Image-based point-spread-function modelling in time-of-flight positron-emission-tomography iterative list-mode reconstruction
A method of calculating a system matrix for time-of-flight (TOF) list-mode reconstruction of positron-emission tomography (PET) images, the method including determining a TOF geometric projection matrix G including effects of object attenuation; estimating an image-blurring matrix R in image space; obtaining a diagonal matrix D that includes TOF-based normalization factor; and calculating the system matrix H as H=DGR.
1. A method of performing time-of-flight (TOF) list-mode reconstruction of a positron-emission tomography (PET) image, the method comprising:
detecting gamma rays by a PET detector;
generating count data based on the detected gamma rays;
determining a TOF geometric projection matrix G including effects of object attenuation;
estimating an image-blurring matrix R in image space;
obtaining a diagonal matrix D that includes TOF-based normalization factors;
calculating a system matrix H as H=DGR; and
reconstructing the PET image from the count data using the calculated system matrix.
2. The method of claim 1 , wherein the estimating step comprises:
estimating R as R=AM,
wherein r j is a j th blurring kernel at voxel j and a j th column of R, A is an analytic blurring matrix and is defined as A={α j } with
α
^
j
=
arg
max
α
j
⩾
0
Φ
(
P
_
non
-
TOF
e
j
,
G
non
-
TOF
α
j
)
P non-TOF is an analytically calculated non-TOF system matrix, e j is a j th unit vector, G non-TOF is a non-TOF geometric projection matrix, and Φ is the Poisson log-likelihood function; and
M is a measurement-based blurring matrix.
3. The method of claim 2 , further comprising calculating the matrix M by computing
x
^
j
=
arg
max
x
j
⩾
0
Φ
(
y
j
,
diag
{
n
non
-
TOF
}
P
_
non
-
TOF
x
j
)
.
for a given point source scan data y j , wherein x j is the reconstructed point source kernel image, diag{n non-TOF } is a diagonal matrix whose diagonal components contain non-TOF-based normalization factors stored in a vector n non-TOF .
4. The method of claim 3 , wherein the step of calculating the matrix M comprises:
fitting the reconstructed point source kernel image to a parametric model to obtain a parameter set;
repeating the computing and fitting steps for a plurality of point sources to obtain a corresponding plurality of parameter sets; and
performing interpolation, using the parameter sets, to obtain kernel images at each voxel j, and using the obtained kernel images to form the matrix M.
5. The method of claim 1 , wherein the estimating step comprises:
estimating r j , wherein r j is a j th blurring kernel image at voxel j and a j th column of R, by reconstructing point source scan data using a non-TOE ordered-subset expectation-maximization (OS-EM) algorithm with a system matrix H m =D m G nonTOF , wherein D m includes detector normalization factors;
fitting a parametric model to the estimated blurring kernel image r j using least-squares estimation to obtain a parameter set p j ;
repeating the estimating and fitting steps for a plurality of point source voxels to obtain a corresponding plurality of parameter sets; and
performing interpolation, using the obtained parameter sets, to obtain kernel images at each voxel j, and using the obtained kernel images to form the matrix R.
6. The method of claim 5 , wherein the parametric model is
f ( r,t,z|c,μ r ,σ r1 ,σ r2 ,μ t ,σ t ,μ z ,σ z )= c ϕ( r|μ r ,σ r1 ,σ r2 )ψ( t|μ t ,σ t )γ( z|μ z ,σ z )
wherein c is a scaling factor indicating a global amplitude of the blurring kernel, ϕ(r|μ r , σ r1 , σ r2 ) is a radial blur function, which is parameterized as an asymmetric Gaussian with mean μ r , left and right standard deviation σ r1 and σ r2 , ψ(t|μ t , σ t ) is a tangential blur function, which is parameterized as a Gaussian with mean μ t and standard deviation σ t , and γ(z|μ z , σ z ) is an axial blur function with mean μ z and standard deviation σ z .
7. The method of claim 1 , further comprising:
displaying, on a display, the reconstructed PET image.
8. An apparatus for performing time-of-flight (TOP) list-mode reconstruction of a positron-emission tomography (PET) image, the apparatus comprising:
a PET detector configured to detect gamma rays; and
a processor configured to
generate count data based on the detected gamma rays;
determine a TOF geometric projection matrix G;
estimate an image-blurring matrix R in image space;
obtain a diagonal matrix D that includes TOF-based normalization factors;
calculate a system matrix H as H=DGR; and
reconstruct the PET image from the count data using, the calculated system matrix.
9. The apparatus of claim 8 , wherein the estimating step comprises:
estimating R as R=AM,
wherein r j is a j th blurring kernel at voxel j and a j th column of R, A is an analytic blurring matrix and is defined as A={α j } a
α
^
j
=
arg
max
α
j
⩾
0
Φ
(
P
_
non
-
TOF
e
j
,
G
non
-
TOF
α
j
)
P non-TOF is an analytically calculated non-TOF system matrix, e j is a j th unit vector, G non-TOF is a non-TOF geometric projection matrix, and Φ is the Poisson log-likelihood function; and
M is a measurement-based blurring matrix.
10. The apparatus of claim 9 , further comprising calculating the matrix M by computing
x
^
j
=
arg
max
x
j
⩾
0
Φ
(
y
j
,
diag
{
n
non
-
TOF
}
P
_
non
-
TOF
x
j
)
.
for a given point source scan data y j , wherein x j is the reconstructed point source kernel image, diag{n non-TOF } is a diagonal matrix whose diagonal components contain non-TOF-based normalization factors stored in a vector n non-TOF .
11. The apparatus of claim 10 , wherein the step of calculating the matrix M comprises:
fitting the reconstructed point source kernel image to a parametric model to obtain a parameter set:
repeating the computing and fitting steps for a plurality of point sources to obtain a corresponding plurality of parameter sets; and
performing interpolation, using the parameter sets, to obtain kernel Images at each voxel j, and using the obtained kernel images to form the matrix M.
12. The apparatus of claim 8 , wherein the estimating step comprises:
estimating r j , wherein r j is a j th blurring kernel image at voxel j and a j th column of R, by reconstructing point source scan data using a non-TOF ordered-subset expectation-maximization (OSEM) algorithm with a system matrix H m =D m G nonTOF , wherein D m includes only detector normalization factors;
fitting a parametric model to the estimated blurring kernel image r j using least-squares estimation to obtain a parameter set p j ;
repeating the estimating and fitting steps for a plurality of point source voxels to obtain a corresponding plurality of parameter sets; and
performing interpolation, using the obtained parameter sets, to obtain kernel images at each voxel j, and using the obtained kernel images to form the matrix R.
13. The apparatus of claim 12 , wherein the parametric model is
f ( r,t,z|c,μ r ,σ r1 ,σ r2 ,μ t ,σ t ,μ z ,σ z )= c ϕ( r|μ r ,σ r1 ,σ r2 )ψ( t|μ t ,σ t )γ( z|μ z ,σ z )
wherein c is a scaling factor indicating a global amplitude of the blurring kernel, ϕ(r|μ r , σ r1 , σ r2 ) is a radial blur function, which is parameterized as an asymmetric Gaussian with mean μ r , left and right standard deviation σ r1 and σ r2 , ψ(t|μ t , σ t ) is a tangential blur function, which is parameterized as a Gaussian with mean μ t and standard deviation σ t , and γ(z|μ z , σ z ) is an axial blur function with mean μ z and standard deviation σ z .