A Stylistic Queueing-Like Model for the Allocation of

A Stylistic Queueing-Like Model for the Allocation of Organs on the Public List Israel David and Michal Moatty-Assa

Supply-Demand Discrepancy 32, 371 40, 000 23, 637 30, 000 24, 690 27, 289 32, 440 32, 589 29, 157 השתלות 20, 000 8, 539 8, 667 9, 357 9, 913 10, 659 10, 588 10, 551 10, 000 0 2002 2003 2004 2005 2006 2007 2008 • Increasing shortage in kidneys for transplant • 4, 252 died waiting (2008) מועמדים חדשים

• • (Kidney offers are thrown away) ~50% refuse 1 st kidney offered!

Whom do I best fit? Who’s the youngest? Who waits the longest? Objectives: Clinical Efficiency: QALY, % survival. Equity: in waiting, across social groups. Matching Criteria: ABO, HLA, PRA, Age, Waiting

The Israeli “Point System” for kidney allocation PRA points Age points 25% - 0% 0 0 – 18 4 50% - 26% 2 19 – 40 2 75% - 51% 4 41 – 60 1 >75% 6 >60 0 HLA mismatches points Waiting time (months) points No MM 4 <24 0 1 MM 3 25 – 48 1 No MM in DR 2 49 – 96 2 >97 4

FIFOf – First In First Offered • FIFO sorting for Offering Decision rule Allocation rule • simplifying assumptions, “stylistic” moel

The decision of the single candidate The future arrival process How good is this offer? population statistics by ABO, HLA How long do I wait? my HLA, ABO donors arrival rate (A continuous, time-dependent, full-info “Secretary”)

Model Assumptions • • Constant lifetime under dialysis (T) What arrival compromising t? Poisson is the of donor kidneys (rate l) Poisson arrival of patients "Aggregate HLA " – only one relevant genetic quality kidney offer a match a mismatch frequency in population p 1 -p gain (life years) R r

First candidate Second candidate n’th candidate Simulation

n = 1, basics X – Offer random value; x = E[X] = Rp + r(1 -p) U(t) – expected optimal value assuming that at t an offer is pending al value from t onwards (exclusive of t if an offer is pending); V(T) = 0. l, T, R, r, p, ant 1

Dynamic Programming 1. U(t , x) = max{x, V(t)} 2. U(t) = EX[U(t , X)] 3. V(t) =

n = 1, depiction of V and U

n = 1, Explicit t* x = E[X] = Rp + r(1 -p)

n = 1, Explicit solution of V(t), U(t) .

(A solvable Volterra)

n = 2, (approx. ) outlook for the second candidate Non-hom. -Poisson stream with 3 stages effective l 0

n = 2, conditional expected gains

n = 2, Explicit t*

n > 1, general input spec ifics o output f can d. n optimization f fics o speci * and t n-1 1 d. (n can ) optimal decision rule (tn*) for cand. n

still… n = 3 effective l 0

n=3 effective l 0

The l-recursion per sub-intervals for all Except for intersections with where or where

leftmost Vn(t)’s for sub-intervals • - optimal value for cand. n in rejecting at the beginning of sub-val l • - arrival probability of an offer during sub-val l • - conditional expected gain if during sub-val l

(explicit expressions for )

The critical subinterval and determining tn* is taken to be such that t is substituted for the beginning of subinterval

blocking and releasing of simultaneous antigen currents 0

Simulation Measures • Long-run proportion of "good" transplants • Long-run death-rate • Long-run Waiting Time for allocated candidate