Analytic derivation of pinhole collimator sensitivity for a general source distribution

Pinhole collimators are widely used for SPECT imaging of small organs and animals. There also has been renewed interest in using pinhole collimation for clinical cardiac SPECT imaging, which uses multiple pinhole arrays 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 overall 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 systems. We derived an analytical formula for the sensitivity of pinhole collimation with a general source distribution model using spherical harmonics. As special cases of this general model, we also provide the pinhole sensitivity formulas for line segment, disk and sphere sources. These results show that the point source model is just the zero-order approximation of the other source models. The point source overestimates or underestimates the sensitivity when the line segment or disk source lies in a plane that is parallel or perpendicular to the pinhole aperture plane, respectively. The sphere source yields the same sensitivity as a point source at the center of the sphere when attenuation is not considered. The calculated sensitivities based on these formulas show good agreement with separate Monte Carlo simulations in simple cases. The general and special sensitivity formulas derived here are useful for the design and optimization of SPECT systems that utilize pinhole collimators.