Design Considerations for the Next-Generation MAPMT-Based Monolithic Scintillation Camera

Abstract

Multi-anode photomultiplier tubes (MAPMTs) offer high spatial resolution with their small size anodes that may range from 64 to 1024 in number per tube. In order to increase detector size, MAPMT modules can be arranged in arrays and combined in a single modular scintillation camera. However, then the large number of channels that require amplification and digitization become practically not feasible unless signals are combined or reduced in some manner. Conventional approaches use resistive charge division readouts with a centroid algorithm (or a variant of it) for simplicity in the electronic circuitry implementation and fast execution. However, coupling signals from many anodes may cause significant information loss and limit achievable resolution. In this study, a new approach for optimizing readout-electronics design for MAPMTs based on an analysis of information content in the signals is presented. An adaptive read-out scheme to be used with maximum-likelihood estimation methods is proposed. This scheme achieves precision in estimating event parameters that is close to what is achieved by retaining all signals.

1. INTRODUCTION

Multi-anode photomultiplier tubes (MAPMTs) offer the opportunity to design a new generation of modular scintillation cameras with improved (<1mm) spatial resolution, including accurate estimates of the depth of interaction (DOI)¹. Unresolved DOI causes parallax errors, especially when thicker scintillators are used to increase detection efficiency.

However, the large number of channels in MAPMTs presents a challenge from the read-out electronics standpoint. Nowadays, commercially available MAPMTs may have 64, 256 or 1024 anodes per tube that require amplification and digitization. Additionally, multiple MAPMT modules can be arranged in arrays and combined in a single modular camera to increase the detector size. If all channels are reported for each event, data management and power consumption become complicated.

One of the conventional approaches to reducing the number of channels is to use resistive charge division readout^2,3 for the cross-wired anode layouts. In this readout scheme, the two ends of each anode wire plane receive a fraction of the total charge produced by the event. The fraction is determined by the inverse distance of the place of generation to the respective amplifiers at both ends. In this way, 2D outputs are projected into four 1D (x, y) outputs; thus, many individual wire signals are coupled into only four signals (2x+2y). The four output signals are usually amplified and shaped by charge-sensitive preamplifiers and then fed into the data acquisition system. Charge division can also be applied to multi-anode layouts^4,5 by connecting individual anodes via chains of resistors. Alternative approaches are to group and sum anode wires into sectors (such as two to four wires per sector)^6,7, use resistive matrix networks^8,9 where the current signal from each anode pad is divided into two equal portions that are collected to the x and y-line, thereby summing signals from each column and row of the array to give two single coordinate outputs. For an anode array with (KxL) elements, the number of outputs is reduced to (K+L).

The charge division method is commonly used with a simple centroid algorithm (or variant of it) for determining the spatial location of a gamma-ray event. Although the implementation of the method in electronic circuitry is fairly simple and high-count rates are achievable, coupling signals from many anodes together limits the achievable spatial resolution, especially for low energy/low scintillation light yield where the total created charge per event is low.

For the sake of improving spatial resolution, more elaborate estimation methods can be applied. In particular, maximum-likelihood (ML) estimation methods have many desirable features that motivate their use^10,11,12. In the limit of many independent measurements, ML estimate is asymptotically efficient and unbiased. In other words, the ML estimator provides better spatial resolution and less distortion than any other estimator in the limit of large counts. Besides requiring no distortion correction, position estimation can be carried out to the edge of the detector face¹³. ML methods have also been applied in estimating DOI¹.

In this study, we propose a new approach for optimizing readout-electronic design for MAPMTs for high performance in cases where it is not feasible to read out all anode signals. Our objective is to develop an adaptive read-out scheme, to be used with ML methods, that achieves precision in estimating event parameters that is close to what is achieved by retaining all signals. This approach is based on an analysis of information content in the anode signals that we carried out using knowledge of the statistical properties of the signals; the likelihood model which we introduce in Section 2. We then present Fisher Information Matrix, a powerful tool we use to quantify information. In Section 3, the Fisher information analysis is applied to the anode signals generated from Monte Carlo simulations of a 64 channel MAPMT coupled to a NaI(TI) scintillator. A possible read-out scheme is proposed and limits on the achievable performance are presented. Implementation aspects of the proposed readout-scheme are discussed in Section 4.

2. METHOD

For SPECT and PET applications, the readout signals are used in producing estimates of the spatial location (x, y, z), deposited energy of interaction (ε) and the time of occurrence (t_o) of an event. We define a vector θ as the set of event parameters θ = {x, y, z, ε, t_o}. A forward model that accurately accounts for gamma ray interaction and scintillation-light photon production, propagation, and detection yields an expression for the likelihood pr (g|θ) of a set of anode signals g as a function of event parameters θ. Forward models, ḡ (θ), or mean detector response function (MDRF), can be derived from experimental measurements^1,14 or first principle simulations. In this work we used knowledge of the stochastic processes that relate the camera signals to the gamma-ray parameters to be estimated. For a scintillation camera, the number of photoelectrons generated at a given anode are well-described by a Poisson distribution¹¹

$$\mathrm{p}\mathrm{r}(g_k|\mathit{r},\epsilon )=\text{Poisson}(g_k|{ḡ}_k(\mathit{r},\epsilon )).$$ (1)

If we ignore the scintillator nonproportionality, the mean detector signal varies linearly with energy;

$${ḡ}_k(\mathit{r},\epsilon )=\frac{\epsilon }{{\epsilon }_o}{ḡ}_k(\mathit{r},{\epsilon }_o)$$ (2)

where ε₀ is the specific energy that the measurements are made at.

The Fisher Information Matrix (FIM) is the tool that we use to quantify the information conveyed by the detector signals. FIM states how well the measured signals g can be used to produce an estimate of the dependent parameters θ, without prior information. FIM analysis gives us an understanding of how sensitive the likelihood is to its parameters; it is equal to the covariance of the score^12,15:

$$F_{nm}={\langle [\frac{\partial }{\partial {\theta }_n}\text{ln}\mathrm{p}\mathrm{r}(\mathit{g}|\mathbf{\theta })\left]\right[\frac{\partial }{\partial {\theta }_m}\text{ln}\mathrm{p}\mathrm{r}(\mathit{g}|\mathbf{\theta })]\rangle }_{\mathit{g}|\mathbf{\theta }}.$$ (3)

This expression indicates that FIM will be larger with the increase in the degree of curvature of the log-likelihood. For a Poisson likelihood model, it reduces to;

$$F_{nm}={\sum }_{k=1}^N\frac{1}{{ḡ}_k(\mathit{r},\epsilon )}\frac{\partial }{\partial {\theta }_n}{ḡ}_k(\mathbf{r},\mathrm{\epsilon })\frac{\partial }{\partial {\theta }_m}{ḡ}_k(\mathbf{r},\mathrm{\epsilon })$$ (4)

It is important to note that the sum in eqn. (4) is over the number of statistically independent anodes N, which allows us to distinguish individual contribution of each anode to the FIM. If we identify and retain the highest terms, we expect to achieve resolution close to what is obtained by including all anodes. As an example, fig. (1) shows a simulated MDRF for a single anode and its contribution to the FIM in the x, y and z directions as a function of the event location on the camera face.

Figure 1.

(a) Simulated MDRF for anode (3, 2) for an 8×8 anode MAPMT (units of photoelectrons), (b) corresponding F_xx, (c) F_yy and (d) F_zz for a 5mm thick NaI(TI) scintillator and a 2mm thick light guide.

For the energy estimation, FIM expression reduces to a simple form;

$$F_{\epsilon \epsilon }={\sum }_{k=1}^N\frac{{ḡ}_k(\mathit{r},{\epsilon }_o)}{\epsilon {\epsilon }_o}$$ (5)

Using the positive-semidefinite property for any covariance matrix it can be shown that¹²

$${[{\mathbf{K}}_{\hat{\mathit{\theta }}}]}_{nn}=\text{Var}\{{\hat{\mathrm{\theta }}}_n\}\ge {[{\mathbf{F}}^{-\mathbf{1}}]}_{nn}$$ (6)

where θ̂ is the ML estimate. This inequality is the Cramér-Rao (CR) lower bound on the variance of an unbiased estimator and it obeys complementarity relation with the Fisher information. For a scintillation camera employing unbiased estimator, the standard deviation of the desired estimate, which we denote as σ, is then;

$$\mathbf{\sigma }=\mathbf{|}\mathbf{F}{\mathbf{|}}^{-1/2}.$$ (7)

This expression represents the best possible resolution that can be achieved for a given MDRF. From the form of F in eqn. (4), it can be said that better resolution can be obtained with MDRFs with the right combinations of means and slopes. Fig. (2) shows the lower bound on the variance on spatial position estimation when all available channels in a 64 anode (8×8) MAPMT are read out, computed using eqn. (5).

Figure 2.

(a) The spatial resolution in x, (b) y and (c) z directions.

3. APPLICATION IN A CAMERA

Signals generated at each anode in a hypothetical gamma camera consisting of an 8×8 MAPMT, coupled to a monolithic NaI(TI) scintillator and fused-silica light guide, were simulated with Monte Carlo techniques. MDRFs were computed for each anode by generating many events at regularly sampled 3D positions and a range of gamma ray energies around 140 keV and then averaging over all data sets. The deflection of rays at the optical interfaces between the scintillator and the light guide was found by Snell’s law, using index of refraction of 1.85 for NaI(TI) and 1.46 for fused silica. The camera design was specified to have no diffuse reflectors; the entrance faces and edges of the detector were modeled to have a black absorber. This way, the scintillation light emitted back toward the entrance window is absorbed and not reflected back toward the collection surface. MDRFs with blackened faces are observed to have more rapid fall-off from the source location compared to diffusely reflected surfaces and thereby encode depth of interaction well. For interactions close to the anode collection surface (larger DOI), the solid angle subtended by each anode is greater; thereby the photoelectrons are collected over a more concentrated area, making MDRFs sharper and, consequently, their slopes steeper. This makes the use of thinner crystals more desirable in terms of achieving better spatial resolution, however, at the expense of reducing detection efficiency and incurring some anisotropy in spatial resolution. A thorough analysis of MDRF forms and their use in camera design can be found in Rogers¹⁶. In our simulations, we modeled NaI(TI) with a 5-mm-thickness and fused silica with different thicknesses ranging between 1–3 mm.

The generated MDRFs were used in computing FIM, as in eqn. (4). The FIM was decomposed into individual anode elements and ranked according to their magnitudes. The results are shown in fig. (3).

Figure 3.

For a central area on the detector face, the four most significant contributions to the overall Fisher information for positioning are identified in descending order in plots (a)-(d). Top surfaces show the relative contribution to FIM. Second surfaces show which (color coded) anodes are contributing at each location on the sampled detector face.

This analysis suggests a readout scheme based on selecting the anode with the largest input signal and adaptively assigning it as the central anode in a specific anode neighborhood read out with each event.

There may be instances where the anode with maximum signal is not uniquely determined. For instance, for events occurring over the corner between four anodes, it is equally likely that any one of the four anodes will be assigned as the central anode. We modeled a source placed at such a location and computed the CR bound for each of the four possible neighborhoods (fig 5). It was found that the lower bound on the variance of the spatial location estimates were the same for all four cases as would be expected by symmetry arguments. Therefore, ambiguity in the choice of central anode doesn’t cause undue problems.

Figure 5.

Illustration of possible 3×3 anode neighborhoods for events occurring over a corner between four anodes. (white spot)

As an example, comparison of the spatial resolution for the simulated camera for exhaustive 8×8 and an adaptive 3×3 readout show that, with an adaptive 3×3 readout scheme it is possible to retain information content and achieve precision in estimation of spatial location very close to what is achieved by including all 64 signals. Fig. (6) shows estimated resolution of the two readout schemes plotted for 140keV gamma ray for a light guide thickness of 2 mm and events occurring at the midplane of a 5mm thick crystal.

Figure 6.

Comparison of spatial resolution in 8×8 and 3×3 readout schemes. The plots are shown for a cut through x-direction, where y is kept fixed across the center of the detector face.

It is observed that for thinner light guides, the standard deviation of the estimate takes a periodic form. This variation of spatial resolution over the corresponding periodic anode locations describes the camera sensitivity to the interaction position. The solid angle subtended by one anode is large when the interaction occurs in the vicinity of that anode; thereby more light is collected by the same anode. For the interactions that occur away from the immediate center of a particular anode, the light collection is shared between different anodes, which results in an increase in the camera sensitivity to interaction position.

4. DISCUSSION AND CONCLUSIONS

Fisher information analysis provides not only a tool for evaluating gamma-camera performance but also for designing read-out strategies. It has been shown that, for an MAPMT-based gamma camera, with optical characteristics as described, a read-out that returns a 3×3 neighborhood centered around the anode with the largest signal has imaging performance very close to what is achieved with an exhaustive 64 channel readout; spatial resolutions are within 0.1–0.2 mm in each dimension in each case. Fisher information will be affected by many camera parameters such as scintillator and light guide thickness, surface treatments for the detector entrance, exit faces and edges, and density of readout elements. Fisher information, thus, provides an effective tool for making all camera decisions¹⁷. Cameras with other parameters than were considered in this study may require larger anode neighborhoods to be read out. Using the methods presented in this study, an adaptive read-out scheme can be found for scintillation cameras with different features.

For small-animal imaging experiments, the scintillation cameras are usually required to have at least 10cm × 10cm of active area. One possible choice for a high-resolution camera is Hamamatsu’s H9500¹⁸ MAPMT with 16 × 16 anodes and a photocathode of 49 mm × 49 mm area. With this MAPMT, arrays of 2×2 or 3×3 need to be employed to meet the active area requirement. Therefore, the readout electronics and the data acquisition system (DAQ) need to support the combination of multiple MAPMTs in a single modular detector. The proposed adaptive read-out scheme could be implemented as an application specific integrated circuit (ASIC)¹⁹. that accommodates tiling. Fig. (7) shows a basic signal processing setup with a mixed-signal approach; signal conditioning and event detection are carried out in the analog domain and selected signals are digitized with high resolution analog-to-digital converters (ADCs). Amplified and shaped analog signals are also summed (fast) and compared to a threshold level to generate the timing trigger. The peaking time and the analog gains should be adjustable to support different types of scintillation crystals. The channel with the largest input signal can be selected with a winner- take-all circuit²⁰ that outputs the digital address of the winner channel. From there, a logic section can control a multiplexer, a flash ADC array, and an event-packet builder to complete the front-end list-mode readout. The overall anode signal sum can be presented as a separate analog input to the multiplexer for energy estimation.

Figure 7.

Potential readout electronics module block diagram.

It is also important to note that, for the cases where gain noise is high, the likelihood model might no longer be described by the Poisson distribution. Fisher information analysis can be carried out for any probability distribution as long as that distribution is well characterized. In that case, FIM expression will in general be more complicated than eqn. (4), but still be straightforward to compute.

Figure 4.

(a) Spatial resolution in x, (b) y and (c) z directions across the detector face when a 3×3 readout scheme is applied. Every anode, in turn, gets selected as having the highest signal and a 3×3 neighborhood is adaptively set around that anode.