Week 4: Mathematical Modeling and Montreal

So I missed a few days of the Immersion program teaching for a Math Camp in Montreal. For the summer camp, I was preparing a fun talk that discusses some ways that math can help in the clinical settings. While preparing this talk, I figured there was no better way than asking the immersion clinicians for ideas that I can talk about. So here it is:

As engineers, we often use mathematics as a set of tools for solving problems. From fluid dynamic models of blood flow, to certain clinical data analysis, there are numerous ways to use math in clinical settings. One topic that caught my attention, and fascinated me was the problem of how organ transplants are optimized at the local, regional, and national levels. There is an increasing trend of government funding for an efficient network called the Organ Procurement and Transplantation Network (OPTN). I've seen an analogous problem, (about the renal transplant network) last year in a mathematical modeling contest, and thought this would be a great problem to think about and ask around.
The topic of establishing an effective network is a difficult one, because of the many factors that must be incorporated in developing this model. As there are more organs sought after than there are available, there would usually be a significant waiting list for patients who need a new organ. One could try making a population dynamics model (using a system of ordinary differential equations) to get a holistic idea of how the waiting list behaves over the long course, but many indications suggest that we are all moving more towards larger and larger waiting lists in the future for almost all organs. Supply simply does not meet the demand.

One novel approach for resolving this situation was looking at donor-patient pairs. It is often true that exists a donor who is more than willing to donate an organ to a specific patient (which made sense in the case of a kidney), but the donor's kidneys are not compatible with the patient. If this is the case, it would make sense to establish a two-way exchange, so the two donors would trade their kidneys.

As a mathematician, one would like to generalize into establishing an n-way exchange, and finding a mathematically elegant solution that solves this optimization problem. However, from a clinical perspective, there are additional factors that need to be considered before we consider a similar exchange. First, suppose we establish an n-way exchange, but the chain breaks (eg. extraction fails) at some point. If this happens, what are the repercussions of such failure to the whole chain? One patient will not receive a sought after kidney due to the failure, while its donor, whose kidney is somewhat like a bargaining chip for acquiring his or her patient's new kidney, may break off; would this result in the entire chain of surgeries to fall apart? If so, will that mean that the all n transplants must happen simultaneously? Second is the feasibility of such cyclic surgeries; after all, these n-way exchanges require a lot of clinical manpower. Do most facilities have such capabilities? While it may be possible to model these intricacies using mathematics, it still won't address all the questions that clinicians may have.

Through summer immersion, I've learned that to address these problems properly, it would require more than mathematical problem-solvers developing models; a collaboration with clinical experts who work hands-on with these issues is a must. It is interesting to see where these considerations may lead to: NYP-Columbia Hospital was successful in having a 3-way exchange for the renal transplant recently in 2004, and this was a huge step forward.