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Optimal Acceptance of Incompatible Kidneys

Xingyu Ren, Michael C. Fu, Steven I. Marcus

TL;DR

The paper addresses the problem of optimally accepting donor kidneys when incompatibility exists, by formulating an infinite-horizon Markov decision process that explicitly tracks compatibility as a state variable. It develops a discrete-time framework with stochastic dynamics for patient health, kidney offers, and mismatch, along with desensitization outcomes and retransplantation, and proves structural properties that yield control-limit-type optimal policies. Through theoretical results and numerical experiments grounded in OPTN data, the authors show that incorporating compatibility and desensitization can meaningfully improve expected life years, providing clear thresholds and insights for clinicians and policymakers. The work offers a practical, decision-support tool that can inform both individual patient decisions and kidney allocation policy design, with potential to expand the donor pool and improve transplant outcomes.

Abstract

Incompatibility between patient and donor is a major barrier in kidney transplantation (KT). The increasing shortage of kidney donors has driven the development of desensitization techniques to overcome this immunological challenge. Compared with compatible KT, patients undergoing incompatible KTs are more likely to experience rejection, infection, malignancy, and graft loss. We study the optimal acceptance of possibly incompatible kidneys for individual end-stage kidney disease patients. To capture the effect of incompatibility, we propose a Markov Decision Process (MDP) model that explicitly includes compatibility as a state variable. The resulting higher-dimensional model makes it more challenging to analyze, but under suitable conditions, we derive structural properties including control limit-type optimal policies that are easy to compute and implement. Numerical examples illustrate the behavior of the optimal policy under different mismatch levels and highlight the importance of explicitly incorporating the incompatibility level into the acceptance decision when desensitization therapy is an option.

Optimal Acceptance of Incompatible Kidneys

TL;DR

The paper addresses the problem of optimally accepting donor kidneys when incompatibility exists, by formulating an infinite-horizon Markov decision process that explicitly tracks compatibility as a state variable. It develops a discrete-time framework with stochastic dynamics for patient health, kidney offers, and mismatch, along with desensitization outcomes and retransplantation, and proves structural properties that yield control-limit-type optimal policies. Through theoretical results and numerical experiments grounded in OPTN data, the authors show that incorporating compatibility and desensitization can meaningfully improve expected life years, providing clear thresholds and insights for clinicians and policymakers. The work offers a practical, decision-support tool that can inform both individual patient decisions and kidney allocation policy design, with potential to expand the donor pool and improve transplant outcomes.

Abstract

Incompatibility between patient and donor is a major barrier in kidney transplantation (KT). The increasing shortage of kidney donors has driven the development of desensitization techniques to overcome this immunological challenge. Compared with compatible KT, patients undergoing incompatible KTs are more likely to experience rejection, infection, malignancy, and graft loss. We study the optimal acceptance of possibly incompatible kidneys for individual end-stage kidney disease patients. To capture the effect of incompatibility, we propose a Markov Decision Process (MDP) model that explicitly includes compatibility as a state variable. The resulting higher-dimensional model makes it more challenging to analyze, but under suitable conditions, we derive structural properties including control limit-type optimal policies that are easy to compute and implement. Numerical examples illustrate the behavior of the optimal policy under different mismatch levels and highlight the importance of explicitly incorporating the incompatibility level into the acceptance decision when desensitization therapy is an option.
Paper Structure (14 sections, 15 theorems, 69 equations, 8 figures, 12 tables, 1 algorithm)

This paper contains 14 sections, 15 theorems, 69 equations, 8 figures, 12 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model Formulation
  3. State of Patient and Kidney Offer Stochastic Processes
  4. Action & Action Space
  5. Dynamics
  6. Reward Functions
  7. Objective Function
  8. Structural Results
  9. Numerical Experiments
  10. Conclusions and Future Research
  11. Proofs
  12. Selection of Parameters in Numerical Experiments
  13. Definition of States and Transition Law
  14. Rewards

Key Result

Theorem 1

The value function $V$ satisfies: $\forall h\in S_H,~m \in S_M$, and $k< K+1$, where Note that $V(H+1,k,m)=0,~\forall k\in S_K, ~m\in S_M$. Moreover, the sequence of functions $\{V_{n}\}_{n\geq 0}$ recursively defined by the value iteration procedure given by $\forall h\in S_H,~m \in S_M,~k<K+1$, converges pointwise to $V$, starting from any bounded function $V_0$, where

Figures (8)

  • Figure 1: For an MDP with state space $S_H\times S_K$, an optimal policy such that states for which optimal actions are $W$ and $T$ are contained in two disjoint connected subsets.
  • Figure 2: For an MDP with state space $S_H\times S_K\times S_M$, suppose that its unique optimal policy partitions ${\mathbb{R}}^3$ into four disjoint decision regions by a vertical plane and a horizontal plane. Both actions $W$ and $T$ are optimal over two disconnected regions. However, all three types of control limit optimal policies exist.
  • Figure 3: The optimal policy $d_1$.
  • Figure 4: Q-function for different mismatch levels.
  • Figure 5: Policy $q_1$: given patient state $h$, the optimal action is to wait ($W$) if and only if the kidney state $k$ is above the curve.
  • ...and 3 more figures

Theorems & Definitions (32)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Theorem 1
  • Definition 1
  • Definition 2
  • Theorem 2
  • ...and 22 more