSRFs and the convolution kernels for the
appropriate geometry in each corresponding direction.
2.2.3. Generation of ESSI. Multiplication of the ISSI with the composition-dependent scatter
factor in a voxel by voxel manner gives the effective scatter source image (ESSI), as shown by
ESSI(x) = R · ISSI(x)
x = 1, 2, . . .
(13)
where R is the voxel-wise multiplication operator by the composition-dependent scatter factor
for each scattering voxel. In the discrete implementation for the computation of the ESSI, the
solid angle subtended by a voxel on the scattering slice with the origin at the point source and
the solid angle subtended by a bin on the detector surface with the origin at the centre of the
scattering voxel must be included in equation (13). A detailed discussion of this is provided
in appendix A.
The composition-dependent scatter factor R at scattering voxel i can be obtained from
equation (1) when the attenuation map is given, by integrating both sides of equation (1) over
4π solid angle and solving for the factor Ri , which gives
Comp
dσi
µi
µi
∼
Ri = =
(14)
=
const(E)
const(E)
F (θ, E) d
where F (θ, E) d = const(E) is a function
of the energy of the incident photons, and has
a value of 5.70 at energy 140 keV, and dσi is the total scatter cross section of the voxel,
Comp
which gives the Compton portion µi
of the linear attenuation coefficient for the voxel. In
this paper we assume that the Compton portion of the total linear attenuation coefficient is
approximately equal to the total linear attenuation coefficient µi for an energy of 140 keV.
This approximation is based on the calculation results provided in section 5.3 which are based
upon data given by Alvarez and Macovski (1976) and Phelps et al (1975). The attenuation
for body tissues (with the exception of bones) is more than 98% due to Compton scatter at a
photon energy of 140 keV. Therefore the assumption of a linear attenuation coefficient based
only upon Compton scatter is acceptable when considering the noise level (several per cent)
in SPECT.
2.2.4. Generation of the scatter response for a point source. After the ESSI is computed for
a point source, the ESSI image is projected using a projector which models the attenuation and
the depth-dependent geometric detector response to obtain the scatter point response function
(SPRF). In the next section, we describe how this is used to calculate the scatter response for
a distributed source. Generally speaking, the scattered photons have lower energy than the
primary photons. Therefore, one should consider the increased attenuation of the scattered
photons when projecting the ESSI. However, the increased attenuation for the scattered photons
is not modelled. We assume that the increase in attenuation is relatively small given the
detection energy window that is used (for example, the linear attenuation coefficient of water
is 0.150 cm−1 at 150 keV and 0.161 cm−1 at 120 keV).
2.2.5. Generation of the scatter response for a distributed source. The scatter response of an
extended source distribution can be obtained by superposition of the individual SPRF over the
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extended source distribution. In SPECT image reconstruction, this is done by first computing
the ISSI for the voxel sources (which are treated as point sources) on the first slice (parallel to
the detector surface, furthest from the detector surface) of the source image, then computing
the ISSI for the voxel sources on the second slice of the source image, and so on, until the
last slice of the source image which is closest to the detector surface. The obtained ISSIs
are added up to form the total ISSI for the extended source distribution, and equation (13) is
used to compute the ESSI. Finally, the ESSI is projected to obtain the scatter response for the
distributed source.
It should be noticed that for a distributed source, in order to generate the ISSI corresponding
to sources on a slice parallel to the detector surface, the sources are convolved with the SSRFs
for all of the succeeding scattering slices. However, convolution can only be used when the
kernels are spatially invariant in the plane parallel to the collimator. This generally is not the
case for fan beam geometry. However, based upon the results of table 1 in section 2.3, we
assume that these SSRFs are spatially invariant in a plane parallel to the detector surface, and
thus one can perform convolutions to obtain the ISSIs.
2.2.6. Observations about the model. Several observations can be made about this method.
First, the proposed model includes the non-uniform attenuation from the point source to the
scattering point during the generation of the ISSI, and handles the non-uniform scatter when
performing the multiplication operation to generate the ESSI. This method also includes the
attenuation from the scattering point to the point of detection and the geometric detector
response effect during the projection operation that generates the SPRFs. One should notice
that the increased attenuation of the scattered photons from the scattering point to the detection
point is not modelled.
Secondly, this method implicitly assumes that: (a) photons scattered by a scattering point
outside the region confined by the fan area (the shadowed area in figure 3) cannot be detected;
(b) photons originating from sources outside the fan area can be scattered, but the scattered
photons cannot be detected due to the narrow energy window (20%); and (c) the backscattered
photons will not be detected. However, for sources near the edge of the fan area, a small portion
of the scattered photons can be detected. Thus, in the situation where the activity source is
truncated, the second assumption leads to under estimation of the scatter effect at the edges of
the detector.
Figure 3. Fan beam geometry. The shadowed area is where scatter is modelled in the slice-by-slice
blurring model for the first-order Compton scatter effect.
A third observation is that equations (8) and (9) are the intermediate steps taken to obtain
equation (10) from equation (7). Several factors should be considered when computing the
small kernels as well as the SSRF for the scattering slice x = 1. In our previous work (Zeng
et al 1999), we used two small orthogonal stationary kernels to perform the convolution in
parallel beam SPECT. In other words, the SSRF for the scattering slice x = 1 and the small
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convolution kernels between the scattering slices were the same in equations (8) and (9), and
the kernels were independent of the position of the point source. In this work though, for a
particular scattering slice, the SSRFs and the small kernels are functions of the distance from
the point source to the scattering slice. For converging beam geometry, the SSRFs and kernels
also depend upon the focal length, the distance from the point source to the focal point, and
the distance to the converging beam central ray. Necessary information can be obtained from
the transformed Klein–Nishina formula, which is obtained by substituting equation (5) into
equation (1).
In this work, we approximate the SSRFs with Gaussians, of which the full width at half
maximum (FWHM) is determined by the energy window and the position of the point source.
By knowing the SSRFs of two neighbouring scattering slices, the two small blurring kernels are
calculated assuming that the SSRF of the slice closer to the detector surface is the convolution
of the SSRF of the slice further from the detector surface with the two small kernels. In
the implementation, these two small kernels are approximated by two small symmetric fivepoint 1D kernels which are calculated on the fly by knowing the FWHMs of the Gaussian
approximations of the SSRFs of the two slices.
Finally, the implementation of equations (6) and (12) to obtain the ISSI for each scattering
slice is the same as the implementation of the slice-by-slice blurring model that models both the
depth-dependent detector response and non-uniform attenuation effects (for example Bai et al
1998a). In this paper, equation (12) is implemented in such a way that the scatter convolution
and the corresponding attenuation are performed at the same time, i.e. the value of a voxel
on slice n + 1 is obtained by first multiplying the values of the voxels on slice n of the ISSI
with their corresponding elements in the convolution kernel, then weighting the results with
the attenuation from the voxels on slice n to the voxel on slice n + 1, and finally summing
them up. In this way, the modelling of non-uniform attenuation involves interpolations of
the attenuation from the voxels in the previous slice to the voxel in the next slice, that is, the
convolved voxels of the previous slice that contribute to the voxel in the slice being calculated
are each weighted by an attenuation factor.
2.3. Computing the convolution kernels from the theoretical SSRFs
In this section we describe how Gaussian functions are used to approximate the theoretical
SSRF F (θ (yd , x, xSS , ySS , f ), E) which is a function of the energy, geometry and position.
(Note: F (θ (yd , x, xSS , ySS , f ), E) reduces to Gfan (yd , x) for a given set of xSS , ySS , E and
f .) From this Gaussian function the convolution kernels are calculated for the slice-by-slice
blurring model. The overall approach is to first calculate the theoretical SSRF, then multiply
the SSRF with the energy detection probability function P (ESC ), where ESC is the energy
of the scattered photon. This procedure results in a modulated SSRF with the assumption in
Step 3 of section 2.1.2. The modulated SSRF has a value zero for an energy outside the given
energy window, a maximum value 1.0 (maximum) for the photopeak energy and a minimum
value (minimum) for the lower edge of the energy window. The full width at the value which
is half the sum of the maximum value and the minimum value of the modulated SSRF is used
as the FWHM of the Gaussian function which is used to approximate the SSRF.
For example, for a fan beam SPECT acquisition in which f = 104, E = 140 keV and a
20% energy window (126–154 keV) is used, the SSRF is computed in the following way. First,
from Compton scatter theory we compute the maximum scatter angle that corresponds to a
scattered photon with energy of 126 keV, since this is the minimum energy that can be detected
for a 20% energy window with the assumption in Step 3 of section 2.1.2. For 126 keV this
angle is 53.53◦ . For this angle the energy- and scatter-dependent factor in the Klein–Nishina
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formula in equation (1) is F (53.53◦ , 140) = 0.55. A Gaussian function is used to approximate
the portion of SSRF Gfan (yd , x) (now only a function of yd ) that will fall within the window
of 0.55 (minimum) to 1.0 (maximum) for a given set of x, xSS , ySS , E and f . In order to
simplify the calculation we simply calculate the width of the SSRF at the value of 0.778, which
is half of the sum of the minimum value of 0.55 and the maximum value of 1.0 and use this
width as the FWHM of the Gaussian approximation of the modulated SSRF. Note that this is
done for point sources that are not too close to the focal line.
Appendix B discusses the approximation of SSRFs using Gaussian functions when the
correct energy detection probability function P (ESC ) is used instead of using the window
function assumption in Step 3 of section 2.1.2. The discussion shows results that are only
slightly different from the results in this section.
The FWHM of the Gaussian approximation is a function of point source position (xSS , ySS ),
the distance of the scattering slice to the point source x, and the focal length. Table 1 gives
examples of the FWHMs of the Gaussians, for varying x, varying xSS and varying ySS . From
table 1 it is apparent that, for a given point source position, the FWHM is approximately
proportional to x, that is
FWHM(x, xSS , ySS ) = FWHM(1, xSS , ySS ) × x.
(15)
Table 1. Computed FWHMs for Gaussian approximation of SSRFs. f = 104 bins, E = 140 keV,
20% energy window; unit: bin (sampling unit on the detector surface). Note: the centre ray of the
fan beam geometry is when the second coordinate is zero.
(xSS , ySS )
x
(10, 0)
(26, 0)
(42, 0)
(58, 0)
(74, 0)
(42, 0)
(42, 5)
(42, 10)
(42, 15)
(42, 20)
1
2
3
4
5
1.44
2.89
4.35
5.84
7.33
1.74
3.49
5.27
7.07
8.89
2.19
4.40
6.66
8.95
11.28
2.95
5.99
9.05
12.21
15.45
4.55
9.26
14.15
19.25
24.64
2.19
4.40
6.66
8.95
11.28
2.21
4.45
6.72
9.04
11.41
2.27
4.58
6.94
9.34
11.80
2.37
4.81
7.30
9.85
12.48
2.54
5.14
7.83
10.62
13.52
For x = 1, the FWHM depends on both xSS and ySS . Therefore, one can compute for a
given focal length, photon energy and detection energy window, a look-up table for the FWHMs
for x = 1 at different xSS and ySS . However, one can also compute the FWHMs and fit them
to a function of (x, xSS , ySS , f, E), thereby avoiding the look-up table and computing the
FWHMs on the fly. Another possibility is to derive the function from the transformed Klein–
Nishina formula by using an appropriate approximation. In the parallel direction, the FWHM
for x = 1 is independent of the point source position, and its value is 1.32 bins.
The small blurring kernels (described in section 2.2.2) are calculated by using the known
expression for the FWHM as a function of distance from the detector. These kernels can be
computed by using a least squares method (Bai et al 1997), or by using Gaussian sampling
with appropriate normalization, considering the fact that convolution of two Gaussian functions
results in another Gaussian function, and the resultant Gaussian function has an integral that
is the product of the integrals of the two original Gaussian functions. One should also note
that the Gaussian functions used to compute the small convolution kernels are normalized (Bai
et al 1997, Zeng et al 1998). However, as shown in equation (15), the Gaussian functions used
to approximate the SSRFs have a maximum value of 1, and FWHMs that are proportional to
x. Therefore, the normalization of the obtained Gaussian functions should be considered.
This is shown in Appendix A.
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It is shown in table 1 that the FWHMs of the Gaussian approximations increase as the
source is moved off the central ray of the fan beam geometry. And it is also observed that
the SSRFs become more asymmetrical as the source is moved further off the central ray.
However, the increase in the width of the Gaussian approximation is comparatively small, and
the asymmetry of the SSRFs occurs when the scattering angle is comparatively large. Thus, one
can assume that the Gaussian approximations for the voxel sources on a source slice (parallel
to the detector surface) of the source image are the same. Therefore, in the implementation of
equations (6) and (10), one can perform the convolution using the same kernels for the voxel
sources in the same source slice.
3. Methods
3.1. Evaluation of the model using Monte Carlo simulations
3.1.1. Scatter point response functions (SPRFs) for parallel beam geometry. Monte Carlo
simulations were performed using PHG Release 2.5b (Vannoy 1994) to generate first-order
Compton scatter for parallel beam geometry, using the non-uniform attenuator of the MCAT
phantom (Terry et al 1990) as the scattering object. The ESSIs generated from the proposed
slice-by-slice blurring model were projected to give the scatter point responses, using a
projector that also modelled attenuation but not geometric detector response as described in
section 2.2.5. Also, the scatter projection was generated using the slice-by-slice blurring model
proposed in our previous work (Zeng et al 1999). First-order Compton scatter point responses
obtained from MC simulations and the slice-by-slice blurring models were compared.
3.1.2. Scatter point response functions (SPRFs) for fan beam geometry. The proposed sliceby-slice blurring model was used to generate a set of first-order Compton scatter projections for
fan beam geometry. A Monte Carlo (MC) simulation was performed to generate the Compton
scattered projections from first-order to sixth-order. The summation of the Compton scatter
projections from first-order to sixth-order was used to approximate the full Monte Carlo (fMC)
result. The scatter projections generated using the proposed model were compared with the MC
first-order Compton scatter projections and also compared with the fMC scatter projections.
3.1.3. Scatter response of a distributed source for fan beam geometry. In order to evaluate
the application of the model to a distributed source, the MCAT phantom was used with activity
present only in the cardiac walls. The proposed slice-by-slice blurring model was used to
generate a set of first-order Compton scatter projections for fan beam geometry. A Monte
Carlo (MC) simulation was performed to generate the Compton scattered projections from
first-order to sixth-order. The summation of the Compton scatter projections from first-order to
sixth-order was used to approximate the full Monte Carlo (fMC) result. The scatter projections
generated using the proposed model were compared with the MC first-order Compton scatter
projections and also compared with the fMC scatter projections.
Figure 4 shows slices of the attenuation map of the MCAT phantom, the detector positions
and the sources. First-order Compton scattered photons were detected from four angles (0◦ ,
90◦ , 180◦ and 270◦ ), which corresponded to the detector positions 1, 2, 3 and 4 respectively.
The geometric detector response was not simulated because our focus was on evaluating how
well the incremental blurring technique performs in modelling only first-order Compton scatter.
The bin size was 0.625 cm, the attenuation map was a 64 × 64 × 64 matrix and a circular orbit
with a radius of 22.0 cm was used. The energy window was ±10%, centred at 140 keV. For
fan beam geometry, the point source was positioned so that, at detector positions 1 and 3, the
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Figure 4. Slices of the attenuation map of the MCAT phantom and sources. Top left: point source
for parallel beam geometry. Top right: point source for fan beam geometry. The point source is
in a different transaxial slice from that for parallel beam geometry. Bottom: distributed source for
fan beam geometry. Scatter projection data are acquired at four detection angles denoted by 1, 2,
3 and 4 for all cases.
Figure 5. Phantom used for physical point source scan using fan-beam SPECT: (a) a transaxial
cross section of the phantom; (b) an axial cross section of the phantom.
point source was almost on the central ray, and at detector positions 0 and 2, the point source
was approximately 10 bins from the central ray. The focal length was 65.0 cm.
3.2. Evaluation of the model using physical fan beam point source data
A point source scan was performed using a Picker PRISM 3000XP three-head SPECT system
(Picker International, Cleveland, OH) with fan beam collimation. The point source, with
an activity of 4.77 MBq (130 µCi) of 99m Tc, was put at approximately the centre of a nonuniform phantom a shown in figure 5. The measured point responses were compared with
the responses generated using slice-by-slice blurring model, which modelled non-uniform
attenuation, geometric point response and non-uniform first-order Compton scatter.
The activity was contained in a capillary glass tube which was approximately 1.5 cm long
and sealed at both ends. The activity was confined within a 1 mm length of the tube. The remainder of the tube was empty. The design of the phantom was based on two considerations: (a) the
presence of non-uniform scattering media and (b) the ease of inserting and removing the point
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(a)
Figure 6. Scatter point responses in parallel beam SPECT. (a) Profiles of scatter point responses.
Vertical axis: absolute scale, arbitrary units. Horizontal axis: bin number. Broad solid line:
the proposed model, varying kernels. Thin solid line: MC. Thin dashed line: stationary kernels
(Zeng et al 1999). The number at the shoulder of each profile corresponds to the detector position
shown in figure 4. (b) Semi-log profiles of scatter point responses. Vertical axis: logarithm of
the reconstructed intensity. Horizontal axis: bin number. Broad solid line: the proposed model,
varying kernels. Thin solid line: MC. Thin dashed line: stationary kernels (Zeng et al 1999). The
number at the shoulder of each profile corresponds to the detector position shown in figure 4.
source. In order to prepare the non-uniform phantom, two slices of Styrofoam were combined
to form a Styrofoam cross. The Styrofoam cross was put into a hollow cylindrical container
(a Deluxe ECT phantom (Data Spectrum, Hillsborough, NC) with the inserts taken out), which
divided the container into four sectors. Then a loaf-like Styrofoam piece of irregular shape
was put into one of the sectors. Half of the intersecting line of the two Styrofoam slices was cut
away, so that the capillary tube could be inserted into the Styrofoam cross at different positions.
After the capillary tube was placed in the Styrofoam cross, the container was filled with water.
A flexible rubber pipe was plugged into the central hole of the lid of the cylindrical
container. The end of the capillary tube that was outside the Styrofoam was tied with string.
The string was guided out of the container through the flexible pipe. In this way, the capillary
tube could be pulled out of the container by pulling the string, without opening the container.
Figure 5 illustrates a transaxial cross section and an axial cross section of the phantom.
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(b)
Figure 6. (Continued)
Table 2. Scatter to primary ratios of the scatter point responses for parallel beam SPECT
(MC, Monte Carlo results; BL, proposed blurring model results). The numbers in the first row
correspond to the detector positions shown in figure 4. The last row is the percentage difference:
(BL − MC)/MC × 100%.
1
2
3
4
MC
0.568
0.565
0.452
0.330
BL
0.555
0.580
0.439
0.336
% −2.3
+ 2.6
−3.0
+ 1.8
The emission scan was performed from four angles (uniformly distributed over 360◦ ),
using one of the three heads of the SPECT system. The scanning time was 30 s for each stop.
The projection bin size was 0.712 cm. The energy window was ±7.5% centred at 140 keV,
and a circular orbit of a radius of 20.9 cm was used.
After the emission scan, the capillary tube was pulled out of the phantom. A transmission
scan was performed using a transmission line source of activity 633 MBq (17.25 mCi) of
99m
Tc that was put at the focal line of the corresponding collimator. The transmission data
were acquired from 120 angles with an angular separation of 3◦ between two consecutive
angles. The total transmission scan time was 20 min.
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(a)
Figure 7. Scatter point responses in fan beam SPECT. (a) In the parallel direction, linear profile.
(b) In the parallel direction, semi-log profile. (c) In the fan direction, linear profile. (d) In the fan
direction, semi-log profiles. Vertical axis: relative intensity. Horizontal axis: bin number. Broad
solid line: the proposed method. Thin solid line: MC simulations for first-order Compton scatter.
Dotted line: full MC simulation result. The number at the shoulder of each profile corresponds to
the detector position shown in figure 4.
Table 3. Scatter to primary ratios of the scatter point responses for fan beam SPECT (fMC, full
Monte Carlo results; MC, Monte Carlo results; BL, proposed blurring model results). The numbers
in the first row correspond to the detector positions shown in figure 4. Row 5 is the percentage
difference: (BL − MC)/MC × 100%, and the last row is (BL − fMC)/MC × 100%.
1
2
3
4
fMC
0.497
0.644
0.536
0.335
MC
0.418
0.502
0.458
0.305
BL
0.442
0.481
0.460
0.296
%
+ 5.4
−4.4
+ 0.4
−3.0
%
−12.4
−33.9
−16.5
−13.2
The position of the point source was obtained by reconstructing the emission projection
data. The scattering media were reconstructed from the transmission data. The reconstructed
scattering media and the point source at the determined location were used to generate the total
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point responses of the SPECT system using the proposed model. The resultant responses were
compared with the emission projection data.
3.3. Application of the model for scatter compensation in image reconstruction
3.3.1. MCAT phantom simulation. A set of essentially noise-free fan beam projection data,
including primary and first-order Compton scattered photons, were generated using Monte
Carlo simulations and the MCAT phantom. The geometric detector response effect was
simulated, and a circular orbit with a radius of 22.0 cm was used when generating the projection
data. The bin size was 0.625 cm and the focal length was 65.0 cm. The energy window used
in the projection simulation was ±10%, centred at the photon peak energy of 140 keV. The
projection data were stored in a 64 × 64 × 120 array. An iterative ordered-subset expectation
maximization (OS-EM) algorithm (Hudson and Larkin 1994, Byrne 1996) was employed to
reconstruct the images, using a rotation and warping based projector/backprojector pair (Zeng
et al 1994). Images were reconstructed from the projection data with and without first-order
Compton scatter. The scatter was modelled only in the projector and not in the backprojector
(Welch and Gullberg 1997). Images reconstructed from scatter-free projection data were used
for comparison.
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(c)
Line profiles were drawn across the left ventricle (LV) wall, and the left ventricle
myocardium-to-chamber contrast (LV contrast) was calculated as
IM − IC
(16)
LV contrast =
IM + I C
where IM was the average reconstructed intensity along the profile line through both myocardial
walls and IC was the average intensity along the line within the left ventricle chamber.
3.3.2. Physical phantom experiment. An anthropomorphic torso phantom with cardiac insert
(Data Spectrum, Hillsborough, NC) was scanned with a Picker PRISM 3000XP three-head
SPECT system with fan beam collimation. The injected activity concentration of 99m Tc in
the cardiac insert was 0.14 MBq (3.83 µCi) cm−3 , and no background activity was injected.
The projection data were acquired from 120 views over 360◦ for 40 stops with 35 s per stop,
and were stored in a 64 × 64 × 120 array in short integer format. The projection bin size
was 0.712 cm, and the reconstruction voxel size was 0.712 × 0.712 × 0.712 cm3 . The energy
window was ±7.5% centred at 140 keV, and a non-circular orbit was used.
All computation tasks were completed on a Sun Ultra Enterprise 3000 (167 MHz, 1G
RAM). Reconstructions were performed with five OS-EM iterations, with four projection
views for each subset.
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Line profiles were drawn in the reconstructed images and the LV contrasts were calculated.
4. Results
4.1. Evaluation of the model using Monte Carlo simulations
4.1.1. Scatter point response functions (SPRFs) for parallel beam geometry. Figure 6 shows
profiles in both absolute scale (a) and log scale (b) of the scatter point responses in parallel
SPECT using the set-up in figure 4. The SPRFs in our previous work (Zeng et al 1999) that were
generated using stationary kernels match extremely well with those of the MC simulations from
the peak region down to one-half of the intensity of the peak for all the detector positions. Using
distance-dependent kernels developed in the proposed blurring model, the SPRFs match well
down to one-tenth of the peak intensity. The tails of the SPRFs generated from the proposed
method match well with those of the MC simulation. Table 2 shows the scatter to primary ratio
values and the percentage difference between the MC results and the results from the proposed
blurring model.
4.1.2. Scatter point response functions (SPRFs) for fan beam geometry. Figure 7 shows
profiles of the scatter point responses in fan beam geometry generated with the proposed model,
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(a)
Figure 8. (a) Scatter responses of a distributed source. From top to bottom, the responses
corresponding to the four detector positions. From left to right, the responses generated with
the proposed model, MC simulation for first-order Compton scatter, and the MC simulation for
Compton scatter up to sixth-order respectively. (b) Linear profiles of the scatter response of a
distributed source in the parallel direction. Vertical axis: relative intensity. Horizontal axis: bin
number. Broad solid line: the proposed method. Thin solid line: MC simulations for first-order
Compton scatter. Dotted line: full MC simulation result. The number at the shoulder of each
profile corresponds to the detector position shown in figure 4. (c) Linear profiles of the scatter
response of a distributed source in the fan direction. Vertical axis: relative intensity. Horizontal
axis: bin number. Broad solid line: the proposed method. Thin solid line: MC simulations for
first-order Compton scatter. Dotted line: full MC simulation result. The number at the shoulder of
each profile corresponds to the detector position shown in figure 4.
MC simulation for first-order Compton scatter, and the MC simulation for Compton scatter up
to sixth-order (fMC) respectively. Figures 7(a) and 7(b) show the linear and semi-log profiles in
the parallel direction and 7(c) and 7(d) show the linear and semi-log profiles in the fan direction.
The comparison of the scatter to primary ratios is shown in table 3. The SPRFs generated using
the proposed model match better with those of MC simulations at the peak region than at the
tails. In the fan beam direction, the peaks of the SPRFs corresponding to detector positions
1 and 3 are shifted slightly from those of the MC simulations. It is hypothesized that this is
introduced by warping and the use of voxels of finite size (section 5.2).
4.1.3. Scatter response of a distributed source for fan beam geometry. Figure 8(a) shows
the scatter responses of a distributed source in fan beam SPECT generated with the proposed
model, MC simulation for first-order Compton scatter, and the MC simulation for Compton
scatter up to sixth-order (fMC) respectively. Figure 8(b) shows the linear profiles of the scatter
point responses of fan beam geometry in the parallel direction and 8(c) shows the profiles in
the fan direction. The comparison of the scatter to primary ratios is shown in table 4.
4.2. Evaluation of the model using physical fan-beam point source data
Figure 9 illustrates a slice of the reconstructed attenuation map, the position of the point source
in the phantom and the detector positions for point response measurements. Figure 10 shows
the line profiles of the point responses obtained by the physical measurement and generated
by the projector that modelled non-uniform attenuation, geometric point response and non-
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Table 4. Scatter to primary ratios of the scatter responses of a distributed source for fan beam
SPECT (fMC, full Monte Carlo results; MC, Monte Carlo results; BL, proposed blurring model
results). The numbers in the first row correspond to the detector positions shown in figure 4. Row 5
is the percentage difference: (BL−MC)/MC×100%, and the last row is (BL−fMC)/MC×100%.
1
2
3
4
fMC
0.559
0.775
0.617
0.362
MC
0.471
0.631
0.514
0.323
BL
0.502
0.610
0.524
0.310
%
+ 6.6
−3.3
+ 2.0
−4.0
%
−11.4
−27.0
−17.7
−16.8
uniform first-order Compton scatter effects. Line profiles were drawn in both parallel and fan
directions.
4.3. Application of the model for scatter compensation in image reconstruction
4.3.1. MCAT phantom simulation. The same transaxial slices of the images are displayed
in figure 11(a) as follows: left and centre images show, respectively, the reconstructions
without and with performing the first-order Compton scatter compensation from the Monte
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(c)
Figure 9. A slice of the reconstructed attenuation map and the two detector positions.
Carlo generated projection data; the image on the right was reconstructed from scatter-free
projection data. The LV contrasts are 0.766 and 0.693 for the images reconstructed with
and without performing the first-order Compton scatter compensation respectively. The LV
contrast of the scatter-free data is 0.784. A slight improvement in resolution is observed in the
line profiles represented in figure 11(b), when scatter compensation is performed.
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(b)
Figure 10. Comparison of the line profiles of the point responses measured with the SPECT system
(thin dashed lines) and generated using the projector (thick solid lines). (a) Detector position 1.
Left: in fan direction. Right: in parallel direction. (b) Detector position 2. Left: in fan direction.
Right: in parallel direction.
4.3.2. Physical phantom experiment. Figure 12(a) shows the same transaxial slices from
the images reconstructed with and without scatter compensation. Line profiles are shown in
figure 12(b). The LV contrast was increased from 0.824 for the image reconstructed without
scatter compensation to 0.903 for the image reconstructed with scatter compensation. Also,
from the line profiles, we observe that the resolution of the reconstructed image is improved
when scatter compensation is performed and myocardial uniformity is also improved.
5. Discussion
5.1. Reconstruction performance of the proposed model
The scatter point response function (SPRF) generated using the proposed method reproduces
well the broad shape of the scatter response (see figures 6 and 7). Thus, performing scatter
compensation is expected to increase the image contrast. This is shown by the results of
improved LV contrast in the reconstructed images (figures 11 and 12). Also, the SPRFs match
those of the MC simulations well in the peak region; that is, a region of 2 bins to each side
of the peak. It is expected that performing scatter compensation using the proposed model
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(a)
(b)
Figure 11. MCAT phantom study. (a) Same slices of the reconstructed images. Left:
reconstructed without scatter compensation. Centre: reconstructed with scatter compensation.
Right: reconstructed from scatter-free data. (b) Line profiles along the line shown in (a). Thin
dashed line: without scatter compensation. Broad solid line: with scatter compensation. Thin
solid line: reconstructed from scatter-free data (gold standard).
gives some improvement in resolution even if geometric response is modelled. In the MCAT
phantom study, images reconstructed with scatter compensation exhibit a slight improvement
in the resolution of the left ventricle walls. Line profiles in figure 12 of the Jaszczak phantom
experiment demonstrate an obvious resolution improvement of the left ventricle walls when
images were reconstructed with scatter compensation in addition to attenuating and detector
response compensation.
5.2. SPRFs for fan beam geometry
The spatial scatter response functions (SSRFs) for fan beam geometry are asymmetric when
the point source is not located on the central ray of the converging beam. One may expect
larger errors when the point source is farther away from the central ray when using Gaussians
to approximate the SSRFs. However, because only the photons that scatter in a small angular
range can be detected due to the narrow energy window, the SSRFs are symmetric in the spatial
region that corresponds to this small angular range. Therefore, Gaussian approximation is a
good approximation (see also appendix B).
In converging beam SPECT, a point source is set on the centre of a voxel to generate the
SPRFs. However, when a rotation and warping based projector is used, the position of the
point source after rotation and warping can be anywhere in a new voxel. The activity image is
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(b)
Figure 12. Physical phantom experiments. The images were zoomed for display. (a) Same
slices from the images reconstructed with (right) and without (left) scatter compensation using the
proposed method. (b) Line profiles along the line shown in (a). Thin dashed line: without scatter
compensation. Broad solid line: with scatter compensation.
voxelized and the projection is performed based on the newly transformed activity distribution.
Therefore a small shift in the SPRFs may result. This is shown by the shift of the peaks in
fan beam SPRFs for detector positions 1 and 3 compared with the MC results displayed in
figure 7. The shift decreases when the voxel size decreases.
5.3. Attenuation effect at different photon energies
In equation (14) it is assumed that Compton scatter dominates the attenuation coefficient in
SPECT when the photon energy is high. In table 5 we list the ratio of the attenuation effect due
to photoelectric absorption to the attenuation effect due to Compton scatter for water, brain and
fat at energies of 140 keV, 90 keV and 10 keV calculated using the equations and the values
given in the work of Alvarez and Macovski (1976). For a variety of other human tissues, the
relative attenuation effect due to the photoelectric absorption to the Compton scatter effect is
nearly the same as for brain and water. One can check the results of water with those given by
Sorenson and Phelps (1987).
For bone, because of its higher effective atomic number (effective Z number), the
photoelectric absorption can be much greater than that of water or other human tissues.
However, bone accounts for only a small portion of the torso, so the error introduced by the high
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Table 5. Ratios of the attenuation effect due to photoelectric absorption to that due to Compton
scatter.
E (keV)
Water
Brain
Fat
Bone
10
90
140
21.85
0.038
0.011
22.03
0.038
0.011
12.52
0.022
0.007
140.57
0.24
0.07
photoelectric effect of bones can be neglected. For lungs, their composition changes in a large
range, and Compton scatter dominates the attenuation effect due to the low atomic number.
5.4. Increased attenuation for the scattered photons
The energies of the scattered photons are generally lower than those of the incident photons,
therefore the attenuation along the same photon path in scattering media is more for the
scattered photons than for the primary ones. This effect has been studied by Frey and Tsui
(1996). However, in this work, this effect was not modelled.
5.5. Computation time
For parallel beam SPECT, the scatter point response functions (SPRFs) simulated using the
distance-dependent kernels in the proposed blurring model and the stationary kernels used in
our previous work (Zeng et al 1999) agree extremely well with the results from MC simulations
that include only first-order Compton scatter from the peak intensity down to one-half of the
peak intensity in the peak regions, and also agree well in the tail regions (figure 6). However,
the time required to generate an SPRF using the varying kernels in the proposed model is
approximately 32 times as long as the time required when using stationary kernels (Zeng et al
1999). Therefore, while it takes approximately 1 s to generate a 64 × 64 SPRF using the
stationary kernels, it takes 32 s when using the varying kernels. It is observed that when x is
greater than 5, the small blurring kernels are essentially stationary. The spatially varying kernels
are significantly different from the stationary kernels only when x is less than or equal to 3.
For a given point source and for the first few scattering slices (x = 1, 2, 3, 4, 5), the central
element of the blurring kernel is different from the side elements, and the difference decreases
as x increases. When x is greater than 5, the central element is essentially the same as the
side elements. That being the case, one can apply stationary kernels when x is greater than
5. Based on this observation, the slice-by-slice blurring can be accelerated by using spatially
varying kernels up to x = 5, and stationary kernels after. In this way the computation time
for generating the SPRF can be reduced by a factor of 5. This is also applicable to fan beam
geometry. The total reconstruction time required when only attenuation compensation and 3D
geometric detector response compensation are performed is 5.2 s per slice per iteration. When
3D scatter compensation is also performed, the reconstruction time per slice per iteration is
6.9 s when using the stationary model (parallel beam SPECT), 60 s when using the spatially
varying kernels and approximately 16 s when stationary kernels are used after x = 5.
The acceleration procedure is only needed for distributed sources. The implementation
of this procedure is as follows: for each source slice which is parallel to the detector surface,
the ISSIs for the scattering slices x = 1, 2, 3, 4 are computed. We call these the first part
of the ISSI for that particular source slice. The first parts of the ISSIs for all the source slices
are generated and appropriately added up to form the first part of the ISSI for the distributed
source. The ISSIs for the scattering slice x = 5 for all the source slices are stored in a new
array to which an incremental blurring using stationary kernels can be applied (Zeng et al
1301
1999) to generate the second part of the ISSI for the distributed source. Combining the first
part and the second part of the ISSI gives the total ISSI for the distributed source, which can
then be used to generate the ESSI. Therefore the time required to generate the ISSI is about six
times longer than when stationary kernels are used (Zeng et al 1999). The acceleration factor
is about five when the transaxial slice of the source image has a dimension of 64 × 64, and ten
when the transaxial slice of the source image has a dimension of 128 × 128.
5.6. Application of the model for different agents
The proposed slice-by-slice blurring model has been developed for the modelling of first-order
Compton scatter in SPECT with the detection energy window centred at the photopeak energy.
For a non-photopeak detection energy window, the energy-detection-probability-modulated
SSRFs (see appendix B) do not have a simple shape (a modulated SSRF usually has two
peaks), thus the proposed model cannot be simply applied to model the scatter for the nonphotopeak energy window. For SPECT imaging using emission agents such as 99m Tc, the
detection energy window is centred at the photopeak energy and the model can be applied
directly. For some agents, 201 Tl for example which has multiple emission energy peaks, the
photons down-scattered from higher energy peaks can be detected in the detection energy
window of the lower energy peaks. These parts of the scattered photons cannot be estimated
by a direct application of the model. For 201 Tl backscatter is a significant contributor to the
scatter response, and our scatter model in the current form cannot be applied. In our future
work, the model will be modified so that it can be used to model the down-scattered events
and also backscattered events in SPECT.
6. Conclusions
In this work we have developed a method to model first-order Compton scatter in parallel and
converging beam SPECT. The scattering angle in the Klein–Nishina formula is expressed in
terms of the positions of the point source, scattering point, detection point and the geometric
focus, so that the spatial scatter response function (SSRF) and the intermediate scatter source
image (ISSI) of a scattering slice to the point source can be computed. By convolving this ISSI
with two orthogonal small 1D kernels, the ISSI of the next scattering slice, which is closer
to the detector surface, can be approximated. An effective scatter source image (ESSI) can
than be obtained from the ISSI. The scatter point response function (SPRF) is generated by
projecting the ESSI to the detector with a projector modelling both attenuation and geometric
detector response effects. The proposed slice-by-slice blurring model has been evaluated
using Monte Carlo simulation studies and physical phantom experiments. The scatter point
response functions (SPRFs) generated using the proposed model match well with those of
the MC simulations that include only first-order Compton scatter for both parallel and fan
beam geometries. For fan beam geometry, images reconstructed from the projection data
generated by MC simulations using the MCAT phantom with the first-order Compton scatter
effect present a higher LV contrast (0.766) when scatter compensation is performed than when
it is not performed (0.693). Slight improvement in the image resolution is also observed.
Physical phantom experiments using a large Jaszczak torso phantom show that the LV contrast
is increased from 0.824, when no scatter compensation is performed, to 0.903 when the scatter
compensation using the proposed model is performed. Image resolution and myocardial
uniformity are also improved by performing scatter compensation. The proposed incremental
blurring model provides an efficient method for first-order Compton scatter compensation in
both parallel and converging beam SPECT.
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C Bai et al
Appendix A. Discrete implementation of the model
In the discrete implementation of the slice-by-slice blurring model for first-order Compton
scatter, one should consider the solid angle subtended by a bin on the detector surface with the
origin at the centre of a scattering voxel in the scattering object, and the solid angle subtended
by the scattering voxel with the origin at the point source. Also, one should consider the
normalization of the Gaussian functions which are used to approximate the spatial scatter
response functions (SSRFs).
Figure A1 shows a scattering voxel on a scattering slice x in the scattering object and a
bin on the detector surface in fan beam geometry. This figure is the same as figure 1, except
that the scattering point SC and the detection point D are now the scattering voxel and the
detection bin respectively.
Figure A1. A plot for the discrete implementation of the slice-by-slice blurring model.
The point source intensity is ISS , the total number of photons that reach the scattering
voxel, can be approximated by (this only considers the decrease of flux due to the increase of
distance from the source)
ISS
A
2
4π x + (ySC − ySS )2
(A1)
where A is the cross section of the voxel to the photon flux. The second term in the expression
is the solid angle subtended by the scattering voxel with the origin at the source. When using
the bin size as the unit of length, the value of A is 1.
The detected scattered photon at the bin on the detector surface is approximated as
ISS
A
A
RSC F (θ, E) 2
2
2
4π x + (ySC − ySS )
xSC + (yd − ySC )2
(A2)
where the last term in the expression is the solid angle subtended by the detection bin with the
origin at the centre of the scattering voxel.
In section 2.3 we indicated that the Gaussian functions used to approximate the SSRFs
have a maximum value of 1 and an FWHM which is proportional to x. Therefore, if we
use the normalized Gaussian functions (and use them to compute the small blurring kernels)
in equations (6) to (12), it is equivalent to using F (θ, E) = Fˆ [θ, E]x 2 in expression (A2),
which gives:
ISS
A
Ax 2
RSC Fˆ (θ, E) 2
.
2
2
4π x + (ySC − ySS )
xSC + (yd − ySC )2
(A3)
1303
For the detection of primary photons one should also consider the solid angle subtended
by the detection bin (0, yp ) with the origin at the source. The flux to the detection bin (without
attenuation) is
ISS
A
.
(A4)
2
4π xSS
+ (yp − ySS )2
Dividing expression (A3) by (A4) and taking into account that A = 1, one can obtain
2
xSS
+ (yp − ySS )2
x 2
(A5)
RSC Fˆ (θ, E).
2
x 2 + (ySC − ySS )2 xSC
+ (yd − ySC )2
This illustrates how to modify equation (13) in the discrete implementation of the slice-byslice blurring model, in order to obtain the correct quantitation (scatter-to-primary ratio). The
modification is done by multiplying each scattering voxel by its corresponding factor given by
the term in the large brackets in expression (A5):
x 2
2
xSS
+ (yp − ySS )2
x 2
.
2
2
+ (ySC − ySS ) xSC + (yd − xSC )2
(A6)
In our implementation, we approximate the first term in expression (A6) by 1, and
2
2
approximate the second term in (A6) by xSS
/xSC
, therefore expression (A6) is simplified
to
2
xSS
.
(A7)
2
xSC
In each projection view, the computation of ISSI is the same as described from equation (6)
to equation (12). Then computation of ESSI using equation (13) is performed by the
voxel-wise multiplication by the composition-dependent scatter factor and the factor given
by expression (A7). For a distributed source, in order to decrease the computation, the
multiplication by the factor in (A7) is performed first for each source position by performing
2
2
a multiplication by xSS
for all the scattering slices, then dividing by xSC
for each slice after
the sum has been performed over all sources. In this way the division step does not have to be
performed in the calculation at each source position.
Another way to simplify expression (A6) is to integrate the first term into the SSRFs to
give spatially modified SSRFs (see also the discussion in appendix B). The modulated SSRFs
can then be used to obtain the Gaussian functions in a manner similar to that described in
section 2.3. Note that the FWHMs of the Gaussian functions obtained from the modulated
SSRFs are also approximately proportional to x, this property makes expression (A3) valid.
Appendix B. Modulated SSRFs and their Gaussian approximations
The proposed slice-by-slice blurring model assumes that energy detection probability P (E)
is a window function of energy E with value 1 when E is within the energy window and 0
when E is outside the energy window. Theoretically this is not correct. A key point of this
model is to compute the spatial scatter response functions (SSRFs) from the Klein–Nishina
formula, and to approximate the SSRFs with Gaussian functions, considering the energy of the
incident photons, the collimation geometry and the detection energy window. The detection
probabilities of the scattered photons with different energies modulate the shape of the SSRFs
and result in the energy-detection-probability modulated SSRFs with shapes that can be very
different from the original SSRFs. Also, when we perform appropriate normalization in the
implementation of the model as discussed in appendix A, we can integrate the first term of
expression (A6) into the SSRFs. This procedure gives the spatially modified SSRFs.
1304
C Bai et al
Now we examine the SSRFs that are calculated directly from the Klein–Nishina formula
(denoted as KN), the spatially modified SSRF (denoted as KN SPL), the energy-detectionprobability-modulated SSRF (denoted as KN ENG), and the energy-detection-probabilitymodulated and spatially modified SSRF (denoted as KN ENG SPL). Figure B1 shows the
plots of KN, KN SPL, KN ENG, and KN ENG SPL in fan beam direction for a scattering
slice x = 1 when the point source is at (42,0). The photopeak energy, detection energy
window, and the collimation geometry are given in section 2.3. KN ENG and KN ENG
SPL have been scaled (divided by the energy detection probability of the primary photons)
so that both of them have a maximum value of 1. Also plotted in figure B1 is the Gaussian
function (denoted as GAU) that is obtained in section 2.3 with the assumption that the energy
detection probability function is a window function. The procedure described in section 2.3 is
used to calculate the HWHM of the Gaussian approximation of each of the above four SSRFs.
The FWHMs of the Gaussian approximations are 2.24 bins, 2.47 bins, 2.15 bins and 2.22 bins
for KN, KN ENG, KN SPL and KN ENG SPL respectively.
Figure B1. The theoretical SSRF in the fan beam direction for the geometry given in
section 2.3 calculated directly from the Klein–Nishina formula (KN), the energy-detectionprobability-modulated SSRF (KN ENG), the spatially modified SSRF (KN SPL), the energydetection-probability-modulated and spatially modified SSRF (KN ENG SPL), and the Gaussian
approximation (GAU) of KN obtained in section 2.3. The point source is at (42,0), the scattering
slice is x = 1. Horizontal axis: fan direction. The SSRFs with energy modulation (KN ENG
an KN ENG SPL) are scaled (divided by the detection probability of primary photons).
This same examination has been performed for a wide range of point source positions and
different scattering slices from those used in table 1. It is observed that the shape of GAU is
close to those of KN SPL, KN ENG and KN ENG SPL, but much different from the shape
of KN, and the FWHM for the Gaussian approximation of KN is only slightly different from
those of the energy-detection-probability modulated and spatially modified SSRFs. Therefore,
even though the shape of the Gaussian approximation obtained with the assumption used in
section 2.1.2 is different from the theoretical SSRF (KN), quantitatively the approximation is
close to the result when the correct energy detection probability function is used.
In SPECT image reconstruction, in order to perform 3D Compton scatter compensation,
only the correct scatter to primary ratio and the correct ‘shape’ of the scatter response are
required. Thus we can perform accurate first-order Compton scatter compensation using the
1305
proposed model without modelling the energy detection probabilities for either the scattered
photons or the primary photons. To generate an absolute first-order Compton scatter projection
we only need to scale the scatter projection obtained using the proposed model by multiplying
by a factor which is the detection probability of the primary photons for the given detection
energy window.
Acknowledgments
The authors would like to thank Dr John Viner, Dr Wayne Springer, Dr Clayton Williams and
Dr James Cronin for discussions on the Compton scatter theory. We thank Dr Dan Kadrmas
for his helpful discussions on this paper and thank Paul Christian for arranging the phantom
scans. We also thank Sean Webb for editing the manuscript. This work is partially supported
by NIH grant R29 HL51462, HL39792, and a grant from Picker International.
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