Abstract: Common segmentation losses aggregate errors voxel-wise, so lesions influence the objective in proportion to their volume, giving small but clinically critical lesions disproportionately little weight. Instance-aware losses aim to address this mismatch by assigning each lesion its own term. However, blob loss and CC-DiceCE derive their regions solely from annotations, so false-positive components receive no instance-level term. This matters in computer-assisted review, where each false-positive component may require separate inspection, making precision and false-positive burden important alongside recall. We introduce the bidirectional connected-component loss (BiCC), which pairs annotation- and prediction-derived partitions to score predicted components on their own scale. By deriving instances from the predictions, this branch directly penalizes false-positive components regardless of their size. The balance parameter $α$ allows control over the lesion-wise precision-recall trade-off. Across five datasets with five-fold cross-validation using nnU-Net, BiCC outperforms CC-DiceCE in lesion-wise F1 on four datasets and blob loss on all five. It significantly improves over DiceCE on three datasets and matches it on two; CC-DiceCE instead loses up to 0.363 precision by favoring recall. Code is available at https://github.com/TIO-IKIM/BiCC-Loss.