HCA, the hospital group taken private in one of the largest leveraged buy-outs ever in 2006, is planning on raising as much as $4.3 billion in an initial public offering next week. It would be the largest US private equity-backed IPO ever, topping the recently completed Nielsen and Kinder Morgan IPOs. The company and its owners plan to sell 124 million shares at a range of $27 to $30 a share, with an option to sell an additional 18 million. The offering is set to price next Tuesday. At the top of the targeted share price range, the company's equity would be valued at $15.4 billion. That would be a nearly $10 billion paper profit for the company's private equity owners and the Frist family, who founded the company and helped take it private for $5.5 billion in equity and $26.1 billion in debt. That is on top of $4.25 billion in dividends, paid for with debt raisings, the company paid to its owners last year.