A Time-dependent Stopping Problem with Application to Live Organ Transplants

Abstract

We consider a time-dependent stopping problem and its application to the decision-making process associated with transplanting a live organ. “Offers” (e.g., kidneys for transplant) become available from time to time. The values of the offers constitute a sequence of independent identically distributed positive random variables. When an offer arrives, a decision is made whether to accept it. If it is accepted, the process terminates. Otherwise, the offer is lost and the process continues until the next arrival, or until a moment when the process terminates by itself. Self-termination depends on an underlying lifetime distribution (which in the application corresponds to that of the candidate for a transplant). When the underlying process has an increasing failure rate, and the arrivals form a renewal process, we show that the control-limit type policy that maximizes the expected reward is a nonincreasing function of time. For non-homogeneous Poisson arrivals, we derive a first-order differential equation for the control-limit function. This equation is explicitly solved for the case of discrete-valued offers, homogeneous Poisson arrivals, and Gamma distributed lifetime. We use the solution to analyze a detailed numerical example based on actual kidney transplant data.

A time-dependent best choice problem with costs and random lifetime in organ transplants

“…This decision is based on the personal experience of the physician and depends on medical judgment regarding the suitability of the organ being offered for specific candidates. In the litera-ture, some models are presented helping physicians to make these decisions (for example see Alagoz et al [1] or David and Yechiali [11]). However, these works do not take into consideration the patient's aversion to risk.…”

“…One can compute the probability of any match-level between a given recipient and a random graft. With each match-level we associate a value, for example, the probability of graft survival for at least one year or expected lifetime of the graft (see David and Yechiali [11] for more details).…”

“…From the mathematical point of view our model is a generalization of the problems considered by Elfving [14], David and Yechiali [11] and Fer-enstein and Krasnosielska [16] to the case of an optimal stopping problem with costs, a function related to aversion to risk and a random horizon. Elfving [14] analyzed an optimal stopping problem with a discount function and Poisson-type arrival of offers.…”

“…He obtained a differential equation which allows one to compute the optimal mean reward. David and Yechiali [11] considered a generalization of the Elfving problem to the case of an optimal stopping problem with a renewal process and an underlying lifetime distribution. They gave an explicit solution for the case with a homogeneous Poisson process, constant discount function and Gamma distribution lifetime.…”

“…In comparison with the model of David and Yechiali [11] and Ferenstein and Krasnosielska [16], we additionally introduce an unobservable random variable and a function related to the changes in patients' risk aversion level. However, the construction of the solution of the model uses similar methods to those in Ferenstein and Krasnosielska [16].…”