to snoo-finity ... and beyond!
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This was the picture that I liked the most for this question. If you imagine that the injury is at the level of the left subclavian vein (LSV) and thoracic duct (TD) crossing each other, maybe it will help you realize that the left breast & left upper extremity bypass the injury. The right lung is drained by the right lymphatic duct, not the TD (see one of the links bellow). And the heart is where eventually all the drainage is going to end up to be pumped again (TD -> LSV -> SVC -> heart).