Optimal rebinning of time-of-flight PET data

Abstract

Time-of-flight (TOF) positron emission tomography (PET) scanners offer the potential for significantly improved signal-to-noise ratio (SNR) and lesion detectability in clinical PET. However, fully 3D TOF PET image reconstruction is a challenging task due to the huge data size. One solution to this problem is to rebin TOF data into a lower dimensional format. We have recently developed Fourier rebinning methods for mapping TOF data into non-TOF formats that retain substantial SNR advantages relative to sinograms acquired without TOF information. However, mappings for rebinning into non-TOF formats are not unique and optimization of rebinning methods has not been widely investigated. In this paper we address the question of optimal rebinning in order to make full use of TOF information. We focus on FORET-3D, which approximately rebins 3D TOF data into 3D non-TOF sinogram formats without requiring a Fourier transform in the axial direction. We optimize the weighting for FORET-3D to minimize the variance, resulting in H(2)-weighted FORET-3D, which turns out to be the best linear unbiased estimator (BLUE) under reasonable approximations and furthermore the uniformly minimum variance unbiased (UMVU) estimator under Gaussian noise assumptions. This implies that any information loss due to optimal rebinning is as a result only of the approximations used in deriving the rebinning equation and developing the optimal weighting. We demonstrate using simulated and real phantom TOF data that the optimal rebinning method achieves variance reduction and contrast recovery improvement compared to nonoptimized rebinning weightings. In our preliminary study using a simplified simulation setup, the performance of the optimal rebinning method was comparable to that of fully 3D TOF MAP.

Fig. 1

Data acquisition geometry: (left) transverse and (right) sagittal view. 3D TOF data are line integrals of a 3D object along an LOR weighted by a TOF kernel h. Each LOR is specified by s, ϕ, z, and δ = tan θ.

Fig. 2

Geometric illustration of the coordinate transformation for the approximate mappings between a TOF oblique sinogram ℘_z, δ, ω_t (ω_s, ϕ) and a non-TOF oblique sinogram ${\wp }_{z,\delta }^{\prime }({\omega }_s^{\prime },\varphi \prime )$.

Fig. 3

Illustration of the noise properties of 3D TOF data in Fourier domain. (a) Covariance $C_m^{\mathit{\text{ij}}}$ of the 2D FFT of the 3D TOF data with respect to the radial coordinate and the TOF variable index. This figure shows profiles along the radial frequency and the TOF frequency direction. In this figure, the zero-frequency component was shifted to the center. (b) Sum of the data variance over the radial and TOF variable index versus transaxial angular index for the oblique plane with the maximum ring difference.

Fig. 4

(a) Axial center plane of the NCAT torso phantom used for Monte Carlo simulation studies. Two ROIs are shown: ROI A with 36 voxels (144 mm²) and ROI B with 23 voxels (92 mm²). (b) Sample mean profiles and (c) sample variance profiles for the rebinned data and the non-TOF data acquired without TOF information in an axial center plane with the maximum ring difference. Note the variance profile in (c) is plotted on log scale. (d) Profiles of the covariance of the rebinned data. The covariance was calculated between a sinogram bin (radial index = 169, angular index= 169) and all other sinogram bins in the same oblique plane. For (b)–(d), the profiles were taken at the 169th angle corresponding to ϕ = π/2.

Fig. 5

Histograms (over sinogram bins) of the ratios of the variance for the non-TOF data, acquired without TOF information, to that for (a) H²-weighted FORET-3D, (b) H-weighted FORET-3D, and (c) unweighted FORET-3D. (d) Histogram of the ratios of the variance for H-weighted to H²-weighted FORET-3D.

Fig. 6

ROI recovery coefficient versus ROI standard deviation trade-off curves for ROI quantitation: (a) ROI A and (b) ROIB. ROI recovery coefficient versus voxel-wise variance across realizations with variances averaged within ROI: (c) ROI A and (d) ROI B. The error bars represent the standard errors estimated by a bootstrap method.

Fig. 7

A transverse plane of fully 3D reconstructed images from the rebinned data by (left) H²-weighted FORET-3D, (middle) H-weighted FORET-3D and (right) from the non-TOF data acquired without TOF information, using real phantom data. The red rectangle represents the background ROI used to calculate the contrast to noise ratio. A 4×4 voxel ROI that includes the lesion in the center was taken as a lesion ROI. Contrast-to-noise ratio (CNR) values were calculated as 17.97 for H²-weighted FORET-3D (left), 16.41 for H-weighted FORET-3D (middle) and 10.95 for the non-TOF data acquisition (right).