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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 
Kidney Paired Donation (KPD) is a system whereby incompatible patientdonor 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 integerprogramming 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 utilityoptimal integer programming methods. In contrast, the chance to obtain a kidney through our method is independent of score, and thus unbiased in this regard.
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D. J. Abraham, A. Blum and T. Sandholm, Clearing algorithms for barter exchange markets: Enabling nationwide kidney exchanges, In: Proceedings of the 8th ACM conference on Electronic commerce. ACM, 2007, 295–304.Google Scholar 
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I. Ashlagi, D. Gamarnik, M. A. Rees and A. E. Roth, The need for (long) chains in kidney exchange, National Bureau of Economic Research, 2012.Google Scholar 
[6] 
G. M. Danovitch, The high cost of organ transplant commercialism, Kidney International, 85 (2014), 248250. Google Scholar 
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J. P. Dickerson, A. D. Procaccia and T. Sandholm, Price of fairness in kidney exchange, In: Proceedings of the 2014 International Conference on Automous Agents and Multiagent Systems. International Foundation for Autonomous Agents and Multiagent Systems, 2014, 1013–1020.Google Scholar 
[8] 
H. Edelsbrunner and J. Harer, Computational Topology: An Introduction, American Mathematical Society, 2010. Google Scholar 
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J. Edmonds, Paths, trees, and flowers, Canadian Journal of Mathematics, 17 (1965), 449467. doi: 10.4153/CJM19650454. Google Scholar 
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W. V. D. Hodge, The Theory and Applications of Harmonic Integrals, Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1989. Google Scholar 
[11] 
X. Jiang, L. H. Lim, Y. Yao and Y. Ye, Statistical ranking and combinatorial Hodge theory, Mathematical Programming, 127 (2011), 203244. doi: 10.1007/s101070100419x. Google Scholar 
[12] 
J. B. Kessler and A. E. Roth, Organ allocation policy and the decision to donate, The American Economic Review, 102 (2012), 20182047. Google Scholar 
[13] 
D. P. Ladner and S. Mehrotra, Methodological challenges in solving geographic disparity in liver allocation, JAMA surgery, 151 (2016), 109110. Google Scholar 
[14] 
L. F. Ross and E. Woodle, Kidney exchange programs: An expanded view of the ethical issues, In: Organ Allocation, Springer, 1998, 285–295.Google Scholar 
[15] 
A. E. Roth, T. Sönmez and Uktu ÜM, Kidney Exchange, National Bureau of Economic Research, 2003.Google Scholar 
[16] 
S. L. Saidman, A. E. Roth, T. Sönmez, M. U. Ünver and F. L. Delmonico, Increasing the opportunity of live kidney donation by matching for twoand threeway exchanges, Transplantation, 81 (2006), 773782. Google Scholar 
[17] 
D. L. Segev, S. E. Gentry, D. S. Warren, B. Reeb and R. A. Montgomery, Kidney paired donation and optimizing the use of live donor organs, Journal of the American Medical Association, 293 (2005), 18831890. Google Scholar 
[18] 
D. J. Taber, M. Gebregziabher, K. J. Hunt, T. Srinivas, K. D. Chavin and P. K. Baliga, et al., Twenty years of evolving trends in racial disparities for adult kidney transplant recipients, Kidney International, 2016.Google Scholar 
[19] 
S. Takemoto, F. K. Port, F. H. Claas and R. J. Duquesnoy, HLA matching for kidney transplantation, Human Immunology, 65 (2004), 14891505. Google Scholar 
[20] 
P.anagiotis Toulis and D. C. Parkes, A random graph model of kidney exchanges: efficiency, individualrationality and incentives, In: Proceedings of the 12th ACM Conference on Electric commerce, ACM, 2011, 323–332.Google Scholar 
[21] 
S. A. Zenios, Optimal control of a pairedkidney exchange program, Management Science, 48 (2002), 328342. Google Scholar 
show all references
References:
[1] 
Ethical Principles to be Considered in the Allocation of Human Organs; 2010. Organ Procurement and Transplantation Network, Available from: http://optn.transplant.hrsa.gov/resources/ethics.Google Scholar 
[2] 
TN and US Kidney Transplant Summary; 2015. Scientific Registry of Transplant Recipients, Available from: http://www.srtr.org.Google Scholar 
[3] 
Spring 2014 Regional Meeting Data; 2014. United Network for Organ Sharing, Available from: https://www.unos.org/wpcontent/uploads/unos/DataSlides_Spring_2014.pdf?75608d.Google Scholar 
[4] 
D. J. Abraham, A. Blum and T. Sandholm, Clearing algorithms for barter exchange markets: Enabling nationwide kidney exchanges, In: Proceedings of the 8th ACM conference on Electronic commerce. ACM, 2007, 295–304.Google Scholar 
[5] 
I. Ashlagi, D. Gamarnik, M. A. Rees and A. E. Roth, The need for (long) chains in kidney exchange, National Bureau of Economic Research, 2012.Google Scholar 
[6] 
G. M. Danovitch, The high cost of organ transplant commercialism, Kidney International, 85 (2014), 248250. Google Scholar 
[7] 
J. P. Dickerson, A. D. Procaccia and T. Sandholm, Price of fairness in kidney exchange, In: Proceedings of the 2014 International Conference on Automous Agents and Multiagent Systems. International Foundation for Autonomous Agents and Multiagent Systems, 2014, 1013–1020.Google Scholar 
[8] 
H. Edelsbrunner and J. Harer, Computational Topology: An Introduction, American Mathematical Society, 2010. Google Scholar 
[9] 
J. Edmonds, Paths, trees, and flowers, Canadian Journal of Mathematics, 17 (1965), 449467. doi: 10.4153/CJM19650454. Google Scholar 
[10] 
W. V. D. Hodge, The Theory and Applications of Harmonic Integrals, Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1989. Google Scholar 
[11] 
X. Jiang, L. H. Lim, Y. Yao and Y. Ye, Statistical ranking and combinatorial Hodge theory, Mathematical Programming, 127 (2011), 203244. doi: 10.1007/s101070100419x. Google Scholar 
[12] 
J. B. Kessler and A. E. Roth, Organ allocation policy and the decision to donate, The American Economic Review, 102 (2012), 20182047. Google Scholar 
[13] 
D. P. Ladner and S. Mehrotra, Methodological challenges in solving geographic disparity in liver allocation, JAMA surgery, 151 (2016), 109110. Google Scholar 
[14] 
L. F. Ross and E. Woodle, Kidney exchange programs: An expanded view of the ethical issues, In: Organ Allocation, Springer, 1998, 285–295.Google Scholar 
[15] 
A. E. Roth, T. Sönmez and Uktu ÜM, Kidney Exchange, National Bureau of Economic Research, 2003.Google Scholar 
[16] 
S. L. Saidman, A. E. Roth, T. Sönmez, M. U. Ünver and F. L. Delmonico, Increasing the opportunity of live kidney donation by matching for twoand threeway exchanges, Transplantation, 81 (2006), 773782. Google Scholar 
[17] 
D. L. Segev, S. E. Gentry, D. S. Warren, B. Reeb and R. A. Montgomery, Kidney paired donation and optimizing the use of live donor organs, Journal of the American Medical Association, 293 (2005), 18831890. Google Scholar 
[18] 
D. J. Taber, M. Gebregziabher, K. J. Hunt, T. Srinivas, K. D. Chavin and P. K. Baliga, et al., Twenty years of evolving trends in racial disparities for adult kidney transplant recipients, Kidney International, 2016.Google Scholar 
[19] 
S. Takemoto, F. K. Port, F. H. Claas and R. J. Duquesnoy, HLA matching for kidney transplantation, Human Immunology, 65 (2004), 14891505. Google Scholar 
[20] 
P.anagiotis Toulis and D. C. Parkes, A random graph model of kidney exchanges: efficiency, individualrationality and incentives, In: Proceedings of the 12th ACM Conference on Electric commerce, ACM, 2011, 323–332.Google Scholar 
[21] 
S. A. Zenios, Optimal control of a pairedkidney exchange program, Management Science, 48 (2002), 328342. Google Scholar 
Blood Type  US waitlist  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 waitlist  Uniform  
Low  81.3%  10%  
Med  11%  70%  
High  7.7%  20% 
Blood Type  US waitlist  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 waitlist  Uniform  
Low  81.3%  10%  
Med  11%  70%  
High  7.7%  20% 
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