  Purpose: The scattering phenomenon creates degrading effects in positron
emission tomography (PET) and the corresponding events are rejected
conventionally. We have proposed a mathematical model to retrace the original
line of response of the single-scattered coincident events with the aim to
incorporate such events in PET.
  Methods: We have devised a new Virtual extrapolation technique based on the
concept of the probability density functions. Through which we transformed the
original two-parameter list mode data for the coincident photon pairs to a
one-parameter data set. The procedure of random sampling and sampling
distribution - by utilizing some unique properties like a collective difference
and the length compensation - was employed in the data analysis. We studied the
effect of finite timing and energy resolution of detectors on the performance
of the proposed model.
  Results: We determined the frequency of occurrence of the data values
corresponding to real as well as fictitious scattering points { placed on the
arc of a circle drawn with the constant scattering angle - to observe the
highest counts. As expected, we found the highest counts at the location of
real scattering points. The model was evaluated for a uniform attenuating
phantom medium. The results were found impressive for the case of ideal time
and energy information, less so for the cases of finite resolutions.
  Conclusions: Our work outlines a completely new approach; though, a lot more
sophistication is required for its practical utility. Nonetheless, the
technique seems promising in developing a potential approach to improve the
sensitivity of PET imaging.
