Preheat oven to 400 degrees F (205 degrees C). Grease 36 cups of 3 large muffin tins.
In a medium saucepan, combine the rice with water. Bring to a boil over high heat, uncovered, and cook 10 minutes. Reduce heat to low, cover, and simmer 20 minutes.
Remove from heat, let cool several minutes, then fluff with a fork and set aside.
In a large bowl, combine the potatoes, carrots, celery, ground beef, and eggs.
Mix ingredients together using your hands or a sturdy spoon.
Add salt, olive oil, rolled oats, and rice; mix well.
Fill each muffin cup with some of the meat mixture, and pat down the to make it firm.
Bake 45 minutes, or until surface feels set. Cool on a rack 10 minutes or longer.
Remove the meat cakes by turning the muffin tin upside down over a sheet of aluminum foil. Tap each muffin cup to release the cake. Refrigerate or freeze in sealed plastic bags. NOTE: Do not feed to dogs while still hot.