From an Islamic point of view, what exactly comes under backbiting? Would talking about someone for the purpose of treatment be considered backbiting ? Are there situations where talking about someone behind their back is permissible?
Re: Backbiting /Gheebah
Good question LP. I would like to know about that as well.
Re: Backbiting /Gheebah
As Muslims we are asked to guard our tongue and avoid the things that are forbidden: Gheebah, Buhtan and Nameemah.
Gheebah or backbiting means speaking about a Muslim in his absence and saying things that he would not like to have spread around or mentioned. Buhtan or slander means saying things about a Muslim that are not true, or in other words telling lies about him. Nameemah or malicious gossip means telling one person what another said in order to cause trouble between them.
Plenty of evidence in Quran and Sunnah that these actions are forbidden.
Allah says in the The Qur’an says:
*“neither backbite one another. Would one of you like to eat the flesh of his dead brother? You would hate it (so hate backbiting). And fear Allah. Verily, Allah is the One Who forgives and accepts repentance, Most Merciful” *[49:12]
It was narrated from Abu Hurayrah (may Allah be pleased with him) that the Messenger of Allah (peace and blessings of Allah be upon him) said: “Do you know what gheebah (backbiting) is?” They said, “Allah and His Messenger know best.” He said, “Saying something about your brother that he dislikes.” It** was said, “What if what I say about my brother is true?” He said, “If what you say is true then you have backbitten about him, and if it is not true, then you have slandered him.” * Sahih Muslim
But your situation is unique, you wish to help a fellow human with their treatment, so I am not sure if that is considered backbiting. It all depends on your intentions, niyyaah.
إِنَّمَا الأَعْمَالُ بِالنِّيَّةِ
You have heard this phrase right? *Innamal aamaalu binniyah.
On the authority of Umar bin Al-Khattab, who said : I heard the Messenger of Allah 
* :**** “Actions are but by intention and every man shall have but that which he intended". **Sahih Bukhaari and Sahih Muslim.
So bottom line is, if your intentions are good, its all good.
and Allah knows best.
PS: If i said something right, then it is from Allah, if I erred somewhere, then it is from me and Shaitaan.