Hello! The sentence is okay, but it would sound more natural if you said: He was taken to the hospital during rush hour WITH a serious case of blood cancer AND an irregular heartbeat.
or: He was taken to the hospital during rush hour. He HAD a serious case of blood cancer with AN irregular heartbeat.
or: He was taken to the hospital during rush hour BECAUSE he had a serious case of blood cancer with AN irregular heartbeat.
But a native speaker would probably say: "He was taken to the hospital during rush hour because of his irregular heartbeat. He has blood cancer, so it's very serious." He probably goes to the hospital a lot for cancer treatment. THIS visit is because of his irregular heartbeat, but it's very serious because he also has cancer.