From the article:

Earlier work by UI redox biology expert Garry Buettner found that at these extremely high levels (in the millimolar range), vitamin C selectively kills cancer cells but not normal cells in the test tube and in mice.

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“In this paper we demonstrate that cancer cells are much less efficient in removing hydrogen peroxide than normal cells. Thus, cancer cells are much more prone to damage and death from a high amount of hydrogen peroxide,” says Buettner […] “This explains how the very, very high levels of vitamin C used in our clinical trials do not affect normal tissue, but can be damaging to tumor tissue.”

Some cancers may be more vulnerable than others:

Buettner says this fundamental information might help determine which cancers and which therapies could be improved by inclusion of high-dose ascorbate in the treatment. “Our results suggest that cancers with low levels of catalase are likely to be the most responsive to high-dose vitamin C therapy, whereas cancers with relatively high levels of catalase may be the least responsive,” he explains.