The clinical management of breast cancer depends on an accurate understanding of the tumor and its anatomical context to adjacent tissues and landmark structures. This context may be provided by semantic segmentation methods; however, previous works have been largely limited to a singular focus on the tumor alone and rarely other tissue types. In contrast, we present a method that exploits tissue-tissue interactions to accurately segment every major tissue type in the breast including: chest wall, skin, adipose tissue, fibroglandular tissue, vasculature and tumor via standard-of-care Dynamic Contrast Enhanced MRI. Comparing our method to prior state-of-the-art, we achieved a superior Dice score on tumor segmentation while maintaining competitive performance on other studied tissues across multiple institutions. Briefly, our method proceeds by localizing the tumor using 2D object detectors, then segmenting the tumor and surrounding tissues independently using two 3D U-nets, and finally integrating these results while mitigating false positives by checking for anatomically plausible tissue-tissue contacts. The object detection models were pre-trained on ImageNet and COCO, and operated on MIP (maximum intensity projection) images in the axial and sagittal planes, establishing a 3D tumor bounding box. By integrating multiple relevant peri-tumoral tissues, our work enables clinical applications in breast cancer staging, prognosis and surgical planning.