Your mother is allowed to take the medication you have mentioned which contains chicken sternum cartilage; even though it may not come from chickens slaughtered by Muslims. Since we are allowed in the Qur’an to eat of the food of the People of the Book, there is no problem with consuming the products containing ingredients from chickens slaughtered in predominantly Christian countries.

It should be pointed out that the Companions of the Prophet, peace be upon him, who, following the Prophet’s death, dispersed in various lands predominantly inhabited by the People of the Book, partook freely of the foods prepared and provided by them. They never questioned the permissibility of consuming such foods. Therefore, we cannot consider foods provided by them as haraam unless of course they contain pork or alcohol or their by products, in which case they are definitely haraam and therefore unlawful for us to consume. Other than that, there was no disagreement among the companions of the Prophet, peace be upon him, on this issue.

Furthermore, especially in the case of your mother, because of her medical condition, the issue is still more flexible. Therefore, she can make use of the above medication without any inhibition whatsoever-- as long as there is no alternative to it.