This work deals with the single photon emission computed tomography (SPECT) reconstruction process. As a SPECT measurement also depends on unknown attenuation properties of the tissue, such a process is challenging. Furthermore, the given attenuation may not be a good approximation to the true attenuation field. Reasons are repositioning or movement of the patient such as relaxation during scan time or even breathing. We propose a novel model for an attenuation correction in SPECT reconstruction, which is a natural extension of an idea of Natterer in that way, as the linear transformation of a so-called attenuation prototype is enhanced to an arbitrary transformation. We present numerical results for a non-linear spline transformation model which clearly indicate the superiority of the proposed reconstruction model compared to the case of no motion correction and the correction with a linear transformation model.
Single photon emission computed tomography (SPECT) is a nuclear medicine
imaging technique which can provide in vivo 3D functional information of
tissue. More precisely, functional information corresponds to the density of an
administered radio-pharmaceutical which has to be reconstructed from a set of
projections also known as sinogram. The reconstruction process is challenging
as the sinogram also depends on unknown attenuation properties of the tissue.
A commonly used simplification is to assume that the attenuation field is given
and to solve only for the density [
Because of these limitations, recent approaches aim to simultaneously
reconstruct the density and attenuation [
This paper pursues the work of Natterer [
This paper is organized a follows. A novel reconstruction model is presented in Section 2 and two numerical examples which clearly demonstrate superiority of the non-linear approach are shown in Section 3. Finally, in Section 4, results are discussed and an outlook is given. 2
In this section we introduce a novel model for the simultaneous reconstruction
of the tracer density f and the attenuation field µ, where we assume that the
unknowns f, µ ∈ L2(R2, R) are compactly supported on a domain Ω ⊂ R2. As
common, the forward projection process is modeled by g = P[f, µ] with the
projection operator P (attenuated ray transform) and the measurement denoted
by g, e.g., [
J [f, µ] = ∥g − P[f, µ]∥L2 + αR[µ] (1) where α denotes a positive regularization parameter and R a regularizer, which is discussed in detail later.
Natterer used a similar approach with the constraint µ(x) = ν(ℓ(x)), where ℓ
denotes an affine linear spatial transformation and ν denotes a given prototype
of the expected attenuation [
The new idea is to extend the class of feasible transformation and to replace the consistency assumption by regularization. To simplify our presentation, we restrict our model to free-form transformations, i.e. the transformation can be expanded in terms of basis functions Bi (e.g. monomials of degree less equal to one and B-splines of a certain order) and coefficients ci. Moreover, we also limit our discussion to plain Tikhonov regularization of the coefficient vector. However, it is noticeable that our model is far more general and allows for non-parametric transformations and arbitrary regularization. With the above specification, we have
R[µ] = ∥c∥22
where
µ ∈ M := {µ(c; x) = ν(x + ∑iciBi(x)), c ∈ Rn}
(2)
The mathematical model is to minimize J 1 subject to µ ∈ M 2. As an
analytical solution to this problem is unknown, we employ numerical optimization
techniques. In particular, we follow the discretize then optimize paradigm and a
straightforward cell-centered discretization of the spatial domain. To eliminate
the constraint, we use a reduction approach and the unknown of the reduced
model are the density values f = (fi)im=1 on cell-centers and the coefficients
c ∈ Rn. The optimization is performed with a generalized Gauss-Newton scheme
and an Armijo Linesearch, e.g. [
The scheme has been implemented using MATLAB R 2010a. In order to have
a gold standard for comparison and validation, we use synthetic data generated
using the XCAT phantom [
The activity and the attenuation images are calculated by the XCAT software based on a 140 KeV photon energy. The two time steps related to the attenuation images assume a respiration period of 5 seconds and are simulated at 0 seconds and 2.5 seconds. This corresponds to a 0 % and 96 % inhale, according to the default respiration of the XCAT phantom. The maximal motion of the chest is parameterizable and choosen to be 1.2 cm. According to the XCAT phantom a movement of 1.2 cm simulates a normal breathing activity. For the sake of testing (a) f0 (b) (c) 0 (d) g of the proposed motion correction in SPECT reconstruction, the comparatively larger non-linear movement of the chest with a 2.4 cm a.-p. movement is choosen as a second test case. In Figure 1 the latter test case is shown exemplarily. In the following the reconstruction results for the test case presented in Figure 1 are illustrated in Figure 2 in comparison with the gold standard provided by the XCAT phantom. In Table 1 a relative error between the gold standard and different reconstructions is shown.
(a) no correction (b) linear (c) non-linear
The proposed model for a motion correction of SPECT reconstruction is a natural extension of the idea of Natterer as the linear transformation model for the attenuation prototype ν is enhanced to an arbitrary transformation. We present results for a non-linear spline transformation model which clearly indicate the superiority of the proposed reconstruction model compared to the case of no motion correction and the correction with a linear transformation model (Tab. 1).
Due to the nature of the addressed SPECT reconstruction problem, the type and weight of the regularization is a crucial issue. In particular, an improved regularization model is under current investigation.
As the current implementation deals with 2D reconstructions of the density of radioactivity and the attenuation, future work involves the implementation of a 3D reconstruction. Furthermore it is planned to explore different transformation models, as well as different regularizations. Because of the promising and superior results of the previous section it would be interesting, to validate the proposed reconstruction model in a clinical setting.