The design of the D-SPECT system is based on compact CZT solid state detectors units that permit movements that would not be achievable with conventional gamma cameras. This new design permits to overcome some inherent SPECT limitations allowing for a region-centric acquisition. By choosing to spend more time directing the detector heads towards a region of interest (ROI), one can allocate more time to collect data from this region at the expense of collecting fewer data from less important regions. However, changing the detector angular movements allowing for a non uniform scanning pattern, the interdependence in the information changes and the D-SPECT system response may be highly shift-variant. In order to be able to compare a set of candidate scanning patterns (for a given activity distribution) a method is needed, that quantifies the information gain. The Fisher Information Matrix (FIM) formalism can be employed to characterise the uncertainty of the reconstruction. Unfortunately, computing, storing and inverting the FIM is not feasible with a 3D imaging system. To tackle this problem we introduce a new approximated expression that relies on a subsampled version of the FIM. This formulation reduces the the computational complexity in inverting the FIM but nevertheless accounts for the global interdependence between the variables. In this paper we adopt this novel algorithm, to investigate the effect of the presence of activity outside the ROI, in determining the optimal D-SPECT acquisition protocol. We perform a set of experiments to determine the optimal scanning pattern for different levels of background. If the activity in the ROI is significantly high with respect to the activity in the background, an acquisition protocol that spends more time acquiring data from the ROI is preferable. Whereas for high levels of activity in the background, an acquisition that scans uniformly the whole field of view may give better results in terms of reconstructed image quality in the ROI.