Analysis of the effects of errors in attenuation maps on PET quantitation in TOF PET

Attenuation correction is critical to positron emission tomography (PET) quantitation. Errors in an attenuation map propagate into PET image reconstruction, subsequently affecting PET quantitation. It is important to understand quantitatively how the errors in the attenuation map impact on PET reconstruction in order to avoid misinterpretation of PET images in the presence of such errors and to set a specification on the accuracy of attenuation maps. In this study we present a computationally efficient method to predict the perturbation in the reconstructed PET image due to a perturbation in the attenuation map for time-of-flight (TOF) PET, which is known to be more robust to modeling inaccuracy than non-TOF PET. Building on existing methods that focus on non-TOF PET, we derive an analytical expression to predict the propagated errors for TOF PET by using a stationary point analysis applied to penalized-likelihood image reconstruction. We calculate the analytical expression efficiently in Fourier space without additional image reconstruction or matrix inversion. In addition, we demonstrate the propagated errors in PET quantitation are proportional to TOF timing resolution. The analytical method enables one to quickly examine the effects of inaccuracies in an attenuation map on PET quantitation. Its applications include analysis of MR-based attenuation correction in PET/MR, where MR-based attenuation maps often have inaccuracies in bones, lungs and implants.