Mathematics of organ transplantation

Summary: Locating and allocating available compatible organs is an important task of surgery, as is determining the likelihood of success and survival.

Organ transplantation involves replacing a damaged organ or body part with an organ taken from another body, a location on the patient’s own body, or sometimes another source. Relatively common organ transplants include hearts, lungs, livers, corneas, bone marrow, and skin. In the twenty-first century, there are increasing instances of transplantations involving parts that have proven more difficult in the past, including a human face in 2010. Transplantation is one of few medical fields where practice is driven by statistical analysis of large-scale national datasets. Collecting comprehensive data about transplantation in the United States is mandatory, and researchers use statistics to inform clinical practice and national policy. Still, there are too few living and deceased organ donors to meet the need. Optimization tools make the best use of scarce resources, like donated organs. With kidney paired donation, optimization can even increase the supply of available organs. An artificial pancreas employing control theory was under development in 2010.

Statistics

Statistical analyses inform transplant policy and individual decisions. The transplant community seeks equity in allocating organs, so the allocation system is frequently analyzed for gender and racial disparities. Understanding outcomes with and without transplantation helps patients decide if they will benefit from a particular transplant.

Survival analysis is the branch of statistics concerned with the distribution of time to an event. Survival analysis is commonly used in medicine to study time-to-death but can also be used to study time to any event, such as time from joining a transplant waitlist until receiving a transplant. The survival function S(t)=Pr(T>t) indicates the probability that the random time of an event T is later than a given time t.

Complications

Survival analysis is complicated by censoring; not all patients in a study have reached the event of interest. In a time-to-death analysis, some patients are likely still alive. The first technique for estimating a survival function with censoring was the product-limit estimator of statisticians Edward Kaplan and Paul Meier.

Confounding is another challenge. One could perform a survival analysis of the association between gender and time-to-transplantation to see whether men and women receive transplants at the same rate. However, not all patients are expected to wait the same amount of time. Other factors (such as age and blood type) confound studies of the effect of the factor of interest (gender) on time-to-transplantation. Cox proportional hazards analysis methods, named for statistician David Cox, can account for confounding, using a regression model based on the hazard function λ(t)dt=Pr(t≤T<t+dt|T≥t, which indicates the instantaneous probability of an event at some time (t) conditional on having survived to at least that time.

Optimization

Donated organs are scarce and each organ must be allocated to one of many potential recipients. Optimization techniques allocate scarce resources by maximizing an objective function. A person’s Lung Allocation Score is largest when the transplant has the largest lifespan benefit, and available lungs are offered to the nearby person with the largest score.

Kidney paired donation in which two living donors who are incompatible with their intended recipients exchange kidneys for compatible transplants requires more complex optimization techniques. More people can obtain better transplants when the paired donations are arranged using either a maximum weight matching in a graph or a maximum weight cycle decomposition (if more than two donors and recipients are involved in each exchange). By optimizing an individual’s outcome rather than the overall good, a Markov decision process model, named for mathematician Andrei Markov, can determine whether it is better for a patient to accept a certain organ offered or wait until a possibly better organ is offered later. Another Markov decision process model can establish the best time for a patient to receive a liver transplant from a living donor.

Control

Control theory studies systems where adjustments over time maintain some desired set point, like a thermostat heating or cooling a room to maintain a comfortable temperature. In transplantation, control theory is used in an experimental artificial pancreas. A healthy person’s pancreas maintains blood glucose levels over time by regulating insulin in response to eating a meal or exercising. An artificial pancreas uses a blood glucose monitor and a mathematical control system to drive an insulin pump. The control algorithms are tested on mathematical models of blood glucose levels before being tested in human subjects.

Bibliography

Cox, David R. “Regression Models and Life Tables.” Journal of the Royal Statistical Society 34, no. 2 (1972).

Harvey, R. A., et al. “Quest for the Artificial Pancreas.” IEEE Engineering in Medicine and Biology Magazine (March/April 2010).

Kaplan, Edward L., and Paul Meier. “Nonparametric Estimation From Incomplete Observations.” Journal of the American Statistical Association 53 (1958).

Segev, D. L., S. E. Gentry, D. Warren, B. Reeb, and R. A. Montgomery. “Kidney Paired Donation: Optimizing the Use of Live Donor Organs.” Journal of the American Medical Association 293 (2005).