in short, depending on your specific application, i.e. the type of tissue or system you are trying to mimic, negative Poisson Ratio (NPR) helps to address problem of compliance mismatch.
some useful literature:
Article Tuning the Poisson's Ratio of Biomaterials for Investigating...
Article A Tubular Biomaterial Construct Exhibiting a Negative Poisson's Ratio
As you probably know, some tissues in the human, as well as animals like cats, have auxetic behavior. In this regard, I can mention the cancellous bone and cat skin.
Cancellous bone has a cellular microstructure that provides energy absorption in the bone, therefore by means of auxetic structure we want to mimic this property to gain faster and more durable healing for the patients.