PET transmission tomography using a novel nonlocal MRF prior

Abstract

In positron emission tomography, transmission scans can be performed to estimate attenuation correction factors (ACFs) which are in turn used to correct the emission scans. And such an attenuation correction is crucial for quantitatively accurate PET reconstructions. The prior model used in this work was based on our assumption that the attenuation values vary smoothly, with occasional discontinuities at anatomical borders. And on the other hand, long acquisition or scan times, although alleviating the noise effect of the count-limited scans, are blamed for patient uncomfortableness and movements. So, transmission tomography often suffers from the noise effect because of the short scan time. Thus reconstruction which is capable of overcoming the noise effect is highly needed. In this article, we apply the nonlocal prior Bayesian reconstruction method in PET transmission tomography. Resulting experimentations validate that the reconstructions using the nonlocal prior can reconstruct better transmission images and overcome noise effect even when the scan time is relatively short.

Introduction

In positron emission tomography (PET), a chemical compound labeled with a positron emitting radioisotope is injected into the bloodstream of the patient. The purpose is to obtain an image of the concentration of that chemical compound which is related to a biological function in the patient’s body. The radiotracer nucleus change from a metastable state to a stable state by emiting a positron (positively charged electron). The emitted positron annihilates with a nearby electron to form a pair of 511 keV photons propagating in opposite directions. A positron-electron annihilation is considered to take place in the line joining two detectors when such a pair of photons are detected almost simultaneously outside the subject. This simultaneous detection is termed coincidence detection. And each coincidence detection increments a counter that represents the line integral specified by two detectors. After a certain scan time, line integral detected sinogram data of the radioisotope density are obtained [1], [2].

However, not all annihilations that result in a photon pair heading towards two detectors are detected. Often, one or both of the photons get scattered or absorbed by the patient body resulting in no detection. The survival probability of an annihilation event is determined by the length and the type of the tissue that the photons traverse. This effect is called attenuation. The attenuation map or attenuation correction factors (ACFs) express spatially the mass absorption coefficients for the transaxial slice of the body. Such attenuation is different for different tissue types, hence the measurements should be compensated for attenuation for each ray (or line integral) [3].

Conventionally, the ACFs are computed from the blank and the transmission sinograms by dividing them. In clinical practice, short scan times with precise attenuation correction are requested for quantitative whole body studies because statistically desired long acquisition times for transmission are clinically impractical and inconvenient to patients. However, short scan times with low total counts levels often bring statistical noise to reconstructions. So, the short scan data are conventionally smoothed with the resulting unfavorable blurring of details propagating to the emission sinogram [3], [4], [5].

Attenuation correction can also be performed by computational methods [3], [6]. The attenuating matter is approximated by an area with a uniform value of linear mass absorption coefficient $f$. The ACFs can be computed by projecting the graphically defined $f$-image and taking the exponential:

$$\text{ACF}=\exp \left({\int }_{l}f(x)\mathrm{d}x\right)$$

For above computational methods, when filtered back projection (FBP) is used in count-limited reconstructions, noise and some bias may be introduced in the image [6]. In order to remove the noise from the transmission image, the segmented attenuation correction method classifies pixels of the image to certain tissue types with typical attenuation values before the projection. But the parameter-relevant segmentations are rather complicated and might be unpredictable especially for the noisy FBP reconstruction. In practice, some details may not properly contribute to the ACFs and noise on the other hand may wrongly contribute to the ACFs especially in the case of short scan [7], [8], [9].

Bayesian methods, which have been widely accepted for producing better reconstructions, are also recommended for transmission tomography [10], [11], [12]. Ollinger proposed one step of Newton’s method for transmission reconstruction [9]. And Lange put forward an one step late expectation-maximization (OSL-EM) iterative reconstruction algorithm for Bayesian reconstruction of transmission tomography [10]. In the past 20 years, Fessler and his group have done a great deal of remarkable work on emission and transmission tompgraphies. They have devised effective convergent iterative algorithms for PET transmission tomography [11], [12], [13], [14].

As to the Bayesian reconstruction for transmission tomography, the simple quadratic membrane (QM) smoothing prior, which smoothes both noise and edge details equally, tends to produce an unfavorable oversmoothing effect. Considering the fact that the transmission attenuation maps are often composed of homogeneous regions with sharp boundaries, edge-preserving nonquadratic priors, which are able to produce sharp edges by choosing a nonquadratic prior energy, have been widely used in the studies of CT or PET transmission tomography [10], [11], [12], [13], [14], [15], [16], [17]. And in 1998, edge-preserving median root prior (MRP) was proposed by Alenius et al. for iterative reconstruction of PET transmission images [18].

However, in the case of short transmission scan when the noise level is relatively significant, edge-preserving nonquadratic priors tend to produce blocky piecewise regions or so-called staircase artifacts. What is more, the application of edge-preserving Bayesian methods are often hindered by the generally needed annealing procedures and complicated parameter estimations for some built-in parameters [10], [15], [16], [17]. Furthermore, the prior energies’ convexities can not be preserved for some edge-preserving nonquadratic priors, which might endanger the whole concavities of their overall objective posterior energy functions and hinder the application of convergent algorithms [10].

The quadratic prior smooth the reconstructed image through an averaging alike operation on pixel densities within a local neighborhood. The edge-preserving nonquadratic priors rely on pixel intensity difference information within a local neighborhood to determine the degrees of regularizing effect on every pixel in image [10], [11], [12], [13], [14], [15], [16], [17]. Both the annoying oversmoothing effect for quadratic priors and staircase effect for nonquadratic priors are the outcomes of their limited power of distinguishing edge information from noise information indiscriminatively. And such limited power can be ascribed to the fact that the simply weighting strategy for pixel density differences within a small fixed local neighborhood can only provide limited prior information. None of above priors addresses the information of global connectivities and continuities in objective image and we term these traditional priors local priors.

Yu et al. devised a boundary-based Bayesian method which incorporates global information of image by level-set methods [19]. But such boundary-based method relies heavily on the level-set operations whose effect in different images is unpredictable and parameter-dependent. Recently, an effective nonlocal MRF quadratic prior model for emission image reconstruction is proposed [20]. This nonlocal quadratic prior can greatly improve the reconstruction by exploiting not only density differences information between individual pixels but also global connectivity and continuity information in the objective image.

In this article, we apply such nonlocal MRF quadratic prior model in transmission image reconstruction. Our aim is to reconstruct high quality transmission images even with relatively short acquisition times. In Section 2, the review of the old local prior model and the theory for the proposed nonlocal prior model are both illustrated. In Sections 3 Joint reconstruction strategy, 4 Experimentations, we give the iterative reconstruction algorithm and perform PET transmission reconstruction using the proposed prior for both simulated data and real scan data. Relevant qualitative and quantitative comparisons show the proposed nonlocal prior’s good properties in lowering noise effect and preserving edges for transmission reconstruction with different scan times. Conclusions and plans for future work are given in Section 5.