One Chance in a Million: An equilibrium Analysis of Bone Marrow Donation Ted Bergstrom, Rod Garratt Damien Sheehan-Connor

University of California, Santa Barbara

Background Bone marrow transplants dramatically improve survival prospects of leukemia patients. For transplants to work, donor must be of same HLA type as recipient. Exact matches outside of family are relatively rare.

Some Genetics Relevant HLA type is controlled by 6 genes, located in three loci, called HLA-A, HLA-B and HLA-DR. You inherit a string of 3 from Mom and another string of 3 from Pop. Diploid reproduction, each parent has two strings, randomly picks one to give to you. String inherited from a single parent called a haplotype.

Possible combinations There are about 30 possible alleles that could go in each of the first two loci, and about 10 possibilities for the third. All that matters is what 6 alleles you have, not who you got them from. Not all combinations are equally likely, nor are genes randomly grouped (linkage disequilibrium)

Your most likely match Probability that two full siblings match is about 1/4. They must receive same string from Mom and also same string from Pop. Chance of this is 1/2x1/2=1/4. Note that chance of a match with a parent is very small. Same for uncles and aunts and cousins, etc.

Frequency data Collected by biologists, using data from the bone marrow registry, based on a sample of about 300,000 people who have been typed. Biologists observed phenotypes, but not full genotypes. That is, they see what 6 genes each person has, but don’t know how they were linked on parental chromosomes.

Clever statistics The sample is not big enough to give good estimates of frequency of rare phenotypes. They do a clever trick. They use phenotype distribution and maximum likelihood techniques to estimate distribution of haplotypes. With estimated haplotype distribution and assumption of random mating w.r.t HLA type, we can estimate distribution of phenotypes.

How rare? About 9 million different types Probability that two random people match –Both US whites : 1/11,000 –Both Afro-American: 1/100,000 –Both Asian-American: 1/30,000 –Afro and Caucasian : 1/110,000 In contrast to blood transfusions.

Distribution of type size is very nonuniform About half the white population are in groups smaller than 1/100,000 of population. About 20 per cent are in groups smaller than 1/1,000,000 of population.

Bone marrow registry Volunteers are DNA typed and names placed in a registry. A volunteer agrees to donate stem cells if called upon when a match is found. Matches are much more likely between individuals of same ethnic background. U.S. registry has about 6 million World registry about 10 million

Costs Cost of tests and maintaining records about $140 per registrant. Paid for by registry. Cost to donor. –Bone marrow—needle into pelvis –Under anesthesia –Some pain in next few days. Physician and hospital costs of transplants Alternate method—blood filtering –Less traumatic but risk to donor from pre-filter treatment, roughly same cost.in total.

Social benefits from an additional donor: Behind the Veil of Ignorance Every person in society faces some small probability of needing a life-saving transplant. Adding a donor increases the probability of a match for any person. We numerically calculate effect of an extra registrant on survival probabilities and value this increment at a “value of statistical life”. VSL estimated at about $6.5 million (Viscusi-Aldy)

Probability of having no match Let p i be fraction of the population that is of HLA type i. Probability that a person in i has no match in the registry is (1-p i ) R. Probability that a randomly selected person has no match in the registry is Sum i p i (1-p i ) R

Gain from extra registrant Calculate the derivative with respect to R of the probability of match. That is the derivative of Sum i p i (1-p i ) R Multiply this by the number of people seeking matches to find the expected annual number of additional matches resulting from one more registrant.

Expected lives saved In a single year, about 6,000 people seek matches. Recipient of a transplant receives a gain of about 1/3 in probability of recovery and a normal life. Expected annual lives saved by one more white registrant is about 1/50,000. By black registrant about 1/25,000.

Annual flow A registrant can remain in registry until age 61. Median age of registrants is 35. We assume that registrants remain in registry for 25 years, on average. We discount benefits appearing in later years and we count VSL at $6.5 million.

Present Value of Lives Saved by Additional Registrant Race of Beneficiary White Afro- AmAsian -AmHispanic White $2431 $1856$1368$2318 Afr-American $84 $2894$82$299 Asian Am $51 $68$1640$106 Hispanic $156 $444$193$1083 Total $2727 $5220$3288$3817 Race of Registrant

Benefit Cost Comparison: Present values of new registrant WhiteAfro_AmAsian AmHispanic Benefit Cost B/C Ratio

Optimal Registry

Differences by racial group in US Race% of Pop# in registry Prob of no match White603 mil.09 Black12500 k.30 Asian k.20

What is going on? A new white donor is much more likely to be just a duplicate, yet new white donors are almost as valuable as new blacks or Asians. All lives saved are valued equally. Difference is in number of people seeking transplants.

Who is seeking transplants? % of pop% of seekers White 6986 Black 124 Asian other 156

Free rider problem for donors Suppose that a person would be willing to register and donate if he new that this would save someone who otherwise would not find a match. But not willing to donate if he knew that somebody else of the same type is in the registry.

Nash equilibrium Need to calculate probability that a donor will be pivotal, given that he is called upon to donate. Probability that you are called on if there are k registrants of your type is 1/k.

Conditional probability of being pivotal if called upon. Probability White-American.06 Afro-American.30 Asian-American.18

Benevolence theory C Cost of donating B Value of being pivotal in saving someone else’s life W Warm glow from donating without having been pivotal. Assume B>C>W. Person will donate if H(x)> (C-V)/(B-V)

Plausible numbers? Suppose V=0 If x=5, then for registrants, C/B<.034 US registry has about 5 million donors or 2% of population. So the most generous 2% of population would need to have C/B< 1/30.