Task-based assessment of binned and list-mode SPECT systems

2.1. Definition of task

We evaluated the LM and binned systems on the task of quantifying activity within a pre-defined region of interest (ROI). In DaT-SPECT, the goal is to estimate the activity in the caudate and putamen regions. The boundaries for these regions can be obtained by either segmenting these regions,^21–23 or from other imaging modalities, such as MRI.²⁴ Thus, the ROI can be defined. We thus frame our task as estimating the activity within the known ROIs from the SPECT images. These SPECT images were acquired using a 2D acquisition protocol, as described in more detail in the next section.

We conducted the quantitative analysis with a synthetic phantom that has similar properties as the brain phantom. Task performance was assessed over multiple realizations. The background was also varied in the different realizations, while the signal properties were constant. Thus, this was a signal-known-exactly/background-known-statistically (SKE/BKS) study in the context of a quantification task except that the signal intensity was not known. The rationale behind conducting this study was to evaluate performance as the background varied on quantification tasks. Variation in background has been known to impact detection-task performance.²⁵

2.2. Object model

To simulate an object similar to the brain, an elliptical shaped synthetic phantom (Fig. 1a) was considered where the background was simulated using a clustered lumpy background model.²⁶ A circular region with 7 mm radius located at an off-center position inside the phantom was considered as the known signal. To analyze the effect of varying signal/tumor-to-background ratio (SBR/TBR) on task performance, the mean SBR value was varied between 1.2:1, 1.5:1, 2:1 and 3:1. The attenuation distribution was considered to be non-uniform (Fig. 1b). The attenuation coefficient was 21m⁻¹ at the rim region of phantom which simulates a skull and 15m⁻¹ at the inner region simulating the soft tissue.

2.3. Simulating the imaging process

We simulated a 2-D SPECT system with configuration similar to GE’s Optima 640 system and a low-energy high resolution (LEHR) parallel-hole collimator. We considered that the phantom has been injected with Ioflupane (I-123) radiotracer, which emits gamma-ray photon at 159 KeV energy. The detector position resolution was set to 4 mm. The detector energy resolution was set to 10% FWHM at 159 KeV and was assumed to be constant in the range of 80 KeV to 175 KeV. LM data, that falls into the energy range 70 KeV to 175 keV, was generated using Monte Carlo simulation and only contained events that went through up to first order of scatter. The data was acquired in 120 fixed angular position over 360°. Approximately 8 × 10⁴ events were detected in a energy window ranging from 80 KeV to 175 KeV, thus simulating a low-dose SPECT protocol.

This LM data was reconstructed as such using the reconstruction algorithm described below. To study the impact of binning on task performance, we binned the energy attribute of this data into multiple bins, namely 2 bins and 3 bins. We set the first bin as same as the photo-peak energy window (143–175 KeV for DaT-SPECT). All the events inside this bin were assigned a fixed energy value, equal to the primary photon energy (159 KeV). Depending on the total number of bins, the scatter-window was then divided into equal-width windows. All photons within a given scatter-energy bin were assigned the energy value at the center of that bin. These binned data were also reconstructed using the reconstruction algorithm described below.

2.4. Process to extract task-specific information

The measured projection data was reconstructed using the maximum-likelihood expectation-maximization-based method as briefly described in Rahman et al.²⁰ The method extends upon an approach originally proposed in^8,27 and allows processing LM data acquired in any arbitrary-sized energy window and incorporate the energy attribute of detected photon. Further, the method compensates for attenuation, scatter, collimator-detector response, and noise in SPECT. Here we briefly describe the technique.

Consider a SPECT system that acquires LM data for a fixed scan time, T. The objective of the reconstruction method is to estimate the discrete activity distribution denoted by the N-dimensional vector λ over Q voxels. Let λ_q denote the mean rate of emission from q^th voxel. Denote the number of detected events by J. The detected events are stored in LM format where for each photon, the attributes of position of interaction of detected photon in the the scintillation crystal, energy deposited by the detected photon and time of detection are recorded. We denote the true and measured attributes of j^th detected event by vectors A_j and ${\widehat{A}}_j$, respectively. We also denote $\mathbb{P}$ as the path traversed by an emitted photon and $\widehat{\mathcal{A}}$ as the full set of measured attributes where $\widehat{\mathcal{A}}=\left\{{\widehat{A}}_j,j=1,2,\dots J\right\}$. We can write the likelihood of the measured LM data as

where, in the second step, we have used the fact that the LM events are all independent, while in the third step, we have decomposed $\text{pr}\left({\widehat{A}}_j\mid \lambda \right)$ as a mixture model in terms of the probability that a photon takes a path and the probability of that path. Here, we point that J is a Poisson distributed random variable. Let $s_{eff}(\mathbb{P})$ denote the sensitivity of a path $\mathbb{P}$ and $\lambda (\mathbb{P})$ denote the activity at the voxel where the path originates from. We can derive the form of $\text{pr}\left({\widehat{A}}_j\mid \mathbb{P}\right)$ using radiative transport equation as²⁷

Thus we can write the log-likelihood of observed LM data as

To maximize this likelihood, an expectation-maximization algorithm is developed. For each event j and each possible path $\mathbb{P}$, we denote a hidden variable $z_{j,\mathbb{P}}$ where

Let us denote the vector z_j by accumulating all the hidden vectors of all possible path for this specific event j. The observed LM data and the hidden vectors form the complete data. We can derive the form of complete data log-likelihood as:

Taking the expectation and maximization steps on this complete data log-likelihood leads to following iterative update:

where, ${\mathbb{P}}_q$ denotes paths that start from voxel q, λ_q denotes the activity rate at that starting voxel and ${\overline{z}}_{j,\mathbb{P}}^{(t+1)}=E\left[z_{j,\mathbb{P}}\mid {\widehat{A}}_j;{\lambda }^{(t)}\right]$ is the expected value of the hidden variable conditioned on observed LM data.

However, the algorithm was still computationally intensive. Thus, in our OAIQ-based studies, we considered only up to first-order scatter events. Further, to ensure that the number of scatter events in the forward and inverse model match, we reconstructed the images using photons that scatter only once.

From the reconstructed images, we estimated the mean activity uptake inside the known region of interest (ROI). Let the N-dimensional vector χ_k denote the binary mask that denotes the ROI for the k^th region. Let ${\widehat{a}}_k$ denote the estimated activity in that region using this procedure. Then, mathematically,

2.5. Figure of merit to measure task performance

The performance of the list-mode and binned-based estimation approaches on the task of activity estimation was quantified on the basis of accuracy and precision of the estimated uptake using the metrics of normalized bias and variance. Further, the overall reliability of the estimated uptake was quantified using the metric of normalized root mean squared error (NRMSE). This is a summary figure of merit that quantifies the effect of both bias and variance, and thus indicates the overall reliability on the quantification task. Let a_k and ${\widehat{a}}_k$ denote the true and estimated activities within the k^th ROIs. Let P denote the number of noise realizations. Then, the NRMSE for the k^th ROI was defined as: