Math meets medicine: optimizing paired kidney donation

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Math meets medicineoptimizing paired kidney
donation

• Sommer Gentry, Ph.D.
• Dorry Segev, M.D.

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Operations Research the most influential
academic discipline youve never heard of

• Optimization versus heuristics
• Diet model
• Kidney paired donation model
• Simulation
• Missing data? Make some.
• Incompatible kidney donors, U.S., model

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Sketch of optimization problems

• Optimize in common usage
• Optimize in the mathematical sense
• Describe precisely
• Decisions to be made variables
• Requirements to be met constraints
• Desired outcome objective
• Solve
• Find decisions that maximize objective, among all
  possible decisions (those that meet the
  constraints)

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The Diet Problem

• Choose the least expensive diet from a set of
  foods that still meets the minimum nutritional
  requirements
• Decisions variables for the amount of each food
  purchasede (eggs), s (spinach), m (macaroni)
• Objective Minimize cost of food 3.00 e
  5.00 s 1.00 m

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Diet Problem continued

• Constraints Eat enough calories to survive and
  enough vitamin C to prevent scurvy300 e 100 s
  450 m 2000 calories5 e 45 s 90 mg
  vitamin C
• Feasible solution any shopping list that
  provides 2000 calories and 90mg of vit. C
• Optimal solution of all feasible solutions,
  shopping list that costs the least is optimal

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Diet problem solved

• The optimal solution is difficult to guess, but a
  computer can find it using a provably correct
  algorithm (Simplex algorithm) s 2 spinach, m
  4 macaroni, cost 14
• Macaroni and spinach? Perhaps variety should be
  among constraints
• Optimization gives provably best solution
• Best in real world? Yes, if the problem was
  expressed accurately

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Heuristics and optimization

• Heuristic rule of thumb that solves a decision
  problem simply
• Buy enough macaroni to meet calorie requirements,
  then add enough spinach to meet Vitamin C
  requirements (not optimal, 14.44)
• Why use a heuristic?
• common-sense appeal
• usually gives acceptable answers
• optimal algorithm is not known, or is not fast
  enough

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Organ allocation

• Decisions who receives which donated organ
• Constraints each solid organ may be donated to
  at most one patient
• Objective many possible
• Utility, maximize sum of life-years gained (QALY)
• Equity, equalize probability of receiving an
  organ between individuals (waiting time)
• Survival, minimize probability of death while
  waiting
• Mixtures of the above or other (DD kidney points)
• Cost

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ESRD and kidney donation

• End-stage-renal-disease (kidney failure) strikes
  tens of thousands of people
• Kidneys can no longer purify the blood
• Treatments include
• Dialysis, artificial kidney support, is extremely
  expensive (60,000 per year) and debilitating
• Kidney transplantation generally restores a
  patient to full health

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Deceased and live donors

• The demand for deceased donor kidneys far
  outstrips the supply (UNOS)
• About 10,000 deceased donor kidneys per year are
  transplanted
• Over 60,000 patients are on the waiting list
• Deceased donation is level
• Live donation is an attractive option
• Operation is very safe, can be laparoscopic
• About 6,000 live donations in 2003
• Spouses, friends, siblings, parents, children
• Siblings sometimes 6-HLA-antigen match

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Donor Incompatibilities

• Many willing live donors are disqualified
• Blood types O(46), A(34), B(16), AB(4)
• O recipients can accept only an O kidney
• A recipients can accept only O, A kidneys
• B recipients can accept only O, B kidneys
• AB recipients can accept any kidney
• Positive crossmatch (XM) predicts rejection
• Even blood compatible donors might have XM
• Crossmatch is difficult to predict
• Highly sensitized patients almost always XM

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Kidney paired donation (KPD)
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Graph recipient / donor pairs
NODE An incompatible donor / recipient pair
EDGE Connects two pairs if an exchange is
possible
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Node 22 (13 edges) Donor Type O Recipient Type
A Willing to travel
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Node 31 (7 edges) Donor Type O Recipient Type
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Node 5 (3 edges) Donor Type A Recipient Type
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Node 6 (1 edge) Donor Type A Recipient Type
O Unwilling to trade with older donors
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Node 35 (1 edge) Donor Type A Recipient Type
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First-accept heuristic
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Maximizing of transplants

• Decisions choose which incompatible pairs
  exchange (select edges in the graph)
• Constraints each incompatible pair involved in
  only one exchange (one edge per node)
• Objective maximize number of transplants
• For 100 donor/recipient pairs, there are millions
  of possible matchings (and the number of these
  grows exponentially) so can not list all the
  options
• Edmonds algorithm, without listing all options,
  finds the exchanges that yield the maximum number
  of transplants

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Maximal (First-Accept heuristic)
matching Accept matches at random until no edges
remain. Only 10 of 40 patients get a transplant
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Maximum cardinality matching Paths, Trees, and
Flowers, Edmonds (1965) 14 of 40 get transplants
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Flexibility edge weights

• All exchanges are equal, but some exchanges are
  more equal than others (edge points)
• Bonus points for disadvantaged groups, like O
  patients or highly sensitized patients
• Points to reward clinical predictors of good
  outcomes HLA antigen mismatch
• Patient priorities donor age versus different
  transplant center
• Maximize sum of edge weights (Edmonds, 1965)
• Fast algorithm finds provably optimal solution

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Edge 5-22 1 HS, within region Edge 5-31 0 HS,
travel required Edge 5-38 6-antigen match for
HS, travel required
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Match and travel in KPD
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Edge weight heuristics

• Edge rank heuristic Take best edge available,
  then next best edge, until no edges remain
• neglects the connection structure of the graph
  might not find an optimal solution

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Deceased donors are a simpler optimization than
KPD
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Deceased donor kidney
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Recipients
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Computational trials in medicine

• Critically important not to waste offers of live
  donor kidneys
• the matching algorithm should be analyzed in
  advance of any national implementation
• Simulated patients and virtual exchanges answer
  questions about the outcomes of matching
  algorithm
• Why run one experiment over several years when
  you can run two hundred experiments in an hour?

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Simulation

• Compile known patient characteristics
• 41 of kidney waitlist is female
• Create a database of virtual patients
• Computer generates a uniform random number f, 0
  lt f lt 100
• If f is lt 41, then patient is female, otherwise
  patient is male
• For simulated database, implement the proposed
  optimization or heuristic

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Simulation prerequisites

• Clinical outcome should be relatively predictable
  based on patients characteristics and clinical
  decisions
• Example required vaccination proportion
• known outcome is susceptibility or immunity to
  disease for an individual
• simulate interactions between individuals
• Not appropriate for investigating the effect of
  new drugs

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Advantages of simulation

• Fast, inexpensive test healthcare allocation
  mechanisms in advance of implementation
• Numerical predictions make an impact
• Run using many simulated databases
• Find confidence intervals, std deviation
• Can re-create missing data, data with known
  biases
• Kidney paired donation Find number and blood
  types of incompatible patient-donor pairs

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Simulated patients and social networks
Mother
Father
Sibling
Friend
Patient
Sibling
Spouse
Child
Child
Each Patient has between 1-4 available donors
Gentry, Segev, et al. 2005. Am J Transplant.
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Blood-type inheritance
Mother AA
Father BO
Friend BO
Recipient AO
Spouse OO
Sibling AB
Daughter OO
Son AO
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Decision tree model of family
(Zenios, Woodle, Ross, Transplantation 724, 2001)
Potential donors
Medical workup (pass 56 or 75), crossmatch
tests (11), blood typing
No willing, healthy donor
Incompatible donor/recipient pairs
Direct donation
Simulate until reach of real live donors (6468)
2406-4443 pairs annually
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Predict cost savings

• Long wait times on dialysis are extremely
  expensive
• 290 million saved over dialysis using any form
  of paired kidney donation
• 50 million additional saved over dialysis using
  maximum edge weight matching for paired kidney
  donation

(Segev, Gentry, et al., JAMA, 2005)