Unfortunately, the German authorities do not view it the same way unless you are severely bedridden due to illness. If your country also has hospitals and medicine for this treatment, the court will not permit you to stay here. I have personally witnessed several cases like this myself.
Anyway, You can appeal to the higher management court like Verwaltungsgericht, in case of rejection later on Oberverwaltungsgericht.
Also in parallel you can apply to the "Härtefallkommission' of your state. They are the people who makes it possible, in exceptional cases, to issue a residence permit to foreigners who are actually obliged to leave the country.