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这句话is often the case with children是定语从句，这句话的主语应该是Joanna was better when the doctor arrived. 所以这是一句非限制性定语从句。按照一般的语序，应该是Joanna was better when the doctor arrived，which is often the case with children. which指代前面整句话。但是因为从句被提前了，所以which需要改变为as。

as is often the case...这是常有的事。

这句话的意思是，小孩子通常就这样,医生一到Joanna就好多了。