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An efficient approach with theoretical guarantees to simultaneously reconstruct activity and attenuation sinogram for TOF-PET

Liyang Hu, Chong Chen

TL;DR

This work addresses attenuation correction in TOF-PET by jointly reconstructing activity and attenuation sinogram from emission data alone. It introduces MLAAS, a maximum-likelihood framework that exploits the exponential attenuation form and a mask-based total-activity constraint, and provides existence, uniqueness (up to a constant), and stability results for the discrete problem. An alternating update algorithm with convergence guarantees solves MLAAS efficiently, and extensive numerical experiments demonstrate convergence, robustness to noise, and superior accuracy/efficiency compared with MLAA and MLACF. The approach offers autonomous attenuation correction, reducing reliance on CT/MRI and enabling more robust, quantitatively accurate PET imaging in the presence of motion and misregistration.

Abstract

In positron emission tomography (PET), it is indispensable to perform attenuation correction in order to obtain the quantitatively accurate activity map (tracer distribution) in the body. Generally, this is carried out based on the estimated attenuation map obtained from computed tomography or magnetic resonance imaging. However, except for errors in the attenuation correction factors obtained, the additional scan not only brings in new radiation doses and/or increases the scanning time but also leads to severe misalignment induced by various motions during and between the two sequential scans. To address these issues, based on maximum likelihood estimation, we propose a new mathematical model for simultaneously reconstructing the activity and attenuation sinogram from the time-of-flight (TOF)-PET emission data only. Particularly, we make full use of the exclusively exponential form for the attenuation correction factors, and consider the constraint of a total amount of the activity in some mask region in the proposed model. Furthermore, we prove its well-posedness, including the existence, uniqueness and stability of the solution. We propose an alternating update algorithm to solve the model, and also analyze its convergence. Finally, numerical experiments with various TOF-PET emission data demonstrate that the proposed method is of numerical convergence and robust to noise, and outperforms some state-of-the-art methods in terms of accuracy and efficiency, and has the capability of autonomous attenuation correction.

An efficient approach with theoretical guarantees to simultaneously reconstruct activity and attenuation sinogram for TOF-PET

TL;DR

This work addresses attenuation correction in TOF-PET by jointly reconstructing activity and attenuation sinogram from emission data alone. It introduces MLAAS, a maximum-likelihood framework that exploits the exponential attenuation form and a mask-based total-activity constraint, and provides existence, uniqueness (up to a constant), and stability results for the discrete problem. An alternating update algorithm with convergence guarantees solves MLAAS efficiently, and extensive numerical experiments demonstrate convergence, robustness to noise, and superior accuracy/efficiency compared with MLAA and MLACF. The approach offers autonomous attenuation correction, reducing reliance on CT/MRI and enabling more robust, quantitatively accurate PET imaging in the presence of motion and misregistration.

Abstract

In positron emission tomography (PET), it is indispensable to perform attenuation correction in order to obtain the quantitatively accurate activity map (tracer distribution) in the body. Generally, this is carried out based on the estimated attenuation map obtained from computed tomography or magnetic resonance imaging. However, except for errors in the attenuation correction factors obtained, the additional scan not only brings in new radiation doses and/or increases the scanning time but also leads to severe misalignment induced by various motions during and between the two sequential scans. To address these issues, based on maximum likelihood estimation, we propose a new mathematical model for simultaneously reconstructing the activity and attenuation sinogram from the time-of-flight (TOF)-PET emission data only. Particularly, we make full use of the exclusively exponential form for the attenuation correction factors, and consider the constraint of a total amount of the activity in some mask region in the proposed model. Furthermore, we prove its well-posedness, including the existence, uniqueness and stability of the solution. We propose an alternating update algorithm to solve the model, and also analyze its convergence. Finally, numerical experiments with various TOF-PET emission data demonstrate that the proposed method is of numerical convergence and robust to noise, and outperforms some state-of-the-art methods in terms of accuracy and efficiency, and has the capability of autonomous attenuation correction.
Paper Structure (22 sections, 14 theorems, 94 equations, 11 figures, 4 tables, 1 algorithm)

This paper contains 22 sections, 14 theorems, 94 equations, 11 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main contributions.
  3. Outline.
  4. The proposed mathematical model and its well-posedness
  5. The proposed mathematical model
  6. Well-posedness
  7. Existence and uniqueness
  8. Stability
  9. Computational Method
  10. The iterative scheme
  11. Convergence analysis
  12. Relationship to existing models
  13. Numerical experiments
  14. Numerical convergence
  15. Numerical comparisons
  16. ...and 7 more sections

Key Result

Lemma 2.1

Given the measured data $\boldsymbol{m} > 0$, the function $L(\,\cdot\,, \,\cdot\,; \boldsymbol{m})$ in neg_log_likelihood is lower bounded in $\mathbb{R}_{+}^{J}\times\mathbb{R}_{+}^{I}$.

Figures (11)

  • Figure 1: Top: The truths of the activity map (left), the attenuation map (middle) and the generated attenuation sinogram (right). Bottom: The reconstructed results of the activity map (left) and the attenuation sinogram (right) by using the proposed MLAAS, and accordingly the reconstructed attenuation image (middle) based on the obtained sinogram.
  • Figure 2: Metrics $\textrm{RE}_{\lambda}^{k}$, $\textrm{RE}_{s}^{k}$ and $\textrm{RE}_{m}^{k}$ are plotted in semi-log scale as functions of iteration numbers for simultaneously reconstructing the activity and the attenuation sinogram of the phantom given in \ref{['fig3_1_XCAT']} by the proposed algorithm MLAAS.
  • Figure 3: The truths of activity image (left), attenuation image (middle) and the corresponding attenuation sinogram (right) of the DRO phantom.
  • Figure 4: The reconstructed results of activity image and attenuation sinogram after 2e4 iterations based on the DRO phantom.
  • Figure 5: Metrics $\textrm{RE}_{\lambda}^{k}$ and $\textrm{RE}_{m}^{k}$ are plotted in a semi-log scale as functions of iteration numbers based on the DRO phantom. Note that the MLAAS is the proposed algorithm, and the MLAA and MLACF are two existing ones.
  • ...and 6 more figures

Theorems & Definitions (35)

  • Remark 2.1
  • Lemma 2.1
  • proof
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • ...and 25 more