Depth-of-interaction encoding techniques for pixelated PET detectors enabled by machine learning methods and fast waveform digitization

Abstract

Objective. Pixelated detectors with single-ended readout are routinely used by commercial positron emission tomography scanners owing to their good energy and timing resolution and optimized manufacturing, but they typically do not provide depth-of-interaction (DOI) information, which can help improve the performance of systems with higher resolution and smaller ring diameter. This work aims to develop a technique for multi-level DOI classification that does not require modifications to the detector designs.Approach. We leveraged high-speed (5 Gs s^-1) waveform sampling electronics with the Domino Ring Sampler (DRS4) and machine learning (ML) methods to extract DOI information from the entire scintillation waveforms of pixelated crystals. We evaluated different grouping schemes for multi-level DOI classification by analyzing the DOI positioning profile and DOI positioning error. We examined trade-offs among crystal configurations, detector timing performance, and DOI classification accuracy. We also investigated the impact of different ML algorithms and input features-extracted from scintillation waveforms-on model accuracy.Main results. The DOI positioning profile and positioning error suggest that 2- or 3-level binning was effective for 20 mm long crystals. 2-level discrete DOI models achieved 95% class-wise accuracy and 83% overall accuracy in positioning events into the correct DOI level and 3-level up to 90% class-wise accuracy for long and narrow crystals (2 × 2 × 20 mm³). Long short-term memory networks trained with time-frequency moments were twice as efficient in training time while maintaining equal or better accuracy compared to those trained with waveforms. Classical ML algorithms exhibit comparable accuracy while consuming one order less training time than deep learning models.Significance. This work demonstrates a proof-of-concept approach for obtaining DOI information from commercially available pixelated detectors without altering the detector design thereby avoiding potential degradation in detector timing performance. It provides an alternative solution for multi-level DOI classification, potentially inspiring future scanner designs.

Keywords: depth-of-interaction (DOI); long short-term memory (LSTM); machine learning (ML); waveform sampling (WFS).

Figure 1.

(a) Illustration of side irradiations and (b) top-view picture of the benchtop experimental setup containing the DOI detector with a single crystal and the reference detector. For acquiring ground truth training and test dataset, side irradiations are performed.

Figure 2.

Demonstration of available features from 3-depth DOI measurements (3 side irradiation depths) used for training ML models. (a) Fast timing signal waveforms from one channel at different DOIs. Note that these waveforms were not normalized. (b) Energy histograms, (c) Rise time histograms, (d) Rise time correlation with energy, (e) instantaneous frequency and (f) spectral entropy extracted from the waveforms at different DOIs. Measurements were performed with a $2\times 2\times 20$ mm³ crystal with Teflon wrapping.

Figure 3.

Class-wise accuracies (labeled inside the heatmaps) for different grouping schemes labeled by red boundaries. Overall accuracy is listed above the heatmaps. All models were LSTM networks trained with the same waveforms along with energy information (W_E), measured on a $2\times 2\times 20$ mm³ crystal with Teflon wrapping.

Figure 4.

(a) Width of DOI positioning profile at 9 DOI layers, calculated from the same 9-level LSTM model used in figure 3. (b) DOI positioning error at 9 DOI layers, and (c) its mean and std error over all depths for different grouping schemes, calculated from corresponding 9-level to 2-level LSTM models used in figure 3.

Figure 5.

The measured timing waveforms from the top DOI layer (1st row), energy histogram (2nd row), and energy-rise time correlation (3rd row) using the 3-level grouping scheme for (a) $3.2\times 3.2\times 20{\text{mm}}^3$ bare crystal, (b) $2\times 2\times 20{\text{mm}}^3$ bare crystal, (c) $3.2\times 3.2\times 20{\text{mm}}^3$ crystal with Teflon wrapping, and (d) $2\times 2\times 20{\text{mm}}^3$ crystal with Teflon wrapping. The same scale was used for energy measurements of different samples..

Figure 6.

Overall accuracy of classical ML and LSTM models for 3-level DOI models with different crystal configurations.

Figure 7.

(a) The best 3-level and (b) 2-level (using 2-level-54 scheme) DOI models, trained with different machine learning algorithms and input features using measurements on a $2\times 2\times 20{\text{mm}}^3$ crystal with Teflon wrapping.

Figure 8.

Overall accuracies of 3-level LSTM models, trained using different input features.TF_E: time–frequency features with energy information, W_E: waveforms with energy information, as defined in section 2.2 and table 2.

Figure 9.

Overall accuracy of 3-level classical ML models for DOI discrimination, trained using rise time, energy, and energy-rise time correlation.

Figure A1.

Overall and class-wise accuracies of classical ML models for 3-level DOI classification using 9-depth measurements on a $2\times 2\times 20{\text{mm}}^3$ crystal with Teflon wrapping. The same measurements were used in figure 7(a). The input features were RT_E.

Figure B1.

Illustration of a bi-directional LSTM network with hyperparameters depicted in section 2.4. intput_size: the number of input features. out_features: the number of output classes. Both forward and backward LSTMs process the input, each producing N-dimensional output where N is defined by hidden_size. Output from forward and backward LSTMs are concatenated to form a 2N-dimension vector. Softmax is applied across output classes.