Non-diverging analytical expression for the sensitivity of converging SPECT collimators

Abstract

Accurate analytical expressions for collimator resolution and sensitivity are important tools in the optimization of SPECT systems. However, presently known expressions for the sensitivity of converging collimators either diverge near the focal point or focal line(s), or are only valid on the collimator axis. As a result, these expressions are unsuitable to calculate volumetric sensitivity for e.g. short-focal length collimators that focus inside the object to enhance sensitivity. To also enable collimator optimization for these geometries, we here present non-diverging sensitivity formulas for astigmatic, cone beam and fan beam collimators that are applicable over the full collimator's field-of-view. The sensitivity was calculated by integrating previously derived collimator response functions over the full detector surface. Contrary to common approximations, the varying solid angle subtended by different detector pixels was fully taken into account which results in a closed-form non-diverging formula for the sensitivity. We validated these expressions using ray-tracing simulations of a fan beam and an astigmatic cone beam collimator and found close agreement between the simulations and the sensitivity expression. The largest differences with the simulation were found close to the collimator, where sensitivity depends on the exact placement of holes and septa, while our expression represents an average over all possible placements as is common practice for analytical sensitivity expressions. We checked that average differences between the analytical expression and simulations reduced to less than 1% of the maximum sensitivity when we averaged our simulations over different septa locations. Moreover, we found that our new expression reduced to the traditional diverging formula under certain assumptions. Therefore, the newly derived sensitivity expression may enable the optimization of converging collimators for a wide range of applications, in particular when the focus is close to, or in, the object of interest.