Alumni Profile

Applying math to engineer more kidney transplants

Sommer Gentry (BS Math and Computational Science, MS EES & OR 1998), an applied mathematics professor at the U.S. Naval Academy, has been interested in applying optimization to scientific and medical problems since she was a student at Stanford. It wasn’t until she married Dr. Dorry Segev, a transplant surgeon at Johns Hopkins, that she found an application that could save hundreds of lives. The Optimized Match system they developed to increase the number of kidney transplants was published last year in the Journal of the American Medical Association. Now the couple is advocating a national registry that can find the maximum number of potential matches.

Why does our current kidney transplant system need improvement?

Well what we typically do to match a patient with a kidney is see if they have a compatible blood type – we’re talking about live donations, so a sister is giving a kidney to a brother or something like that. Then also you do what’s called a cross-match test. If you are sensitized to the potential donor then that means you will definitely reject that kidney so they don’t do the transplant. When someone has a willing donor but they aren’t compatible, they are usually sent home. We think that is really too bad because it is possible for them to exchange with another family in the same situation with a complimentary incompatibility. For example, my brother gives to a stranger but that stranger’s loved one gives to me.

So what does Optimized Match do?

The great thing about paired donation is that any paired donation is recapturing willing donors who in today’s system basically just don’t donate. There is a small paired donation program at Johns Hopkins and at a few other institutions but they’re local and there isn’t a lot of awareness. We could be doing a whole lot more nationally for paired donations than we do.

So Part A is that if you do any paired donations that’s excellent because those people, right now aren’t donating at all. But beyond that there’s a subtle decision problem underlying kidney paired donation. If you write down everybody’s blood types and everybody’s sensitivities you can figure out who is able to trade with whom. But pairs could have many potential matches so which of those should you pick? It is important to do something mathematically sophisticated at that point because you could get 10 to 20 percent more kidney transplants if you smartly pair them. We can write that information as a graph, which is basically nodes and lines connecting them in a network.

Would you describe the math in lay terms?

Each node in the graph is a pair of one recipient and the person who’d like to donate to them but can’t. Then you draw a line from that node to someone else if those people could exchange kidneys. The graph represents all the possible exchanges. Then what we’re looking for is called a maximum matching—the maximum number of people who could get transplanted. If you choose one particular line and say that’s a trade, we’re going to do these two transplants, then those two nodes go away and so do all of the lines connected to them. At the end of this process there are no lines left because if there were still lines left then there would still be compatible trades you haven’t done. You actually run this until there are no lines left.

If we take away lines at random, if we say let’s trade those guys and those guys and those guys, that’s actually how you’d do it if you had no optimization step. That would still be a good thing, because some people would get kidneys but you wouldn’t get as many as you could. You have to look at all the lines on the graph and choose the best matching, so the best subset of lines. There’s a really simple case where you have four nodes and a U shaped line through them. Basically now if you match the bottom two nodes, then the other two nodes have no one to trade with. You’ve got only two transplants out of that. Alternatively if you use optimization you are going to want to choose the uprights of the U. Then you can get four transplants.

Beyond physiology what other constraints that can make this more complicated?

You might not want people to have to travel too far in order to do these donations or you want to give people who are under 18 a benefit. You’d give a benefit for very young recipients because their development is being affected for every day that they can’t get a transplant. Another might be if you were a prior live donor so you gave a kidney to someone earlier in your life it might be nice to get a benefit at this stage. That’s a very small number of people but if that were to happen that’s something that people would like to give a bonus for. Another thing are people who have perhaps run out of dialysis access. It’s impossible for them to be dialyzed anymore. For them a transplant is really urgent.

How widely is optimized matching used?

We have started to work with the Canadian Council on Donation and Transplantation to help them design a paired donation system. We’ve basically made a blanket offer that if anyone sends us de-identified data that we will run a matching for them. We’re hoping that there will be a U.S. national system soon. The more people that are listed the better you can do and the more important it is to use optimization. There are some people out there who are highly sensitized. There is almost no one with whom they have a negative cross match. There is almost no one that they can accept a kidney from. They are looking for that needle in a haystack, and for them in particular a national system would really increase their chances.

We have definitely been using this to engineer matches at Johns Hopkins. Hopkins has done something like 24 transplants through this procedure. There have only been about 100 in the U.S. to date so Hopkins has by far done the most. There are more than 60,000 people on the waiting list. We think that possibly 4,000 of them already have a live donor but aren’t compatible. They are waiting on a list but they could get a transplant tomorrow if we could find a paired donation match for them. It’s very sad. We think also that there might be as many as 3,500 additional pairs like this every year.

How did you end up doing this?

Operations research is a really broad field and I always found applications that had to do with science or with medicine or something that people needed in their lives to be more compelling than the financial applications. I think operations research has been widely applied in finance or business but it’s just now starting to make an impact on health care decision making. It’s really exciting to see that you can use operations research in people’s lives. I’m a professor at the Naval Academy and I tell my students they should major in math if they want to save lives. I can actually say that now. That’s really cool.

Did your studies at Stanford have an influence on your work?

I credit Stanford with all of the training that led to this work. The Masters degree that I did in OR was really the core of the way I still think. There were two people that I worked with very closely. One was my advisor, Arthur Veinott. He was always cheery. He always had a Diet Coke in his hand. He definitely encouraged me to join the Masters program. My Bachelors was in mathematical and computational science. There were a few required OR classes and those were what captivated me and so I kept taking Operations Research with all of my electives. I also had a really great time working with Sam Savage. He does decision analysis modeling in spreadsheets.