In the world of kidney transplants, people often find themselves in a frustrated position of being willing, ready, and able to donate a kidney to a family member, yet cannot do so because the kidney does not match the recipient. To address the issue, multi-way kidney donations are being used to find more matches, and mathematicians within the field of number theory are working on algorithms that will best match the largest groups possible.
Finding the best matches is a complex task. Doctors tend to consider some patients higher priority than others. For example, although dialysis allows most patients to survive after their kidneys have failed, patients who cannot be treated with dialysis will die without a transplant. If a patient had donated a kidney when healthy, a sense of justice calls for that patient to receive one when sick. Doctors also favor giving higher priority to children.
Furthermore, some matches are better than others. Donor and recipient must have compatible blood types and sufficient immunological compatibility for a transplant to be possible. But in addition, there are degrees of immunological compatibility: the better the match, the longer a transplanted kidney is likely to keep working. Another consideration is that transplants are least expensive and easiest on doctors and families when the donors and patients live near one another, so that all the surgeries can take place in the same locality.