# American Institute of Mathematical Sciences

March  2019, 1(1): 87-101. doi: 10.3934/fods.2019004

## Combinatorial Hodge theory for equitable kidney paired donation

 1 Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI 48824, USA 2 Department of Mathematics, University of Tennessee, Knoxville TN 37996, USA

* Corresponding author

Published  March 2019

Kidney Paired Donation (KPD) is a system whereby incompatible patient-donor pairs (PD pairs) are entered into a pool to find compatible cyclic kidney exchanges where each pair gives and receives a kidney. The donation allocation decision problem for a KPD pool has traditionally been viewed within an economic theory and integer-programming framework. While previous allocation schema work well to donate the maximum number of kidneys at a specific time, certain subgroups of patients are rarely matched in such an exchange. Consequently, these methods lead to systematic inequity in the exchange, where many patients are rejected a kidney repeatedly. Our goal is to investigate inequity within the distribution of kidney allocation among patients, and to present an algorithm which minimizes allocation disparities. The method presented is inspired by cohomology and describes the cyclic structure in a kidney exchange efficiently; this structure is then used to search for an equitable kidney allocation. Another key result of our approach is a score function defined on PD pairs which measures cycle disparity within a KPD pool; i.e., this function measures the relative chance for each PD pair to take part in the kidney exchange if cycles are chosen uniformly. Specifically, we show that PD pairs with underdemanded donors or highly sensitized patients have lower scores than typical PD pairs. Furthermore, our results demonstrate that PD pair score and the chance to obtain a kidney are positively correlated when allocation is done by utility-optimal integer programming methods. In contrast, the chance to obtain a kidney through our method is independent of score, and thus unbiased in this regard.

Citation: Joshua L. Mike, Vasileios Maroulas. Combinatorial Hodge theory for equitable kidney paired donation. Foundations of Data Science, 2019, 1 (1) : 87-101. doi: 10.3934/fods.2019004
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A three-way kidney exchange cycle. The PD pairs are numbered 1, 2, and 3 with P for patient and D for donor indicating an individual's role. Potential donations are indicated by red arrows from the donor to patient
A graph with vertices $a$ through $e$ and an edge flow $h$ defined on it via weights. For example, $h(a,b) = 5$ and $h(b,e) = 11$
An edge flow and the process used to Hodge decompose it. (a) Original edge flow $U$. (b) Space of cyclic flows, $\ker(\Delta_1)$. (c) The globally cyclic portion $V$ minimizing $\left\| U-V \right\|$. (d) The gradient portion, $W = U-V$. (e) A scoring made by assigning 0 to one vertex value. (f) A new scoring with mean 0
The progression of Hodge Cycle on a tiny KEG. Green nodes are patients while red nodes are donors. Recorded cycles are dark and blue. Removed edges are unlabeled. (a) Initial KEG, $U$. (b) Initial cyclic KEG, $V_0$. (c) One cycle allocated, $V_1$. (d) Two cycles allocated (empty)
(Left) Patient score and CPRA. (Right) Donor score and representing patient CPRA. A comparison of vertex scores with patient CPRA values in KEGs with US proportions (Table 1). Both plots represent 50 random KEGs with 100 PD pairs each and show every PD pair. Underdemanded PD pairs with AB donors are marked with red Xs
(Left) Patient score and CPRA. (Right) Donor score and representing patient CPRA. A comparison of vertex scores with patient CPRA values in KEGs with uniform proportions (Table 1). Both plots represent 10 random KEGs with 100 PD pairs each, and show every PD pair. Underdemanded PD pairs with AB donors are marked with red Xs
(Left) Score pdfs for US average blood type and CPRA. (Right) Scoring probability densities for uniform blood type and CPRA. Probability density functions for the score of a random PD pair, given allocation by HC, TTCC, rCM, or WF. Each plot represents 50 random KEGs with (left) 50 PD pairs each or (right) 100 PD pairs each. Pairs without an obtainable cycle have been removed from the score determination
(Left)US average blood type and CPRA. (Right) Uniform blood type and CPRA. Both plots show the chance to obtain a kidney via HC, TTCC, rCM, or WF as a function of PD pair scores (patient score minus donor score). The solid blue line indicates the range of occuring PD pair scores. These plots are likelihood ratios of the densities seen in Fig 7; specifically, these values are the conditional probabilities of being allocated a kidney given a particular score
(Left) KEGs with US proportions and varying numbers of PD pairs. (Right) KEGs with 150 PD pairs and varying patient sensitivity. Both plots describe the average time for Hodge Cycle to find an allocation with one standard deviation error bars. Non-high CPRA patients are split evenly between low (0-10%) and medium (10-80%) CPRA
(Left) KEGs with US proportions and 50 PD pairs each. (Right) KEGs with uniform proportions and 100 PD pairs each. Both plots show cross comparison of the percentage of patients who were allocated kidneys by HC to TTCC, rCM, and WF algorithms. Each point represents a single randomly generated KEG and its solution with the marked method
(Left) KEGs with US proportions and 50 PD pairs each. (Right) KEGs with uniform proportions and 100 PD pairs each. Both plots compare average donation utility to number of patients allocated. Each point represents a single randomly generated KEG and its solution with the marked method. 0.75 is the expected average of all donation utilities in a KEG, since they are chosen uniformly between 0.5 and 1. The same KEGs are used for each solution method, and are the same as those used in Fig 10
(Top) Blood type proportions for generated KEGs. (Bottom) CPRA level proportions for generated KEGs. These are the multinomial probabilities used in practice. US averages from [2]. Specific CPRA is chosen uniformly within level range (0-10, 10-80, or 80-100%)
 Blood Type US wait-list US whole Uniform O 48.6% 44% 25% A 32.7% 42% 25% B 14.9% 10% 25% AB 3.8 % 4% 25% CPRA level US wait-list Uniform Low 81.3% 10% Med 11% 70% High 7.7% 20%
 Blood Type US wait-list US whole Uniform O 48.6% 44% 25% A 32.7% 42% 25% B 14.9% 10% 25% AB 3.8 % 4% 25% CPRA level US wait-list Uniform Low 81.3% 10% Med 11% 70% High 7.7% 20%
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