短语讲解：

1. Much of the improvement came from lower death rates from lung cancer in men and breast cancer in women.

come from来自，源出；产自
例句：From this area come most of our best potatoes.
我们最好的马铃薯大部分是这个地区生产的。

2. Those increases mainly came in the less-common cancers, such as pancreas and liver, plus some kinds of skin and esophageal cancers and those associated with the human papillomavirus (HPV).

associate with与…交往, 联系
例句：The doctor was associated in the child's mind with injections and pains.
在孩子的心里, 医生总是使他联想到打针与疼痛。

3. It's not entirely clear why some cancer rates are on the rise, but he says there may be a link in some cases to increasing rates of obesity.

on the rise在上涨[增长]
例句：The prices are on the rise.
物价在上涨。