Combine the pineapple juice, vinegar, ginger, soy sauce and brown sugar in a small saucepan and bring to a boil. Reduce the heat to low and simmer until the volume is reduced by half, about 30 minutes. Add the ketchup and cook an additional 5 minutes. Remove from the heat and add the lime juice, and white pepper.
To prepare the burgers:
Shape the ground tuna firmly into 4 round uniform patties about 1 1/2inches thick. Place in the refrigerator for 30 minutes. (The burgers must be very cold to hold their shape when cooking) Brush both sides of the burgers lightly with the pineapple glaze and season with salt and pepper to taste. Heat a large cast iron skillet over medium high heat until just smoking. Cook the burgers, basting often with the glaze until cooked to desired doneness