Analytic derivation of pinhole collimation sensitivity for a general source model using spherical harmonics.

Pinhole collimators are widely used for single photon emission computed tomography (SPECT) imaging of small organs and animals. There has also been renewed interest in using pinhole arrays for clinical cardiac SPECT imaging to achieve high sensitivity and complete data sampling. Overall sensitivity of a pinhole array is critical in determining a system's performance. Conventionally, a point source model has been used to evaluate the sensitivity and optimize the system design. This model is simple but far from realistic. This work addresses the use of more realistic source models to assess the sensitivity performance of pinhole collimation. We have derived an analytical formula for pinhole collimation sensitivity with a general source distribution model using spherical harmonics. As special cases of this general model, we provided the pinhole sensitivity formulae for line, disk and sphere sources. These results show that the point source model is just the zeroth-order approximation of the other source models. The point source model overestimates or underestimates the sensitivity relative to the more realistic model. The sphere source model yields the same sensitivity as a point source located at the center of the sphere when attenuation is not taken into account. In the presence of attenuation, the average path length of emitted gamma rays is 3/4 of the radius of the sphere source. The calculated sensitivities based on these formulae show good agreement with separate Monte Carlo simulations in simple cases. The general and special sensitivity formulae derived here can be useful for the design and optimization of SPECT systems that utilize pinhole collimators.

The geometric response function for convergent slit-slat collimators.

We have derived an analytic geometric transfer function (GTF) for a convergent slit-slat collimator that treats the parallel slit-slat collimator as a special case. The effective point spread function (EPSF) is then derived from the GTF through the Fourier transform. The results of these derivations give an accurate description of the complete geometric response for a slit-slat collimator that includes the effects of the shape and orientation of the slit and slats. We have also derived exact and approximate sensitivity formulae and spatial resolution formulae using the EPSF.