In this work, we propose a new criterion based on spatial context to select relevant nodes in a max-tree representation of an image, dedicated to the detection of 3D brain tumors for \textsuperscript{18}$F$-FDG PET images. This criterion prevents the detected lesions from merging with surrounding physiological radiotracer uptake. A complete detection method based on this criterion is proposed, and was evaluated on five patients with brain metastases and tuberculosis, and quantitatively assessed using the true positive rates and positive predictive values. The experimental results show that the method detects all the lesions in the PET.