There is nothing wrong with having an operation to construct an artificial breast, because that comes under the heading of removing a defect and restoring the shape to the way it was created by Allah. 

The basic principle concerning that is the report narrated by Abu Dawood (4232), al-Tirmidhi (1770) and al-Nasaa’i (5161), from ‘Abd al-Rahmaan ibn Tarafah that his grandfather ‘Arfajah ibn As’ad had his nose cut off in the battle of al-Kulaab, and he wore a nose made of silver, but it began to stink, so the Prophet (blessings and peace of Allah be upon him) told him to get a nose made of gold. This hadeeth was classed as hasan by al-Albaani in Saheeh Abi Dawood. 

It says in Fataawa al-Lajnah al-Daa’imah (25/62): I am a young man, eighteen years old. Four years ago my breast became prominent and that was accompanied by some pain. After a while, the pain went away, praise be to Allaah, but the prominence remained. This prominence of my breast is obvious even under clothing. I asked the specialist about that and he said that this can easily be removed by means of cosmetic surgery. Is it permissible to have this surgery? Please note that this problem causes me embarrassment in front of other people.  

They replied: It is permissible for you to have cosmetic surgery to remove this prominence if it is thought most likely that the surgery will be successful and that the harms will not outweigh or be equal to the benefits.  

And Allaah is the Source of strength. May Allaah send blessings and peace upon our Prophet Muhammad and his family and companions. End quote.  

‘Abd al-‘Azeez ibn ‘Abd-Allaah ibn Baaz, ‘Abd al-Razzaaq ‘Afefi, ‘Abd-Allaah ibn Ghadyaan, ‘Abd-Allaah ibn Qa’ood  

End quote. 

They were also asked (25/59): one of my colleagues had gotten married, praise be to Allaah, but he came to me and said that his wife wants to have cosmetic surgery on her face and breasts, because her nose is large and wide, and she wants to reduce it by means of the easy methods that modern science has developed. Is there any doubt about this surgery or is there any sin involved? Please note that not doing it may lead to psychological distress because this fault appears so prominently on her face.  

They replied: If the situation is as described, and there is the hope that the surgery will be successful and will not lead to a greater or equal harm, then it is permissible to do it in order to achieve the desired purpose, otherwise it is not permissible.  

And Allaah is the Source of strength. May Allaah send blessings and peace upon our Prophet Muhammad and his family and companions. End quote.   

‘Abd al-‘Azeez ibn ‘Abd-Allaah ibn Baaz, ‘Abd al-Razzaaq ‘Afefi, ‘Abd-Allaah ibn Ghadyaan, ‘Abd-Allaah ibn Qa’ood  

End quote. 

For more information, please see the answer to question number 47694. 

There is nothing wrong with taking some skin from the back to complete the operation. 

And Allah knows best.