Requires making something more general than before---the aff deepens, NOT expands, the program
Stack Exchange 15, obscureed, Bronze contributor, “Usage: deepen, broaden, expand, and extend,” 8/9/15, https://english.stackexchange.com/questions/265872/usage-deepen-broaden-expand-and-extend
As John Lawler has pointed out in comments, the different verbs have different meanings. So, "deepened" might be appropriate if the new problems are very similar to the previous problems, but somehow more difficult or intricate. "Broadened" is probably more common, when the new problems are related to the previous ones, and perhaps of similar difficulty, but with new aspects. "Extended" is similar to "broadened", but gives more of an air of progress into new territory. "Expanded" is also similar, but gives an air of making the technique more general.
If I had to pick one, I would probably choose "extended". ("Expanded" would be a close contender if the original technique was highly specialised and the new work helped to make it more general.) This is mainly personal preference, and also dependent on the nature of these new studies, so I have no references to cite. I have not attempted to answer the question of which is more common. You could try Google Ngram for that.