Baxter International Inc.’s in-house team capitalized on its story-telling talent to win approval of last year’s $3.9 billion acquisition involving complicated medical technology to treat kidney disease.

The legal team had to explain the complex details of peritoneal dialysis and hemodialysis businesses to antitrust officials in the United States, European Union and other jurisdictions around the globe, clearing the way for the company’s purchase of Swedish medical-device maker Gambro A.B.