A part per trillion is about the same as one teaspoon in a lake fifteen feet deep and a mile square. This is far below the threshold that EPA believes puts consumers at risk of cancer. There isn't an easy answer on how to avoid contact with trace amounts of dioxins. Dioxin are a byproduct of the bleaching process most paper products are subject to, but it also gets into the environment through the burning of chemicals. Some tampons and sanitary pads contain minute amounts of this chemical, but so do milk cartons.
Dioxins also turn up in fish, meat, and dairy products.
Those concerned about exposure to dioxins through their feminine products should make the move to unbleached ones. They should keep in mind though that dioxins are all around them, and changing their brand of tampon or napkin won't eliminate all exposures to dioxins. Would that it were that simple. To round this up for those in the bleachers, there is no asbestos in tampons. Thinking consumers might want to reduce their contact with dioxins anyway, trading comfort and absorbency for a bit more peace of mind.