Literal versus contextual translation of 斷腸
I think 斷腸 should be translated as "gut wrenching (pain/sadness)" but "heart broken" is also correct because 斷腸 and 碎心 basically have the same metaphorical meaning -- "extreme emotional pain"


斷腸 (gut wrenching): an intense sadness/ emotional pain like one's intestine is being twisted apart
心碎 (heart broken): an intense sadness/ emotional pain like one's heart is being broke to pieces 


心碎 is more common than 斷腸 when it come to describe emotional pain because "心" also represent "emotion/mind", therefore,"心碎" expresses "emotional damage" in a more literally way, while 斷腸 is completely metaphorical

Another difference between 斷腸 and 心碎 is 斷腸 can be used as an adjective directly, for example: 斷腸人(heart broken one); 斷腸夢 (gut wrenching dream); 斷腸痛楚 (gut wrenching pain)

But we only say "碎心人" (heart broken one), not 心碎人