Matching theory for kidney transplantation...Kidney disease Kidney exchange Pairwise exchange Top...

transcript

Kidney disease Kidney exchange Pairwise exchange Top Trading Cycles and Chains

Matching theory for kidneytransplantation

Monica Salvioli

May 21st 2015

Every ten minutes, someone is added to thenational transplant waiting list in the USA

Organ Procurement and Transplantation Network

On average, 21 people die each day whilewaiting for a transplant in the USA

Despite advances in medicine and technology the gap betweensupply and demand continues to widen

Transplant waiting list in Italy

Centro Nazionale Trapianti

Kidney disease

Garibotto G, Manuale di nefrologia, 2012Harrison’s Principles of Internal Medicine, 17th Ed, 2008

Functions

• removal of waste products of metabolism through theproduction of urine

• regulation of electrolytes

• maintenance of acid–base balance

• regulation of blood pressure

• secretion of hormones

Chronic kidney disease

renal function <60 ml/min/1,73 m2

for over 3 months

(normal value: 120-130 ml/min/1,73 m2)

In the USA:

• 13% of the population• 15 - 30% of old people• 50% of people who suffer from metabolic or

cardiovascular diseases

have a kidney damage

Complications

• Fatigue• Anemia• Diminished urine output• Hypertension• Heart failure• Bone disorder• Death

Diseases associated to the kidney and urinary tract causeapproximately 830,000 deaths annually

(12th highest cause of death)

Approximately 1.8 million people currently have access torenal replacement therapy

Renal replacement therapy

• Dialysis

• Kidney transplant

Dialysis

• replaces only 10% of renal function

• does not correct the compromised endocrine functions ofthe kidney

• 4 hours, 3 times a week

• 17000 - 40000 e per year

Kidney transplant

• total replacement of renal function

• longer life expectancy with respect to people on dialysis

• good quality of life

Kidney transplant

90% of kidney transplant patients are able to return towork

Kidney transplant

Women can get pregnant

Kidney transplant

Sean Elliott and Alonzo Mourning made history by returning to play inthe NBA following the surgery

Sources of kidneys

• Deceased donors• Living donors

Mean rates of graft and patient survival for kidneys transplanted in the USAfrom 1992 to 2002

Transplant activity in Italy

Compatibility

The major problem in organ transplantation is rejection,during which the body has an immune response to thetransplanted organ, possibly leading to transplant failureand the need to immediately remove the organ from therecipient.

Transplant rejection can be reduced by use ofimmunosuppressant drugs after transplant and bydetermining the compatibility between donor andrecipient.

Compatibility

There are actually three tests that are done to evaluatecompatibility between recipient and donor (living or deceased):

• ABO blood type test

• Comparison of HLA antigens

• Screen for antibodies

Compatibility

There are actually three tests that are done to evaluatecompatibility between recipient and donor (living or deceased):

• ABO blood type test

• Comparison of HLA antigens

• Screen for antibodies

Blood typing

There are 4 different blood types: the most common blood typein the population is type O (40%), the next most common isblood type A (36%), then B (17%), and the rarest is blood typeAB (7%).

The blood type of the donor must be compatible with therecipient. The rules for blood type in transplantation are thesame as they are for blood transfusion.

Blood typing

Blood type O is considered the universal donor, becausepeople with blood type O can give to any other blood type.

Blood type AB is called the universal recipient because theycan receive an organ or blood from people with any blood type.

HLA typing

HLA stands for human leukocyte antigen

HLA are proteins that are located on the surface of the whiteblood cells and other tissues in the body.

HLA typingIdentified antigens

Marsh, Nomenclature for factors of the HLA system, 2010

HLA typing

Out of over 100 different antigens that have been identified,there are six that have been shown to be the most important inorgan transplantation. Of these six antigens, we inherit threefrom each parent.

Except in cases of identical twins and some siblings, it is rare toget a six-antigen match between two people, especially if theyare unrelated. The chance of a perfect or six-antigen matchbetween two unrelated people is about one in 100,000.

HLA typing

We count HLA matches/mismatches, with a possible rangefrom 0 to 6

HLA typing

Kidneys have been transplanted between two people with nomatching antigens without a rejection episode. In other caseswhere all six antigens matched, recipients have suffered fromrejection.

There is no way to predict who will experience a rejectionepisode, but it has been observed a higher incidence of graftfailure due to rejection with a lower degree of HLA compatibility.Living donors with a 6 antigen match also allow the opportunityfor decreased immunosuppression.

Mean rate of graft survival5-Year-Follow-Up

Cellular and Molecular Immunology, 2007

Preformed antibodies

Immunologic incompatibility occurs when transplant candidatesare exposed to foreign (nonself) human leukocyte antigens(HLA) through blood transfusion, pregnancy, and/or priortransplantation. Exposure to foreign HLA leads many patientsto develop anti-HLA antibodies, which cause reactivity againstpotential organ donors.

Recipient serum is tested against donor cells to determine if therecipient has preformed antibodies against any antigens on thedonor’s cells.

A positive result means that the recipient has responded to thedonor and that the transplant should not be carried out.A negative result means that the recipient has not responded tothe donor and therefore transplantation should be safe.

There is a test you need to take that will determine how easy ordifficult it will be to find a compatible person. The test is calledPRA (panel reactive antibody).

PRA is a blood test that measures the level of antibodies in therecipients blood. The more antibiodies you have, the moredifficult it will be to find a compatible donor.

A person’s PRA can be anywhere from 0% to 99%.Your PRA represents the percent of the population that theantibodies in your blood would react to and reject the kidney.

For example, having a PRA of 25 means that 25% of thepopulation will not be able to donate a kidney to you, becausethe antibodies present in your blood would attack thetransplanted kidney and can cause immediate rejection.

PRA’s levels

• LOW PRA: <10%

• MEDIUM PRA: 10-80%

• HIGH PRA: > 80% (highly sensitized patients)

Patients with a high degree of sensitization often have greatdifficulty in finding an organ donor to whom they will not have asignificant immunologic reaction, which could lead to early andsevere rejection of the allograft if transplantation were to occur.

Approximately 16 percent of patients currently on the waitinglist are highly presensitized to HLA. Many of these patientshave potential living donors that are excluded because of thepresence of preformed HLA antibodies.

Similarly, based upon the distribution of blood groups in theUnited States, approximately one-third of potential living donorsare excluded because they are ABO incompatible with theirintended recipient.

Aull MJ, Kidney Paired Donation and Its Potential Impact on Transplantation, 2013

Solutions

• densensitization protocols

• kidney exchange

Kidney exchange, as a mechanism to facilitate living donorkidney transplantation, is a concept first introduced byRapaport in 1986 as a solution to the shortage of deceaseddonor organs.

Rapaport proposed an international registry whereby eligibleand willing but ABO-incompatible living donors could donate viaexchange facilitated through this registry.

History

Wallis CB, Kidney Paired Donation, 2011

Kidney exchange in the USAPercentage of living donor transplants from paired donation

Aull MJ, Kidney Paired Donation and Its Potential Impact on Transplantation, 2013

Transplants from kidney exchange in Italy

Centro Nazionale Trapianti

"Conventional" Kidney Exchange

List Exchange

Closed Chain

Open Chain

A kidney exchange problem is a matching problem consistingof:

• a set of incompatible patient-donor pairs N = {1, ...,n}

• a profile (%) of ordered lists of all donors’ kidneys, one listfor each patient

Characteristics of the model

• Constraints on the size of exchanges

• List exchange

• Compatible pairs

• Patients’ preferences

Pairwise exchange

Roth AE, Pairwise kidney exchange, 2005

• Constraints on the size of exchanges: exchanges involvetwo patients

• List exchange: no

• Compatible pairs: no

• Patients’ preferences: 0–1 preferences

0–1 preferences

For any recipient i ∈ N:

• for any patient j with a compatible donor for patient i wehave j �i i ,

• for any patient j without any compatible donor for patient iwe have i �i j ,

• for any patients j, h each of whom has a compatible donorfor patient i we have j ∼i h,

• for any patients j, h neither of whom has a compatibledonor for patient i we have j ∼i h.

(�i denotes the strict preference relation and ∼i denotes theindifference relation induced by %i )

Each patient is indifferent between all compatible donors andbetween all incompatible donors, except she strictly prefers herdonor to any other incompatible donor, and any compatibledonor to her own donor.

A pairwise kidney exchange problem is a pair (N,%), where%= (%i)i∈N denotes the list of patient preferences.

Compatibility

We consider the case in which an exchange can involve onlytwo pairs.

DefinitionPatients i , j ∈ N are mutually compatible if i �j j and j �i i .That is, two patients are mutually compatible if each one has adonor whose kidney is compatible for the other patient.

Matching

DefinitionA matching µ : N → N is a function such that: µ(i) = j if andonly if µ(j) = i for any pair of patients i , j ∈ N.

For each matching µ and patient i ∈ N, µ(i) = i means that thepatient i remains unmatched. For any matching µ and pair ofpatients i , j ∈ N, µ(i) = j means that patient i receives acompatible kidney from the donor of patient j and patient jreceives a compatible kidney from the donor of patient i .

DefinitionThe matrix R = [ri,j ]i∈N,j∈N defined by

{1 if j �i i and i �j j0 otherwise

for any pair of (not necessarily distinct) patients i , j ∈ N is calledmutual compatibility matrix.We will refer to the pair (N,R) as the reduced problem of(N,%).

Occasionally it will be helpful to think of the reduced problem asa graph.

G = (N,R) is a graph whose vertices N are the patients (andtheir incompatible donors), and whose edges R are theconnections between mutually compatible pairs of patients; i.e.there is an edge (i , j) ∈ R if and only if ri,j = 1.

Graphs

A matching then can be thought of as a subset of the set ofedges such that each patient can appear in at most one of theedges.

With this alternative representation if (i , j) is an edge in thematching µ, patients i and j are matched by µ and, if patient idoes not appear in any edge in the matching µ, she remainsunmatched.

We need a mechanism, a systematic procedure that selects amatching for each problem.

Individual rationality

DefinitionA matching µ is individually rational if for any patient i ∈ N,µ(i) 6= i implies µ(i) �i i .

LetM be the set of individually rational matchings for theproblem (N,%).

Efficiency

DefinitionA matching µ ∈M is Pareto-efficient if there exists no othermatching η ∈M such that η(i) %i µ(i) for all i ∈ N andη(i) �i µ(i) for some i ∈ N.In the present setting, µ is Pareto-efficient if and only if the setMµ = i ∈ N : µ(i) 6= i of patients matched by µ is maximal, i.e. ifthere does not exist any other matching η ∈M such thatMη ⊃ Mµ.Let E be the set of Pareto-efficient matchings for the problem(N,%).

Non-efficient matching

Efficient matching

For any matching µ ∈M, let |µ| = |Mµ| = |{i ∈ N : µ(i) 6= i}|denote the number of patients who are matched with anotherpatient.

LemmaFor any pair of Pareto-efficient matchings µ, η ∈ E , |µ| = |η|.

If exchange is possible among more than two pairs, theconclusion of the Lemma no longer holds.

D �A A �A B ∼A C, D �C C �C B ∼C A,C �B B �B A ∼B D, B �D A �D D �D C.

The experience of transplant centers is mostly with the priorityallocation systems used to allocate cadaver organs. It istherefore natural to consider how priority mechanisms wouldfunction in the context of live kidney exchange.

DefinitionA priority ordering is a permutation of patients such that thekth patient in the permutation is the patient with the kth priority.

Priorities may depend on quantifiable patient characteristicssuch as the patient’s PRA, which is correlated with how difficultit will be to find a compatible kidney for that patient.(So it might be desirable, for example, for a high PRA patient tohave a high priority for a compatible kidney in the relatively rareevent that one becomes available).

DefinitionA non-negative function π : N → R+ is a priority function if itis increasing in priority, i.e. if π(i) ≥ π(i + 1).

A priority mechanism produces a matching as follows, for anyproblem (N,R) and priority ordering (1,2, ...,n) among thepatients:

• STEP 0: E0 =M

• STEP k: for every k ≤ n we define Ek ⊆ Ek−1:

{{µ ∈ Ek−1 : µ(k) 6= k} if ∃µ ∈ Ek−1 : µ(k) 6= kEk−1 otherwise

We refer to each matching in En as a priority matching, and apriority mechanism is a function which selects a prioritymatching for each problem.

A priority matching matches as many patients as possiblestarting with the patient with the highest priority and followingthe priority ordering, never “sacrificing”a higher priority patientbecause of a lower priority patient.By construction, a priority matching is maximal, and hencePareto-efficient, i.e. En ⊂ E .

Example

STEP 0 = {(1,5)(2,4); (1,5)(3,4);(2,5)(3,4); (2,4)(3,5)}

STEP 1 = {(1,5)(2,4); (1,5)(3,4)}STEP 2 = {(1,5)(2,4)}STEP 3 = {(1,5)(2,4)}STEP 4 = {(1,5)(2,4)}STEP 5 = {(1,5)(2,4)}

Example

STEP 0 = {(1,5)(2,4); (1,5)(3,4);(2,5)(3,4); (2,4)(3,5)}

STEP 1 = {(1,5)(2,4); (1,5)(3,4)}STEP 2 = {(1,5)(2,4)}STEP 3 = {(1,5)(2,4)}STEP 4 = {(1,5)(2,4)}STEP 5 = {(1,5)(2,4)}

Strategy-proofness

The first issue has to do with patients who have multipleincompatible donors willing to donate on their behalf. Thesecond issue involves revealing which compatible kidneys thepatient is willing to accept.

TheoremA priority mechanism makes it a dominant strategy for a patientto reveal both:• her full set of acceptable kidneys,• her full set of available donors.

Strategy-proofness

A patient maximizes her chance of being included in anexchange by revealing all of her willing donors and by acceptingher full set of compatible kidneys.

These two conclusions have the same cause. A patientenlarges the set of other patients with whom she is mutuallycompatible by coming to the exchange with more donors, andby being able to accept a kidney from more of those otherpatients’ donors.

Gallai–Edmonds decompositionThe following partition of the set of patients is key to thestructure of the set of Pareto-efficient matchings. Partition N as{NU ,NO,NP} such that:

NU = {i ∈ N : ∃µ ∈ E such that µ(i) = i},NO = {i ∈ N \ NU : ∃j ∈ NU such thatri,j = 1},NP = N \ (NU ∪ NO).

NU is the set of patients for each of whom there is at least onePareto-efficient matching which leaves her unmatched.NO is the set of patients each of whom is not in NU (i.e., each ofwhom is matched with another patient at each Pareto-efficientmatching) but is mutually compatible with at least one patient in NU .NP is the set of remaining patients (i.e., the set of patients who arematched with another patient at each Pareto-efficient matching andwho are not mutually compatible with any patient in NU ).

DefinitionConsider the reduced problem (N,R). For I ⊂ N, letRI = [ri,j ]i∈I,j∈I . We refer to the pair (I,RI) as the reducedsubproblem restricted to I.

Let (I,RI) be the reduced subproblem of (N,R) with I = N \NO.

Gallai-Edmonds Decomposition Theorem

Let (I,RI) be the reduced subproblem with I = N \ NO and let µbe a Pareto-efficient matching for the original problem (N,R).We have:

1. For any patient i ∈ NO, µ(i) ∈ NU ;2. For any even component (J,RJ) of (I,RI), J ⊆ NP and for

any patient i ∈ J , µ(i) ∈ J \ {i};3. For any odd component (J,RJ) of (I,RI), J ⊆ NU and for

any patient i ∈ J it is possible to match all remainingpatients in J with each other (so that any patient j ∈ J \ {i}can be matched with a patient in J \ {i , j}). Moreover forany odd component (J,RJ), either

• one and only one patient i ∈ J is matched with a patient inNO under the Pareto-efficient matching µ whereas allremaining patients in J are matched with each other sothat µ(j) ∈ J \ {i , j} for any patient j ∈ J \ {i}, or

• one patient i ∈ J remains unmatched under thePareto-efficient matching µ whereas all remaining patientsin J are matched with each other so that µ(j) ∈ J \ {i , j} forany patient j ∈ J \ i .

Top trading cycles and chains

Roth AE, Kidney exchange, 2004

• Constraints on the size of exchanges: no constraints

• List exchange: yes

• Compatible pairs: yes

• Patients’ preferences: strict preference relation overD⋃{w}

Preferences

p = patient (recipient)d = donorw = waiting list

Matching

The outcome of a kidney exchange problem is a matching µ ofkidneys/waitlist option to patients such that:

1. each patient is either assigned a kidney or the waitlistoption w , and

2. no kidney can be assigned to more than one patientalthough the waitlist option w can be assigned to severalpatients.

Algorithm

1. Initially all kidneys are available and all agents are active. Ateach stage of the procedure:• each remaining active patient points to the best remaining

unassigned kidney or to the waitlist option w , whichever ismore preferred, based on his preferences

• each remaining passive patient continues to point to hisassignment

Step 1

2. There is either a cycle, or a w-chain, or both. By definition, acycle can neither intersect with another cycle nor with aw-chain.

(2a) Proceed to Step 3 if there are no cycles. Otherwise locateeach cycle and carry out the corresponding exchange.Remove all patients in a cycle together withtheir assignments.(2b) Each remaining patient points to its top choice amongremaining kidneys. There is either a cycle, or a w-chain, orboth.Proceed to Step 3 if there are no cycles. Otherwise locate allcycles, carry out the corresponding exchanges, and removethem.(2c) Repeat Step 2b until no cycle exists.

Step 2

3. If there are no pairs left, then we are done. Otherwise, eachremaining pair initiates a w-chain. Some of these w-chains mayintersect and others may not.

(3a) If each remaining w-chain is minimal, then each remainingpatient points to the wait list option w. In this case carry our thebasic indirect exchanges and we are done.

(3b) Otherwise select only one of the chains with the chainselection rule. The assignment is final for the patients in theselected w-chain. In addition to selecting a w-chain, the chainselection rule also determines:• whether the selected w-chain is removed and the

associated exchange is immediately carried out, or• the selected w-chain remains in the procedure although

each patient in it is passive henceforth.

4. Each time a w-chain is selected, a new series of cycles mayform. Repeat Steps 2 and 3 with the remaining active patientsand unassigned kidneys until no patient is left.

Examples of Chain Selection Rules

a. Choose minimal w-chains and remove them.

b. Choose the longest w-chain and remove it.

c. Choose the longest w-chain and keep it.

d. Prioritize patient-donor pairs in a single list. Choose thew-chain starting with the highest priority pair and remove it.

e. Prioritize patient-donor pairs in a single list. Choose thew-chain starting with the highest priority pair and keep it.

f. Prioritize the patient-donor pairs so that pairs with Oblood-type donor have higher priorities than those who do not.Choose the w-chain starting with the highest priority pair;remove it in case the pair has an O blood-type donor but keep itotherwise.

Step 3

«60 lives, 30 kidneys, all linked »

New York Times, 2012

Matching theory for kidney transplantation...Kidney disease Kidney exchange Pairwise exchange Top...

Documents