Patent Publication Number: US-11385362-B2

Title: Scintillation detector and associated scintillation detector ring and method

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application U.S. National Stage Application filed under 35 U.S.C. § 371 of International Application No. PCT/US2017/058501, filed Oct. 26, 2017, which claims the benefit of claims priority from United States Provisional Patent Application number 62/414,469 filed Oct. 28, 2016. Both of these applications are hereby incorporated by reference in their entireties. 
    
    
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
     This invention was made with government support under Grant No. P41 EB002035, awarded by NIH. The government has certain rights in the invention. 
    
    
     BACKGROUND OF THE INVENTION 
     Positron Emission Tomography (PET) has become a mature and reliable technology. It has broad applications in neurology and oncology due to its ability to monitor metabolism of glucose and the uptake of other targeting radiotracers in specific organs and tissues. Compared with other alternative options (functional MRI, CT, and SPECT), the sensitivity of PET is several orders of magnitude higher, which provides more reliable results (lower noise, better spatial resolution, better contrast to noise ratio, etc.). 
       FIG. 1  shows a conventional PET detector  100 . It includes a pixelated scintillator block and associated photon detection sensors (e.g., photomultiplier tubes or silicon photomultipliers). If a gamma-ray photon interacts inside one of the pixels, visible scintillation photons are produced and propagate to photon sensors via reflections by crystal surfaces. By the proportion of light detected by the photon sensors, the gamma-ray photon&#39;s energy, interaction pixel and interaction time may be estimated. 
       FIG. 2  depicts a detector ring  200  formed from hundreds of PET detectors  100 . In medical practice, detector ring  200  surrounds a patient.  FIG. 3A  illustrates an idealized version of PET, where a radioactive atom emits a positron  301  that, before traveling any distance, encounters an electron resulting in an annihilation event occurring at an annihilation position  303 . The annihilation results in emission of two gamma-ray photons  305 (1, 2) collinearly travelling in opposite directions from annihilation position  303 . 
       FIG. 3B  depicts a non-ideal PET scenario that radiologists encounter, where a positron is emitted by a radioactive atom, and after the positron travels a short distance  307  (positron range is mentioned in the limitation 1 below), it combines with an electron which results in annihilation. The electron and positron combine and convert to two gamma-ray photons  309 (1, 2) emitted from the annihilation position. However, the two gamma-ray photons do not travel in exactly opposite directions. Rather, there is generally a very small angular deviation, from 180°, between the two gamma-ray photons&#39; directions. This is called non-collinearity effect as mentioned in the limitation 1. 
     If both gamma-ray photons are detected, the annihilation position is typically somewhere along the line connecting the two gamma-ray photons&#39; interaction positions on the detector ring. This line is called the line of response (LOR). By collecting enough lines of response, an image of the isotope map marking a certain organ or tumor may be reconstructed by certain algorithms such as filtered back projection (FBP) or maximum likelihood estimation (MLE). 
       FIG. 4  (panels a, b and c) shows the influence of positron range, non-collinearity and depth of interaction (DOI) for accurate estimation of LOR. According to  FIG. 4 , panel c, when DOI information is not available, the LOR is typically assigned to the fixed positions of the scintillation crystals. However, this may result in an incorrectly assigned LOR due to parallax error. If accurate DOI information is available, the true LOR may be determined. 
     Compared with traditional SPECT technology, PET has much greater sensitivity since it does not need collimators. Compared with functional MRI or CT, the sensitivity is usually several orders of magnitude higher. However, PET technology still has some limitations. These limitations include: 
     Limitation 1: The positron range and non-collinearity degrades the spatial resolution. Since the annihilation position is different from the radioactive isotope&#39;s position because the positron travels a small distance before it combines with an electron (positron range effect), there is error or blur in the reconstructed image. Also, due to the non-collinearity effect, the annihilation position further deviates from the ideal line of response. 
     Limitation 2: Making the pixelated crystal&#39;s cross section small can be very challenging. Due to the high energy of gamma-ray photons, they are very difficult to stop. A thicker crystal should be utilized to efficiently stop the majority of the gamma-ray photons. However, this makes the pixelated crystal&#39;s aspect ratio low. The smaller the pixel&#39;s pitch size, the larger the number of reflections required for a photon to travel to the photon sensors, which will decrease the light output amount due to the losses from reflections. 
     Limitation 3: The depth of interaction estimation is critical for accurate measurement. The depth of interaction is important, since it affects the spatial resolution at the edge of the field of view. As shown in  FIG. 5 , if the two detector elements on opposite sides of the detector ring detects two gamma-ray photons simultaneously, the annihilation position is within region a. However, if the two detector elements on the right simultaneously detects two gamma-ray photons, the annihilation positon is within region b. Compared with the annihilation in region a, the one in region b creates more ambiguity due to the lack of depth of interaction information. Moreover, as can be seen in the plot in the right side of  FIG. 5 , reconstructed spatial resolution worsens (size of smallest resolvable feature increases) as the DOI resolution gets worse, and this effect is obvious as the spatial resolution decreases with the distance from the center of field of view increases. 
     Limitation 4: The cost of manufacturing pixelated crystals is extremely high. Since pixelated crystals need special surface treatment (insulating, polishing) between individual pixels, the cost of manufacturing such crystals is high. Moreover, if the pixels are smaller than around 1 mm, the difficulty in special treatments drives the cost even higher. 
     Limitation 5: The sensitivity is reduced due to the dead-area gap caused by crystal pixelating. Due to the pixels, a gap is needed to properly insulate the scintillation photons from propagating into adjacent pixels. This gap will decrease the detection efficiency of the detector to gamma-ray photons. Especially for smaller pixel pitches, since the gap width almost remains constant, the fraction of the gap width/pixel pitch will be larger and the detection efficiency is even lower. 
     SUMMARY OF THE INVENTION 
     The present invention provides improved scintillation-crystal based gamma-ray detectors, stacked detectors, and PET detector rings and arrays that take advantage of total internal reflection within the scintillation detector substrate. The detectors and methods described herein provide improved spatial resolution including depth-of-interaction (DOI) resolution while preserving energy resolution and detection efficiency, which is especially useful in small-animal or human positron emission tomography (PET) or other techniques that depend on high-energy gamma-ray detection. Additionally, embodiments of the present invention reduce the total number of readout channels required and reduce the need to do complicated and repetitive cutting and polishing operations to form pixelated crystal arrays as is the standard in current PET detector modules. 
     An embodiment of the present invention provides a scintillation detector comprising a scintillator substrate having an interior bound by a front surface having a front perimeter, a back surface opposite the front surface and having a back perimeter, and an edge surface between the front and back perimeters; and a plurality of photodetectors each having a respective photosensitive region in optical communication with one of a respective plurality of regions of the edge surface. Optionally, one or more of the photosensitive regions face one of a respective plurality of regions of the edge surface. Optionally, the front and back surfaces are planar surfaces or substantially planar surfaces. The front and back surfaces may also be substantially parallel to one another although curvature between the two surfaces may help increase the number of photons reaching the photodetectors at the edges of the substrate (for example, if the two surfaces are slightly concave with respect to one another, the number of reflections necessary for a photon to reach the edges may decrease). Optionally, the front and back surfaces are planar surfaces or substantially planar surfaces but are not parallel to each other. Optionally, the front surface, back surface, or front and back surface, of at least a portion of the scintillation detectors have non-planar surfaces and/or are non-parallel to one another. The front surface and back surface, independently from one another, of at least a portion of the scintillation detectors may have a concave shape or convex shape. For example, the front surface may be concave and the back surface convex, or the front surface is convex and the back surface is concave, for at least a portion of the scintillation detectors. 
     In an embodiment, the scintillator substrate has a uniform thickness between 0.1 mm and 25 mm, between 0.5 mm and 20 mm, between 0.5 mm and 10 mm, or between 1 mm and 4 mm, for example, although the thickness is decided by the desired DOI. 
     Optionally, the front perimeter has a front shape that is geometrically similar to the back shape of the back perimeter; however, in certain embodiments the shape of the front perimeter may be different from the shape of the back perimeter. The shapes of the front and back perimeters may be polygons. Optionally, the scintillation detector has a circular, elliptical, triangular, rectangular, hexagonal, or octagonal shape. 
     In a further embodiment, the scintillation substrate has a refractive index exceeding 1.1, exceeding 1.4, or exceeding 1.8 at an emission maximum of the scintillator material. The front and back surfaces may also comprise optically reflective surfaces, including but not limited to a high reflectivity mirror, for facilitating the transportation of photons without total internal reflection (TIR). The scintillation substrate may include both organic and inorganic scintillator substrates. 
     Suitable photodetectors include, but are not limited to, silicon photomultipliers (SiPMs), photomultiplier tubes (PMTs), and complementary metal-oxide semiconductor (CMOS) sensors. The photodetectors may further comprise an avalanche photodiode (APD). In an embodiment, the photodetectors comprise SiPMs, APDs, or combinations thereof. In an embodiment, the photodetectors comprise SiPMs. 
     An embodiment of the present invention provides a plurality of scintillation detectors arranged as a stack, the plurality of scintillation detectors comprising a first scintillation detector and a second scintillation detector, the back substrate-surface of the first scintillation detector being adjacent to, opposing, and optionally spatially separated from the front substrate-surface of the second scintillation detector. In further embodiments, the stack comprises two or more, three or more, four or more, five or more, six or more, seven or more, eight or more, nine or more, ten or more, fifteen or more, or twenty or more scintillation detectors. In a further embodiment, optically opaque materials or optically reflecting materials are positioned between portions of two or more of the scintillation detectors to reduce transmission of light between different scintillation detectors, thereby limiting optical cross-talk between the different detectors. Suitable optically opaque materials include but are not limited to thin metal layers (such as aluminum foil), ceramics, plastic, cloth, and paper. Optically opaque materials may be painted, dyed, or otherwise colored to be black or another color so as to reduce or eliminate transmission of light. 
     The stacking or tiling of layers of scintillation detectors may be offset by a small distance to compensate for the gaps between detector layers. For example, the second scintillation detector may be positioned in relation to the first scintillation detector such that a center region of the second scintillation detector is aligned with a center region of the first scintillation detector relative to an axis perpendicular to the front substrate-surface of the first scintillation detector. Alternatively, the second scintillation detector may be positioned in relation to the first scintillation detector such that a center region of the second scintillation detector is laterally offset from center region of the first scintillation detector relative to an axis perpendicular to the front substrate-surface of the first scintillation detector. Alternatively, the second scintillation detector is positioned in relation to the first scintillation detector such that a center region of the second scintillation detector is aligned with an edge surface of the first scintillation detector relative to an axis perpendicular to the front substrate-surface of the first scintillation detector. 
     In a further embodiment, multiple stacks of scintillation detectors may be arranged adjacent to one another. The scintillation detectors within a given stack may have different lateral lengths to allow multiple stacks to be arranged adjacent to one another in a non-linear geometry. 
     An embodiment of the present invention provides a positron emission tomography (PET) detector ring having a plurality of scintillation detectors comprising a scintillator substrate and a plurality of photodetectors. The scintillator substrate for each scintillation detector has an interior bound by a front surface having a front perimeter, a back surface opposite the front surface and having a back perimeter, and an edge surface between the front and back perimeters. Each of the plurality of photodetectors is disposed on one of a respective plurality of regions of the edge surface, and each having a field of view that includes a portion of the interior. In an embodiment, the PET detector ring comprises two or more, three or more, four or more, five or more, six or more, seven or more, eight or more, nine or more, ten or more, fifteen or more, twenty or more, fifty or more, or a hundred or more scintillation detectors. At least a first sub-plurality of the plurality of scintillation detectors form a first array having (i) a first concave inward-facing surface and including each front surface of the first sub-plurality and (ii) a first convex outward-facing surface and including each back surface of the first sub-plurality. The second array may be spatially shifted from the first array such that photons transmitted between photodetectors of adjacent scintillation detectors of the first array are incident on a substrate of a scintillation detector of the second array. 
     In a further embodiment, the plurality of scintillation detectors comprise a second sub-plurality of scintillation detectors forming a second array having (i) a second concave inward-facing surface, opposite the first convex outward-facing surface, and include each front surface of the second sub-plurality of scintillation detectors and (ii) and a second convex outward-facing surface that includes each back surface of the second sub-plurality of scintillation detectors. 
     In further embodiments, the substrates of the scintillation detectors, stacks, and detector rings comprise a plurality of optical scatterers (also referred to herein as optical barriers). Optical scatterers may be any inclusion, treatment, or modification to the inside of the substrate or on the surface of the substrate, where the inclusion, treatment, or modification is able to absorb or redirect light to reshape the mean detector response function (MDRF). For example, the optical scatterers may comprise, but are not limited to, a through hole, a blind hole, an object on the front surface, an object on the back surface, a volume with modified optical properties generated by laser etching or laser engraving, or combinations thereof. The optical scatterers may have any shape or pattern, as long as they are able to reshape the MDRF. In certain embodiments, each of the plurality of optical scatterers may have an extent exceeding a wavelength corresponding to an emission maximum of the scintillator material. In certain embodiments, the optical scatterers may utilize Rayleigh scattering, where the size of particles forming the optical scatterers is smaller than the dimension of the wavelength of light. In certain embodiments, the dimension or size of the optical scatterers may be close to the wavelength of the scintillation emission maximum, in which case certain interference patterns may form on the edges of the detector, which helps determine the positioning of the gamma-ray photons. For example, a plurality of small optical scatterers with size or spacing close to the emission wavelength can create interference similar to that formed by an optical lens. This “virtual” lens can be used to reshape the MDRF of the detector, which helps to increase the capability for determining the position of the gamma-ray photons. 
     The scintillation detectors, stacks, and detector rings described herein may further comprise a memory storing non-transitory computer-readable instructions; and a microprocessor adapted to execute the instructions to estimate lateral coordinates, in a plane of the substrate, of a scintillation-event interaction-position based on a plurality of electrical signal amplitudes and respective plurality of locations of the plurality of photodetectors. Optionally, the microprocessor is configured to receive electrical signals generated by one or more scintillation events from the plurality of photodetectors, and is configured to estimate lateral coordinates of a scintillation-event interaction-position in a plane of the substrate of the at least one of the plurality of scintillation detectors, based on a plurality of electrical signal amplitudes and respective plurality of locations of the plurality of photodetectors. Optionally, the microprocessor is further configured to estimate depth coordinates of a scintillation-event interaction-position based on the plurality of electrical signals and respective plurality of locations of the plurality of photodetectors within the stack. 
     An embodiment of the present invention provides a method for determining the interaction position of a scintillation event occurring in a substrate resulting from electromagnetic radiation incident on a front surface of the substrate. This method comprises the steps of detecting light from the gamma-ray interaction event and estimating lateral coordinates of the interaction position. 
     The step of detecting comprises detecting, with a plurality of photodetectors, scintillation light from the gamma-ray interaction event resulting in a respective plurality of electrical signal amplitudes, the plurality of photodetectors being disposed at a respective region around a perimeter of the substrate and having a photosensitive region in optical communication with an edge surface of the substrate located between the front surface and a back surface of the substrate opposite the front surface. In a further embodiment, one or more of the photosensitive regions face an edge surface of the substrate. 
     The step of estimating comprises estimating lateral coordinates, in a plane of the substrate, of the interaction position based on the plurality of electrical signal amplitudes and respective regions. In a further embodiment, the estimating step comprises comparing the plurality of electrical signal amplitudes to a mean-detector response function of the plurality of photodetectors at the respective regions around the substrate perimeter. In an embodiment, the comparing step comprises performing at least one of a maximum likelihood estimate, a least-squares estimate, an iterative estimate, a LN norm comparison (where N is 0, 1, 2 . . . , and combinations of thereof), or an Anger method. For example, the least-square method is approximately equal to minimizing the L2 norm between the electrical signals and the reference data (MDRF). 
     Optionally, the substrate is one of a plurality of stacked scintillator substrates each having: (a) a respective edge surface and (b) a respective plurality of photodetectors having a photosensitive region in optical communication with the respective edge surface, the scintillation event occurring in one of the plurality scintillator substrates. The method further comprises determining, from zero and non-zero electrical signal amplitudes from the photodetectors, (i) the one of the plurality of substrates containing the scintillation event and (ii) an associated depth-of-interaction of the scintillation event in a direction perpendicular to the plane of the substrate. In a further embodiment, one or more of the photosensitive regions of the substrate will face the respective edge surface of the substrate. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  illustrates a conventional positron emission tomography (PET) detector. 
         FIG. 2  by illustrates a detector ring formed by a plurality of the PET detectors of  FIG. 1 . 
         FIG. 3A  and  FIG. 3B  illustrate PET line of response with a zero positron range without non-collinearity effect and a non-zero positron range with non-collinearity effect, respectively. 
         FIG. 4  (panels a, b and c) shows, respectively, the importance of positron range, non-collinearity and depth of interaction information (DOI) for accurate estimation of the annihilation position on the correct line of response. 
         FIG. 5  illustrates two possible positron-electron annihilation regions within the detector ring of  FIG. 2 , which result in different measurement ambiguity due to the lack of depth of interaction (DOI) information. 
         FIG. 6  is a plot showing spatial resolution of a reconstructed image as a function of depth-of-interaction (DOI) resolution. 
         FIG. 7  shows a scintillation detector without any optical scatterers, in an embodiment. 
         FIG. 8  shows an example of the scintillation detector  FIG. 7  that has a plurality of optical scatterers, in an embodiment. 
         FIG. 9  shows an example of the scintillation detector  FIG. 7  that includes a light spreader, in an embodiment. 
         FIG. 10  is a cross-sectional view of the stacked scintillation detectors, in an embodiment. 
         FIG. 11  is a graphical projection of a stacked scintillation detector, in an embodiment. 
         FIG. 12  shows a PET detector ring in an embodiment of the present invention. 
         FIG. 13  shows a further illustration of total-internal reflection of a scintillation photon in a slab of scintillation crystal. 
         FIG. 14  shows a spectral reflectance curve of a silicon photomultiplier. 
         FIG. 15  shows simulated mean detector response functions (MDRFs) of a thin layer detector formed of lutetium-yttrium oxyorthosilicate (LYSO). 
         FIG. 16  shows an estimated interaction position derived from the MDRFs of  FIG. 15 . Central points cannot be resolved due to poor resolution. 
         FIG. 17  shows simulated MDRFs of a thin layer LYSO detector with optical barriers in an embodiment of the invention. The shadows of the rods reshape the MDRF. 
         FIG. 18  shows an estimated interaction position derived from the MDRFs of  FIG. 17 . Central points are more resolvable due to improved spatial resolution. 
         FIG. 19  illustrates simulated MDRFs of a thin layer LYSO detector with laser-etched (or laser engraved) optical barriers in one embodiment of the invention. 
         FIG. 20  shows an estimated interaction position derived from the MDRFs of  FIG. 19 . 
         FIG. 21  and  FIG. 22  depict, respectively, a conventional modular detector and a pixelated detector. 
         FIG. 23  and  FIG. 24  illustrate a detector package where the scintillation detector comprises a slab substrate with the photon sensors positioned behind the back of the scintillator and light-guide substrate surface instead of the side edges. 
         FIGS. 25 and 26  illustrate, respectively, MDRFs and interaction-position estimates of the detector package of  FIG. 23 . 
         FIG. 27  and  FIG. 28  illustrate cross-sectional views of stacked scintillation detectors in embodiments of the present invention where the scintillation detectors in each stack are laterally offset from one another. 
         FIG. 29  and  FIG. 30  are cross-sectional views of stacked scintillation detectors (aligned and offset, respectively) in embodiments of the present invention where the scintillation detectors are curved. 
         FIG. 31  is a cross-sectional view of stacked scintillation detectors in an embodiment, where multiple tapered scintillation detectors are arranged laterally adjacent to each other in a configuration other than a flat planar configuration. 
         FIG. 32  is a plan view of twelve stacked scintillation detectors arranged to form a PET detector ring, in an embodiment. 
         FIGS. 33A-E  illustrate cross-sectional views of scintillation substrates having different shapes and surface curvatures, in respective embodiments. 
         FIG. 34  is a schematic block diagram of a PET scanner configured to operate with one or more scintillator detectors in each of  FIGS. 7-11 , in an embodiment. 
         FIG. 35  is a flowchart illustrating a method for determining interaction position of a gamma-ray interaction event, in an embodiment. 
         FIG. 36  is a simulated projection image by a slit having a width corresponding to a width of a gamma-ray photon beam used to illuminate an embodiment of a scintillation detector without optical barriers. 
         FIG. 37  is a profile of the projection image of  FIG. 36 . 
         FIG. 38  is a simulated projection image by a slit having a width corresponding to a width of a gamma-ray photon beam used to illuminate an embodiment of a scintillation detector with drilled-hole optical barriers. 
         FIG. 39  is a profile of the projected image of  FIG. 38 . 
         FIG. 40  includes photographs of components of a prototype detector, in an embodiment. 
         FIG. 41  illustrates measured MDRF of the prototype detector of  FIG. 40  when it lacks optical barriers. 
         FIG. 42  is a measured projection image of a slit having a width equal to that of a gamma-ray photon beam used to evaluate positioning performance of the prototype detector of  FIG. 40  when it lacks optical barriers. 
         FIG. 43  is a profile of the projection image of  FIG. 42 . 
         FIG. 44  illustrates measured MDRF of prototype detector of  FIG. 40  when it includes drilled-hole optical barriers in the scintillation crystal. 
         FIG. 45  is a measured projection image of a slit having a width equal to that of a gamma-ray photon beam used to evaluate positioning performance of the prototype detector of  FIG. 40  when the detector includes optical barriers. 
         FIG. 46  is a profile of the projection image of  FIG. 44 . 
         FIG. 47  is a position-corrected (method described below) energy spectrum of a flood image by  137 Cs, using crystal CsI(TI). 
         FIG. 48  illustrates an MDRF for refractive index of edge optical-gel/SiPM-window set to 1.5 and a corresponding point-grid image. 
         FIG. 49  illustrates an MDRF for refractive index of edge optical-gel/SiPM-window set to 1.82 and a corresponding point-grid image. 
         FIG. 50  illustrates an MDRF for refractive index of edge optical-gel/SiPM-window set to 1.5, but with optical barriers in the scintillation crystal, and a corresponding point-grid image. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Modern hard X-ray and gamma-ray detectors generally employ either scintillation-based or semiconductor-based techniques. During the last two decades, much progress has been made in the field of room-temperature semiconductor detectors such as CdZnTe, but there still remain major challenges compared with scintillation detectors. These include high cost to manufacture large-area detectors (Takahashi, Tadayuki, and Shin Watanabe. “Recent progress in CdTe and CdZnTe detectors.”  Nuclear Science, IEEE Transactions  on 48.4 (2001): 950-959), relatively lower detection efficiency (Chen, H., et al. “CZT device with improved sensitivity for medical imaging and homeland security applications.”  SPIE Optical Engineering+Applications.  International Society for Optics and Photonics, 2009), relatively poor timing resolution (Meng, Ling J., and Zhong He. “Exploring the limiting timing resolution for large volume CZT detectors with waveform analysis.”  Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment  550.1 (2005): 435-445) and complicated readout ASICs (Peng, Hao, and Craig S. Levin. “Recent developments in PET instrumentation.”  Current Pharmaceutical Biotechnology  11.6 (2010): 555). The scintillation-detection technique thus remains dominant in modern clinical and preclinical SPECT and PET systems. However, as described herein, it is still possible to enhance the performance of scintillator detectors. Most scintillation detectors apply monolithic crystals or pixelated crystals and then attach photon sensors on the front or back surfaces to collect light and use the information to calculate the event&#39;s position, energy, and time of interaction. Recently, silicon photomultipliers (SiPMs) have made great progress in achieving more compact size, smaller dark count rate, less cross talk, higher photon detection efficiency, higher gain, and broader spectral response. These advances enable new scintillation detector geometries as described herein. For example, the thin sides of monolithic scintillation crystals may be utilized as the photon readout surfaces. 
     For example,  FIG. 7  illustrates a graphical projection of a scintillation detector  700 , where the scintillation detector  700  includes a scintillator substrate  715  and a plurality of photodetectors  710 . Scintillator substrate  715  has an edge surface  706 . Each photodetector  710  has a photosensitive region in optical communication with one of a respective plurality of regions of edge surface  706 . 
       FIG. 7  includes callouts for photodetectors  710 (1) and  710 (3), where photodetector  710 (2) therebetween is spatially separated from scintillator substrate  715  at least for illustration of edge surface  706 (2).  FIG. 7  includes a coordinate system  798  that defines directions x, y, and z. Herein and unless stated otherwise, references to directions or planes denoted by at least one of x, y, or z refer to coordinate system  798 . Scintillation detector  700  may also include a readout circuit and data processing unit communicatively coupled to the plurality of photodetectors. 
     Edge surfaces  706  are between a front surface  715 F and a back surface  715 B of scintillator substrate  715 , which are, for example, parallel to the x-y plane. Each of front surface  715 F and back surface  715 B may be one of planar, concave, and convex. Scintillator substrate  715  has spatial dimensions  715 X,  715 Y, and  715 Z in respective directions x, y, and z. Spatial dimensions  715 X and  715 Y are, for example, between twenty millimeters and one hundred millimeters. Spatial dimension  715 Z is, for example, between two millimeters and ten millimeters. In an embodiment, spatial dimensions  715 X,  715 Y, and  715 Z are 15-150 mm, 15-150 mm, and 0.5-20 mm, respectively. In an embodiment, spatial dimensions  715 X,  715 Y, and  715 Z are 50.4 mm±Δ, 50.4 mm±Δ, and 3.0 mm±δ, respectively, where Δ=5 mm and δ=2 mm. In an embodiment, spatial dimensions  715 X,  715 Y, and  715 Z are 27.4 mm±Δ, 27.4 mm±Δ, and 3.0 mm±δ, respectively. 
     Candidate materials for scintillator substrate  715  include, but are not limited to, lutetium-yttrium oxyorthosilicate (LYSO), lutetium oxyorthosilicate (LSO), cesium iodide (CsI), sodium iodide (NaI), lanthanum(III) bromide (LaBr 3 ), yttrium aluminum perovskite (YAP), yttrium aluminum garnet (YAG), bismuth germinate (BGO), calcium fluoride activated by europium, lutetium aluminum garnet, gadolinium silicate doped with cerium, cadmium tungstate, lead tungstate, double tungstate of sodium and bismuth, zinc selenide activated with tellurium (ZnSe(Te)), and combinations thereof. 
     Each photodetector  710  may be a photodetector known in the art, for example, a photon sensor. Types of photon sensors include, but are not limited to, photomultipliers, such as silicon photomultipliers and photomultiplier tubes. In an embodiment, scintillator substrate  715  is monolithic. Photodetectors  710  may be attached to an edge surface  706  via an adhesive, such as, but not limited to, an optical gel or by room-temperature-vulcanization (RTV) silicone. In a further embodiment, the light sensor comprises a layer of window material, such as silicone epoxy. In an embodiment, the refractive index of the adhesive, window material, or combination thereof, should be between that of the photon sensor material and the scintillation crystal. 
     Edge surfaces  706  may have a thin-film coating thereon situated between photodetectors  710  and edge surfaces  706 . The thin-film coating may have at least one of an angular-dependent transmittance and an angular-dependent reflectance, which may be engineered to reshape the MDRF of detector  700  and hence increase accuracy of scintillation localization. Candidate materials for the thin film coating include, but are not limited to, metals, ceramic materials, cermats, alloys, and combinations thereof. In certain embodiments, the thin film coating has an effect on reflection, refraction properties (such as angular dependence), and polarization properties. 
     One or more edge surfaces  706  may also have a surface roughness. Surface roughness may reduce measurement inaccuracies, e.g. high bias and variance, resulting from total internal reflection at edges of scintillator substrate  715 . Part or all of a surface  706  may have a patterned surface roughness, such as cross-hatching, hatching or stripes, as illustrated in regions of edge surface  706 (4), which may be beneath a detector  710  and/or between adjacent detectors  710 . The hatching or stripes may be oriented at an angle (e.g.,) 90° with respect to at least one of surfaces  715 F and  715 B. In an embodiment, the surface roughness is sufficient for each edge surface  706  to be a Lambertian surface. In an embodiment, the surface roughness has an Ra value higher than the wavelength of the emission peak. As known in the art, “Ra” refers to average roughness and is the arithmetic average of the absolute values of the profile height deviations from a mean line. 
     At least one photodetector  710  may be attached to scintillator substrate  715  such that its photosensitive region faces edge surface  706 . Photodetectors  710  may include a photodetector spatially separated from scintillator substrate  715  and in optical communication with a region of edge surface  706 . For example, detector  710 (2) may be in optical communication with edge surface  706 (1) via an optical coupler  708 . Optical coupler  708  may include a waveguide, such an optical fiber, prism, lens, mirror or combinations thereof. 
       FIG. 8  illustrates a graphical projection of a scintillation detector  800  that includes a scintillator substrate  815  and photodetectors  710 . Scintillation detector  800  and scintillator substrate  815  are examples of scintillation detector  700  and scintillator substrate  715 , respectively. Scintillator substrate  815  includes one or more optical scatterers  820 . An optical scatterer  820  is anything that can reflect, refract, absorb or scatter photons that help to shape the MDRFs, including but not limited to a through hole or a blind hole drilled into the scintillator substrate  815 , a laser-etched site, or a photon scatterer attached on the front and/or back surfaces of scintillator substrate  815 . When optical scatterers  820  are through holes or blind holes, they may be lined with a polymer coating, such as a fluoropolymer. 
     In the x-y plane, one optical scatterer may have a cross-section having a closed shape. The closed shape is, for example, at least one of a conic section, a polygon, a concave shape, and a convex shape. Each optical scatter has a width  820 W, which is 2.0±1.0 mm, for example. In an embodiment, each optical scatterer  820  is a cylindrical hole having a diameter 2.0±0.5 mm. 
     Scintillator substrate  815  may include at least one peripheral optical scatterer  822  with a distance  824  from an edge surface. In an embodiment, peripheral optical scatterers  822  are used to increase spatial resolution near the edges. 
       FIG. 9  is a plan view of a scintillation detector  900 , which includes a scintillator substrate  915 , photodetectors  710 , and a light spreader  918 . Scintillation detector  900  is an example of scintillation detector  700 . Scintillator substrate  915  may be either scintillator substrate  715  or  815 , and has edge surfaces  906 , which are equivalent to edge surfaces  706  of scintillator substrate  715 . Light spreader  918  at least partially surrounds scintillator substrate  915  between edge surfaces  906  and detectors  710 . Light spreader  918  may have a thickness, in the z direction, equal to a distance between front surface  715 F and back surface  715 B at edge surface  706  of scintillator substrate  715 . Light spreader  918  has thicknesses  918 X and  918 Y in the x direction and the y directions. In an embodiment, the min and max of  918 X and  918 Y will depend on the size of the substrate  915 , and typically will not exceed 0.25 W, where W is the width of the substrate  915  (if it is square). In an embodiment, the light spreader  918  comprises glasses, polymers or other transparent materials with good light-transmitting or optical coupling properties. 
     Due to the total internal reflection effect and the large index of refraction difference between typical scintillator crystals (n≥1.8) and air (n=1), if the front and back surfaces are well polished, at least 83% of photons will be transported to the edges of the thin slab with almost with no loss. With an embodiment described herein, scintillator crystals with a lower refractive index of n=1.1 would still yield 41.6% of photons being transported to the edges of scintillator substrate  715 . 
     Since photodetectors  710  (consider silicon photon multipliers (SiPMs) as an example below of a photodetector) are attached on edge surfaces  706 , once the scintillation photons arrive at the edges, photodetectors  710  will absorb a portion of the scintillation photons and generate signals proportional to the number of photons absorbed. The summation of the signals gives the energy information (if the photodetectors are well calibrated), while the signal amplitude differences between different SiPMs give the position information. For example, if a gamma-ray photon interacts inside scintillator substrate  715  at a position close (e.g., closest) to photodetector  710 (1), photodetector  710 (1) will probably produce a relatively larger signal output compared with other photodetectors  710 ; this signal difference indicates that the interaction happened at a position close to photodetector  710 (1). 
     A technical benefit of optical barriers  820  is that they will cast shadows on photodetectors  710  by reflection/deflection of scintillation photons. The shadow&#39;s position of an optical barrier on the thin surfaces will move a significant distance even when the interaction position moves slightly with respect to one or more optical barriers  820 . The significant shift of a shadow will produce a significant change of signal amplitude generated by the photodetector  710  where the shadow is cast. Thus, the spatial resolution across the plane of the scintillator substrate  815  will be enhanced greatly. In addition, since the photons are well confined within scintillator substrate  815 , the contrast created by the shadows is significantly large, which means it is suitable to combine the thin slab edge readout technique with optical scatterers  820 . 
       FIG. 10  is a cross-sectional view of a stacked scintillation detector  1070 . Scintillation detector  1070  includes a stack of scintillation detectors  1000 (1-N), where N is a positive integer. Each scintillation detector  1000  is, for example, a scintillation detector  700 , and includes respective scintillator substrates  1015 (1-N), each of which are examples of scintillator substrate  715 . Each scintillator substrate  1015  may be formed of a different scintillation crystal. For example, “odd” scintillation substrates  1015 (1, 3, . . . ) may be formed of a first scintillation crystal and “even” scintillation substrates  1015 (2, 4, . . . ) may be formed of a second scintillation crystal.  FIG. 11  illustrates an graphical projection of a scintillation detector  1170 , which is an example of scintillation detector  1070  where N=6. Adjacent scintillator substrates  1015  are separated by a distance  1015 D, which is between one millimeters and five millimeters, for example. 
     Stacked scintillation detectors  1070  and  1170  are capable of detecting more scintillation events than a single scintillation detector  1000 . In scintillation detector  1070  it is straight forward to determine in which layer the gamma interaction occurs. This gives the depth of interaction (DOI) information. Therefore, this makes the detector a 3D gamma-ray detector. Some readouts of photodetector  710  may be combined to decrease the total number of readouts without significantly affecting the result. Scintillation detector  1170  is illustrated with six scintillation detectors  1000  (approximately 10-20 mm of scintillation material) for illustrative purposes, and may include more or fewer scintillation detectors  700  without departing from the scope hereof. 
       FIG. 12  is a plan view of a PET detector ring  1280 . PET detector ring  1280  includes a first plurality of scintillation detectors  1200  arranged to form an inner ring enclosed by dashed circle  1282 . Each scintillation detector  1200  is for example of scintillation detector  700  or a stacked scintillation detector  1070 . 
     PET detector ring  1280  may also include a second plurality of scintillation detectors  1200  arranged to form an outer ring enclosed between dashed circles  1282  and  1284 . Inner and outer rings may have a common center  1286 . Scintillation detectors  1200  in the outer ring may be collectively rotated around common center  1286  by an angle θ such that photons propagating through a gap between detectors  1200  in the inner right also propagate through a detector  1200  in the outer ring. 
     Each scintillation detector  1200  has a back surface  1204  opposite a front surface  1202 . Surfaces  1202  and  1204  are, for example, a front surface  715 F and back surface  7156  of scintillator substrate  715 . Surfaces  1202  and  1204  may be planar and parallel. Alternatively, surfaces  1202  and  1204  may each be a portion of respective cylindrical surfaces. For example, the cylindrical surfaces may be concentric and have a common center of curvature being an axis through common center  1286  and normal to a plane of  FIG. 12 . 
     Embodiments herein are capable of utilizing thin slabs of scintillation crystals to acquire each photon&#39;s interaction position and energy information; there is no need to pixelate the crystal, which addresses limitations 2, 4, and 5 described in the Background. Stacking multiple identical layers of scintillation crystal slabs together increases detector stopping power, which facilities determination of which layer the photon interaction occurs (direct readout), and the depth of interaction may be acquired. The depth of interaction resolution depends on the thickness of the scintillation crystal, which makes it easy to achieve a depth of interaction resolution of around one millimeter. Embodiments herein hence overcome limitation 3 described above. Moreover, the traditional pixelated detector has a chance to wrongly decode the pixels—mistakenly assigning an interaction event from a certain pixel to an adjacent pixel. By contrast, embodiments herein overcome this weakness since there is no need for pixel decoding. Additionally, embodiments of the present invention also do not contain pixel gaps, which results in increased gamma-ray sensitivity. In additional embodiments, the scintillation detectors of the present invention are used to detect scattering of photons by charged particles (i.e., Compton scattering). 
     EXAMPLE 1 
     An analysis of side readouts of monolithic scintillation crystals. A method of using the side surfaces of a thin monolithic scintillation crystal for reading out scintillation photons is described. A Monte-Carlo simulation was carried out for an LYSO crystal of 50.8 mm×50.8 mm×3 mm with five silicon photomultipliers attached on each of the four side surfaces. With 511 keV gamma-rays, X-Y spatial resolution of 2.10 mm was predicted with an energy resolution of 9%. The addition of optical barriers was also explored to improve the X-Y spatial resolution. An X-Y spatial resolution of 786 μm was predicted with an energy resolution of 9.2%. Multiple layers may be stacked together and readout channels may be combined. Depth of interaction information (DOI) may be directly read out. This method provides an attractive detector module design for positron emission tomography (PET). 
       FIG. 13  illustrates a scintillation event  1301  (or equivalently, a gamma-ray interaction event) within scintillator substrate  1315  of a scintillation detector  1300 . Scintillator substrate  1315  has a high refractive index, e.g., exceeding 1.7, compared to its surrounding medium, e.g., air. Scintillator substrate  1315  is, for example, one of scintillator substrates  715 ,  815  and  915 . Similarly, scintillation detector  1300  is, for example, one of scintillation detectors  700 ,  800 , and  900 . 
     If substrate  1315  has no surface coatings, most of the scintillation photons will be reflected by top surface  1315 F and back surface  1315 B due to total internal reflection (TIR). Only a small portion of scintillation photons with incident angles smaller than a critical angle θ c  have a chance to escape. For the LYSO crystal surrounded by a medium with unity refractive index, θ c =33.8° at the gamma-ray frequencies of interest. The majority (&gt;80%) of the photons will impinge on the front and back surfaces with angles larger than the critical angle, and will propagate, via TIR, to the edges of the scintillator. 
     In these simulations, the photons reaching the four thin sides are collected by photon sensors attached to them. The information in the amplitudes of the photon sensor signals allows the position and energy information of the gamma-ray to be estimated. 
       FIGS. 15 and 16  illustrate results of a Monte Carlo simulation of scintillation substrate  700 . The simulation was in part based on spectral reflectance of a spectrometer shown in  FIG. 14 , as described below. The simulated scintillator substrate was a 50.8-mm×50.8-mm LYSO slab, with thickness of three millimeters. Five silicon photomultipliers (SiPM) are attached on each side surface of the thin slab, so there are a total of twenty SiPMs. The spectrum-averaged photon detection efficiency (PDE) was chosen to be 20%. The dark current rate was chosen as 1.0 MHz/mm 2 , the excess noise factor (ENF) of the SiPMs was assumed to be 1.21. These parameters were chosen conservatively, within the limits of performance parameters provided by several manufacturers. 
     A portion of the photons that arrive at the side surfaces of the scintillation crystal will be reflected due to the refractive index differences of the SiPM&#39;s window material, silicon substrate, optical gel, and scintillation crystal. The reflected photons are deleterious to the interaction-position estimation. To measure the reflectance of SiPM, Hamamatsu 11827-3344 MG was scanned in a Cary 5000 spectrometer. The spectral reflectance curve is shown in  FIG. 14 . The reflectance in the simulation was chosen to be 0.4 as a conservative estimate, and the type of reflection was chosen to be specular reflection. 
     The mean-detector-response function (MDRF) was acquired by simulating a scan of a 511 keV gamma-ray beam normal to the LYSO scintillator surface in a matrix of 252×252 points across the 50.8 mm×50.8 mm scintillator area (depth=3.0 mm), with a step size of 0.2 mm.  FIG. 15  illustrates the simulated MDRF, MDRF  1500 , which includes twenty 50.8-mm×50.8-mm frames. Each frame corresponds to one of twenty SiPMs each attached to one of the four sides of the detector. 
     The mean-detector-response function (MDRF) is used to find the gamma-ray interaction position and energy with maximum likelihood estimation. If the detector is sampled by a tightly collimated beam, at positions of a 20×20 square array having 2.5-mm spacing for example, the estimated interaction positions are shown in point-grid image  1600 ,  FIG. 16 . Point-grid image  1600  may be derived from MDRF  1500  using a maximum-likelihood estimation (Hesterman, Jacob Y. et al. “Maximum-Likelihood Estimation With a Contracting-Grid Search Algorithm.”  IEEE Transactions on Nuclear Science  57.3 (2010): 1077-1084). 
     The average spatial resolution of the points in point-grid image  1600  as defined by FWHM is 1.49 mm, and the average absolute values of biases in x, y directions are 56 μm and 52 μm, respectively. However, points near the plane center cannot be resolved. As used in this context, “bias” refers to the difference between true positions and estimated positions averaged over many events at different positions on the detector plane. 
     The energy resolution of this detector is given by equation: 
                 (       Δ   ⁢           ⁢   E     E     )     2     =         (     δ   sc     )     2     +       (     δ   p     )     2     +       (     δ   st     )     2     +         (     δ   n     )     2     .             
The symbol δ represents the 2.355 times the standard deviation of each component normalized by the estimated energy in unit of photon number. δ sc  is the intrinsic resolution of the scintillation crystal, δ p  is the transfer resolution, δ st  is the statistical resolution and δ n  is the energy deviation caused by dark current rate. For this simulation, δ sc ≈7.8%, δ p ≈0.91%, since the MDRF is quite uniform
 
                 δ   st     =       2.35   ×       ENF   N         ≈     4.1   ⁢   %         ,       δ   n     ≈     1.43   ⁢   %       ,       (       Δ   ⁢           ⁢   E     E     )     ≈     9.0   ⁢   %       ,         
and ENF denotes the excess noise factor.
 
     Optical barriers  820  (also referred to herein as optical scatterers) may be introduced to improve interaction position estimation. They may be holes mechanically drilled through the scintillator, photon absorbers or scatterers attached on the two major surfaces (for breaking the TIR condition) or even laser-etched patterns. Shadows will be created due to the block or redirection of scintillation photons. Optical barriers created in the plane of the crystal layer, may result in improved spatial resolution. For example, adding 16-square 2-mm Lambertian rods of reflectance 0.95 inside the crystal layer in a 4×4 grid results in a modified MDRF  1700 , shown in  FIG. 17 . 
     As shown in  FIG. 18 , the modified MDRF yields improved spatial resolution, at a cost of reduced sensitivity due to inclusion of the rods (about 2.48% reduction). The average spatial resolution is 786 μm, the average absolute values of biases in X, Y directions are 23 μm and 22 μm, respectively, and ΔE/E≈9.2%. The energy resolution is worse than that without barriers, due to the photon loss and unevenness caused by the rod barriers. 
     Laser etching may create Lambertian surfaces to form the optical barrier pattern. For example, a 4×4 grid of 2-mm diameter rods may be created inside the crystal layer, at the same positions of  FIG. 17 , resulting in an MDRF  1900 ,  FIG. 19 . As shown in  FIG. 19 , there is no dead area compared with  FIG. 17 , so gamma-ray interaction positions within the rod areas may be determined. The interaction-position estimations of 20×20 scan is shown in  FIG. 20 . The laser-etched barrier yields an improved average spatial resolution of 867 μm, the average absolute values of biases in x, y directions are 30 μm and 33 μm, respectively, ΔE/E≈9.4%. 
       FIG. 21  and  FIG. 22  depict, respectively, a conventional modular detector  2100  and a pixelated detector  2200 . These conventional detectors have several significant limitations. For example, the crystal layers in traditional modular cameras have a finite thickness, which results in lower detection efficiency for higher energy rays. Additionally, it is hard to make calibrations for depth of interaction (DOI). X-Y resolution is coupled with DOI position and is usually biased at edges or corners. Pixelated detector modules, such as used in PET detectors, typically have higher cost for cutting and treating crystal surfaces and making arrays of small pixelated crystals can be very challenging. Additionally, it is complicated to achieve DOI information and there is a loss of sensitivity due to the gaps between pixels. 
     However, the improved performance of the present invention is not due to just the use of a slab of scintillation material. Positioning the photon sensors along the edges of the scintillation detectors provides additional benefits.  FIG. 23  and  FIG. 24  are schematics of detector package  2300  where the scintillation detector comprises a slab substrate with the photon sensors positioned behind the back substrate surface. This configuration results in MDRF  2500 ,  FIG. 25 , and point-grid image  2600 ,  FIG. 26 .  FIGS. 23-26  illustrate that the positioning seen in a monolithic crystal detector is relatively poorer due to a larger point spread function. Additionally, the positioning at the center of each PMT degrades (cannot be resolved), and the positioning close to detector edges has poorer spatial resolution. 
       FIG. 27  illustrates a stacked scintillation detector  2780 , which includes a stack of scintillation detectors  700 (7-12). Scintillation detectors  700 (8-12) are horizontally offset from scintillation detector  700 (7) by respective distances  2752 (8-12). Any two distances  2752  may be equal. 
       FIG. 28  illustrates a stacked scintillation sensor  2880  that includes a plurality of scintillation detectors  2800 , which are each examples of scintillation detector  700 . Scintillation detectors  2800  are stacked in the z direction. As shown in  FIG. 28 , the scintillation detectors  2800  in each row are laterally offset, in the x direction, to scintillation detectors in a different row. Each scintillation detector  2880  includes a plurality of photodetectors  710 . 
       FIG. 29  illustrates a stacked scintillation sensor  2980  A that includes a plurality of scintillation detectors  2900  in two curved rows  2983  and  2984 . Each scintillation detector  2900  is an example of scintillation detector  700 . Each scintillation detector  2900  includes a plurality of photodetectors  710 . Each scintillation detector  2900  has a front surface  2902  and a back surface  2904 , which in this example have the same respective concavity. Each photodetectors  710  in each row (e.g., row  2983 ) are aligned to a respective photodetector  710  in a different row (e.g., row  2984 ). For example, photodetector  710 (291) is aligned with photodetector  710 (292). 
       FIG. 30  illustrates a stacked scintillation sensor  3080  that includes a plurality of scintillation detectors  3000  in two curved rows  3083  and  3084 . Scintillation detectors  3000  in row  3083  are laterally offset from scintillation detectors  3000  in row  3084  relative to an axis perpendicular to the front substrate-surfaces of the first scintillation detector (not shown). 
       FIG. 31  illustrates a stacked scintillation sensor  3175  that includes a plurality of scintillation detectors  3100  in three curved or angled rows  3181 - 3183 . Each scintillation detector  3100  includes a plurality of photodetectors  710  surrounding a scintillator substrate  3115 . Scintillator substrate  3115  is an example of a scintillator substrate  715 , and has a trapezoidal cross-section, which facilitates orienting adjacent scintillation detectors  3100  in a same row. Scintillator substrate  3115  has a front surface  3115 F and a back surface  3115 B, which are planar in the embodiment of  FIG. 31 . 
       FIG. 32  is a plan view of twelve stacked scintillation detectors  3270  arranged to form a PET detector ring  3280 . Each stacked scintillation detector  3270  includes four scintillation detectors  3100 , and is an example of stacked scintillation detector  1170 ,  FIG. 11 . Stacked scintillation detectors  3270  are arranged to form a aperture  3210  defined by twelve front surfaces  3115 F of inner-most scintillation detectors  3100 . When surfaces  3115 F are planar, aperture  3210  is a polygonal aperture. Without departing from the scope hereof, PET detector ring  3280  may include fewer than or more than twelve stacked scintillation detectors  3270 . 
       FIGS. 33A-E  are cross-sectional views of different possible scintillation substrates  3315 A- 3315 E, which are example of scintillation substrate  715 . Each scintillation substrate  3315  has opposing surfaces  3302  and  3304 , which independently from one another may be planar, concave, or convex. Each scintillation substrate  3315  has edge surfaces  3306 , which are in optical communication with photodetectors  710  when a scintillation substrate  3315  is part of a scintillation detector  700 . It is understood that  FIGS. 33A-E  are non-limiting examples and that scintillation substrates having different or modified cross-sections are also suitable and encompassed by the present invention. 
       FIG. 34  is a schematic block diagram of a PET scanner  3490 . PET scanner  3490  includes a scintillation detector  3400  communicatively coupled to a data processor  3402 . Data processor  3402  includes a microprocessor  3432  communicatively coupled to a memory  3460 , which stores software  3420 . Microprocessor  3432  performs functions of PET scanner  3490  described herein when executing machine-readable instructions of software  3420 . 
     Scintillation detector  3400  is, for example, one of scintillation detectors  700 ,  800 ,  900 ,  1070 ,  1170 ,  2800 ,  2900 ,  3000 ,  3100 , and  3270 . Scintillation detector  3400  includes a scintillator substrate  715  and plurality of photodetectors  710 (1-N) each in optical communication with one of a respective plurality of edge region locations  3412 (1-N) on one of edge surfaces  706  of scintillator substrate  715 . Edge region locations  3412  are stored in memory  3460  of data processor  3402  and may be distinct and non-overlapping. 
     Memory  3460  may be transitory and/or non-transitory and may include one or both of volatile memory (e.g., SRAM, DRAM, or any combination thereof) and nonvolatile memory (e.g., FLASH, ROM, magnetic media, optical media, or any combination thereof). Part or all of memory  3460  may be integrated into microprocessor  3432 . 
     Memory  3460  also stores detector parameters  3405  associated with scintillation detector  3400 . Examples of detector parameters  3405  include the detectors MDRF (e.g. MDRFs  1500 ,  1700 ,  1900 , and  2500 ) and in some embodiments grid-point images (e.g., grid-point images  1600  and  2600 ) and resolution maps derived from a grid-point image. Detector parameters  3405  may also include detector geometry, such as spatial dimensions of scintillator substrate  715  and detectors  710 . Software  3420  includes an estimator  3422 , which may include at least one of the following estimation methods: maximum likelihood, contracting-grid maximum likelihood search, least-squares, iterative, LN norm comparisons (where N is 0, 1, 2 . . . ), and an Anger method. In an embodiment, an energy window or timing coincidence window is applied before the estimation of the gamma-ray interaction position. These window limits are also stored in memory  3460 . In an embodiment, a likelihood threshold (lower bound, if using Maximum likelihood estimation), LN norm threshold (upper bound, if using LN norm comparison) are also stored inside the memory  3460 . 
     In a use scenario, a scintillation event  3401  occurs in substrate  715 , as illustrated in  FIG. 34 . When PET scanner  3490  detects scintillation event  3401 , detectors  710 (1-N) of scintillation detector  3400  generate respective electrical signal amplitudes  3400 D(1-N) which data processor  3402  receives and stores in memory  3460 . Estimator  3422  generates lateral coordinates  3491  of the scintillation event from detector parameters  3405 , edge region locations  3412 , and electrical signal amplitudes  3400 D. 
       FIG. 35  is a flowchart illustrating a method  3500  for determining the interaction position of a gamma-ray interaction event, occurring in a substrate resulting from electromagnetic radiation incident on a front surface of the substrate. Method  3500  includes at least one of steps  3510 ,  3520 ,  3522 , and  3524 . Method  3500  is, for example, implemented by microprocessor  3432  executing computer-readable instructions of software  3420 . 
     Step  3510  comprises detecting, with a plurality of photodetectors, light from the gamma-ray interaction event resulting in a respective plurality of electrical signal amplitudes. Each of the plurality of photodetectors having a respective photosensitive region in optical communication with one of a respective plurality of regions of an edge surface of the substrate located between a front perimeter of the front surface and a back perimeter of a back surface of the substrate opposite the front surface. In an example of step  3510 , scintillation detector  3400  detects light from scintillation event  3401  having occurred within scintillator detector  3400 . 
     Step  3520  comprises estimating lateral coordinates, in a plane of the substrate, of the interaction position based on the plurality of electrical signal amplitudes and respective regions. In an example of step  3520 , estimator  3422  generates lateral coordinates  3491  from electrical signal amplitudes  3400 D. 
     Step  3520  may include step  3522  of comparing the plurality of electrical signal amplitudes to a mean-detector response function of the plurality of photodetectors at the respective regions around the substrate perimeter. In an example of step  3520 , estimator  3422  generates lateral coordinates  3491  from mean detector parameters  3405 , electrical signal amplitudes  3400 D, and optionally edge region locations  3412 . 
     Step  3522  may include step  3524  of performing at least one of a maximum likelihood estimate, a least-squares estimate, an iterative estimate, LN norm comparison (where N can be 0, 1, 2, 3, 4, 5 . . . , and a combination of L1 with L2, L2 with L4, etc.), and an Anger method. In an example of step  3522 , estimator  3422  uses at least one of a maximum likelihood estimate, a least-squares estimate, an iterative estimate, and an Anger method to generate lateral coordinates  3491  from detector parameters  3405 , edge region locations  3412 , and electrical signal amplitudes  3400 D. 
     EXAMPLE 2 
     Simulation of a CsI/LYSO Detector Layer. This section describes simulation of crystals that that were tested experimentally. The geometry of the scintillation crystal was set as 27.4 mm×27.4 mm×3 mm. Sixteen SiPMs were attached to the edges of the scintillation crystal with 4 SiPMs on each edge. Several factors influencing the spatial resolution were studied:
         I. gamma-ray photon energy and scintillator material (662 keV for CsI(TI) vs 511 keV for LYSO);   II. optical barriers (with/without);   III. optical-gel &amp; SiPM-window refractive indices (1.5 vs 1.82).
 
The dark-count rate was set to 2 MHz/SiPM in accordance with the Hamamatsu S13360-6050 PE MPPC data sheet. The readout shaping time was set to 4 μs for CsI(TI) and 150 ns for LYSO, based on their published decay times (Mao R, Zhang L, Zhu, R. Optical and Scintillation Properties of Inorganic Scintillators in High Energy Physics.  IEEE Transactions on Nuclear Science.  2008; 55 (4): 2425-2431). The optical-gel/SiPM-window materials&#39; refractive indices were studied to evaluate the effect of reflectance at the edge boundaries. The simulated resolutions (evaluated at 20×20 positions by a narrow beam of gamma-ray photons) are shown in Table I.
       

     
       
         
           
               
             
               
                 TABLE I 
               
             
            
               
                   
               
               
                 Factors that influence spatial resolution based on simulated data. 
               
            
           
           
               
               
               
               
               
            
               
                   
                   
                   
                 Edge-coupling 
                 Edge-coupling 
               
            
           
           
               
               
               
            
               
                 Spatial Resolution Factors 
                 index 1.5 
                 index 1.82 
               
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 without 
                 CsI(TI) @ 662 
                 Whole 
                 0.69 ± 0.31 mm 
                 0.53 ± 0.16 mm 
               
               
                 optical 
                 keV 
                 Center 
                 0.55 ± 0.08 mm 
                 0.75 ± 0.08 mm 
               
               
                 barrier 
                 LYSO @ 511 
                 Whole 
                 0.98 ± 0.43 mm 
                 0.75 ± 0.23 mm 
               
               
                   
                 keV 
                 Center 
                 0.78 ± 0.11 mm 
                 1.06 ± 0.11 mm 
               
               
                 with 
                 CsI(TI) @ 662 
                 Whole 
                 0.38 ± 0.20 mm 
                 0.23 ± 0.07 mm 
               
               
                 optical 
                 keV 
                 Center 
                 0.22 ± 0.08 mm 
                 0.22 ± 0.09 mm 
               
               
                 barrier 
                 LYSO @ 511 
                 Whole 
                 0.57 ± 0.31 mm 
                 0.33 ± 0.09 mm 
               
               
                   
                 keV 
                 center 
                 0.34 ± 0.12 mm 
                 0.32 ± 0.13 mm 
               
               
                   
               
               
                 *Whole: average spatial resolution of the whole detector region (27.4 mm × 27.4 mm) 
               
               
                 *Center: average spatial resolution of the center region (15 mm × 15 mm) 
               
               
                 *Edge-coupling index: the refractive index of the optical-gel and SiPM-window material 
               
            
           
         
       
     
     A slit beam of 662-keV gamma photons was also simulated to verify the experiment described below. The width of the simulated beam was 0.44 mm (FWHM of a Gaussian profile). The resulting projection images for detectors without and with optical barriers are shown in  FIGS. 36-37 and 38-39 , respectively.  FIG. 36  is a simulated projection image by a slit of width 0.44 mm.  FIG. 37  is a profile of the projection image of  FIG. 36 . with indicated FWHM, on the prototype detector (27.4 mm×27.4 mm×3 mm CsI(TI), with sixteen SiPMs) without optical barriers.  FIG. 38  is a simulated projection image by a slit of width 0.44 mm.  FIG. 39  is a profile of the projected image of  FIG. 38  with indicated FWHM, on the prototype detector with drilled-hole optical barriers. 
     Experimental Verification 
       FIG. 40  includes photographs of components of a prototype detector  4001  and  4002 . The prototype detector includes the following components, labelled (a) through (e) in  FIG. 40 :
         (a) Sixteen Hamamatsu S13360-6050PE MPPCs (SiPMs);   (b) CsI(TI) scintillation crystal: CsI(TI) crystal of dimension 27.4 mm×27.4 mm×3 mm (obtained from Proteus, without mechanical holes as in detector  4001 , and with mechanical holes, as in detector  4002 );   (c) pre-amplifier circuit;   (d) amplifier and analog-to-digital-converting circuit;   (e) FPGA data acquisition circuit (AiT 16-channel-readout circuit).
 
The photograph labelled (f) is prototype detector  4001  or  4002 .
       

     The scintillator was chosen to be CsI(TI) for convenience (no background activity from the crystal and low hygroscopicity). The MDRFs were acquired by scanning the detector with a crossed-slit-collimated beam of 662 keV photons from a  137 Cs source. The beam size on the detector surface was measured to be about 0.44 mm×0.44 mm (FWHM) with an intensified quantum imaging detector (iQID) camera (Miller B W, Gregory S J, Fuller E S, Barrett H H, Barber H B, Furenlid L R, The iQID camera: An ionizing-radiation quantum imaging.  Nucl Instrum Methods Phys Res A.  2014; 11 (767): 146-152). Due to a limited dynamic range in the AiT readout electronics, only the center area of the detector was calibrated (15.0 mm×15.0 mm) and used for imaging. 
     Experiment Case 1: Without Optical Barrier 
     The measured MDRFs of the CsI(TI) crystal without optical barriers are shown in  FIG. 41 .  FIG. 41  illustrates measured MDRF of prototype detector  4001  without optical barriers, created by scanning a thin beam of 662-keV gamma-ray photons (0.44 mm×0.44 mm, FWHM), in a regular array of positions across the detector 
     A slit projection of 662-keV photons of width 0.44 mm (FWHM of a Gaussian profile, as measured with the iQID camera) was used to evaluate the positioning performance. The resulting projection image and cross-sectional profile are shown in  FIGS. 42 and 43 , respectively.  FIG. 42  is a measured projection image by a slit of width 0.44 mm.  FIG. 43  is a profile of the projection image of  FIG. 42  with indicated FWHM, on the prototype detector (27.4 mm×27.4 mm×3 mm CsI(TI), with sixteen SiPMs) without optical barriers. 
     Experiment Case 2: With Optical Barrier 
     With optical barriers, created by mechanically-drilled holes (with Teflon lining the holes), the resolution can be greatly improved.  FIG. 44  illustrates measured MDRF of prototype detector  4002 , when the scintillation crystal includes drilled-hole optical barriers. The MDRF was created by scanning a thin beam of 662-keV gamma photons across the central area of the detector. 
     Again, the performance was evaluated by a slit beam of 662-keV photons of width 0.44 mm (FWHM). The resulting projection image and cross-sectional profile is shown in  FIGS. 45 and 46 , respectively. A position-corrected (method described below) energy spectrum of a flood image by  137 Cs is shown in  FIG. 47 . 
     The Monte-Carlo simulation indicates that the PET-block-detector-sized design (50.8 mm×50.8 mm=3 mm LYSO crystal) can provide reasonable spatial resolution even without optical barriers (the average FWHM is 1.49 mm), and excellent spatial resolution with drilled-hole optical barriers (the average FWHM is 0.56 mm) or laser-etched pattern optical barriers (the average FWHM is 0.62 mm). It is worth mentioning that the introduction of mechanically drilled-hole optical barriers will decrease the sensitivity a little bit, which is less than 2% in this simulation, while the majority of the detection efficiency loss is due to the gaps between different detector modules, since the SiPMs and associated readout circuitry will occupy the gaps. For example, for a crystal area of 50.8 mm×50.8 mm with 3 mm gap between the detector modules, the sensitivity reduction is about 11%. So, the key to minimize the sensitivity loss is to minimize the gaps between detector modules. 
     A further Monte-Carlo simulation suggests that the optical-gel/SiPM-window materials&#39; refractive indices have a great impact on the detector&#39;s spatial resolution. Whether a photon has a chance to be reflected at the detector edge interface by undesired TIR is determined by the minimum refractive index on its path from the scintillation crystal to the silicon substrate of the SiPM. As TABLE I shows, if the minimum refractive index on its path is 1.5, the overall spatial resolution will degrade by a few hundred um. However, the central area spatial resolution may actually be improved due to the relatively steeper gradient features in the MDRFs in the center area, created by TIR at the edges. 
     To understand this effect, the detector&#39;s (without optical barriers) simulated central-region MDRFs for minimum refractive indices of 1.5 and 1.82 were compared, as shown in  FIGS. 48-50 .  FIG. 48  illustrates an MDRF  4810  for refractive index of edge optical-gel/SiPM-window set to 1.5 and a corresponding point-grid image  4820 .  FIG. 49  illustrates an MDRF  4910  for refractive index of edge optical-gel/SiPM-window set to 1.82 and a corresponding point-grid image  4920 .  FIG. 50  illustrates an MDRF  5010  for refractive index of edge optical-gel/SiPM-window set to 1.5, but with optical barriers in the scintillation crystal, and a corresponding point-grid image  5020 . 
     If the optical-gel/SiPM-window material had the lower index of 1.5 and a line connecting the gamma-ray interaction position and the center of the SiPM exceeded the TIR critical angle, then a steep drop in the MDRF value was observed. The spatial resolution will then be worse at proximities close to each SiPM and corner regions of the crystal. If the optical-gel/SiPM-window material had the larger refractive index of 1.82, there was no TIR, and all of the MDRFs&#39; position dependence was due to the solid angle subtended by the area of the SiPM relative to origin of scintillation photons at the gamma-ray interaction location. If optical barriers are added, which are close to crystal boundaries, then the spatial resolution close to the crystal boundaries can be improved. Overall, high reflectance at the crystal-to-SiPM coupling layer can decrease the spatial resolution near crystal edges, but optical barriers can help to offset this effect, as shown in  FIGS. 48-50 . 
     The experimental results demonstrated the potential of this detector design. The spatial resolution for the crystal without optical barriers was calculated based on the slit projection image. By summing along the vertical direction of  FIG. 42 , the resulting profile plot is shown in  FIG. 43 . The FWHM was measured to be 0.80 mm. Assuming the line-spread function of the slit approximates a Gaussian profile, the spatial resolution (FWHM) is estimated to be: δ=(0.80 2 −0.44 2 ) 0.5 =0.67 mm. For comparison, the simulated slit image of the prototype detector without optical barriers ( FIGS. 35-36 ) yielded a spatial resolution of: δ simulation =0.65 mm. The experimental result was quite close to that predicted by simulation. 
     The procedure was repeated for the detector with optical barriers. The spatial resolution of the prototype detector with optical barriers was measured to be: δ experiment ≈0.34 mm, while the simulated resolution was predicted to be: δ simulation ≈0.31 mm. The measured spatial resolution is a little worse than that predicted by simulation, which is expected, since the simulation does not include all electronic noise sources. 
     The energy resolution of the prototype detector with optical barriers was measured by exposing the detector to the  137 Cs source without collimation. First a flood image was acquired, and each event&#39;s position in the flood image was estimated using the measured MDRF. Each event&#39;s summed-SiPM-signal value was also binned according to its estimated position. Then, the average energy at each position was calculated and stored in a 61×61 matrix (0.25 mm step size). This matrix was used for position-dependent energy correction. A second flood image was acquired that generated the energy spectrum including position-dependent correction ( FIG. 47 ). The FWHM energy resolution (ΔE/E) was estimated to be 6.4% at 662 keV, which is very good for a CsI(TI) detector (see Kazuch et al., Non-Proportionality and Energy Resolution of CsI(TI).  IEEE Transactions on Nuclear Science.  2007; 54 (5): 1836-1841; and Becker et al., Small Prototype Gamma Spectrometer Using CsI(TI) Scintillator Coupled to a Solid-State Photomultiplier.  IEEE Transactions on Nuclear Science,  2013; 60 (2): 968-972)). 
     Discussion 
     The edge readout detector design described in the above examples can easily reach &lt;1 mm resolution with the help of optical barriers, while simultaneously achieving good gamma-ray detection efficiency via multiple layers, thus avoiding the usual tradeoff between spatial resolution and gamma-ray detection sensitivity. The energy resolution of this design is also excellent due to the large fraction of photons arriving at crystal edges. Also, due to the monolithic design, there are no pixel gaps with reflective material, which further increases the gamma-ray detection sensitivity. The direct DOI estimation gives this design even more advantages over traditional PET detectors, and multi-position energy-deposition events across different layers caused by Compton scatter can, in principle, be identified. The first interaction position can often be estimated, from Compton kinematics which can help improve the spatial resolution. The effort of scintillator cutting and surface treatment is reduced due to the non-pixelated design. The detector design is also capable of working as a high-sensitivity Compton camera for high-energy gamma-ray photons when outfitted with fast timing electronics. Additionally, multiple modules can be tiled to produce cameras for SPECT. 
     In addition to the above, additional features or operating conditions may be utilized to improve detector performance. For instance, a large number of readout channels may be present when multiple layers are combined.  FIG. 11  shows a 6-layer design in which there may be a total of 120 readout channels for a complete detector module. Accordingly, if necessary, the number of readouts can be reduced or compact FPGA-based readout circuits may be utilized to handle large numbers of light sensors. When optical barriers are used, the MDRFs contain many sharp features that can create local likelihood-maxima. Thus, algorithms can be further utilized to carry out the positon estimation of each event quickly and accurately. In addition, since multiple SiPMs&#39; signals are added to produce a pulse for time stamp, the noise in different channels will also be added together, thus potentially affecting the coincidence timing resolution. The noise from dark count and circuit can be suppressed or reduced by cooling if necessary. The gaps between detector modules should also be minimized to preserve the detection efficiency. This can be achieved by making SiPMs and the pre-amplifiers thinner. 
     The edge-readout detector design described in the above experiments exhibits many appealing features. These include excellent spatial resolution, good energy resolution, and the ability to recover DOI information. The reduced fabrication effort is further appealing for clinical applications, including but not limited to human brain PET. 
     Combinations of Features 
     Features described above as well as those claimed below may be combined in various ways without departing from the scope hereof. The following examples illustrate some possible, non-limiting combinations: 
     (A1) A scintillation detector comprising a scintillator substrate and a plurality of photodetectors. The scintillator substrate comprises an interior bound by a front surface having a front perimeter, a back surface opposite the front surface and having a back perimeter, and an edge surface between the front and back perimeters. Each of the plurality of photodetectors has a respective photosensitive region facing or otherwise in optical communication with one of a respective plurality of regions of the edge surface. 
     (A2) In the scintillation detector denoted by (A1), the front surface may be a planar surface, and the back surface being a planar surface substantially parallel to the front surface. 
     (A3) In any scintillation detector denoted by one of (A1) and (A2), the scintillator substrate may have a uniform thickness between 0.5 millimeters and twenty millimeters, optionally between one millimeter and four millimeters. 
     (A4) In any scintillation detector denoted by one of (A1) through (A3), the front perimeter may have a front shape that is geometrically similar to a back shape of the back perimeter, the front shape being a polygon. 
     (A5) In any scintillation detector denoted by one of (A1) through (A4), the scintillation detector may have one of a rectangular, hexagonal, and octagonal shape. 
     (A6) In any scintillation detector denoted by one of (A1) through (A5), the substrate may include a scintillator material and may have a refractive index exceeding 1.1 at an emission maximum of the scintillator material. 
     (A7) In any scintillation detector denoted by one of (A1) through (A6), each of the plurality of photodetectors may be attached to the scintillator substrate 
     (A8) In any scintillation detector denoted by one of (A1) through (A7), each of the plurality of photodetectors may include a silicon photomultiplier, photomultiplier tube, avalanche photodiode, complementary metal-oxide semiconductor, or a combination thereof. 
     (A9) In any scintillation detector denoted by one of (A1) through (A8), each of the plurality of photodetectors may include a silicon photomultiplier. 
     (A10) Any scintillation detector denoted by one of (A1) through (A9), may further include a memory and a microprocessor. The memory stores non-transitory computer-readable instructions. The microprocessor is adapted to execute the instructions to estimate lateral coordinates, in a plane of the substrate, of a scintillation-event interaction-position based on a plurality of electrical signal amplitudes and respective plurality of locations of the plurality of photodetectors. 
     (A11) In any scintillation detector denoted by one of (A1) through (A10), the substrate may include a plurality of optical scatterers. 
     (A12) In any scintillation detector denoted by (A11), in which the substrate comprising a scintillator material, each of the plurality of optical scatterers may have an extent exceeding a wavelength corresponding to an emission maximum of the scintillator material, or may have a size or extent close to the wavelength corresponding to maximum emission. 
     (A13) In any scintillation detector denoted by one of (A11) and (A12), the plurality of optical scatterers comprise a through hole, a blind hole, an object on the front surface, an object on the back surface, a volume with modified optical properties generated by laser etching or laser engraving, or combinations thereof. 
     (B1) A stacked scintillation detector comprising a plurality of scintillation detectors, denoted by (A1), arranged as a stack. The plurality of scintillation detectors comprises a first scintillation detector and a second scintillation detector, the back substrate-surface of the first scintillation detector being adjacent to, opposing, and spatially separated from the front substrate-surface of the second scintillation detector. The stack may comprise two or more, three or more, four or more, five or more, six or more, seven or more, eight or more, nine or more, ten or more, fifteen or more, or twenty or more scintillation detectors. 
     (B2) In the stacked scintillation detector denoted by (B1), the second scintillation detector may be positioned in relation to the first scintillation detector such that a center region of the second scintillation detector is aligned with a center region of the first scintillation detector relative to an axis perpendicular to the front substrate-surface of the first scintillation detector. 
     (B3) In the stacked scintillation detector denoted by (B1), the second scintillation detector may be positioned in relation to the first scintillation detector such that a center region of the second scintillation detector is laterally offset from center region of the first scintillation detector relative to an axis perpendicular to the front substrate-surface of the first scintillation detector. 
     (B4) In any stacked scintillation detector denoted by one of (B1) and (B3), the second scintillation detector may be positioned in relation to the first scintillation detector such that a center region of the second scintillation detector is aligned with an edge surface of the first scintillation detector relative to an axis perpendicular to the front substrate-surface of the first scintillation detector. 
     (B5) Any stacked scintillation detector denoted by one of (B1) through (B4) may further include a microprocessor configured to receive electrical signals generated by one or more scintillation events from the plurality of photodetectors of at least one of the plurality of scintillation detectors, and configured to estimate lateral coordinates of a gamma-ray interaction position in a plane of the substrate of the at least one of the plurality of scintillation detectors, based on a plurality of electrical signal amplitudes and respective plurality of locations of the plurality of photodetectors. 
     (B6) In any stacked scintillation detector denoted by (B5), the microprocessor may be further configured to estimate depth coordinates of a scintillation-event interaction-position based on the plurality of electrical signal and respective plurality of locations of the plurality of photodetectors within the stack. 
     (C1) A PET detector ring comprising a plurality of scintillation detectors denoted by (A1). At least a first sub-plurality of the plurality of scintillation detectors form a first array having (i) a first concave inward-facing surface and including each front surface of the first sub-plurality and (ii) and a first convex outward-facing surface and including each back surface of the first sub-plurality. 
     (C2) In the PET detector ring denoted by (C1), the plurality of scintillation detectors may comprise a second sub-plurality of scintillation detectors forming a second array having (i) a second concave inward-facing surface, opposite the first convex outward-facing surface, and including each front surface of the second sub-plurality of scintillation detectors and (ii) a second convex outward-facing surface that includes each back surface of the second sub-plurality of scintillation detectors. The PET detector ring may comprise two or more, three or more, four or more, five or more, six or more, seven or more, eight or more, nine or more, ten or more, fifteen or more, twenty or more, fifty or more, or a hundred or more scintillation detectors. 
     (C3) In the PET detector ring denoted by (C2), the second array may be spatially shifted from the first array such that gamma-ray photons transmitted between photodetectors of adjacent scintillation detectors of the first array are incident on a substrate of a scintillation detector of the second array. 
     (D1) denotes a method for determining interaction position of a scintillation event, occurring in a substrate resulting from electromagnetic radiation incident on a front surface of the substrate. The method comprises a step of detecting, with a plurality of photodetectors, scintillation light from the gamma-ray interaction event resulting in a respective plurality of electrical signal amplitudes, the plurality of photodetectors being disposed at a respective region around a perimeter of the substrate and having a photosensitive region in optical communication with an edge surface of the substrate located between the front surface and a back surface of the substrate opposite the front surface. The method also includes a step of estimating lateral coordinates, in a plane of the substrate, of the interaction position based on the plurality of electrical signal amplitudes and respective regions. 
     (D2) In the method denoted by (D1), the substrate may be one of a plurality of stacked scintillator substrates. Each scintillator substrate has (a) a respective edge surface and (b) a respective plurality of photodetectors having a photosensitive region in optical communication with the respective edge surface, the gamma-ray interaction event occurring in one of the plurality of scintillator substrates, the method further comprising determining, from zero and non-zero electrical signal amplitudes from the photodetectors, (i) the one of the plurality of substrates and (ii) an associated depth-of-interaction of the gamma-ray interaction event in a direction perpendicular to the plane of the substrate. 
     (D3) In any method denoted by one of (D1) and (D2), the step of estimating may include comparing the plurality of electrical signal amplitudes to a mean-detector response function of the plurality of photodetectors at the respective regions around the substrate perimeter. 
     (D4) In the method denoted by (D3), the step of comparing may include performing at least one of a maximum likelihood estimate, a least-squares estimate, an iterative estimate, a LN norm comparison (where N is 0, 1, 2 . . . ), and an Anger method. 
     Having now fully described the present invention in some detail by way of illustration and examples for purposes of clarity of understanding, it will be obvious to one of ordinary skill in the art that the same may be performed by modifying or changing the invention within a wide and equivalent range of conditions, formulations and other parameters without affecting the scope of the invention or any specific embodiment thereof, and that such modifications or changes are intended to be encompassed within the scope of the appended claims. 
     When a group of materials, compositions, components, or compounds is disclosed herein, it is understood that all individual members of those groups and all subgroups thereof are disclosed separately. Every formulation or combination of components described or exemplified herein may be used to practice the invention, unless otherwise stated. Whenever a range is given in the specification, for example, a temperature range, a time range, or a composition range, all intermediate ranges and subranges, as well as all individual values included in the ranges given are intended to be included in the disclosure. Additionally, the end points in a given range are to be included within the range. In the disclosure and the claims, “and/or” means additionally or alternatively. Moreover, any use of a term in the singular also encompasses plural forms. 
     As used herein, “comprising” is synonymous with “including,” “containing,” or “characterized by,” and is inclusive or open-ended and does not exclude additional, unrecited elements or method steps. As used herein, “consisting of” excludes any element, step, or ingredient not specified in the claim element. As used herein, “consisting essentially of” does not exclude materials or steps that do not materially affect the basic and novel characteristics of the claim. Any recitation herein of the term “comprising,” particularly in a description of components of a composition or in a description of elements of a device, is understood to encompass those compositions and methods consisting essentially of and consisting of the recited components or elements. 
     One of ordinary skill in the art will appreciate that starting materials, device elements, analytical methods, mixtures and combinations of components other than those specifically exemplified may be employed in the practice of the invention without resort to undue experimentation. All art-known functional equivalents, of any such materials and methods are intended to be included in this invention. The terms and expressions which have been employed are used as terms of description and not of limitation, and there is no intention that in the use of such terms and expressions of excluding any equivalents of the features shown and described or portions thereof, but it is recognized that various modifications are possible within the scope of the invention claimed. The invention illustratively described herein suitably may be practiced in the absence of any element or elements, limitation or limitations which is not specifically disclosed herein. Headings are used herein for convenience only. 
     All publications referred to herein are incorporated herein to the extent not inconsistent herewith. Some references provided herein are incorporated by reference to provide details of additional uses of the invention. All patents and publications mentioned in the specification are indicative of the levels of skill of those skilled in the art to which the invention pertains. References cited herein are incorporated by reference herein in their entirety to indicate the state of the art as of their filing date and it is intended that this information may be employed herein, if needed, to exclude specific embodiments that are in the prior art. 
     Changes may be made in the above methods and systems without departing from the scope hereof. It should thus be noted that the matter contained in the above description or shown in the accompanying drawings should be interpreted as illustrative and not in a limiting sense. The following claims are intended to cover all generic and specific features described herein, as well as all statements of the scope of the present method and system, which, as a matter of language, might be said to fall therebetween. 
     REFERENCES 
     
         
         [1] Peterson T E, Furenlid L R, SPECT detectors: the Anger Camera and beyond.  Physics in Medicine and Biology.  2011; 56 (17): R145-R182. 
         [2] Barrett H H, Hunter W C J, Miller B W, Moore S K, Chen Y, Furenlid L R. Maximum-Likelihood Methods for Processing Signals From Gamma-Ray Detectors.  IEEE Transactions on Nuclear Science.  2009; 56 (3): 725-735. 
         [3] Borghi G, Tabacchini V, Seifert S, Schaart D R. Experimental Validation of an Efficient Fan-Beam Calibration Procedure for k-Nearest Neighbor Position Estimation in Monolithic Scintillator Detectors.  IEEE Transactions on Nuclear Science.  2015; 62 (1): 57-67. 
         [4] Li X, Lockhart C, Lewellen T K, Miyaoka R S. Study of PET Detector Performance with Varying SiPM Parameters and Readout Schemes.  IEEE Transactions on Nuclear Science.  2011; 58 (3): 590-596. 
         [5] Lewellen T K. Recent developments in PET detector technology.  Physics in Medicine and Biology.  2008; 53: R287-R317. 
         [6] Peng H, Levin C S. Recent developments in PET instrumentation.  Current Pharmaceutical Biotechnology.  2010; 11 (6); 555-571. 
         [7] Groll A, Kim K, Bhatia H, et al. Hybrid Pixel-Waveform (HPWF) Enabled CdTe Detectors for Small Animal Gamma-Ray Imaging Applications.  IEEE Transactions on Nuclear Science.  2016; 
         [8] Pratx G, Levin C S. Accurately positioning events in a high-resolution PET system that uses 3D CZT detectors,  IEEE Nuclear Science Symposium Conference Record,  2007; 4:2660-2664, 
         [9] Mosset J B, Devroede O, Krieguer M, et al. Development of an optimized LSO/LuYAP phoswich detector head for the Lausanne ClearPET demonstrator.  IEEE Transactions on Nuclear Science.  2006; 53 (1): 25-29. 
         [10] Roncali E, Viswanath V, Cherry S R. Design Considerations for DOI-encoding PET Detectors Using Phosphor-Coated Crystals.  IEEE Transactions on Nuclear Science.  2014; 61 (1): 67-73. 
         [11] Nishikido F, Inadama N, Oda I, et al, Four-layer depth-of-interaction PET detector for high resolution PET using a multi-pixel 38550 avalanche photodiode,  Nuclear Instruments and Methods in Physics Research A.  2010; 621: 570-575. 
         [12] Kishimoto A, Kataoka J, Kato T, et al. Development of a Dual-Sided Readout DOI-PET Module Using Large-Area Monolithic MPPC-Arrays,  IEEE Transactions on Nuclear Science,  2013; 60 (1): 38-43. 
         [13] Vandenberghe S, Mikhaylova E, D&#39;Hoe E, Mollet P, Karp J S. Recent developments in time-of-flight PET.  EJNMMI Physics.  2016; 3 (1): 3. 
         [14] Borghi G, Peet B J, Tabacchini V, Schaart D R, A 32 mm×32 mm×22 mm monolithic LYSO:Ce detector with dual-sided digital photon counter readout for ultrahigh-performance TOF-PET and TOF-PET/MRI.  Physics in Medicine  &amp;  Biology.  2016; 61:4929-4949. 
         [15] Surti S. Update on Time-of-Flight PET Imaging,  Journal of Nuclear Medicine,  2015; 56 (1); 98-105. 
         [16] Levin C S, Maramraju S H, Khalighi M M, Deller T W, Delso G, Jansen F. Design Features and Mutual Compatibility Studies of the Time-of-Flight PET Capable GE SIGNA PET/MR System.  IEEE Transactions on Medical Imaging.  2016; 35 (8); 1907-1914. 
         [17] Minamimoto R, Levin C, Jamali M, et al. Improvements in PET Image Quality in Time of Flight (TOF) Simultaneous PET/MRI.  Molecular Imaging and Biology.  2016; 18: 776-781. 
         [18] Karp J S, Surti S, Daube-Witherspoon M E, Muehllehner G. The benefit of time-of-flight in PET imaging: Experimental and clinical results.  Journal of Nuclear Medicine,  2008; 49 (3): 462-470. 
         [19] Mao R, Zhang L, Zhu, R. Optical and Scintillation Properties of Inorganic Scintillators in High Energy Physics,  IEEE Transactions on Nuclear Science.  2008; 55 (4): 2425-2431. 
         [20] Konstantinou G, Chil R, Udias J M, Desco M, Vaquero J J. Simulation, development and testing of a PET detector prototype using monolithic scintillator crystals treated with the Sub-Surface Engraving Technique.  IEEE Nuclear Science Symposium and Medical imaging Conference  ( NSS/MIC ). 2015; 1-4. 
         [21] Sabet H, Bläckberg L, Uzun-Ozsahin D, El-Fakhri G. Novel laser-processed CsI:TI detector for SPECT.  Medical Physics.  2016; 43 (5): 2630-2638. 
         [22] H W C J, Miyaoka R S, MacDonald L, McDougald W, Lewellen T K, Light-Sharing Interface for dMiCE Detectors Using Sub-Surface Laser Engraving.  IEEE Transactions on Nuclear Science.  2015; 62 (1): 27-35. 
         [23] Moriya T, Fukumitsu K, Yamashita T, and Watanabe M. Fabrication of Finely Pitched LYSO Arrays Using Subsurface Laser Engraving Technique with Picosecond and Nanosecond Pulse Lasers.  IEEE Transactions on Nuclear Science.  2014; 61 (2): 1032-1038. 
         [24] Hesterman J Y, Caucci L, Kupinski M A, Barrett H H, and Furenlid L R. Maximum-Likelihood Estimation With a Contracting-Grid Search Algorithm.  IEEE Transactions on Nuclear Science,  2010; 57 (3): 1077-1084. 
         [25] Miller B W, Gregory S J, Fuller E S, Barrett H H, Barber H B, Furenlid L R. The iQID camera: An ionizing-radiation quantum imaging.  Nucl Instrum Methods Phys Res A.  2014; 11 (767): 146-152. 
         [26] Kazuch S A, Swiderski L, Czarnacki W, et al, Non-Proportionality and Energy Resolution of CsI(TI).  IEEE Transactions on Nuclear Science.  2007; 54 (5): 1836-1841. 
         [27] Becker E M, Farsoni A T, Alhawsawi A M, Alemayehu B. Small Prototype Gamma Spectrometer Using CsI(TI) Scintillator Coupled to a Solid-State Photomultiplier.  IEEE Transactions on Nuclear Science.  2013; 60 (2): 968-972. 
         [28] Takahashi, Tadayuki, and Shin Watanabe. “Recent progress in CdTe and CdZnTe detectors.” Nuclear Science,  IEEE Transactions on  48.4 (2001): 950-959 
         [29] Chen, H., et al. “CZT device with improved sensitivity for medical imaging and homeland security applications.”  SPIE Optical Engineering+Applications.  International Society for Optics and Photonics, 2009 
         [30] Meng, Ling J., and Zhong He. “Exploring the limiting timing resolution for large volume CZT detectors with waveform analysis.”  Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment  550.1 (2005): 435-445.