Transcript One Chance in a Million: An equilibrium Analysis of Bone Marrow

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 pi 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-pi)R.
• Probability that a randomly selected person
has no match in the registry is
Sumi pi (1-pi)R
Gain from extra registrant
• Calculate the derivative with respect to R of
the probability of match. That is the
derivative of Sumi pi (1-pi)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 Registrant
Race of
Beneficiary
White
Afro- Am
Asian -Am
Hispanic
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
Benefit Cost Comparison:
Present values of new registrant
White
Afro_Am
Asian Am
Hispanic
Benefit
2727
5220
3238
2717
Cost
391
703
509
452
B/C Ratio
7.0
7.5
6.5
7.1
Optimal Registry
Differences by racial group in US
Race
% of Pop # in
registry
Prob of
no match
White
60
3 mil
.09
Black
12
500 k
.30
Asian
3.2
400 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
69
86
Black
12
4
Asian
3.6
3.3
other
15
6
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
Afro-American
Asian-American
.06
.30
.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.