1. Depth Encoding of Point-of-interaction in Thick Scintillation Cameras

Takehiro Tomitani, Yasuyuki Futami, Yasushi Iseki¹, Shigeru Kouda², Teiji Nishio³, Takeshi Murakami, Atsushi Kitagawa, Mitsutaka Kanazawa, Eriko Urakabe, Munefumi Shinbo and Tatsuaki Kanai
(¹Toshiba Co., ²Instit. Molecular Science, ³Nat. Cancer Center)

Keywords: gamma camera, position arithmetic, maximum likelihood estimate

The secondary beam generator/separator was built last year at HIMAC, from which positron emitting nuclei beams such as ¹⁰C, ¹¹C, ¹⁸Ne and ¹⁹Ne are available. We intend to develop a positron camera to image end point distribution of positron emitting beams. Scintillation cameras have been adopted in such fields as PET, positron cameras and gamma ray astronomy. In these applications, thick NaI(Tl) crystals are adopted to image high energy gamma rays and collimators are not used, which causes a two-dimensional position error that depends on depth-of-interaction. In principle, depth-of-interaction can be obtained from the depth dependency of light distribution. The present study theoretically analyzed the possibility of the detection for depth encoding of point-of-interaction.

Three-dimensional position arithmetic can be derived from maximum likelihood estimation. Here we assume that the number of photoelectrons generated at the photocathode of a photomultiplier (PMT) follows Poisson distribution. Photons emitted at position (x, y, z) in the crystal are shared among PMTs. Let n_k denote the number of photoelectrons generated at the k-th PMT and _k(x, y, z) denote its ensemble average. Then the likelihood, L, that m PMTs detect {n₁, n₂, ....., nm} photoelectrons is

where p_k, Poisson distribution, is by definition

A sufficient condition which maximum likelihood estimate (MLE), must satisfy is

MLE, , is determined from this set of equations. Note that each equation is implicitly dependent on x, y and z through (x, y, z) and transcendental, and hence can only be solved numerically by Newton-Raphson's approximation.

Depth resolution depends on the light distribution, that strongly depends on the mechanism of reflection on the reflector. In the case of perfectly diffusive reflection, the light distribution has a broad maximum near the reflector and the estimate of depth is two-valued, so that the depth cannot be determined. If the reflector is a mirror whose reflection coefficient ranges from 0 to 1, then the depth dependency of the light distribution is monotonic and depth information is available. 3D spatial resolutions are shown in Fig. 1 as function of depth, in which reflection coefficients are 1 (left side graphs), and 0 (right side graphs), respectively. In view of 2D position arithmetic, perfectly diffusive reflection is superior to mirror reflection, yet depth information cannot be derived. Depth information can be obtained in the case of mirror reflection with reflection coefficients ranging from 0 to 1, that is, opaque to mirror reflection at a cost of 2-D spatial resolution. The choice seems dependent on the pposes of a particular application. For parallax correction, moderate depth information may suffice, while in other applications such as gamma ray astronomy, depth information may be more important to determine the gamma ray direction.

Fig.1. Spatial resolution as a function of light source depth. The ordinate is spatial resolution in cm FWHM and the abscissa is depth in cm. Curves a, b, c and d refer to the positions just under the center of a PMTs in the x-direction, midway between two adjacent PMTs in the y-direction and midway among three OMTs, respectively.