Run (るん) is Shiro's original owner, she is a young girl that looks similar to Shinnosuke.
Shiro's parent is Borscht, a female dog which gave birth to 4 puppies, Shiro was one of them. Run's father started to suffer from "Cataplexy to allergy to dogs" (a fake disease) and was hospitalized, after this they couldn't keep the dogs anymore. 3 puppies were adopted safely, but Shiro and Borscht were going to be taken to the public health center because nobody were found to adopt them.
Run was against handing over Shiro and Borscht, so she secretly put Shiro in a cardboard box so that someone might pick him up in the street (even though her older brother and mother already knew this). She is seen hiding in a corner when Shinnosuke's friends found Shiro.
After these events, Run's father gets better but Borscht was taken to the public health center.
Some time later, Run runs into her puppy, who's now grown up - she hears Shin call him "Shiro" and remarks "you look happy, Shiro." Her dad shows up and he buys her cake.