Preheat oven to 375°F (190°C).
Remove and dice mandarin orange slices into quarters; set aside.
Mix together wheat bran and liquid from mandarin oranges; let stand. Combine oil, egg, brown sugar and vanilla and add to bran mixture.
Sift together flour, baking soda, baking powder, salt and nutmeg. Add to first mixture and stir just until blended.
Fold in diced oranges and spoon into prepared muffin cups. Bake 20 – 25 minutes.