The Nashville-based Frist family, which helped start the for-profit hospital chain HCA Inc., are launching a company to take the U.S. system of managing and operating hospitals to China. China Healthcare Corp. said it has an initial agreement with a city southeast of Shanghai to build and operate a hospital that would replace an existing location. The startup would own a 70% stake in the joint venture, while the government of Ningbo would own the remaining 30%. The agreement requires approval from the Chinese ministries of health and commerce, a process that takes up to nine months on average.