Characterization and Reduction of Noise in Dynamic PET Data Using Masked Volumewise Principal Component Analysis

Phantom Study

A phantom study was performed to investigate the differences in expectation value and SD of PET image volumes reconstructed with FBP and OSEM.

A phantom with 2 cylindric inserts was used, both 15 mm in diameter. The insert in the upper part of the gantry was filled with 223 kBq of ¹⁸F per milliliter, the insert in the left part of the gantry with 73 kBq of ¹⁸F per milliliter, and the rest of the phantom with water. The duration of the emission scan was 90 min.

The study was performed on an eXplore VISTA small-animal PET scanner (GE Healthcare). The unit contained 2 rings of 18 phosphor sandwich (phoswich) detector modules capable of acquiring 3-dimensional data with an axial field of view of 48 mm and an effective transaxial field of view of 67 mm (5).

Acquired data were reconstructed to 2 different datasets, using Fourier rebinning followed by 2-dimensional OSEM (2 iterations and 16 subsets) as well as FBP with ramp and Hanning filters, both using 5-min frames. The dimensions of the reconstructed datasets were 175 × 175 × 61 × 18, 175 × 175 pixels with 0.3875-mm sides per transaxial slice, 61 slices with 0.775-mm spacing, and 18 frames of 5 min each.

Three circular regions of interest (ROIs) of equal size were calculated and drawn, one for each insert and one in a region where no radioactive substance was present (Fig. 1). Each ROI had a diameter of 11 mm, which gave 633 pixels per ROI. To avoid most of the spillover effects, the diameter was reduced by 4 mm. This value was chosen because the spatial resolution measured in full width at half maximum is less than 2 mm for the eXplore VISTA scanner.

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FIGURE 1.

Cross-section of phantom reconstructed with FBP. The 3 ROIs are outlined in white.

The arithmetic mean and sample SD within the ROIs were calculated for each slice and frame for data reconstructed with FBP and OSEM. Furthermore, the arithmetic means of these estimates for all ROIs at a given slice and frame were calculated.

Data Analysis

All the presented clinical data were analyzed using masked volumewise PCA with the prenormalization and PCA methods described in the following sections. The data in the clinical study were masked using a previously described masking method (6). In the phantom study, a volume of interest (VOI) was created using a simple cylinder of the size of the phantom. Because the phantom was perfectly symmetric and had known dimensions, no image analytic methods were required to produce a VOI.

Prenormalizations

Two methods are commonly used to prenormalize data before ordinary PCA. The first is to remove the mean of each observation, in this article referred to as removal-of-mean prenormalization. The second is to use standardized variables, which are acquired by removing the mean and dividing each observation by its SD.

In PET image volumes reconstructed with FBP, the background contains a large amount of noise. In 2-dimensional reconstructions of PET data, each slice tends to have varying levels of noise. Because PCA cannot separate variance due to signal from variance due to noise, it is desirable to have each slice scaled with the SD of the noise to get unit noise variance in each slice.

Background noise prenormalization uses a mask to separate the background from the object to estimate the SD of the background in each slice. The actual prenormalization calculations are similar to those used in standardized variable prenormalization except that the SD of the background in each slice is used instead of the SD of the whole signal in each frame. This method is used only on PET data reconstructed with FBP, because image volumes reconstructed with OSEM have values close to zero in the background (2).

Higher-Order PC Noise Prenormalization

Higher-order PC noise prenormalization is a novel method presented for the first time in this article. Much like background noise prenormalization, it estimates the SD of the noise in each slice, which is used for prenormalizing the slices.

Because most of the expectation value in dynamic PET data is accounted for by the lower-order masked volumewise PCs, the SD of the noise can be approximated from the reconstruction of higher-order PCs.

Higher-order PC noise prenormalization has 3 steps. The first is to perform masked volumewise PCA on the whole field of view or the VOI without any prenormalization. The second step is to reconstruct the dataset using only the higher-order masked volumewise PCs and estimate the SD in each slice in the reconstructed data. The third step is to calculate the prenormalized variables using the equationzik=(xik−x¯i)sw.x_ik is the kth sample in frame i, x¯i is the arithmetic mean in frame i, and s_w is the sample SD in slice w where the sample x_ik is located within the reconstruction of higher-order PCs. The whole procedure is illustrated in Figure 2.

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FIGURE 2.

Schematic of higher-order PC noise prenormalization.

Reconstruction from Selected PCs

Because the signal in PET datasets is temporally correlated and the noise is not, masked volumewise PCA can be used to reduce the dimensionality of datasets. In the space spanned by the masked volumewise PCs, the signal is described mostly by lower-order masked volumewise PCs whereas noise is described by higher-order masked volumewise PCs. It is therefore useful to be able to separate data spanned by the lower-order masked volumewise PCs from data spanned by the higher-order masked volumewise PCs in the original frame space.

Because PCA and masked volumewise PCA with nonmasked data changes merely the basis of the prenormalized data, no quantitative information is lost during the masked volumewise PCA and the full original dataset or parts of it can be reconstructed from the masked volumewise PCs.

A PET dataset with p frames is defined asX=[X1,X2,…,Xp]T.

Prenormalizing this matrix yields the matrix Z. To remove all but the k lowest-order PCs from Z, Z is projected onto the corresponding eigenvectors retrieved from PCA of Z. Instead of representing this signal as p frames, one can represent it by k PCs, by projecting the data onto the eigenvectors retrieved from PCA of the data. The PCs are calculated fromY=EZ=[y1,y2,…,yk]T,where Z is the prenormalized matrix X and E is the eigenvector matrix retrieved from the PCA. To project a selected number of PCs back to frame space, all unwanted PCs in Y are set to zero, creating the new matrix Y~. The prenormalized signal in frame space is then calculated asZ˜=E−1Y˜=ETY˜.

The inverse prenormalization is then performed on Z~, creating the modified input matrix X~. A flow chart of the procedure is shown in Figure 3.

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FIGURE 3.

Removal of masked volumewise (MVW) PCs from X.

Clinical Study to Trim a Higher-Order PC Noise Prenormalization Parameter

To determine the optimal number of masked volumewise PCs used for performing the higher-order PC noise prenormalization, the sample SDs in different reconstructions were calculated. These estimates were then compared with the sample SD of the background in a dataset reconstructed with FBP.

The mean squared error (MSE) was used to retrieve a quantitative measure of the differences between the estimates from the reconstructions and the estimate from the background. Because PCA picks the same eigenvectors no matter the scale of the input data, a multiplication of the prenormalization coefficients by a scalar will not affect the PCA results. Therefore, each estimate from the reconstructions was multiplied by the scalar value that minimized the MSE to the estimate from the background, to get a fair comparison of MSEs. These comparisons were made between 2 clinical datasets reconstructed with FBP, where the first dataset was retrieved from a full-body study performed with ¹⁸F-FDG and the second was a brain study performed with ¹¹C-Pittsburgh compound B.

Masked Volumewise PCA

PCA is a method that explains the variance–covariance structure through linear combinations of the original variables (7). Each linear combination, known as PC, is picked in such a way that it maximizes the variance, which is the same as minimizing the MSE, under the constraint that the norm of its weight vector equals 1 and that the new PC is uncorrelated to any previous PCs (8). The weight vectors are the eigenvectors of the covariance matrix of the multivariate data and are usually calculated by singular value decomposition.

Because it is common that only a limited part of a dataset contains the object, the dataset can be masked to include only data within this VOI. This procedure reduces memory use and computation time and has an advantage in that the directions of the eigenvectors are dependent only on data inside the VOI and are not influenced by noise or other disturbing signals in the background. Masked volumewise PCA is performed in several steps: A mask representing the scanned object is created using transmission images either from PET or, if a PET/CT study is performed, from CT. The mask is used to extract the object from the background and can also be used to perform background noise prenormalization. PCA is performed on the data within the VOI representing the object. PCs created with masked volumewise PCA are referred to as masked volumewise PCs (2). To be viewed, the masked volumewise PCs are placed in a matrix of the same size as the input data, creating the unmasked masked volumewise PCs seen in Figure 4.

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FIGURE 4.

Illustration of masked volumewise PCA procedure.

Clinical Study for Evaluating Higher-Order PC Noise Prenormalization

To evaluate the higher-order PC noise prenormalization, we used a clinical dynamic PET dataset from a previous study (3). This dataset contained 14 frames and was acquired from a full-body study performed with ¹¹C-metomidate. Reconstructed datasets generated using both FBP and OSEM were available.

Two VOIs were selected in the OSEM reconstructed data: one over an adrenal gland and another over the stomach.

Both the FBP and the OSEM datasets were prenormalized with the removal-of-mean, standardized-variable, background, and higher-order PC prenormalizations, creating 8 new datasets. Masked volumewise PCA was applied to all datasets, and reconstructions were performed with masked volumewise PC 1; with masked volumewise PC 1 and masked volumewise PC 2; with masked volumewise PC 1, masked volumewise PC 2, and masked volumewise PC 3; and so on until all but the highest-order masked volumewise PC (masked volumewise PC 14) were used in a reconstruction. In this way, 104 datasets (8 × 13 = 104) were created.

Usually, when a VOI is drawn within an organ, the volume is not completely homogeneous. There are variations in scale between the time–activity curves, yet it can often be assumed that they all share a similar kinetic behavior. Since the fact that time–activity curves differ only in scale can be described by a single component, masked volumewise PCA was performed on the VOI to find the masked volumewise PC that optimally represented the time–activity curve within the VOI in a mean square sense. This optimal component was reconstructed and referred to as x^. To measure the deviation between a dimension-reduced dataset x˜ and the optimal component x^, the MSE between the samples was calculated asMSEx^=1p.NVOI∑n=1p∑i=1NVOI(x^[i,n]−x˜[i,n])2,where p is the number of frames and N_VOI is the number of samples within the VOI.