Revisiting Kidneys: More Transplants through Matching

There is no current legal market for kidneys in Singapore. As of the huge furore over the matter sparked by a tycoon's attempt to buy one in in mid-2008 (he has since gotten a kidney), the Singapore government remains in the process of studying the possible implementation of such a market. Is there a fair price for a kidney? While questions like that and others are being debated, there remains a constant shortage of donor kidneys for transplant and it may be more socially beneficial to also look into incremental ameliorative solutions that are morally repugnant to fewer people.

Patients requiring transplants can only accept kidneys from donors with certain blood types. Simplistically, acceptance and rejection depends on the blood types of patient and donor. Other than ABO compatibility issues, additional immunological factors may impinge on compatibility, but in a nutshell, a patient may have a relative or friend willing to donate a kidney to him or her, but is unable to accept that kidney. (Friends of patients are unable to donate kidneys in Singapore due to possible abuse of such a provision.) In the USA, schemes such as paired donation combat the problem of immunological incompatibility and allow more patients to obtain kidneys. In a paired donation, a friend or relation of patient 1 donates to patient 2, and a friend or relation of patient 2 donates to patient 1. More can be done in this vein and is being studied.

In the spirit of the medical student-residency matching in the USA, a central clearing house for matching kidneys is being studied. Issues of moral hazard relating to information revelation do exist. For instance, with a central clearing house, a patient at a transplant centre may end up not getting a kidney even though a suitable donor is available at the same centre possibly due to another centre playing the priority game. Should that transplant centre reveal information about the existence of that donor? In Singapore, fewer of such issues exist. In fact, the constraint of distance between patients and donors for kidney swaps is essentially a non-issue. Furthermore, a central clearing house is eminently in line with the culture of Singapore.

Suppose K gives the set of available kidneys, P gives the set of patients, and C(j) gives the subset of K that is compatible with patient j. Consider the following optimization problem:

Decision Variables:

x_ij = 1 if kidney i is allocated to patient j and 0 otherwise. For all i in K and all j in P. (These are {0, 1} variables.)

Parameters:

w_j is a subjective priority for patient j.

Objective:

Maximize ∑_{j in P} w_j ∑_{i in K} x_ij
This represents a weighted sum of the number of patients who get kidneys.

Constraints:

Each kidney is allocated to at most one patient:

∑_{j in P} x_ij ≤ 1

for all i in K.

Only a compatible kidney is allocated:

∑_{i in C(j)} ≤ 1
x_ij = 0 if i is not in C(j).

for all j in P.

A system for deciding patient priority w_j, depending on the patient's circumstances, must be agreed upon to ensure fairness or some modicum of uniform unfairness. Other matters may need to be addressed. For instance, patients with compatible directed donors must get kidneys to prevent information from being hidden from the central clearing house. The participation of these donors in the system represents a positive externality on the system.

To represent patients with compatible directed donors, we consider the set of such patients D and add the constraints:

∑_{i in C(j)} = 1

for all j in D.

Some patients may be willing to consider ABO incompatible kidneys (these have an 85% to 90% success rate in comparison to 98%). This may be done by replacing the objective function with:

Maximize ∑_{j in P} w_j ∑_{i in K} c_ij x_ij

where c_ij is a parameter representing the compatibility of kidney i with patient j. Again, a systematic means for choosing this parameter must be developed that captures the tradeoff of the risk of the transplant failing and the value of additional transplants being possible.

More issues may and will arise should such a system be implemented but I would like to close. Singapore is in a unique position, due to our circumstances, to implement what may be the most effective transplant infrastructure in the world. Aside from being an education to the world, a few patients may find themselves with a kidney, and that alone will be sufficient cause to implement a simple clearing house.