Transcript BonemarrowJapan - University of California, Santa Barbara

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.
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.
Rarer half: Magnified view
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.
• Alternate method—blood filtering
– Somewhat less traumatic for donor
– Somewhat more risky for recipient
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 every person.
• We numerically calculate the effect of an
extra registrant on survival probability for
each individual and value this increment at a
“value of statistical life” .
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
Value of a New Registrant
• With 3 million in registry, an extra white
registrant increases the probability that a
given white person finds a match by about
1.8 x 10-8.
What is this worth?
Increase in expected matches
• In a single year, about 6,000 people seek
matches. Thus a new registrant increases
expected number of matches by about
6 x 10-8 x 103=6 x 10-5
• But a new registrant remains in registry for
several years. Suppose 10 years.
• Then over 10 years, expected number of
matches found increases by 6 x 10-4.
Little things add up
• Recipient of transplant gains 1/3 in
probability of healthy life.
• Thus expected number of “lives saved” by
an extra registrant is about
1/3x6x10-4= 2x10-4.
• With “Value of life of $5 x 106, the value
of an additional registrant is about $1000.
Benefit cost for white US
registrants
• Estimated Cost of new registrant $140
• Estimated benefit $1000
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
Dollar values of added registrant
Race of New Registrant
White
$1187
Black
$588
Asian
$477
$27
$510
$21
$18
$17
$351
Others
$58
$101
$50
Total
$1291
$1219
$900
White
Expected
value in Black
lives
saved
Asian
by race
What is going on?
• A new white donor is much more likely to
be just a duplicate, yet new white donors are
at least 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.